Housing Price Forecasting via Time Series & VAR Analysis
Case Study Background
In this case study we are attempting to forecast house price growth.
We will structure these forecasts using the natural temporal persistence in this series, as well as leveraging the information contained in exogenous drivers such as housing starts.
Outline
We will categorize our research process into 3 broad steps: 1) Explore, 2) Explain, 3) Forecast.
Explore
Housekeeping
Standard best practice to clear workspace:
rm(list=ls()) # clear work space
cat("\014") # clear console Data
Our object of interest is housing starts. An important feature is housing prices. There are undoubtedly many other relevant features, but we will focus on this manageable subset so that we can focus on the time series methods. The specific features are chosen to highlight certain time series techniques, not necessarily their explanatory power.
Our data source will be FRED (https://fred.stlouisfed.org/) from the St. Louis Federal Reserve.
We can obtain our data via FRED’s API using the fredr
package. To do so, you need to obtain a free API key by visiting (https://fred.stlouisfed.org/docs/api/api_key.html).
After you’ve obtained the key, save it to a sample txt file and store in your working directory.
Connect to FRED API
#install.packages("fredr")
library(fredr)
fred_api_key = read.delim("API_Key.txt", header = FALSE, sep="")[[1]]
fredr_set_key(fred_api_key) #Store the API key as an environment variableDownload Housing Prices
Set start/end dates
startdate = "1959-01-01"
enddate = "2023-12-01"Notice that the FRED database uses the first of the month as a generic placeholder for the monthly value. That is in contrast to 1) the date at which the data for that month was released, and 2) in contrast for the date at which the activity took place
Download Prices
In order to download data via FREDR we need the FRED series_id. These are easily obtained by searching directly on FRED’s website.
Price: S&P/Case-Shiller U.S. National Home Price Index (CSUSHPINSA)
CSUSHPINSA<-fredr(
series_id = "CSUSHPINSA",
frequency = "m", # monthly
observation_start = as.Date(startdate),
observation_end = as.Date(enddate)
)Download Housing Starts
Housing starts: New Privately-Owned Housing Units Started: Total Units (HOUSTNSA)
HOUSTNSA<-fredr(
series_id = "HOUSTNSA",
frequency = "m", # monthly
observation_start = as.Date(startdate),
observation_end = as.Date(enddate)
)
HOUSTNSA$value <- HOUSTNSA$value*1000 # Convert raw data "Thousands of Units" to "Units" Combine
Let’s combine the data into a single data frame for ease of analysis.
library(dplyr)
HOUSTNSA<-HOUSTNSA %>%
dplyr::select(date,value)
CSUSHPINSA<-CSUSHPINSA %>%
dplyr::select(date,value)
temp_df<-merge(HOUSTNSA,CSUSHPINSA,by="date",all=T)
colnames(temp_df) <- c("Date","HousingStart","PriceIndex")Check for missing
head(temp_df)## Date HousingStart PriceIndex
## 1 1959-01-01 96200 NA
## 2 1959-02-01 99000 NA
## 3 1959-03-01 127700 NA
## 4 1959-04-01 150800 NA
## 5 1959-05-01 152500 NA
## 6 1959-06-01 147800 NAsummary(is.na(temp_df))## Date HousingStart PriceIndex
## Mode :logical Mode :logical Mode :logical
## FALSE:780 FALSE:780 FALSE:444
## TRUE :336idxHousingStarts<-is.na(temp_df$HousingStart)
idxPrices<-is.na(temp_df$PriceIndex)
par(mfrow=c(1,2))
plot(idxPrices,main="Prices")
plot(idxHousingStarts,main="HousingStarts")
Clean
We should NOT simply eliminate the rows with missing data. This (usually) works fine in cross sectional data. But in time series, each row is connected to the nearby rows. Omitting a row deletes critical information, which most likely will influence your results.
We must think WHY the data is missing. If it is random throughout the series, then we could use a local mean between adjacent points, or some other method of interpolation.
Thankfully, in our case, the data is missing at the edges of the dataset for a structural reason we noted above. So, we can trim the sample to make the series balanced.
df <- na.omit(temp_df) # remove the rows with NA
head(df)## Date HousingStart PriceIndex
## 337 1987-01-01 105100 63.732
## 338 1987-02-01 102800 64.132
## 339 1987-03-01 141200 64.467
## 340 1987-04-01 159300 64.972
## 341 1987-05-01 158000 65.547
## 342 1987-06-01 162900 66.218Basic EDA
EDA in time series looks much like typical EDA, but the temporal nature does offer some differences in interpretation and other avenues to explore.
Full sample correlation matrix
cor(df[,c(-1)]) # get grid of the date column## HousingStart PriceIndex
## HousingStart 1.00000000 0.01090519
## PriceIndex 0.01090519 1.00000000- The correlation of ~.01 implies that on average the movements of
housing starts over time don’t track well the movements of housing
prices.
- This might not be an appropriate measure of the relationship between the series for several reasons.
- The two series might not move together contemporaneously. e.g. perhaps prices this month correlate to starts in five months from now (lead/lag).
- Also, the correlation only captures the average relationship through time. Perhaps the the series move together only in up/down trends, but not in the middle of their respective distributions.
- Or perhaps their relationships have changed through time.
- We will need more instructive EDA for time series.
Descriptive Stats
summary(df)## Date HousingStart PriceIndex
## Min. :1987-01-01 Min. : 31900 Min. : 63.73
## 1st Qu.:1996-03-24 1st Qu.: 86975 1st Qu.: 82.03
## Median :2005-06-16 Median :113050 Median :141.88
## Mean :2005-06-16 Mean :111370 Mean :143.06
## 3rd Qu.:2014-09-08 3rd Qu.:136850 3rd Qu.:180.26
## Max. :2023-12-01 Max. :197900 Max. :312.83Time Series Plots
library(tidyverse)
plot_df <- df %>%
`colnames<-`(c("Date","HousingStart","PriceIndex \n(Jan 2000=100)")) %>%
pivot_longer(c("HousingStart","PriceIndex \n(Jan 2000=100)")) # Convert wide dataframe to long
ggplot(data = plot_df, aes(x = Date,y = value, group = name, color = name)) +
geom_line() +
scale_x_date(date_labels = "%Y (%b)") + # format datetime
facet_grid(name~., scales = "free_y") +
labs(x = "", y = '') +
theme(legend.position = "none")
We need to address each before we proceed with modeling.
Seasonality
Understand Seasonality
Let’s zoom in on housing starts.
library(ggplot2)
ggplot(data = df, aes(x = Date,y = HousingStart)) +
geom_line() +
scale_x_date(date_labels = "%Y (%b)") + # format datetime
labs(x = "", y = "Housing Start Units")
Why might this seasonality exist?
There are many ways to adjust for these seasonal patterns.
Prepare the df for analysis
library(xts)
FirstObs=df$Date[1]
#We could parse the FirstObs chr string, but for our purposes it's easier to input by hand the start date
df_ts <- ts(xts(df[,2:3], order.by = df$Date), frequency = 12, start = c(1987,1)) Moving Avg
library(TTR)
SMA24<-SMA(df_ts[,"HousingStart"],n=24) # Create a simple 24 period moving avg using `SMA()`
toplot<-cbind(as.xts(df_ts[,"HousingStart"]),as.xts(SMA24))
colnames(toplot)[1]="Level" # Add column name
colnames(toplot)[2]="24mth MovAvg" # Add column name
plot(toplot,main="Housing Starts",legend.loc="topright", auto.legend=TRUE)
- The red line (i.e. the SMA) takes the trailing local average over 24months.
- The smoother average can give us a sense of the underlying trend of housing starts, abstracting away from the seasonal variation.
- In our case, housing starts have been declining in the recent period, but not by as much as the raw series suggests, because we are in a seasonal down turn near the end of the series (Winter 2023).
Seasonal Decomposition
We could decompose the housing starts series into 3 pieces: 1) trend, 2) seasonal, 3) noise. The decomposition can be additive or multiplicative. There are numerous techniques for identifying and extracting the 3 components. Let’s demonstrate a few.
Classical Decomposition
Classical decomposition methods compute the seasonal component by essentially including dummy variables for each periodic unit (e.g. month, quarters, etc..)
library(stats)
decomp<-decompose(df_ts[,"HousingStart"], type="additive")
plot(decomp) # plot components
Panel 1: Raw data
Panel 2: Trend of the housing starts series, which is extracted using a moving average
Panel 3: Seasonal captures the seasonal variations of the Raw data around the trend
Panel 4: Is the noise in the Raw data that can not be explained by the Trend nor the Seasonal components
After the decomposition is implemented we can easily construct a seasonally adjusted series by subtracting the seasonal components from the raw series.
We often report time series as seasonally adjusted values to abstract away from the seasonal variations and uncover the structural behavior of the time series.
For instance, policy makers might be able to influence the underlying trend.
SA = decomp$x-decomp$seasonal # Raw Minus SeasonalCompare the SA and NSA
Trend = decomp$trend
library(forecast)
forecast::autoplot(df_ts[,"HousingStart"], series="Raw Data") +
forecast::autolayer(decomp$x-decomp$seasonal, series="Seasonally Adjusted") +
forecast::autolayer(decomp$trend, series="Trend") +
xlab("Year") + ylab("") + labs(color = "") +
ggtitle("Classical Additive Seasonally Adjusted") +
scale_colour_manual(values=c("black","blue","red"), breaks=c("Raw Data","Seasonally Adjusted","Trend")) +
theme(legend.position="bottom")
X11 Decomposition
The X11 approach is a variant of the classical decomposition.
library(seasonal)
x11decomp <- seas(df_ts[,"HousingStart"], x11="")
forecast::autoplot(x11decomp) +
ggtitle("X11 Decomposition")
Compare SA and NSA
forecast::autoplot(df_ts[,"HousingStart"], series="Raw Data") +
autolayer(seasadj(x11decomp), series="Seasonally Adjusted") +
autolayer(trendcycle(x11decomp), series="Trend") +
xlab("Year") + ylab("") + labs(color = "") +
ggtitle("X11 Seasonally Adjusted") +
scale_colour_manual(values=c("black","blue","red"), breaks=c("Raw Data","Seasonally Adjusted","Trend")) +
theme(legend.position="bottom")
STL Decomposition
STL is an acronym for “Seasonal and Trend decomposition using Loess”, wherein Loess is a method for estimating nonlinear relationships.
stldecomp <- stl(df_ts[,"HousingStart"], s.window="periodic", robust=TRUE)
forecast::autoplot(stldecomp) +
ggtitle("STL Decomposition")
Loess is data intensive method and traditionally two-sided, meaning that we need data on either side of each point in order to apply the adjustment.
forecast::autoplot(df_ts[,"HousingStart"], series="Raw Data") +
autolayer(seasadj(stldecomp), series="Seasonally Adjusted") +
autolayer(trendcycle(stldecomp), series="Trend") +
xlab("Year") + ylab("") + labs(color = "") +
ggtitle("STL Seasonally Adjusted") +
scale_colour_manual(values=c("black","blue","red"), breaks=c("Raw Data","Seasonally Adjusted","Trend")) +
theme(legend.position="bottom")
Compare the Decomp Methods
forecast::autoplot(df_ts[,"HousingStart"], series="Raw Data") +
autolayer(decomp$x-decomp$seasonal, series="Classical Additive Seasonally Adjusted") +
autolayer(seasadj(x11decomp), series="X11 Seasonally Adjusted") +
autolayer(seasadj(stldecomp), series="STL Seasonally Adjusted") +
xlab("Year") + ylab("") + labs(color = "") +
ggtitle("Seasonally Adjusted") +
scale_colour_manual(values=c("black","darkgreen","steelblue","darkred"),
breaks=c("Raw Data","Classical Additive Seasonally Adjusted",
"X11 Seasonally Adjusted", "STL Seasonally Adjusted")) +
theme(legend.position="bottom") + guides(colour = guide_legend(ncol = 2))
Note that this is NOT a forecasting exercise, so we don’t typically use some type of out of sample MSFE to determine which decomposition is best. We can investigate in sample behavior among the ``Noise” terms using MSE, AIC, and the like. Caution: x11 standardizes the noise and seasonal terms, so you can’t compare the 3 models directly, without some adjustments. (beyond our purview)
mean(decomp$random^2,na.rm = TRUE)## [1] 61361365mean(resid(x11decomp)^2,na.rm = TRUE)## [1] 0.005967311mean((stldecomp$time.series[3])^2,na.rm = TRUE)## [1] 242347.5We could proceed exclusively with the seasonally adjusted series. But for instructional purposes, we’ll work with both the raw and seasonally adjusted series.
Stationarity
Stationarity is a time series concept regarding the behavior of the data in small windows of time.
- Consider a series that is not ``variance stationary”. It might have been relatively calm, with low variability. But then during COVID, volatility spikes. We should accommodate for this structural change within our model.
- Likewise, consider a series that is not ``mean stationary”. It might
have a persistent upward trend. We should accommodate for this trend
when attempting to find drivers of the series.
- Most standard time series modeling techniques require stationarity before implementing.
- Even those that don’t technically require stationarity could benefit from this type of behavior.
- Therefore, it’s essential we are able to detect and remedies any violations to this assumption.
Detect Non-Stationarity w/ADF
There are several tests to detect (non) stationarity.
The Augmented Dickey Fuller (ADF) test focuses on a manifestation of non stationarity called a unit root. A particular case of unit roots is the `Random Walk’ wherein today’s value is equal to yesterday’s + random noise.
ADF Test
Ho: Non Stationary (Unit Root aka Random Walk)
Ha: Stationary
library(tseries)
adf.test(df_ts[,"HousingStart"])##
## Augmented Dickey-Fuller Test
##
## data: df_ts[, "HousingStart"]
## Dickey-Fuller = -2.3521, Lag order = 7, p-value = 0.4289
## alternative hypothesis: stationaryadf.test(SA)##
## Augmented Dickey-Fuller Test
##
## data: SA
## Dickey-Fuller = -1.5802, Lag order = 7, p-value = 0.7552
## alternative hypothesis: stationary- Notice the high pvalue implies we fail to reject (i.e. we have a non
stationary series).
- This is true for both the raw series as well as the seasonally adjusted series.
- We saw previosly that these series have several changes in trend (dramatic drop and then a steady up turn).
Detect Non-Stationarity w/ KPSS
The Kwiatkowski-Phillips-Schmidt-Shin (KPSS) test can be used in several ways. We’ll focus on its importance for testing trend stationarity (i.e. is the series stationary after adjusting for its trend).
H0: Trend Stationary
Ha: Not Trend Stationary
kpss.test(df_ts[,"HousingStart"], null="Trend")##
## KPSS Test for Trend Stationarity
##
## data: df_ts[, "HousingStart"]
## KPSS Trend = 0.50217, Truncation lag parameter = 5, p-value = 0.01kpss.test(SA, null="Trend")##
## KPSS Test for Trend Stationarity
##
## data: SA
## KPSS Trend = 0.54187, Truncation lag parameter = 5, p-value = 0.01- Note the low pvalues imply that we reject and that we have both the raw and SA series that are not trend stationary.
- This implies that the seasonal osciallations were likely relatively
steady over time, and thereby not engendering a non stationarity.
- If we can accommodate for the trend, we might be able to have a stationary process, which is ready for further modeling.
Detect Non-Stationarity w/ACF
Autocorrelation Function (ACF) tells us the correlation of a series with its own past.
The Partial Autocorrelation Function (PACF) tells us the autocorrelation of a series with its own past after correlation for the impact of other lags of the series.
The visualizations are collectively referred to as the (auto)correlogram.
library(astsa)
acf2(df_ts[,"HousingStart"], main = "ACF and PACF of Housing Starts Units")
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13]
## ACF 0.93 0.84 0.75 0.67 0.63 0.61 0.61 0.63 0.69 0.76 0.83 0.85 0.81
## PACF 0.93 -0.12 -0.10 0.09 0.21 0.07 0.08 0.20 0.37 0.26 0.15 0.01 -0.32
## [,14] [,15] [,16] [,17] [,18] [,19] [,20] [,21] [,22] [,23] [,24] [,25]
## ACF 0.72 0.62 0.54 0.50 0.47 0.47 0.48 0.53 0.59 0.65 0.66 0.61
## PACF -0.28 -0.11 -0.09 0.06 -0.07 0.02 -0.11 0.09 0.03 -0.01 -0.02 -0.11
## [,26] [,27] [,28] [,29] [,30] [,31] [,32] [,33] [,34] [,35] [,36] [,37]
## ACF 0.53 0.42 0.34 0.30 0.27 0.26 0.27 0.31 0.37 0.42 0.43 0.38
## PACF -0.06 -0.16 -0.03 0.05 0.01 0.01 -0.05 -0.01 0.01 0.03 0.02 -0.15
## [,38] [,39] [,40] [,41] [,42] [,43] [,44] [,45] [,46] [,47] [,48]
## ACF 0.29 0.20 0.13 0.08 0.06 0.05 0.06 0.11 0.17 0.23 0.25
## PACF -0.03 0.02 0.08 -0.02 0.04 0.03 -0.06 0.07 0.13 0.08 0.08First Bar: The correlation between housing starts in month t has a roughly .9 correlation with housing starts in month t-1.
Second Bar: The correlation between housing start in month t has a roughly .8 correlation wth housing starts in month t-2.
First Bar: The correlation between housing starts in month t has a roughly .9 correlation with housing starts in month t-1. (i.e. it’s the same as the ACF)
Second Bar: After controlling for the effects of month t-1, housing starts in month t have a slight negative correlation with housing starts in month t-2.
The ACF depicts a seasonal pattern (notice the wave cresting every 12mhts).
The PACF exhibits strong persistence near the 1st lags, which might be helpful when we model the process with its own past lags (e.g. ARMA).
Remedy w/Differencing
A common approach to overcoming non stationarity is by taking the (percent) difference in the series. Often, if a series is non-stationary, then computing the \(\Delta x_{t}=x_{t}-x_{t-1}\) is stationary. If the first difference is not stationary, we can compute a second difference analogously.
Often in our modeling process these growth rates are the objects of interest, which aids our interpretation of any results.
Let’s first difference the Housing Starts series and recompute the ADF test.
FirstDiff <- diff(df_ts[,"HousingStart"]) # compute the first difference in housing
FirstDiff <- na.omit(FirstDiff) #clean up the data
plot(FirstDiff) 
adf.test(FirstDiff) ##
## Augmented Dickey-Fuller Test
##
## data: FirstDiff
## Dickey-Fuller = -16.453, Lag order = 7, p-value = 0.01
## alternative hypothesis: stationary- Notice that the low pvalue suggests we now have a stationary
process.
- This is confirmed by visual inspection of the time series plot, which has roughly similar mean and variance throughout the time series.
We can also accommodate for seasonality within our differencing method by taking year-over-year growth rates.
SeasDiffFirstDiff <- diff(FirstDiff,lag = 12)
plot(SeasDiffFirstDiff)
adf.test(SeasDiffFirstDiff) ##
## Augmented Dickey-Fuller Test
##
## data: SeasDiffFirstDiff
## Dickey-Fuller = -8.7369, Lag order = 7, p-value = 0.01
## alternative hypothesis: stationaryPro Tip: Converting your series to logs and then computing the difference approximates a growth RATE.
Converting to growth rates is also helpful to standardize our features. So, let’s do that for housing starts.
library("quantmod")
HousingStartGrowth = na.omit(Delt(df_ts[,"HousingStart"]))*100 # convert to growth rateadf.test(HousingStartGrowth)##
## Augmented Dickey-Fuller Test
##
## data: HousingStartGrowth
## Dickey-Fuller = -15.47, Lag order = 7, p-value = 0.01
## alternative hypothesis: stationaryHousing Prices: Stationarity
adf.test(df_ts[,"PriceIndex"])##
## Augmented Dickey-Fuller Test
##
## data: df_ts[, "PriceIndex"]
## Dickey-Fuller = -1.0697, Lag order = 7, p-value = 0.9271
## alternative hypothesis: stationaryplot(df_ts[,"PriceIndex"])
- The high pvalue suggests non-stationarity.
- There is a clear trend and a structural change around the 2008 crash.
Let’s compute a 1st difference
PriceIndexGrowth = na.omit(Delt(df_ts[,"PriceIndex"]))*100
adf.test(PriceIndexGrowth)##
## Augmented Dickey-Fuller Test
##
## data: PriceIndexGrowth
## Dickey-Fuller = -3.445, Lag order = 7, p-value = 0.04782
## alternative hypothesis: stationaryplot(PriceIndexGrowth) 
So let’s adjust that Growth series for seasonality.
#PriceIndexGrowth_decomp<-decompose(df_ts[,"PriceIndex"], type="additive")
PriceIndexGrowth_decomp<-decompose(PriceIndexGrowth, type="additive")
PriceIndexGrowthSA = PriceIndexGrowth - PriceIndexGrowth_decomp$seasonal
plot(PriceIndexGrowthSA)
adf.test(PriceIndexGrowthSA)##
## Augmented Dickey-Fuller Test
##
## data: PriceIndexGrowthSA
## Dickey-Fuller = -3.185, Lag order = 7, p-value = 0.09059
## alternative hypothesis: stationary- The low pvalue now suggests a stationary series.
- There are several structural features in the price series (the crash, the trend, the varying seasonality) that need to be addressed before we can model this series.
