Testing for time-invariant unobserved heterogeneity in nonlinear panel-data models

Transcription

1 Testing for time-invariant unobserved heterogeneity in nonlinear panel-data models Francesco Bartolucci University of Perugia Federico Belotti University of Rome Tor Vergata Franco Peracchi University of Rome Tor Vergata August 29, 2012 Abstract Recent literature on nonlinear panel data has emphasized the importance of accounting for timevarying unobserved heterogeneity, which may stem either from time-varying unit-specific omitted variables or macro-level shocks that affect each individual unit differently. In this paper, we propose a Hausman-like procedure to test the null hypothesis of time-invariant individual effects. The test can be used when the dependent variable is discrete and is based on a comparison between standard and pairwise conditional likelihood estimators. It requires no assumptions on the distribution on the time-varying individual effects, in particular it does not require them to be independent of the covariates in the model. We investigate the finite sample properties of the test by a simulation study. The results of this study show good size and power properties of the proposed test, especially with ordinal outcomes. A health economics example based on a sample from the Health and Retirement Study is used to illustrate the test. Keywords: Categorical response models; conditional likelihood; Hausman-like test; self-reported health; Health and Retirement Study. JEL: C12, C33, C35. Preliminary draft. We thank seminar participants at the University of Padua, Department of Statistical Sciences for useful comments. We also thank Florian Heiss for allowing us to use his arldv Stata package. Corresponding author: Federico Belotti, CEIS and Department of Economics and Finance, University of Rome Tor Vergata, via Columbia 2, Rome, Italy. federico.belotti@uniroma2.it, phone:

2 1 Introduction A distinctive feature of panel data modeling is the treatment of unobserved heterogeneity, which is typically interpreted as the effect of unobservable factors on the outcome of interest. The simplest way of dealing with this form of heterogeneity consists of including in the model time-invariant individual effects. For a detailed treatment see Hsiao (2005), Wooldridge (2010), and Arellano & Bonhomme (2011). However, assuming that these effects are constant over time may be difficult to justify in certain applications. For example, Stowasser et al. (2011) argue that the dynamic pattern of self reported health status (SRHS) can be better modeled by introducing a latent time-varying individual-specific true health component. Obviously, biased parameter estimates may result if the individual effects are time-varying but they are assumed to be time-invariant. This is especially true in the case of long panels. A few studies try to relax the assumption of time-invariant individual effects by using a dynamic latent process for the unobserved heterogeneity. In the case of nonlinear panel data models, one strategy is to include time-varying random effects, which are assumed to be independent of the covariates and are treated as either continuous or discrete. In particular, Heiss (2008) proposes a limited dependent variable model which, for every sample unit, relies on a sequence of timevarying effects which is assumed to follow an AR(1) process with common parameters. Bartolucci & Farcomeni (2009) propose a multivariate extension of the dynamic logit model based on timevarying individual effects which are assumed to follow a time-homogeneous Markov chain for every sample unit. The approaches mentioned above to account for time-varying unobserved heterogeneity have pros and cons. Though the Heiss (2008) s formulation is parsimonious (it uses only one more parameter with respect to a standard random-effects model) and more easily justifiable (continuous random effects are more naturally conceivable in most applications), the discrete approach, formulated as in Bartolucci & Farcomeni (2009), usually results in a more flexible model which may reach a better fit; see Bartolucci et al. (2012) for more detailed comments. On the other hand, both approaches require to formulate some assumptions on the distribution of the random effects and are computationally demanding. Therefore, practitioners find it useful to test for the presence of time-invariant unobserved heterogeneity before estimating these types of models. In this paper, we propose a Hausman-like specification test for the null hypothesis of time- 1

