Panel Data: Linear Models
Laura Magazzini
University of Verona
[email protected]://dse.univr.it/magazzini
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Introduction
Outline
What is Panel Data?
Motivation: the omitted variable problem
An example: Production function
Model specification
Estimation
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Introduction
What is panel (or longitudinal) data?
It is a time-series of cross-section, where the same unit is observedover a number of periods
. Units can be individuals, firms, households, industries, markets, regions,countries, ...
Micro- vs. Macro-panels: different techniques are required forestimation
. Bank of Italy, European panel: large N & small T
. OECD: large N & small/medium/large T
We work on micro-panel (large N & small T )
. Random sampling over the cross-sectional dimension
Micro & Macro-panel: one of the most active bodies of literature ineconometrics
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Introduction
Basic model and notation
We will consider the linear model
yit = x ′itβ + vit
with i = 1, ...,N (sample units), t = 1, ...,T (time periods)
For each sample units, we have the following T equations:
yi1 = x ′i1β + vi1
yi2 = x ′i2β + vi2...
yiT = x ′iTβ + viT
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Introduction
Advantages of panel data
Greater flexibility in the study of dynamics than CS or TS(ex.1) Repeated CS: in two points in time you observed 50% of women appear
working. One-half of the women will be working? Or the some one-halfof women will be working over all time periods? [Ben-Porath (1973)]
(ex.2) Production function: economies of scale (ES) versus technical change(TC). CS only provides information about ES. TS muddle the twoeffects.
Greater precision in estimation (greater number of observations dueto pooling)
Heterogeneity across units: it is possible to disentangle differentsources of variance of the units of interest (permanent versustransitory factors)Can solve the omitted variables bias (fixed effects). Consistent estimates can be obtained in the presence of omitted
variables, if the omitted variable vary across sample units, but it isconstant over time, e.g. preferences, individual ability, propensity topatent, ...
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Introduction
Example 1: Production function
Max output given the value of the inputs
Consider the case of agricultural production:
Q = φ(L,V )
o Q: Outputo L: Input that varies over time (labor)o V : Input that remains constant over time (soil quality). You can also think of a firm production function where V represents
managerial capability
Typically, V is known to the farmer/manager, but unknown to theeconometrician
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Introduction
Example 1: Econometric specification
Let us consider a Cobb-Douglas production function:
φ(L,V ) = ALαV β
Taking logs (and adding an error term, summarizing all inputs outsidethe farmer’s control, e.g. rainfall):
q = a + αl + βv + u
Parameter of interest: α, i.e. the (%) increase in Q driven by a 1percent increase in L, holding V constant
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Introduction
Example 1: Data availability
q = a + αl + βv + u
Ideal world
You measure Q, L, and V on a sample of N farmers
If standard hypotheses hold, the relationship can be estimated by OLS
Real world
V is not observable: you measure only Q and L on a sample of Nfarmers
q = a + αl + (βv + u) = a + αl + ε
Omitted variable bias?
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Introduction
Example 1: Estimation by OLS?
E [q|l ] = a + αl + (βE [v |l ] + E [u|l ]) = a + αl + E [ε|l ]
OLS regression of q on l allows the identification of the parameter ofinterest α if and only if E [ε|l ] = 0
We assume E [u|l ] = 0, therefore we need the omitted variable v
(1) not to affect q once l is controlled for, i.e. β = 0 or(2) uncorrelated with l : E [v |l ] = 0
We do not believe (1): soil quality affects harvest (managerialcapabilities affect firm output)
What does economic theory tell us about hypothesis (2)?
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Introduction
Example 1: Relationship between L and V
According to economic theory, a farmer/firm chooses L thatmaximizes the expected profit
Let pl the cost of a unit of L, and p the price of the output Q
π = ALαV βp − Lpl
Taking first derivatives and solving first order condition, the optimal Ldepends on V
As a consequence, L is correlated with V : firms choose the optimal Lon the basis of characteristics that are unobservable for the researcherbut known to the farmer/firm!
cov(v , l) 6= 0⇒ E [v |l ] 6= 0 and, therefore, E [ε|l ] 6= 0: OLS isinconsistent
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Introduction
Example 1: The panel solution (1)
The omitted variable bias is linked to the problem of endogeneity
Instrumental Variable can be applied for estimation (need to searchfor “external” instruments)
What if...?
