Financial Econometrics: Problem Set # 3 Solutions N Vera Chau The University of Chicago: Booth February 9, 219 1 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.4.2. -.2 -.4 -.6 199 1992 1994 1996 1998 2 22 24 26 RETURN.6.4.2. -.2 -.4 -.6 199 1992 1994 1996 1998 2 22 24 26 1
b. I ve included full summary stats: 7 6 5 4 3 2 1 -.4 -.2..2.4.6 Series: RETURN Sample 1988M12 27M2 Observations 216 Mean.27685 Median.1 Maximum.58 Minimum -.45 Std. Dev..13737 Skewness.814536 Kurtosis 6.79536 Jarque-Bera 153.524 Probability. 7 6 5 4 3 2 1 -.4 -.2..2.4.6 Series: R Sample 1988M12 27M2 Observations 218 Mean.3111 Median.5 Maximum.58 Minimum -.45 Std. Dev..131437 Skewness.694191 Kurtosis 6.692837 Jarque-Bera 141.3789 Probability. c. For ARMA(3,2): r =.311, s =.1314 = t =.311.1314/ 218 3.49 For ARMA(1,3):.277.137/ 218 3.13 Thus, for both, we reject the null that average returns are zero for both cases. February 9, 219 2 Winter, 219
2 a. Here are the returns plotted together: 1 8 6 4 2-2 1992 1993 1994 1995 1996 1997 1998 1999 2 21 22 R_AR1MA3 R_AR4MA3 And here are they are individually R_AR1MA3 R_AR4MA3 4 9 8 3 2 1-1 1992 1993 1994 1995 1996 1997 1998 1999 2 21 22 7 6 5 4 3 2 1 1992 1993 1994 1995 1996 1997 1998 1999 2 21 22 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, 219 3 Winter, 219
2 16 Series: R_AR1MA3 Sample 1982M1 22M5 Observations 121 2 16 Series: R_AR4MA3 Sample 1982M1 22M5 Observations 121 12 8 4-1. -.5..5 1. 1.5 2. 2.5 3. Mean.56455 Median.47 Maximum 3.4 Minimum -.92 Std. Dev..788725 Skewness.765777 Kurtosis 3.62984 Jarque-Bera 13.7721 Probability.123 12 8 4 1 2 3 4 5 6 7 8 Mean 1.66753 Median 1.256531 Maximum 8.112964 Minimum.28119 Std. Dev. 1.42675 Skewness 1.637625 Kurtosis 6.18215 Jarque-Bera 15.735 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 = 12.77 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, 219 4 Winter, 219
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 1.13.13.213 2.33.33.1729 3 -.28 -.28.277 4 -.83 -.83 1.2226.269 5 -.65 -.62 1.874.45 6.1.17 3.1981.362 7 -.92 -.96 4.3961.355 8.89.77 5.537.355 9 -.87 -.93 6.6195.357 1.116.131 8.5685.285 11.15.155 11.859.158 12 -.112 -.153 13.716.133 13 -.8 -.66 14.678.144 14.49.59 15.34.181 15 -.86 -.15 16.157.184 16.24 -.34 16.247.236 17.153.147 19.865.134 18 -.13 -.1 19.892.176 19.25.2 19.99.221 2 -.21 -.11 2.61.271 21 -.2 -.11 2.62.329 22.64.73 2.723.352 23.54.15 21.21.385 24 -. -.1 21.21.447 25.22 -.39 21.282.53 26 -.254 -.26 32.66.99 27 -.4 -.17 32.68.125 28.47.31 32.451.145 29 -.37 -.2 32.69.171 3.2 -.17 32.69.27 31.81.65 33.848.26 32.58.137 34.45.223 33.122.35 37.117.174 34.39.21 37.396.199 35 -.66 -.82 38.188.29 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.2932.529.562965.5745 AR(1) 1.177567.71511 16.46696. AR(2) -.58477.1171 -.57857.5639 AR(3) -.158279.68461-2.311968.224 SIGMASQ 4.47E-6 4.59E-7 9.743478. R-squared.954247 Mean dependent var.2693 Adjusted