Pooling Space and Time

Transcription

1 Pooling Space and Time Jamie Monogan University of Georgia March 21, 2012 Jamie Monogan (UGA) Pooling Space and Time March 21, / 47

2 Objectives By the end of this meeting participants should be able to: Diagnose potential problems with panel data by studying residuals. Evaluate the trade-offs between fixed effects and random effects models. Estimate fixed effects and random effects models. Correct for serial correlation of error in panel data analysis with either GLS-ARMA or OLS with a lagged dependent variable. Correct for contemporaneous correlation in standard errors of estimates. Jamie Monogan (UGA) Pooling Space and Time March 21, / 47

3 Introduction Getting a feel for panel data Focus thus far: a single series (data corrections, method corrections). What if we have multiple series? This week: the econometric perspective, principally for long panels. Alternatively: the multilevel modeling approach, principally for short panels. Long v. wide form. More time points or cross-sectional observations? Recommended Reading Beck, Nathaniel Time-Series Cross-Section Data: What Have We Learned in the Past Few Years? Annual Review of Political Science 4: Jamie Monogan (UGA) Pooling Space and Time March 21, / 47

4 Introduction Suggested Comprehensive Exam Question What is the best estimator for pooled data? Suggested answer: The author of this question doesn t understand that the choice of estimator must be tailored to knowledge of the particular error structure problems of the current analysis. Of course, you will probably fail if you impugn the intelligence of an exam author. Introduction and Point of View Many reported pooled regressions are wrong. They are wrong because they have a problem, say a, which produces biased and inconsistent estimates and they employ an estimator which is designed to deal with a totally unrelated problem b. As in: Have a headache? Try chemotherapy. Got a case of arthritis, take Ambien. Unit effect bias? Use panel corrected standard errors. Jamie Monogan (UGA) Pooling Space and Time March 21, / 47

5 Introduction A Guiding Principle Error assumption problems are proportional to the size of errors. A well fit model can tolerate error assumption violations and produce good estimates. (Observational wisdom: second digit of s.e.) A specification where most of the variance remains in the error term will be highly sensitive to error assumptions. The Implication for Procedure Spend your time getting the theory and model specification right, not in worrying about the error term. Do what you can (e.g., dummies, where needed) to move variance into the structural part of the model. And only after you have done your best on structure should you begin to think about modeling error. Jamie Monogan (UGA) Pooling Space and Time March 21, / 47

6 Introduction A Point of View on Theory Instead of the dichotomy between theory and empirical knowledge of error, assume three components: 1 Genuine a priori theory (not tainted by encounter with data). 2 Knowledge about context and data that you bring to your research, call it quasi-theory (i.e., unit effects). 3 Error: that which is totally inexplicable. Advice My procedural advice amounts to saying maximize the variance accounted for by 1 and 2 before concerning yourself with 3. It is not smart to fool yourself or your reader that (2) is actually an explanation. But it is a lot less smart to leave that variance in the error term, where it will surely cause mischief. Jamie Monogan (UGA) Pooling Space and Time March 21, / 47

7 Introduction Our Theoretical Prior Our units are homogeneous in the sense that the same causal theory applies to all. (Though in multilevel models you may have random coefficients.) That does not require that our y have the same mean level across units. That issue, to be called unit effects, can be modeled. Why unit effects? Everything that happened to the units before the period of our observation history is likely to have altered mean levels and it is not appropriate to include history in our theory and model. Jamie Monogan (UGA) Pooling Space and Time March 21, / 47

8 A Typology Introduction Cross Sectionally Time Serially Notes & Issues Variable Variable Jointly Variable Yes Yes No problems Unit Constants Yes No Perfect collinearity with unit dummies Time Constants No Yes Non-problematic in typical designs Quasi-Unit Constants Quasi-Time Constants Yes Trivial relative to unit variance Trivial relative to time variance Yes Common. Potential for serious bias with unmodelled unit effects. Relatively uncommon Jamie Monogan (UGA) Pooling Space and Time March 21, / 47

9 Substantive Example Substantive Example Cite DeBoef, Suzanna and James A. Stimson The Dynamic Structure of Congressional Elections. Journal of Politics 57: Structural Model Dependent variable: Democratic percent of two-party House vote, (dempct) Public Policy Mood, (mood) National Macropartisanship, time-wise, (dmp) State Partisanship (WEM), x-sectional, (d2p) Midterm dummy, (midterm) Net incumbency advantage, % Democratic incumbents - % Republican incumbents, (inc) South Dummy, (south) Jamie Monogan (UGA) Pooling Space and Time March 21, / 47

