Empirical Methods in Finance Homework 6
Prof. . Lochstoer TA:
February 12, 2022
Problem 1: ARCH, GARCH, and Realized Variance
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From Kenneth French’s web page, download the monthly and daily returns to the invest- ment (CMA) factor from 19630701 to 20191231. CMA is the last factor in the FF 5-factor model. Using the monthly return data:
1. Estimate an ARMA(1,1) for the return series. Report the results. What is the esti- mated monthly persistence of expected returns to CMA? What is the half-life of the expected return series in months?
Suggested Solution: Shown in Table 1. The estimated monthly persistence of ex- pected returns to CMA is 0.425. The half-life of the expected return series in months is 0.81.
2. EstimateanARCH(12)andaGARCH(1,1)processfortheresidualsfromthisARMA(1,1). Report the results and plot the time series of the conditional variance from each model
on the same plot. Are the estimated variance processes stationary?
Suggested Solution: The estimation for ARCH(12) and GARCH(1,1) are shown in Table2. The plots of conditional variance are shown in Figure 1. The blue line is the GARCH(1,1) and the black line is the ARCH(12). They are pretty close to each other.
3. Plot the absolute values of the normalized residuals, ηt = εt/σt, for each model on two separate plots. Using eyeball econometrics, do the models do a good job of accounting for clustering of volatility? Plot the autocorrelation functions of |ηt|.
Table 1: Estimation results of ARMA(1,1)
Observations Log Likelihood
Akaike Inf. Crit.
Dependent variable:
0.473∗∗ (0.236)
−0.364 (0.249)
0.269∗∗∗ (0.089)
702 −1,470.366 3.862 2,948.732
∗p<0.1; ∗∗p<0.05; ∗∗∗p<0.01
Figure1: Conditional variance
Estimation results: ARCH(12) and GARCH(1,1)
Dependent variable:
0.151∗ (0.088)
0.201 (0.358)
0.156∗ (0.084)
0.155 (0.328)
−0.021 (0.335)
0.160∗∗ (0.063)
0.145∗∗∗ (0.030)
ma1 −0.077 (0.370)
Observations
Log Likelihood Akaike Inf. Crit. Bayesian Inf. Crit.
1.049∗∗∗ (0.246)
0.141∗∗∗ (0.051)
0.157∗∗∗ (0.055)
0.049 (0.049)
0.033 (0.046)
0.074 (0.048)
0.130∗∗ (0.057)
0.020 (0.046)
0.044 (0.057)
0.034 (0.046)
0.000 (0.042)
0.045 (0.040)
0.003 (0.039)
702 1,409.740 4.062 4.166
0.817∗∗∗ (0.035)
702 1,411.338 4.038 4.077
∗p<0.1; ∗∗p<0.05; ∗∗∗p<0.01 3
Figure 2: Absolute values of the normalized residuals for ARCH(12)
Figure 3: Absolute values of the normalized residuals for GARCH(1,1)
Figure 4: Autocorrelation function (ARCH(12))
Suggested Solution: Shown in Figure 2, Figure 3, Figure 4 and Figure 5. The models do a better job of accounting for clustering of volatility.
4. Using the daily data on CMA, estimate monthly realized variance for month t as RV=Nt r2, (1)
where r2 is the squared return of day d in month t and where Nt is the number of
days in month t. Plot the resulting monthly time series of RVt.
Suggested Solution: Shown in Figure 6.
5. What are the first order autocorrelations of RVt and ε2t ? What is the correlation between RVt and ε2t ? What is the correlation between RVt and σt2 from the GARCH model? What is the correlation between ε2t and σt2 from the GARCH model? Suggested Solution: The first-order autocorrelation of RVt and ε2t is 0.684 and 0.221. The correlation between RVt and ε2t is 0.474. The correlation between RVt and σt2 is 0.621. The correlation between ε2t and σt2 is 0.367.
6. Run an ARMA(1,1) on RVt assuming normally distributed errors (which strictly speak- ing can’t be correct). Report the results. Let vt = Et−1 [RVt] where the expectation is obtained from the estimated ARMA. What is the correlation between vt and RVt?
Figure 5: Autocorrelation function (GARCH(1,1))
Figure 6: Monthly times series of RV
Estimation for ARMA(1,1) on RV
Dependent variable:
0.923∗∗∗ (0.018)
−0.506∗∗∗ (0.039)
2.811∗∗∗ (0.704)
702 −1,759.714 8.795 3,527.429
∗p<0.1; ∗∗p<0.05; ∗∗∗p<0.01
Figure7: Times series of vt and σt2
Observations Log Likelihood
Akaike Inf. Crit.
Plot on the same graph the time series of vt and σt2 from the GARCH.
Suggested Solution: Shown in Figure 7. The correlation between vt and RVt is 0.74.
library(stargazer)
library(forecast)
library(tseries)
mydata<-data.frame(F_F_Research_Data_5_Factors_2x3)
mydata$Date<-format(as.Date(mydata$X1),"%Y-%m")
m1<-arima(mydata$CMA,order=c(1,0,1))
stargazer(m1)
half_life<-log(0.5)/log(0.4252)
library(fGarch)
library(data.table)
mydata$CMA<-ts(mydata$CMA)
m2<-garchFit(~arma(1,1)+garch(12,0),data=mydata$CMA,trace=F)
m3<-garchFit(~arma(1,1)+garch(1,1),data=mydata$CMA,trace=F)
plot(abs(eita2),type="l")
plot(abs(eita3),type="l")
Acf(abs(eita2))
Acf(abs(eita3))
newdata<-data.frame(F_F_Research_Data_5_Factors_2x3_daily)
newdata$Date<-format(as.Date(newdata$X1),"%Y-%m")
RV<-aggregate(newdata$CMA^2, by=list(Category=newdata$Date), FUN=sum)
plot(RV$x,type="l")
acf(RV$x)[1]
m4<-arima(RV$x,order=c(1,0,1))
fcast <- forecast(m4, h=1)
plot(fcast$fitted)
cor(fcast$fitted,RV$x)
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