Prep the df
Let’s group all of these adjusted series together
df_adj <- data.frame(cbind(HousingStartGrowth, PriceIndexGrowthSA)) %>%
`colnames<-`(c("HousingStartGrowth","PriceIndexGrowthSA"))
df_adj$Date <- df$Date[c(-1)]
#df_adj<-na.omit(df_adj)Time Series EDA
Now that we have adjusted our data series, let’s examine their time series properties.
AutoCorrelogram
tsdisplay(df_adj$HousingStartGrowth, main = "Time Series Plot of Housing Start Units Growth")
Let’s inspect the price series.
tsdisplay(df_adj$PriceIndexGrowthSA, main = "Time Series Plot of SA Price Index Growth")
- The oscillation in the ACF suggests a bit of seasonality remains.
- The decline in the ACF and the two spikes in the PACF suggest short term persistence in the series. We can leverage this when we model the series.
Cross Correlations
Sometimes series lead/lag one another for `economic’ reasons. We can visualize and detect this cross auto correlations with a CCF (Cross Correlation Function).
Here we’ll examine whether growth of housing starts lead or lag price growth.
ccfvalues = ccf(x=df_adj$HousingStartGrowth, y=df_adj$PriceIndexGrowthSA,
main = "CCF of Housing Start Growth and Price Index Growth", lag.max = 24)
When lag = -1 we are computing the cor(HousingStart_t-1, Price_t); i.e. Starts LEAD Prices.
When lag = +1 we are computing the cor(HousingStart_t+1,Price_t); i.e. Starts LAG Prices.
Positive values indicate that the series move together. Negative values indicate the series move in opposite directions.
The blue lines provide a 95% CI.
- The spikes are more prominent in the negative lag area, suggesting possible leading effects for housing starts.
- A model such as \(Prices_{t} = \alpha + \beta HousingStarts{t-1} + e_{t}\) might be useful for explaining/forecast prices.
- However, there is not as much information in prices to forecast future housing starts.
- Regardless of the direction, neither series has statistically significant lead/lag effects, and thus may not be helpful for modeling.
Here is an alternative visualization.
library(astsa)
HSGrowth <- df_adj$HousingStartGrowth
PriceGrowth <- df_adj$PriceIndexGrowthSA
lag2.plot(series1 = HSGrowth,
series2 = PriceGrowth,
max.lag = 11) 
# Note: series1 is the one that gets lagged. Explain
Now that we’ve explored our data’s temporal patterns let’s attempt to explain its movements. We’ll do so with traditional and time series econometric techniques.
Regression
Since we are dealing with time series data we need to be careful
about what information we are using and WHEN that occurs. Let’s focus on
a CONTEMPORANEOUS regression, wherein we are using data at time t to
explain our outcome variable (housing prices) at time t.
\[ Price_{t}=\alpha + \beta Starts_{t} +
u_{t}\] Q: What do we assume about the errors of this model?
- Among other things, we assume that the errors are independent of
each other.
- In our time series setting, that means the errors in January are not related to the errors in February, etc..
- This is a sickness called serial correlation, wherein the errors exhibit autocorrelation.
- If your model is plagued by serial correlation your estimators typically remain unbiased, but are not efficient.
LinearModel = lm(PriceIndexGrowthSA ~ HousingStartGrowth, data = df_adj)
summary(LinearModel)##
## Call:
## lm(formula = PriceIndexGrowthSA ~ HousingStartGrowth, data = df_adj)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.17237 -0.26748 0.03394 0.32352 2.01820
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.359393 0.027740 12.956 <2e-16 ***
## HousingStartGrowth 0.000659 0.002121 0.311 0.756
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.5827 on 441 degrees of freedom
## Multiple R-squared: 0.0002189, Adjusted R-squared: -0.002048
## F-statistic: 0.09655 on 1 and 441 DF, p-value: 0.7562- The impact is not statistically significant.
- i.e. the growth rate of housing starts in this month does not help to explain the seasonally adjusted growth rate of prices in this month.
- If our model is ``sick” with serial correlation, this t-stat could be contaminated, and give us misleading conclusions.
Detect Serial Correlation
There are myriad visual and formal tests for serial correlation. Most operate on the residuals of the model.
Visual inspection
Let’s inspect our residuals for troublesome patterns.
checkresiduals(LinearModel)
##
## Breusch-Godfrey test for serial correlation of order up to 10
##
## data: Residuals
## LM test = 398.15, df = 10, p-value < 2.2e-16- Notice the spikes in the residual ACF.
- These are indicative of some type of omitted temporal pattern, which
might suggest the presence of serial correlation.
- Also notice that the model does not seem to work well around the 2008 crash and COVID. These structural changes could impede our ability to detect the underlying relation between starts and prices.
Durbin Watson Test
The Durbin Watson (DW) test asks whether the model residuals are auto correlated.
Ho: No auto correlation in residuals
Ha: Auto correlation in residuals
library(lmtest)
dwtest(formula = LinearModel, alternative = "two.sided")##
## Durbin-Watson test
##
## data: LinearModel
## DW = 0.14586, p-value < 2.2e-16
## alternative hypothesis: true autocorrelation is not 0BG Test
Breusch Godfrey (BG) Test
Ho: Model residuals not autocorrelated
Ha: Model residuals are autocorrelated
bgtest(formula = LinearModel)##
## Breusch-Godfrey test for serial correlation of order up to 1
##
## data: LinearModel
## LM test = 380.36, df = 1, p-value < 2.2e-16LB Test
Ljung Box Test
Ho: Model residuals not autocorrelated
Ha: Model residuals are autocorrelated
Box.test(residuals(LinearModel), type = "Ljung-Box") ##
## Box-Ljung test
##
## data: residuals(LinearModel)
## X-squared = 382.5, df = 1, p-value < 2.2e-16Remediating Serial Correlation
If we are lucky enough to know the precise form of the serial correlation, then we can include it directly in our modeling via GLS. However, that is seldom the case in practice. So, we rely upon HAC (heteroscedasticity and autocorrelation consistent) standard errors.
library(sandwich)
NW_VCOV_LinearModel <- NeweyWest(LinearModel, prewhite = F, adjust = T) # The pre-whitening and adjust are set to F, and T respectively to ensure the proper formula and small sample adjustments are made.
# Compute Standard Errors
HAC_err=sqrt(diag(NW_VCOV_LinearModel))
HAC_err## (Intercept) HousingStartGrowth
## 0.057184133 0.002616771Let’s see if this new std error is meaningful.
summary(LinearModel)$coefficients## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.3593925674 0.027739609 12.9559350 9.330969e-33
## HousingStartGrowth 0.0006589791 0.002120743 0.3107303 7.561525e-01LinearTValue = summary(LinearModel)$coefficients[2,1]/summary(LinearModel)$coefficients[2,2]
LinearTValue## [1] 0.3107303HACTValue = summary(LinearModel)$coefficients[2,1]/HAC_err[2]
HACTValue## HousingStartGrowth
## 0.2518291- Notice the (slight) differences in the standard errors.
- The t value has not changed much.
- The implication is that eventhough there was a statistical presence of serial correlation, this remedy did not change the outcome (i.e. housing start growth is not a significant driver of future price growth)
- That could mean that HAC hasn’t adequately dealt with the serial correlation, or that the problem was not pervasive.
- We can iterate our test and remediation process, but will forgo that for the sake of this exercise.
Rolling Regression
Consider running a regression from Jan87 through Dec97 and grabbing \(\beta\).
Then run the regression again on Feb87 through Jan98, and grabbing the \(\beta\).
- An important feature of time series models is coefficient stability. i.e. is the `economic’ process stable through time?
- We can investigate by running our model over small windows through time.
- If the coefficients change often/suddenly, it implies something is misspecified in our model.
Moving Window
Let’s run our regression over a fixed window of 30 periods, grab the results, then move the 30 period window forward by one month, rerun the regression, and repeat until the end of the sample.
library(tidyfit)
library(tidyr)
library(purrr)
library(lubridate)
library(stringr)
library(ggplot2)
RollingReg<-df_adj %>%
regress(PriceIndexGrowthSA~HousingStartGrowth,
m("lm",vcov.="HAC"),
.cv = "sliding_index", .cv_args=list(lookback=months(30),index = "Date"),
.force_cv= TRUE, .return_slices=TRUE)
beta.df<-coef(RollingReg)
beta.df <- beta.df %>%
unnest(model_info) %>%
filter(term=="HousingStartGrowth") %>%
mutate(upper = estimate + 2 * std.error, lower = estimate - 2 * std.error) %>%
mutate(beta = estimate)
beta.df %>%
mutate(Dates = as.Date(slice_id)) %>%
filter(term == "HousingStartGrowth") %>%
ggplot(aes(Dates)) +
geom_ribbon(aes(ymax = upper, ymin = lower), alpha = 0.25) +
geom_line(aes(y = beta)) +
theme_bw(8)
- beta is the explanatory power of housing start growth in month t
upon SA price growth in month t.
- There seems to be structure changes during the housing crash of 2008 and the COVID spike. - We might want to include dummies / control variables to accommodate for these periods.
- 30 periods is rather arbitrary. We should tune that parameter, but we won’t for the sake of time.
Forecast
Now that we’ve explored the data and have explained its movements, we can attempt to forecast. There are myriad models to forecast a time series. Some use exogenous features (e.g. housing starts lags to forecast housing prices), while others use the series’ own temporal persistence (i.e. today is a good predictor for tomorrow). We will explore a few common approaches.
Training and Test Samples
- The temporal persistence of the data means that each observation is connected to the others. Randomly breaking them apart loses valuable information
- There is danger of leakage. We can’t train on 2023 and test on 2022.
The DGP of 2023 is built from the behavior of 2022. This is
cheating.
- There is danger of leakage. We can’t train on 2023 and test on 2022.
The DGP of 2023 is built from the behavior of 2022. This is
cheating.
Relatedly, typical cross validation techniques are in danger of similar issues with time series.
We typically order the observations by time (they already are) and chose the most recent “x” observations as testing and all dates prior to train.
The size of the test/train split remains an choice that we should stress test, just as we do with cross sectional models.
Moreover, we need to ensure that our training sample has similar structural characteristics as our test sample. e.g. we want to avoid training housing prices 2010-2019 and then test in 2020 amid the COVID shock.
For our example we will use
Train: Jan1987-Dec2013
Test: Jan2014-end of sample
We could create separate dataframes from each subsample, but we will avoid that in order to illustrate some subseting features of the functions we will use.
Let’s adjust the dataframe into a time series object so that we can take advantage of certain time series features.
df_adjts <- ts(df_adj, start = c(1987,02), frequency = 12)Distributed Lag
A distributed lag model forecasts a series with past values of some exogenous series (depicted here with one lag). \(Price_{t}=\alpha + \beta Starts_{t-1} + u_{t}\).
An autoregressive distributed lag (ADL) also includes lagged values of Price (depicted here with one lag). \(Price_{t}=\alpha + \beta Starts_{t-1} + \gamma Price_{t-1} + u_{t}\).
We can use the CCF to detect leads/lags in our training sample.
ccf(x=window(df_adjts[,"HousingStartGrowth"], end = c(2013,12)),
y=window(df_adjts[,"PriceIndexGrowthSA"], end = c(2013,12)),
main = "CCF of Housing Start Growth and Price Index Growth") 
- The spikes within the negative lag regions suggests that Starts
appear to have some predictive power over future prices.
- However, these spikes aren’t strongly significant, so we proceed with caution.
- The reverse doesn’t seems to be true (But, we’ll explore such situations with tools later)
- Given the temporal persistence of the price series, let’s try an ADL model. We’ll demonstrate with 2 lags on price growth and 12 on starts. We can also use information criterion to guide lag selection.
ADL_alldata <- ts.intersect(df_adjts,
PriceIndexGrowthLag1 = stats::lag(df_adjts[,"PriceIndexGrowthSA"], -1),
PriceIndexGrowthLag2 = stats::lag(df_adjts[,"PriceIndexGrowthSA"], -2),
HousingStartGrowthLag1 = stats::lag(df_adjts[,"HousingStartGrowth"], -1),
HousingStartGrowthLag2 = stats::lag(df_adjts[,"HousingStartGrowth"], -2),
HousingStartGrowthLag3 = stats::lag(df_adjts[,"HousingStartGrowth"], -3),
HousingStartGrowthLag4 = stats::lag(df_adjts[,"HousingStartGrowth"], -4),
HousingStartGrowthLag9 = stats::lag(df_adjts[,"HousingStartGrowth"], -9),
HousingStartGrowthLag10 = stats::lag(df_adjts[,"HousingStartGrowth"], -10),
HousingStartGrowthLag11 = stats::lag(df_adjts[,"HousingStartGrowth"], -11),
HousingStartGrowthLag12 = stats::lag(df_adjts[,"HousingStartGrowth"], -12)) %>%
`colnames<-`(c(colnames(df_adjts),
"PriceIndexGrowthLag1","PriceIndexGrowthLag2","HousingStartGrowthLag1",
"HousingStartGrowthLag2","HousingStartGrowthLag3","HousingStartGrowthLag4",
"HousingStartGrowthLag9","HousingStartGrowthLag10","HousingStartGrowthLag11","HousingStartGrowthLag12"))
ADL_traindata= window(ADL_alldata, end = c(2013,12))
ADL_testdata= window(ADL_alldata, start = c(2014,1))
### Estimate ADL Model
ADL <- lm(PriceIndexGrowthSA ~ PriceIndexGrowthLag1+PriceIndexGrowthLag2+
HousingStartGrowthLag1 + HousingStartGrowthLag2 + HousingStartGrowthLag3 + HousingStartGrowthLag4 +
HousingStartGrowthLag9 + HousingStartGrowthLag10 + HousingStartGrowthLag11+HousingStartGrowthLag12,
data = ADL_traindata)
summary(ADL)##
## Call:
## lm(formula = PriceIndexGrowthSA ~ PriceIndexGrowthLag1 + PriceIndexGrowthLag2 +
## HousingStartGrowthLag1 + HousingStartGrowthLag2 + HousingStartGrowthLag3 +
## HousingStartGrowthLag4 + HousingStartGrowthLag9 + HousingStartGrowthLag10 +
## HousingStartGrowthLag11 + HousingStartGrowthLag12, data = ADL_traindata)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.56159 -0.07812 0.01070 0.09129 1.15650
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.0221263 0.0123032 1.798 0.07311 .
## PriceIndexGrowthLag1 1.2788007 0.0528036 24.218 < 2e-16 ***
## PriceIndexGrowthLag2 -0.3816805 0.0522783 -7.301 2.58e-12 ***
## HousingStartGrowthLag1 0.0021030 0.0009023 2.331 0.02043 *
## HousingStartGrowthLag2 0.0025162 0.0008662 2.905 0.00395 **
## HousingStartGrowthLag3 0.0005049 0.0008867 0.569 0.56952
## HousingStartGrowthLag4 0.0009072 0.0008871 1.023 0.30729
## HousingStartGrowthLag9 0.0022299 0.0008776 2.541 0.01156 *
## HousingStartGrowthLag10 0.0010170 0.0008828 1.152 0.25023
## HousingStartGrowthLag11 -0.0007679 0.0008658 -0.887 0.37582
## HousingStartGrowthLag12 -0.0009210 0.0009003 -1.023 0.30716
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1946 on 300 degrees of freedom
## Multiple R-squared: 0.8932, Adjusted R-squared: 0.8896
## F-statistic: 250.8 on 10 and 300 DF, p-value: < 2.2e-16acf2(residuals(ADL), main = "ACF and PACF of ADL Model Residuals") 
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13]
## ACF -0.01 0.1 -0.18 0.05 -0.04 -0.09 -0.04 -0.05 0.04 0.01 0.25 0.32 0.13
## PACF -0.01 0.1 -0.18 0.05 -0.01 -0.14 -0.02 -0.05 0.01 0.02 0.24 0.35 0.13
## [,14] [,15] [,16] [,17] [,18] [,19] [,20] [,21] [,22] [,23] [,24] [,25]
## ACF 0.01 -0.11 -0.09 0.04 -0.07 -0.03 -0.02 0.02 0.03 0.23 0.15 0.05
## PACF 0.07 -0.03 -0.12 0.10 -0.01 -0.01 0.06 -0.04 -0.09 0.08 0.01 -0.05
## [,26] [,27] [,28]
## ACF -0.07 -0.02 -0.05
## PACF -0.04 0.09 -0.01The model residuals look nice and clean.
There appears to be just a bit of temporal persistence left in the two month lag, but that’s minor
The first 2 lags of prices seem significant. The first is >0, implying a rise in prices this month are associated with a rise in prices next month. The opposite is true for 2 months prior.
Moreover, the first 2 lags of housing starts growth are significant and positive; i.e. if starts grow faster this month, prices are expected to rise next month.
Let’s compute the predicted value for our first observation in the training sample:
predict(ADL, newdata = head(ADL_testdata,1), interval = "confidence", level=0.95)## fit lwr upr
## 1 0.3760576 0.3064077 0.4457075- The “fit” is the prediction by the ``fitted’’ model
- the lwr and upr indicate the confidence bounds on that forecast.
Let’s compare to the observed value
window(df_adjts[,"PriceIndexGrowthSA"], start = c(2014,1), end = c(2014,1))#observation## Jan
## 2014 0.4279801Holt Winters
Holt Winters is a class of time series models that act entirely on the trend-cycle decomposition we highlighted earlier. It attempts to fit a model for the raw data by optimizing equations to represent three aspects of the additive time series: the level of the series, the trend of the series, and the seasonal component of the series.
plot(decompose(window(df_adjts[,"PriceIndexGrowthSA"], end = c(2013,12)), type="additive"))
A preliminary diagnostic to determine of this class of model will be useful for you is whether the random component is very large in magnitude relative to the trend and seasonal. If so, the model likely won’t be very good.
Holt Winters is a flexible decomposition model well suited for forecasting. Within the R implementation there are several settings we can tune if desired:
• alpha: the “base value”. Higher alpha puts more weight on the most recent observations.
• beta: the “trend value”. Higher beta means the trend slope is more dependent on recent trend slopes.
• gamma: the “seasonal component”. Higher gamma puts more weighting on the most recent seasonal cycles.
### Fitting with Holt-Winters
HW1 <- HoltWinters(window(df_adjts[,"PriceIndexGrowthSA"], end = c(2013,12)), seasonal = "additive")
### Plot fitted value of two models
{plot(window(df_adjts[,"PriceIndexGrowthSA"], end = c(2013,12)),
ylab="Price Index Growth") # Visually evaluate the fits
lines(HW1$fitted[,1], lty=2, col="red")
legend("bottomleft",legend=c("Raw Data", "R Fit Holt-Winters"),
col=c("black","red"), lty=c(1,2), cex=0.8, box.lty=0)} 
optalpha = HW1$alpha
optbeta = HW1$beta
optgamma = HW1$gamma
rbind(optalpha, optbeta, optgamma)## alpha
## optalpha 0.7339494
## optbeta 0.0000000
## optgamma 1.0000000Tuning Holt Winters
### Fitting with Holt-Winters
HW1 <- HoltWinters(window(df_adjts[,"PriceIndexGrowthSA"], end = c(2013,12)), seasonal = "additive")
HW2 <- HoltWinters(window(df_adjts[,"PriceIndexGrowthSA"], end = c(2013,12)), seasonal = "additive",alpha=optalpha*.1,beta=optbeta,gamma=optgamma)
### Plot fitted value of two models
{plot(window(df_adjts[,"PriceIndexGrowthSA"], end = c(2013,12)),
ylab="Price Index Growth") # Visually evaluate the fits
lines(HW1$fitted[,1], lty=2, col="blue")
lines(HW2$fitted[,1], lty=2, col="red")
legend("bottomleft",legend=c("Raw Data", "Optimal Holt-Winters","Less Weight to Recent"),
col=c("black","blue", "red"), lty=c(1,2,2), cex=0.8, box.lty=0)} 
Forecasting with Holt Winters
Let’s predict one step ahead
predict(HW1, 1, prediction.interval = TRUE, level=0.95)## fit upr lwr
## Jan 2014 0.4136086 0.870189 -0.04297183And compare to the observed value in the testing set
window(df_adjts[,"PriceIndexGrowthSA"], start = c(2014,1), end = c(2014,1))#observation## Jan
## 2014 0.4279801- The forecast of .415 seems closer to the actual .429
- But the confidence bound seems wider.
ARMA
AutoRegressive Moving Average (ARMA) models are the bedrock of time series modeling. The basic idea is that every time series has its own temporal `fingerprint”. That fingerprint may depend upon its own past (AR), and/or the errors from that specification may also depend upon their own past (MA).
The further in the past that is relevant, the higher the “order” of the model. For instance, if housing prices in March depend upon housing prices in February and January, we have and order lag of 2, yielding AR(2).
- There could be economic inertia. e.g. higher prices generate higher demand, which prompts higher prices. This could continue until supply catches up, which could take some time.
- There could be a latent driver that itself was temporal persistent. Being latent, we don’t have that driver.
