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dynRecursive.Rmd
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---
title: "Dynamic Regression 3: Parameter Stability"
author: "Miguel A. Arranz"
date: " November 2016 "
output:
html_document:
toc: yes
---
```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
```
# Goals
Perform stability analysis of dynamic linear regression model coefficients
# Packages and data
```{r packages, warning=F, message=F, error=F}
library(quantmod)
library(dynlm)
library(tseries)
library(forecast)
library(lmtest)
library(sandwich)
library(ggplot2)
library(plotly)
library(tsoutliers)
source("myfuncs.R")
library(gets)
library(dyn)
```
Download data
```{r datadownload, message=FALSE, warning=FALSE}
symbols.vec <- c("INDPRO", "GS10")
getSymbols(symbols.vec, src="FRED")
INDPRO <- INDPRO['2000-01-01/2016-06-30']
GS10 <- GS10['2000-01-01/2016-06-30']
y <- as.zoo(diff(log(INDPRO)))
x <- as.zoo(diff(GS10))
```
# Recursive Model Estimation
```{r model1, error=FALSE}
Xregs <- merge(lag(x, -1), lag(x,-3))
Mod1.arx <-arx(y, ar = 2:4, mxreg = Xregs, plot=F, vcov.type = "white", verbose = T)
#print(Mod1.arx)
```
```{r recursive}
recModel1.arx <- recursive(Mod1.arx)
```
I will start in 2003-08-01, which is observation 44
```{r N0}
N0 <- which(time(x)==as.Date("2003-08-01"))
N0
```
The number of recursive estimations is going to be
```{r}
N <- length(y) - N0 + 1
N
```
So that I create a matrix for the coefficients and a vector for the recursive sum of squared residuals
```{r betamatrix}
betamatrix <- matrix(ncol=5, nrow = N)
betamatrix <- zoo(betamatrix, time(y)[N0:length(y)])
RSSR <- array(dim = N)
sigma12 = RSSR
RSSR <- zoo(RSSR, time(y)[N0:length(y)])
```
I will construct the starting points as $N+i$ so that the first one will be $N+i = 164$. That means that I have to reduce it by 1
```{r}
N0 <- N0-1
```
Now we make the estimations
```{r recgets}
for(i in 1:N)
{
k <- N0 + i
vy <- y[1:k]
vx <- x[1:k]
tempmodel <- dynlm(vy ~ L(vy, 2:4) + L(vx, 1) + L(vx,3) - 1)
betamatrix[i,] <- tempmodel$coefficients
ssr <- drop(crossprod(tempmodel$residuals))
RSSR[i] <- ssr
sigma1 <- summary(tempmodel)$sigma
sigma12[i] <- sigma1*sigma1
}
colnames(betamatrix) <- names(tempmodel$coefficients)
```
Make graphs
```{r recgraphs}
p1 <- autoplot(betamatrix[,1])+ ylab("") + xlab("") + labs(title=expression(Y[t-2]),
subtitle="Recursive Estimation")
p2 <- autoplot(betamatrix[,2])+ ylab("") + xlab("") + labs(title=expression(Y[t-3]),
subtitle="Recursive Estimation")
p3 <- autoplot(betamatrix[,3])+ ylab("") + xlab("") + labs(title=expression(Y[t-4]),
subtitle="Recursive Estimation")
p4 <- autoplot(betamatrix[,4])+ ylab("") + xlab("") + labs(title=expression(X[t-1]),
subtitle="Recursive Estimation")
p5 <- autoplot(betamatrix[,5])+ ylab("") + xlab("") + labs(title=expression(X[t-3]),
subtitle="Recursive Estimation")
```
Interactive graphs
```{r inrec}
ggplotly(p1)
ggplotly(p2)
ggplotly(p3)
ggplotly(p4)
ggplotly(p5)
```
Possible changes of regime:
+ $\beta_1$ : 2005-09-01/10-01, 2005-11/2007-01-01, 2007-01/2008-06, 2008-09-01 (AO),
2008-12-01/End
+ $\beta_2$ 2005-11/2006-01, 2006-02/2008-07, 2008-12 (AO), 2009-10/End
+ $\beta_3$ 2004-11/2005-09, 2006-01/2008-07, 2009-09/End
+ $\beta_4$ 2004-12/2005-10, 2005-12/2008-10, 2009-07/End
+ $\beta_5$ 2008-09/2009-07
# Chow Tests
## 1-step Ahead Chow Test
For each sample $Y_1, Y_2, \ldots, Y_m$, it is tested whether the next observation comes from the same model as $Y_0, Y_1, \ldots, Y_{m-1}$. The unrestricted model is
$$
\begin{cases}
Y_t = \beta_1 Y_{t-2} + \ldots + \epsilon_t & \forall t =1, 2, \ldots , m-1 \\
Y_m \text{ unrestricted}
\end{cases}
$$
whereas the restricted model is that the model governs the full sample.
