1. Chapter 15
Panel Data Models

2. 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

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

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

5. 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

6. 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

7. The Fixed Effects Model
15.3
The Fixed Effects Model

8. 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 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

9. 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.

10. The Random Effects Model
15.4
The Random Effects Model

11. 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)

12. Comparing Fixed and Random Effects Estimators
15.5
Comparing Fixed and Random Effects Estimators

13. 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.

14. 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

15. 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

16. 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

17. 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

18. 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)