- Thankfully, we can use a recursive substitution trick to capture the missing feature. (See lecture notes for details). Intuitively, …
- Suppose \(HousePrice_{t}=f(CPI_{t})\) and \(CPI_{t}=f(CPI_{t-1})\).
- This implies that \(HousePrice_{t}=f(CPI_{t-1})\), but if we don’t have CPI (in this silly example) this we can’t estimate this model.
- Thankfully, \(HousePrice_{t-1}=f(CPI_{t-1})\), so we can proxy the persistent latent driver CPI by using housing prices own temporal persistence \(HousePrice_{t-1}=f(HousePrice_{t-1})\)
- This is the foundation for ARMA-type models
Detect ARMA patterns
acf2(window(df_adjts[,"PriceIndexGrowthSA"], end = c(2013,12)),
main = "ACF and PACF of Price Index Growth") 
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13]
## ACF 0.93 0.81 0.68 0.59 0.53 0.49 0.49 0.52 0.57 0.63 0.67 0.67 0.59
## PACF 0.93 -0.37 -0.01 0.22 -0.03 0.11 0.18 0.12 0.18 0.15 0.05 -0.23 -0.26
## [,14] [,15] [,16] [,17] [,18] [,19] [,20] [,21] [,22] [,23] [,24] [,25]
## ACF 0.49 0.38 0.31 0.27 0.25 0.26 0.28 0.31 0.34 0.36 0.33 0.26
## PACF -0.03 0.00 0.02 0.03 -0.10 0.03 -0.03 -0.03 0.04 0.04 -0.13 -0.02
## [,26] [,27] [,28] [,29] [,30] [,31] [,32] [,33] [,34] [,35] [,36] [,37]
## ACF 0.18 0.11 0.07 0.04 0.04 0.06 0.08 0.10 0.11 0.11 0.07 0.01
## PACF 0.07 0.00 -0.10 0.05 0.05 0.00 -0.05 -0.02 -0.05 0.03 -0.05 -0.02
## [,38] [,39] [,40] [,41] [,42] [,43] [,44] [,45] [,46] [,47] [,48]
## ACF -0.05 -0.10 -0.13 -0.15 -0.15 -0.13 -0.11 -0.10 -0.1 -0.11 -0.13
## PACF 0.00 -0.06 -0.06 0.01 -0.01 -0.03 -0.01 -0.03 0.0 0.07 0.06- The periodic swings in the ACF indicate some type of seasonality.
- The seasonal wave cresting roughly every twelve months is suggestive of an annual cycle.
- The spikes in the PACF suggest the possibility of a moving average
MA component or order 1 or 2.
- Reading the correlogram for this purpose is usually referred to as the Box Jenkins approach. There are automatic search tools as well.
We can consider a 12month lag.
acf2(window(df_adjts[,"PriceIndexGrowthSA"], end = c(2013,12))%>%diff(lag = 12),
main = "ACF and PACF of Price Index Growth Diff 12", max.lag = 36)
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13]
## ACF 0.88 0.75 0.62 0.54 0.46 0.40 0.35 0.31 0.28 0.21 0.14 0.05 0.05
## PACF 0.88 -0.11 -0.10 0.18 -0.08 0.04 0.03 -0.02 0.03 -0.20 -0.03 -0.08 0.30
## [,14] [,15] [,16] [,17] [,18] [,19] [,20] [,21] [,22] [,23] [,24] [,25]
## ACF 0.05 0.04 0.05 0.07 0.06 0.04 0.02 0.00 0.00 -0.01 -0.03 -0.04
## PACF -0.09 -0.11 0.23 -0.03 -0.15 0.04 0.02 -0.01 -0.08 -0.01 -0.08 0.11
## [,26] [,27] [,28] [,29] [,30] [,31] [,32] [,33] [,34] [,35] [,36]
## ACF -0.05 -0.03 -0.03 -0.03 -0.01 0.02 0.04 0.05 0.03 -0.01 -0.06
## PACF -0.03 -0.01 0.09 0.04 -0.02 0.07 -0.01 -0.02 -0.14 -0.07 -0.15- Notice the seasonal wave of autocorrelation is gone.
- The remaining ACF decay and the single PACF indicate a remaining MA component.
Fit ARMA
AR(1)
We can fit an arma model by indicating the number of AR lags and how many MA lags. In this case, let’s first an AR(1,0,0) \[often written as AR(1)\] \(Price_{t} = \alpha + \beta Price_{t-1}+u_{t}\)
ARMA1 <- arima(window(df_adjts[,"PriceIndexGrowthSA"], end = c(2013,12)),
order=c(1, 0, 0))
summary(ARMA1)##
## Call:
## arima(x = window(df_adjts[, "PriceIndexGrowthSA"], end = c(2013, 12)), order = c(1,
## 0, 0))
##
## Coefficients:
## ar1 intercept
## 0.9261 0.3062
## s.e. 0.0204 0.1569
##
## sigma^2 estimated as 0.04676: log likelihood = 35.34, aic = -64.68
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set -0.002324819 0.2162413 0.1499008 164.1931 262.1761 0.4686845
## ACF1
## Training set 0.3587264- A 1 unit change in house price growth this month is associated with a .9261 unit increase in house price growth next month; i.e. this series is highly persistent.
- The intercept is the expected level of house price growth next month if prices didn’t grow at all this month.
ARMA(1,1)
ARMA11 <- arima(window(df_adjts[,"PriceIndexGrowthSA"], end = c(2013,12)),
order=c(1, 0, 1))
summary(ARMA11)##
## Call:
## arima(x = window(df_adjts[, "PriceIndexGrowthSA"], end = c(2013, 12)), order = c(1,
## 0, 1))
##
## Coefficients:
## ar1 ma1 intercept
## 0.8905 0.3250 0.3054
## s.e. 0.0259 0.0444 0.1329
##
## sigma^2 estimated as 0.04099: log likelihood = 56.51, aic = -105.01
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set -0.002101165 0.2024476 0.1377998 147.0238 224.0294 0.4308494
## ACF1
## Training set 0.06232922Seasonal ARMA
The ARMA structure is similar in spirit to our seasonal decomposition. They both use a series’ own past for modeling. We can combine the two with a Seasonal ARMA.
The c(p,d,q) elements in the ARMA function indicate the AR(p), MA(q), and the number (d) of differences. Similarly, seasonal components can be added analogously.
Consider an ARMA(1,0,0)(1,1,2)12 would be \(Price_{t}=\alpha+\beta_{1} Price_{t-1}+\beta_{2}Price_{t-12}+\gamma_{1}u_{t-12}+\gamma_{2}u_{t-24}+u_{t}\)
We could apply the seasonal ARMA directly to the Price Growth or even the Price series (with differencing), but we’ve been working with the SA price growth series, which has a meaningful interpretation for us, so we’ll proceed with that. Moreover, we did note that some seasonality did remain in that SA series, so we can attempt to address here.
ARMA11S <- arima(window(df_adjts[,"PriceIndexGrowthSA"], end = c(2013,12)),
order=c(1, 0, 0), seasonal = list(order = c(0,1,2), period = 12))
summary(ARMA11S)##
## Call:
## arima(x = window(df_adjts[, "PriceIndexGrowthSA"], end = c(2013, 12)), order = c(1,
## 0, 0), seasonal = list(order = c(0, 1, 2), period = 12))
##
## Coefficients:
## ar1 sma1 sma2
## 0.9249 -0.4969 -0.1044
## s.e. 0.0219 0.0572 0.0607
##
## sigma^2 estimated as 0.03701: log likelihood = 68.25, aic = -128.5
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set -0.0005814306 0.1887761 0.1213594 -74.6149 307.0408 0.3794462
## ACF1
## Training set 0.1688262- We want the smallest AIC (even the most negative)
- The Seasonal model appears to have the smallest -128.53
Auto Detect Lag Length
We can search among the evaluation criterion to find the optimal ARMA order.
ARIMA_auto <- auto.arima(window(df_adjts[,"PriceIndexGrowthSA"], end = c(2013,12)), seasonal = TRUE)
summary(ARIMA_auto)## Series: window(df_adjts[, "PriceIndexGrowthSA"], end = c(2013, 12))
## ARIMA(2,0,2)(0,1,2)[12]
##
## Coefficients:
## ar1 ar2 ma1 ma2 sma1 sma2
## 0.2877 0.5216 0.8491 0.3394 -0.5532 -0.0983
## s.e. 0.1592 0.1483 0.1540 0.0570 0.0601 0.0621
##
## sigma^2 = 0.03436: log likelihood = 81.97
## AIC=-149.94 AICc=-149.57 BIC=-123.76
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set -0.001796371 0.1801185 0.1149167 -87.20039 294.0825 0.3593022
## ACF1
## Training set -0.002396919- Here, R found an ARMA(2,0,2) with a 2 month seasonal
component.
\(Price_{t}=\alpha+\beta_{1} Price_{t-1}+\beta_{2}Price_{t-2}+\gamma_{1}u_{t-1}+\gamma_{2}u_{t-2}+\gamma_{t-2}u_{t-12}+\gamma_{4}u_{t-24}+u_{t}\)
Post Estimation Diagnostics
Box.test(residuals(ARIMA_auto), type = "Ljung-Box") ##
## Box-Ljung test
##
## data: residuals(ARIMA_auto)
## X-squared = 0.001873, df = 1, p-value = 0.9655- Recall the Lung Box Ho: No Autocorrelation in the errors.
- The very high pvalue suggests we fail to reject (ie no remaining serial correlation).
- Suggest we fit the series reasonably well with its own temporal dynamics
tsdisplay(residuals(ARIMA_auto), main='Model Residuals')
- The visual diagnostics look pretty good.
- There are some annual spikes in the residuals
- There some issues with the model surrounding the GFC, which is not surprising.
Forecast Seasonal ARMA
Let’s use this ARMA model to predict one step ahead into the testing sample.
predict(ARIMA_auto, 1, prediction.interval = TRUE, level=0.95)## $pred
## Jan
## 2014 0.4861484
##
## $se
## Jan
## 2014 0.1853573And recall the observed value:
window(df_adjts[,"PriceIndexGrowthSA"], start = c(2014,1), end = c(2014,1))#observation## Jan
## 2014 0.4279801- The fit is pretty good within relatively small standard error.
ARMA-GARCH
So far we’ve been focused on the price growth series. We can use ARMA to model or forecast that growth rate using its own temporal persistence.
The variability of the series may ALSO be temporally persistent.
- There could be an excluding driver that is swinging widely (e.g. lumber prices might be rising dramatically amid a sudden forest fire)
- This variability feed in directly to our confidence intervals.
- Fundamentally, that variability matters because it might impact decision making (e.g. if house prices are very variable, I may not know if now is a good time to invest in a home)
- Variability of time series is often itself autoregressive. So we have persistence in the “level” and “variance” of a series.
- By imposing an autoregressive structure onto the residuals of the model we can capture this characteristic.
Visualize GARCH Effects
Let’s visualize the variance of the model residuals.
residualSqr = residuals(ARIMA_auto)^2
tsdisplay(residualSqr, main = "Residual Square") 
- Residual variance is proxied with the square of the residual
- Notice the persistence in the residual variance visa vis the ACF and PACF
- This is indicative of autoregressive volatility
- Notice also the spikes around the 2008 housing crisis, where uncertainty in the housing market was terribly high
Test for GARCH
We can test formally for (G)ARCH type effects with an Engle LM Test, which determines if the residual variance is indeed related to its own past.
Ho: No ARCH effects (white noise)
Ha: ARCH effects
library(FinTS)
ArchTest(window(df_adjts[,"PriceIndexGrowthSA"], end = c(2013,12)), lags = 1, demean = TRUE)##
## ARCH LM-test; Null hypothesis: no ARCH effects
##
## data: window(df_adjts[, "PriceIndexGrowthSA"], end = c(2013, 12))
## Chi-squared = 229.53, df = 1, p-value < 2.2e-16- Notice that the small p value suggests we reject the null
- Implying we have ARCH-type effects
- Our model fit the ``level” of the series well, but something remains in the residual that is persistent.
Fitting an ARMA-GARCH
Augmenting our ARMA model with GARCH effects is easy in R. One limitation is the inability to forecast seasonal ARIMA-GARCH directly. So, we will demonstrate without seasonality.
Find approprate ARMA
ARIMA_nonseasonal<-auto.arima(window(df_adjts[,"PriceIndexGrowthSA"], end = c(2013,12)),
seasonal = FALSE)
predict(ARIMA_nonseasonal,1, prediction.interval = TRUE, level=0.95)## $pred
## Jan
## 2014 0.5224658
##
## $se
## Jan
## 2014 0.1916616Fit ARMA-GARCH
We’ll use that optimal ARMA structure as the basis for the “mean” specification. We’ll also use the “workhorse” GARCH specification of a GARCH(1,1), which includes one lag of the variance of the errors (GARCH) as well as one lag of the model residuals (ARCH).
library(rugarch)
spec <- ugarchspec(variance.model=list(model="sGARCH", garchOrder=c(1,1)),
mean.model=list(include.mean = T, armaOrder=c(3,2)),
distribution.model="norm")
garch_fit <- ugarchfit(spec = spec, data = window(df_adjts[,"PriceIndexGrowthSA"], end = c(2013,12)))
coef(garch_fit)## mu ar1 ar2 ar3 ma1
## 0.3868365484 0.8108704370 -0.0811580412 0.1094611286 0.5506303108
## ma2 omega alpha1 beta1
## 0.4649426613 0.0001840998 0.0754721723 0.92352782751-step Ahead ARMGARCH
ugarchforecast(fitORspec = garch_fit, n.ahead = 1)##
## *------------------------------------*
## * GARCH Model Forecast *
## *------------------------------------*
## Model: sGARCH
## Horizon: 1
## Roll Steps: 0
## Out of Sample: 0
##
## 0-roll forecast [T0=Dec 2013]:
## Series Sigma
## T+1 0.4295 0.2556For comparison, the observed testing sample observation was
window(df_adjts[,"PriceIndexGrowthSA"], start = c(2014,1), end = c(2014,1))#observation## Jan
## 2014 0.4279801- Notice the fit is quite good.
NNET
A feed forward neural network with hidden layers can leverage lagged inputs for forecasting.
Fit the NNET
set.seed(123) # set random seed so that we can replicate the example
NNET <- nnetar(window(df_adjts[,"PriceIndexGrowthSA"], end = c(2013,12))) Forecast the NNET
In the following, h = # of periods ahead to forecast. PI = True computes prediction intervals.
forecast(NNET, h = 1, PI=TRUE) ## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Jan 2014 0.4763293 0.2509084 0.7021914 0.1535827 0.8470947As a reminder
window(df_adjts[,"PriceIndexGrowthSA"], start = c(2014,1), end = c(2014,1))#observation## Jan
## 2014 0.4279801Random Forest
We must use caution when implementing a random forest model with time series data. As you may know, random forests are built by randomly selecting training data, upon which we estimate a regression tree. The process is repeated and then averaged together in an ensemble. The process of randomly selecting the observations to be used in the training set will disrupt the temporal persistence of the data. We need to ensure that the we are always using training data that occurred prior to the testing data. As such, we use forward-validation rather than k-fold cross validation.
Prep data
This step of prepping the data is sometimes called time embedding.
lag_order <- 12 # the lags number you want to include in your model
RF_data <- embed(df_adjts[,"PriceIndexGrowthSA"], lag_order + 1) %>% # embedding matrix
`colnames<-`(c("Raw",paste("Lag",1:lag_order, sep=''))) %>% ts(end = end(df_adjts), frequency = 12)
head(RF_data)## Raw Lag1 Lag2 Lag3 Lag4 Lag5 Lag6
## Feb 1988 0.6735593 0.7251563 0.8027668 0.6896900 0.7843665 0.8086141 0.7738515
## Mar 1988 0.2661027 0.6735593 0.7251563 0.8027668 0.6896900 0.7843665 0.8086141
## Apr 1988 0.1211820 0.2661027 0.6735593 0.7251563 0.8027668 0.6896900 0.7843665
## May 1988 0.3176342 0.1211820 0.2661027 0.6735593 0.7251563 0.8027668 0.6896900
## Jun 1988 0.4735772 0.3176342 0.1211820 0.2661027 0.6735593 0.7251563 0.8027668
## Jul 1988 0.6659990 0.4735772 0.3176342 0.1211820 0.2661027 0.6735593 0.7251563
## Lag7 Lag8 Lag9 Lag10 Lag11 Lag12
## Feb 1988 0.6555892 0.5613103 0.3156842 0.2352270 0.1949953 0.8141863
## Mar 1988 0.7738515 0.6555892 0.5613103 0.3156842 0.2352270 0.1949953
## Apr 1988 0.8086141 0.7738515 0.6555892 0.5613103 0.3156842 0.2352270
## May 1988 0.7843665 0.8086141 0.7738515 0.6555892 0.5613103 0.3156842
## Jun 1988 0.6896900 0.7843665 0.8086141 0.7738515 0.6555892 0.5613103
## Jul 1988 0.8027668 0.6896900 0.7843665 0.8086141 0.7738515 0.6555892Assign Train and Test Samples
RF_traindata <- window(RF_data, end = c(2013, 12))
RF_testdata <- window(RF_data, start = c(2014, 1))
y_train <- RF_traindata[, 1] # the target
X_train <- RF_traindata[, -1] # everything but the target
y_test <- RF_testdata[, 1]
X_test <- RF_testdata[, -1]Fit the Random Forest
library(randomForest)
set.seed(123) # set random seed so that we can replicate the example
RF_fit <- randomForest(X_train, y_train)Forecast the Random Forest
predict(RF_fit, head(X_test,1)) ## 1
## 0.3714989And a reminder of the testing sample
window(df_adjts[,"PriceIndexGrowthSA"], start = c(2014,1), end = c(2014,1))## Jan
## 2014 0.4279801- We get a relatively accurate forecast.
- Unfortunately, a confidence interval isn’t provided natively.
Prophet
library(prophet)Prep Data
Prophet_traindata <- df_adj[1:323,]
Prophet_testdata <- df_adj[324:nrow(df_adj) ,]
df_prophet <- Prophet_traindata %>%
select(-HousingStartGrowth) %>%
rename(ds = Date) %>%
rename(y = PriceIndexGrowthSA)Forecast with Prophet
m <- prophet(df_prophet)
future <- make_future_dataframe(m, periods = 365, freq = "day", include_history = TRUE) # periods: Int number of periods to forecast forward.
