panel

PanelsFixed effects

Random effectsDifference-in-differences and panels

Panel Data

Walter Sosa-Escudero

Walter Sosa-Escudero Panel Data

PanelsFixed effects


A quick introduction to standard panel methods.

Mostly and application of FWL and GLS theory.


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Why panels?

Example (Cronwell y Trumbull): Determinants of crime

y = g(I), y = crime, I = criminal justice variables.

Cross sectoin: (yi, Ii) for several regions i = 1, . . . , n

I is important

Criticism: I captures the effect of other regional effects.


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There is an omitted variable, likely related to I.

OLS that regressesa y on I is biased.

Solution? Control for this omitted variable.

Panels to the rescue: a feasible solution without using othervariables


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A simple linear panel model

yit = xit + uit

uit = i + it

i = 1, . . . , N, t = 1, . . . , T . xit, a vector of K explanatoryvariables, including a constant.

i captures the effect of non-measured effects that vary byindividuals only. it is the standard errror term.


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Fixed effects estimation

yit = xit + i + it

Estimates and i as extra parameters.

It can be seen as a standard linear model where each individual hasits own intercept:

yit = i + 1 +2 x2,it + + K xK,it + itAdd N 1 individual level dummy variables.


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In matrix termsY = X +D+ u

Y is NT 1, X is NT K, X includes and intercept.D is a matrix of N 1 individual level dummy variables.

1N

11...1

, Z =

1N 0 00 1N 0

......

. . ....

0 0 1N

NT(N1)


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Write the model as:

Y = X +D+ u = X + u

with X [X D] y [ ].Then, the fixed effects estimator is:

EF =

(EFEF

)= (X X)1X Y.

Just the OLS estimator adding N 1 individual level dummies.


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Fixed effects and the within transformation

If the interest is in estimating , by the Frisch-Waugh-Lovell result,we can do

= (XX)1X

Y

where X MDX, and MD = I D(DD)1D, the matrix thatprojects X on the orthogonal complement of D. Y is definedaccordingly.

As a simple exercise, show

Xit = Xit XiThat is, getting rid of the dummies in the first step amounts tosubstracting individual level means. This is the within tranformation.


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The FWL Theorem suggests two forms of obtaining the fixedeffects estimator

1 Regress Y on X and D.

2 First demean all data with respect to individual means. Thendo OLS with demeaned data


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Fixed effects and unobserved heterogeneity

FE is unbiased, independently of whether X is correlatedwith D (FE controls for D).

If the ommited variable in our example varies only at theindividual level, it is as if we had controlled for it withoutobserving it.

Intuition: the within transformation kills every variable thatvaries at the individual level only, observed or not.


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Biases: graphical representation


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Verbalization

Y = crime rate.

X = ineficiency of judicial system (more inefficiency, morecrime).

Two regions

An omitted crime determinant that varies only by region andpositively correlated with judicial inefficienty (populationdensity?).


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OLS


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Fixed effects


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Random effects estimator

Same modelY = X +D+

If seen as a random variable D es orthogonal to X, and ifE(i|X) = 0, then the OLS estimator that regresses Y on X isunbiased.

That is, if D is orthogonal to X, omitting the dummy variablesdoes not bias OLS.


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Fixed or random?

Careful. It is a matter of treatment/estimation.

Y = X +D+

Fixed effects (controls for D)

Y = X +D+

Random effects (treats D as an omitted variable)

Y = X +D+


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Y = X +D+

Y = X + u, u D+ Problem: u does not satisfy the classical assumptions, even whenD and do.

Simple proof: assume classical assumptions separately (zero expectedvalue, no serial correlation/heteroskedasticity). Also D and uncorrelated. Then

V (u) = V (D+ )

= DV ()D + V ()= 2DD

+ 2 INT ,

certainly non-spherical.


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Intuition: uit = i + it

Trivially, uit is correlated with ui,t1 since both share i: thepersistent presence of i implies that random effects induceserial correlation.

Though not biased, OLS is inefficient (in the sense discussedin class).

Efficient? GLS.


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GLS random effects

RecallGLS = (X

1X)1X 1Y

In our caseV (u) = 2DD

+ 2 INT ()with = (2, 2 )

FGLS requires to estimate first (variance components).

Random effects estimator: GLS for random effects.


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Summary

Y = X +D+

X D: OLS, FE, RE are unbiased for . RE is efficient.X D: only EF is unbiased for .Most practitioners use FE.


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Hausman test

H0 : X D, HA : X D

Hausman Test: under H0

H = (EA EF )(EF EA)1(EA EF ) 2(K)

Reject if H is significantly high.

Intuition: under H0, EA y EF are consistent, H should be small.Under HA, only EF is consistent, H should be lasrge.


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Panels and impact evaluation

Motivation: effect of minimum wages on employment (Card,Krueger (1994)).

Intuition: minimum wage (MW) reduces employment.

Compare employment in McDonalds before and after MWwage changes? Confounds MW effect with temporal changes.

Compare employment in McDonalds, same period, two stateswith different MW policies? Confounds MW with regionaldeterminants.


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A very simple context

Unit of analysis: restaurant i in state s in period t.

Variable of interest: Yits : employment in restaurant its.

Two periods: t = 1, 2

Two states: A,B.

N restaurants per state.

MW is a state policy.

State A does not change MW. Only B does it, in period 2.

Dist = 1 if state s changes MW in t, 0 otherwise.

Note that in our case Dist = 1 iff s = B y t = 2.


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Additive structure:

Yits = s + t + Dist + ist

is the key parameter: effect of MW controlling for regional(s) and temporal (t) factors that confound MW in thedetermination of employment.

We will assume that given s and t, Dst is exogenous.


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Estimation 1: Differences in differences


Note

E(Y |B, 2) E(Y |B, 1) = 2 1 + E(Y |A, 2) E(Y |A, 1) = 2 1

Substracting[E(Y |B, 2) E(Y |B, 1)

]

[E(Y |A, 2) E(Y |A, 1)

]=

Change in B Change in A =

= Average change in B Average change in A


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Estimation 2: Panels


DBist = 1 iff i is in state B

D2ist = 0 iff t = 2.

Then, by construction Dist = DBist D2ist (change occursonly in state B and in period 2).

Replacing:

Yits = s + t + (DBist D2ist) + ist


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Yits = s + t + (DBist D2ist) + ist

It is a panel of N restaurants in 2 regions and 2 periods, withregional and period specific fixed effects.

Regress Yits on 1) dummy for region B, 2) dummy for period2) interaction between both.

The parameter of interest is the coefficient of the interactionterm.


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Comments

Panel estimation facilitates computation of standard errorsand hypothesis tests.

Common trends: crucial. Both states share t: thetemporal evolution of employment in both states is identical.Treatment (MW change) implies departing away from thiscommon trend.

No serial correlation: key assumption for inference. SeeBertrand, Duflo, y Mullainathan (2004) on cluster correlation.


PanelsFixed effectsRandom effectsDifference-in-differences and panels

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