3 invariant individual effects in nonlinear panel data models against the alternative that these effects are time-varying. Our test is a pure specification test because it leaves the alternative deliberately vague. The test is based on a comparison between standard and pairwise conditional likelihood estimators. The two estimators converge to the same point in the parameter space under the null, but they diverge under the alternative. The proposed test is attractive because: (i) it does not require assumptions on the distribution of the unit-specific effects; (ii) it allows them to depend on observed explanatory variables; (iii) it can be used when the dependent variable is binary or ordinal; and (iv) it can be easily implemented using existing software for conditional likelihood inference in panel data models. 1 The test can also be viewed as a specification test against time-varying omitted variables that are possibly correlated with the covariates included in the model. In order to investigate the finite sample properties of the proposed test, we performed a simulation study. The results of this study show that the test performs quite well, with small size distortion and good power properties, especially when n 2000 and T 7 (common sample sizes in economics) and when the dependent variable is ordinal. The remainder of this article is organized as follows. Section 2 presents the statistical framework and the proposed test. Section 3 investigates the small sample properties of the test by simulation, while Section 4 provides an illustration based on a health economics application. Finally, Section 5 offers some conclusions. 2 The proposed test We consider a general model based on a latent continuous random variable yit for the ith unit at time t. In particular, we assume that y it = G(yit), (1) where G(.) is a parametric function which may depend on specific parameters according to the nature of y it, and yit = α it + x itβ + ε it, i = 1,..., n, t = 1,..., T, (2) where x it is a vector of c covariates, β is a parameter vector, α it represents unobservable individual time-varying characteristics, and ε it is a random error. Without loss of generality, we will focus 1 The specification test described in this article has been implemented in a series of R and Stata functions which are available from the authors upon request. 2

4 our attention on the case of binary and ordinal outcomes assuming that y it is tied to the latent variable y it through the following G(.) function y it = G(y it) = J 1 j=0 j 1{τ j y it < τ j+1 }, (3) where J is the number of response categories and = τ 0 < τ 1 < < τ J 1 < τ J = is a set of thresholds. When J > 2, y it is an ordinal outcome variable while, when J = 2, y it becomes a binary 0-1 indicator and model (1) collapses to a simple binary outcome model. The null hypothesis of interest is that the effect of unobserved heterogeneity, accounted for by the parameters α it, is in fact constant over time, that is, H 0 : α it = α i, i = 1,..., n, t = 1,..., T, (4) whereas the alternative hypothesis (H 1 ) is that unobserved heterogeneity is time-varying, without a priori assumptions on how it evolves over time. In order to test H 0, we propose a procedure based on the comparison of two asymptotically normal estimators which are both consistent under the null, but diverge under the alternative. In the following, we provide some details on these estimators, introduce the test statistic based on them and study its null asymptotic distribution. When J = 2 (case of binary outcomes), a consistent estimator of β under the null hypothesis of time-invariant heterogeneity may be obtained from the Condition Maximum Likelihood (CML) approach proposed by Andersen (1970) and Chamberlain (1980). This approach assumes that, under the null and conditionally on α i and x it, the errors ε it are independently and identically distributed (IID) with a standard logistic distribution. The resulting estimator is based on the maximization of the following conditional log-likelihood function n l 1 (β) = l 1i (β), l 1i (β) = log p(y i y i+ ), i=1 where y i = (y i1,..., y it ) is the sequence of responses for unit i, y i+ = t y it is a sufficient statistic for the time-invariant individual effect α i, and p(y i y i+ ) is the conditional probability of the observed sequence of responses given the sufficient statistic. More explicitly, we have t p(y i y i+ ) = exp(y itx it β) z:z + =y i+ t exp(z tx (5) itβ), where z = (z 1,..., z T ) denotes a vector of zeros and ones of the same dimension as y i, and the sum at the denominator is over all the vectors z whose elements sum up to y i+. 3