forecast <- predict(m, future)
forecast## ds trend additive_terms additive_terms_lower
## 1 1987-02-01 0.4028826 -0.0356499451 -0.0356499451
## 2 1987-03-01 0.4022385 -0.1196667639 -0.1196667639
## 3 1987-04-01 0.4015253 -0.0844410783 -0.0844410783
## 4 1987-05-01 0.4008352 -0.0200013993 -0.0200013993
## 5 1987-06-01 0.4001220 0.0297437350 0.0297437350
## 6 1987-07-01 0.3994319 0.0609799763 0.0609799763
## 7 1987-08-01 0.3987187 0.0666822180 0.0666822180
## 8 1987-09-01 0.3980056 0.0366828934 0.0366828934
## 9 1987-10-01 0.3973154 0.0124852690 0.0124852690
## 10 1987-11-01 0.3966023 0.0030460452 0.0030460452
## 11 1987-12-01 0.3959122 0.0169109489 0.0169109489
## 12 1988-01-01 0.3951990 0.0079288625 0.0079288625
## 13 1988-02-01 0.3944859 -0.0332513176 -0.0332513176
## 14 1988-03-01 0.3938187 -0.1104020156 -0.1104020156
## 15 1988-04-01 0.3931056 -0.0844846277 -0.0844846277
## 16 1988-05-01 0.3924154 -0.0201696967 -0.0201696967
## 17 1988-06-01 0.3917023 0.0280727129 0.0280727129
## 18 1988-07-01 0.3910121 0.0612855477 0.0612855477
## 19 1988-08-01 0.3902990 0.0673855969 0.0673855969
## 20 1988-09-01 0.3895858 0.0361204385 0.0361204385
## 21 1988-10-01 0.3888957 0.0137233500 0.0137233500
## 22 1988-11-01 0.3881826 0.0058882113 0.0058882113
## 23 1988-12-01 0.3874924 0.0133089833 0.0133089833
## 24 1989-01-01 0.3867793 0.0060550578 0.0060550578
## 25 1989-02-01 0.3860661 -0.0406241970 -0.0406241970
## 26 1989-03-01 0.3854220 -0.1134952892 -0.1134952892
## 27 1989-04-01 0.3847088 -0.0845261697 -0.0845261697
## 28 1989-05-01 0.3840187 -0.0201315617 -0.0201315617
## 29 1989-06-01 0.3833055 0.0286350603 0.0286350603
## 30 1989-07-01 0.3826154 0.0612087408 0.0612087408
## 31 1989-08-01 0.3819022 0.0671680630 0.0671680630
## 32 1989-09-01 0.3811891 0.0363152097 0.0363152097
## 33 1989-10-01 0.3804990 0.0133117282 0.0133117282
## 34 1989-11-01 0.3797858 0.0049075936 0.0049075936
## 35 1989-12-01 0.3790957 0.0144860745 0.0144860745
## 36 1990-01-01 0.3783825 0.0066836927 0.0066836927
## 37 1990-02-01 0.3776694 -0.0381081177 -0.0381081177
## 38 1990-03-01 0.3770252 -0.1165847501 -0.1165847501
## 39 1990-04-01 0.3763121 -0.0845117476 -0.0845117476
## 40 1990-05-01 0.3756219 -0.0200751150 -0.0200751150
## 41 1990-06-01 0.3749088 0.0291925114 0.0291925114
## 42 1990-07-01 0.3742187 0.0611069758 0.0611069758
## 43 1990-08-01 0.3735055 0.0669334627 0.0669334627
## 44 1990-09-01 0.3727924 0.0365025579 0.0365025579
## 45 1990-10-01 0.3721022 0.0128987062 0.0128987062
## 46 1990-11-01 0.3713891 0.0039598955 0.0039598955
## 47 1990-12-01 0.3706989 0.0156872258 0.0156872258
## 48 1991-01-01 0.3699858 0.0073087629 0.0073087629
## 49 1991-02-01 0.3692726 -0.0356499451 -0.0356499451
## 50 1991-03-01 0.3686285 -0.1196667639 -0.1196667639
## 51 1991-04-01 0.3679153 -0.0844410783 -0.0844410783
## 52 1991-05-01 0.3672252 -0.0200013993 -0.0200013993
## 53 1991-06-01 0.3665121 0.0297437350 0.0297437350
## 54 1991-07-01 0.3658219 0.0609799763 0.0609799763
## 55 1991-08-01 0.3651088 0.0666822180 0.0666822180
## 56 1991-09-01 0.3643956 0.0366828934 0.0366828934
## 57 1991-10-01 0.3637055 0.0124852690 0.0124852690
## 58 1991-11-01 0.3629923 0.0030460452 0.0030460452
## 59 1991-12-01 0.3623022 0.0169109489 0.0169109489
## 60 1992-01-01 0.3615890 0.0079288625 0.0079288625
## 61 1992-02-01 0.3608759 -0.0332513176 -0.0332513176
## 62 1992-03-01 0.3602087 -0.1104020156 -0.1104020156
## 63 1992-04-01 0.3594956 -0.0844846277 -0.0844846277
## 64 1992-05-01 0.3588055 -0.0201696967 -0.0201696967
## 65 1992-06-01 0.3580923 0.0280727129 0.0280727129
## 66 1992-07-01 0.3574022 0.0612855477 0.0612855477
## 67 1992-08-01 0.3566890 0.0673855969 0.0673855969
## 68 1992-09-01 0.3559759 0.0361204385 0.0361204385
## 69 1992-10-01 0.3552857 0.0137233500 0.0137233500
## 70 1992-11-01 0.3545726 0.0058882113 0.0058882113
## 71 1992-12-01 0.3538824 0.0133089833 0.0133089833
## 72 1993-01-01 0.3531693 0.0060550578 0.0060550578
## 73 1993-02-01 0.3524561 -0.0406241970 -0.0406241970
## 74 1993-03-01 0.3518120 -0.1134952892 -0.1134952892
## 75 1993-04-01 0.3510989 -0.0845261697 -0.0845261697
## 76 1993-05-01 0.3504087 -0.0201315617 -0.0201315617
## 77 1993-06-01 0.3496956 0.0286350603 0.0286350603
## 78 1993-07-01 0.3490054 0.0612087408 0.0612087408
## 79 1993-08-01 0.3482923 0.0671680630 0.0671680630
## 80 1993-09-01 0.3475791 0.0363152097 0.0363152097
## 81 1993-10-01 0.3468890 0.0133117282 0.0133117282
## 82 1993-11-01 0.3461758 0.0049075936 0.0049075936
## 83 1993-12-01 0.3454857 0.0144860745 0.0144860745
## 84 1994-01-01 0.3447725 0.0066836927 0.0066836927
## 85 1994-02-01 0.3440594 -0.0381081177 -0.0381081177
## 86 1994-03-01 0.3434153 -0.1165847501 -0.1165847501
## 87 1994-04-01 0.3427021 -0.0845117476 -0.0845117476
## 88 1994-05-01 0.3420120 -0.0200751150 -0.0200751150
## 89 1994-06-01 0.3412988 0.0291925114 0.0291925114
## 90 1994-07-01 0.3406087 0.0611069758 0.0611069758
## 91 1994-08-01 0.3398955 0.0669334627 0.0669334627
## 92 1994-09-01 0.3391824 0.0365025579 0.0365025579
## 93 1994-10-01 0.3384922 0.0128987062 0.0128987062
## 94 1994-11-01 0.3377791 0.0039598955 0.0039598955
## 95 1994-12-01 0.3370890 0.0156872258 0.0156872258
## 96 1995-01-01 0.3363758 0.0073087629 0.0073087629
## 97 1995-02-01 0.3356627 -0.0356499451 -0.0356499451
## 98 1995-03-01 0.3350185 -0.1196667639 -0.1196667639
## 99 1995-04-01 0.3343054 -0.0844410783 -0.0844410783
## 100 1995-05-01 0.3336152 -0.0200013993 -0.0200013993
## 101 1995-06-01 0.3329021 0.0297437350 0.0297437350
## 102 1995-07-01 0.3322119 0.0609799763 0.0609799763
## 103 1995-08-01 0.3314988 0.0666822180 0.0666822180
## 104 1995-09-01 0.3307856 0.0366828934 0.0366828934
## 105 1995-10-01 0.3300955 0.0124852690 0.0124852690
## 106 1995-11-01 0.3293824 0.0030460452 0.0030460452
## 107 1995-12-01 0.3286922 0.0169109489 0.0169109489
## 108 1996-01-01 0.3279791 0.0079288625 0.0079288625
## 109 1996-02-01 0.3272659 -0.0332513176 -0.0332513176
## 110 1996-03-01 0.3265988 -0.1104020156 -0.1104020156
## 111 1996-04-01 0.3258856 -0.0844846277 -0.0844846277
## 112 1996-05-01 0.3251955 -0.0201696967 -0.0201696967
## 113 1996-06-01 0.3244823 0.0280727129 0.0280727129
## 114 1996-07-01 0.3237922 0.0612855477 0.0612855477
## 115 1996-08-01 0.3230790 0.0673855969 0.0673855969
## 116 1996-09-01 0.3223659 0.0361204385 0.0361204385
## 117 1996-10-01 0.3216758 0.0137233500 0.0137233500
## 118 1996-11-01 0.3209626 0.0058882113 0.0058882113
## 119 1996-12-01 0.3202725 0.0133089833 0.0133089833
## 120 1997-01-01 0.3195593 0.0060550578 0.0060550578
## 121 1997-02-01 0.3188462 -0.0406241970 -0.0406241970
## 122 1997-03-01 0.3182020 -0.1134952892 -0.1134952892
## 123 1997-04-01 0.3174889 -0.0845261697 -0.0845261697
## 124 1997-05-01 0.3167987 -0.0201315617 -0.0201315617
## 125 1997-06-01 0.3160856 0.0286350603 0.0286350603
## 126 1997-07-01 0.3153955 0.0612087408 0.0612087408
## 127 1997-08-01 0.3146823 0.0671680630 0.0671680630
## 128 1997-09-01 0.3139691 0.0363152097 0.0363152097
## 129 1997-10-01 0.3132790 0.0133117282 0.0133117282
## 130 1997-11-01 0.3125659 0.0049075936 0.0049075936
## 131 1997-12-01 0.3118757 0.0144860745 0.0144860745
## 132 1998-01-01 0.3111626 0.0066836927 0.0066836927
## 133 1998-02-01 0.3104494 -0.0381081177 -0.0381081177
## 134 1998-03-01 0.3098053 -0.1165847501 -0.1165847501
## 135 1998-04-01 0.3090921 -0.0845117476 -0.0845117476
## 136 1998-05-01 0.3084018 -0.0200751150 -0.0200751150
## 137 1998-06-01 0.3076885 0.0291925114 0.0291925114
## 138 1998-07-01 0.3069981 0.0611069758 0.0611069758
## 139 1998-08-01 0.3062848 0.0669334627 0.0669334627
## 140 1998-09-01 0.3055715 0.0365025579 0.0365025579
## 141 1998-10-01 0.3048812 0.0128987062 0.0128987062
## 142 1998-11-01 0.3041678 0.0039598955 0.0039598955
## 143 1998-12-01 0.3034775 0.0156872258 0.0156872258
## 144 1999-01-01 0.3027642 0.0073087629 0.0073087629
## 145 1999-02-01 0.3020508 -0.0356499451 -0.0356499451
## 146 1999-03-01 0.3014000 -0.1196667639 -0.1196667639
## 147 1999-04-01 0.3006794 -0.0844410783 -0.0844410783
## 148 1999-05-01 0.2999821 -0.0200013993 -0.0200013993
## 149 1999-06-01 0.2992615 0.0297437350 0.0297437350
## 150 1999-07-01 0.2985641 0.0609799763 0.0609799763
## 151 1999-08-01 0.2978435 0.0666822180 0.0666822180
## 152 1999-09-01 0.2971229 0.0366828934 0.0366828934
## 153 1999-10-01 0.2964256 0.0124852690 0.0124852690
## 154 1999-11-01 0.2957050 0.0030460452 0.0030460452
## 155 1999-12-01 0.2950076 0.0169109489 0.0169109489
## 156 2000-01-01 0.2942771 0.0079288625 0.0079288625
## 157 2000-02-01 0.2935465 -0.0332513176 -0.0332513176
## 158 2000-03-01 0.2928631 -0.1104020156 -0.1104020156
## 159 2000-04-01 0.2921326 -0.0844846277 -0.0844846277
## 160 2000-05-01 0.2914256 -0.0201696967 -0.0201696967
## 161 2000-06-01 0.2906950 0.0280727129 0.0280727129
## 162 2000-07-01 0.2899880 0.0612855477 0.0612855477
## 163 2000-08-01 0.2892575 0.0673855969 0.0673855969
## 164 2000-09-01 0.2885269 0.0361204385 0.0361204385
## 165 2000-10-01 0.2878199 0.0137233500 0.0137233500
## 166 2000-11-01 0.2870807 0.0058882113 0.0058882113
## 167 2000-12-01 0.2863652 0.0133089833 0.0133089833
## 168 2001-01-01 0.2856260 0.0060550578 0.0060550578
## 169 2001-02-01 0.2848867 -0.0406241970 -0.0406241970
## 170 2001-03-01 0.2842190 -0.1134952892 -0.1134952892
## 171 2001-04-01 0.2834797 -0.0845261697 -0.0845261697
## 172 2001-05-01 0.2827643 -0.0201315617 -0.0201315617
## 173 2001-06-01 0.2820251 0.0286350603 0.0286350603
## 174 2001-07-01 0.2813096 0.0612087408 0.0612087408
## 175 2001-08-01 0.2805704 0.0671680630 0.0671680630
## 176 2001-09-01 0.2798311 0.0363152097 0.0363152097
## 177 2001-10-01 0.2791037 0.0133117282 0.0133117282
## 178 2001-11-01 0.2783520 0.0049075936 0.0049075936
## 179 2001-12-01 0.2776245 0.0144860745 0.0144860745
## 180 2002-01-01 0.2768728 0.0066836927 0.0066836927
## 181 2002-02-01 0.2761211 -0.0381081177 -0.0381081177
## 182 2002-03-01 0.2754422 -0.1165847501 -0.1165847501
## 183 2002-04-01 0.2746905 -0.0845117476 -0.0845117476
## 184 2002-05-01 0.2739630 -0.0200751150 -0.0200751150
## 185 2002-06-01 0.2732113 0.0291925114 0.0291925114
## 186 2002-07-01 0.2724839 0.0611069758 0.0611069758
## 187 2002-08-01 0.2717171 0.0669334627 0.0669334627
## 188 2002-09-01 0.2709504 0.0365025579 0.0365025579
## 189 2002-10-01 0.2702083 0.0128987062 0.0128987062
## 190 2002-11-01 0.2694416 0.0039598955 0.0039598955
## 191 2002-12-01 0.2686996 0.0156872258 0.0156872258
## 192 2003-01-01 0.2679328 0.0073087629 0.0073087629
## 193 2003-02-01 0.2671660 -0.0356499451 -0.0356499451
## 194 2003-03-01 0.2664735 -0.1196667639 -0.1196667639
## 195 2003-04-01 0.2657067 -0.0844410783 -0.0844410783
## 196 2003-05-01 0.2649647 -0.0200013993 -0.0200013993
## 197 2003-06-01 0.2641818 0.0297437350 0.0297437350
## 198 2003-07-01 0.2634242 0.0609799763 0.0609799763
## 199 2003-08-01 0.2626413 0.0666822180 0.0666822180
## 200 2003-09-01 0.2618584 0.0366828934 0.0366828934
## 201 2003-10-01 0.2611007 0.0124852690 0.0124852690
## 202 2003-11-01 0.2603178 0.0030460452 0.0030460452
## 203 2003-12-01 0.2595602 0.0169109489 0.0169109489
## 204 2004-01-01 0.2587773 0.0079288625 0.0079288625
## 205 2004-02-01 0.2579944 -0.0332513176 -0.0332513176
## 206 2004-03-01 0.2572620 -0.1104020156 -0.1104020156
## 207 2004-04-01 0.2564791 -0.0844846277 -0.0844846277
## 208 2004-05-01 0.2557145 -0.0201696967 -0.0201696967
## 209 2004-06-01 0.2549245 0.0280727129 0.0280727129
## 210 2004-07-01 0.2541599 0.0612855477 0.0612855477
## 211 2004-08-01 0.2533698 0.0673855969 0.0673855969
## 212 2004-09-01 0.2525798 0.0361204385 0.0361204385
## 213 2004-10-01 0.2518152 0.0137233500 0.0137233500
## 214 2004-11-01 0.2510252 0.0058882113 0.0058882113
## 215 2004-12-01 0.2502606 0.0133089833 0.0133089833
## 216 2005-01-01 0.2494705 0.0060550578 0.0060550578
## 217 2005-02-01 0.2486805 -0.0406241970 -0.0406241970
## 218 2005-03-01 0.2479668 -0.1134952892 -0.1134952892
## 219 2005-04-01 0.2471768 -0.0845261697 -0.0845261697
## 220 2005-05-01 0.2464121 -0.0201315617 -0.0201315617
## 221 2005-06-01 0.2456221 0.0286350603 0.0286350603
## 222 2005-07-01 0.2448575 0.0612087408 0.0612087408
## 223 2005-08-01 0.2440674 0.0671680630 0.0671680630
## 224 2005-09-01 0.2432773 0.0363152097 0.0363152097
## 225 2005-10-01 0.2425127 0.0133117282 0.0133117282
## 226 2005-11-01 0.2417226 0.0049075936 0.0049075936
## 227 2005-12-01 0.2409580 0.0144860745 0.0144860745
## 228 2006-01-01 0.2401679 0.0066836927 0.0066836927
## 229 2006-02-01 0.2393778 -0.0381081177 -0.0381081177
## 230 2006-03-01 0.2386641 -0.1165847501 -0.1165847501
## 231 2006-04-01 0.2378741 -0.0845117476 -0.0845117476
## 232 2006-05-01 0.2371094 -0.0200751150 -0.0200751150
## 233 2006-06-01 0.2363194 0.0291925114 0.0291925114
## 234 2006-07-01 0.2355548 0.0611069758 0.0611069758
## 235 2006-08-01 0.2347647 0.0669334627 0.0669334627
## 236 2006-09-01 0.2339746 0.0365025579 0.0365025579
## 237 2006-10-01 0.2332100 0.0128987062 0.0128987062
## 238 2006-11-01 0.2324199 0.0039598955 0.0039598955
## 239 2006-12-01 0.2316553 0.0156872258 0.0156872258
## 240 2007-01-01 0.2308652 0.0073087629 0.0073087629
## 241 2007-02-01 0.2300751 -0.0356499451 -0.0356499451
## 242 2007-03-01 0.2293614 -0.1196667639 -0.1196667639
## 243 2007-04-01 0.2285713 -0.0844410783 -0.0844410783
## 244 2007-05-01 0.2278067 -0.0200013993 -0.0200013993
## 245 2007-06-01 0.2270166 0.0297437350 0.0297437350
## 246 2007-07-01 0.2262520 0.0609799763 0.0609799763
## 247 2007-08-01 0.2254619 0.0666822180 0.0666822180
## 248 2007-09-01 0.2246718 0.0366828934 0.0366828934
## 249 2007-10-01 0.2239072 0.0124852690 0.0124852690
## 250 2007-11-01 0.2231171 0.0030460452 0.0030460452
## 251 2007-12-01 0.2223525 0.0169109489 0.0169109489
## 252 2008-01-01 0.2215624 0.0079288625 0.0079288625
## 253 2008-02-01 0.2207723 -0.0332513176 -0.0332513176
## 254 2008-03-01 0.2200332 -0.1104020156 -0.1104020156
## 255 2008-04-01 0.2192431 -0.0844846277 -0.0844846277
## 256 2008-05-01 0.2184785 -0.0201696967 -0.0201696967
## 257 2008-06-01 0.2176884 0.0280727129 0.0280727129
## 258 2008-07-01 0.2169238 0.0612855477 0.0612855477
## 259 2008-08-01 0.2161337 0.0673855969 0.0673855969
## 260 2008-09-01 0.2153436 0.0361204385 0.0361204385
## 261 2008-10-01 0.2145790 0.0137233500 0.0137233500
## 262 2008-11-01 0.2137889 0.0058882113 0.0058882113
## 263 2008-12-01 0.2130242 0.0133089833 0.0133089833
## 264 2009-01-01 0.2122342 0.0060550578 0.0060550578
## 265 2009-02-01 0.2114441 -0.0406241970 -0.0406241970
## 266 2009-03-01 0.2107304 -0.1134952892 -0.1134952892
## 267 2009-04-01 0.2099403 -0.0845261697 -0.0845261697
## 268 2009-05-01 0.2091757 -0.0201315617 -0.0201315617
## 269 2009-06-01 0.2083856 0.0286350603 0.0286350603
## 270 2009-07-01 0.2076210 0.0612087408 0.0612087408
## 271 2009-08-01 0.2068309 0.0671680630 0.0671680630
## 272 2009-09-01 0.2060408 0.0363152097 0.0363152097
## 273 2009-10-01 0.2052762 0.0133117282 0.0133117282
## 274 2009-11-01 0.2044861 0.0049075936 0.0049075936
## 275 2009-12-01 0.2037215 0.0144860745 0.0144860745
## 276 2010-01-01 0.2029314 0.0066836927 0.0066836927
## 277 2010-02-01 0.2021413 -0.0381081177 -0.0381081177
## 278 2010-03-01 0.2014277 -0.1165847501 -0.1165847501
## 279 2010-04-01 0.2006376 -0.0845117476 -0.0845117476
## 280 2010-05-01 0.1998730 -0.0200751150 -0.0200751150
## 281 2010-06-01 0.1990829 0.0291925114 0.0291925114
## 282 2010-07-01 0.1983182 0.0611069758 0.0611069758
## 283 2010-08-01 0.1975281 0.0669334627 0.0669334627
## 284 2010-09-01 0.1967381 0.0365025579 0.0365025579
## 285 2010-10-01 0.1959734 0.0128987062 0.0128987062
## 286 2010-11-01 0.1951833 0.0039598955 0.0039598955
## 287 2010-12-01 0.1944187 0.0156872258 0.0156872258
## 288 2011-01-01 0.1936286 0.0073087629 0.0073087629
## 289 