R-squared.95286 S.D. dependent var.9921 S.E. of regression.2155 Akaike info criterion -9.38829 Sum squared resid.59 Schwarz criterion -9.271632 Log likelihood 624.1347 Hannan-Quinn criter. -9.336456 F-statistic 662.1937 Durbin-Watson stat 1.96488 Prob(F-statistic). Inverted AR Roots.93.55 -.31 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 1 -.3 -.3.9 2.9.9.118 3.28.28.1226 4 -.24 -.24.25 5.23.22.2739 6 -.13 -.13.2981 7 -.15 -.14 1.8445 8.12.12 3.3429 9 -.46 -.44 3.6462.56 1.46.51 3.9582.138 11.12.114 6.639.19 12 -.121 -.118 8.212.84 13 -.35 -.47 8.3953.136 14.35.31 8.5773.199 15 -.76 -.55 9.4528.222 16.9 -.13 9.464.35 17.125.163 11.88.22 18 -.26 -.23 11.985.286 19.5 -.2 12.371.336 2.1.31 12.371.416 21 -.2 -.39 12.437.492 22.44.12 12.745.547 23.34.12 12.934.67 24 -.5. 12.937.677 25.32 -.16 13.17.729 26 -.263 -.234 24.627.136 27.5 -.31 24.63.173 28.52.25 25.84.198 29 -.41.25 25.374.231 3.3.11 25.376.279 31.64.77 26.91.297 32.49.67 26.58.328 33.18.27 28.573.282 34.29.54 28.729.324 35 -.67 -.78 29.545.335 36 -.13 -.119 31.512.295 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.34.5749.52888.5979 AR(1).18359.23343.532887.5951 AR(2).19336.229441.829565.484 AR(3).1459.23485.59874.9524 AR(4).53514.174833 3.6651.27 MA(1) 1.9483.229518 4.766866. MA(2) 1.7965.224138 4.814289. MA(3).99576.154982 6.424994. MA(4).287254.122181 2.35157.23 SIGMASQ 4.22E-6 4.81E-7 8.783574. R-squared.956784 Mean dependent var.2693 Adjusted R-squared.953595 S.D. dependent var.9921 S.E. of regression.2137 Akaike info criterion -9.35736 Sum squared resid.557 Schwarz criterion -9.138966 Log likelihood 627.5857 Hannan-Quinn criter. -9.268615 F-statistic 3.1112 Durbin-Watson stat 1.995111 Prob(F-statistic). Inverted AR Roots.95.2-.8i.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, 219 5 Winter, 219
AR(3).16.12.8.4. -.4 -.8 I II III IV I II III IV I II III IV I II III IV I II III IV I II III IV 211 212 213 214 215 216 CS_GROWTH CS_GROWTHF_AR3 ERROR_AR3.16.12.8.4. -.4 -.8 I II III IV I II III IV I II III IV I II III IV I II III IV I II III IV 211 212 213 214 215 216 And for the ARMA(4,4) model February 9, 219 6 Winter, 219
.2.15.1.5. -.5 -.1 I II III IV I II III IV I II III IV I II III IV I II III IV I II III IV 211 212 213 214 215 216 CS_GROWTH CS_GROWTHF ERROR_ARMA44.2.15.1.5. -.5 -.1 I II III IV I II III IV I II III IV I II III IV I II III IV I II III IV 211 212 213 214 215 216 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, 219 7 Winter, 219