. The soil quality/managerial ability V is constant over time
. Q and L are observed for (at least) T = 2 time periods
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Introduction
Example 1: The panel solution (2)
When t = 1: qi1 = a + αli1 + βvi1 + ui1
When t = 2: qi2 = a + αli2 + βvi2 + ui2
Taking the difference (we assume V constant over time ⇒ vi1 = vi2):
qi2 − qi1︸ ︷︷ ︸∆qi
= α(li2 − li1︸ ︷︷ ︸∆li
) + ui2 − ui1︸ ︷︷ ︸∆ui
The equation ∆qi = α∆li + ∆ui does not depend from theunobserved variable v
If ∆ui satisfies classic assumptions, the regression of ∆qi on ∆li canprovide an estimate of the parameter of interest α.
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Introduction
Example 1: The panel solution (3)
Advantages: repeated observations over time on the same unit allowsto use estimation methods that are robust to the presence of omittedvariables in the model, if these variables are constant over time.
Any transformation of the initial model that eliminates theunobservable variable v is a good starting point
The linearity and additivity of the model are necessary in this context.
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Introduction
Example 2: Return to schooling
Aim: Study the variation in income associated to a change in theyears of schooling
The model of interest is:
wi = α + ρsi + ai + εi
with wi indicates the income, si is the number of years of schooling,ai represents individual ability (i = 1, ...,N).
Likely, individual ability affects income (cov(w , a) > 0) and iscorrelated with the years of schooling (cov(s, a) > 0)
Unfortunately, ai is typically unobservable!
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Introduction
Example 2: Identification and estimation
Let us suppose we observe (w , s) for the same unit at two points intime
Typically, si does not vary over time, i.e. we look at the relationshipbetween w and s when choices about s have already been done
At time 1: wi1 = α + ρsi1 + ai + εi1
At time 2: wi2 = α + ρsi2 + ai + εi2
Taking differences (since si1 = si2):
wi2 − wi2 = εi2 − εi1
The availability of repeated observations does not improve theidentification of ρ
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The Omitted Variables Problem
Motivation: The omitted variables problem
Panel data can be used to obtain consistent estimators in thepresence of omitted variables
Let y and x = (x1, ..., xK ) be observable random variables
Let c be an unobservable random variable
We are interested in the partial effect of the observable explanatoryvariables xj in the population regression function: E [y |x1, ..., xK , c]
Assuming a linear model: E [y |x1, ..., xK , c] = β0 + x′β + c , i.e.
y = β0 + x′β + c + ε
- Interest lies in the (K × 1) vector β- c is called unobserved effect
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The Omitted Variables Problem
What if cov(x, c) 6= 0?
y = β0 + x′β + c + ε
1 Find a proxy for c and estimate β using OLS
2 Find an “external” instrument for x and apply 2SLS
3 If we can observe the same units at different points in time (i.e. wecan collect a panel data set), we can get consistent estimates of β aslong as we can assume c to be constant over time
. Accomplished by transforming the original data (“internal”instruments)
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The Omitted Variables Problem
The panel solution to omitted variable bias (T = 2)
Assume we can observe (y , x) at two different points in time:
t = 1: (y1, x1) & t = 2: (y2, x2)
The population regression function is:
E [yt |xt , c] = β0 + x′tβ + c or yt = β0 + x′tβ + c + ut
where by definition E [ut |xt , c] = 0 (t = 1, 2).
