Christopher Dougherty EC220 - Introduction to econometrics (chapter 9) Slideshow: simultaneous equations estimation: Durbin-Wu-Hausman test Original citation:

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: http://learningresources.lse.ac.uk/135/

Available in LSE Learning Resources Online: May 2012

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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

wwUU

ppUUb

ii

iiIV2

wuUpw 321 puwp 21

puwp 5.05.1 wuUpw 4.05.05.2

22

23121

1

wp uuUw

OLS IV

b1 s.e.(b1) b2 s.e.(b2) b1 s.e.(b1) b2 s.e.(b2)

1 0.36 0.49 1.11 0.22 2.33 0.97 0.16 0.452 0.45 0.38 1.06 0.17 1.53 0.57 0.53 0.263 0.65 0.27 0.94 0.12 1.13 0.32 0.70 0.154 0.41 0.39 0.98 0.19 1.55 0.59 0.37 0.305 0.92 0.46 0.77 0.22 2.31 0.71 0.06 0.356 0.26 0.35 1.09 0.16 1.24 0.52 0.59 0.257 0.31 0.39 1.00 0.19 1.52 0.62 0.33 0.328 1.06 0.38 0.82 0.16 1.95 0.51 0.41 0.229 –0.08 0.36 1.16 0.18 1.11 0.62 0.45 0.33

10 1.12 0.43 0.69 0.20 2.26 0.61 0.13 0.29

wuUpw 321 puwp 21

puwp 5.05.1 wuUpw 4.05.05.2

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.


OLS IV

b1 s.e.(b1) b2 s.e.(b2) b1 s.e.(b1) b2 s.e.(b2)

1 0.36 0.49 1.11 0.22 2.33 0.97 0.16 0.452 0.45 0.38 1.06 0.17 1.53 0.57 0.53 0.263 0.65 0.27 0.94 0.12 1.13 0.32 0.70 0.154 0.41 0.39 0.98 0.19 1.55 0.59 0.37 0.305 0.92 0.46 0.77 0.22 2.31 0.71 0.06 0.356 0.26 0.35 1.09 0.16 1.24 0.52 0.59 0.257 0.31 0.39 1.00 0.19 1.52 0.62 0.33 0.328 1.06 0.38 0.82 0.16 1.95 0.51 0.41 0.229 –0.08 0.36 1.16 0.18 1.11 0.62 0.45 0.33

10 1.12 0.43 0.69 0.20 2.26 0.61 0.13 0.29

wuUpw 321 puwp 21

puwp 5.05.1 wuUpw 4.05.05.2

3

We will now perform a Durbin–Wu–Hausman test using the first sample.


4

ivregress 2sls p (w=U)Instrumental variables (2SLS) regression Number of obs = 20 Wald chi2(1) = 0.15 Prob > chi2 = 0.7018 R-squared = 0.1606 Root MSE = 1.187------------------------------------------------------------------------------ p | Coef. Std. Err. z P>|z| [95% Conf. Interval]-------------+---------------------------------------------------------------- w | .1619431 .4230184 0.38 0.702 -.6671577 .9910439 _cons | 2.328433 .9202004 2.53 0.011 .5248734 4.131993------------------------------------------------------------------------------Instrumented: wInstruments: 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.


5

ivregress 2sls p (w=U)Instrumental variables (2SLS) regression Number of obs = 20 Wald chi2(1) = 0.15 Prob > chi2 = 0.7018 R-squared = 0.1606 Root MSE = 1.187------------------------------------------------------------------------------ p | Coef. Std. Err. z P>|z| [95% Conf. Interval]-------------+---------------------------------------------------------------- w | .1619431 .4230184 0.38 0.702 -.6671577 .9910439 _cons | 2.328433 .9202004 2.53 0.011 .5248734 4.131993------------------------------------------------------------------------------Instrumented: wInstruments: U

estimates store REGIV


The next command is ‘estimates store’ followed by a name for the IV regression. Here it has been called ‘REGIV’.

6

reg p w

Source | SS df MS Number of obs = 20---------+------------------------------ F( 1, 18) = 26.16 Model | 19.8854938 1 19.8854938 Prob > F = 0.0001Residual | 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------------------------------------------------------------------------------

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'.


7

reg p w



estimates store REGOLShausman REGIV REGOLS, constant


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.

8

reg p w



estimates store REGOLShausman REGIV REGOLS, constant


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.

9

This produces the output shown. The top half reproduces the coefficients from the IV and OLS regressions.


---- Coefficients ---- | (b) (B) (b-B) sqrt(diag(V_b-V_B)) | REGIV REGOLS Difference S.E.-------------+---------------------------------------------------------------- w | .1619431 1.107448 -.9455052 .3634014 _cons | 2.328433 .3590688 1.969364 .7780495------------------------------------------------------------------------------ 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 = 0.0339

10




The null hypothesis is that the OLS estimators are consistent and that the differences between the OLS and IV coefficients are random.

11




The IV estimates are in column b. They will be consistent both under the null hypothesis and the alternative.

12




The OLS estimates are in column B. They will be unbiased and efficient under the null hypothesis and inconsistent under the alternative.

13


---- Coefficients ---- | (b) (B) (b-B) sqrt(diag(V_b-V_B)) | REGIV REGOLS Difference S.E.-------------+---------------------------------------------------------------- w | .1619431 1.107448 -.9455052 .3634014 _cons | 2.328433 .3590688 1.969364 .7780495------------------------------------------------------------------------------ b = consistent unde) Ho and Ha; obtained from ivregress B = inconsistent under Ha, effic)ent under Ho; obtained from regress


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.

14




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).

99.5)2( %5 crit,2

21.9)2( %1 crit,2

15




99.5)2( %5 crit,2

21.9)2( %1 crit,2

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.

Copyright Christopher Dougherty 2011.

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

http://www.oup.com/uk/orc/bin/9780199567089/.

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

http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx

or the University of London International Programmes distance learning course

20 Elements of Econometrics

www.londoninternational.ac.uk/lse.

11.07.25

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Christopher Dougherty EC220 - Introduction to econometrics (chapter 9) Slideshow: simultaneous equations estimation: Durbin-Wu-Hausman test Original citation: