Chapter 7 - Modeling Issues

1. Chapter 7 - Modeling Issues • 7.1 Heterogeneity • 7.2 Comparing fixed and random effects estimators • 7.3 Omitted variables • Models of omitted variables • Augmented regression estimation • 7.4 Sampling, selectivity bias, attrition • Incomplete and rotating panels • Unplanned nonresponse • Non-ignorable missing data

2. 7.1 Heterogeneity • Also think of clustering • different observations from the same subject (observational unit) tend to be related to one another. • Methods for handling this • variables in common • jointly dependent distribution functions

3. Variables in Common • These are latent (unobserved) variables • May be fixed (mean structure) or random (covariance structure) • May be organized by the cross-section or by time • If by cross-section, may be subject oriented (by i) or spatial • May be nested

4. Jointly Dependent Distribution Functions • For the covariance structures, this is a more general way to think about random variables that are common to subjects. • Also includes additional structures not suggested by the common variable approach • Example: For the error components model, we have Corr (yi1, yi2)= where  >0. However, we need not require positive correlations for a general uniform correlation model.

5. Practical Identification with Heterogeneity may be difficult – Jones (1993) Example

6. Theoretical Identification with Heterogeneity may be Impossible– Neyman Scott (1948) Example • Example - Identification of variance components • Consider the fixed effects panel data model yit = i + it , i=1, …, n, t=1,2, • where Var it = 2 and Cov (i1, i2) = 2. • The ordinary least squares estimator of iis = (yi1+ yi2)/2. • Thus, the residuals are = (yi1- yi2)/2 and = (yi2- yi1)/2= - ei1 . • Thus,  cannot be estimated, despite having 2n - n = n degrees of freedom available for estimating the variance components.

7. Estimation of regression coefficients without complete identification is possible • If our main goal is to estimate or test hypotheses about the regression coefficients, then we do not require knowledge of all aspects of the model. • For example, consider the one-way fixed effects model yi = i1i + Xiβ + i . • Apply the common transformation matrix Q = I – T -1J to each equation to get yi* = Q yi = QXi β + Qi = Xi*β + i*, • because Q 1 = 0. • Use ols on the transformed data. For T = 2, • Note that can estimate the quantity s 2 (1-r) yet cannot separate the terms s 2 and r.

8. 7.2 Comparing fixed and random effects estimators • Sometimes, the context of the problem does not make clear the choice of fixed or random effects estimators. • It is of interest to compare the fixed effects to the random effects estimators using the data. • In random effects models, we assume that { ai } are independent of {eit} . • Think instead of drawing {xit } at random and performing inference conditional on {xit }. • Interpret {ai} to be the unobserved (time-invariant) characteristics of a subject. • This assumes that individual effects { ai} are independent of other individual characteristics {xit}, our strict conditional exogeneity assumption SEC6.

9. A special case • Consider the error components model with K = 2 so that yit = i + 0+ 1xit,1 + it , • We can express (Fuller-Battese) the gls estimator as • where • As20, we have that xit* xit,,yit* yit and b1,ECb1,OLS • As2 , we have that b1,ECb1,FE

10. A special case • Define the so-called “between groups” estimator, • This estimator can be motivated by averaging all observations from a subject and then computing an ordinary least squares estimator using the data . • The following decomposition due to Maddala (1971), b1,EC = (1- ) b1,FE + b1,B • where • measures the relative precision of the two estimators of  .

11. A special case • To express the relationship between iand xi, we consider E [i| xi ]. • Specifically, we assume that i= i + , where {i} is i.i.d. • Thus, the model of correlated effects is yit = i + 0+ 1xit,1 +  + it . • Surprisingly, one can show that the generalized least squares estimator of 1 is b1,FE. • Intuitively, by considering deviations from the time series means, , the fixed effects estimator “sweeps out” all time-constant omitted variables. • In this sense, the fixed effects estimator is robust to this type of model aberration. • Under the correlated effects model, • the estimator b1,FE is unbiased, consistent and asymptotically normal. • the estimators b1,HOM, b1,B and b1,EC are biased and inconsistent.

12. A special case • To test whether or not to use the random or fixed estimator, we need only examine the null hypothesis H0:  = 0. • This is customarily done using the Hausman (1978) test statistic • This test statistic has an asymptotic (as n) chi-square distribution with 1 degree of freedom. • Test statistic is large, go with the fixed effects estimator • Test statistic is small, go with the random effects estimator

13. General Case • Assume that E αi = α and re-write the model as yi = Ziα + Xiβ + i* ,   • where i* = i + Zi (αi - α) and • Var i* = ZiD Zi + Ri = Vi. • This re-writing is necessary to make the beta’s under fixed and random effects formulations comparable. • With the appropriate definitions, the extension of Maddala’s (1971) result is • where • The extension of the Hausman test statistic is

14. Case study: Income tax payments • Consider the model with Variable Intercepts but no Variable Slopes (Error Components) • The test statistic is = 6.021 • With K = 8 degrees of freedom, the p-value associated with this test statistic is . • For the model with Variable Intercepts and two Variable Slopes • The test statistic is = 13.628. • With K = 8 degrees of freedom, the p-value associated with this test statistic is Prob( >13.628)=0.0341. .

