Department of Economics Eric Zivot University of Washington Spring 6 Economics 584 Computer Lab # Suggested Solutions Empirical Exercises Comparing forecasting models Simulated values from the model y y y iid N t = 1. t 1.4 t + εt, εt ~ (,(.5) ) y = y = 1 are illustrated below. 3 1-1 - -3 5 5 75 1 15 15 175 5 5 Y The series looks stationary with a high degree of persistence (note: the sum of the AR coefficients is.8). The SACF and PACF are illustrated below.
Date: 5/4/5 Time: 9:6 Sample: 1 5 Included observations: 5 Autocorrelation Partial Correlation AC PAC Q-Stat Prob. *******. ******* 1.94.94 6.59.. ****** ***..738 -.431 344.76.. ****.. 3.565.6 46... ***.. 4.44.65 47... ***. * 5.334.11 5.96.. ** *. 6.7 -.93 519.84.. ** *. 7.1 -.69 531.3.. *.. 8.145 -.6 536.7.. *.. 9.84.15 538.57... *. 1.6 -.64 538.75..... 11 -. -.17 538.88..... 1 -.55.7 539.69. The SACF decays geometrically to zero and the PACF cuts off at lag. This is consistent with an AR() model. Descriptive statistics are given below 16 1 8 4-3 - -1 1 Series: Y Sample 1 5 Observations 5 Mean -.58697 Median -.35574 Maximum.531798 Minimum -.916817 Std. Dev. 1.86649 Skewness -.66333 Kurtosis.577 Jarque-Bera 9.53879 Probability.8519
The mean is close to zero. Interestingly, the JB statistic rejects normality for the data. This could be due to the fact that the JB statistic was designed for iid data. 3. Using the first observations to fit the AR() model gives Dependent Variable: Y Method: Least Squares Date: 5/4/5 Time: 9:1 Sample (adjusted): 3 Included observations: 198 after adjustments Convergence achieved after 3 iterations Variable Coefficient Std. Error t-statistic Prob. C.4171.4487.167769.8669 AR(1) 1.3754.6353.951. AR() -.47386.6377-7.43491. R-squared.853581 Mean dependent var.471 Adjusted R-squared.8579 S.D. dependent var 1.39917 S.E. of regression.538 Akaike info criterion 1.481761 Sum squared resid 49.49381 Schwarz criterion 1.531583 Log likelihood -143.6943 F-statistic 568.3973 Durbin-Watson stat 1.9585 Prob(F-statistic). Inverted AR Roots.66-.18i.66+.18i The estimated results are similar to the actual values. The inverted roots of the characteristic polynomial ˆ( φ z) = 1 1.38z+.474z = are complex and have modulus inside the complex unit circle so that the fitted model is stationary and ergodic. The plot of the actual, fitted and residuals indicate that the model tracks the simulated data well. The correlogram of the residuals (not shown) reveals no omitted serial correlation.
4 1 - -4-1 - 5 5 75 1 15 15 175 Residual Actual Fitted 4. Using the first observations to fit a mis-specified MA(1) gives Dependent Variable: Y Method: Least Squares Date: 5/4/5 Time: 9:3 Sample: 1 Included observations: Convergence achieved after 13 iterations Backcast: Variable Coefficient Std. Error t-statistic Prob. C.49147.1946.47745.6336 MA(1).84146.3971 1.46. R-squared.633518 Mean dependent var.46649 Adjusted R-squared.631667 S.D. dependent var 1.3336 S.E. of regression.79994 Akaike info criterion.378897 Sum squared resid 13.889 Schwarz criterion.41188 Log likelihood -35.8897 F-statistic 34.76 Durbin-Watson stat.783957 Prob(F-statistic). Inverted MA Roots -.84
The MA coefficient is close to one which is required to capture the large first order sample autocorrelation. The small DW statistic indicates omitted positive serial correlation in the residuals. The SACF and PACF of the residuals (not shown) indicates omitted serial correlation. The modified Q-statistics are large for all lags. The plot of the actual, fitted and residuals below indicates that the model does not track the simulated data as well as the AR() model. 3 1 1-1 -1 - -3 - -3 5 5 75 1 15 15 175 Residual Actual Fitted 5. Forecasts from the rolling 1-step ahead forecasts from the AR() and MA(1) are displayed in the tables below. Forecast: YF AR Actual: Y Forecast sample: 1 5 Included observations: 5 Root Mean Squared Error.49495 Mean Absolute Error.411 Mean Absolute Percentage Error 15.4196 Theil Inequality Coefficient.18 Bias Proportion.586 Variance Proportion.356 Covariance Proportion.944139
Forecast: YF MA1 Actual: Y Forecast sample: 1 5 Included observations: 5 Root Mean Squared Error.746679 Mean Absolute Error.63343 Mean Absolute Percentage Error 17.985 Theil Inequality Coefficient.47445 Bias Proportion.155979 Variance Proportion.5477 Covariance Proportion.319314 The RMSE and RAE are both smaller for the AR() model indicating a superior fit. 6. To statistically compare the forecasting accuracy of the AR() and MA(1) models, we may compute Diebold-Mariano (DM) statistics using the squared error and absolute error loss functions. The DM statistics are based on the following loss differentials d d MA1 AR ( ˆ ε ) ( ˆ ε ) sq, t = t t = ˆ ε ˆ ε MA1 AR abs, t t t computed using the rolling 1-step ahead forecast errors from the AR() and MA(1) models, respectively. A time plot of these loss differentials are shown below