10 Substantive Example What are threats to inference? Do we have probable unit effects here? ANOVA of the outcome by state: F = 19.98, p < Thus we probably would consider fixed effects, with 47 state dummy variable estimates. But if we do, then we can t model state-level constants such as state partisanship. And that is a lot of extra parameters. Jamie Monogan (UGA) Pooling Space and Time March 21, / 47

11 Substantive Example Two Conceptions of Causality in One Model? Cross-Sectional Causation: Coleman When we say X causes Y, we mean that at some time before the time of observation some dynamic process occurred which aligned Y with X and furthermore, that process had reached an equilibrium level when we observed x and y. Time series: Granger When we say that x causes y, we mean that E(y y t k, x t k ) differs from E(y y t k ) Jamie Monogan (UGA) Pooling Space and Time March 21, / 47

12 Substantive Example Panel Data Organization: It Matters for Matrix Structure Stacked Pooled Data Case Ordering (AKA Long Form ) unit 1: time 1 unit 1: time 2 unit 1: time 3 unit 1: time 4... unit 1: time T unit 2: time 1 unit 2: time 2... Jamie Monogan (UGA) Pooling Space and Time March 21, / 47

13 Substantive Example Informing Software of the Panel Structure R For many commands (lm, lme), R does not get bent out of shape. For others (plm), there are options to specify in the command, e.g.: index=c( state, year ) Stata Stata has very powerful pooled data capabilities, none of which work, unless your file is tsset for panel data. The syntax is: tsset unitvar timevar (and order matters) E.g., tsset state year Once you have tsset it, Stata respects the stacked structure of your data (e.g. it won t resort) and opens up all the xt commands, e.g., xtreg, xtgls, xtpcse,xtregar Do help xt Jamie Monogan (UGA) Pooling Space and Time March 21, / 47

14 OLS Matrix Structure OLS for Panel Data N is number of units. T is number of time points for each. y = xβ + u, where: y is a (column) vector of length N*T, x is a matrix N*T rows and k+1 columns, β is a (row) vector of length k+1, and u is a (column) vector of disturbances, N*T. Jamie Monogan (UGA) Pooling Space and Time March 21, / 47

15 OLS Matrix Structure I (Identity) Restated as Partitioned Matrix I = P O O... O O P O... O O O P... O O O O... P Here, the diagonal P s are identity partitions, T T, and the off-diagonal O s are T T partitions of zeros. The P s and O s are on the following slide. Jamie Monogan (UGA) Pooling Space and Time March 21, / 47

16 OLS Matrix Structure Matrix Partitions for OLS Each P is: No time-wise autocorrelation Each O is: No cross-unit (spatial) autocorrelation. Jamie Monogan (UGA) Pooling Space and Time March 21, / 47

17 Assumption Violations and Models That Deal With Them Customary Problems of Pooled Data Unit effects, i.e., each unit requires an offset, C i, so that E(y) is the same across units. Autocorrelation in the time dimension of residuals. Heteroscedasticity across units. Contemporaneous correlation: errors at time t are correlated across units. Spatial autocorrelation: geographically proximate units have correlated errors. (Spring 2013: Spatial Data Analysis.) Jamie Monogan (UGA) Pooling Space and Time March 21, / 47

18 Assumption Violations and Models That Deal With Them The expected effects for each are: Unit effects: β is biased and inconsistent. Autocorrelation: β is inefficient (with biased standard errors) Heteroscedasticity: β is inefficient (with biased standard errors). Spatial Autocorrelation: β is inefficient (with biased standard errors). Contemporaneous Correlation: β is inefficient (with biased standard errors). Jamie Monogan (UGA) Pooling Space and Time March 21, / 47

19 Assumption Violations and Models That Deal With Them Observing Error Problems with R Perform OLS: mod.name < lm(y x1+x2+ +xk) and capture residuals (mod.name$residuals) Unit effects: Are the means of y the same across units? anova.mod < lm(y as.factor(unitvar), data=dataset) anova(anova.mod) Are residual means the same across units? anova.resids < lm(main.model$residuals as.factor(dataset$unitvar)) anova(anova.resids) Unexplained variance attributable to units: ercomp(re.mod) Autocorrelation: pbgtest (lmtest package) Heteroscedasticity: by(data=main.model$residuals, INDICES=dataset$factor,FUN=sd) Jamie Monogan (UGA) Pooling Space and Time March 21, / 47

20 Assumption Violations and Models That Deal With Them Observing Error Problems with Stata Perform OLS: reg y x1-xk and capture residuals (predict res,r) Unit effects: anova y unit, (are y means the same across units?) or anova res unit, (are residual means same across units? i.e., are they 0?) perform xtreg and observe rho and F test on between variance. Rho (ρ) here is the proportion of all residual variance attributable to units (Between). F is the significance of rho. Autocorrelation: Stata is no help Heteroscedasticity: ta unit,s(res) Jamie Monogan (UGA) Pooling Space and Time March 21, / 47