What about E [c |xt ]?. If E [x′tc] = 0, we can apply OLS
. If E [x′tc] 6= 0, pooled OLS is biased and inconsistent
But we can take first difference and eliminate c:
y2 − y1︸ ︷︷ ︸∆y
= (x2 − x1︸ ︷︷ ︸∆x
)′β + u2 − u1︸ ︷︷ ︸∆u
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The Omitted Variables Problem
Can we apply OLS for estimation? (T = 2)
∆y = ∆x′β + ∆u
Exogeneity: E [∆x′∆u] = 0. E (x′2u2) + E (x′1u1)− E (x′1u2)− E (x′2u1) = 0. Stronger than E (x′tut) = 0 (t=1,2). Strict exogeneity: cov(xt , us) = 0 for all t and s. No restrictions on the correlation between xt and c
Rank condition: rankE (∆x′∆x) = K
. If xt contains a variable that is constant across time for every memberof the population, then ∆x contains an entry that is identically zero,and rank condition fails
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Linear Model Notation
The basic linear panel data model (1)
For a randomly drawn cross-section i , we assume (i = 1, ...,N,t = 1, ..,T ):
yit = x′itβ + ci + uit
. ci : individual effect or individual heterogeneity
. uit : idiosyncratic errors/disturbances
Assume ci uncorrelated with uit
Assume uit homeschedastic and serially uncorrelated
We consider a “balanced panel”: each cross-section i is observed Ttimes (total of N × T observations)
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Linear Model Notation
The basic linear panel data model (2)
In compact form we can write:
yi = x′iβ + ci ιT + ui
where vectors have dimension T × 1
yi = (yi1, ..., yiT )′ xi = (xi1, ..., xiT )′
ui = (ui1, ..., uiT )′ ιT = (1, ..., 1)′
Different estimators are available on the basis of underlyingassumptions on the correlation structure of ci
Asymptotics rely on N →∞, for fixed T
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Linear Model OLS estimation
When pooled OLS?
yit = x′itβ + ci + uit = x′itβ + vit
vit : composite error, sum of the unobserved effect and idiosyncraticerror
OLS is consistent if E [x′itvit ] = 0:
. E [x′ituit ] = 0
. E [x′itci ] = 0, t = 1, 2, ...,T
Robust standard errors: the presence of ci induces correlation overtime for the same individual
OLS is not efficient
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Linear Model Random effect estimation
Random effects structure
yit = x′itβ + ci + uit = x′itβ + vit
uit homoschedastic and serially uncorrelated: E [uiu′i |xi , ci ] = σ2
uIT
ci homoschedastic: E [c2i |xi ] = σ2
c
As a result, the error structure has the following form:
Ωi = E [viv′i ] =
σ2c + σ2
u σ2c . . . σ2
c
σ2c σ2
c + σ2u . . . σ2
c...
.... . .
...σ2c . . . . . . σ2
c + σ2u
(T×T )
= σ2c ιT ι
′T + σ2
uIT
E [vv′] = IN ⊗ Ωi = Ω
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Linear Model Random effect estimation
GLS estimation (unfeasible)
βRE(GLS) =
(N∑i=1
X′iΩ−1i Xi
)−1( N∑i=1
Xi′Ω−1
i yi
)
The estimator can be obtained by applying OLS regression to Ω−1/2Xon Ω−1/2y
Ω−1/2 = [IN ⊗ Ωi ]−1/2 = IN ⊗ Ω
−1/2i
Ω−1/2i = 1
σu
[IT − θ
T ιT ι′T
]with θ = 1− σu√
σ2u+Tσ2
c
The GLS estimator can be obtained by the OLS regression of(yit − θyi ) on (xit − θxi )
. If σ2c = 0, θ = 0: RE = OLS (no unobs. heterogeneity; Breusch Pagan
LM statistic)
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Linear Model Random effect estimation
GLS estimation (feasible)
In order to implement the RE procedure, we need to obtain σ2c and σ2
u
βRE(FGLS) =
(N∑i=1
X′i Ω−1Xi
)−1( N∑i=1
Xi′Ω−1yi
)
To get Ω (get σ2c and σ2
u), Wooldridge suggests:
. σ2c + σ2
u from pooled OLS residuals. As σ2
c = E [vitvis ], autocorrelation in OLS residuals can be exploited toobtain an estimate of σ2
c
. σ2u can be recovered by taking the difference σ2
c + σ2u − σ2
c
. Alternative procedure described in Greene (Maddala and Mount, 1973)
. In small sample you can have σ2c < 0!