16. 7.3 Omitted Variables • I call these models of “correlated effects.” • Section 7.2 described the Hausman/Mundlak model of time-constant omitted variables. • Chamberlain (1982) – an alternative hypothesis • Omitted variables need not be time-constant • Hausman and Taylor (1981) – another alternative hypothesis • Some of the explanatory variables are not correlated with ai . • To estimate these models, Arellano (1993) used an “augmented” regression model. We will also use this approach. • For a different approach, Stata has programmed an instrumental variable approach introduce by Hausman and Taylor as well as Amemiya and Mac Curdy.

17. Unobserved Variables Models • Introduced by Palta and Yao (1991) and co-workers. • Let oi =(zi´, xi´)´ be observed variables and ui be “unobserved” variables. • Assuming multivariate normality, we can express:

18. Unobserved Variables Models • The unobserved variables enter the likelihood through: • linear conditional expectations (g) • correlation between observed and unobserved variables (Suo) • The fixed effects estimator may be biased, unlike the correlated effects model case. • By examining certain special cases, we again arrive at the Mundlak, Chamberlain and Hausman/Taylor alternative hypotheses. • Other alternatives are also of interest. Specifically, an “extended” Mundlak alternative is:

19. Examples of Correlated Effects Models • Assuming q = 1 and zit =1, this is Chamberlain’s alternative. • Chamberlain used the hypothesis: • Thus, • Assume an error components design for the x’s. That is, • Assuming q = 1 and zit =1, this is Mundlak’s alternative. That is • Further assume that the first K-rx-variables are uncorrelated with ai . This is the Hausman/Taylor alternative.

20. Augmented Regression Estimation and Testing • I advocate the “augmented” regression approach that uses the model: E [yi | hioi] = Zi hi + Xib + Gig. • Random slopes hithat do not affect the conditional regression function. • Thus, so that E [yi | oi] = Xib + Gig. • Choose Gi = G(Xi, Zi) to be a known function of the observed effects. • Choice of G depends on the alternative model you consider. • The test for omitted variables is thus H0: g = 0. • Define bAR and gAR to be the corresponding weighted least squares estimators.

21. Some Results • The estimator bAR is unbiased (even in the presence of omitted variables). • The weights corresponding to gls (Wi = Vi) and • yields bAR = bFE. • This is an extension of Mundlak’s alternative. • The chi-square test for H0: g = 0 is:

22. Determinants of Tax Liability • I examine a 4 % sample (258) of taxpayers from the University of Michigan/Ernst & Young Tax Data Base • The panel consists of tax years 1982-84, 86, 87 • Tax Liability data, we use • xit’b = linear function of demographic and earning characteristics of a taxpayer • zit’ai= a1i + a2i LNTPIit + a3i MRit • yit = logarithmic tax liability for the ith taxpayer in the tth year

23. Empirical Fits • I present fits of four different models • Random effects • includes variable intercept plus two variable slopes • +omitted variable corrections • Random coefficients • With AR1 parameter • effects +omitted variable corrections (“Extended Mundlak alternative”)

24. Results • Section 7.2 indicated, with only variable intercepts, that the fixed effects estimator is preferable to the random effects estimator. • For random effects, • two additional variable slope terms were useful • the random coefficients model did not yield a positive definite estimate of Var a, I used a third order factor analytic model • New tests indicate that both the fixed effects model with 3 variable components and the extended Mundlak model are preferable to the random effects model with 3 variable components • Comparing fixed effects model with 3 variable components and the extended Mundlak model, the AIC favors the former yet I advocate the latter (parsimony and so on).

26. 7.4 Sampling, selectivity bias and attrition • 7.4.1 Incomplete and rotating panels • Early longitudinal and panel data methods assumed balanced data, that is, Ti = T. • This suggests techniques from multivariate analysis. • Data may not be available due to: • Delayed entry • Early exit • Intermittent nonresponse • If planned, then there is generally no difficulty. • See the text for the algebraic transformation needed. • Planned incomplete data is the norm in panel surveys of people.

28. 7.4.2 Unplanned nonresponse • Types of panel survey nonresponse (source Verbeek and Nijman, 1996) • Initial nonresponse. A subject contacted cannot, or will not, participate. Because of limited information, this potential problem is often ignored in the analysis. • Unit nonresponse. A subject contacted cannot, or will not, participate even after repeated attempts (in subsequent waves) to include the subject. • Wave nonresponse. A subject does not respond for one or more time periods but does respond in the preceding and subsequent times (for example, the subject may be on vacation). • Attrition. A subject leaves the panel after participating in at least one survey.