21 Assumption Violations and Models That Deal With Them Assume Unit Effects Thus, the true model is often y it = β 0 + k β j x jit + C i + u it j=1 where C i is the effect of being in unit i at all times. In Matrix Notation: y = xβ + Zγ + u, where Z is a matrix of unit dummy variables. But with OLS we fit: y it = β 0 + which clearly is underspecified. k β j x jit + u it j=1 Jamie Monogan (UGA) Pooling Space and Time March 21, / 47

22 Assumption Violations and Models That Deal With Them The Consequence of Ignoring Unit Effects If unit effects exist, Then OLS estimates will be biased and inconsistent This is a potentially serious bias, quite likely to produce false inferences. Why? If C i is not estimated, then unit variables which are accidentally correlated with it will proxy the effect. This will produce false inferences that unit variables explain between unit differences that are actually due to history. Jamie Monogan (UGA) Pooling Space and Time March 21, / 47

23 Assumption Violations and Models That Deal With Them Solutions to Unit Effects 1 Fixed effects (i.e., unit dummy variables). 2 Random effects (GLS). 3 Lagged dependent variable. 4 Recentering approaches. 5 (list non-exhaustive) Jamie Monogan (UGA) Pooling Space and Time March 21, / 47

24 Assumption Violations and Models That Deal With Them Fixed Effects Least Squares Dummy Variables (LSDV) Fixed Effects y it = k βx jit + γ i + u it j=1 R command: fe.mod < plm(y x, data=dataset, index=c( unitvar, timevar ), model= within ) Stata command: xtreg depvar varlist,fe fe implies fixed effects. Stata estimates, but does not report, the unit constants, γ i. Jamie Monogan (UGA) Pooling Space and Time March 21, / 47

25 Assumption Violations and Models That Deal With Them Fixed Effects Properties of Fixed Effects Estimator The fixed effects estimator solves the unit effects problem and produces unbiased β. But two issues: 1 If the number of units is large, the estimation of all those parameters is inefficient. 2 Unit dummies are perfectly collinear with unit constants (and highly collinear with quasi-unit constants), making it impossible to estimate a model with both f.e. and unit constants. Jamie Monogan (UGA) Pooling Space and Time March 21, / 47

26 Assumption Violations and Models That Deal With Them Random Effects Error Components, aka GLSE, aka Random Effects For pooled data we can apportion error variance sources as σ 2 = σ 2 units (+σ2 time ) + σ2 within That is, some proportion of the total error is due to differences between units, some proportion to differences between times, and some is pure error, due to neither units or time. or, dropping time σ 2 = σ 2 units + σ2 within Jamie Monogan (UGA) Pooling Space and Time March 21, / 47

27 Assumption Violations and Models That Deal With Them Random Effects The Logic of Random Effects In the fixed effects approach we estimate parameters for fixed effects between units and thereby remove that variance from the error term. In random effects (GLSE, error components, etc.) we leave the between unit variance in the error term and change our assumptions to reflect the knowledge that between unit variance is in the error term. Note: These terms have different meaning in other contexts. That is, we specify a (particular) GLS model which assumes between unit variance in the errors. The empirical key to this specification is estimating ρ, conceptually the proportion of all error that is due to unit effects. Jamie Monogan (UGA) Pooling Space and Time March 21, / 47

28 Assumption Violations and Models That Deal With Them Random Effects Estimating ρ ρ = σ2 Between σ 2 Total ˆρ = (OLS Residual Variance) - (LSDV Residual Variance) (OLS Residual Variance) Many ways to estimate ρ in the r.e. model: Maddala & Mount (1973): Monte Carlo analyses with 11 different estimators. Baltagi (1981, 2005): Considers additional options. These analyses suggest that the choice between estimators essentially doesn t matter. Hence, software defaults are usually fine. More recent analysis by Clark & Linzer suggests that the choice of estimator does matter. (E.g., plm [R] yields different results from lmer [R] and BUGS.) Jamie Monogan (UGA) Pooling Space and Time March 21, / 47

29 Assumption Violations and Models That Deal With Them The GLSE Normal Equations Random Effects ˆβ GLS = [x Ω 1 x] 1 x Ω 1 y where Ω = P O O... O O P O... O O O P... O O O O... P Jamie Monogan (UGA) Pooling Space and Time March 21, / 47