2011-02-01 0.1928385 -0.0356499451 -0.0356499451
## 290 2011-03-01 0.1921249 -0.1196667639 -0.1196667639
## 291 2011-04-01 0.1913348 -0.0844410783 -0.0844410783
## 292 2011-05-01 0.1905702 -0.0200013993 -0.0200013993
## 293 2011-06-01 0.1897801 0.0297437350 0.0297437350
## 294 2011-07-01 0.1890155 0.0609799763 0.0609799763
## 295 2011-08-01 0.1882254 0.0666822180 0.0666822180
## 296 2011-09-01 0.1874353 0.0366828934 0.0366828934
## 297 2011-10-01 0.1866707 0.0124852690 0.0124852690
## 298 2011-11-01 0.1858806 0.0030460452 0.0030460452
## 299 2011-12-01 0.1851160 0.0169109489 0.0169109489
## 300 2012-01-01 0.1843259 0.0079288625 0.0079288625
## 301 2012-02-01 0.1835358 -0.0332513176 -0.0332513176
## 302 2012-03-01 0.1827967 -0.1104020156 -0.1104020156
## 303 2012-04-01 0.1820066 -0.0844846277 -0.0844846277
## 304 2012-05-01 0.1812419 -0.0201696967 -0.0201696967
## 305 2012-06-01 0.1804519 0.0280727129 0.0280727129
## 306 2012-07-01 0.1796872 0.0612855477 0.0612855477
## 307 2012-08-01 0.1788971 0.0673855969 0.0673855969
## 308 2012-09-01 0.1781070 0.0361204385 0.0361204385
## 309 2012-10-01 0.1773424 0.0137233500 0.0137233500
## 310 2012-11-01 0.1765523 0.0058882113 0.0058882113
## 311 2012-12-01 0.1757877 0.0133089833 0.0133089833
## 312 2013-01-01 0.1749976 0.0060550578 0.0060550578
## 313 2013-02-01 0.1742075 -0.0406241970 -0.0406241970
## 314 2013-03-01 0.1734939 -0.1134952892 -0.1134952892
## 315 2013-04-01 0.1727038 -0.0845261697 -0.0845261697
## 316 2013-05-01 0.1719392 -0.0201315617 -0.0201315617
## 317 2013-06-01 0.1711491 0.0286350603 0.0286350603
## 318 2013-07-01 0.1703845 0.0612087408 0.0612087408
## 319 2013-08-01 0.1695944 0.0671680630 0.0671680630
## 320 2013-09-01 0.1688043 0.0363152097 0.0363152097
## 321 2013-10-01 0.1680397 0.0133117282 0.0133117282
## 322 2013-11-01 0.1672496 0.0049075936 0.0049075936
## 323 2013-12-01 0.1664850 0.0144860745 0.0144860745
## 324 2013-12-02 0.1664595 0.0099362179 0.0099362179
## 325 2013-12-03 0.1664340 0.0058573769 0.0058573769
## 326 2013-12-04 0.1664085 0.0023220897 0.0023220897
## 327 2013-12-05 0.1663830 -0.0006133726 -0.0006133726
## 328 2013-12-06 0.1663575 -0.0029103283 -0.0029103283
## 329 2013-12-07 0.1663320 -0.0045485019 -0.0045485019
## 330 2013-12-08 0.1663066 -0.0055263119 -0.0055263119
## 331 2013-12-09 0.1662811 -0.0058606221 -0.0058606221
## 332 2013-12-10 0.1662556 -0.0055859637 -0.0055859637
## 333 2013-12-11 0.1662301 -0.0047532525 -0.0047532525
## 334 2013-12-12 0.1662046 -0.0034280375 -0.0034280375
## 335 2013-12-13 0.1661791 -0.0016883342 -0.0016883342
## 336 2013-12-14 0.1661536 0.0003778938 0.0003778938
## 337 2013-12-15 0.1661282 0.0026755305 0.0026755305
## 338 2013-12-16 0.1661027 0.0051052964 0.0051052964
## 339 2013-12-17 0.1660772 0.0075668428 0.0075668428
## 340 2013-12-18 0.1660517 0.0099618578 0.0099618578
## 341 2013-12-19 0.1660262 0.0121970912 0.0121970912
## 342 2013-12-20 0.1660007 0.0141872071 0.0141872071
## 343 2013-12-21 0.1659752 0.0158573774 0.0158573774
## 344 2013-12-22 0.1659497 0.0171455371 0.0171455371
## 345 2013-12-23 0.1659243 0.0180042321 0.0180042321
## 346 2013-12-24 0.1658988 0.0184020009 0.0184020009
## 347 2013-12-25 0.1658733 0.0183242474 0.0183242474
## 348 2013-12-26 0.1658478 0.0177735738 0.0177735738
## 349 2013-12-27 0.1658223 0.0167695622 0.0167695622
## 350 2013-12-28 0.1657968 0.0153480074 0.0153480074
## 351 2013-12-29 0.1657713 0.0135596200 0.0135596200
## 352 2013-12-30 0.1657458 0.0114682367 0.0114682367
## 353 2013-12-31 0.1657204 0.0091485888 0.0091485888
## 354 2014-01-01 0.1656949 0.0066836927 0.0066836927
## 355 2014-01-02 0.1656694 0.0041619395 0.0041619395
## 356 2014-01-03 0.1656439 0.0016739719 0.0016739719
## 357 2014-01-04 0.1656184 -0.0006905608 -0.0006905608
## 358 2014-01-05 0.1655929 -0.0028462675 -0.0028462675
## 359 2014-01-06 0.1655674 -0.0047152020 -0.0047152020
## 360 2014-01-07 0.1655420 -0.0062298707 -0.0062298707
## 361 2014-01-08 0.1655165 -0.0073359663 -0.0073359663
## 362 2014-01-09 0.1654910 -0.0079947394 -0.0079947394
## 363 2014-01-10 0.1654655 -0.0081849281 -0.0081849281
## 364 2014-01-11 0.1654400 -0.0079041795 -0.0079041795
## 365 2014-01-12 0.1654145 -0.0071699089 -0.0071699089
## 366 2014-01-13 0.1653890 -0.0060195616 -0.0060195616
## 367 2014-01-14 0.1653635 -0.0045102533 -0.0045102533
## 368 2014-01-15 0.1653381 -0.0027177912 -0.0027177912
## 369 2014-01-16 0.1653126 -0.0007350877 -0.0007350877
## 370 2014-01-17 0.1652871 0.0013299980 0.0013299980
## 371 2014-01-18 0.1652616 0.0033573399 0.0033573399
## 372 2014-01-19 0.1652361 0.0052176799 0.0052176799
## 373 2014-01-20 0.1652106 0.0067761383 0.0067761383
## 374 2014-01-21 0.1651851 0.0078960041 0.0078960041
## 375 2014-01-22 0.1651596 0.0084427060 0.0084427060
## 376 2014-01-23 0.1651342 0.0082878518 0.0082878518
## 377 2014-01-24 0.1651087 0.0073132261 0.0073132261
## 378 2014-01-25 0.1650832 0.0054146298 0.0054146298
## 379 2014-01-26 0.1650577 0.0025054528 0.0025054528
## 380 2014-01-27 0.1650322 -0.0014801282 -0.0014801282
## 381 2014-01-28 0.1650067 -0.0065844205 -0.0065844205
## 382 2014-01-29 0.1649812 -0.0128240437 -0.0128240437
## 383 2014-01-30 0.1649558 -0.0201883831 -0.0201883831
## 384 2014-01-31 0.1649303 -0.0286387261 -0.0286387261
## 385 2014-02-01 0.1649048 -0.0381081177 -0.0381081177
## 386 2014-02-02 0.1648793 -0.0485019530 -0.0485019530
## 387 2014-02-03 0.1648538 -0.0596993063 -0.0596993063
## 388 2014-02-04 0.1648283 -0.0715549732 -0.0715549732
## 389 2014-02-05 0.1648028 -0.0839021874 -0.0839021874
## 390 2014-02-06 0.1647773 -0.0965559495 -0.0965559495
## 391 2014-02-07 0.1647519 -0.1093168928 -0.1093168928
## 392 2014-02-08 0.1647264 -0.1219755932 -0.1219755932
## 393 2014-02-09 0.1647009 -0.1343172185 -0.1343172185
## 394 2014-02-10 0.1646754 -0.1461264003 -0.1461264003
## 395 2014-02-11 0.1646499 -0.1571922091 -0.1571922091
## 396 2014-02-12 0.1646244 -0.1673131027 -0.1673131027
## 397 2014-02-13 0.1645989 -0.1763017236 -0.1763017236
## 398 2014-02-14 0.1645734 -0.1839894210 -0.1839894210
## 399 2014-02-15 0.1645480 -0.1902303784 -0.1902303784
## 400 2014-02-16 0.1645225 -0.1949052399 -0.1949052399
## 401 2014-02-17 0.1644970 -0.1979241379 -0.1979241379
## 402 2014-02-18 0.1644715 -0.1992290429 -0.1992290429
## 403 2014-02-19 0.1644460 -0.1987953684 -0.1987953684
## 404 2014-02-20 0.1644205 -0.1966327895 -0.1966327895
## 405 2014-02-21 0.1643950 -0.1927852475 -0.1927852475
## 406 2014-02-22 0.1643695 -0.1873301383 -0.1873301383
## 407 2014-02-23 0.1643441 -0.1803767011 -0.1803767011
## 408 2014-02-24 0.1643186 -0.1720636451 -0.1720636451
## 409 2014-02-25 0.1642931 -0.1625560728 -0.1625560728
## 410 2014-02-26 0.1642676 -0.1520417739 -0.1520417739
## 411 2014-02-27 0.1642421 -0.1407269839 -0.1407269839
## 412 2014-02-28 0.1642166 -0.1288317111 -0.1288317111
## 413 2014-03-01 0.1641911 -0.1165847501 -0.1165847501
## 414 2014-03-02 0.1641657 -0.1042185066 -0.1042185066
## 415 2014-03-03 0.1641402 -0.0919637607 -0.0919637607
## 416 2014-03-04 0.1641147 -0.0800445029 -0.0800445029
## 417 2014-03-05 0.1640892 -0.0686729668 -0.0686729668
## 418 2014-03-06 0.1640637 -0.0580449843 -0.0580449843
## 419 2014-03-07 0.1640382 -0.0483357759 -0.0483357759
## 420 2014-03-08 0.1640127 -0.0396962776 -0.0396962776
## 421 2014-03-09 0.1639872 -0.0322500930 -0.0322500930
## 422 2014-03-10 0.1639618 -0.0260911406 -0.0260911406
## 423 2014-03-11 0.1639363 -0.0212820497 -0.0212820497
## 424 2014-03-12 0.1639108 -0.0178533358 -0.0178533358
## 425 2014-03-13 0.1638853 -0.0158033703 -0.0158033703
## 426 2014-03-14 0.1638598 -0.0150991338 -0.0150991338
## 427 2014-03-15 0.1638343 -0.0156777254 -0.0156777254
## 428 2014-03-16 0.1638088 -0.0174485794 -0.0174485794
## 429 2014-03-17 0.1637833 -0.0202963238 -0.0202963238
## 430 2014-03-18 0.1637579 -0.0240841985 -0.0240841985
## 431 2014-03-19 0.1637324 -0.0286579365 -0.0286579365
## 432 2014-03-20 0.1637069 -0.0338500030 -0.0338500030
## 433 2014-03-21 0.1636814 -0.0394840762 -0.0394840762
## 434 2014-03-22 0.1636559 -0.0453796503 -0.0453796503
## 435 2014-03-23 0.1636304 -0.0513566403 -0.0513566403
## 436 2014-03-24 0.1636049 -0.0572398671 -0.0572398671
## 437 2014-03-25 0.1635795 -0.0628633106 -0.0628633106
## 438 2014-03-26 0.1635540 -0.0680740208 -0.0680740208
## 439 2014-03-27 0.1635285 -0.0727355933 -0.0727355933
## 440 2014-03-28 0.1635030 -0.0767311244 -0.0767311244
## 441 2014-03-29 0.1634775 -0.0799655788 -0.0799655788
## 442 2014-03-30 0.1634520 -0.0823675185 -0.0823675185
## 443 2014-03-31 0.1634265 -0.0838901603 -0.0838901603
## 444 2014-04-01 0.1634010 -0.0845117476 -0.0845117476
## 445 2014-04-02 0.1633756 -0.0842352415 -0.0842352415
## 446 2014-04-03 0.1633501 -0.0830873539 -0.0830873539
## 447 2014-04-04 0.1633246 -0.0811169642 -0.0811169642
## 448 2014-04-05 0.1632991 -0.0783929757 -0.0783929757
## 449 2014-04-06 0.1632736 -0.0750016837 -0.0750016837
## 450 2014-04-07 0.1632481 -0.0710437390 -0.0710437390
## 451 2014-04-08 0.1632226 -0.0666307996 -0.0666307996
## 452 2014-04-09 0.1631971 -0.0618819710 -0.0618819710
## 453 2014-04-10 0.1631717 -0.0569201404 -0.0569201404
## 454 2014-04-11 0.1631462 -0.0518683085 -0.0518683085
## 455 2014-04-12 0.1631207 -0.0468460239 -0.0468460239
## 456 2014-04-13 0.1630952 -0.0419660183 -0.0419660183
## 457 2014-04-14 0.1630697 -0.0373311336 -0.0373311336
## 458 2014-04-15 0.1630442 -0.0330316223 -0.0330316223
## 459 2014-04-16 0.1630187 -0.0291428906 -0.0291428906
## 460 2014-04-17 0.1629933 -0.0257237388 -0.0257237388
## 461 2014-04-18 0.1629678 -0.0228151399 -0.0228151399
## 462 2014-04-19 0.1629423 -0.0204395795 -0.0204395795
## 463 2014-04-20 0.1629168 -0.0186009647 -0.0186009647
## 464 2014-04-21 0.1628913 -0.0172850922 -0.0172850922
## 465 2014-04-22 0.1628658 -0.0164606505 -0.0164606505
## 466 2014-04-23 0.1628403 -0.0160807157 -0.0160807157
## 467 2014-04-24 0.1628148 -0.0160846872 -0.0160846872
## 468 2014-04-25 0.1627894 -0.0164005959 -0.0164005959
## 469 2014-04-26 0.1627639 -0.0169477106 -0.0169477106
## 470 2014-04-27 0.1627384 -0.0176393584 -0.0176393584
## 471 2014-04-28 0.1627129 -0.0183858708 -0.0183858708
## 472 2014-04-29 0.1626874 -0.0190975652 -0.0190975652
## 473 2014-04-30 0.1626619 -0.0196876712 -0.0196876712
## 474 2014-05-01 0.1626364 -0.0200751150 -0.0200751150
## 475 2014-05-02 0.1626109 -0.0201870801 -0.0201870801
## 476 2014-05-03 0.1625855 -0.0199612695 -0.0199612695
## 477 2014-05-04 0.1625600 -0.0193478073 -0.0193478073
## 478 2014-05-05 0.1625345 -0.0183107263 -0.0183107263
## 479 2014-05-06 0.1625090 -0.0168290019 -0.0168290019
## 480 2014-05-07 0.1624835 -0.0148971085 -0.0148971085
## 481 2014-05-08 0.1624580 -0.0125250875 -0.0125250875
## 482 2014-05-09 0.1624325 -0.0097381299 -0.0097381299
## 483 2014-05-10 0.1624070 -0.0065756937 -0.0065756937
## 484 2014-05-11 0.1623816 -0.0030901877 -0.0030901877
## 485 2014-05-12 0.1623561 0.0006547332 0.0006547332
## 486 2014-05-13 0.1623306 0.0045862059 0.0045862059
## 487 2014-05-14 0.1623051 0.0086244634 0.0086244634
## 488 2014-05-15 0.1622796 0.0126853346 0.0126853346
## 489 2014-05-16 0.1622541 0.0166828601 0.0166828601
## 490 2014-05-17 0.1622286 0.0205319459 0.0205319459
## 491 2014-05-18 0.1622032 0.0241509724 0.0241509724
## 492 2014-05-19 0.1621777 0.0274642819 0.0274642819
## 493 2014-05-20 0.1621522 0.0304044695 0.0304044695
## 494 2014-05-21 0.1621267 0.0329144112 0.0329144112
## 495 2014-05-22 0.1621012 0.0349489688 0.0349489688
## 496 2014-05-23 0.1620757 0.0364763260 0.0364763260
## 497 2014-05-24 0.1620502 0.0374789147 0.0374789147
## 498 2014-05-25 0.1620247 0.0379539110 0.0379539110
## 499 2014-05-26 0.1619993 0.0379132868 0.0379132868
## 500 2014-05-27 0.1619738 0.0373834208 0.0373834208
## 501 2014-05-28 0.1619483 0.0364042831 0.0364042831
## 502 2014-05-29 0.1619228 0.0350282214 0.0350282214
## 503 2014-05-30 0.1618973 0.0333183889 0.0333183889
## 504 2014-05-31 0.1618718 0.0313468625 0.0313468625
## 505 2014-06-01 0.1618463 0.0291925114 0.0291925114
## 506 2014-06-02 0.1618208 0.0269386798 0.0269386798
## 507 2014-06-03 0.1617954 0.0246707550 0.0246707550
## 508 2014-06-04 0.1617699 0.0224736946 0.0224736946
## 509 2014-06-05 0.1617444 0.0204295851 0.0204295851
## 510 2014-06-06 0.1617189 0.0186153048 0.0186153048
## 511 2014-06-07 0.1616934 0.0171003594 0.0171003594
## 512 2014-06-08 0.1616679 0.0159449522 0.0159449522
## 513 2014-06-09 0.1616424 0.0151983442 0.0151983442
## 514 2014-06-10 0.1616170 0.0148975496 0.0148975496
## 515 2014-06-11 0.1615915 0.0150664022 0.0150664022
## 516 2014-06-12 0.1615660 0.0157150165 0.0157150165
## 517 2014-06-13 0.1615405 0.0168396553 0.0168396553
## 518 2014-06-14 0.1615150 0.0184230042 0.0184230042
## 519 2014-06-15 0.1614895 0.0204348389 0.0204348389
## 520 2014-06-16 0.1614640 0.0228330640 0.0228330640
## 521 2014-06-17 0.1614385 0.0255650853 0.0255650853
## 522 2014-06-18 0.1614131 0.0285694744 0.0285694744
## 523 2014-06-19 0.1613876 0.0317778701 0.0317778701
## 524 2014-06-20 0.1613621 0.0351170597 0.0351170597
## 525 2014-06-21 0.1613366 0.0385111746 0.0385111746
## 526 2014-06-22 0.1613111 0.0418839351 0.0418839351
## 527 2014-06-23 0.1612856 0.0451608758 0.0451608758
## 528 2014-06-24 0.1612601 0.0482714864 0.0482714864
## 529 2014-06-25 0.1612346 0.0511512052 0.0511512052
## 530 2014-06-26 0.1612092 0.0537432064 0.0537432064
## 531 2014-06-27 0.1611837 0.0559999329 0.0559999329
## 532 2014-06-28 0.1611582 0.0578843296 0.0578843296
## 533 2014-06-29 0.1611327 0.0593707456 0.0593707456
## 534 2014-06-30 0.1611072 0.0604454823 0.0604454823
## 535 2014-07-01 0.1610817 0.0611069758 0.0611069758
## 536 2014-07-02 0.1610562 0.0613656126 0.0613656126
## 537 2014-07-03 0.1610308 0.0612431897 0.0612431897
## 538 2014-07-04 0.1610053 0.0607720396 0.0607720396
## 539 2014-07-05 0.1609798 0.0599938524 0.0599938524
## 540 2014-07-06 0.1609543 0.0589582327 0.0589582327
## 541 2014-07-07 0.1609288 0.0577210417 0.0577210417
## 542 2014-07-08 0.1609033 0.0563425747 0.0563425747
## 543 2014-07-09 0.1608778 0.0548856340 0.0548856340
## 544 2014-07-10 0.1608523 0.0534135563 0.0534135563
## 545 2014-07-11 0.1608269 0.0519882550 0.0519882550
## 546 2014-07-12 0.1608014 0.0506683388 0.0506683388
## 547 2014-07-13 0.1607759 0.0495073600 0.0495073600
## 548 2014-07-14 0.1607504 0.0485522479 0.0485522479
## 549 2014-07-15 0.1607249 0.0478419691 0.0478419691
## 550 2014-07-16 0.1606994 0.0474064561 0.0474064561
## 551 2014-07-17 0.1606739 0.0472658301 0.0472658301
## 552 2014-07-18 0.1606484 0.0474299397 0.0474299397
## 553 2014-07-19 0.1606230 0.0478982249 0.0478982249
## 554 2014-07-20 0.1605975 0.0486599049 0.0486599049
## 555 2014-07-21 0.1605720 0.0496944802 0.0496944802
## 556 2014-07-22 0.1605465 0.0509725279 0.0509725279
## 557 2014-07-23 0.1605210 0.0524567617 0.0524567617
## 558 2014-07-24 0.1604955 0.0541033192 0.0541033192
## 559 2014-07-25 0.1604700 0.0558632329 0.0558632329
## 560 2014-07-26 0.1604445 0.0576840366 0.0576840366
## 561 2014-07-27 0.1604191 0.0595114535 0.0595114535
## 562 2014-07-28 0.1603936 0.0612911127 0.0612911127
## 563 2014-07-29 0.1603681 0.0629702391 0.0629702391
## 564 2014-07-30 0.1603426 0.0644992633 0.0644992633
## 565 2014-07-31 0.1603171 0.0658333019 0.0658333019
## 566 2014-08-01 0.1602916 0.0669334627 0.0669334627
## 567 2014-08-02 0.1602661 0.0677679362 0.0677679362
## 568 2014-08-03 0.1602407 0.0683128412 0.0683128412
## 569 2014-08-04 0.1602152 0.0685528005 0.0685528005
## 570 2014-08-05 0.1601897 0.0684812333 0.0684812333
## 571 2014-08-06 0.1601642 0.0681003580 0.0681003580
## 572 2014-08-07 0.1601387 0.0674209093 0.0674209093
## 573 2014-08-08 0.1601132 0.0664615841 0.0664615841
## 574 2014-08-09 0.1600877 0.0652482367 0.0652482367
## 575 2014-08-10 0.1600622 