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Christopher Dougherty EC220 - Introduction to econometrics (chapter 9) Slideshow: simultaneous equations estimation: Durbin-Wu-Hausman test Original citation: Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 9). [Teaching Resource] © 2012 The Author This version available at: Available in LSE Learning Resources Online: May 2012 This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user credits the author and licenses their new creations under the identical terms.

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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

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OLS IV b 1 s.e.(b 1 ) b 2 s.e.(b 2 ) b 1 s.e.(b 1 ) b 2 s.e.(b 2 ) – 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

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OLS IV b 1 s.e.(b 1 ) b 2 s.e.(b 2 ) b 1 s.e.(b 1 ) b 2 s.e.(b 2 ) – We will now perform a Durbin–Wu–Hausman test using the first sample. SIMULTANEOUS EQUATIONS ESTIMATION: DURBIN–WU–HAUSMAN TEST

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4 ivregress 2sls p (w=U) Instrumental variables (2SLS) regression Number of obs = 20 Wald chi2(1) = 0.15 Prob > chi2 = R-squared = Root MSE = p | Coef. Std. Err. z P>|z| [95% Conf. Interval] w | _cons | Instrumented: w Instruments: U estimates store REGIV 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. SIMULTANEOUS EQUATIONS ESTIMATION: DURBIN–WU–HAUSMAN TEST

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5 ivregress 2sls p (w=U) Instrumental variables (2SLS) regression Number of obs = 20 Wald chi2(1) = 0.15 Prob > chi2 = R-squared = Root MSE = p | Coef. Std. Err. z P>|z| [95% Conf. Interval] w | _cons | Instrumented: w Instruments: U estimates store REGIV SIMULTANEOUS EQUATIONS ESTIMATION: DURBIN–WU–HAUSMAN TEST The next command is ‘ estimates store ’ followed by a name for the IV regression. Here it has been called ‘ REGIV ’.

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6 reg p w Source | SS df MS Number of obs = F( 1, 18) = Model | Prob > F = Residual | R-squared = Adj R-squared = Total | Root MSE = p | Coef. Std. Err. t P>|t| [95% Conf. Interval] w | _cons | estimates store REGOLS We then run the OLS regression, and follow with the command ‘ estimates store ' followed by a name for the OLS regression. Here it has been called ‘ REGOLS '. SIMULTANEOUS EQUATIONS ESTIMATION: DURBIN–WU–HAUSMAN TEST

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7 reg p w Source | SS df MS Number of obs = F( 1, 18) = Model | Prob > F = Residual | R-squared = Adj R-squared = Total | Root MSE = p | Coef. Std. Err. t P>|t| [95% Conf. Interval] w | _cons | estimates store REGOLS hausman REGIV REGOLS, constant SIMULTANEOUS EQUATIONS ESTIMATION: DURBIN–WU–HAUSMAN TEST To perform the test, we give the command ‘hausman’ followed by the name you gave to the IV regression, then the name of the OLS regression, followed by a comma, and then ‘ constant ’, as shown.

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8 reg p w Source | SS df MS Number of obs = F( 1, 18) = Model | Prob > F = Residual | R-squared = Adj R-squared = Total | Root MSE = p | Coef. Std. Err. t P>|t| [95% Conf. Interval] w | _cons | estimates store REGOLS hausman REGIV REGOLS, constant SIMULTANEOUS EQUATIONS ESTIMATION: DURBIN–WU–HAUSMAN TEST By default, the test does not include the constant in the comparison of the coefficients because sometimes the constant has different meanings in the IV and LS regressions. Here the constant has the same meaning (we just get different estimates), so we include it.

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9 This produces the output shown. The top half reproduces the coefficients from the IV and OLS regressions. SIMULTANEOUS EQUATIONS ESTIMATION: DURBIN–WU–HAUSMAN TEST ---- Coefficients ---- | (b) (B) (b-B) sqrt(diag(V_b-V_B)) | REGIV REGOLS Difference S.E w | _cons | b = consistent under Ho and Ha; obtained from ivregress B = inconsistent under Ha, efficient under Ho; obtained from regress Test: Ho: difference in coefficients not systematic chi2(2) = (b-B)'[(V_b-V_B)^(-1)](b-B) = 6.77 Prob>chi2 =

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10 SIMULTANEOUS EQUATIONS ESTIMATION: DURBIN–WU–HAUSMAN TEST ---- Coefficients ---- | (b) (B) (b-B) sqrt(diag(V_b-V_B)) | REGIV REGOLS Difference S.E w | _cons | b = consistent under Ho and Ha; obtained from ivregress B = inconsistent under Ha, efficient under Ho; obtained from regress Test: Ho: difference in coefficients not systematic chi2(2) = (b-B)'[(V_b-V_B)^(-1)](b-B) = 6.77 Prob>chi2 = The null hypothesis is that the OLS estimators are consistent and that the differences between the OLS and IV coefficients are random.