5 When J > 2 (case of ordinal outcomes), a number of estimators have been proposed in the literature. Chamberlain (1980) proposes to reduce the ordered model to a binary one by dichotomizing the ordinal outcome at a specific cut-off point j. However, the resulting estimator, say β j, does not exploit all the variation in the response, as units for which either y it < j or y it j for every t do not contribute to the log-likelihood. Following Chamberlain (1980) idea, Das & van Soest (1999), Ferrer-i-Carbonell & Frijters (2004) and Baetschmann et al. (2011) propose different strategies to exploit all the information available in the data. In this paper, we follow Baetschmann et al. (2011) by considering all possible dichotomizations y (j) it sample, where y (j) it = 1{y it > j 1}, j = 1,..., J 1. of the ordered outcome y it for each unit in the Under the assumption that the unknown parameter vector is the same for all y (j) it, the quasi-loglikelihood of this restricted CML estimator is l 1 (β) = n J 1 i=1 j=1 log p(y (j) i y (j) i+ ). (6) At this stage, in order to construct our test we need an alternative estimator of the parameter vector β, which is also consistent under the null but has different convergence properties under the alternative. When J = 2, one such estimator may be obtained by maximizing the pairwise version of l 1 (β), that is n l 2 (β) = l 2i (β), T l 2i (β) = log p(y i,t 1, y it y i,t 1 + y it ), i=1 t=2 where the conditional probability is defined as in (5) for each single pair of consecutive observations. This estimator is not ensured to be consistent under the alternative, but it has a different asymptotic bias than the estimator based on l 1 (β). When J > 2, a natural estimator to consider is the maximizer of l 2 (β) = n J 1 T i=1 j=1 t=2 log p(y (j) i,t 1, y(j) it y(j) i,t 1 + y(j) it ). (7) Regardless of the specific nature of the discrete outcome, let ˆβ 1 denote the estimator of β obtained by maximizing l 1 (β) and let ˆβ 2 denote the estimator obtained by maximizing l 2 (β). Under H 0, it may be simply proved that ˆβ 1 p β0 and ˆβ 2 p β0, where β 0 denotes the true value of β, and that n ( ˆβ1 β 0 ˆβ 2 β 0 ) d N(0, W 0 ), 4

6 where the joint variance-covariance matrix W 0 may be consistently estimated by a sandwich formula. More specifically, when J = 2 (binary case) the consistent estimator of W 0 is ( H1 O ) 1 ( S11 S 12 ) ( H1 O ) 1, Ŵ 0 = O H 2 S 21 S 22 O H 2 with H a = 1 n 2 l ai (β) n i=1 ˆβ, a = 1, 2, a β S ab = 1 n l ai (ˆβ a ) l bi (ˆβ b ) n β β, a, b = 1, 2. i=1 A similar expression must be used when J > 2 (ordinal outcomes), which is based on the loglikelihood defined in (6) and (7). On the other hand, if H 0 does not hold, then ˆβ 1 p β1 and ˆβ 2 p β2, with β 1 β 2. This second result follows from the fact that, since l 2 (β) is maximized using only consecutive pairs of observations, ˆβ 1 and ˆβ 2 will have different probability limits in general. The asymptotic results above suggests the following Hausman-like test statistic for H 0 : ˆδ = n(ˆβ 1 ˆβ 2 ) ˆV 1 0 (ˆβ 1 ˆβ 2 ), (8) where ˆV 0 is a consistent estimator of the variance-covariance matrix of n(ˆβ 1 ˆβ 2 ). This estimator is here computed as ˆV 0 = DŴ 0D, where D = (I c, I c ) and I c denotes the identity matrix of size c. The null asymptotic distribution of this test statistic is of χ 2 type with a number of degrees of freedom equal to the number of covariates, that is, ˆδ d χ 2 c. On the basis of this result we can test H 0 in the usual way and we can compute an asymptotic p-value measuring the evidence provided by the sample against this hypothesis. If the variancecovariance matrix ˆV 0 is singular, its inverse may be substituted in (8) with a generalized inverse V 0, and in this case the distribution to use in testing for H 0 is χ 2 c, where c < c is the rank of ˆV 0. It is worth emphasizing that the proposed testing procedure should have good properties even for other choices of the G(.) function in model (1). Our conjecture is motivated by the fact that, for instance in the case of count and polychotomous unordered outcome variables, the CML approach 5

