1. VARIABLE MISSPECIFICATION II: INCLUSION OF AN IRRELEVANT VARIABLE In this sequence we will investigate the consequences of including an irrelevant variable in a regression model. 1 True model Correct specification, no problems Correct specification, no problems Coefficients are biased (in general). Standard errors are invalid. Consequences of variable misspecification Fitted model

2. The effects are different from those of omitted variable misspecification. In this case the coefficients in general remain unbiased, but they are inefficient. The standard errors remain valid, but are needlessly large. 2 VARIABLE MISSPECIFICATION II: INCLUSION OF AN IRRELEVANT VARIABLE True model Correct specification, no problems Correct specification, no problems Coefficients are biased (in general). Standard errors are invalid. Consequences of variable misspecification Fitted model Coefficients are unbiased (in general), but inefficient. Standard errors are valid (in general)

3. These results can be demonstrated quickly. 3 VARIABLE MISSPECIFICATION II: INCLUSION OF AN IRRELEVANT VARIABLE

4. Rewrite the true model adding X 3 as an explanatory variable, with a coefficient of 0. Now the true model and the fitted model coincide. Hence b 2 will be an unbiased estimator of  2 and b 3 will be an unbiased estimator of 0. 4 VARIABLE MISSPECIFICATION II: INCLUSION OF AN IRRELEVANT VARIABLE

5. However, the variance of b 2 will be larger than it would have been if the correct simple regression had been run because it includes the factor 1 / (1 – r 2 ), where r is the correlation between X 2 and X 3. 5 VARIABLE MISSPECIFICATION II: INCLUSION OF AN IRRELEVANT VARIABLE

6. The estimator b 2 using the multiple regression model will therefore be less efficient than the alternative using the simple regression model. 6 VARIABLE MISSPECIFICATION II: INCLUSION OF AN IRRELEVANT VARIABLE

7. The intuitive reason for this is that the simple regression model exploits the information that X 3 should not be in the regression, while with the multiple regression model you find this out from the regression results. 7 VARIABLE MISSPECIFICATION II: INCLUSION OF AN IRRELEVANT VARIABLE

8. The standard errors remain valid, because the model is formally correctly specified, but they will tend to be larger than those obtained in a simple regression, reflecting the loss of efficiency. 8 VARIABLE MISSPECIFICATION II: INCLUSION OF AN IRRELEVANT VARIABLE

9. These are the results in general. Note that if X 2 and X 3 happen to be uncorrelated, there will be no loss of efficiency after all. 9 VARIABLE MISSPECIFICATION II: INCLUSION OF AN IRRELEVANT VARIABLE

10. . reg LGFDHO LGEXP LGSIZE Source | SS df MS Number of obs = 868 ---------+------------------------------ F( 2, 865) = 460.92 Model | 138.776549 2 69.3882747 Prob > F = 0.0000 Residual | 130.219231 865.150542464 R-squared = 0.5159 ---------+------------------------------ Adj R-squared = 0.5148 Total | 268.995781 867.310260416 Root MSE =.388 ------------------------------------------------------------------------------ LGFDHO | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+-------------------------------------------------------------------- LGEXP |.2866813.0226824 12.639 0.000.2421622.3312003 LGSIZE |.4854698.0255476 19.003 0.000.4353272.5356124 _cons | 4.720269.2209996 21.359 0.000 4.286511 5.154027 ------------------------------------------------------------------------------ The analysis will be illustrated using a regression of LGFDHO, the logarithm of annual household expenditure on food eaten at home, on LGEXP, the logarithm of total annual household expenditure, and LGSIZE, the logarithm of the number of persons in the household. 10 VARIABLE MISSPECIFICATION II: INCLUSION OF AN IRRELEVANT VARIABLE

11. The source of the data was the 1995 US Consumer Expenditure Survey. The sample size was 868. 11 VARIABLE MISSPECIFICATION II: INCLUSION OF AN IRRELEVANT VARIABLE. reg LGFDHO LGEXP LGSIZE Source | SS df MS Number of obs = 868 ---------+------------------------------ F( 2, 865) = 460.92 Model | 138.776549 2 69.3882747 Prob > F = 0.0000 Residual | 130.219231 865.150542464 R-squared = 0.5159 ---------+------------------------------ Adj R-squared = 0.5148 Total | 268.995781 867.310260416 Root MSE =.388 ------------------------------------------------------------------------------ LGFDHO | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+-------------------------------------------------------------------- LGEXP |.2866813.0226824 12.639 0.000.2421622.3312003 LGSIZE |.4854698.0255476 19.003 0.000.4353272.5356124 _cons | 4.720269.2209996 21.359 0.000 4.286511 5.154027 ------------------------------------------------------------------------------