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11 SIMULTANEOUS EQUATIONS ESTIMATION: DURBIN–WU–HAUSMAN TEST ---- Coefficients ---- | (b) (B) (b-B) sqrt(diag(V_b-V_B)) | REGIV REGOLS Difference S.E w | _cons | b = consistent under Ho and Ha; obtained from ivregress B = inconsistent under Ha, efficient under Ho; obtained from regress Test: Ho: difference in coefficients not systematic chi2(2) = (b-B)'[(V_b-V_B)^(-1)](b-B) = 6.77 Prob>chi2 = The IV estimates are in column b. They will be consistent both under the null hypothesis and the alternative.

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12 SIMULTANEOUS EQUATIONS ESTIMATION: DURBIN–WU–HAUSMAN TEST ---- Coefficients ---- | (b) (B) (b-B) sqrt(diag(V_b-V_B)) | REGIV REGOLS Difference S.E w | _cons | b = consistent under Ho and Ha; obtained from ivregress B = inconsistent under Ha, efficient under Ho; obtained from regress Test: Ho: difference in coefficients not systematic chi2(2) = (b-B)'[(V_b-V_B)^(-1)](b-B) = 6.77 Prob>chi2 = The OLS estimates are in column B. They will be unbiased and efficient under the null hypothesis and inconsistent under the alternative.

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13 SIMULTANEOUS EQUATIONS ESTIMATION: DURBIN–WU–HAUSMAN TEST ---- Coefficients ---- | (b) (B) (b-B) sqrt(diag(V_b-V_B)) | REGIV REGOLS Difference S.E w | _cons | b = consistent unde) Ho and Ha; obtained from ivregress B = inconsistent under Ha, effic)ent under Ho; obtained from regress Test: Ho: difference in coefficients not systematic chi2(2) = (b-B)'[(V_b-V_B)^(-1)](b-B) = 6.77 Prob>chi2 = Under the null hypothesis, the test statistic is in principle distributed as a chi-squared statistic with degrees of freedom equal to the number of coefficients being compared. However for finite samples the degrees of freedom may be fewer. Stata gives the number.

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14 SIMULTANEOUS EQUATIONS ESTIMATION: DURBIN–WU–HAUSMAN TEST ---- Coefficients ---- | (b) (B) (b-B) sqrt(diag(V_b-V_B)) | REGIV REGOLS Difference S.E w | _cons | b = consistent unde) Ho and Ha; obtained from ivregress B = inconsistent under Ha, effic)ent under Ho; obtained from regress Test: Ho: difference in coefficients not systematic chi2(2) = (b-B)'[(V_b-V_B)^(-1)](b-B) = 6.77 Prob>chi2 = The critical value of chi-squared with 1 degree of freedom at the 5 percent level is 5.99, so in this case we reject the null hypothesis at this significance level. However, we do not reject it at the 1 percent level (critical value 9.21).

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15 SIMULTANEOUS EQUATIONS ESTIMATION: DURBIN–WU–HAUSMAN TEST ---- Coefficients ---- | (b) (B) (b-B) sqrt(diag(V_b-V_B)) | REGIV REGOLS Difference S.E w | _cons | b = consistent unde) Ho and Ha; obtained from ivregress B = inconsistent under Ha, effic)ent under Ho; obtained from regress Test: Ho: difference in coefficients not systematic chi2(2) = (b-B)'[(V_b-V_B)^(-1)](b-B) = 6.77 Prob>chi2 = This result is expected because we know that OLS yields inconsistent estimates in a model of this type and so we know the null hypothesis is false.

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Copyright Christopher Dougherty These slideshows may be downloaded by anyone, anywhere for personal use. Subject to respect for copyright and, where appropriate, attribution, they may be used as a resource for teaching an econometrics course. There is no need to refer to the author. The content of this slideshow comes from Section 9.3 of C. Dougherty, Introduction to Econometrics, fourth edition 2011, Oxford University Press. Additional (free) resources for both students and instructors may be downloaded from the OUP Online Resource Centre Individuals studying econometrics on their own and who feel that they might benefit from participation in a formal course should consider the London School of Economics summer school course EC212 Introduction to Econometrics or the University of London International Programmes distance learning course 20 Elements of Econometrics

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