7 may be used to construct two asymptotically normal estimators which are both consistent under the null and are likely to diverge under the alternative. As far as the drawbacks of the test are concerned, we are forced to assume a logistic distribution in order to exploit the CML approach, implying that no time-invariant regressors can be included in the model (the inclusion of time-invariant covariates causes the variance-covariance matrix ˆV 0 to be singular) and within-unit variation in the dependent variable is required for estimation. 2 Moreover, being a pure specification test, our procedure may lack power in some cases (Holly, 1982) such as, for instance, in presence of heteroskedastic errors. 3 Simulation study To assess the the size and power of the proposed test described in Section 2, we perform a Monte Carlo simulation study, which is fully described subsequently. 3.1 Simulation design We consider the following data-generating process (DGP) for the latent variable yit = α it + x it β + ε it, (9) where x it N(0, 1) and every error term ε it is homoskedastic with a standard logistic distribution. As far as the unit-specific parameters α it are concerned, we consider the following two cases 1. a Gaussian stationary AR(1) process with unit variance. More precisely, we assume α it = { vit, t = 1, ρα it 1 + (1 ρ 2 ) 1/2 v it, t = 2,..., T, (10) with v it N(0, 1). We consider a set of plausible values for the autocorrelation coefficient ρ (1, 0.9, 0.8, 0.7, 0.5, 0.25). 3 Notice that ρ = 1 does not represent the random walk case since, given the above formulation of the autoregressive process, if ρ = 1 then α i1 = α i2 = = α it and the unobserved heterogeneity becomes time-invariant (H 0 ); 2 Even if this is the standard practice, we have to highlight also that unit-specific effects enter additively in the model. 3 It is worth noting that for an AR(1) process, the value of the ρ coefficient equals the first-order autocorrelation. 6

8 2. a three states first-order Homogeneous Markov Chain (HMC) with mean equal to 1.5, unit variance, initial probabilities equal to 1/3 and transition probabilities matrix 1 π π/2 π/2 Π = π/2 1 π π/2. (11) π/2 π/2 1 π where the parameter π = (0, , 0.135, , ) is chosen so that the empirical first-order autocorrelation of the discrete process reflects that of the continuous AR(1) process at point 1. As above, if π = 0, then α i1 = α i2 = = α it and the unit-specific parameters become time-invariant (H 0 ). Furthermore, when the unit-specific parameters follow a Gaussian stationary AR(1) process, we study the properties of the test when the error terms in equation (12) are heteroskedastic. More precisely, we consider the following case y it = α it + x it β + σ it ε it, (12) where σ it = exp(z it δ) is a vector of observation-specific scale parameters for the distribution of every error term ε it. In particular, we set z it N(0, 0.25) and δ = 0.5. Given the aforementioned design, we consider both binary and ordinal outcomes using specific thresholds in equation (3). The former is obtained setting τ 1 = 0; while an ordinal outcome with J = 5 categories is generated setting τ 1 = 2, τ 2 = 0.75, τ 3 = 0.75, τ 4 = 2. We investigate the effect of different sample size n (1000, 2000, 4000) and panel length T (5, 7, 10). In each experiment we fix β = 1 in the linear model for the latent variable. Size at 5% level and power of the test were computed using 1000 replications per experiment. 3.2 Results Tables from 1 to 4 summarize simulation results by reporting the average and the standard deviation of the test statistic (over replications) as well as the power of the test. The first panel of each table reports the size of the test when the unobserved heterogeneity is time-invariant, i.e. ρ = 1 or π = 0. Overall, the test shows very small size distortions, regardless of sample size, panel length and nature of the outcome and unobserved heterogeneity, although the test tend to be a bit oversized in presence of heteroskedastic errors, especially in the ordinal case. The remaining panels of each table report the power of the test for different values of ρ and π. We start our discussion from Table 1, where the outcome is binary, the unit-specific effects follow 7