The idea is to see whether the model based on $Y_1, Y_2, \ldots, Y_m$ is good at forecasting $Y_m$ for every feasible $m$. We use an $F$-type statistic
$$\dfrac{RSS_m - RSS_{m-1}}{RSS_{m-1}/(m-1-k)} \sim \chi^2(1)$$
where the denominator is the estimate of $\sigma^2$ for the recursive residuals up to $m-1$.
```{r chow1}
chow1 <- (coredata(RSSR)[-1] - coredata(RSSR)[-N])/sigma12[-N]
cvalue <- qchisq(0.99, df=1)
chow1 <- chow1/cvalue
chow1 <- zoo(chow1, time(RSSR))
autoplot(chow1) + xlab("") + ylab("") + labs(title="1-step ahead Chow test",
subtitle="Normalized by the 1% critical value")
which(chow1>1)
w <- which(chow1>1)
time(RSSR)[w]
```
Notice that we have normalized by the 1% critical value, so that the horizontal line at 1 gives the critical value of the test.
At least 2 or 3 rejections would be needed to reject the model based on this type of test.
## Break-Point Chow Test
This is a variation of the 1-step Chow test. The idea is to see whether a model based on $Y_1, Y_2, \ldots, Y_{m-1}$ is good at forecasting all the remaining variables $Y_m, \ldots, Y_T$. The forecast horizon $T_m+1$ therefore decreases with $m$.
The unrestricted residual sum of squares is thus $RSS_{m-1}$ and the restricted model residual sum of squares is $RSS_T$. Hence, the test is based on the statistic
$$ \dfrac{RSS_T - RSS_{m-1}}{RSS_{m-1}/(T-m+1)} \sim \chi^2(T_m+1)$$
or, if we think as the problem of dividing the sample in two subsamples, the first with $T_1$ and the second with $T_2$ observations, the test statistic is
$$ \dfrac{RSS_T - RSS_1}{\sigma_1^2} \sim \chi^2 (T_2)$$
```{r chow2}
chow2 <- (coredata(RSSR)[N] - coredata(RSSR)[-N])/sigma12[-N]
cvalues <- qchisq(.99, (N-1):1)
chow2 <- chow2/cvalues
chow2 <- zoo(chow2, time(RSSR)[1:(N-1)])
autoplot(chow2) + xlab("") + ylab("") +labs(title="Break-Point Chow Test",
subtitle="Normalized by 1% Critical Value")
time(chow2)[which(chow2>1)]
```
What is the point with the maximum value for the test?
```{r maxchow2}
time(RSSR)[which(chow2==max(chow2))]
```
## Forecast Chow Test
The idea is to see if a model based on the shortest sample $Y_1, \ldots , Y_{M-1}$, can forecast $Y_M, \ldots, Y_m$, where $m$ increases from $M$ to $T$. The unrestricted residual sum of squares is $RSS_{M-1}$, while the restricted residuals sum of squares is $RSS_m$ and hence the tests are based on
$$ \dfrac{RSS_m - RSS_{M-1}}{\hat{\sigma}^2_{M-1}} \sim \chi^2(m-M+1)$$
Let's change the notation to make it clear.