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Linear Model Random effect estimation
Random effect estimationyit = x′itβ + ci + uit
Obtained from the OLS regression of (yit − θyi ) on (xit − θxi ) (in themore general case: OLS regression of Ω−1/2y on Ω−1/2X)
Assumptions (stronger than OLS):
(1) Strict exogeneity: E [x′isuit ] = 0 for each s, t = 1, ...,T
(2) Orthogonality between ci and each xit : E [ci |xi] = E [ci ] = 0
(3) Rank condition: rank E [X ′i Ω−1Xi ] = K , where Ω = E [viv′i ]
Why REE? Exploit serial correlation of the error term in a GLSframework: efficient
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Linear Model Random effect estimation
The strict exogeneity assumption
yit = x′itβ + ci + uit
E [yit |xi1, xi2, ..., xiT , ci ] = E [yit |xit , ci ] = x′itβ + ci
Once xit and ci are controlled for, xis has no partial effect on yit fors 6= t
xit , t = 1, ...,T are strictly exogenous conditional on theunobserved effect ci
The strict exogeneity assumption can be stated in terms of theidiosyncratic error term: E [uit |xi1, xi2, ..., xiT , ci ] = 0
This implies that explanatory variables in each time period areuncorrelated with the idiosyncratic error in each time period:E [x′isuit ] = 0 for each s, t = 1, ...,T
. Stronger than zero contemporaneous correlation: E [x′ituit ] = 0
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Linear Model Fixed effect estimation
Fixed effect framework
We maintain the strict exogeneity assumption: E [uit |xi, ci ] = 0
Allow ci to be arbitrarily correlated with xiFE is more robust than RE
. We can consistently estimate partial effects in the presence oftime-constant omitted variable, that can be related to the observablesxi
. BUT we cannot include time-constant factors in xi (e.g. gender, racein the analysis of individuals; foundation year for firms; ...)
To get estimates we transform the equation to remove ci and applyOLS
. Dummy variable regression
. Within transformation
. First difference
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Linear Model Fixed effect estimation
Dummy variable regressionLeast Squares Dummy Variables (LSDV)
yi = xiβ + ci ιT + ui
Collecting the terms over the N units gives:y1
y2...yN
=
x1
x2...xN
β +
ιT 0 . . . 00 ιT . . . 0
...0 0 . . . ιT
c1
c2...
cN
+
u1
u2...uN
Or, letting di be a dummy variable indicating unit i
y = [X d1 d2 . . .dN ]
[βc
]+ u = Xβ + Dc + u
Classical regression model with K + N parameters
What if N is thousands?
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Linear Model Fixed effect estimation
Dummy variable regressionDiscussion
The parameter of interest is β
ci : nuisance parameters that only increase the computationalcomplexity of estimation
Incidental parameter problem: increasing N also increases the numberof ci to be estimated
Solution: use the within gruop (WG) transformation
. Numerically, LSDV and WG transformation lead to the same estimatefor β (result of partitioned regression – just algebra)
. Estimate of β “easier” to compute with WG (an important issue someyears ago...)
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Linear Model Fixed effect estimation
Within group (WG) transformation
We transform the model in order to remove the term ci
For individual i at time t: yit = x ′itβ + ci + uit
For individual i , the average over the T periods is: yi = x ′iβ + ci + ui
Therefore by taking deviations from group means, we get:
yit − yi = (xit − xi )′β + (uit − ui )
Under the assumption of strict exogeneity, we can apply OLS the thetransformed data to get a consistent estimate of β
Estimates of ci can be computed by ci = yi − βxi (unbiased; notconsistent for fixed T and N →∞)
The F test can be applied for the joint significance of ci
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Linear Model Fixed effect estimation
Fixed effect estimationyit = x′itβ + ci + uit
WG: OLS regression of yit − yi on xit − xi (removes ci )
Assumptions:
(1) Strict exogeneity: E [x′isuit ] = 0 for each s, t = 1, ...,T
(2) Rank condition: rank(∑T
t=1 E [x ′it xit ])
= rank E [X ′i Xi ] = K , where
xit = xit − xi
No assumption about the correlation of ci and each xit : consistenteven if E [ci |xi] 6= 0
. More robust than RE, but effect of time-invariant variables cannot beidentified
Efficient if uit homoschedastic and uncorrelated over time
