ECG 752: Econometrics II Spring Assessed Computer Assignment 3: Answer Key

Transcription

1 ECG 752: Econometrics II Spring 2005 Assessed Computer Assignment 3: Answer Key Question 1 The time series plots of x(d), x(bw) and x(m) are presented below. 1

2 A common characteristic of all series is that they fluctuate randomly around zero. This should be expected since we are working with the first differences of the exchange rates logarithms which are known to follow the weak form of the Efficient Market Hypothesis. In terms of the volatility of the time series, some interesting differences emerge. First, note that the variation of the daily data appears to be fluctuating and we can observe many clusters of large and small movements. As we move to bi-weekly data, the volatility clustering is less obvious. We could argue that the period has the least volatility, whereas the has the most. However, apart from these two extremes, it is hard to observe any other differences in the series variation. Finally, the monthly data show the least volatility-clustering; it is much harder to distinguish between low and high volatility periods. Overall, the daily data 2

3 exhibit the most volatility clustering, whereas the monthly exhibit the least (see Question 4). Question 2 The following table presents the AIC and SBC values for all time series and specifications. GARCH(0,0) GARCH(1,1) GARCH(2,2) GARCH(3,3) AIC SBC AIC SBC AIC SBC AIC SBC Daily Bi-weekly Monthly The above results suggest that: -The daily data are best explained by a GARCH(3,3) -The biweekly data are best explained by a GARCH(1,1), in the case of AIC, and a GARCH(0,0) in the case of SBC. -The monthly data are best explained by a GARCH(0,0) The estimation results of these specifications are following. DAILY DATA GARCH(3, 3) Fit statistics SSE Observations 2973 MSE Uncond Var Log Likelihood Total R-Square. SBC AIC Normality Test Pr > ChiSq <.0001 GARCH(3, 3) Parameter Estimates Variable DF Estimate Standard Error t Value ApproxPr > t Intercept ARCH E E <.0001 ARCH <.0001 ARCH

4 Variable DF Estimate Standard Error t Value ApproxPr > t ARCH <.0001 GARCH <.0001 GARCH <.0001 GARCH <.0001 BI-WEEKLY DATA GARCH(0, 0) Fit statistics Ordinary Least Squares Estimates SSE DFE 281 MSE Root MSE SBC AIC Regress R-Square Total R-Square Durbin-Watson GARCH(0, 0) Parameter Estimates Variable DF Estimate Standard Error t Value ApproxPr > t Intercept GARCH(1, 1) Fit statistics GARCH Estimates SSE Observations 282 MSE Uncond Var Log Likelihood Total R-Square. SBC AIC Normality Test Pr > ChiSq

5 GARCH(1, 1) Parameter Estimates Variable DF Estimate Standard Error t Value ApproxPr > t Intercept ARCH ARCH GARCH MONTHLY DATA GARCH(0, 0) Fit statistics Ordinary Least Squares Estimates SSE DFE 137 MSE Root MSE SBC AIC Regress R-Square Total R-Square Durbin-Watson GARCH(0, 0) parameter Estimates Variable DF Estimate Standard Error t Value ApproxPr > t Intercept

6 Question 3 The plots of the estimated conditional variances of the three series are presented below. 6

7 The estimated conditional variances exhibit the most time variation for the daily data. On the basis of AIC/SIC, there is mixed evidence about whether the conditional variance is time varying at the bi-weekly frequency but even for the GARCH(1,1) model chosen by AIC, the estimated conditional variances have a far narrower range than their daily counterparts and the peaks in conditional variance for the bi-weekly data are an order of magnitude smaller than those exhibited in the daily data. The monthly data are conditionally homoscedastic and so the conditional variances are constant over time. 7

8 Question 4 Pulling together the results from Questions 1-3, it is clear that as the sampling frequency decreases the time dependence in the conditional variance dies out. The daily data clearly exhibit time variation in the conditional variances. At the bi-weekly frequency, the evidence is mixed as to whether the conditional variances are time varying. However, this series exhibits less time variation in its conditional variance than the daily data no matter which information criteria is used to select the model. Finally, at the monthly frequency, there is no evidence of time variation in the conditional variance. 8

9 SAS CODE /* IMPORT THE DATASET */ PROC IMPORT OUT=daily DATAFILE= "comprb305_data.xls" DBMS=EXCEL REPLACE;SHEET="DAILY$"; GETNAMES=YES;USEDATE=YES;RUN; PROC IMPORT OUT=biweekly DATAFILE= "comprb305_data.xls" DBMS=EXCEL REPLACE;SHEET="BIWEEKLY$"; GETNAMES=YES;USEDATE=YES;RUN; PROC IMPORT OUT=monthly DATAFILE= "comprb305_data.xls" DBMS=EXCEL REPLACE;SHEET="MONTHLY$"; GETNAMES=YES;USEDATE=YES;RUN; /*CREATE FIRST DIFFERENCES*/ data daily;set daily; uk_day=dif(uk_day);run; data biweekly;set biweekly; uk_biweek=dif(uk_biweek);run; data monthly;set monthly; uk_month=dif(uk_month);run; /*PLOT THE VARIABLES IF INTEREST*/ *ods graphics on;*ods latex; title1 UK vs US Spot exchange rate ; symbol c=blue i=join v=none; proc gplot data=daily; title2 Daily data: Data in 1st differences of logs ; plot uk_day*date;run;quit; proc gplot data=biweekly; title2 Biweekly data: Data in 1st differences of logs ; plot uk_biweek*date;run;quit; proc gplot data=monthly; title2 Monthly data: Data in 1st differences of logs ; plot uk_month*date;run;quit; goptions reset=all; /*GARCH estimation*/ /*DAILY DATA*/ proc autoreg data=daily; GARCH_11: model uk_day=/ garch=(p=1, q=1); 9

10 GARCH_22: GARCH_33: run; model uk_day=/ garch=(p=2, q=2); model uk_day=/ garch=(p=3, q=3);output out=final_daily cev=volatility; /*BIWEEKLY DATA*/ proc autoreg data=biweekly; GARCH_11: model uk_biweek=/ garch=(p=1, q=1);output out=final_biweekly cev=volatility; GARCH_22: model uk_biweek=/ garch=(p=2, q=2); GARCH_33: model uk_biweek=/ garch=(p=3, q=3); run; /*MONTHLY DATA*/ proc autoreg data=monthly; GARCH_11: model uk_month=/ garch=(p=1, q=1);output out=final_monthly cev=volatility; GARCH_22: model uk_month=/ garch=(p=2, q=2); GARCH_33: model uk_month=/ garch=(p=3, q=3); run; /* Estimator of the variance in the GARCH(0,0) case */ data final_biweekly;set final_biweekly; garch00= ;run; data final_monthly;set final_monthly; garch00= ;run; /*Plot the estimated conditional variances*/ title1 Estimated Conditional Variance ; symbol c=blue i=join v=none; proc gplot data=final_daily; title2 Daily data: GARCH(3,3) ; plot volatility*date;run;quit; proc gplot data=final_biweekly; title2 Biweekly data: AIC ---> GARCH(1,1) ; plot volatility*date;run;quit; proc gplot data=final_biweekly; title2 Biweekly data: SBC ---> GARCH(0,0) ; plot garch00*date;run;quit; proc gplot data=final_monthly; title2 Monthly data: GARCH(0,0) ; plot garch00*date;run;quit; goptions reset=all; 10