Booth School of Business, University of Chicago Business 41202, Spring Quarter 2016, Mr. Ruey S. Tsay. Midterm

Transcription

1 Booth School of Business, University of Chicago Business 41202, Spring Quarter 2016, Mr. Ruey S. Tsay Midterm ChicagoBooth Honor Code: I pledge my honor that I have not violated the Honor Code during this examination. Signature: Name: ID: Notes: Open notes and books. Exam time: 180 minutes. You may use a calculator or a PC. However, turn off Internet connection and cell phones. Internet access and phone communication are strictly prohibited during the exam. The exam has 8 pages and the R output has 12 pages. Please check that you have all 20 pages. For each question, write your answer in the blank space provided. Manage your time carefully and answer as many questions as you can. For simplicity, if not specifically given, use 5% Type-I error in hypothesis testings. Round your answer to 3 significant digits. Problem A: (30 pts) Answer briefly the following questions. Each question has two points. 1. Give two reasons by which the return series of an asset tend to contain outliers. 2. Describe two differences between an AR(1) model and an MA(1) model of a time series. 1

2 3. Give two characteristics of the return r t if it follows the model r t = a t, a t = σ t ɛ t, where ɛ t are iid N(0, 1) and σ 2 t = a 2 t (Questions 4 to 6): Suppose that the asset return r t follows the model r t = a t a t = σ t ɛ t, ɛ t iid t 6 σ 2 t = a 2 t σ 2 t 1. Does the unconditional variance of r t exist? Why? 5. Suppose that r 100 = 0.05 and σ 100 = 0.3. Compute 1-step and 2-step ahead volatility forecasts at the forecast origin t = 100. (Note that it is volatility, not σ 2.) 6. Compute the 22-step ahead mean and volatility forecasts (one month ahead). 7. Give an advantage of Spearman s ρ over the Pearson correlation. 8. Give a feature that GARCH-M models have, but the GARCH models do not. 9. Suppose that r t follows the model r t = r t 1 + a t 0.9a t 1, and we have r 1001 = 1.2 and r 1000 (1) = 1.0, where r t (1) denotes the 1- step ahead prediction of r t+1 at the forecast origin t. Compute r 1001 (1). 2

3 10. Why is the usual R 2 measure not proper in time series analysis? 11. Give two real applications of seasonal time series models in finance. 12. (Questions 12-13) Suppose that the daily simple returns of an asset in week 1 were -0.5%, 1.2%, 2.5%, -1.0%, and 0.6%. What are the corresponding daily log returns? 13. What is the weekly simple return of the asset? 14. (Questions 14-15): The summary statistics of daily simple returns of an asset are given below: > basicstats(rtn) rtn nobs Mean SE Mean???????? LCL Mean UCL Mean Stdev Skewness Kurtosis What is the standard deviation of the mean? Is the expected return of the asset significantly different from zero? Why? 15. Based on the summary statistics, are the returns normally distributed? Perform a statistical test to justify your conclusion. 3

4 Problem B. (23 points) Consider the monthly U.S. unemployment rates from January 1947 to March Due to strong serial dependence, we analyze the differenced series x t = r t r t 1, where r t is the seasonally adjusted unemployment rate. Answer the following questions, using the attached R output. Note: A fitted ARIMA model should include residual variance. 1. (2 points) The auto.arima command specifies an ARIMA(2,0,2) model for x t. The fitted model is referred to as m1 in the output. Write down the fitted model. 2. (3 points) Model checking shows two large outliers. An ARIMA(2,0,2) model with two outliers are then specified, m3. Write down the fitted model. 3. (3 points) Model checking shows some serial correlations at lags 12 and 24. A seasonal model is then employed and called m4. Write down the fitted model. 4. (3 points) The outliers remain in the seasonal model. Therefore, a refined model is used and called m5. Write down the fitted model. 5. (2 points) Based on the model checking statistics provided, are there serial correlations in the residuals of model m5? Why? 6. (2 points) Among models m1,m3, m4 and m5, which model is preferred under the in-sample fit? Why? 7. (2 points) If root mean squares of forecast errors are used in out-ofsample prediction, which model is preferred? Why? 4