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Linear Model Fixed effect estimation
First difference (FD)
Another way to remove the term ci from the equation is to take firstdifferences:
yit − yit−1 = (xit − xit−1)′β + (uit − uit−1)
OLS can be applied for estimation if ∆xit is uncorrelated with ∆uit
(satisfied under strict exogeneity)
However it is not efficient, due to the correlation introduced amongthe error terms ∆uit and ∆uit−1 (if uit is uncorrelated over time)
For example, for T = 3
∆yi2 = ∆x ′i2β + (ui2 − ui1)
∆yi3 = ∆x ′i3β + (ui3 − ui2)
GLS estimation could be employed to solve the problem: you get the“within-group” estimator
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Linear Model Fixed effect estimation
First difference estimation
yit = x′itβ + ci + uit
FD: OLS regression of ∆yit on ∆xit (removes ci )
Assumptions:
(1) E [∆x′it∆uit ] = 0, that is E [x′isuit ] = 0 for eacht = 1, ...,T ; s = t − 1, t, t + 1
. satisfied under strict exogeneity
(2) Rank condition: rank E [∆X ′i ∆Xi ] = K
No assumption about the correlation of ci and each xit : consistenteven if E [ci |xi] 6= 0
. More robust than RE, but effect of time-invariant variables cannot beidentified
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Linear Model Fixed effect estimation
Non-spherical uit
What if Ωi 6= σ2c ιT ι
′T + σ2
uIT ?. That is, uit heteroskedastic and/or correlated over time
If E (ci |xi ) 6= 0, then the FE estimator is still consistent (under strictexogeneity); it is no longer efficient. Robust formulas should be employed for the computation of the
standard errors!. βFD is efficient if uit is a random walk (∆uit serially uncorrelated)
If E (ci |xi ) = 0 (the orthogonality condition holds), then the REestimator remains consistent (under strict exogeneity); it is no longerefficient. A more general estimator of Ωi can be obtained as:
Ωi = N−1N∑i=1
vi v′i
with vi pooled OLS residuals (efficient in the more general case). Assume alternative specifications: parametric assumptions about the
correlation structure in uit , e.g. AR(1) and perform GLS estimation
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Linear Model Which one to choose?
WG vs. FDWhich one to choose?
WG: OLS regression of (yit − yi ) on (xit − xi )
FD: OLS regression of ∆yit on ∆xitBoth WG and FD produces unbiased and consistent estimates of theparameter of interest β, as ci is removed from the regression. The estimate of β is not affected by the correlation (if any) between ci
and xi. Generally, if the two estimators are different, this can be interpreted as
evidence against the assumption of strict exogeneity
When T = 2, βWG = βFDIf T ≥ 3, under homoschedasticity of u, βWG is to be preferredbecause efficientIf uncorrelation and homoschedasticity of u is not satisfied, the choicedepends on the assumptions about uit :. If uit is a random walk, then ∆uit is serially uncorrelated: βFD is
efficient. In the more general set up, use FD or WG with robust s.e.!
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Linear Model Which one to choose?
FE vs. RE (1)Which one to choose?
Traditional approach: ci treated either as parameter to be estimatedvs. random disturbance
. “Philosophical” issue
. Wrongheaded in microeconometrics applications
Modern terminology: fixed effects estimation vs. random effectsestimation
. The difference is in the assumptions about E [ci |xi ]
. FE allows consistent estimation of β even in cases where ci iscorrelated with xi
. RE requires ci to be uncorrelated with xi
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Linear Model Which one to choose?
FE vs. RE (2)Which one to choose?
FE: OLS regression of (yit − yi ) on (xit − xi )
. Only “within” variation is considered
RE: OLS regression of (yit − θyi ) on (xit − θxi )
. Both “within” and “between” variation are employed for estimation
. It is possible to show that βRE = ΛβB + (IK − Λ)βFE with βB obtainedfrom the OLS regression of yi on xi
. θ = 1− σu√σ2u+Tσ2
c
: if T →∞, RE = FE – you need a different
framework!
Key: correlation between ci and xit. If E [ci |xit ] = E [ci ] (= 0): RE is consistent and efficient, FE consistent. If E [ci |xit ] 6= E [ci ]: FE consistent, but RE is not
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Linear Model Which one to choose?
