Transcript Chapter 15

Chapter 15
Panel Data Analysis
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.
What is in this Chapter?
• 1. Fixed effects models.
• 2. Random effects models.
• 3. Seemingly unrelated regression (SUR)
model
• 4. Random coefficient model.
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
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.
The LSDV or Fixed Effects Model
The LSDV or Fixed Effects Model
• Define
1
1
xi   xit , yi   yit
T t
T t
The LSDV or Fixed Effects Model
The LSDV or Fixed Effects Model
• In the case of several explanatory
variables, Wxx is a matrix and β and Wxy
are vectors.
The OLS model
• If we consider the hypothesis
then the model is
Alternative method for the fixed
effects model
yit  i  xit '  uit
• where αi (i=1, 2…, N) and β (KX1 vector)
are unknown parameters to be estimated.
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:
Alternative method for the fixed
effects model
y  x '  u
*
it
*
it
*
it
• where * denotes variables deviated from
the group mean (an example)
_
y  yit  y i
*
it
_
x  xit  x i
*
it
_
u  uit  u i
*
it
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
Industry and year dummies
• Yi,t = 0 + 1 Xi,t + control variables + year
dummies + industry dummies
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.
The Random Effects Model
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
The Random Effects Model
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
The Random Effects Model
The Random Effects Model
• W refers to within-group
• B refers to between-group
• T refers to total
The Random Effects Model
Thus the OLS and LSDV estimators
are special cases of the GLS estimator with
θ = 1 and θ =0, respectively.
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 crosssection 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:
The SUR Model
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 uit .
• Then we compute ˆ ij  1/(T  k ) uˆituˆ jt where k
is the number of regressors.
• After we get the estimates ˆ ij we use GLS on all
the N equations jointly.

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."
The Random Coefficient Model
The Random Coefficient Model
The Random Coefficient Model
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.
The Random Coefficient Model
• In practice the GLS estimator cannot be
computed because the parameters  2 and  i2 in
equation (15.8 ) are not known.
• To obtain these we estimate equations for the N
cross-section units and get the residuals uˆit .
• Then