3..5. 1.5 1..5. -.5-1. 5 1 15 5 3 35 4 45 5 D DABS In general both loss differentials are positive indicating that the MA(1) model produces a larger forecast error than the AR() model. The DM statistic DM = d SE ( d ) may be computed by regressing the loss differential on a constant and choosing the NW correction to the standard error. Dependent Variable: D Method: Least Squares Date: 5/4/5 Time: 9:53 Sample: 1 5 Included observations: 5 Newey-West HAC Standard Errors & Covariance (lag truncation=3) Variable Coefficient Std. Error t-statistic Prob. C.316.13399 3.337.4
Dependent Variable: DABS Method: Least Squares Date: 5/4/5 Time: 1:1 Sample: 1 5 Included observations: 5 Newey-West HAC Standard Errors & Covariance (lag truncation=3) Variable Coefficient Std. Error t-statistic Prob. C.113.66393 3.17858.6 The DM statistic has an asymptotic standard normal distribution. Using both the squared and absolute value loss functions we reject the null hypothesis that the AR() and MA(1) models have equally forecasting accuracy. Since the t-statistics are positive we conclude that the AR() model is more accurate than the MA(1) model. Working with State Space Models In this exercise, a simple AR() model is estimated by conditional MLE and by exact MLE via state space methods. The AR() model has the form y y y iid N t = φ1 t 1+ φ t + εt, εt ~ (, σ ) The model is fit to detrended quarterly observations on log real GDP over the period 1947:1 through 1999:4, and then dynamic forecasts are produced over the period 1:1 through 3:4. Q1 and Q. The conditional MLEs for the AR() are produced using the following Eviews commands LS dtlrgdp ar(1) ar() and are given in the table below.
Dependent Variable: DTLRGDP Method: Least Squares Date: 5/4/5 Time: 1:4 Sample (adjusted): 1947Q3 1999Q4 Included observations: 1 after adjustments Convergence achieved after 3 iterations Variable Coefficient Std. Error t-statistic Prob. AR(1) 1.311783.6467.341. AR() -.358569.644-5.56591. R-squared.945366 Mean dependent var.55 Adjusted R-squared.94513 S.D. dependent var.4677 S.E. of regression.9531 Akaike info criterion -6.45911 Sum squared resid.18894 Schwarz criterion -6.4735 Log likelihood 68.68 Durbin-Watson stat.78946 Inverted AR Roots.9.39 3. The exact MLEs for the AR() are produced by first creating a state space form. The Kalman filter is used to create the prediction error decomposition of the log-likelihood, and this likelihood is maximized to give the MLEs. The state space set up allows for the marginal likelihood to be created for the first two initial values. In Eviews, the state space form for the AR() model (without a constant) is @signal dtlrgdp = sv1 @state sv1 = c()*sv1(-1) + c(3)*sv(-1) + [var = exp(c(1))] @state sv = sv1(-1) The coefficient c(1) denotes the variance of the error term, c() denotes the first AR term and c(3) denotes the second AR term. Notice that there is no constant in the specification because we are modeling the detrended data. The exact MLEs are
Sspace: SSAR Method: Maximum likelihood (Marquardt) Date: 5/4/5 Time: 1:54 Sample: 1947Q1 1999Q4 Included observations: 1 Estimation settings: tol=.1, derivs=accurate numeric Initial Values: C(1)=., C()=1.31777, C(3)=-.36195 Convergence achieved after 7 iterations Coefficient Std. Error z-statistic Prob. C(1) -9.313497.7384-17.4359. C() 1.31768.55531 3.7799. C(3) -.361841.5758-6.944. Final State Root MSE z-statistic Prob. SV1 -.518.9497 -.539963.589 SV -.8845. NA. Log likelihood 684.837 Akaike info criterion -6.4347 Parameters 3 Schwarz criterion -6.38498 Diffuse priors Hannan-Quinn criter. -6.4139 The exact MLEs are close the conditional MLEs. The estimate of the standard deviation of the error term is sqrt(exp(-9.313497)) =.9497 which is close to the standard error of the regression reported in the conditional MLE output. The exact log-likelihood is slightly higher than the conditional log-likelihood. Remark: Good starting values are important for the estimation of state space models. By default, for nonlinear least squares type problems, EViews uses the values in the coefficient vector at the time you begin the estimation procedure as starting values. If you wish to change the starting values, first make certain that the spreadsheet view of the coefficient vector is in edit mode, then enter the coefficient values. When you are finished setting the initial values, close the coefficient vector window and estimate your model. You may also set starting coefficient values from the command window using the
PARAM command. Simply enter the PARAM keyword, followed by pairs of coefficients and their desired values: param c(1) 153 c().68 c(3).15 sets C(1)=153, C()=.68, and C(3)=.15. All of the other elements of the coefficient vector are left unchanged. The forecasts from the state space model, the conditional AR() and the actual values are illustrated below. Notice that the forecasts from the state space model are essentially identical to those from the conditional AR() model..1. -.1 -. -.3 -.4 -.5 -.6 1 3 4 DTLRGDP DTLRGDPF DTLRGDPF_AR 4. The filtered estimates of the state vector from the state space model are illustrated below.