30 Assumption Violations and Models That Deal With Them Each Partition, P, is: Random Effects P = 1 ρ ρ... ρ ρ 1 ρ... ρ ρ ρ 1... ρ ρ ρ ρ... 1 Notice that ρ is constant at all lags. There is no decay at longer lags. That is because we are controlling for a cross sectional effect. ρ is autocorrelation, but it is not the time serial variety. Jamie Monogan (UGA) Pooling Space and Time March 21, / 47

31 Assumption Violations and Models That Deal With Them Random Effects Estimation of GLSE R re.mod < plm(y x, data=dataset, index=c( unitvar, timevar ),model= random ) Stata xtreg depvar varlist,re re (random effects) is the default and doesn t need to be specified, but it is useful to remind yourself what model you are estimating. Jamie Monogan (UGA) Pooling Space and Time March 21, / 47

32 Assumption Violations and Models That Deal With Them Random Effects Properties of Random effects Estimator Does not deal with correlated error of various kinds or with heteroscedasticity. But these error assumption violations produce unbiased but inefficient β and are generally much less serious in their consequences than are unit effects. Produces BLUE estimates if the problem of unit effects is the only assumption violation and the unit effects do not correlate with the covariates. Running a random effects model? A Hausman test is a common tool for evaluating model specification. In R: phtest from the plm library. Jamie Monogan (UGA) Pooling Space and Time March 21, / 47

33 Dealing With ARMA Error Structures Modeling Priorities Our prior consideration of the structural underspecification issue (unit effects) was the most serious threat to valid inference. If that issue is resolved, then it makes sense to be concerned about correlated error issues, which are a cause of inefficiency, but not bias to β. We have two such issues: 1 Time-serial autocorrelation of error, and 2 contemporaneous correlation. Jamie Monogan (UGA) Pooling Space and Time March 21, / 47

34 Dealing With ARMA Error Structures GLS-ARMA Also known as the Kmenta model for heteroscedastic and AR(1) errors ˆβ GLS = [x Ω 1 x] 1 x Ω 1 y where Ω = σ 2 1 P O O... O O σ 2 2 P O... O O O σ 2 3 P... O O O O... σ 2 N P The σ 2 terms on the main diagonal now represent the assumption that error variances differ between units. Jamie Monogan (UGA) Pooling Space and Time March 21, / 47

35 Dealing With ARMA Error Structures GLS-ARMA Matrix Partition P Specification The AR(1) Case P = 1 ρ ρ 2... ρ T 1 ρ 1 ρ... ρ T 2 ρ 2 ρ 1... ρ T ρ T 1 ρ T 2 ρ T Estimation Issues Autocorrelation may be estimated (a) from OLS residuals (i.e, feasible GLS, not true GLS) or (b) from an interative estimator such as Cochrane-Orcutt or Prais-Winston. Heteroscedastic variance estimates can come directly from computed residual variances by unit. Jamie Monogan (UGA) Pooling Space and Time March 21, / 47

36 Notation: Dealing With ARMA Error Structures Kronecker product, A B, of A (m n) and B (p q). A B is of order mp nq: A B = a 11 B a 12 B... a 1n B a 21 B a 22 B... a 2n B.... a m1 B a m2 B... a mn B Kronecker Product The Kronecker product notation is useful in the pooled case because it captures the idea that our Ω is the product of assumptions about cross sections and time series. Thus Ω = Σ P represents Ω as a product of Σ the cross-sectional variance/covariance and P the time-serial autocorrelation. Jamie Monogan (UGA) Pooling Space and Time March 21, / 47

37 Dealing With ARMA Error Structures Example: GLS-ARMA (Kmenta) Ω = Σ P for: P = Σ = σ 2 1 O O... O O σ 2 2 O... O O O σ O O O O... σ 2 N 1 ρ ρ 2... ρ T 1 ρ 1 ρ... ρ T 2 ρ 2 ρ 1... ρ T ρ T 1 ρ T 2 ρ T Jamie Monogan (UGA) Pooling Space and Time March 21, / 47

38 Dealing With ARMA Error Structures R estimation of GLS-ARMA Accounding for first-order serial correlation, option in lme or gls : correlation = corar1(0, form = time state) Two-stage estimator with a unique panel heteroscedasticity and autocorrelation structure: pggls Panelwise heteroscedasticity? Hard to combine with other panel features. Jamie Monogan (UGA) Pooling Space and Time March 21, / 47