9 a Gaussian AR(1) process and the errors are homoskedastic. The results for the power of the test can be summarized as follows. First, the power and the autocorrelation coefficient seem to be linked by an inverse U-shaped relationship. Indeed, for given values of n and T, the power improves when ρ departs from one but starts to decrease when ρ drops below 0.5. This behavior is coherent with the fact that, as ρ approaches one, both the standard and the pairwise estimators are consistent, so their difference is small and the test may have poor power. On the other hand, when ρ approaches zero, α it + ε it degenerates towards an iid sequence (which is obtained if ρ = 0). In this case, the standard estimator is inconsistent but also the robustness of the pairwise estimator begins to falter and, once again, their difference gets smaller. Second, the power rapidly increases as n and T increase, exceeding 80% when n 4000 and T 10, regardless of the value of ρ. The same qualitative behavior, but with a higher power, is observed when the dependent variable is ordinal (Table 2). This result follows from the fact that in the ordinal case more information is exploited in the estimation process. The presence of heteroskedasticity does not seem to significantly reduce the power of the test. As shown in Tables 3 and 4, the test still shows good power properties in both binary and ordinal cases. This result might be related to the fact that heteroskedasticity adversely affects both the standard and the pairwise estimators in a similar way, leaving their difference almost unchanged. Finally, as shown in Tables 5 and 6, the performance of the test is very similar when the process representing the evolution of the unobserved heterogeneity is discrete rather than continuous. The same inverse U-shaped relationship is also observed between the power of the test and the parameter that indexes the first-order HMC transition probabilities matrix, i.e. π. 4 Empirical example In this section, we illustrate our procedure through an empirical application on SRHS of the elderly American population. 4.1 Data Our data are from the Health and Retirement Study (HRS), a longitudinal panel study that surveys a representative sample of more than 26,000 Americans over the age of 50 every two years. We 8

10 employ the RAND HRS Data File (Version L), a user-friendly version of the data produced by the RAND Center for the Study of Aging, which contains all waves from 1992 to After selecting respondents aged 50 and older in the first wave, our sample consists of a balanced panel of 4,094 individuals (a total of 40,940 observations). The outcome variable is the (5-categories) SRHS that, as in other surveys, is measured on a five-point ordered scale (poor, fair, good, very good, excellent). As covariates, we consider a typical set of socio-demographic characteristics (gender, age, education and ethnicity), the number of doctor visits and the body mass Index (BMI). Definitions and summary statistics for these variables are presented in Table 7. 4 Following Heiss (2008), we estimate an ordered logit model for SRHS in wave 10 where the covariates consist of a typical set of socio-demographics plus lagged values of SRHS (Table 8). This simple exercise gives an idea of the SRHS correlation pattern over a longer period of time highlighting two interesting results: i) coefficients of most lags are highly significantly different from zero, and ii) they get smaller the further away the respective observation is from wave 10. These stylized facts suggest that the most plausible model for this application is one in which SRHS depends on the true health status and this unobserved variable follows some random process over time with decreasing correlation Results We consider two model specifications: M1) age splines, BMI, number of GP visits and M2) M1 + wave dummies. Table 9 reports the two set of estimates used to construct the test statistic: the top panel shows the standard CML fixed-effects ordered logit estimates for the two model specifications, while the bottom panel reports the corresponding pairwise estimates. The key point here is that the two set of estimates diverge producing a test statistic of and , respectively. Given the corresponding degrees of freedom (6 and 15), we strongly reject the null hypothesis of time-invariant unobserved heterogeneity confirming Heiss (2008) results even in a longer panel. 4 Notice that, due to a failure in the AR(1) ordered logit model convergence, we are forced to drop outliers in the distribution of BMI and doctor visits. We use the method of percentiles. Since it does not seem to be outliers in the left tail, we drop out only values > 99.9 percentile, losing 37 individuals (370 observations). 5 See Heiss (2008) for a detailed discussion. An alternative estimator to Heiss (2008) approach would be a random effects model with state dependence. However, state-dependence of SRHS is not very convincing from a theoretical point of view. While for example in a model of female labor force participation, lagged outcomes can causally affect current outcome (Hyslop, 1999), this causality is not so clear in a SRHS model: why a simple perception of health should affect the future true health status? 9