.25.2.15.1.5. -.5 -.1 Forecast: GROWTHF Actual: GROWTH Forecast sample: 217M1 221M1 Included observations: 49 Root Mean Squared Error.1115 Mean Absolute Error.1115 Mean Abs. Percent Error 15.89817 Theil Inequality Coefficient.73637 Bias Proportion 1. Variance Proportion NA Covariance Proportion NA Theil U2 Coefficient NA Symmetric MAPE 14.72747 -.15 IV I II III IV I II III IV I II III IV I II III IV 217 218 219 22 221 GROWTHF 2 S.E..25.2.15.1.5. -.5 -.1 Forecast: GROWTH2F Actual: GROWTH2 Forecast sample: 217M1 222M1 Included observations: 61 Root Mean Squared Error.2262 Mean Absolute Error.2262 Mean Abs. Percent Error 37.2619 Theil Inequality Coefficient.15747 Bias Proportion 1. Variance Proportion NA Covariance Proportion NA Theil U2 Coefficient NA Symmetric MAPE 31.4936 -.15 IV I II III IV I II III IV I II III IV I II III IV I II III IV 218 219 22 221 222 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, 219 8 Winter, 219
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-5 5.78E-5 1.49458.1355 INFLATION(-1).473442.279267 1.6953.95 INFLATION(-2).498848.271128 1.839897.662 MA(1) -.3546.277264 -.127894.8983 MA(2) -.49318.157992-3.121546.19 MA(3) -.23952.66649-3.688.23 SIGMASQ 5.36E-6 1.38E-7 38.9424. R-squared.45913 Mean dependent var.361 Adjusted R-squared.454362 S.D. dependent var.3151 S.E. of regression.2327 Akaike info criterion -9.27712 Sum squared resid.3781 Schwarz criterion -9.231843 Log likelihood 3277.178 Hannan-Quinn criter. -9.259613 F-statistic 98.7558 Durbin-Watson stat 1.995868 Prob(F-statistic). Inverted MA Roots.87 -.42+.24i -.42-.24i February 9, 219 9 Winter, 219
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* 1.. 4.E-5 2 -.4 -.4.115 3 -.17 -.17.2137 4 -.21 -.21.5281.467 5 -.4 -.4.537.765 6 -.21 -.22.8588.835 7.12.11.9539.917 8 -.28 -.28 1.496.914 9.65.65 4.5418.64 1.58.57 6.956.433 11.126.127 18.36.19 12 -.91 -.9 24.218.4 13 -.71 -.66 27.84.2 14 -.48 -.46 29.467.2 15.13.114 37.188. 16.3.28 37.852. 17.2.6 37.854.1 18.6.51 4.452. 19 -.17 -.13 4.65.1 2.35.13 41.563.1 21 -.4 -.5 41.574.1 22 -.3 -.1 41.581.2 23 -.8 -.41 46.35.1 24 -.93 -.93 52.692. 25.31.13 53.389. 26.29 -.9 54.9. 27 -.62 -.72 56.862. 28 -.56 -.52 59.148. 29.23.2 59.522. 3 -.45 -.47 61.2. 31.11.1 61.117. 32 -.25 -.21 61.595. 33.1.8 61.597.1 34 -.16.5 61.78.1 35 -.19 -.18 62.37.1 36.16 -.16 62.221.2 *Probabilities may not be valid for this equation specification. b. Here are the forecasts. February 9, 219 1 Winter, 219
.8.6.4.2. -.2 Forecast: INFLATIONF Actual: INFLATION Forecast sample: 218M12 219M4 Included observations: 5 Root Mean Squared Error.187 Mean Absolute Error.187 Mean Abs. Percent Error 33.7315 Theil Inequality Coefficient 1. Bias Proportion 1. Variance Proportion NA Covariance Proportion NA Theil U2 Coefficient NA Symmetric MAPE 2. -.4 M12 M1 M2 M3 M4 218 219 INFLATIONF 2 S.E. c I got.167 +.199 +.183 +.1946 =.7373 5 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 2 1 + 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 11 75 = 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, 219 11 Winter, 219
c. By the same kind of logic, the distribution 1 years from now is N (11, 1 1 2 ) so the probability of below 5 is about.2 or 2%. Note: The variance is 1 2 + 1 2 +... so it is k 1 2, not k 2 1 2 and the standard deviation is 1 12. February 9, 219 12 Winter, 219