5 8. (2 points) If mean absolute forecast errors are used in out-of-sample comparison, which model is selected? 9. (2 points) Consider models m1 and m3. State the impact of outliers on in-sample fitting. 10. (2 points) Again, consider models m1 and m3. State the impact of outliers on out-of-sample predictions. Problem C. (27 points) Consider the daily log returns of Amazon (AMZN) stock obtained via quantmod. Statistical analysis is included in the attached R output. Answer the following questions. Note, a model should include both mean and volatility equations and the innovation distribution used. 1. (2 points) Are there serial correlations in the daily log returns? Why? Write down the proper null hypothesis for testing. 2. (3 points) A standard GARCH(1,1) model is fitted. Write down the fitted model. 3. (3 points) Model checking shows the normality is rejected. A skew standardized Student-t distribution is used. Write down the fitted model. Model m3. 5

6 4. (2 points) Based on the fitted model m3. Does the model support that the innovation is skewed? Perform a test to support your conclusion. 5. (2 points) Compute the 95% interval forecasts for 1-step and 2-step ahead predictions using model m3. 6. (2 points) An IGARCH(1,1) model is also entertained. Write down the fitted model. Model m4. 7. (2 points) Why are the 1-step to 5-step ahead volatility forecasts of the IGARCH(1,1) model not constant? 8. (2 points) An EGARCH model is also entertained. Write down the fitted model? Model m5. 9. (2 points) Based on the fitted EGARCH model, is the leverage effect significant? Why? 10. (3 points) The lag-1 VIX index is used as an explanatory variable for volatility. Write down the fitted model. Model m6 6

7 11. (2 points) Based on the fitted model, does the lag-1 VIX index affect significantly the AMZN volatility? Why? 12. (2 points) Among all volatility models entertained, which model provides best in-sample fit? Why? Problem D. (10 points) Consider the monthly log returns of Procter and Gamble stock from January 1960 to March Use the R output to answer the following questions. 1. (2 points) An IGARCH(1,1) model is entertained. Write down the fitted model. 2. (2 points) Based on the statistics provided, is the model adequate? Why? 3. (4 points) Based on the fitted IGARCH(1,1) model, compute the 1-step and 2-step ahead forecasts for mean and volatility of the log returns. 4. (2 points) A GARCM-M model is entertained. Based on the fitted model, is the risk premium statistically significant? Perform a test to justify your answer. 7

8 Problem E. (10 points) Consider the monthly log returns of value-weighted index and the S&P composite index from January 1960 to March Our goal is to study the relationship between the volatility of the two market indexes. Based on the output provided, answer the following questions: 1. (1 points) A GARCH(1,1) model with skew standardized Student-t innovations is employed for the S&P index returns. Does the fitted model support the use of skew innovations? Why? 2. (2 points) A similar GARCH(1,1) model is also employed for the valueweighted index returns. Let the resulting volatility be volvw t. Let volsp t be the corresponding volatility of the S&P index return. Write down the fitted simple linear regression model for the dependent variable volsp t. Is this simple linear regression model adequate? Why? 3. (2 points) A refined model is employed. Write down the fitted linear regression model with time series errors. 4. (3 points) Alternatively, one can use volvw t as an explanatory variable in volatility modeling of the S&P index return. Write down the fitted volatility model. 5. (2 point) Does volvw t significantly contribute to the volatility modeling of the S&P index returns? Why? 8