0.0638128540 0.0638128540
## 576 2014-08-11 0.1600368 0.0621923473 0.0621923473
## 577 2014-08-12 0.1600113 0.0604272030 0.0604272030
## 578 2014-08-13 0.1599858 0.0585600394 0.0585600394
## 579 2014-08-14 0.1599603 0.0566341174 0.0566341174
## 580 2014-08-15 0.1599348 0.0546918577 0.0546918577
## 581 2014-08-16 0.1599093 0.0527734126 0.0527734126
## 582 2014-08-17 0.1598838 0.0509153410 0.0509153410
## 583 2014-08-18 0.1598583 0.0491494298 0.0491494298
## 584 2014-08-19 0.1598329 0.0475017016 0.0475017016
## 585 2014-08-20 0.1598074 0.0459916393 0.0459916393
## 586 2014-08-21 0.1597819 0.0446316543 0.0446316543
## 587 2014-08-22 0.1597564 0.0434268132 0.0434268132
## 588 2014-08-23 0.1597309 0.0423748326 0.0423748326
## 589 2014-08-24 0.1597054 0.0414663390 0.0414663390
## 590 2014-08-25 0.1596799 0.0406853849 0.0406853849
## 591 2014-08-26 0.1596545 0.0400102021 0.0400102021
## 592 2014-08-27 0.1596290 0.0394141644 0.0394141644
## 593 2014-08-28 0.1596035 0.0388669270 0.0388669270
## 594 2014-08-29 0.1595780 0.0383357027 0.0383357027
## 595 2014-08-30 0.1595525 0.0377866281 0.0377866281
## 596 2014-08-31 0.1595270 0.0371861756 0.0371861756
## 597 2014-09-01 0.1595015 0.0365025579 0.0365025579
## 598 2014-09-02 0.1594760 0.0357070779 0.0357070779
## 599 2014-09-03 0.1594506 0.0347753759 0.0347753759
## 600 2014-09-04 0.1594251 0.0336885281 0.0336885281
## 601 2014-09-05 0.1593996 0.0324339579 0.0324339579
## 602 2014-09-06 0.1593741 0.0310061256 0.0310061256
## 603 2014-09-07 0.1593486 0.0294069700 0.0294069700
## 604 2014-09-08 0.1593231 0.0276460832 0.0276460832
## 605 2014-09-09 0.1592976 0.0257406093 0.0257406093
## 606 2014-09-10 0.1592721 0.0237148670 0.0237148670
## 607 2014-09-11 0.1592467 0.0215997047 0.0215997047
## 608 2014-09-12 0.1592212 0.0194316057 0.0194316057
## 609 2014-09-13 0.1591957 0.0172515716 0.0172515716
## 610 2014-09-14 0.1591702 0.0151038173 0.0151038173
## 611 2014-09-15 0.1591447 0.0130343192 0.0130343192
## 612 2014-09-16 0.1591192 0.0110892639 0.0110892639
## 613 2014-09-17 0.1590937 0.0093134479 0.0093134479
## 614 2014-09-18 0.1590683 0.0077486814 0.0077486814
## 615 2014-09-19 0.1590428 0.0064322506 0.0064322506
## 616 2014-09-20 0.1590173 0.0053954912 0.0053954912
## 617 2014-09-21 0.1589918 0.0046625239 0.0046625239
## 618 2014-09-22 0.1589663 0.0042491966 0.0042491966
## 619 2014-09-23 0.1589408 0.0041622742 0.0041622742
## 620 2014-09-24 0.1589153 0.0043989080 0.0043989080
## 621 2014-09-25 0.1588898 0.0049464084 0.0049464084
## 622 2014-09-26 0.1588644 0.0057823365 0.0057823365
## 623 2014-09-27 0.1588389 0.0068749177 0.0068749177
## 624 2014-09-28 0.1588134 0.0081837734 0.0081837734
## 625 2014-09-29 0.1587879 0.0096609529 0.0096609529
## 626 2014-09-30 0.1587624 0.0112522411 0.0112522411
## 627 2014-10-01 0.1587369 0.0128987062 0.0128987062
## 628 2014-10-02 0.1587114 0.0145384433 0.0145384433
## 629 2014-10-03 0.1586859 0.0161084645 0.0161084645
## 630 2014-10-04 0.1586605 0.0175466777 0.0175466777
## 631 2014-10-05 0.1586350 0.0187938951 0.0187938951
## 632 2014-10-06 0.1586095 0.0197958085 0.0197958085
## 633 2014-10-07 0.1585840 0.0205048687 0.0205048687
## 634 2014-10-08 0.1585585 0.0208820077 0.0208820077
## 635 2014-10-09 0.1585330 0.0208981478 0.0208981478
## 636 2014-10-10 0.1585075 0.0205354434 0.0205354434
## 637 2014-10-11 0.1584820 0.0197882131 0.0197882131
## 638 2014-10-12 0.1584566 0.0186635238 0.0186635238
## 639 2014-10-13 0.1584311 0.0171814030 0.0171814030
## 640 2014-10-14 0.1584056 0.0153746604 0.0153746604
## 641 2014-10-15 0.1583801 0.0132883188 0.0132883188
## 642 2014-10-16 0.1583546 0.0109786593 0.0109786593
## 643 2014-10-17 0.1583291 0.0085119012 0.0085119012
## 644 2014-10-18 0.1583036 0.0059625485 0.0059625485
## 645 2014-10-19 0.1582782 0.0034114431 0.0034114431
## 646 2014-10-20 0.1582527 0.0009435776 0.0009435776
## 647 2014-10-21 0.1582272 -0.0013542747 -0.0013542747
## 648 2014-10-22 0.1582017 -0.0033960454 -0.0033960454
## 649 2014-10-23 0.1581762 -0.0050989029 -0.0050989029
## 650 2014-10-24 0.1581507 -0.0063857556 -0.0063857556
## 651 2014-10-25 0.1581252 -0.0071876551 -0.0071876551
## 652 2014-10-26 0.1580997 -0.0074460258 -0.0074460258
## 653 2014-10-27 0.1580743 -0.0071146530 -0.0071146530
## 654 2014-10-28 0.1580488 -0.0061613679 -0.0061613679
## 655 2014-10-29 0.1580233 -0.0045693748 -0.0045693748
## 656 2014-10-30 0.1579978 -0.0023381743 -0.0023381743
## 657 2014-10-31 0.1579723 0.0005159484 0.0005159484
## 658 2014-11-01 0.1579468 0.0039598955 0.0039598955
## 659 2014-11-02 0.1579213 0.0079441951 0.0079441951
## 660 2014-11-03 0.1578958 0.0124039031 0.0124039031
## 661 2014-11-04 0.1578704 0.0172599499 0.0172599499
## 662 2014-11-05 0.1578449 0.0224209019 0.0224209019
## 663 2014-11-06 0.1578194 0.0277850932 0.0277850932
## 664 2014-11-07 0.1577939 0.0332430750 0.0332430750
## 665 2014-11-08 0.1577684 0.0386803166 0.0386803166
## 666 2014-11-09 0.1577429 0.0439800869 0.0439800869
## 667 2014-11-10 0.1577174 0.0490264384 0.0490264384
## 668 2014-11-11 0.1576920 0.0537072107 0.0537072107
## 669 2014-11-12 0.1576665 0.0579169713 0.0579169713
## 670 2014-11-13 0.1576410 0.0615598098 0.0615598098
## 671 2014-11-14 0.1576155 0.0645519071 0.0645519071
## 672 2014-11-15 0.1575900 0.0668238057 0.0668238057
## 673 2014-11-16 0.1575645 0.0683223154 0.0683223154
## 674 2014-11-17 0.1575390 0.0690119972 0.0690119972
## 675 2014-11-18 0.1575135 0.0688761820 0.0688761820
## 676 2014-11-19 0.1574881 0.0679174923 0.0679174923
## 677 2014-11-20 0.1574626 0.0661578471 0.0661578471
## 678 2014-11-21 0.1574371 0.0636379484 0.0636379484
## 679 2014-11-22 0.1574116 0.0604162600 0.0604162600
## 680 2014-11-23 0.1573861 0.0565675036 0.0565675036
## 681 2014-11-24 0.1573606 0.0521807124 0.0521807124
## 682 2014-11-25 0.1573351 0.0473568942 0.0473568942
## 683 2014-11-26 0.1573096 0.0422063682 0.0422063682
## 684 2014-11-27 0.1572842 0.0368458480 0.0368458480
## 685 2014-11-28 0.1572587 0.0313953528 0.0313953528
## 686 2014-11-29 0.1572332 0.0259750323 0.0259750323
## 687 2014-11-30 0.1572077 0.0207019944 0.0207019944
## 688 2014-12-01 0.1571822 0.0156872258 0.0156872258
## additive_terms_upper yearly yearly_lower yearly_upper
## 1 -0.0356499451 -0.0356499451 -0.0356499451 -0.0356499451
## 2 -0.1196667639 -0.1196667639 -0.1196667639 -0.1196667639
## 3 -0.0844410783 -0.0844410783 -0.0844410783 -0.0844410783
## 4 -0.0200013993 -0.0200013993 -0.0200013993 -0.0200013993
## 5 0.0297437350 0.0297437350 0.0297437350 0.0297437350
## 6 0.0609799763 0.0609799763 0.0609799763 0.0609799763
## 7 0.0666822180 0.0666822180 0.0666822180 0.0666822180
## 8 0.0366828934 0.0366828934 0.0366828934 0.0366828934
## 9 0.0124852690 0.0124852690 0.0124852690 0.0124852690
## 10 0.0030460452 0.0030460452 0.0030460452 0.0030460452
## 11 0.0169109489 0.0169109489 0.0169109489 0.0169109489
## 12 0.0079288625 0.0079288625 0.0079288625 0.0079288625
## 13 -0.0332513176 -0.0332513176 -0.0332513176 -0.0332513176
## 14 -0.1104020156 -0.1104020156 -0.1104020156 -0.1104020156
## 15 -0.0844846277 -0.0844846277 -0.0844846277 -0.0844846277
## 16 -0.0201696967 -0.0201696967 -0.0201696967 -0.0201696967
## 17 0.0280727129 0.0280727129 0.0280727129 0.0280727129
## 18 0.0612855477 0.0612855477 0.0612855477 0.0612855477
## 19 0.0673855969 0.0673855969 0.0673855969 0.0673855969
## 20 0.0361204385 0.0361204385 0.0361204385 0.0361204385
## 21 0.0137233500 0.0137233500 0.0137233500 0.0137233500
## 22 0.0058882113 0.0058882113 0.0058882113 0.0058882113
## 23 0.0133089833 0.0133089833 0.0133089833 0.0133089833
## 24 0.0060550578 0.0060550578 0.0060550578 0.0060550578
## 25 -0.0406241970 -0.0406241970 -0.0406241970 -0.0406241970
## 26 -0.1134952892 -0.1134952892 -0.1134952892 -0.1134952892
## 27 -0.0845261697 -0.0845261697 -0.0845261697 -0.0845261697
## 28 -0.0201315617 -0.0201315617 -0.0201315617 -0.0201315617
## 29 0.0286350603 0.0286350603 0.0286350603 0.0286350603
## 30 0.0612087408 0.0612087408 0.0612087408 0.0612087408
## 31 0.0671680630 0.0671680630 0.0671680630 0.0671680630
## 32 0.0363152097 0.0363152097 0.0363152097 0.0363152097
## 33 0.0133117282 0.0133117282 0.0133117282 0.0133117282
## 34 0.0049075936 0.0049075936 0.0049075936 0.0049075936
## 35 0.0144860745 0.0144860745 0.0144860745 0.0144860745
## 36 0.0066836927 0.0066836927 0.0066836927 0.0066836927
## 37 -0.0381081177 -0.0381081177 -0.0381081177 -0.0381081177
## 38 -0.1165847501 -0.1165847501 -0.1165847501 -0.1165847501
## 39 -0.0845117476 -0.0845117476 -0.0845117476 -0.0845117476
## 40 -0.0200751150 -0.0200751150 -0.0200751150 -0.0200751150
## 41 0.0291925114 0.0291925114 0.0291925114 0.0291925114
## 42 0.0611069758 0.0611069758 0.0611069758 0.0611069758
## 43 0.0669334627 0.0669334627 0.0669334627 0.0669334627
## 44 0.0365025579 0.0365025579 0.0365025579 0.0365025579
## 45 0.0128987062 0.0128987062 0.0128987062 0.0128987062
## 46 0.0039598955 0.0039598955 0.0039598955 0.0039598955
## 47 0.0156872258 0.0156872258 0.0156872258 0.0156872258
## 48 0.0073087629 0.0073087629 0.0073087629 0.0073087629
## 49 -0.0356499451 -0.0356499451 -0.0356499451 -0.0356499451
## 50 -0.1196667639 -0.1196667639 -0.1196667639 -0.1196667639
## 51 -0.0844410783 -0.0844410783 -0.0844410783 -0.0844410783
## 52 -0.0200013993 -0.0200013993 -0.0200013993 -0.0200013993
## 53 0.0297437350 0.0297437350 0.0297437350 0.0297437350
## 54 0.0609799763 0.0609799763 0.0609799763 0.0609799763
## 55 0.0666822180 0.0666822180 0.0666822180 0.0666822180
## 56 0.0366828934 0.0366828934 0.0366828934 0.0366828934
## 57 0.0124852690 0.0124852690 0.0124852690 0.0124852690
## 58 0.0030460452 0.0030460452 0.0030460452 0.0030460452
## 59 0.0169109489 0.0169109489 0.0169109489 0.0169109489
## 60 0.0079288625 0.0079288625 0.0079288625 0.0079288625
## 61 -0.0332513176 -0.0332513176 -0.0332513176 -0.0332513176
## 62 -0.1104020156 -0.1104020156 -0.1104020156 -0.1104020156
## 63 -0.0844846277 -0.0844846277 -0.0844846277 -0.0844846277
## 64 -0.0201696967 -0.0201696967 -0.0201696967 -0.0201696967
## 65 0.0280727129 0.0280727129 0.0280727129 0.0280727129
## 66 0.0612855477 0.0612855477 0.0612855477 0.0612855477
## 67 0.0673855969 0.0673855969 0.0673855969 0.0673855969
## 68 0.0361204385 0.0361204385 0.0361204385 0.0361204385
## 69 0.0137233500 0.0137233500 0.0137233500 0.0137233500
## 70 0.0058882113 0.0058882113 0.0058882113 0.0058882113
## 71 0.0133089833 0.0133089833 0.0133089833 0.0133089833
## 72 0.0060550578 0.0060550578 0.0060550578 0.0060550578
## 73 -0.0406241970 -0.0406241970 -0.0406241970 -0.0406241970
## 74 -0.1134952892 -0.1134952892 -0.1134952892 -0.1134952892
## 75 -0.0845261697 -0.0845261697 -0.0845261697 -0.0845261697
## 76 -0.0201315617 -0.0201315617 -0.0201315617 -0.0201315617
## 77 0.0286350603 0.0286350603 0.0286350603 0.0286350603
## 78 0.0612087408 0.0612087408 0.0612087408 0.0612087408
## 79 0.0671680630 0.0671680630 0.0671680630 0.0671680630
## 80 0.0363152097 0.0363152097 0.0363152097 0.0363152097
## 81 0.0133117282 0.0133117282 0.0133117282 0.0133117282
## 82 0.0049075936 0.0049075936 0.0049075936 0.0049075936
## 83 0.0144860745 0.0144860745 0.0144860745 0.0144860745
## 84 0.0066836927 0.0066836927 0.0066836927 0.0066836927
## 85 -0.0381081177 -0.0381081177 -0.0381081177 -0.0381081177
## 86 -0.1165847501 -0.1165847501 -0.1165847501 -0.1165847501
## 87 -0.0845117476 -0.0845117476 -0.0845117476 -0.0845117476
## 88 -0.0200751150 -0.0200751150 -0.0200751150 -0.0200751150
## 89 0.0291925114 0.0291925114 0.0291925114 0.0291925114
## 90 0.0611069758 0.0611069758 0.0611069758 0.0611069758
## 91 0.0669334627 0.0669334627 0.0669334627 0.0669334627
## 92 0.0365025579 0.0365025579 0.0365025579 0.0365025579
## 93 0.0128987062 0.0128987062 0.0128987062 0.0128987062
## 94 0.0039598955 0.0039598955 0.0039598955 0.0039598955
## 95 0.0156872258 0.0156872258 0.0156872258 0.0156872258
## 96 0.0073087629 0.0073087629 0.0073087629 0.0073087629
## 97 -0.0356499451 -0.0356499451 -0.0356499451 -0.0356499451
## 98 -0.1196667639 -0.1196667639 -0.1196667639 -0.1196667639
## 99 -0.0844410783 -0.0844410783 -0.0844410783 -0.0844410783
## 100 -0.0200013993 -0.0200013993 -0.0200013993 -0.0200013993
## 101 0.0297437350 0.0297437350 0.0297437350 0.0297437350
## 102 0.0609799763 0.0609799763 0.0609799763 0.0609799763
## 103 0.0666822180 0.0666822180 0.0666822180 0.0666822180
## 104 0.0366828934 0.0366828934 0.0366828934 0.0366828934
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## yhat_lower yhat_upper trend_lower trend_upper yhat
## 1 -0.3818331 1.1303419 0.4028826 0.4028826 0.367232655
## 2 -0.4512743 0.9973696 0.4022385 0.4022385 0.282571703
## 3 -0.4498207 1.0505244 0.4015253 0.4015253 0.317084241
## 4 -0.3131183 1.0998785 0.4008352 0.4008352 0.380833777
## 5 -0.2847160 1.1770354 0.4001220 0.4001220 0.429865763
## 6 -0.2360160 1.1579718 0.3994319 0.3994319 0.460411861
## 7 -0.2907294 1.1812013 0.3987187 0.3987187 0.465400955
## 8 -0.2832166 1.2007270 0.3980056 0.3980056 0.434688483
## 9 -0.3227517 1.1642614 0.3973154 0.3973154 0.409800715
## 10 -0.3901949 1.0937038 0.3966023 0.3966023 0.399648343
## 11 -0.3427855 1.1448020 0.3959122 0.3959122 0.412823104
## 12 -0.3155179 1.1616491 0.3951990 0.3951990 0.403127870
## 13 -0.3869529 1.0140782 0.3944859 0.3944859 0.361234542
## 14 -0.4505241 0.9839954 0.3938187 0.3938187 0.283416705
## 15 -0.4555459 1.0299648 0.3931056 0.3931056 0.308620945
## 16 -0.4186624 1.1515838 0.3924154 0.3924154 0.372245733
## 17 -0.3200412 1.1932323 0.3917023 0.3917023 0.419774995
## 18 -0.2237229 1.2451131 0.3910121 0.3910121 0.452297687
## 19 -0.2710631 1.1901486 0.3902990 0.3902990 0.457684588
## 20 -0.3280764 1.1182006 0.3895858 0.3895858 0.425706282
## 21 -0.3708194 1.1174016 0.3888957 0.3888957 0.402619050
## 22 -0.3168899 1.1127853 0.3881826 0.3881826 0.394070763
## 23 -0.2900406 1.1384566 0.3874924 0.3874924 0.400801392
## 24 -0.2809530 1.1777475 0.3867793 0.3867793 0.392834319
## 25 -0.4092270 1.0519898 0.3860661 0.3860661 0.345441916
## 26 -0.4615635 1.0017538 0.3854220 0.3854220 0.271926690
## 27 -0.4261968 0.9919610 0.3847088 0.3847088 0.300182662
## 28 -0.3544392 1.0753378 0.3840187 0.3840187 0.363887127
## 29 -0.3203031 1.1696344 0.3833055 0.3833055 0.411940601
## 30 -0.3402867 1.2109813 0.3826154 0.3826154 0.443824138
## 31 -0.2894191 1.1832106 0.3819022 0.3819022 0.449070312
## 32 -0.3637890 1.1528081 0.3811891 0.3811891 0.417504311
## 33 -0.3350428 1.1253533 0.3804990 0.3804990 0.393810686
## 34 -0.2988870 1.0825241 0.3797858 0.3797858 0.384693404
## 35 -0.3194962 1.0859761 0.3790957 0.3790957 0.393581742
## 36 -0.3540353 1.0933113 0.3783825 0.3783825 0.385066212
## 37 -0.3626027 1.0677764 0.3776694 0.3776694 0.339561254
## 38 -0.4397389 0.9922619 0.3770252 0.3770252 0.260440487
## 39 -0.4510489 1.0088515 0.3763121 0.3763121 0.291800342
## 40 -0.3424846 1.0153265 0.3756219 0.3756219 0.355546831
## 41 -0.3429128 1.1339243 0.3749088 0.3749088 0.404101310
## 42 -0.2233057 1.1400679 0.3742187 0.3742187 0.435325631
## 43 -0.3110798 1.1646001 0.3735055 0.3735055 0.440438970
## 44 -0.3790476 1.1430250 0.3727924 0.3727924 0.409294917
## 45 -0.3456035 1.0908957 0.3721022 0.3721022 0.385000922
## 46 -0.2909968 1.1348736 0.3713891 0.3713891 0.375348963
## 47 -0.3274980 1.1471253 0.3706989 0.3706989 0.386386151
## 48 -0.3342901 1.1037375 0.3699858 0.3699858 0.377294540
## 49 -0.4599574 1.0780207 0.3692726 0.3692726 0.333622684
## 50 -0.4460268 0.9220766 0.3686285 0.3686285 0.248961731
## 51 -0.4556097 1.0326082 0.3679153 0.3679153 0.283474269
## 52 -0.3733440 1.0344002 0.3672252 0.3672252 0.347223804
## 53 -0.3849496 1.1059852 0.3665121 0.3665121 0.396255791
## 54 -0.3126781 1.1917263 0.3658219 0.3658219 0.426801889
## 55 -0.2579194 1.1780422 0.3651088 0.3651088 0.431790983
## 56 -0.3128351 1.1353677 0.3643956 0.3643956 0.401078510
## 57 -0.3252480 1.1002434 0.3637055 0.3637055 0.376190742
## 58 -0.3536016 1.0871171 0.3629923 0.3629923 0.366038371
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## 582 -0.4911714 0.9400533 0.1598717 0.1598941 0.210799176