11 Since H 0 has been rejected, we estimate the latent AR(1) ordered logit model proposed by Heiss (2008) in order to confirm the presence of a lower than one statistically significant autoregressive coefficient. 6 Table 10 shows the estimates for the two model specifications. Being a random effects model, we also include the same time-invariant socio-demographic covariates as in Heiss (2008) application. It is worth emphasizing that the lack of these covariates in performing our test (Table 9) does not affect its power since, being time-invariant, they are conditioned out from the likelihood function as fixed-effects. The estimated ρ = 0.95 and appears to be highly statistically significant (basically the same value found by Heiss (2008)). Hence, a more plausible model for the data is one where SRHS depends on true unobservable health, and this latent variable follows a time-series process with decaying autocorrelation. 5 Concluding remarks We propose a specification test for the null hypothesis of time-invariant unobserved heterogeneity in nonlinear panel data models against the alternative of time-varying heterogeneity. Our test is based on a comparison between standard and pairwise conditional likelihood estimators and can be considered as a pure specification test because it leaves the alternative deliberately vague. The finite-sample properties of the test are investigated via a set of Monte Carlo experiments. The results suggest that the test generally performs well and shows small size distortions and good power properties regardless of how the unit-specific effects evolve over time, especially for ordinal outcomes and when N 2000 and T 7. Moreover, our simulation results suggest that the adverse effect of heteroskedasticity on both size and power of the test is negligible. It is worth emphasizing that our test does not require to formulate assumptions on the distribution of unobserved heterogeneity, and it can be easily implemented using standard software for CML estimation. On the other hand, the use of the CML approach implies that no time-invariant regressors can be included in the model and within-unit variation in the dependent variable is required for estimation. We provide an empirical illustration using data from the HRS. We estimate the same model as Heiss (2008) using a longer balanced panel. The hypothesis of time-invariant unobserved hetero- 6 The estimation is performed using the arldv Stata package produced by Florian Heiss. The likelihood function of this model does not have a closed-form solution, so estimation involves numerical integration. We use 50 integration points. 10

12 geneity is rejected, thus confirming Heiss (2008) results: a more plausible model for the selected sample is one in which SRHS depends on true unobservable health, and this latent variable follows an autoregressive process with decreasing autocorrelation. In future work, we plan to extend our approach to the cases of count and polychotomous unordered outcome variables, for which we conjecture that the proposed test should provide equally good results. Our conjecture is driven by the fact that, even in these cases, the CML approach should guarantee to obtain two asymptotically normal estimators which are both consistent under the null and are likely to diverge under the alternative. 11

13 References Andersen, E. (1970). Asymptotic properties of conditional maximum likelihood estimators. Journal of the Royal Statistical Society Series B, 32, Arellano, M., & Bonhomme, S. (2011). Nonlinear panel data analysis. Annual Review of Economics, 3. Baetschmann, G., Staub, K., & Winkelmann, R. (2011). Consistent Estimation of the Fixed Effects Ordered Logit Model. IZA Discussion Papers 5443 Institute for the Study of Labor (IZA). Bartolucci, F., & Farcomeni, A. (2009). A multivariate extension of the dynamic logit model for longitudinal data based on a latent markov heterogeneity structure. Journal of the American Statistical Association, 104, Bartolucci, F., Silvia, B., & Pennoni, F. (2012). Mixture latent autoregressive models for longitudinal data. arxiv: v1,. Chamberlain, G. (1980). Analysis of covariance with qualitative data. Review of Economic Studies, 47, Das, M., & van Soest, A. (1999). A panel data model for subjective information on household income growth. Journal of Economic Behavior and Organization, 40, Ferrer-i-Carbonell, A., & Frijters, P. (2004). How important is methodology for the estimates of the determinants of happiness. Economic Journal, 114, Heiss, F. (2008). Sequential numerical integration in nonlinear state space models for microeconometric panel data. Journal of Applied Econometrics, 23, Holly, A. (1982). A remark on hausman s specification test. Econometrica, 50, Hsiao, C. (2005). Analysis of Panel Data. New York: Cambridge University Press. Hyslop, D. R. (1999). State dependence, serial correlation and heterogeneity in intertemporal labor force participation of married women. Econometrica, 67,

