Chapter 15 Panel Data Models

Balanced and unbalanced panel 15.1 A Microeconomic Panel Panel Data Panel data combines Cross-Sectional and Time Series data and looks at multiple subjects and how they change over the course of time. Balanced and unbalanced panel In microeconomic panels, the individuals are not always interviewed the same number of times, leading to an unbalanced panel → In an unbalanced panel, the number of time series observations is different across individuals In a balanced panel, each individual has the same number of observations

(National Longitudinal Surveys) Panel Data 15.1 A Microeconomic Panel Table 15.1 Representative Observations from NLS (National Longitudinal Surveys) Panel Data

A Microeconomic Panel: Pooled Least Squares Estimates 15.2 A Microeconomic Panel: Pooled Least Squares Estimates

15.2 Pooled Model A pooled least squares model is one where the data on different individuals are simply pooled together with no provision for individual differences that may lead to different coefficients. Notice that the coefficients (β1, β2, β3) do not have i or t subscripts The data for different individuals are pooled together, and the equation is estimated using least squares Eq. 15.1

Cluster-Robust Standard Errors 15.2 Pooled Model 15.2.1 Cluster-Robust Standard Errors What are the consequences of using pooled least squares? The least squares estimator is consistent but its standard errors are typically too small overstating the reliability of the least squares estimator Unrealistic assumptions All individuals have the same coefficients The errors for different individuals are uncorrelated

The Fixed Effects Model 15.3 The Fixed Effects Model

𝑦 𝑖𝑡 = 𝛽 1𝑖 + 𝛽 2𝑖 𝑥 2𝑖𝑡 + 𝛽 3𝑖 𝑥 3𝑖𝑡 + 𝑒 𝑖𝑡 15.3 The Fixed Effects Model We can extend the model in Eq. 15.1 to relax the assumption that all individuals have the same coefficients → Add an i subscript to the equation 𝑦 𝑖𝑡 = 𝛽 1𝑖 + 𝛽 2𝑖 𝑥 2𝑖𝑡 + 𝛽 3𝑖 𝑥 3𝑖𝑡 + 𝑒 𝑖𝑡 A legitimate panel data model above (Eq. 15.7) is not suitable for short (only 5 time-series observations) and wide panels → Need a simplification The intercepts β1i are different for different individuals but the slope coefficients β2 and β3 are assumed to be constant for all individuals Eq. 15.7 Eq. 15.8

The Fixed Effects Model 15.3 The Fixed Effects Model All behavioral differences between individuals, referred to as individual heterogeneity, are captured by the intercept Individual intercepts are included to ‘‘control’’ for individual-specific and time-invariant characteristics (i.e., gender, race and education) A model with these features is called a fixed effects model The intercepts are called fixed effects Fixed effects estimator is unable to estimate coefficients on time-invariant variables like race and gender.

The Random Effects Model 15.4 The Random Effects Model

The Random Effects Model 15.4 The Random Effects Model In the random effects model, we recognize that the individuals were randomly selected and thus we treat the individual differences as random rather than fixed. We can include random individual differences by specifying the intercept parameters to consist of a fixed part that represents the population average and by specifying the random individual differences from the population average: The random individual differences ui are called random effects (i.e., random individual effects or random error terms)

Comparing Fixed and Random Effects Estimators 15.5 Comparing Fixed and Random Effects Estimators

Comparing Fixed and Random Effects Estimators 15.5 Comparing Fixed and Random Effects Estimators If random effects are present, then the random effects estimator is preferred for several reasons: The random effects estimator takes into account the random sampling process by which the data were obtained The random effects estimator permits us to estimate the effects of variables that are individually time-invariant The random effects estimator is a generalized least squares estimation procedure, and the fixed effects estimator is a least squares estimator  In large samples, the GLS estimator has a smaller variance than the LSE.

15.5 Comparing Fixed and Random Effects Estimators 15.5.1 Endogeneity in the Random Effects Model If the random error vit = ui + eit is correlated with any of the right-hand-side explanatory variables in a random effects model, then the GLS estimators of the parameters are biased and inconsistent The problem is common in random effects models, because the individual specific error component ui may well be correlated with some of the explanatory variables A person’s ability and industriousness are variables not explicitly included in the wage eq’n, and thus these factors are included in ui. → These characteristics may be correlated with his education level and/or job experiences → RE estimator is inconsistent

Comparing Fixed and Random Effects Estimators 15.5 Comparing Fixed and Random Effects Estimators 15.5.3 The Hausman Test To check for any correlation between the error component ui (random individual effects) and the regressors in a random effects model, we can use a Hausman test The Hausman test can be carried out for specific coefficients, using a t-test, or jointly, using an F-test or a chi-square test The idea for the Hausman test is that both the random effects and fixed effects estimators are consistent if there is no correlation b/w error terms and the explanatory variables Null hypothesis of the Hausman test is that the difference b/w the estimators is zero (two-tail test)  if reject the null, then use the fixed effects estimator

Comparing Fixed and Random Effects Estimators 15.5 Comparing Fixed and Random Effects Estimators 15.5.3 The Hausman Test Let the parameter of interest be βk Denote the fixed effects estimate as bFE,k and the random effects estimate as bRE,k The t-statistic for testing that there is no difference between the estimators is: Eq. 15.37

Comparing Fixed and Random Effects Estimators 15.5 Comparing Fixed and Random Effects Estimators 15.5.3 The Hausman Test Applying the t-test to the SOUTH we get: Using the standard 5% large sample critical value of 1.96, we reject the hypothesis that the estimators yield identical results Our conclusion is that the random effects estimator is inconsistent, and that we should use the fixed effects estimator, or should attempt to improve the model specification

Comparing Fixed and Random Effects Estimators 15.5 Comparing Fixed and Random Effects Estimators Types of the Model for Panel Analysis Pooled Least Squares Estimation Model Selection Hausman Test: FE and RE estimators are consistent (No correlation between ui and the explanatory variables) Accept the null: FE=RE ⇒ Select the RE model Reject the null: FE≠RE ⇒ Select the FE model Fixed Effects Model (OLS) Error terms are correlated with the explanatory variables Random Individual Differences No individual heterogeneity=β1 Uncorrelated errors Random Effects Model (GLS)