9 R output: edited to shorten the output ### Problem B ######### > rate <- as.numeric(unrate[,1]) > xt <- diff(rate) ### Differenced series > require(forecast) > auto.arima(xt) Series: xt ARIMA(2,0,2) with zero mean > m1 <- arima(xt,order=c(2,0,2),include.mean=f) > m1 Call:arima(x=xt,order=c(2,0,2),include.mean=F) Coefficients: ar1 ar2 ma1 ma s.e sigma^2 estimated as : log likelihood = , aic = > which.min(m1$residuals) [1] 22 > i22[22]=1; i22 <- rep(0,818) > m2 <- arima(xt,order=c(2,0,2),xreg=i22,include.mean=f) > m2 ar1 ar2 ma1 ma2 i s.e sigma^2 estimated as : log likelihood = , aic = > which.max(m2$residuals) [1] 21 > i21 <- rep(0,818) > i21[21]=1 > out <- cbind(i22,i21) > m3 <- arima(xt,order=c(2,0,2),xreg=out,include.mean=f) > m3 Call: arima(x = xt, order = c(2, 0, 2), xreg = out, include.mean = F) Coefficients: ar1 ar2 ma1 ma2 i22 i s.e sigma^2 estimated as : log likelihood = , aic = > Box.test(m3$residuals,lag=12,type= Ljung ) Box-Ljung test data: m3$residuals X-squared = 31.83, df = 12, p-value = > m4 <- arima(xt,order=c(2,0,2),seasonal=list(order=c(1,0,1),period=12), 9

10 include.mean=f) > m4 Call:arima(x = xt,order=c(2,0,2),seasonal=list(order=c(1,0,1),period=12), include.mean = F) Coefficients: ar1 ar2 ma1 ma2 sar1 sma s.e sigma^2 estimated as : log likelihood = , aic = > m5 <- arima(xt,order=c(2,0,2),seasonal=list(order=c(1,0,1),period=12), include.mean=f,xreg=out) > m5 Call:arima(x=xt,order=c(2,0,2),seasonal=list(order=c(1,0,1),period=12), xreg = out, include.mean = F) Coefficients: ar1 ar2 ma1 ma2 sar1 sma1 i22 i s.e sigma^2 estimated as : log likelihood = 263.2, aic = > Box.test(m5$residuals,lag=24,type= Ljung ) Box-Ljung test data: m5$residuals X-squared = , df = 24, p-value = > source("backtest.r") > backtest(m1,xt,750,include.mean=f) [1] "RMSE of out-of-sample forecasts" [1] [1] "Mean absolute error of out-of-sample forecasts" [1] > backtest(m3,xt,750,include.mean=f,xre=out) [1] "RMSE of out-of-sample forecasts" [1] [1] "Mean absolute error of out-of-sample forecasts" [1] > backtest(m4,xt,750,include.mean=f) [1] "RMSE of out-of-sample forecasts" [1] [1] "Mean absolute error of out-of-sample forecasts" [1] > backtest(m5,xt,750,include.mean=f,xre=out) [1] "RMSE of out-of-sample forecasts" [1]

11 [1] "Mean absolute error of out-of-sample forecasts" [1] ##### Problem C > getsymbols("amzn") [1] "AMZN" > getsymbols("^vix") ## to be used later. [1] "VIX" > vix <- as.numeric(vix[,6]) > vixm1 <- vix[-1] > amzn <- diff(log(as.numeric(amzn[,6]))) > Box.test(amzn,lag=10,type= Ljung ) Box-Ljung test data: amzn X-squared = , df = 10, p-value = > require(rugarch) > spec1 <- ugarchspec(variance.model=list(model="sgarch"), mean.model=list(armaorder=c(0,0))) > m1 <- ugarchfit(data=amzn,spec=spec1) > m1 * * * GARCH Model Fit * * * Conditional Variance Dynamics GARCH Model : sgarch(1,1) Mean Model : ARFIMA(0,0,0) Distribution : norm Optimal Parameters mu omega alpha beta Information Criteria Akaike Bayes Shibata Hannan-Quinn Weighted Ljung-Box Test on Standardized Residuals 11

12 statistic p-value Lag[1] Lag[4*(p+q)+(p+q)-1][5] d.o.f=0 H0 : No serial correlation Weighted Ljung-Box Test on Standardized Squared Residuals statistic p-value Lag[1] Lag[4*(p+q)+(p+q)-1][9] d.o.f=2 > spec2 <- ugarchspec(variance.model=list(model="sgarch"), mean.model=list(armaorder=c(0,0)),distribution.model="std") > m2 <- ugarchfit(data=amzn,spec=spec2) > m2 Conditional Variance Dynamics GARCH Model : sgarch(1,1) Mean Model : ARFIMA(0,0,0) Distribution : std Optimal Parameters mu omega alpha beta shape Information Criteria Akaike Bayes Shibata Hannan-Quinn > spec3 <- ugarchspec(variance.model=list(model="sgarch"),mean.model= list(armaorder=c(0,0)),distribution.model="sstd") > m3 <- ugarchfit(data=amzn,spec=spec3) > m3 Conditional Variance Dynamics GARCH Model : sgarch(1,1) Mean Model : ARFIMA(0,0,0) 12

