PS 271B: Quantitative Methods II Lecture Notes

Transcription

1 PS 271B: Quantitative Methods II Lecture Notes (Part 6: Panel/Longitudinal Data; Multilevel/Mixed Effects models) Langche Zeng

2 Panel/Longitudinal Data; Multilevel Modeling; Mixed effects 2 Multilevel data can have nested or crossed grouping structure, with time series cross sectional data (longitudinal data) as a special case. e.g. Student test data: individual students nested within classes and schools. e.g. Dyad year design in IR. Country-year data in comparative. may have both country/dyad and year specific effects. Observe N units over T time periods. Some make distinction between TSCS data and panel data. Panel data have large N (which are typically random samples) and small, fixed T, asymptotics in N (estimation of covariance structures w.r.t. T gets better as N

3 increases). TSCS data has reasonable sized T and fixed N. We are likely to know more and care more about the units in TSCS data, not panel data (thus fixed effects are more meaningful in TSCS data) and have more info about the time series structure in TSCS data. The structure presents special opportunities and challenges. On one hand, unobserved heterogeneities confounding the key relationship may be dealt with without the use of instrumental variables (fixed effects models); on the other hand, error structure is almost always more complex than cross sectional data. 3 y i,t = x i,t β + α i + ɛ i,t (1) Obs for the same individual are likely correlated overtime. so the N T points likely have less info than N T uncorrelated points.

4 α i represents individual (or unit) specific unmeasured heterogeneity. A fundamental distinction is between fixed effects and random effects. For the issue at this point, better terms should be related random effects or unrelated random effects related or unrelated to X. If fixed (related) effects, then there are unmeasured heterogeneity that can confound the effects of x on y that are not controlled for (omitted variable error), so OLS estimation will be biased. Random (unrelated) effects in contrast are indepedent of X and so can be less harmful if relegated into the error term. But the presence of individual specific α i induces a correlation struction that, if not accounted for, causes inefficiency in OLS estimation. Hausman test can be used to see whether there is fixed effects or just random effects. H 0 is no fixed effects. Estimation of fixed effects model: using unit dummies is possible but estimator is inconsistent in N (the number of parameters increases along with N). Better: demean the data within units and model y it ȳ i ( within effects) or model the first difference y it y i(t 1). the differencing in either method gets rid of α i. for T > 2, whthin estimator is more efficient than the FD 4

5 estimator. 5 Practical concerns: FE can wipe out X s that vary slowly within units (effects of those that don t vary at all over time cann t be estimated.) If α i = α, this would be complete pooling assuming the same parameters for all units. (The other extreme is to assume complete heterogeneity so that analysis must be done unit-by-unit. Something in between is likely more realistic.) If instead of having α i fixed/related to x, we can assume they come from an unrelated distribution, we have random effect model which can be estimated either by GLS (generalized least square; by treating the error structure and making it spherical), or MLE (model the resulting distribution for y), or gee which assumes certain working correlation structure (random intercepts correspond to exchangable or constant within-unit correlation), or Baysian methods.

6 Stata has an extensive collection of commands in the xt family that accommodate various specific assumptions on the error structure (within unit correlation, between unit correlation, heteroscedasticity, etc.) In addition to xtgls and xtgee, xtpcse implements Beck&Katz s correction of OLS error to allow for between unit contemporaneous correlation and groupwise heteroscedasticity (essentially estimating structure of N xn covariance matrix that doesn t change with T.) xtregar is xtreg with ar(1) error. xtgee also works for Y in generalized linear models. (Also see corresponding xt commands for various nonlinear models, such as xtprobit, xtlogit, xttobit, xtpoisson, xtnbreg, etc.) In R, package plm (panel linear model) implements many similar functions as the stata xt family. Zelig has.gee versions for several linear and generalized linear models. If β is also allowed to vary w.r.t i, we have random coefficient model, or mixed effect model. Mixed since β i can be decomposed into its mean, 6

7 which is a fixed effect (note that the meaning of fixed here differs from earlier usage on effects related to X), plus a random part with mean 0. random intercept alone implies the exchangable correlation structure which may not be realistic; random coefficients relax this. General form of the linear mixed effects model: Y = Xβ + W b + ɛ (2) where β is the vector of fixed effects (do not vary w.r.t. i or t), and b is a vector of random effects (can be individual specific or time specific or twoway effects.) X and W may overlap (random coefficients mode is a special case when X = W.) The random intercepts or coefficients can in turn be modeled as functions of independent variables that vary at the higher level (e.g., school level, country level), thus multi-level modeling (or hierarchical modeling). Can show that these can be expressed in the general form of (2), with various interaction terms entering the model. Estimation by MLE, REMLE (restricted 7

8 version, takes care of some bias issue such as degree of freedom corrections) or Bayesian. In R, the lme4 package (see also nlme and plm) provides utilities for modeling mixed effects models for linear, generalized linear and nonlinear models by MLE or REMLE (we ll look at examples). Zelig has for example ls.mixed, logit.mixed. for stata see xtmixed and the gllamm package. Assuming Y b N(Xβ + W b, Σ ɛ ), and b N(0, Σ b ), marginal distribution of Y is obtained by integrating b out (generally requires numerical/simulation methods.) The variances (and covariances) of the random effects are estimated and reported. (if the variance is 0, the corresponding RE is 0). Predictions for specific unit are obtained from the posterior mean of the random effect (which is the likelihood of the relevant subset of data times the prior.) In the classical framework, these empirical Bayesian (parameters in the prior estimated from data) predictions are Blup (Best linear unbiased predictor). 8

9 Dynamics in TSCS data: Before we move on to examples using lme4 for mixed effects/multi-level models, some discussion on the dynamics issue in TSCS data. Typically T isn t huge so not very complex time series structure to model. Usual choices are AR(1) or lagged Y or ADL (autoregressive distributed lags, with both lagged x and lagged Y in the model). Can show that AR(1) is a special case of ADL too. (Beck&Katz 2004) The lagged Y model assumes that the effects of all variables die out over time (substitute expression for Y t 1 into Y t, and Y t 2 into Y t 1...); the AR1 error model assumes that the measured variables have only immediate impact but the unmeasured variables have impacts that die out over time. differencing to get rid of α i : y it = ρy it 1 + x it β + α i + ɛ it (3) y it = ρ y it 1 + x it β + ɛ it (4) 9

10 OLS will still be inconsistent because y it 1 is correalted with ɛ it (though some argues that it s not a huge issue for T >> 20.). So instrumental variable methods are needed. GMM is the estimation method used to deal with IV estimation is this model, where past values of y (previous lags) are used as instruments. GMM (generalized methods of moments): recall idea of methods of moments. when number of moment conditions is greater than number of unknown parameters, there s no exact solution to the equations. The idea then is to find the parameter values so that a quadratic function of the moments is minimized. this is the idea of GMM. (some similarity with OLS: only need two data points to solve for two parameters by setting the error to 0. with N data points too many constraints for exact solution. instead minimize sum of squared errors.) In IV estimation, the moment conditions are those that require that the 10

11 IV varaibles are independent of the errors (E(Z ɛ) = 0) GMM is akin to weighted sum of squares on these moment conditions, with the weight matrix chosen to improve efficiency (with two step GMM taking better care on the error structure and more efficient than one step GMM.) In plm pgmm implements the GMM estimation of the dynamic model. (stata has xtabond) Now go over lme4.r. 11