12. Now add LGHOUS, the logarithm of annual expenditure on housing services. It is safe to assume that LGHOUS is an irrelevant variable and, not surprisingly, its coefficient is not significantly different from zero. 12 VARIABLE MISSPECIFICATION II: INCLUSION OF AN IRRELEVANT VARIABLE. reg LGFDHO LGEXP LGSIZE LGHOUS Source | SS df MS Number of obs = 868 ---------+------------------------------ F( 3, 864) = 307.22 Model | 138.841976 3 46.2806586 Prob > F = 0.0000 Residual | 130.153805 864.150640978 R-squared = 0.5161 ---------+------------------------------ Adj R-squared = 0.5145 Total | 268.995781 867.310260416 Root MSE =.38812 ------------------------------------------------------------------------------ LGFDHO | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+-------------------------------------------------------------------- LGEXP |.2673552.0370782 7.211 0.000.1945813.340129 LGSIZE |.4868228.0256383 18.988 0.000.4365021.5371434 LGHOUS |.0229611.0348408 0.659 0.510 -.0454214.0913436 _cons | 4.708772.2217592 21.234 0.000 4.273522 5.144022 ------------------------------------------------------------------------------

13. . reg LGFDHO LGEXP LGSIZE LGHOUS Source | SS df MS Number of obs = 868 ---------+------------------------------ F( 3, 864) = 307.22 Model | 138.841976 3 46.2806586 Prob > F = 0.0000 Residual | 130.153805 864.150640978 R-squared = 0.5161 ---------+------------------------------ Adj R-squared = 0.5145 Total | 268.995781 867.310260416 Root MSE =.38812 ------------------------------------------------------------------------------ LGFDHO | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+-------------------------------------------------------------------- LGEXP |.2673552.0370782 7.211 0.000.1945813.340129 LGSIZE |.4868228.0256383 18.988 0.000.4365021.5371434 LGHOUS |.0229611.0348408 0.659 0.510 -.0454214.0913436 _cons | 4.708772.2217592 21.234 0.000 4.273522 5.144022 ------------------------------------------------------------------------------ It is however highly correlated with LGEXP (correlation coefficient 0.81), and also, to a lesser extent, with LGSIZE (correlation coefficient 0.33). 13 VARIABLE MISSPECIFICATION II: INCLUSION OF AN IRRELEVANT VARIABLE. cor LGHOUS LGEXP LGSIZE (obs=869) | LGHOUS LGEXP LGSIZE --------+--------------------------- lGHOUS| 1.0000 LGEXP| 0.8137 1.0000 LGSIZE| 0.3256 0.4491 1.0000

14. . reg LGFDHO LGEXP LGSIZE ------------------------------------------------------------------------------ LGFDHO | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+-------------------------------------------------------------------- LGEXP |.2866813.0226824 12.639 0.000.2421622.3312003 LGSIZE |.4854698.0255476 19.003 0.000.4353272.5356124 _cons | 4.720269.2209996 21.359 0.000 4.286511 5.154027 ------------------------------------------------------------------------------ Its inclusion does not cause the coefficients of those variables to be biased. 14 VARIABLE MISSPECIFICATION II: INCLUSION OF AN IRRELEVANT VARIABLE. reg LGFDHO LGEXP LGSIZE LGHOUS ------------------------------------------------------------------------------ LGFDHO | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+-------------------------------------------------------------------- LGEXP |.2673552.0370782 7.211 0.000.1945813.340129 LGSIZE |.4868228.0256383 18.988 0.000.4365021.5371434 LGHOUS |.0229611.0348408 0.659 0.510 -.0454214.0913436 _cons | 4.708772.2217592 21.234 0.000 4.273522 5.144022 ------------------------------------------------------------------------------

15. But it does increase their standard errors, particularly that of LGEXP, as you would expect, reflecting the loss of efficiency. 15 VARIABLE MISSPECIFICATION II: INCLUSION OF AN IRRELEVANT VARIABLE. reg LGFDHO LGEXP LGSIZE ------------------------------------------------------------------------------ LGFDHO | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+-------------------------------------------------------------------- LGEXP |.2866813.0226824 12.639 0.000.2421622.3312003 LGSIZE |.4854698.0255476 19.003 0.000.4353272.5356124 _cons | 4.720269.2209996 21.359 0.000 4.286511 5.154027 ------------------------------------------------------------------------------. reg LGFDHO LGEXP LGSIZE LGHOUS ------------------------------------------------------------------------------ LGFDHO | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+-------------------------------------------------------------------- LGEXP |.2673552.0370782 7.211 0.000.1945813.340129 LGSIZE |.4868228.0256383 18.988 0.000.4365021.5371434 LGHOUS |.0229611.0348408 0.659 0.510 -.0454214.0913436 _cons | 4.708772.2217592 21.234 0.000 4.273522 5.144022 ------------------------------------------------------------------------------

16. Copyright Christopher Dougherty 2012. 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 6.3 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/http://www.oup.com/uk/orc/bin/9780199567089/. Individuals studying econometrics on their own 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 http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx or the University of London International Programmes distance learning course EC2020 Elements of Econometrics www.londoninternational.ac.uk/lsewww.londoninternational.ac.uk/lse. 2012.11.09