FE vs. REThe Hausman test
Both FE and RE assume strict exogeneity
If E [ci |x] = E [ci ] (= 0)
. Both βFE and βRE are consistent for β: βFE − βRE ≈ 0
. βRE is efficient: Var(βFE ) is “greater than” Var(βRE )
If E [ci |x] 6= E [ci ]
. βFE is consistent, but βRE is biased: βFE − βRE 6= 0
We can apply the Hausman test
(βFE − βRE )′(Var(βFE )− Var(βRE ))−1(βFE − βRE ) ∼ χ2K
Remark: Two maintained hypotheses (not tested!): (i) strictexogeneity; (ii) random effect structure of the covariance (under thenull, RE has to be efficient: valid under spherical uit)
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Linear Model Which one to choose?
Between FE and RE: Correlated random effects(Mundlak, 1978; Chamberlain, 1982, 1984)
RE assumes no correlation between ci and xit
Richer models can be specified that relax this assumption
Mundlak (1978): ci = x ′iπ + wi with wi i.i.d.
. GLS estimation of the regression of yit on xit and xi produces the fixedeffect estimator
Chamberlain (1982, 1984): ci = x ′i1π1 + ...+ x ′iTπT + wi
. Estimation of the extended model by minimum distance methodproduces the fixed effect estimator
In nonlinear models, fixed effect models are not always estimable andricher RE models provide an alternative approach
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Linear Model Which one to choose?
FE vs. REA robust version of the Hausman test
Starting from the Mundlak (1978) definition (linear projection):ci = x ′iπ + wi with wi i.i.d. we can write:
yit = x ′itβ + ci + uit = x ′itβ + x ′iπ + (wi + uit)
GLS estimation produces: βGLS = βFE and πGLS = βBET − βFE(βBET : OLS estimate in the regression of yi on xi )
Hausman test can be carried out by testing H0: π = 0 in theextended regression
Robust version of the Hausman test: use a robust Wald statistic inthe context of pooled OLS (strict exo is still needed, but we can relaxon efficiency of RE under the null)
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Goodness of fit
The R2 with panel data
R2 as the square of correlation coefficient between observed and fittedvalues
Total variability can be decomposed into within and betweenvariability:
1
NT
∑i ,t
(yit − y)2 =1
NT
∑i ,t
(yit − yi )2 +
1
NT
∑i ,t
(yi − y)2
STATA provides three “R2” statistics:
. R2within = corr 2((xit − xi )
′βFE , yit − yi )
. R2between = corr 2(x ′i βB , yi )
. R2overall = corr 2(x ′it βOLS , yit)
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Discussion
DiscussionSource of the examples: Wooldridge
Two questions:. Is the unobserved effect ci uncorrelated with xit for all t?. Is the strict exogeneity assumption (conditional on ci ) reasonable?
Examples:
(a) Program evaluation
log(wageit) = θt + z′itγ + δ1progit + ci + uit
(b) Distributed Lag Model (Hausman, Hall, Griliches, 1984)
patentsit = θt + z′itγ + δ0RDit + δ1RDit−1 + ...+ δ5RDit−5 + ci + uit
(c) Lagged Dependent Variable
log(wageit) = β1 log(wageit−1) + ci + uit
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Main References
Main References
Baltagi BH (2001): Econometric Analysis of Panel Data, John Wiley& Sons Ltd.
Chamberlein G (1984): Panel Data, in Griliches and Intriligator, (eds.)Handbook of Econometrics, Vol.2, Elsevier Science, Amsterdam
Greene, WH (2003): Econometric Analysis, Prentice Hall, ch.13
Hsiao C (2003): Analysis of Panel Data, Cambridge University Press
Mundlak Y (1978): “On the Pooling of Time Series and CrossSection Data”, Econometrica, 46(1), 69-85
Verbeek M (2006): A Guide to Modern Econometrics, ch. 10
Wooldridge, JM (2002): Econometric Analysis of Cross Section andPanel Data, MIT Press: Cambridge, ch.10
Laura Magazzini (@univr.it) Panel Data: Linear Models 44 / 45
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