39 Dealing With ARMA Error Structures Stata estimation of GLS-ARMA xtgls depvar varlist (no autocorrelation, no heteroscedasticity) xtgls depvar varlist,p(h) (heteroscedastic, no autocorrelation) xtgls depvar varlist,corr(ar1) (homoscedastic, AR(1) error) xtgls depvar varlist,corr(ar1) p(h) (heteroscedastic and autocorrelated) Stata Xtgls Options Panels (abbreviated p) p(i) iid (homoscedastic) p(h) heteroscedastic, σi 2 σj 2 Corr (abbreviated c, for autocorrelation) c(i) no autocorrelation c(ar1) AR(1) with single ρ estimate for all panels c(psar1) Panel specific ρ estimates (very complex model) Jamie Monogan (UGA) Pooling Space and Time March 21, / 47

40 Dealing With ARMA Error Structures When to Use GLS-ARMA? Like the PCSE model to come, GLS-ARMA deals with relatively small problems, inefficiency and modest bias to standard errors. It should be considered only when more serious issues are known to be under control. Particularly if the key issue is autocorrelated errors, consider instead the dynamic (Koyck) setup which controls both history and time-wise autocorrelation. That is, as Beck and Katz argue, a lagged dependent variable with OLS may often be the best estimator. Jamie Monogan (UGA) Pooling Space and Time March 21, / 47

41 Beck & Katz Perspective Beck and Katz Perspective on Dynamics We have a choice between static formulations, which then have expected unit effects and autocorrelation in the time dimension, or dynamic specification, where y t 1 on the right hand side will usually control history (hence unit effects) and eliminate autocorrelated error. Static requires some feasible GLS approach to deal with error assumptions and, because it is static, may be an inferior model of (inherently dynamic) causality. Dynamic is biased and inconsistent, but is probably the right structure and can then be estimated with OLS. Conclusion: Dynamic specification with OLS is better. Note: we do have options on how we dynamically specify a model. One problem remains: contemporaneous correlation. This was a possibility with all the other models too, but was so far down the list of problems that most analysts chose not to deal with it. Jamie Monogan (UGA) Pooling Space and Time March 21, / 47

42 Beck and Katz Panel Corrected Standard Errors Contemporaneous Correlation Example Contemporaneous (error) correlation arises from some time-related process which affects most units at the same time similarly And is not part of the model structure. In a state analysis where some economic indicator (e.g., budget shares) is dependent, the national economy is perturbing most states to have similar errors at particular times. You could model this effect (if you have theory) You could dummy time points (which might be sensible if you had many states and few time points). But if you did neither, you would have contemporaneous correlation of errors across units. Jamie Monogan (UGA) Pooling Space and Time March 21, / 47

43 Beck and Katz Panel Corrected Standard Errors The Impact β estimates will be inefficient. Standard error estimates will be biased downward. Therefore, t tests are overstated and p estimates smaller than they should be. Monte Carlo analysis suggests that these effects are quite small, that even high levels of contemporaneous correlation cause only modest distortions. Jamie Monogan (UGA) Pooling Space and Time March 21, / 47

44 Beck and Katz Panel Corrected Standard Errors Panel Corrected Standard Errors Assume Σ, a N N matrix of cross unit contemporary correlations: T e it e jt t=1 ˆΣ ij = T I T is a T T identity for the time dimension, then ˆΩ = ˆΣ I T where Ω is NT NT and standard errors are taken from the diagonals of: (X X ) 1 X ΩX (X X ) 1 This is the panel correction. Jamie Monogan (UGA) Pooling Space and Time March 21, / 47

45 Beck and Katz Panel Corrected Standard Errors Software R Estimate a linear model Input the model name into the pcse function from the pcse library. Stata xtpcse y varlist Produces OLS β s And panel corrected standard errors. Jamie Monogan (UGA) Pooling Space and Time March 21, / 47

46 Beck and Katz Panel Corrected Standard Errors Responses to Beck and Katz Carole Wilson Monte Carlo Study The fundamental result is that on average PCSE standard errors are not larger than OLS standard errors. And often smaller! Why? Probably because the effect of contemporaneous correlation is too small to observe under typical conditions. Jamie Monogan (UGA) Pooling Space and Time March 21, / 47

47 Homework For Next Time Read: Box-Steffensmeier & Jones Time is of the Essence. American Journal of Political Science. Software Work File panelhw.dta is a (fake) panel of 8 regions in the US for 30 years. It has variables region, year, x, dep1, dep2, dep3 (3 different dependent variables with differing properties). Estimate regressions of dep1 as a function of x and dep2 as a function of x, diagnosing observed error properties and reporting justification for model selection. For unit effect problems entertain both fixed and random effects; compare the results with a Hausman test. Estimate a model for dep3 with a lagged dependent variable and panel-corrected standard errors. Jamie Monogan (UGA) Pooling Space and Time March 21, / 47