eco5316sp2017_hw8_2
Seonjun Kang
library(Quandl)
Quandl.api_key("KmxULt3z1Vz1neVxGioB")
library("vars")
library("urca")
library("gdata")
library("stargazer")
library("KFAS")(a) Construct the time series for monthly returns
First, we load data.
SP500 <- Quandl("YAHOO/INDEX_GSPC/CLOSE", collapse = "monthly", type = 'zoo')
IBM <- Quandl("GOOG/NYSE_IBM/CLOSE", collapse = "monthly", type = 'zoo')
Tbill3m <- Quandl("FRED/TB3MS", type = 'zoo')rsp500 <- 100*diff(log(SP500), differences = 1)
rIBM <- 100*diff(log(IBM), differences = 1)
rTbill3m <- Tbill3m / 12par(mfrow=c(3,2))
plot(log(SP500), main=expression("logged SP500"))
plot(rsp500, main=expression("100*differentiated logged SP500"))
plot(IBM, main=expression("logged IBM"))
plot(rIBM, main=expression("100*differentiated logged IBM"))
plot(Tbill3m, main=expression("Tbill 3month"))
plot(rTbill3m, main=expression("100*differentiated logged Tbill 3month"))
summary(rsp500)## Index rsp500
## Min. :1950 Min. :-24.5428
## 1st Qu.:1967 1st Qu.: -1.7583
## Median :1984 Median : 0.9057
## Mean :1984 Mean : 0.6118
## 3rd Qu.:2001 3rd Qu.: 3.4312
## Max. :2017 Max. : 15.1043summary(rIBM)## Index rIBM
## Min. :1981 Min. :-30.3834
## 1st Qu.:1990 1st Qu.: -3.7874
## Median :1999 Median : 0.4004
## Mean :1999 Mean : 0.5231
## 3rd Qu.:2008 3rd Qu.: 4.8545
## Max. :2017 Max. : 30.2914summary(rTbill3m)## Index rTbill3m
## Min. :1934 Min. :0.0008333
## 1st Qu.:1955 1st Qu.:0.0316667
## Median :1976 Median :0.2533333
## Mean :1976 Mean :0.2931917
## 3rd Qu.:1996 3rd Qu.:0.4460417
## Max. :2017 Max. :1.3583333- Since all three data set have different periods, we trim the data from April/1981 to April/2017
rsp5002 <- window(rsp500, start = 1981 + 3/12, end = 2017 + 3/12)
rIBM2 <- window(rIBM, start = 1981 + 3/12, end = 2017 + 3/12)
rTbill3m2 <- window(rTbill3m, start = 1981 + 3/12, end = 2017 + 3/12)(b) Estimate a simple OLS relating excess returns for IBM and excess returns for S&P 500
- We estimate a simple OLS
excessIBM <- rIBM2 - rTbill3m2
excessSP500 <- rsp5002 - rTbill3m2
sOLS <- lm(excessIBM ~ excessSP500)
plot(excessIBM, excessSP500, main="A simple OLS relating excess returns for IBM S&P500")
abline(lm(excessSP500~excessIBM))
summary(sOLS)##
## Call:
## lm(formula = excessIBM ~ excessSP500)
##
## Residuals:
## Min 1Q Median 3Q Max
## -31.2343 -3.1235 -0.1644 3.4677 24.5022
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.09957 0.29559 -0.337 0.736
## excessSP500 0.92509 0.06804 13.597 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 6.134 on 431 degrees of freedom
## Multiple R-squared: 0.3002, Adjusted R-squared: 0.2986
## F-statistic: 184.9 on 1 and 431 DF, p-value: < 2.2e-16According to the results, \(\alpha\) (intercept) is -0.09957 and \(\beta\) is 0.92509.
(c) Estimate a CAPM model with time-varying coefficients with time varying coefficient alpha and beta using Kalman filter. Create a plot showing how the smoothed states alpha and beta changed over time.
excesssp500ts <- as.ts(excessSP500)
excessIBMts <- as.ts(excessIBM)
T <- length(excesssp500ts)
# construct system matrices for state-space model
Zt <- array(rbind(rep(1,T), excesssp500ts), dim=c(1,2,T))
Ht <- matrix(NA)
Tt <- diag(2)
Rt <- diag(2)
Qt <- matrix(c(NA,0,0,NA), 2,2)
# use diffuse prior
a1 <- matrix(c(0, 1), 2, 1)
P1 <- matrix(0, 2, 2)
P1inf <- diag(2)
# define state-space CAPM model
y.SS <- SSModel(excessIBMts ~ -1 + SSMcustom(Z=Zt, T=Tt, R=Rt, Q=Qt, a1=a1, P1=P1, P1inf=P1inf), H=Ht)
# estimate variances of innovations using maximum likelihood
y.SS.ML <- fitSSM(y.SS, inits = c(0.001,0.001,0.001), method="BFGS")
# Kalman filtering and smoothing, with parameters in Q and H set to maximum likelihood estimates
y.SS.KFS <- KFS(y.SS.ML$model)Then, we get coefficients.
alpha.KFS <- ts( cbind(y.SS.KFS$a[,1], y.SS.KFS$alphahat[,1]), start=c(1981,4), frequency=12)
beta.KFS <- ts( cbind(y.SS.KFS$a[,2], y.SS.KFS$alphahat[,2]), start=c(1981,4), frequency=12)- Next, we plot the filtered and smoothed states.
par(mfrow=c(2,1), mar=c(4,4,2,1))
# plot filtered and smoothed state alpha and betta
plot.ts(alpha.KFS, plot.type="single", xlab="",ylab="Alpha", col=c("blue","red"), lwd=2)
legend("topright", c("filtered","smoothed"), col=c("blue","red"), lwd=2, cex=0.7, bty="n")
plot.ts(beta.KFS, plot.type="single", xlab="",ylab="Beta", col=c("blue","red"), lwd=2)
legend("topright", c("filtered","smoothed"), col=c("blue","red"), lwd=2, cex=0.7, bty="n")
par(mfrow=c(2,1), mar=c(4,4,2,1))
# plot smoothed state alpha and betta
plot.ts(alpha.KFS[,2], plot.type="single", xlab="",ylab="Alpha", col="red", lwd=2)
legend("topright", "smoothed", col="red", lwd=2, cex=0.7, bty="n")
plot.ts(beta.KFS[,2], plot.type="single", xlab="",ylab="Beta", col="red", lwd=2)
legend("topright", "smoothed", col="red", lwd=2, cex=0.7, bty="n")
(d) Compare alpha and beta from (b) and (c)
#A simple OLS alpha, beta
mean(sOLS$coefficients[1])## [1] -0.09956639mean(sOLS$coefficients[2])## [1] 0.9250887#Kalman filtered alpha, beta
mean(y.SS.KFS$a[, 1])## [1] -0.06942577mean(y.SS.KFS$a[, 2])## [1] 0.9198105#Kalman filtered smoothed alpha, beta
mean(y.SS.KFS$alphahat[, 1])## [1] -0.07471126mean(y.SS.KFS$alphahat[, 2])## [1] 0.9228255- All beta look almost the same (around 0.92). The alpha of OLS(- 0.09956639) is smaller than other two models’ alpha (- 0.06942577 and - 0.07471126). However, as we see the plot, the beta and alpha of IBM fluctuated in the past, which sometimes IBM stock is more risky or less risky. Therefore, if alpha and beta are not constant, state space model looks better than a simple OLS which assumes that alpha and beta for CAPM are constant.