29. Missing data models • Let rij be an indicator variable for the ijth observation, with a one indicating that this response is observed and a zero indicating that the response is missing. • Let ri = (ri1, …, riT) and r = (r1, …, rn). • The interest is in whether or not the responses influence the missing data mechanism. • Use yi = (yi1, …, yiT) to be the vector of all potentially observed responses for the ith subject • Let Y = (y1, …, yn)to be the collection of all potentially observed responses.

30. Rubin’s (1976) Missing data models • Missing completely at random (MCAR). • The case where Y does not affect the distribution of r. • Specifically, the missing data are MCAR if f(r | Y) = f(r), where f(.) is a generic probability mass function. • Little (1995) - the adjective “covariate dependent” is added when • Y does not affect the distribution of r, conditional on the covariates. • If the covariates are summarized as {X, Z}, then the condition corresponds to the relation f(r | Y, X, Z) = f(r| X, Z). • Example: x=age, y=income. Missingness could vary by income but is really due to age (young people don’t respond)

31. General advice on missing at random • One option is to treat the available data as if nonresponses were planned and use unbalanced estimation techniques. • Another option is to utilize only subjects with a complete set of observations by discarding observations from subjects with missing responses. • A third option is to impute values for missing responses. • Little and Rubin note that each option is generally easy to carry out and may be satisfactory with small amounts of missing data. • However, the second and third options may not be efficient. • Further, each option implicitly relies heavily on the MCAR assumption.

32. Selection Model • Partition the Y vector into observed and missing components: Y = (Yobs, Ymiss). • Selection model is given by f(r | Y). • With parameters θ and ψ, assume that the log likelihood of the observed random variables is L(θ,ψ) = log f(r, Yobs,θ,ψ) = log f(Yobs,θ) + log f(r | Yobs,ψ). • MCAR case • f(r | Yobs,ψ) = f(r | ψ) does not depend on Yobs. • Data missing at random (MAR) • if selection mechanism model distribution does not depend on Ymiss but may depend on Yobs. • That is, f(r | Y) = f(r | Yobs). • For both MAR and MCAR, the likelihood may be maximized over the parameters separately in term. • For inference about θ, the selection model mechanism may be “ignored.” • MAR and MCAR are referred to as the ignorable case.

33. Example – Income tax payments • Let y = tax liability and x = income. • The taxpayer is not selected (missing) with probability . • The selection mechanism is MCAR. • The taxpayer is not selected if tax liability < $100. • The selection mechanism depends on the observed and missing response. The selection mechanism cannot be ignored. • The taxpayer is not selected if income < $20,000. • The selection mechanism is MCAR, covariate dependent. • Assuming that the purpose of the analysis is to understand tax liabilities conditional on knowledge of income, stratifying based on income does not serious bias the analysis. • The probability of a taxpayer being selected decreases with tax liability. For example, suppose the probability of being selected is logit (-yi). • In this case, the selection mechanism depends on the observed and missing response. The selection mechanism cannot be ignored. • The taxpayer is followed over T = 2 periods. In the second period, a taxpayer is not selected if the first period tax < $100. • The selection mechanism is MAR. That is, the selection mechanism is based on an observed response.

34. Example - correction for selection bias • Historical heights. y = the height of men recruited to serve in the military. • The sample is subject to censoring in that minimum height standards were imposed for admission to the military. • The selection mechanism is non-ignorable because it depends on the individual’s height. • The joint distribution for observables is • f(r, Yobs, , ) = f(Yobs, , )  f(r | Yobs ) • This is easy to maximize in  and . • If one ignored the censoring mechanisms, then the “log likelihood” is • MLEs based on this are different, and biased.

35. Non-ignorable missing data • There are many models of missing data mechanisms - see Little and Rubin (1987). • Heckman two-stage procedure • Heckman (1976) developed for cross-sectional data but also applicable to fixed effects panel data models. • Thus, use yit = i + xit β + it . • Further, assume that the sampling response mechanism is governed by the latent (unobserved) variable rit* rit* = wit γ+ hit . • We observe

36. Assume {yit, rit} is multivariate normal to get E (yit | rit* 0) = i + xit β + (wit γ), • where =  and . • Heckman’s two-step procedure • Use the data {( rit, wit)} and a probit regression model to estimate γ. Call this estimator gH. • Use the estimator gH to create a new explanatory variable, xit,K+1 = (wit gH). • Run a one-way fixed effects model using the K explanatory variables xit as well as the additional explanatory variable xit,K+1. • To test for selection bias, test H0: = 0 . • Estimates of  give a “correction for the selection bias.”

37. Hausman and Wise procedure • Use an error components model, yit = ai + xit β + it. • The sampling response mechanism is governed by the latent variable error components model rit* = xi + wit γ + hit . • The variances are: • if sax = sh = 0, then the selection process is independent of the observation process.

38. Hausman and Wise procedure • Again, assume joint normality. With this assumption, one can check that: • where git = E (xi + hit | ri). • Calculating this quantity is computationally intensive, requiring numerical integration of multivariate normals. • if sax = sh = 0, then E (yit | ri) = xit β.