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930—Chapter 29. Panel Estimation

Cross-section random effects test comparisons:

Variable	Fixed	Random	Var(Diff.)	Prob.


F	0.110124	0.109781	0.000031	0.9506
K	0.310065	0.308113	0.000006	0.4332

The bottom portion of the output contains the results from the corresponding fixed effects estimation:

Cross-section random effects test equation:

Dependent Variable: I

Method: Panel Least Squares

Date: 11/24/04 Time: 12:51

Sample: 1935 1954

Cross-sections included: 10

Total panel (balanced) observations: 200

Variable	Coefficient	Std. Error	t-Statistic	Prob.


C	-58.74394	12.45369	-4.716990	0.0000
F	0.110124	0.011857	9.287901	0.0000
K	0.310065	0.017355	17.86656	0.0000

Effects Specification

Cross-section fixed (dummy variables)

In some cases, EViews will automatically drop non-varying variables in order to construct the test statistic. These dropped variables will be indicated in this latter estimation output.

Estimation Background

The basic class of models that can be estimated using panel techniques may be written as:

Yit = f( Xit, β) + δi + γt + it

(29.1)

The leading case involves a linear conditional mean specification, so that we have:

Yit = α + Xit′β + δi + γt + it	(29.2)
where Yit is the dependent variable, and Xit is a k -vector of regressors, and it	are the
error terms for i = 1, 2, …, M cross-sectional units observed for dated periods

t = 1, 2, …, T . The α parameter represents the overall constant in the model, while the δi and γt represent cross-section or period specific effects (random or fixed).

Estimation Background—931

Note that in contrast to the pool specifications described in Equation (27.2) on page 860, EViews panel equations allow you to specify equations in general form, allowing for nonlinear coefficients mean equations with additive effects. Panel equations do not automatically allow for β coefficients that vary across cross-sections or periods, but you may, of course, create interaction variables that permit such variation.

Other than these differences, the pool equation discussion of “Estimation Background” on page 859 applies to the estimation of panel equations. In particular, the calculation of fixed and random effects, GLS weighting, AR estimation, and coefficient covariances for least squares and instrumental variables is equally applicable in the present setting.

Accordingly, the remainder of this discussion will focus on a brief review of the relevant econometric concepts surrounding GMM estimation of panel equations.

GMM Details

The following is a brief review of GMM estimation and dynamic panel estimators. As always, the discussion is merely an overview. For detailed surveys of the literature, see Wooldridge (2002) and Baltagi (2001).

Background

The basic GMM panel estimators are based on moments of the form,

g( β) = Σ gi( β)	= Σ Zi′ i( β)	(29.3)
i = 1	i = 1
where Zi is a Ti × p matrix of instruments for cross-section i , and,
i( β) = ( Yi − f( Xit, β) )		(29.4)

In some cases we will work symmetrically with moments where the summation is taken over periods t instead of i .

GMM estimation minimizes the quadratic form:

M	′ M
S( β) = Σ Zi′ i( β)	H Σ Zi′ i( β)	(29.5)
i = 1	i = 1	(29.5)

= g( β)′Hg( β)

with respect to β for a suitably chosen p × p weighting matrix H .

Given estimates of the coefficient vector, β , an estimate of the coefficient covariance matrix is computed as,

ˆ	−1	( G′HΛHG) ( G′HG )	−1	(29.6)
V( β) = ( G′HG)		( G′HΛHG) ( G′HG )		(29.6)

932—Chapter 29. Panel Estimation

where Λ is an estimator of E(gi( β) gi( β) ′) = E(Zi′ i( β) i( β) ′Zi) , and G is a
Ti × k derivative matrix given by:
	M

G( β) =	− Σ Zi′ fi( β)	(29.7)
	i = 1

In the simple linear case where f( Xit, β) = Xit′ β , we may write the coefficient estimator in closed form as,

′

−1 M

′ M

β = Σ Zi′Xi

Σ Zi′Xi

H Σ Zi′Yi

(29.8)

i = 1

= ( M

′HM ) −1( M ′ HM

)

with variance estimator,

−1

( MZX′HΛHMZX) ( MZX′ HMZX)

−1

(29.9)

V( β) = ( MZX′HMZX)

for MAB of the general form:

MAB = M

−1

Σ Ai′Bi

(29.10)

i = 1

The basics of GMM estimation involve: (1) specifying the instruments Z , (2) choosing the weighting matrix H , and (3) determining an estimator for Λ .

It is worth pointing out that the summations here are taken over individuals; we may equivalently write the expressions in terms of summations taken over periods. This symmetry will prove useful in describing some of the GMM specifications that EViews supports.

