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1、Chapter 9,More on Specification and Data Issues,Wooldridge:Introductory Econometrics:A Modern Approach,5e,Tests for functional form misspecificationOne can always test whether explanatory should appear as squares or higher order terms by testing whether such terms can be excludedOtherwise,one can us
2、e general specification tests such as RESETRegression 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
3、 for omitted higher order terms and interactions,i.e.for misspecification of functional form.,Multiple Regression Analysis:Specification and Data Issues,Example:Housing price equationDiscussionOne may also include higher order terms,which implies complicated interactions and higher order terms of al
4、l explanatory variablesRESET provides little guidance as to where misspecification comes from,Evidence formisspecification,Less evidence formisspecification,Multiple Regression Analysis:Specification and Data Issues,Testing against nonnested alternativesDiscussionCan always be done;however,a clear w
5、inner need not emergeCannot 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 specificationis more appropriate?,Multiple Regression Analysis:Specification and Data Issues,Using proxy variabl
6、es for unobserved explanatory variablesExample:Omitted ability in a wage equationGeneral 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
7、 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,
8、Multiple Regression Analysis:Specification and Data Issues,Assumptions necessary for the proxy variable method to workThe proxy is just a proxy“for the omitted variable,it does not belong into the population regression,i.e.it is uncorrelated with its errorThe proxy variable is a good“proxy for the o
9、mitted 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 om
10、itted variable,Multiple Regression Analysis:Specification and Data Issues,Under these assumptions,the proxy variable method works:Discussion of the proxy assumptions in the wage exampleAssumption 1:Should be fullfilled as IQ score is not a direct wage determinant;what matters is how able the person
11、proves at workAssumption 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
12、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 meas
13、ured 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
14、 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 variablesIn
15、 many cases,omitted unobserved factors may be proxied by the value of the dependent variable from an earlier time periodExample:City crime ratesIncluding the past crime rate will at least partly control for the many omitted factors that also determine the crime rate in a given yearAnother way to int
16、erpret 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,R
17、andom component,Average slope,Random component,Assumptions:,Error term,The individual random com-ponents are independent of the explanatory variable,The model has a random intercept and a random slope,WLS or OLS with robust standard errors will consistently estimate the average intercept and average
18、 slope in the population,Multiple Regression Analysis:Specification and Data Issues,Properties of OLS under measurement errorMeasurement error in the dependent variableConsequences of measurement error in the dependent variableEstimates will be less precise because the error variance is higherOtherw
19、ise,OLS will be unbiased and consistent(as long as the mea-surement error is unrelated to the values of the explanatory variables),Mismeasured value=True value+Measurement error,Population regression,Estimated regression,Multiple Regression Analysis:Specification and Data Issues,Measurement error in
20、 an explanatory variable,Mismeasured value=True value+Measurement error,Population regression,Estimated regression,Classical errors-in-variables assumption:,The mismeasured variable x1 is cor-related with the error term!,Error unrelated to true value,Multiple Regression Analysis:Specification and Da
21、ta Issues,Consequences of measurement error in an explanatory variableUnder the classical errors-in-variables assumption,OLS is biased and inconsistent because the mismeasured variable is endogenousOne can show that the inconsistency is of the following form:The effect of the mismeasured variable su
22、ffers from attenuation bias,i.e.the magnitude of the effect will be attenuated towards zeroIn addition,the effects of the other explanatory variables will be biased,This factor(which involves the error variance of a regression of the true value of x1 on the other explanatory variables)will always be
23、 between zero and one,Multiple Regression Analysis:Specification and Data Issues,Missing data and nonrandom samplesMissing data as sample selectionMissing data is a special case of sample selection(=nonrandom samp-ling)as the observations with missing information cannot be usedIf the sample selectio
24、n is based on independent variables there is no problem as a regression conditions on the independent variablesIn general,sample selection is no problem if it is uncorrelated with the error term of a regression(=exogenous sample selection)Sample selection is a problem,if it is based on the dependent
25、 variable or on the error term(=endogenous sample selection),Multiple Regression Analysis:Specification and Data Issues,Example for exogenous sample selectionExample for endogenous sample selection,If the sample was nonrandom in the way that certain age groups,income groups,or household sizes were o
26、ver-or undersampled,this is not a problem for the regression because it examines the savings for subgroups defined by income,age,and hh-size.The distribution of subgroups does not matter.,If the sample is nonrandom in the way individuals refuse to take part in the sample survey if their wealth is pa
27、rticularly high or low,this will bias the regression results because these individuals may be systematically different from those who do not refuse to take part in the sample survey.,Multiple Regression Analysis:Specification and Data Issues,Outliers and influential observationsExtreme values and ou
28、tliers may be a particular problem for OLS because the method is based on squaring deviationsIf outliers are the result of mistakes that occured when keying in the data,one should just discard the affected observationsIf outliers are the result of the data generating process,the decision whether to
29、discard the outliers is not so easyExample:R&D intensity and firm size,Multiple Regression Analysis:Specification and Data Issues,Example:R&D intensity and firm size(cont.),The regression without the outlier makes more sense.,The outlier is not the result of a mistake:One of the sampled firms is muc
30、h larger than the others.,Multiple Regression Analysis:Specification and Data Issues,Least absolute deviations estimation(LAD)The least absolute deviations estimator minimizes the sum of absolute deviations(instead of the sum of squared deviations,i.e.OLS)It may be more robust to outliers as deviati
31、ons are not squaredThe least absolute deviations estimator estimates the parameters of the conditional median(instead of the conditional mean with OLS)The least absolute deviations estimator is a special case of quantile regression,which estimates parameters of conditional quantiles,Multiple Regression Analysis:Specification and Data Issues,