1. We take a first sample to estimate the model $Y_1, \ldots , Y_{M}$ and obtain the residual sum of squares $(RSS_M)$ and the variance of the residuals $(\hat{\sigma}^2_{M})$.
2. Take a second sample $Y_1, \ldots , Y_{M}, \ldots Y_{M+H}$ and obtain the sum of squared residuals $RSS_{M+H}$.
3. The test statistic is
$$ \dfrac{RSS_{M+H} - RSS_{M}}{\hat{\sigma}^2_{M}} \sim \chi^2(H)$$
We can:
a. take $M$ as fixed and change the values of $H$ and thus obtain a forecast deterioration test with a fixed origin and changing forecast horizon, or
b. increase the value of $M$ while keeping $H$ constant.
I am taking the second approach here. This code is for $H=12$ so that I am applying the test to the forecasts of the following year.
```{r chow3}
H <- 12
k1 <- N-H
k2 <- H + 1
den <- sigma12[1:k1]
num <- coredata(RSSR)[k2:N] - coredata(RSSR)[1:k1]
chow3 <- num/den
chow3 <- zoo(chow3, time(RSSR)[1:k1])
cval <- qchisq(0.99, H)
chow3 <- chow3/cval
autoplot(chow3) + xlab("") + ylab("") +
labs(title="Chow Forecast Test H=12" ,
subtitle="Normalized by 1% Critical Value")
```
Tell me when I am rejecting $H_0$
```{r rejectchow3}
time(RSSR)[chow3>1]
```
# Dummies and Interactions
Is there a break in Sept. 2008?
That is observation
```{r obs200809}
which(time(y)==as.Date("2008-09-01"))
```
I am creating a dummy
$$ D1_t =
\begin{cases}
1 & t>2008-09-01 \\
0 & \text{elsewhere}
\end{cases}$$
```{r d1}
d1 <- array(0, dim=length(y))
d1[105:length(y)] <- 1
d1<- zoo(d1, time(y))
```
and estimate a model with interactions
```{r interac1}
modinter1 <- dynlm(y ~ d1*(L(y, 2:4) + L(x,1) + L(x,3)))
print(summary(modinter1))
coeftest(modinter1, vcov=vcovHC)
```
```{r interac1b}
modinter1 <- dynlm(y ~ d1*(L(y, 2) + L(y,3) + L(y,4) + L(x,1) + L(x,3) - 1 ) - d1
- d1:L(x,3) - d1:L(y,4) - d1:L(y,2))
print(summary(modinter1))
coeftest(modinter1, vcov=vcovHC)
```
Dubious!
Is there a break in January 2004?
```{r Jan2004}
t1 <- which(time(y)==time(RSSR)[which(chow2==max(chow2))])
d1 <- array(0, dim=length(y))
d1[1:t1] <- 1
d1 <- zoo(d1, time(y))
modinter3 <- dynlm(y ~ d1*(0+ L(y,2) + L(y,3) + L(y,4) + L(x,1) + L(x,3)))
summary(modinter3)
coeftest(modinter3, vcov=vcovHC)
```
```{r modinter3b}
modinter3 <- dynlm(y ~ d1*(0+ L(y,2) + L(y,3) + L(y,4) + L(x,1) + L(x,3))
- d1 - d1:L(x,1) - d1:L(x,3) - d1:L(y,2) - d1:L(y,4))
summary(modinter3)
coeftest(modinter3, vcov=vcovHC)
```
Dubious!
Was it in September 2009?
```{r obs2009-09}
d2 <- array(0, dim=length(y))
d2[which(time(y)==as.Date("2009-09-01")):length(N)] <- 1
d2 <- zoo(d2, time(y))
modinter2 <- dynlm(y ~ d2*(L(y, 2:4) + L(x,1) + L(x,3)) - 1 - d2)
print(summary(modinter2))
coeftest(modinter2, vcov=vcovHC)
```
```{r modinter2}
modinter2 <- dynlm(y ~ 0+ L(y,2) + L(y,3) + L(y,4) + L(x,3) + d2:L(x,1))
summary(modinter2)
coeftest(modinter2, vcov=vcovHC)
```
2005-12/2008-09?
