1. 1 In the Monte Carlo experiment in the previous sequence we used the rate of unemployment, U, as an instrument for w in the price inflation equation. SIMULTANEOUS EQUATIONS ESTIMATION: DURBIN-WU-HAUSMAN TEST

2. 2 OLS IV a s.e.(a) b s.e.(b) a s.e.(a) b s.e.(b) 10.360.491.110.222.330.970.160.45 20.450.381.060.171.530.570.530.26 30.650.270.940.121.130.320.700.15 40.410.390.980.191.550.590.370.30 50.920.460.770.222.310.710.060.35 60.260.351.090.161.240.520.590.25 70.310.391.000.191.520.620.330.32 81.060.380.820.161.950.510.410.22 9-0.080.361.160.181.110.620.450.33 101.120.430.690.202.260.610.130.29 We ran OLS and IV regressions for 10 samples. As far as we could tell, the IV estimates were distributed around the true value, while the OLS estimates were clearly upwards biased. SIMULTANEOUS EQUATIONS ESTIMATION: DURBIN-WU-HAUSMAN TEST

3. 3 OLS IV a s.e.(a) b s.e.(b) a s.e.(a) b s.e.(b) 10.360.491.110.222.330.970.160.45 20.450.381.060.171.530.570.530.26 30.650.270.940.121.130.320.700.15 40.410.390.980.191.550.590.370.30 50.920.460.770.222.310.710.060.35 60.260.351.090.161.240.520.590.25 70.310.391.000.191.520.620.330.32 81.060.380.820.161.950.510.410.22 9-0.080.361.160.181.110.620.450.33 101.120.430.690.202.260.610.130.29 We will now perform a Durbin-Wu-Hausman test using the first sample. SIMULTANEOUS EQUATIONS ESTIMATION: DURBIN-WU-HAUSMAN TEST

4. 4. ivreg p (w=U) Instrumental variables (2SLS) regression Source | SS df MS Number of obs = 20 ---------+------------------------------ F( 1, 18) = 0.13 Model | 5.39052472 1 5.39052472 Prob > F = 0.7207 Residual | 28.1781361 18 1.565452 R-squared = 0.1606 ---------+------------------------------ Adj R-squared = 0.1139 Total | 33.5686608 19 1.76677162 Root MSE = 1.2512 ------------------------------------------------------------------------------ p | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+-------------------------------------------------------------------- w |.1619431.4459005 0.363 0.721 -.7748591 1.098745 _cons | 2.328433.9699764 2.401 0.027.2905882 4.366278 ------------------------------------------------------------------------------ Instrumented: w Instruments: U ------------------------------------------------------------------------------. hausman, save We begin by running the IV regression. In the command, the instrumented variable(s) and instrument(s) are placed in parentheses, with an = sign separating them. Here w is the instrumented variable and U is the instrument. The next command is "hausman, save". SIMULTANEOUS EQUATIONS ESTIMATION: DURBIN-WU-HAUSMAN TEST

5. 5. reg p w Source | SS df MS Number of obs = 20 ---------+------------------------------ F( 1, 18) = 26.16 Model | 19.8854938 1 19.8854938 Prob > F = 0.0001 Residual | 13.683167 18.760175945 R-squared = 0.5924 ---------+------------------------------ Adj R-squared = 0.5697 Total | 33.5686608 19 1.76677162 Root MSE =.87188 ------------------------------------------------------------------------------ p | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+-------------------------------------------------------------------- w | 1.107448.2165271 5.115 0.000.6525417 1.562355 _cons |.3590688.4913327 0.731 0.474 -.673183 1.391321 ------------------------------------------------------------------------------. hausman, constant sigmamore We then run the OLS regression, and follow with the command "hausman, constant sigmamore". SIMULTANEOUS EQUATIONS ESTIMATION: DURBIN-WU-HAUSMAN TEST