Filtered State SV1 Estimate Filtered State SV Estimate.1.1.8.8.4.4.. -.4 -.4 -.8 -.8 -.1 5 55 6 65 7 75 8 85 9 95 -.1 5 55 6 65 7 75 8 85 9 95 SV1 ± RMSE SV ± RMSE For the AR() model, the first state variable is y(t) and the second state variable is y(t-1). Estimate Simple Unobserved Components Model 1. The state space representation for the Clark model is @signal lrgdp*1 = sv1 + sv @state sv1 = c(1) + sv1(-1) + [var = exp(c())] @state sv = c(3)*sv(-1) + c(4)*sv3(-1) + [var = exp(c(5))] @state sv3 = sv(-1) To improve numerical stability, the log of real GDP is multiplied by 1. This is done so that the derivatives of the log-likelihood are more closely scaled. The starting values for the estimation are set using param c(1) c() -1 c(3) 1. c(4) -.4 c(5) -1 The MLEs are given in the table below Sspace: SSCLARK Method: Maximum likelihood (Marquardt) Date: 5/3/5 Time: 11:59 Sample: 1947Q1 3Q4 Included observations: 8 Estimation settings: tol=.1, derivs=accurate numeric
Initial Values: C(1)=., C()=-1., C(3)=1., C(4)= -.4, C(5)=-1. Convergence achieved after iterations Coefficient Std. Error z-statistic Prob. C(1).8618.46946 17.59837. C() -1.3773.663-1.869559.615 C(3) 1.441194.13548 1.6375. C(4) -.493771.137166-3.59989.3 C(5) -.644815.45481-1.417796.1563 Final State Root MSE z-statistic Prob. SV1 98.768.34589 396.7918. SV -1.33.311634 -.4451.6583 SV3-1.189.77786 -.536437.5917 Log likelihood -35.196 Akaike info criterion.896458 Parameters 5 Schwarz criterion.971663 Diffuse priors 3 Hannan-Quinn criter..9681 The MLEs for the AR coefficients are 1.441 and -.494, respectively. The roots of the characteristic equation φ ( z) = 1 1.441z+.494z = are 1.779 and 1.137, respectively. Since these values are greater than 1, the AR component is covariance stationary. The variance of the permanent component is.95, and the variance of the transitory component is.5476. Notice that the transitory component has a higher variance than the permanent component. The ratio of the permanent component variance to the stationary component variance is.553 indicating that the stationary component is almost twice as important as the permanent component for explaining the variation of log real GDP. The filtered state estimates are given below
96 Filtered State SV1 Estimate 15 Filtered State SV Estimate 15 Filtered State SV3 Estimate 9 1 1 88 5 5 84 8-5 -5 76-1 -1 7 5 55 6 65 7 75 8 85 9 95-15 5 55 6 65 7 75 8 85 9 95-15 5 55 6 65 7 75 8 85 9 95 SV1 ± RMSE SV ± RMSE SV3 ± RMSE Notice that the filtered trend estimate is very close to a linear trend, and the filtered state estimates are very similar to the filtered estimates of the AR() for the linearly detrended data. The graphs have been modified since the initial states are not estimated very precisely, and this results in very large SE values that distort the graphs. The filtered cycle state without the SE bars and omitting the initial state estimates is illustrated below. This model shows boom periods during the late 6s and late 9s, with recessions in the late 5s, mid 7s, early 8s, early 9s and early s. 6 4 - -4-6 -8 5 55 6 65 7 75 8 85 9 95 SVF The 1-step ahead response (signal) is given below. Notice that the Clark model tracks actual output fairly well.
96 One-step-ahead LRGDP*1 Signal Prediction 9 88 84 8 76 7 5 55 6 65 7 75 8 85 9 95 LRGDP*1 ± RMSE