Financial Econometrics: Problem Set # 3 Solutions

Transcription

1 Financial Econometrics: Problem Set # 3 Solutions N Vera Chau The University of Chicago: Booth February 9, a. You can generate the returns using the exact same strategy as given in problem 2 below. Here are the returns for my two models, ARMA(3,2) and ARMA(1,3) respectively..6 R RETURN

2 b. I ve included full summary stats: Series: RETURN Sample 1988M12 27M2 Observations 216 Mean Median.1 Maximum.58 Minimum -.45 Std. Dev Skewness Kurtosis Jarque-Bera Probability Series: R Sample 1988M12 27M2 Observations 218 Mean.3111 Median.5 Maximum.58 Minimum -.45 Std. Dev Skewness Kurtosis Jarque-Bera Probability. c. For ARMA(3,2): r =.311, s =.1314 = t = / For ARMA(1,3): / Thus, for both, we reject the null that average returns are zero for both cases. February 9, Winter, 219

3 2 a. Here are the returns plotted together: R_AR1MA3 R_AR4MA3 And here are they are individually R_AR1MA3 R_AR4MA b. The question only asks for the mean and standard deviation but I ve included the full summary stats for both here. NOTICE, the mean returns are higher for the ARMA(4,3) but so is the standard deviation. (Weird Eviews note. The sample here says it starts in 1982 but I ve made sure that all the in-sample values here are NA so the mean does not include these values in counting the denominator - I checked mannually and the observations clearly says 121) February 9, Winter, 219

4 2 16 Series: R_AR1MA3 Sample 1982M1 22M5 Observations Series: R_AR4MA3 Sample 1982M1 22M5 Observations Mean Median.47 Maximum 3.4 Minimum -.92 Std. Dev Skewness Kurtosis Jarque-Bera Probability Mean Median Maximum Minimum Std. Dev Skewness Kurtosis Jarque-Bera Probability. c. For the ARMA(1,3) model, I had r =.56, s =.79, T = 121 so t = r se( r) = so we can reject the null that the average return is zero. r s T = 7.81 For the ARMA(4,3) model, I had r = 1.66, s = 1.43, T = 121 so t = so we can again reject the null. 3 a. I went out to 211m1 for the estimate and I did this for the growth rate not levels but I won t deduct for the latter. I tried the AR(1) model that we fit in the first class and the residual ACFs were horrible. I m going with AR(3) and ARMA(4,4). Here are the estimates and residual ACFs AR(3) February 9, Winter, 219

5 Correlogram of Residuals Date: 1/24/17 Time: 2:33 Sample: 2M1 211M1 Included observations: 132 Q-statistic probabilities adjusted for 3 ARMA terms Autocorrelation Partial Correlation AC PAC Q-Stat Prob Dependent Variable: CS_GROWTH Method: ARMA Maximum Likelihood (OPG - BHHH) Date: 1/24/17 Time: 2:32 Sample: 2M2 211M1 Included observations: 132 Convergence achieved after 17 iterations Coefficient covariance computed using outer product of gradients Variable Coefficient Std. Error t-statistic Prob. C AR(1) AR(2) AR(3) SIGMASQ 4.47E E R-squared Mean dependent var.2693 Adjusted R-squared S.D. dependent var.9921 S.E. of regression.2155 Akaike info criterion Sum squared resid.59 Schwarz criterion Log likelihood Hannan-Quinn criter F-statistic Durbin-Watson stat Prob(F-statistic). Inverted AR Roots And for the ARMA(4,4) Correlogram of Residuals Date: 1/24/17 Time: 2:43 Sample: 2M1 211M1 Included observations: 132 Q-statistic probabilities adjusted for 8 ARMA terms Autocorrelation Partial Correlation AC PAC Q-Stat Prob Dependent Variable: CS_GROWTH Method: ARMA Maximum Likelihood (OPG - BHHH) Date: 1/24/17 Time: 2:43 Sample: 2M2 211M1 Included observations: 132 Convergence achieved after 44 iterations Coefficient covariance computed using outer product of gradients Variable Coefficient Std. Error t-statistic Prob. C AR(1) AR(2) AR(3) AR(4) MA(1) MA(2) MA(3) MA(4) SIGMASQ 4.22E E R-squared Mean dependent var.2693 Adjusted R-squared S.D. dependent var.9921 S.E. of regression.2137 Akaike info criterion Sum squared resid.557 Schwarz criterion Log likelihood Hannan-Quinn criter F-statistic Durbin-Watson stat Prob(F-statistic). Inverted AR Roots i.2+.8i -.88 b. Here are the one-step-ahead, out of sample forecast values for the AR(3) model graphed alongside the actual, observed data as well as the errors (remember, forecast errors are just the difference between the actual and forecasted values). February 9, Winter, 219

