Chapter 9,More on Specification and Data Issues,Wooldridge: Introductory Econometrics: A Modern Approach, 5e,Tests for functional form misspecification One can always test whether explanatory should appear as squares or higher order terms by testing whether such terms can be excluded Otherwise, one can use general specification tests such as RESET Regression specification error test (RESET) The idea of RESET is to include squares and possibly higher order fitted values in the regression (similarly to the reduced White test),Test for the exclusion of these terms. If they cannot be exluded, this is evidence for omitted higher order terms and interactions, i.e. for misspecification of functional form.,Multiple Regression Analysis: Specification and Data Issues,Example: Housing price equation Discussion One may also include higher order terms, which implies complicated interactions and higher order terms of all explanatory variables RESET provides little guidance as to where misspecification comes from,Evidence for misspecification,Less evidence for misspecification,Multiple Regression Analysis: Specification and Data Issues,Testing against nonnested alternatives Discussion Can always be done; however, a clear winner need not emerge Cannot be used if the models differ in their definition of the dep. var.,Model 1:,Model 2:,Define a general model that contains both models as subcases and test:,Which specification is more appropriate?,Multiple Regression Analysis: Specification and Data Issues,Using proxy variables for unobserved explanatory variables Example: Omitted ability in a wage equation General approach to using proxy variables,In general, the estimates for the returns to education and experience will be biased because one has omit the unobservable ability variable. Idea: find a proxy variable for ability which is able to control for ability differences between individuals so that the coefficients of the other variables will not be biased. A possible proxy for ability is the IQ score or similar test scores.,Replace by proxy,Omitted variable, e.g. ability,Regression of the omitted variable on its proxy,Multiple Regression Analysis: Specification and Data Issues,Assumptions necessary for the proxy variable method to work The proxy is just a proxy“ for the omitted variable, it does not belong into the population regression, i.e. it is uncorrelated with its error The proxy variable is a good“ proxy for the omitted variable, i.e. using other variables in addition will not help to predict the omitted variable,If the error and the proxy were correlated, the proxy would actually have to be included in the population regression function,Otherwise x1 and x2 would have to be included in the regression for the omitted variable,Multiple Regression Analysis: Specification and Data Issues,Under these assumptions, the proxy variable method works: Discussion of the proxy assumptions in the wage example Assumption 1: Should be fullfilled as IQ score is not a direct wage determinant; what matters is how able the person proves at work Assumption 2: Most of the variation in ability should be explainable by variation in IQ score, leaving only a small rest to educ and exper,In this regression model, the error term is uncorrelated with all explanatory variables. As a consequence, all coefficients will be correctly estimated using OLS. The coefficents for the explanatory variables x1 and x2 will be correctly identified. The coefficient for the proxy va-riable may also be of interest (it is a multiple of the coefficient of the omitted variable).,Multiple Regression Analysis: Specification and Data Issues,As expected, the measured return to education decreases if IQ is included as a proxy for unobserved ability. The coefficient for the proxy suggests that ability differences between indivi-duals are important (e.g. + 15 points IQ score are associated with a wage increase of 5.4 percentage points). Even if IQ score imperfectly soaks up the variation caused by ability, inclu-ding it will at least reduce the bias in the measured return to education. No significant interaction effect bet-ween ability and education.,Multiple Regression Analysis: Specification and Data Issues,Using lagged dependent variables as proxy variables In many cases, omitted unobserved factors may be proxied by the value of the dependent variable from an earlier time period Example: City crime rates Including the past crime rate will at least partly control for the many omitted factors that also determine the crime rate in a given year Another way to interpret this equation is that one compares cities which had the same crime rate last year; this avoids comparing cities that differ very much in unobserved crime factors,Multiple Regression Analysis: Specification and Data Issues,Models with random slopes (= random coefficient models),Average intercept,Random component,Average slope,Random component,Assumptions:,Error term,The individual r