## 583 -0.5155655 0.9319221 0.1598457 0.1598688 0.209007778
## 584 -0.5557130 0.9713718 0.1598201 0.1598433 0.207334563
## 585 -0.5062524 0.9469979 0.1597947 0.1598179 0.205799014
## 586 -0.5754240 0.9456679 0.1597691 0.1597925 0.204413542
## 587 -0.5096788 0.9595398 0.1597435 0.1597670 0.203183214
## 588 -0.4993420 0.9001144 0.1597181 0.1597416 0.202105746
## 589 -0.5207744 0.9660126 0.1596926 0.1597162 0.201171765
## 590 -0.5275781 0.9630873 0.1596672 0.1596908 0.200365324
## 591 -0.5103441 0.9724532 0.1596417 0.1596655 0.199664654
## 592 -0.6095161 0.9644899 0.1596163 0.1596401 0.199043130
## 593 -0.4788258 0.9368033 0.1595908 0.1596147 0.198470405
## 594 -0.5716568 0.9165739 0.1595652 0.1595893 0.197913694
## 595 -0.5359011 0.9966561 0.1595396 0.1595639 0.197339132
## 596 -0.5322461 0.8921910 0.1595140 0.1595384 0.196713193
## 597 -0.5184776 0.9467605 0.1594884 0.1595130 0.196004088
## 598 -0.4921784 0.8818579 0.1594629 0.1594876 0.195183121
## 599 -0.5797272 0.9722432 0.1594374 0.1594622 0.194225932
## 600 -0.5882679 0.9633366 0.1594119 0.1594368 0.193113597
## 601 -0.4952434 0.9101217 0.1593864 0.1594114 0.191833540
## 602 -0.5751621 0.8604020 0.1593609 0.1593860 0.190380221
## 603 -0.5364030 0.9487377 0.1593354 0.1593607 0.188755578
## 604 -0.5557771 0.8864101 0.1593099 0.1593355 0.186969204
## 605 -0.5557191 0.9108800 0.1592844 0.1593103 0.185038244
## 606 -0.5724824 0.8820509 0.1592589 0.1592851 0.182987014
## 607 -0.5094620 0.9401142 0.1592334 0.1592596 0.180846365
## 608 -0.6028502 0.9484418 0.1592079 0.1592343 0.178652779
## 609 -0.5854712 0.9152418 0.1591824 0.1592089 0.176447258
## 610 -0.6414936 0.9002968 0.1591569 0.1591835 0.174274016
## 611 -0.5550791 0.9214380 0.1591313 0.1591581 0.172179031
## 612 -0.5893485 0.8861669 0.1591057 0.1591327 0.170208489
## 613 -0.5782180 0.8933262 0.1590802 0.1591073 0.168407186
## 614 -0.6043493 0.9249696 0.1590546 0.1590819 0.166816933
## 615 -0.5599348 0.9134240 0.1590291 0.1590565 0.165475015
## 616 -0.5905323 0.9198569 0.1590035 0.1590312 0.164412768
## 617 -0.5604199 0.9030936 0.1589779 0.1590059 0.163654314
## 618 -0.5382699 0.8879327 0.1589523 0.1589806 0.163215500
## 619 -0.6033328 0.8897617 0.1589268 0.1589553 0.163103090
## 620 -0.5503368 0.8842989 0.1589012 0.1589300 0.163314237
## 621 -0.5583171 0.8723632 0.1588757 0.1589047 0.163836251
## 622 -0.5957849 0.8757417 0.1588501 0.1588794 0.164646692
## 623 -0.5856622 0.8253821 0.1588246 0.1588540 0.165713786
## 624 -0.5263585 0.9273954 0.1587989 0.1588286 0.166997154
## 625 -0.5139808 0.8866048 0.1587732 0.1588032 0.168448847
## 626 -0.5699460 0.8911871 0.1587475 0.1587778 0.170014648
## 627 -0.5388074 0.9231821 0.1587218 0.1587524 0.171635626
## 628 -0.5706308 0.9087309 0.1586961 0.1587271 0.173249876
## 629 -0.5562014 0.9421774 0.1586704 0.1587018 0.174794411
## 630 -0.5276893 0.9412829 0.1586447 0.1586764 0.176207137
## 631 -0.5944831 0.8884611 0.1586190 0.1586510 0.177428867
## 632 -0.5336222 0.8899166 0.1585933 0.1586259 0.178405294
## 633 -0.5880717 0.8629506 0.1585677 0.1586006 0.179088867
## 634 -0.5483692 0.9654929 0.1585421 0.1585753 0.179440519
## 635 -0.6156674 0.9359662 0.1585166 0.1585500 0.179431172
## 636 -0.5461085 0.9448577 0.1584911 0.1585247 0.179042980
## 637 -0.5251375 0.9061647 0.1584654 0.1584993 0.178270263
## 638 -0.5567971 0.9370289 0.1584399 0.1584738 0.177120087
## 639 -0.5662854 0.9805512 0.1584143 0.1584484 0.175612479
## 640 -0.5616574 0.8891374 0.1583888 0.1584229 0.173780249
## 641 -0.5503176 0.9435786 0.1583632 0.1583974 0.171668421
## 642 -0.5856633 0.8697456 0.1583376 0.1583720 0.169333274
## 643 -0.4767406 0.9094218 0.1583121 0.1583465 0.166841029
## 644 -0.5820039 0.9007177 0.1582865 0.1583211 0.164266189
## 645 -0.5761965 0.8379205 0.1582609 0.1582956 0.161689597
## 646 -0.6065975 0.8916328 0.1582348 0.1582702 0.159196244
## 647 -0.5909231 0.9052272 0.1582090 0.1582447 0.156872905
## 648 -0.5795101 0.8831566 0.1581834 0.1582193 0.154805647
## 649 -0.5354932 0.8611684 0.1581579 0.1581939 0.153077303
## 650 -0.5966608 0.8984774 0.1581323 0.1581686 0.151764963
## 651 -0.5994659 0.8679342 0.1581068 0.1581434 0.150937577
## 652 -0.5280173 0.9278521 0.1580812 0.1581181 0.150653719
## 653 -0.5641934 0.8943722 0.1580557 0.1580929 0.150959605
## 654 -0.5500859 0.8988430 0.1580301 0.1580676 0.151887403
## 655 -0.5963897 0.8169213 0.1580046 0.1580424 0.153453909
## 656 -0.6102627 0.8976412 0.1579790 0.1580172 0.155659622
## 657 -0.5329804 0.9147805 0.1579535 0.1579919 0.158488258
## 658 -0.5938349 0.9335203 0.1579279 0.1579667 0.161906718
## 659 -0.5996207 0.9214044 0.1579024 0.1579414 0.165865531
## 660 -0.5909088 0.8966596 0.1578768 0.1579161 0.170299752
## 661 -0.5674646 0.9274865 0.1578513 0.1578909 0.175130312
## 662 -0.5419987 0.9065447 0.1578257 0.1578654 0.180265777
## 663 -0.5264319 0.9519975 0.1578002 0.1578400 0.185604481
## 664 -0.5592906 0.9593293 0.1577745 0.1578147 0.191036976
## 665 -0.5201085 0.9576053 0.1577489 0.1577894 0.196448730
## 666 -0.5259965 0.9276794 0.1577232 0.1577640 0.201723014
## 667 -0.5512463 0.9560963 0.1576975 0.1577387 0.206743878
## 668 -0.4860706 1.0163822 0.1576718 0.1577135 0.211399163
## 669 -0.4810182 1.0033869 0.1576461 0.1576883 0.215583437
## 670 -0.4945883 0.9543052 0.1576205 0.1576630 0.219200788
## 671 -0.5462045 0.9370739 0.1575948 0.1576378 0.222167399
## 672 -0.4970972 0.9600835 0.1575692 0.1576125 0.224413810
## 673 -0.5027753 0.9497459 0.1575436 0.1575872 0.225886833
## 674 -0.4953010 0.9425997 0.1575180 0.1575617 0.226551028
## 675 -0.5227915 0.9712026 0.1574924 0.1575362 0.226389726
## 676 -0.4795773 0.9556975 0.1574668 0.1575107 0.225405549
## 677 -0.4885552 0.9512014 0.1574413 0.1574854 0.223620417
## 678 -0.4954378 0.9506308 0.1574157 0.1574600 0.221075031
## 679 -0.5487619 0.9284838 0.1573901 0.1574349 0.217827856
## 680 -0.5234347 0.9792548 0.1573645 0.1574098 0.213953612
## 681 -0.5430687 0.9789816 0.1573390 0.1573846 0.209541334
## 682 -0.5171387 0.9140475 0.1573134 0.1573598 0.204692029
## 683 -0.6169784 0.9386734 0.1572878 0.1573350 0.199516016
## 684 -0.5521998 0.9238891 0.1572623 0.1573098 0.194130009
## 685 -0.5642818 0.9080580 0.1572365 0.1572845 0.188654026
## 686 -0.5461206 0.8955717 0.1572108 0.1572593 0.183208219
## 687 -0.5550928 0.9148290 0.1571851 0.1572340 0.177909694
## 688 -0.5252612 0.9000022 0.1571593 0.1572088 0.172869438Multi Step Ahead Forecasts
Background
Let’s define Static and Dynamic multi step ahead forecasts.
- Static Forecast:
- Train = 1990-2000. Test = 2001 (1 step ahead)
- Train = 1990-2000. Test = 2002 (2 step ahead)
- Dynamic Forecast:
- Train = 1990-2000. Test = 2001 (1 step ahead)
- Train = 1990-2000 & 2001 Forecast. Test = 2002 (1 step ahead)
Notice that static forecasts successively further into the future, which can be difficult. Meanwhile, dynamic forecasts build upon potentially error-prone prior forecasts, which can be equally troublesome.
Notice pure distributed lag models (i.e. \(Y_{t}=\alpha + \beta X_{t-1}+u_{T]\)}) don’t permit dynamic forecasts since they require a forecast of \(X\). Moreover, distributed lag models only permit static forecasts up util the horizon of the \(X\) lag (i.e. one in this case).
Forecasts
Let’s use our training set, which ended in Dec2013 to create a series of 24 static forecasts (ie through the end of 2015).
### ARIMA
ARIMA_mulpred <- ts(forecast(ARIMA_auto, h = 24)$mean, start = c(2014,1), frequency =12)
### ARMA-ARCH
Garch_mulpred <- ts(fitted(ugarchforecast(fitORspec = garch_fit, n.ahead = 24)), start = c(2014,1), frequency =12)
### Holt-Winter
HW1_mulpred <- ts(forecast(HW1, h = 24)$mean, start = c(2014,1), frequency =12)
### NNET
NNET_mulpred <- ts(forecast(NNET, h = 24)$mean, start = c(2014,1), frequency =12)
### Random Forest
# We will need to write a loop to get static multiple step ahead forecast
h <- 24
RF_mulpred <- matrix(NA, h, 1)
X_multest <- head(X_test,1)
for (i in 1:h){
set.seed(123)
RF_mulpred[i] <- predict(RF_fit, X_multest)
X_multest[2:ncol(X_multest)] <- X_multest[-ncol(X_multest)] # update the "regressor" with predicted value
X_multest[1] <- RF_mulpred[i]
}
RF_mulpred <- ts(RF_mulpred, start = c(2014,1), frequency = 12)
### Prophet
df_prophet <- Prophet_traindata %>%
select(-HousingStartGrowth) %>%
rename(ds = Date) %>%
rename(y = PriceIndexGrowthSA)
m <- prophet(df_prophet)
future <- make_future_dataframe(m, periods = 24, freq = "month")
forecast <- predict(m, future)
fcst <- forecast[324:nrow(forecast),]
Prophet_mulpred <- ts(fcst[[16]], start = c(2014, 1), frequency = 12)
### Plot Forecast
Test<-window(df_adjts[,"PriceIndexGrowthSA"], start = c(2014, 1), end=c(2015,12))
mulpreds <- ts.union(ARIMA_mulpred, Garch_mulpred, HW1_mulpred, NNET_mulpred, RF_mulpred, Prophet_mulpred, Test) %>%
`colnames<-`(c("ARMA","ARMA-GARCH","Holt Winters", "NNETAR", "Random Forest", "Prophet", "TestSample"))
autoplot(mulpreds) +
labs(color ='', y = "Forecast", x = '', title = "Static 24-Step Ahead Forecast")
Forecast evaluation
Let’s compute the out of sample (OOS) forecast errors
errOOS <- ts((window(df_adjts[,"PriceIndexGrowthSA"], start = c(2014, 1)) - mulpreds[,-7]),
start = c(2014,1), frequency =12)%>%
`colnames<-`(c("ARMA","ARMA-GARCH","Holt Winters", "NNETAR", "Random Forest", "Prophet"))We can summarize the errors
summary(errOOS)## ARMA ARMA-GARCH Holt Winters NNETAR
## Min. :-0.73175 Min. :-0.245126 Min. :-0.50072 Min. :-0.66700
## 1st Qu.:-0.42927 1st Qu.:-0.085938 1st Qu.:-0.09069 1st Qu.:-0.46159
## Median :-0.06388 Median : 0.013576 Median : 0.05912 Median :-0.29957
## Mean :-0.14647 Mean :-0.007118 Mean : 0.07576 Mean :-0.28807
## 3rd Qu.: 0.10868 3rd Qu.: 0.066888 3rd Qu.: 0.36287 3rd Qu.:-0.13048
## Max. : 0.41137 Max. : 0.180397 Max. : 0.60061 Max. : 0.02024
## Random Forest Prophet
## Min. :-0.53100 Min. :-0.03696
## 1st Qu.:-0.36493 1st Qu.: 0.12346
## Median :-0.24063 Median : 0.25432
## Mean :-0.25351 Mean : 0.24154
## 3rd Qu.:-0.15272 3rd Qu.: 0.34758
## Max. : 0.05648 Max. : 0.51311Let’s construct the Mean Squared Forecast Error (MSFE).
MSFE<-colMeans(errOOS^2,na.rm = TRUE)
MSFE## ARMA ARMA-GARCH Holt Winters NNETAR Random Forest
## 0.14347940 0.01353426 0.09154525 0.12124769 0.08934739
## Prophet
## 0.08317619- ARMA-GARCH is a clear winner to the other models.
- The complex Neural Net, Prophet, and RF behave about equally to each other.
Simulation
There are many uses of simulation in a time series context.
Parametric simulation (i.e. Monte Carlo) is often used when you have a good sense of the DGP. Meanwhile, non parametric simulation (i.e. Bootstrapping) relies upon successive draws of the empirical distribution.
In the following we will demonstrate using bootstrapping for construction forecast confidence intervals.
Bootstrapping Time Series
Recall that bootstrapping randomly resamples from the observed dataset in order to create many pseudo samples for analysis. The model of interest is implemented on each pseudo sample. The range of the resulting estimates provides a confidence interval for the estimation on the observed sample.
Unfortunately, random resampling disrupts the temporal persistence of the process. So, we rely upon block bootstrapping. Briefly, suppose your EDA suggests that the series has a temporal persistence of 12 months. Then we resample from the observed data by drawing blocks of 12 successive months. We can stitch these blocks together to generate a a pseudo sample of the desired size.
Create Block Bootstrap Samples
Note that due to the simulation process the pseudo samples you draw may be different each time. Best practice is to draw several hundred (or thousand) pseudo samples. For illustratives purpose, we’re only drawing for 4 pseudo samples here.
nT = nrow(df_adjts)
Train = window(df_adjts[,"PriceIndexGrowthSA"], end = c(2013,12))
Test = window(df_adjts[,"PriceIndexGrowthSA"], start = c(2014, 1))
TrainNum <- round(length(Train))
BootpathNum = 4
BootSamples <- bld.mbb.bootstrap(Train, BootpathNum,block_size=12) %>% data.frame() %>% ts(end= end(Train), frequency=12)
colnames(BootSamples) <- c(lapply(1:(BootpathNum), FUN =function(x){paste("Path",x,sep="")}))
plot(BootSamples, main = "Bootstrapped Samples")
Bootstrapping for a Confidence Interval
Our goal here is to form a confidence interval around our forecast. We want to avoid making distributional assumptions, so we rely upon bootstraps.
Fit an ARIMA to Training Sample
ARMA_fit <- auto.arima(Train)
arimafc <- forecast(ARMA_fit, h=nT-TrainNum, level=95)
arimafc_err <- df_adjts[,"PriceIndexGrowthSA"] - arimafc$meanForecast on each Pseudo Sample
Use that fitted model specification and parameter estimates to construct forecasts for each bootstrap sample.
Bootfcast <- matrix(0, nrow=BootpathNum, ncol=nT-TrainNum)
for(i in seq(BootpathNum)) {
ARMA_refit <- Arima(BootSamples[,i], model=ARMA_fit) # fit model to bootstrapped samples without re-estimating
Bootfcast[i,] <- forecast(ARMA_refit, h = nT-TrainNum)$mean}Gather the Forecasts
We will gather all of the forecasts for Jan2014, and compute the 2.5% and 97.5% quantiles, which will serve to form a 95% confidence interval. Then repeat that for Feb2014, etc…
bootfc <- structure(list(
mean = ts(colMeans(Bootfcast), start = start(Test), frequency=12),
lower = ts(apply(Bootfcast, 2, quantile, prob=0.025), start = start(Test), frequency=12),
upper = ts(apply(Bootfcast, 2, quantile, prob=0.975), start = start(Test), frequency=12),
level = 95),
class="forecast"){plot(bootfc, xlim = c(1988,2023),main="Bootstrap Forecasts", shadecols = "gray70",fcol = NA)
lines(df_adjts[,"PriceIndexGrowthSA"], col = "black", lty = 2)
legend("bottomleft",legend=c("Raw Data","Bootstrapped Confidence Interval"),
col=c("black","grey70"), lty=c(2,1), lwd = c(1,4), cex=0.8, box.lty=0)}
- The gray region illustrates the range of the 95% forecast interval.
- Notice the large spike of the black line outside of the gray near the end of the sample after 2020.
- That implies our series experienced a “rare” event (e.g. price changes around COVID)
Bootstrap Aggregating (Bagging)
Aggregate forecasts across bootstrap sample (aka “bagging”) can be used to improve forecast accuracy by reducing one or more sources of uncertainty. Consider:
- Parameter Uncertainty — e.g. is \(\hat{\beta}\) correct?
- Model Uncertainty — e.g should I use ARMA(1,1) or ARMA(3,2) when I train?
Note: the term model uncertainty sometimes is decomposed to mean the proper model within the training period and stability of the DGP into the out of sample period.
Bagging can help to ameliorate parameter and model uncertainty.
Let’s say we found that an ARMA(1,1) was optimal for our series in the training period and we estimate the parameters \(\Theta_{ARMA(1,1)}\)
Case 1: Produce 1000s of bootstrap paths. For each path, forecast the series using the same \(\Theta_{ARMA(1,1)}\).
Case 2: Produce 1000s of bootstrap paths. For each path \(h\), refit an optimal ARMA(1,1), yielding \(\Theta^{h}_{ARMA(1,1)}\)
Case 3: Produce 1000s of bootstrap paths. For each path \(h\), refit an optimal ARMA(p,q), yielding \(\Theta^{h}_{ARMA(p,q)}\)
In the following we will illustrate Case 2 and Case 3 and compare to the single ARMA fit as our proxy for Case 1.
Bagging-Case 2
We will find the optimal functional form via information criterion.
We will then fit this functional form to each bootstrap sample, and then forecast. In doing so, the ARMA order remains the same, but the parameter estimates are optimized for each bootstrap path.
