1. Chapter 15
Panel Data Analysis

2. What is in this Chapter?
This chapter discusses analysis of panel data.
This is a situation where there are observations on individual cross-section units over a period of time.
The chapter discusses several models for the analysis of panel data.

3. What is in this Chapter? 1. Fixed effects models.
2. Random effects models.
3. Seemingly unrelated regression (SUR) model
4. Random coefficient model.

4. Introduction
One of the early uses of panel data in economics was in the context of estimation of production functions.
The model used is now referred to as the "fixed effects" model and is given by

5. Introduction
This model is also referred to as the "least squares with dummy variables" (LSDV) model.
The αi are estimated as coefficients of dummy variables.

6. The LSDV or Fixed Effects Model

7. The LSDV or Fixed Effects Model
Define

8. The LSDV or Fixed Effects Model

9. The LSDV or Fixed Effects Model
In the case of several explanatory variables, Wxx is a matrix and β and Wxy are vectors.

10. The OLS model
If we consider the hypothesis then the model is

11. Alternative method for the fixed effects model
where αi (i=1, 2…, N) and β (KX1 vector) are unknown parameters to be estimated.

12. Alternative method for the fixed effects model
As part of this study’s focus on the dynamic relationships between yit and xit (i.e. the β parameters) we take the ‘group difference’ between variables and redefine the equation as follows:

13. Alternative method for the fixed effects model
where * denotes variables deviated from the group mean (an example)

14. Industry and year dummies
Industry dummies
Using the first one-digit (or two-digit) of the firm’s SIC code.
Control for the potential variation across industries
Year dummies
Panel structure data
Year effect refers to the aggregate effects of unobserved factors in a particular year that affect all the companies equally

15. Industry and year dummies
Yi,t = 0 + 1 Xi,t + control variables + year dummies + industry dummies

18. The Random Effects Model
In the random effects model, the αi are treated as random variables rather than fixed constants.
The αi are assumed to be independent of the errors uu and also mutually independent.
This model is also known as the variance components model.
It became popular in econometrics following the paper by Balestra and Nerlove on the demand for natural gas.

19. The Random Effects Model

20. The Random Effects Model
For the sake of simplicity we shall use only one explanatory variable.
The model is the same as equation (15.1) except that αi are random variables.
Since αi are random, the errors now are vit = αi + uit

21. The Random Effects Model

22. The Random Effects Model
Since the errors are correlated, we have to use generalized least squares (GLS) to get efficient estimates.
However, after algebraic simplification the GLS estimator can be written in the simple form

23. The Random Effects Model

24. The Random Effects Model
W refers to within-group
B refers to between-group
T refers to total

25. The Random Effects Model
Thus the OLS and LSDV estimators
are special cases of the GLS estimator with θ = 1 and θ =0, respectively.

26. The SUR Model
Zeilner suggested an alternative method to analyze panel data, the seemingly unrelatedregression (SUR) estimation
In this model a GLS method is applied to exploit the correlations in the errors across cross-section units
The random effects model results in a particular type of correlation among the errors. It is an equicorrelated model.
In the SUR model the errors are independent over time but correlated across cross-section units:

27. The SUR Model

28. The SUR Model
This type of correlation would arise if there are omitted variables that are common to all equations .
The estimation of the SUR model proceeds as follows.
We first estimate each of the N equations (for the cross-section units) by OLS.
We get the residuals
Then we compute where k is the number of regressors.
After we get the estimates we use GLS on all the N equations jointly.

29. The SUR Model
If we have large N and small T this method is not feasible.
Also, the method is appropriate only if the errors are generated by a true multivariate distribution.
When the correlations are due to common omitted variables it is not clear whether the GLS method is superior to OLS.
The argument is similar to the one mentioned in Section 6.9. See "autocorrelation caused by omitted variables."

30. The Random Coefficient Model

31. The Random Coefficient Model

32. The Random Coefficient Model

33. The Random Coefficient Model
If δ2 is large compared with υi, then the weights in equation (15.8) are almost equal and the weighted average would be close to simple average of the βi.

34. The Random Coefficient Model
In practice the GLS estimator cannot be computed because the parameters in equation (15.8 ) are not known.
To obtain these we estimate equations for the N cross-section units and get the residuals
Then