6 AR(3) I II III IV I II III IV I II III IV I II III IV I II III IV I II III IV CS_GROWTH CS_GROWTHF_AR3 ERROR_AR I II III IV I II III IV I II III IV I II III IV I II III IV I II III IV And for the ARMA(4,4) model February 9, Winter, 219

7 I II III IV I II III IV I II III IV I II III IV I II III IV I II III IV CS_GROWTH CS_GROWTHF ERROR_ARMA I II III IV I II III IV I II III IV I II III IV I II III IV I II III IV c. I re-set the sample using proc to 4 years ahead and used dynamic forecasts. Recall that my models are AR(3) and ARMA(4,4) February 9, Winter, 219

8 Forecast: GROWTHF Actual: GROWTH Forecast sample: 217M1 221M1 Included observations: 49 Root Mean Squared Error.1115 Mean Absolute Error.1115 Mean Abs. Percent Error Theil Inequality Coefficient Bias Proportion 1. Variance Proportion NA Covariance Proportion NA Theil U2 Coefficient NA Symmetric MAPE IV I II III IV I II III IV I II III IV I II III IV GROWTHF 2 S.E Forecast: GROWTH2F Actual: GROWTH2 Forecast sample: 217M1 222M1 Included observations: 61 Root Mean Squared Error.2262 Mean Absolute Error.2262 Mean Abs. Percent Error Theil Inequality Coefficient Bias Proportion 1. Variance Proportion NA Covariance Proportion NA Theil U2 Coefficient NA Symmetric MAPE IV I II III IV I II III IV I II III IV I II III IV I II III IV GROWTH2F 2 S.E. 4 I tried a few different ones but ARMA(2,3) did the best. a. Here are the outputs and residual diagnostics. February 9, Winter, 219

9 Dependent Variable: INFLATION Method: ARMA Maximum Likelihood (OPG - BHHH) Date: 2/8/19 Time: 13:27 Sample: 196M4 218M12 Included observations: 75 Convergence achieved after 21 iterations Coefficient covariance computed using outer product of gradients Variable Coefficient Std. Error t-statistic Prob. C 8.64E E INFLATION(-1) INFLATION(-2) MA(1) MA(2) MA(3) SIGMASQ 5.36E E R-squared Mean dependent var.361 Adjusted R-squared S.D. dependent var.3151 S.E. of regression.2327 Akaike info criterion Sum squared resid.3781 Schwarz criterion Log likelihood Hannan-Quinn criter F-statistic Durbin-Watson stat Prob(F-statistic). Inverted MA Roots i i February 9, Winter, 219

10 Correlogram of Residuals Date: 2/8/19 Time: 13:27 Sample: 196M1 218M12 Included observations: 75 Q-statistic probabilities adjusted for 3 ARMA terms and 2 dynamic regressors Autocorrelation Partial Correlation AC PAC Q-Stat Prob* E *Probabilities may not be valid for this equation specification. b. Here are the forecasts. February 9, Winter, 219

11 Forecast: INFLATIONF Actual: INFLATION Forecast sample: 218M12 219M4 Included observations: 5 Root Mean Squared Error.187 Mean Absolute Error.187 Mean Abs. Percent Error Theil Inequality Coefficient 1. Bias Proportion 1. Variance Proportion NA Covariance Proportion NA Theil U2 Coefficient NA Symmetric MAPE M12 M1 M2 M3 M INFLATIONF 2 S.E. c I got = a. Recall that V t+k = V t + k β + k 1 ɛ t+j = Yt k + k 1 ɛ t+j. Then, ( ) k 1 f(v t+k V t = 1, ) = f V t + k β + ɛt+j V t = 1, = 1, + 1k + f ( k 1 ) ɛt+j V t = 1, where I ve taken out the values that are constant once we condition on V t = 1,. Now, remember that ɛ t N (, 1 2 ) and that they are iid. Remember also from hw1 that when X N (µ 1, σ 2 x), Y N (µ 2, σ 2 2) then X + Y N (µ 1 + µ 2, σ σ 2 2). This gives us, f(v t+k V t = 1, ) = 1, + 1k + N (, (k 1)1 2 ) Final step, add the constant to the mean of the normal distribution and we get, = N ( 1, + 1k, k1 2) Note that the variance is multiplied by k and not k-1 because we are summing from ɛ t, ɛ t+1,... ɛ t+k 1 which gives us k total ɛ. b. In year 1, the distribution is given by, N (1, 1, 1 2 ) Notice that = 26. Which is just short of 3 sd from the mean. If you plug this into excel or eviews or any stat program, this turns out to be about.45, so about.45% probability. February 9, Winter, 219

12 c. By the same kind of logic, the distribution 1 years from now is N (11, ) so the probability of below 5 is about.2 or 2%. Note: The variance is so it is k 1 2, not k and the standard deviation is February 9, Winter, 219