6. 6. hausman, constant sigmamore ---- Coefficients ---- | (b) (B) (b-B) sqrt(diag(V_b-V_B)) | Prior Current Difference S.E. ---------+------------------------------------------------------------- w |.1619431 1.107448 -.9455052.2228577 _cons | 2.328433.3590688 1.969364.4641836 ---------+------------------------------------------------------------- b = less efficient estimates obtained previously from ivreg. B = more efficient estimates obtained from regress. Test: Ho: difference in coefficients not systematic chi2( 1) = (b-B)'[(V_b-V_B)^(-1)](b-B) = 18.00 Prob>chi2 = 0.0000 This produces the output shown. The top half reproduces the coefficients from the IV and OLS regressions. SIMULTANEOUS EQUATIONS ESTIMATION: DURBIN-WU-HAUSMAN TEST

7. 7. hausman, constant sigmamore ---- Coefficients ---- | (b) (B) (b-B) sqrt(diag(V_b-V_B)) | Prior Current Difference S.E. ---------+------------------------------------------------------------- w |.1619431 1.107448 -.9455052.2228577 _cons | 2.328433.3590688 1.969364.4641836 ---------+------------------------------------------------------------- b = less efficient estimates obtained previously from ivreg. B = more efficient estimates obtained from regress. Test: Ho: difference in coefficients not systematic chi2( 1) = (b-B)'[(V_b-V_B)^(-1)](b-B) = 18.00 Prob>chi2 = 0.0000 The null hypothesis is that the OLS estimators are consistent and that the differences between the OLS and IV coefficients are random. Note that if the null hypothesis is true, IV will be less efficient than OLS. SIMULTANEOUS EQUATIONS ESTIMATION: DURBIN-WU-HAUSMAN TEST

8. 8. hausman, constant sigmamore ---- Coefficients ---- | (b) (B) (b-B) sqrt(diag(V_b-V_B)) | Prior Current Difference S.E. ---------+------------------------------------------------------------- w |.1619431 1.107448 -.9455052.2228577 _cons | 2.328433.3590688 1.969364.4641836 ---------+------------------------------------------------------------- b = less efficient estimates obtained previously from ivreg. B = more efficient estimates obtained from regress. Test: Ho: difference in coefficients not systematic chi2( 1) = (b-B)'[(V_b-V_B)^(-1)](b-B) = 18.00 Prob>chi2 = 0.0000 Under the null hypothesis, the test statistic is distributed as a chi-squared statistic with degrees of freedom equal to the number of instrumented variables. SIMULTANEOUS EQUATIONS ESTIMATION: DURBIN-WU-HAUSMAN TEST

9. 9. hausman, constant sigmamore ---- Coefficients ---- | (b) (B) (b-B) sqrt(diag(V_b-V_B)) | Prior Current Difference S.E. ---------+------------------------------------------------------------- w |.1619431 1.107448 -.9455052.2228577 _cons | 2.328433.3590688 1.969364.4641836 ---------+------------------------------------------------------------- b = less efficient estimates obtained previously from ivreg. B = more efficient estimates obtained from regress. Test: Ho: difference in coefficients not systematic chi2( 1) = (b-B)'[(V_b-V_B)^(-1)](b-B) = 18.00 Prob>chi2 = 0.0000 The critical value of chi-squared with 1 degree of freedom at the 0.1 percent level is 10.83, so we reject the null hypothesis and conclude that the OLS estimators are inconsistent. SIMULTANEOUS EQUATIONS ESTIMATION: DURBIN-WU-HAUSMAN TEST

10. 10. hausman, constant sigmamore ---- Coefficients ---- | (b) (B) (b-B) sqrt(diag(V_b-V_B)) | Prior Current Difference S.E. ---------+------------------------------------------------------------- w |.1619431 1.107448 -.9455052.2228577 _cons | 2.328433.3590688 1.969364.4641836 ---------+------------------------------------------------------------- b = less efficient estimates obtained previously from ivreg. B = more efficient estimates obtained from regress. Test: Ho: difference in coefficients not systematic chi2( 1) = (b-B)'[(V_b-V_B)^(-1)](b-B) = 18.00 Prob>chi2 = 0.0000 Of course, this result is exactly what we expected because we know that OLS yields inconsistent estimates in a model of this type. SIMULTANEOUS EQUATIONS ESTIMATION: DURBIN-WU-HAUSMAN TEST

11. Copyright Christopher Dougherty 2001. This slideshow may be freely copied for personal use.