A wide range of specifications may be viewed as specific cases in the GMM framework. For example, the simple 2SLS estimator, using ordinary estimates of the coefficient covariance, specifies:

ˆ	2	MZZ)	−1
H = ( σ		MZZ)	(29.11)
ˆ 2			(29.11)
ˆ 2	MZZ
Λ = σ	MZZ

Substituting, we have the familiar expressions,

ˆ	ˆ 2	MZZ)	−1	MZX)	−1	ˆ 2	MZZ)	−1
β = ( MZX′( σ		MZZ)		MZX)		( MZX′( σ	MZZ)	MZY)
	= ( MZX′MZZ−1MZX)−1( MZX′MZZ−1MZY)							(29.12)
	= ( MZX′MZZ−1MZX)−1( MZX′MZZ−1MZY)

and,

						Estimation Background—933

ˆ	ˆ	2	( MZX′MZZ	−1	MZX)	−1
V( β) =	σ		( MZX′MZZ		MZX)	(29.13)

Standard errors that are robust to conditional or unconditional heteroskedasticity and contemporaneous correlation may be computed by substituting a new expression for Λ ,

Λ = T	−1	T	ˆ ˆ		(29.14)
		Σ Zt′ t t′Zt

t = 1

so that we have a White cross-section robust coefficient covariance estimator. Additional robust covariance methods are described in detail in “Robust Coefficient Covariances” on page 869.

In addition, EViews supports a variety of weighting matrix choices. All of the choices available for covariance calculation are also available for weight calculations in the standard panel GMM setting: 2SLS, White cross-section, White period, White diagonal, Cross-sec- tion SUR (3SLS), Cross-section weights, Period SUR, Period weights. An additional differenced error weighting matrix may be employed when estimating a dynamic panel data specification using GMM.

The formulae for these weights are follow immediately from the choices given in “Robust Coefficient Covariances” on page 869. For example, the Cross-section SUR (3SLS) weighting matrix is computed as:

	−1	T	ˆ	−1
H = T		Σ Zt′ΩMZt		(29.15)
		t = 1

ˆ
where ΩM is an estimator of the contemporaneous covariance matrix. Similarly, the
White period weights are given by:
H =		−1	M	ˆ ˆ	−1
H =	M		Σ Zi′ i i′Zi		(29.16)

i = 1

These latter GMM weights are associated with specifications that have arbitrary serial correlation and time-varying variances in the disturbances.

GLS Specifications

EViews allows you to estimate a GMM specification on GLS transformed data. Note that the moment conditions are modified to reflect the GLS weighting:

M	M	ˆ	−1
g( β) = Σ gi( β) =				( β)	(29.17)
	Σ Zi′ Ω		i
i = 1	i = 1

Dynamic Panel Data

Consider the linear dynamic panel data specification given by:

934—Chapter 29. Panel Estimation

p
Yit = Σ ρjYit − j + Xit′β + δi + it	(29.18)

j = 1

First-differencing this specification eliminates the individual effect and produces an equation of the form:

p
∆Yit = Σ ρj∆Yit − j + ∆Xit′ β + ∆ it	(29.19)

j = 1

which may be estimated using GMM techniques.

Efficient GMM estimation of this equation will typically employ a different number of instruments for each period, with the period-specific instruments corresponding to the different numbers of lagged dependent and predetermined variables available at a given period. Thus, along with any strictly exogenous variables, one may use period-specific sets of instruments corresponding to lagged values of the dependent and other predetermined variables.

Consider, for example, the motivation behind the use of the lagged values of the dependent variable as instruments in Equation (29.19). If the innovations in the original equation are i.i.d., then in t = 3 , the first period available for analysis of the specification, it is obvious that Yi1 is a valid instrument since it is correlated with ∆Yi2 , but uncorrelated with ∆ i3 . Similarly, in t = 4 , both Yi2 and Yi1 are potential instruments. Continuing in this vein, we may form a set of predetermined instruments for individual i using lags of the dependent variable:

	Yi1	0	0	…	…	…	…	0
Wi =	0	Yi1	Yi2	… … … … 0					(29.20)
Wi =	…	…	…	…	…	…	…	…	(29.20)
	…	…	…	…	…	…	…	…
	0	0	0	… Yi1		Yi2	… YiTi − 2

Similar sets of instruments may be formed for each predetermined variables.

Assuming that the it are not autocorrelated, the optimal GMM weighting matrix for the differenced specification is given by,

	d		−1	M		−1
H		= M		Σ Zi′ΞZi		(29.21)

i = 1

where Ξ is the matrix,

									Estimation Background—935

			− 1		… 0
	2		− 1	0	… 0	0
1	−1		2	0	… 0	0	σ	2	(29.22)
Ξ = --	… … … … … …						σ		(29.22)
2	0		0	0	… 2	− 1
	0		0	0	… 2	− 1
		0	0	0	… −1	2

and where Zi contains a mixture of strictly exogenous and predetermined instruments. Note that this weighting matrix is the one used in the one-step Arellano-Bond estimator.

Given estimates of the residuals from the one-step estimator, we may replace the Hd weighting matrix with one estimated using computational forms familiar from White period covariance estimation:

	−1	M		−1
H= M		Σ Zi′∆ i∆ i′Zi		(29.23)

i = 1

This weighting matrix is the one used in the Arellano-Bond two-step estimator.

Lastly, we note that an alternative method of transforming the original equation to eliminate the individual effect involves computing orthogonal deviations (Arellano and Bover, 1995). We will not reproduce the details on here but do note that residuals transformed using orthogonal deviations have the property that the optimal first-stage weighting matrix for the transformed specification is simply the 2SLS weighting matrix:

	−1	M		−1
H = M		Σ Zi′Zi		(29.24)

i = 1

936—Chapter 29. Panel Estimation

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