Problem Set #9: Panel Data

Economics 835: Econometrics

Fall 2012

1 Estimating the fixed effects

Consider a standard fixed effects model:

yit = ai + xitβ + uit where E(uit|xi1, . . . ,xiT ) = 0

and let β be a consistent estimator of β.

We might “estimate” the fixed effects (note that the fixed effects are random variables and not parameters,and so this doesn’t fit in our standard estimation framework) by:

ai = yi − xiβ

In order to talk about the properties of ai, we need to extend our notion of convergence in probabilityto include convergence in probability to a random variable. Let wn, n = 1, 2 . . . be a sequence of randomvariables, and let w be a random variable. Then we say that wn →p w if and only if (wn − w)→p 0.

a) Find plim ai.

b) Is ai a consistent estimator of ai (in the sense that plim ai = ai)?

2 Fixed effects with measurement error

Consider the following model of fixed effects with measurement error.

Suppose that we have a panel data set consisting of a random sample of individuals (indexed by i ∈{1, 2, . . . , n}) observed at two points in time (indexed by t ∈ {0, 1}. For each individual and point in time,we observe the scalar outcome yit, which is a function of an individual-specific fixed effect ai, a time-specificfixed effect δt, a scalar explanatory variable varying across individuals and time xit, and a time-varyingunobserved component uit:

yit = ai + βxit + uit

where uit is mean-zero, IID across time and individuals, and strictly exogenous.

To keep things simple, we suppose that:

xit = xi + vit

where vit is mean-zero, IID across time and individuals, and independent of all other exogenous variables.

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However, we do not get to observe xit, we only get to observe a noisy measure of it called xit. Suppose that:

xit = xit + εit

where εit is mean-zero, IID across time and individuals, and independent of all other exogenous variables.

Let:

µx = E(xi) = E(xit)

µa = E(ai)

σ2a = var(ai)

σ2x = var(xi)

σ2v = var(vit)

σ2ε = var(εit)

ρa,x = corr(ai, xi)

We are interested in estimating β, the effect of xit on yit. However, there are two problems with doing so:correlation between xit and the individual-specific fixed effect (ai), and measurement error in xit.

Let the OLS estimator for β be defined as

βOLS ≡ ˆcov(yit, xit)

ˆvar(xit)

and let βOLS = plim βOLS . Let the FD (first difference) estimator be defined as:

βFD ≡ ˆcov(∆yi,∆xi)

ˆvar(∆xi)

where ∆xi = xi1 − xi0 and ∆yi = yi1 − yi0. Also let βFD = plim βFD.

a) Find the probability limit of βOLS in terms of the parameters:

(µx, µa, β, σ2a, σ

2x, σ

2u, σ

2v , σ

2ε , ρa,x)

b) Find the probability limit of βFD in terms of these same parameters.

c) Suppose (for this part of the question only) that there is no fixed effect, i.e., σ2a = 0. Find each of the

two probability limits. Which estimator produces a smaller (or at least no larger) asymptotic bias?

d) Suppose (for this part of the question only) that there is a fixed effect, but that it is uncorrelated withxi, i.e., ρa,x = 0. Find each of the two probability limits. Which estimator produces a smaller (or at leastno larger) asymptotic bias?

e) Suppose (for this part of the question only) that there is no measurement error, i.e., var(εit) = 0. Findeach of the two probability limits. Which estimator produces a smaller (or at least no larger) asymptoticbias?

f) Consider the following two statements (difference between the two is in boldface):

When estimating a model using panel data in which the explanatory variable is measured witherror, a fixed effects estimator will be preferable (in terms of having smaller asymptotic bias) to asimple OLS estimator when the covariance between the fixed effect and the explanatory variablesis large relative to the measurement error.

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When estimating a model using panel data in which the explanatory variable is measured witherror, a fixed effects estimator will be preferable (in terms of having smaller asymptotic bias) to asimple OLS estimator when the covariance between the fixed effect and the explanatory variablesis small relative to the measurement error.

Based on your results, which of the two statements is correct?