4. Examples. Results: Example 4.1 Implementation of the Example 3.1 in SAS. In SAS we can use the Proc Model procedure.

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1 4. Examples Example 4.1 Implementation of the Example 3.1 in SAS. In SAS we can use the Proc Model procedure. Simulate data from t-distribution with ν = 6. SAS: data tdist; do i = 1 to 500; y = tinv(ranuni(158),6); z = 1; output; end; run; proc model data = tdist; endogenous y; parms nu 5; eq.h1 = y** - nu/(nu-); eq.h = y**4-3*nu**/((nu-)*(nu-4)); fit h1 h/gmm; instruments z / noint; run; Results: The MODEL Procedure Nonlinear GMM Summary of Residual Errors DF DF Equation Model Error SSE MSE Root MSE h h Nonlinear GMM Parameter Estimates Approx Approx Parameter Estimate Std Err t Value Pr > t nu <.0001 Number of Observations Statistics for System Used 500 Objective Missing 0 Objective*N.4584 Thus ˆν = 5.37 with standard error J =.4584 with 1 degree of freedom and p-value Thus the data does not reject the moment conditions implied by the model. 1

2 Example 4. Normality of SP500 (daily) returns, y t. If normality holds, the moment conditions: y t μ = 0 (y t μ) σ = 0 (y t μ) 3 /σ 3 = 0 (y t μ) 4 /σ 4 3 = 0 where μ =E[y t ], σ =Var[y t ]. Data is obtained from finance.yahoo.com web site with sample period Jan, 1995 to May 19, 005. In EViews open Object New object... System System and write commands (c(1) = mean, c() = standard c param c(1) 0 c() 1.0 spret - c(1) (spret - c(1))^ - c()^ ((spret - c(1))/c())^3 ((spret - c(1))/c())^4-3 Sample statistics Series: SPRET Sample /01/1995 4/19/005 Observations 570 Mean Median Maximum Minimum Std. Dev Skewness Kurtosis Jarque-Bera Probability

3 Select Estimate GMM to get results System: GMM_NORMALITY Estimation Method: Generalized Method of Moments Date: 04/0/05 Time: 0:51 Sample: /01/1995 4/19/005 Included observations: 570 Total system (balanced) observations 1080 Kernel: Bartlett, Bandwidth: Fixed (8), No prewhitening Iterate coefficients after one-step weighting matrix Convergence not achieved after: 1 weight matrix, 506 total coef iterations Coefficient Std. Error t-statistic Prob. Note that the J-statistic in EViews is not multiplied by number of observations. Thus J = , which is highly statistically significant (df = 4 = ), and thus rejects the normality hypothesis. C(1) C() Determinant residual covariance J-statistic Equation: SPRET - C(1) S.E. of regression Sum squared resid Durbin-Watson stat.0964 Equation: (SPRET - C(1))^ - C()^ S.E. of regression Sum squared resid Durbin-Watson stat Equation: ((SPRET - C(1))/C())^3 S.E. of regression Sum squared resid 160. Durbin-Watson stat Equation: ((SPRET - C(1))/C())^4-3 S.E. of regression Sum squared resid Durbin-Watson stat

4 Let us next test whether a t-distribution with location parameter μ, scale parameter σ, and degrees of freedom parameter ν fits better. The density function is with f(y) = Γ ( ) ν+1 πνσ Γ ( ν (y μ) ) (1+ σ (ν ) E[y] =μ, Var[y] = νσ ν, and E [ (y μ) 4] 3ν σ 4 = (ν )(ν 4). The implied moment conditions are E [y μ] = 0 [ ] E (y μ) σ ν = 0 ν E [ (y μ) 3] = 0 [ ] E (y μ) 4 3ν σ 4 = 0 (ν )(ν 4) ) 1 (ν+1) EViews estimation produces with commands (c(1) = mean, c() = scale, c(3) = c param c(1) 0 c() 1.0 c(3) 7.0 spret - c(1) (spret - c(1))^ - c()^*c(3)/(c(3)-) (spret - c(1))^3 (spret - c(1))^4 - (3*c()^4)*(c(3)^)/((c(3)-)*(c(3)-4)) System: GMM_T Estimation Method: Generalized Method of Moments Date: 04/5/05 Time: 00:9 Sample: /01/1995 4/19/005 Included observations: 570 Total system (balanced) observations 1080 Kernel: Bartlett, Bandwidth: Fixed (8), No prewhitening Iterate coefficients after one-step weighting matrix Convergence achieved after: 1 weight matrix, 8 total coef iterations Coefficient Std. Error t-statistic Prob. C(1) C() C(3) Determinant residual covariance J-statistic Equation: SPRET - C(1) S.E. of regression Sum squared resid Durbin-Watson stat.035 Equation: (SPRET - C(1))^ - C()^*C(3)/(C(3)-) S.E. of regression Sum squared resid Durbin-Watson stat Equation: (SPRET - C(1))^3 S.E. of regression Sum squared resid Durbin-Watson stat Equation: (SPRET - C(1))^4 - (3*C()^4)*(C(3)^)/((C(3)-)*(C(3)-4)) S.E. of regression Sum squared resid Durbin-Watson stat The J-statistic is J = T J EViews = = with p-value (df = 1), which indicates 7 8

5 the return distribution seems to behave like a the t- distribution at least up to the first four moments with estimates ˆμ =0.041, ˆσ =0.934, and ˆν =6.13. Example 4.4 Estimation of Dynamic Rational Expectations Model. Hansen and Singleton (198). Generalized instrumental variables method of nonlinear rational expectations models. Econometrica 50, Errata: Econometrica 5, Theoretical background, data and SAS-code for a similar problem can be found from. 9