13 Distribution : sstd Optimal Parameters mu omega alpha beta skew shape Information Criteria Akaike Bayes Shibata Hannan-Quinn Weighted Ljung-Box Test on Standardized Residuals statistic p-value Lag[1] Lag[4*(p+q)+(p+q)-1][5] d.o.f=0 H0 : No serial correlation Weighted Ljung-Box Test on Standardized Squared Residuals statistic p-value Lag[1] Lag[4*(p+q)+(p+q)-1][9] d.o.f=2 > ugarchforecast(m3,n.ahead=5) ** * GARCH Model Forecast * ** Model: sgarch Horizon: 5 0-roll forecast [T0= :00:00]: Series Sigma T T T T T

14 > spec4 <- ugarchspec(variance.model=list(model="igarch"), mean.model=list(armaorder=c(0,0)),distribution.model="sstd") > m4 <- ugarchfit(data=amzn,spec=spec4) > m4 Conditional Variance Dynamics GARCH Model : igarch(1,1) Mean Model : ARFIMA(0,0,0) Distribution : sstd Optimal Parameters mu omega alpha beta NA NA NA skew shape Information Criteria Akaike Bayes Shibata Hannan-Quinn Weighted Ljung-Box Test on Standardized Residuals statistic p-value Lag[1] Lag[4*(p+q)+(p+q)-1][5] d.o.f=0 H0 : No serial correlation Weighted Ljung-Box Test on Standardized Squared Residuals statistic p-value Lag[1] Lag[4*(p+q)+(p+q)-1][9] d.o.f=2 > ugarchforecast(m4,n.ahead=5) ** * GARCH Model Forecast * ** Model: igarch 14

15 Horizon: 5 0-roll forecast [T0= :00:00]: Series Sigma T T T T T > spec5 <- ugarchspec(variance.model=list(model="egarch"), mean.model=list(armaorder=c(0,0))) > m5 <- ugarchfit(data=amzn,spec=spec5) > m5 Conditional Variance Dynamics GARCH Model : egarch(1,1) Mean Model : ARFIMA(0,0,0) Distribution : norm Optimal Parameters mu omega alpha beta gamma Information Criteria Akaike Bayes Shibata Hannan-Quinn Weighted Ljung-Box Test on Standardized Residuals statistic p-value Lag[1] Lag[4*(p+q)+(p+q)-1][5] d.o.f=0 H0 : No serial correlation Weighted Ljung-Box Test on Standardized Squared Residuals statistic p-value 15

16 Lag[1] Lag[4*(p+q)+(p+q)-1][9] d.o.f=2 > spec6 <- ugarchspec(variance.model=list(model="sgarch",external.regressors= as.matrix(vixm1)),mean.model=list(armaorder=c(0,0)),distribution.model="sstd") > m6 <- ugarchfit(data=amzn,spec=spec6) > m6 Conditional Variance Dynamics GARCH Model : sgarch(1,1) Mean Model : ARFIMA(0,0,0) Distribution : sstd Optimal Parameters mu omega alpha beta vxreg skew shape Information Criteria Akaike Bayes Shibata Hannan-Quinn Weighted Ljung-Box Test on Standardized Residuals statistic p-value Lag[1] Lag[2*(p+q)+(p+q)-1][2] Lag[4*(p+q)+(p+q)-1][5] d.o.f=0 H0 : No serial correlation Weighted Ljung-Box Test on Standardized Squared Residuals statistic p-value Lag[1] 3.139e Lag[4*(p+q)+(p+q)-1][9] 1.149e d.o.f=2 16