```{r inter3}
t1 <- which(time(y)==as.Date("2005-12-01"))
t2 <- which(time(y)==as.Date("2008-09-01"))
d3 <- array(0, dim=length(y))
d3[t1:t2] <- 1
d3 <- zoo(d3, time(y))
modinter3 <- dynlm( y ~d3*(0 + L(y,2) + L(y,3) + L(y,4) + L(x,1) + L(x,3)))
summary(modinter3)
coeftest(modinter3, vcov=vcovHC)
```
```{r inter3b}
modinter3 <- dynlm( y ~ d3*(0 + L(y,2) + L(y,3) + L(y,4) + L(x,1) + L(x,3))
- d3:L(x,1) - d3:L(y,3) - d3:L(y,4) + d3:L(y,1))
summary(modinter3)
coeftest(modinter3, vcov=vcovHC)
```
Combining some dummies
```{r combinter}
modinter4 <- dynlm( y ~ d3*(0 + L(y,2) + L(y,3) + L(y,4) + L(x,1) + L(x,3))
- d3:L(x,1) - d3:L(y,3) - d3:L(y,4) + d3:L(y,1) +
L(x, 1)*d2 - d2 - L(x,1) - L(y,2))
summary(modinter4)
coeftest(modinter4, vcov=vcovHC)
```
So that we can compare our models:
```{r modcompare, warning=FALSE, error=FALSE, message=FALSE}
library(stargazer)
mod0 <- dynlm(y ~0 + L(y,2) + L(y,3) + L(y,4) + L(x,1) + L(x,3))
modinter2 <- dynlm(y ~ 0 + L(y,2) + L(y,3) + L(y,4) + L(x,3) + d2:L(x,1))
modinter3 <- dynlm(y ~0 + d3 +L(y,2) + L(y,3) + L(y,4) + L(x,1) + L(x,3)
+ d3:L(y,1) + d3:L(y,2) + d3:L(x,3) )
modinter4 <- dynlm(y ~0 + d3 + L(y,3) + L(y,4) + L(x,3)
+ d3:L(y,1) + d3:L(x,3)+ d3:L(y,2) + d2:L(x,1) )
c0 <- coeftest(mod0, vcov=vcovHC)
c2 <- coeftest(modinter2, vcov=vcovHC)
c3 <- coeftest(modinter3, vcov=vcovHC)
c4 <- coeftest(modinter4, vcov=vcovHC)
stargazer(c0, c2, c3, c4, type="text")
```
Let's make a graph for the largest models
```{r}
mydata <- merge(y, mod0$fitted.values, modinter2$fitted.values,
modinter3$fitted.values, modinter4$fitted.values, all=F)
names(mydata) <- c("Actual", "Model1", "Model2", "Model3", "Model4")
p <- autoplot(mydata, facets = NULL) + xlab("") + ylab("") +
labs(title="Models with Interactions", subtitle="Fitted Values")
ggplotly(p)
```
# Structural Change (strucchange package and lm)
```{r lm}
mydata <- cbind(y, lag(y, -2), lag(y, -3), lag(y, -4), lag(x, -1), lag(x, -3))
names(mydata) <- c("Ylag0", "Ylag2", "Ylag3", "Ylag4", "Xlag1", "Xlag3")
mydata <- as.xts(mydata)
mydata <- as.ts(mydata)
modlm <- lm(Ylag0 ~ Ylag2 + Ylag3 + Ylag4 + Xlag1 + Xlag3 - 1, data=mydata)
print(summary(modlm))
```
```{r strucchange}
library(strucchange)
plot(Fstats(modlm, data = mydata))
bp.modlm <- breakpoints(Ylag0 ~ Ylag2 + Ylag3 + Ylag4 + Xlag1 + Xlag3 - 1, data=mydata)
```