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

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)

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

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

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

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

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

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

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

. reg LGFDHO LGEXP LGSIZE Source | SS df MS Number of obs = F( 2, 865) = Model | Prob > F = Residual | R-squared = Adj R-squared = Total | Root MSE = LGFDHO | Coef. Std. Err. t P>|t| [95% Conf. Interval] LGEXP | LGSIZE | _cons | 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

The source of the data was the 1995 US Consumer Expenditure Survey. The sample size was VARIABLE MISSPECIFICATION II: INCLUSION OF AN IRRELEVANT VARIABLE. reg LGFDHO LGEXP LGSIZE Source | SS df MS Number of obs = F( 2, 865) = Model | Prob > F = Residual | R-squared = Adj R-squared = Total | Root MSE = LGFDHO | Coef. Std. Err. t P>|t| [95% Conf. Interval] LGEXP | LGSIZE | _cons |

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 = F( 3, 864) = Model | Prob > F = Residual | R-squared = Adj R-squared = Total | Root MSE = LGFDHO | Coef. Std. Err. t P>|t| [95% Conf. Interval] LGEXP | LGSIZE | LGHOUS | _cons |

. reg LGFDHO LGEXP LGSIZE LGHOUS Source | SS df MS Number of obs = F( 3, 864) = Model | Prob > F = Residual | R-squared = Adj R-squared = Total | Root MSE = LGFDHO | Coef. Std. Err. t P>|t| [95% Conf. Interval] LGEXP | LGSIZE | LGHOUS | _cons | 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| LGEXP| LGSIZE|

. reg LGFDHO LGEXP LGSIZE LGFDHO | Coef. Std. Err. t P>|t| [95% Conf. Interval] LGEXP | LGSIZE | _cons | 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 | LGSIZE | LGHOUS | _cons |

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 | LGSIZE | _cons | reg LGFDHO LGEXP LGSIZE LGHOUS LGFDHO | Coef. Std. Err. t P>|t| [95% Conf. Interval] LGEXP | LGSIZE | LGHOUS | _cons |

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 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 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 or the University of London International Programmes distance learning course EC2020 Elements of Econometrics