17 #### Problem D ## > da=read.table("m-pg3dx-6015.txt",header=t) > head(da) PERMNO date RET vwretd ewretd sprtrn > pg <- log(da[,3]+1) > source( Igarch.R ) > m3 <- Igarch(pg) Estimates: Maximized log-likehood: Coefficient(s): beta < 2.22e-16 *** --- > names(m3) [1] "par" "volatility" > r3 <- pg/m3$volatility > Box.test(r3,lag=12,type= Ljung ) Box-Ljung test data: r3 X-squared = , df = 12, p-value = > Box.test(r3^2,lag=12,type= Ljung ) Box-Ljung test data: r3^2 X-squared = , df = 12, p-value = > length(pg) [1] 663 > pg[663] [1] > m3$volatility[663] [1] > source("garchm.r") > m4 <- garchm(pg,type=1) Maximized log-likehood: Coefficient(s): mu gamma omega alpha *** beta e-12 *** 17

18 #### Problem E > da=read.table("m-pg3dx-6015.txt",header=t) > head(da) PERMNO date RET vwretd ewretd sprtrn > sp <- log(da[,6]+1) > vw <- log(da[,4]+1) > m1 <- garchfit(~garch(1,1),data=sp,trace=f,cond.dist="sstd") > summary(m1) Title: GARCH Modelling Call: garchfit(formula = ~garch(1, 1), data = sp, cond.dist = "sstd", trace = F) Mean and Variance Equation: data ~ garch(1, 1)[data = sp] Conditional Distribution: sstd Error Analysis: mu 6.092e e e-05 *** omega 9.338e e * alpha e e e-05 *** beta e e < 2e-16 *** skew 7.699e e < 2e-16 *** shape 7.901e e *** --- Standardised Residuals Tests: Statistic p-value Ljung-Box Test R Q(10) Ljung-Box Test R Q(20) Ljung-Box Test R^2 Q(10) Ljung-Box Test R^2 Q(20) > volsp <- volatility(m1) > n1 <- garchfit(~garch(1,1),data=vw,trace=f,cond.dist="sstd") > summary(n1) Title: GARCH Modelling Call:garchFit(formula = ~garch(1, 1), data = vw, cond.dist = "sstd", trace = F) Mean and Variance Equation: data ~ garch(1, 1) [data = vw] 18

19 Conditional Distribution: sstd Error Analysis: mu 8.787e e e-09 *** omega 9.953e e * alpha e e e-05 *** beta e e < 2e-16 *** skew 7.383e e < 2e-16 *** shape 7.428e e *** --- > volvw <- volatility(n1) > k1 <- lm(volsp~volvw) > summary(k1) Call: lm(formula = volsp ~ volvw) Coefficients: (Intercept) ** volvw <2e-16 *** --- Residual standard error: on 661 degrees of freedom Multiple R-squared: , Adjusted R-squared: F-statistic: 2.315e+04 on 1 and 661 DF, p-value: < 2.2e-16 > k2 <- ar(k1$residuals) > k2$order [1] 1 > k3 <- arima(volsp,order=c(1,0,0),xreg=volvw) > k3 Call:arima(x = volsp, order = c(1, 0, 0), xreg = volvw) Coefficients: ar1 intercept volvw s.e sigma^2 estimated as 7.603e-07: log likelihood = , aic = > tsdiag(k3) > Box.test(k3$residuals,lag=12,type= Ljung ) Box-Ljung test data: k3$residuals X-squared = , df = 12, p-value = > spec1 <- ugarchspec(variance.model=list(model="sgarch",external.regressors= as.matrix(volvw)),mean.model=list(armaorder=c(0,0)),distribution.model="sstd") > n4 <- ugarchfit(data=sp,spec=spec1) > n4 19

20 Conditional Variance Dynamics GARCH Model : sgarch(1,1) Mean Model : ARFIMA(0,0,0) Distribution : sstd Optimal Parameters mu omega alpha beta vxreg skew shape Weighted Ljung-Box Test on Standardized Residuals statistic p-value Lag[1] Lag[4*(p+q)+(p+q)-1][5] d.o.f=0 H0 : No serial correlation 20