14 Stowasser, T., Heiss, F., McFadden, D., & Winter, J. (2011). Healthy, Wealthy and Wise? Revisited: An Analysis of the Causal Pathways from Socio-economic Status to Health. NBER Working Papers National Bureau of Economic Research, Inc. Wooldridge, J. M. (2010). Econometric Analysis of Cross Section and Panel Data. MIT press. 13

15 Table 1: Binary outcome, AR(1) unit-specific effects and homoskedastic errors T=5 T=7 T=10 N mean sd power mean sd power mean sd power ρ = ρ = ρ = ρ = ρ = ρ = Table 2: Ordinal outcome, AR(1) unit-specific effects and homoskedastic errors T=5 T=7 T=10 N mean sd power mean sd power mean sd power ρ = ρ = ρ = ρ = ρ = ρ =

16 Table 3: Binary outcome, AR(1) unit-specific effects and heteroskedastic errors T=5 T=7 T=10 N mean sd power mean sd power mean sd power ρ = ρ = ρ = ρ = ρ = Table 4: Ordinal outcome, AR(1) unit-specific effects and heteroskedastic errors T=5 T=7 T=10 N mean sd power mean sd power mean sd power ρ = ρ = ρ = ρ = ρ =

17 Table 5: Binary outcome, first-order HMC unit-specific effects and homoskedastic errors T=5 T=7 T=10 N mean sd power mean sd power mean sd power π = π = π = π = π = Table 6: Ordinal outcome, first-order HMC unit-specific effects and homoskedastic errors T=5 T=7 T=10 N mean sd power mean sd power mean sd power π = π = π = π = π =

18 Table 7: Summary statistics n = 4094; T = 10. Variable Description Mean Std. Dev. Min. Max. Self-rated Health (1 = poor, 2 = fair, 3 = good, 4 = very good, 5 = excellent) Age Age of the respondent (year) Female Dummy for female High school Dummy for high school (raeduc = 3) Some college Dummy for college degree (raeduc = 4) College degree+ Dummy for higher education (raeduc = 5) Non white Dummy for hispanic and black BMI Body mass index GP visits Number of GP visits (prv two years)

19 Table 8: Ordered Logit of SRHS in wave 10 on past SRHS SRHS wave 10 Age *** Female High school Some college College degree Non white SRHS wave *** SRHS wave *** SRHS wave *** SRHS wave *** SRHS wave ** SRHS wave SRHS wave SRHS wave ** SRHS wave cut-off ** cut-off *** cut-off *** cut-off *** Obs 4,094 Log-lik -4, Significance levels: * p < 10%; ** p < 5%, *** p < 1% 18

20 Table 9: Test implementation for both model specifications. S1 S2 Standard Age splines: *** Age splines: *** *** Age splines: *** *** Age splines: BMI *** *** GP visits *** *** Pairwise Age splines: *** * Age splines: *** Age splines: * *** Age splines: BMI ** ** GP visits *** *** Wave dummies No Yes H 0 = time-invariant individual effects Test statistic P-value Significance levels: * p < 10%; ** p < 5%, *** p < 1% 19

21 Table 10: AR(1) random effects ordered logit model (Heiss, 2008). S1 S2 Age splines: *** *** Age splines: *** *** Age splines: *** *** Age splines: BMI *** *** GP visits *** *** Female ** High school *** *** Some college *** *** College degree *** *** Non white *** *** Wave dummies No Yes σ *** *** ρ *** *** Log-lik Significance levels: * p < 10%; ** p < 5%, *** p < 1% 20