ARMA_fit <- auto.arima(Train) # optimal ARIMA is ARIMA(2,0,2)(0,1,2)[12]
bagfcast1 <- matrix(0, nrow=BootpathNum, ncol=nT-TrainNum)
for(i in seq(BootpathNum)) {
ARMA_refit <- Arima(BootSamples[,i],
order=c(2, 0, 2),
seasonal = list(order = c(0,1,2), period = 12)) # fitted ARIMA(2,0,2)(0,1,2)[12] with re-estimating
bagfcast1[i,] <- forecast(ARMA_refit, h = nT-TrainNum)$mean} %>% ts(frequency=12, start = start(Test))Bagging-Version 2
We will find the optimal ARIMA structure for each bootstrap path, fit the ARMA model accordingly, and then forecast.
bagfcast2 <- matrix(0, nrow=BootpathNum, ncol=nT-TrainNum)
for(i in seq(BootpathNum)) {
ARMA_refit <- auto.arima(BootSamples[,i]) # fitted optimal arima model with each sample
bagfcast2[i,] <- forecast(ARMA_refit, h = nT-TrainNum)$mean} Compare with (Non) Bagging
Let’s generate bagging forecast errors in the testing period.
bagfc1 <- colMeans(bagfcast1) %>% ts(frequency=12, start = start(Test))
bagfc2 <- colMeans(bagfcast2) %>% ts(frequency=12, start = start(Test))
bagfc1_err <- df_adjts[,"PriceIndexGrowthSA"] - bagfc1
bagfc2_err <- df_adjts[,"PriceIndexGrowthSA"] - bagfc2Now compare the forecast errors built from the optimal arima forecast, bootstrapped forecast, bagging1 and bagging2 forecast errors
{plot(arimafc_err, col = "red", xlab = "", ylab = "OOS Error", main = "ARIMA OOS Forecast Error")
lines(bagfc1_err, col = "green", lty = 3)
lines(bagfc2_err, col = "purple", lty = 4)
legend("bottomleft",legend=c("ARIMA Forecast OOS Error",
"Bagging Case 2 (Reduce Parameter Uncertainty)",
"Bagging case 3 (Reduce Parameter and Model Uncertainty)"),
col=c("red", "green","purple"), lty=c(1,2,3,4), cex=0.6, box.lty=0)
}
- Bagging was able to improve upon the ARMA model
- We were able to reduce parameter and model uncertainty in Cases 2 and 3, respectively
- Notice again the poor fit around COVID. Simply using temporal persistence is not able to capture this exogenous shock
VAR
Recall our earlier discussion where housing starts may both lead and lag price growth.
Economically, it might be that rapid housing starts signal more supply, which depresses prices.
Conversely, starts may be rising because prices are rising, which signal more profitability.
This simultaneity will bias coefficient estimates of our univariate specifications. Instead, we might consider a simultaneous system of autoregressive specifications; i.e. a vector auto regression VAR.
VAR tools are helpful for
- Causal estimation / inference
- Multivariate forecasting
- Stress testing
A VAR is an Autoregressive model across multiple series. e.g. housing starts are a function of past starts and prices; meanwhile prices may be a function of past prices and starts.
Lag Selection
We can rely upon information criterion to guide our choice of lag structure.
library(vars)
VAR_data<-window(df_adjts[,c("PriceIndexGrowthSA","HousingStartGrowth")], end = c(2013,12))
VARselect(VAR_data, lag.max = 12, type = "const", season = 12)$selection## AIC(n) HQ(n) SC(n) FPE(n)
## 12 2 2 12The AIC suggests 12 lags.
- The number of parameters proliferate quickly in a VAR.
- In our case, we will start with 2 lags, as indicated by other criterion above
Fitting a VAR
VAR_est <- VAR(y = VAR_data, p = 2, type = "const")
summary(VAR_est)##
## VAR Estimation Results:
## =========================
## Endogenous variables: PriceIndexGrowthSA, HousingStartGrowth
## Deterministic variables: const
## Sample size: 321
## Log Likelihood: -1213.124
## Roots of the characteristic polynomial:
## 0.8272 0.3958 0.3958 0.1464
## Call:
## VAR(y = VAR_data, p = 2, type = "const")
##
##
## Estimation results for equation PriceIndexGrowthSA:
## ===================================================
## PriceIndexGrowthSA = PriceIndexGrowthSA.l1 + HousingStartGrowth.l1 + PriceIndexGrowthSA.l2 + HousingStartGrowth.l2 + const
##
## Estimate Std. Error t value Pr(>|t|)
## PriceIndexGrowthSA.l1 1.2889478 0.0504579 25.545 < 2e-16 ***
## HousingStartGrowth.l1 0.0010583 0.0008132 1.301 0.1941
## PriceIndexGrowthSA.l2 -0.3872130 0.0503939 -7.684 1.96e-13 ***
## HousingStartGrowth.l2 0.0022784 0.0008133 2.801 0.0054 **
## const 0.0263956 0.0121799 2.167 0.0310 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.1948 on 316 degrees of freedom
## Multiple R-Squared: 0.8889, Adjusted R-squared: 0.8875
## F-statistic: 632.2 on 4 and 316 DF, p-value: < 2.2e-16
##
##
## Estimation results for equation HousingStartGrowth:
## ===================================================
## HousingStartGrowth = PriceIndexGrowthSA.l1 + HousingStartGrowth.l1 + PriceIndexGrowthSA.l2 + HousingStartGrowth.l2 + const
##
## Estimate Std. Error t value Pr(>|t|)
## PriceIndexGrowthSA.l1 -3.19578 3.46608 -0.922 0.3572
## HousingStartGrowth.l1 0.14333 0.05586 2.566 0.0108 *
## PriceIndexGrowthSA.l2 3.53848 3.46168 1.022 0.3075
## HousingStartGrowth.l2 0.02815 0.05587 0.504 0.6147
## const 0.39305 0.83667 0.470 0.6388
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 13.38 on 316 degrees of freedom
## Multiple R-Squared: 0.02705, Adjusted R-squared: 0.01474
## F-statistic: 2.197 on 4 and 316 DF, p-value: 0.06923
##
##
##
## Covariance matrix of residuals:
## PriceIndexGrowthSA HousingStartGrowth
## PriceIndexGrowthSA 0.03794 -0.1156
## HousingStartGrowth -0.11557 179.0473
##
## Correlation matrix of residuals:
## PriceIndexGrowthSA HousingStartGrowth
## PriceIndexGrowthSA 1.00000 -0.04434
## HousingStartGrowth -0.04434 1.00000Interpreting a VAR
Notice that there are 2 “regression” tables. That’s because we have a system of two equations.
The parameter on HousingStartGrowth.l2 is 0.0022832
This means that a one unit increase in the housing starts growth rate this month is associated with a 0.0022832 unit increase in housing prices in two months from now
i.e. supply of homes is associated with future price increases (e.g. Say’s law, supply breeds demand???)
We’ll view the second output table
Most of the coefficients are not statistically significant. The R2 is low and the F stat is above 5%.
The implication is that after controlling for the simultaneity of prices, explaining housing is driven only by its own temporal dynamics (witness only HousingStartGrowth.l1 is significant)
Note: that does NOT mean that the model is perfect. We could/should consider other features and model orders.
VAR Diagnostics
tsdisplay(VAR_est$varresult$PriceIndexGrowthSA$residuals, main = "Residuals of Price Index Growth")
tsdisplay(VAR_est$varresult$HousingStartGrowth$residuals, main = "Residuals of Housing Start Growth")
- Some annual seasonality seems to remain in each series, which you can see in the (P)ACF spikes near 12
- The model shows particular trouble modeling prices around COVID.
- Housing starts are worse in general (larger residuals than prices), but nothing unique about the COVID period
Fit Seasonal VAR
We can attempt to address that possible remaining seasonality with a seasonal VAR.
VAR_est2 <- VAR(y = VAR_data, p = 2, season = 12,type = "const")
summary(VAR_est2)##
## VAR Estimation Results:
## =========================
## Endogenous variables: PriceIndexGrowthSA, HousingStartGrowth
## Deterministic variables: const
## Sample size: 321
## Log Likelihood: -1048.708
## Roots of the characteristic polynomial:
## 0.8256 0.4466 0.395 0.395
## Call:
## VAR(y = VAR_data, p = 2, type = "const", season = 12L)
##
##
## Estimation results for equation PriceIndexGrowthSA:
## ===================================================
## PriceIndexGrowthSA = PriceIndexGrowthSA.l1 + HousingStartGrowth.l1 + PriceIndexGrowthSA.l2 + HousingStartGrowth.l2 + const + sd1 + sd2 + sd3 + sd4 + sd5 + sd6 + sd7 + sd8 + sd9 + sd10 + sd11
##
## Estimate Std. Error t value Pr(>|t|)
## PriceIndexGrowthSA.l1 1.285511 0.051566 24.929 < 2e-16 ***
## HousingStartGrowth.l1 0.002655 0.001341 1.980 0.04858 *
## PriceIndexGrowthSA.l2 -0.387227 0.051465 -7.524 6e-13 ***
## HousingStartGrowth.l2 0.003736 0.001354 2.760 0.00614 **
## const 0.025073 0.012285 2.041 0.04211 *
## sd1 -0.097567 0.056789 -1.718 0.08680 .
## sd2 -0.138604 0.064941 -2.134 0.03362 *
## sd3 -0.126514 0.087349 -1.448 0.14854
## sd4 -0.169843 0.091958 -1.847 0.06572 .
## sd5 -0.114447 0.072260 -1.584 0.11427
## sd6 -0.092925 0.066281 -1.402 0.16194
## sd7 -0.085448 0.061604 -1.387 0.16644
## sd8 -0.096723 0.059431 -1.627 0.10467
## sd9 -0.081146 0.059838 -1.356 0.17607
## sd10 -0.077203 0.061370 -1.258 0.20936
## sd11 -0.037761 0.058308 -0.648 0.51773
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.1961 on 305 degrees of freedom
## Multiple R-Squared: 0.8913, Adjusted R-squared: 0.886
## F-statistic: 166.7 on 15 and 305 DF, p-value: < 2.2e-16
##
##
## Estimation results for equation HousingStartGrowth:
## ===================================================
## HousingStartGrowth = PriceIndexGrowthSA.l1 + HousingStartGrowth.l1 + PriceIndexGrowthSA.l2 + HousingStartGrowth.l2 + const + sd1 + sd2 + sd3 + sd4 + sd5 + sd6 + sd7 + sd8 + sd9 + sd10 + sd11
##
## Estimate Std. Error t value Pr(>|t|)
## PriceIndexGrowthSA.l1 0.28991 2.16996 0.134 0.893805
## HousingStartGrowth.l1 -0.40220 0.05643 -7.128 7.41e-12 ***
## PriceIndexGrowthSA.l2 1.24854 2.16570 0.577 0.564699
## HousingStartGrowth.l2 -0.16060 0.05698 -2.819 0.005138 **
## const 0.70157 0.51695 1.357 0.175743
## sd1 12.01779 2.38973 5.029 8.43e-07 ***
## sd2 38.13026 2.73281 13.953 < 2e-16 ***
## sd3 31.31391 3.67576 8.519 7.49e-16 ***
## sd4 22.63754 3.86970 5.850 1.27e-08 ***
## sd5 14.72243 3.04078 4.842 2.05e-06 ***
## sd6 7.26792 2.78917 2.606 0.009617 **
## sd7 6.27819 2.59236 2.422 0.016027 *
## sd8 4.91979 2.50092 1.967 0.050067 .
## sd9 9.13871 2.51806 3.629 0.000333 ***
## sd10 -5.44701 2.58254 -2.109 0.035745 *
## sd11 -9.10430 2.45369 -3.710 0.000246 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 8.254 on 305 degrees of freedom
## Multiple R-Squared: 0.6427, Adjusted R-squared: 0.6251
## F-statistic: 36.57 on 15 and 305 DF, p-value: < 2.2e-16
##
##
##
## Covariance matrix of residuals:
## PriceIndexGrowthSA HousingStartGrowth
## PriceIndexGrowthSA 0.03847 0.08792
## HousingStartGrowth 0.08792 68.12423
##
## Correlation matrix of residuals:
## PriceIndexGrowthSA HousingStartGrowth
## PriceIndexGrowthSA 1.00000 0.05431
## HousingStartGrowth 0.05431 1.00000- The model fits are much improved. Witness the higher R2 and F stats.
tsdisplay(VAR_est2$varresult$PriceIndexGrowthSA$residuals, main = "Residuals of Price Index Growth")
tsdisplay(VAR_est2$varresult$HousingStartGrowth$residuals, main = "Residuals of Housing Start Growth")
- Appears to be gone for housing starts, but not prices.
- Again, prices are plagued by COVID. We should attempt to model that feature directly. Ignoring it could impact all of our estimates. We will avoid that for the sake of this tutorial.
Recall the impact of possible serial correlation. So, it’s best practice to consider our robust standard errors.
library(lmtest)
library(sandwich)
coeftest(VAR_est2, vcov. = sandwich) ##
## t test of coefficients:
##
## Estimate Std. Error t value
## PriceIndexGrowthSA:(Intercept) 0.0250726 0.0161273 1.5547
## PriceIndexGrowthSA:PriceIndexGrowthSA.l1 1.2855112 0.0743666 17.2861
## PriceIndexGrowthSA:HousingStartGrowth.l1 0.0026552 0.0013821 1.9212
## PriceIndexGrowthSA:PriceIndexGrowthSA.l2 -0.3872265 0.0720220 -5.3765
## PriceIndexGrowthSA:HousingStartGrowth.l2 0.0037364 0.0016213 2.3046
## PriceIndexGrowthSA:sd1 -0.0975667 0.0553393 -1.7631
## PriceIndexGrowthSA:sd2 -0.1386039 0.0823705 -1.6827
## PriceIndexGrowthSA:sd3 -0.1265143 0.0926034 -1.3662
## PriceIndexGrowthSA:sd4 -0.1698430 0.0969352 -1.7521
## PriceIndexGrowthSA:sd5 -0.1144473 0.0723069 -1.5828
## PriceIndexGrowthSA:sd6 -0.0929247 0.0582006 -1.5966
## PriceIndexGrowthSA:sd7 -0.0854480 0.0566530 -1.5083
## PriceIndexGrowthSA:sd8 -0.0967228 0.0544400 -1.7767
## PriceIndexGrowthSA:sd9 -0.0811456 0.0512582 -1.5831
## PriceIndexGrowthSA:sd10 -0.0772026 0.0555088 -1.3908
## PriceIndexGrowthSA:sd11 -0.0377606 0.0579757 -0.6513
## HousingStartGrowth:(Intercept) 0.7015672 0.5784665 1.2128
## HousingStartGrowth:PriceIndexGrowthSA.l1 0.2899129 2.4765210 0.1171
## HousingStartGrowth:HousingStartGrowth.l1 -0.4021989 0.0566167 -7.1039
## HousingStartGrowth:PriceIndexGrowthSA.l2 1.2485351 2.3867528 0.5231
## HousingStartGrowth:HousingStartGrowth.l2 -0.1606013 0.0648886 -2.4750
## HousingStartGrowth:sd1 12.0177861 2.7442141 4.3793
## HousingStartGrowth:sd2 38.1302559 3.4915496 10.9207
## HousingStartGrowth:sd3 31.3139071 3.9627792 7.9020
## HousingStartGrowth:sd4 22.6375404 4.1175356 5.4978
## HousingStartGrowth:sd5 14.7224330 3.3386093 4.4098
## HousingStartGrowth:sd6 7.2679164 2.8217627 2.5757
## HousingStartGrowth:sd7 6.2781916 2.7815838 2.2571
## HousingStartGrowth:sd8 4.9197920 2.7365118 1.7978
## HousingStartGrowth:sd9 9.1387057 2.8416286 3.2160
## HousingStartGrowth:sd10 -5.4470132 2.7869083 -1.9545
## HousingStartGrowth:sd11 -9.1042990 2.8827970 -3.1581
## Pr(>|t|)
## PriceIndexGrowthSA:(Intercept) 0.121061
## PriceIndexGrowthSA:PriceIndexGrowthSA.l1 < 2.2e-16 ***
## PriceIndexGrowthSA:HousingStartGrowth.l1 0.055637 .
## PriceIndexGrowthSA:PriceIndexGrowthSA.l2 1.513e-07 ***
## PriceIndexGrowthSA:HousingStartGrowth.l2 0.021859 *
## PriceIndexGrowthSA:sd1 0.078891 .
## PriceIndexGrowthSA:sd2 0.093459 .
## PriceIndexGrowthSA:sd3 0.172884
## PriceIndexGrowthSA:sd4 0.080757 .
## PriceIndexGrowthSA:sd5 0.114504
## PriceIndexGrowthSA:sd6 0.111384
## PriceIndexGrowthSA:sd7 0.132521
## PriceIndexGrowthSA:sd8 0.076617 .
## PriceIndexGrowthSA:sd9 0.114441
## PriceIndexGrowthSA:sd10 0.165295
## PriceIndexGrowthSA:sd11 0.515333
## HousingStartGrowth:(Intercept) 0.226143
## HousingStartGrowth:PriceIndexGrowthSA.l1 0.906886
## HousingStartGrowth:HousingStartGrowth.l1 8.596e-12 ***
## HousingStartGrowth:PriceIndexGrowthSA.l2 0.601277
## HousingStartGrowth:HousingStartGrowth.l2 0.013866 *
## HousingStartGrowth:sd1 1.638e-05 ***
## HousingStartGrowth:sd2 < 2.2e-16 ***
## HousingStartGrowth:sd3 5.036e-14 ***
## HousingStartGrowth:sd4 8.135e-08 ***
## HousingStartGrowth:sd5 1.436e-05 ***
## HousingStartGrowth:sd6 0.010475 *
## HousingStartGrowth:sd7 0.024712 *
## HousingStartGrowth:sd8 0.073192 .
## HousingStartGrowth:sd9 0.001439 **
## HousingStartGrowth:sd10 0.051556 .
## HousingStartGrowth:sd11 0.001747 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1- After controlling for the dynamic interactions, it appears that
seasonality is more apparent in the housing starts model.
- Witness the bottom half of the table, coefficients such as HousingStartGrowth:sd1 have low pvalues indicating a significant seasonal difference.
- The pvalues from the coefficients above suggest some indication of
today’s housing starts to explain tomorrow’s prices, but not vice
versa.
- We need a test of combine the aggregated effects of these forces.
Determining Direction of causality
If Housing Starts today have an impact on Prices tomorrow, then Housing Starts are said to Granger Cause Prices.
causality(VAR_est2, cause = "HousingStartGrowth")$Granger##
## Granger causality H0: HousingStartGrowth do not Granger-cause
## PriceIndexGrowthSA
##
## data: VAR object VAR_est2
## F-Test = 4.4035, df1 = 2, df2 = 610, p-value = 0.01263Let’s examine the other direction.
causality(VAR_est2, cause = "PriceIndexGrowthSA")$Granger##
## Granger causality H0: PriceIndexGrowthSA do not Granger-cause
## HousingStartGrowth
##
## data: VAR object VAR_est2
## F-Test = 1.802, df1 = 2, df2 = 610, p-value = 0.1658- The relatively high pvalue suggests that we fail to reject.
- Prices don’t “cause” starts
- The causal direction is only one way (housing starts to prices).
Impulse Response Function (IRF)
Another way to understand and use the VAR estimates is via a stress test. For instance, what happens to Prices over the next 2yrs if we get a shock to housing starts today?
irf_PriceIndexGrowth <- irf(VAR_est2, impulse = "HousingStartGrowth",
response = "PriceIndexGrowthSA",
cumulative = FALSE,
n.ahead = 20, boot = TRUE)
irf_PriceIndexGrowth$irf$HousingStartGrowth[3]## [1] 0.05012329plot(irf_PriceIndexGrowth, ylab = "Price Index Growth", main = "Cumulative Impact of 1 sigma shock to Housing Starts")
irf_PriceIndexGrowth <- irf(VAR_est2, impulse = "HousingStartGrowth",
response = "PriceIndexGrowthSA",
cumulative = TRUE,
n.ahead = 20, boot = TRUE)
irf_PriceIndexGrowth$irf$HousingStartGrowth[3]## [1] 0.07200637plot(irf_PriceIndexGrowth, ylab = "Price Index Growth", main = "Cumulative Impact of 1 sigma shock to Housing Starts")
- From the first graph we can see that a 1 sigma rise in housing starts this month will trigger a rise in house price growth next month
- This shock’s effect will grow more in the second and third months.
- By the 4th and 5th months, the shock’s impact is fading.
- Within roughly 1.5-2 yrs, the shock’s impact is gone.
- The second graph shows the CUMULATIVE impacts, which tells the same story, but from a different visual perspective.
irf_HousingStartGrowth <- irf(VAR_est2, impulse = "PriceIndexGrowthSA",
cumulative = FALSE,
response = "HousingStartGrowth",
n.ahead = 20, boot = TRUE)
plot(irf_HousingStartGrowth, ylab = "Housing Start Growth", main = "Cumulative Impact from 1 sigma shock to Prices")
irf_HousingStartGrowth <- irf(VAR_est2, impulse = "PriceIndexGrowthSA",
cumulative = TRUE,
response = "HousingStartGrowth",
n.ahead = 20, boot = TRUE)
plot(irf_HousingStartGrowth, ylab = "Housing Start Growth", main = "Cumulative Impact from 1 sigma shock to Prices")
- A one unit increase in prices triggers a drop in housing starts for a couple of months
- Then the impact rises and fades quickly
- Notice that throughout the CI bands in red surround the 0 line, indicating little statistical significance.
Forecast with VAR
Given the causality findings, we may consider refining the model. For now, we’ll use that estimated structure to predict both Prices and Starts 24-months head.
VAR_mulpred <- predict(VAR_est2, n.ahead = 24)
plot(VAR_mulpred, names = "PriceIndexGrowthSA", main = "Forecast of Price Growth")
plot(VAR_mulpred, names = "HousingStartGrowth", main = "Forecast of Housing Starts Growth")
- The red dashed lines are 95% CIs.
- The Price Growth CIs are very wide. We saw evidence of this earlier with the poor explanatory power especially during COVID.
- Meanwhile, the housing starts forecast seems much more stable.
- It detects the seasonality and accommodates the dynamic linkages with prices.