Chapter 10: Basic Linear Unobserved Effects Panel Data. Models:

Chapter 10: Basic Linear Unobserved Effects Panel Data Models:

Microeconomic Econometrics I Spring 2010

10.1 Motivation: The Omitted Variables Problem

We are interested in the partial effects of the observable explanatory variables x_jin the popu- lation regression function

E( y|x1, x₂, ..., x_k, c) (10.1)

• An unobserved,time-constant variable is called an unobserved effect in panel data analy- sis.

• Such a data set is usually called a balanced panel because the same time periods are available for all cross section units.

10.2 Assumptions about the Unobserved Effects and Explanatory Variables

10.2.1 Random or Fixed Effects?

• The basic unobserved effects model (UEM) can be written, for a randomly drawn cross section observation i , as

y_it= xitβ+ ci+ uit, t = 1,2,..., T (10.11)

• In addition to unobserved effect, there are many other names given to c_iin applications:

unobserved component, latent variable, and unobserved heterogeneity are common.

• If i indexes individuals, then c_i is sometimes called an individual effect or individual heterogeneity; analogous terms apply to families, firms, cities, and other cross-sectional units.

• The u_it are called the idiosyncratic errors or idiosyncratic disturbances because these change across t as well as across i .

• Nevertheless, later we will label two different estimation methods random effects esti- mation and fixed effects estimation.

10.2.2 Strict Exogeneity Assumptions on the Explanatory Variables

With an unobserved effect the most revealing form of the strict exogeneity assumption is

E( y_it|xi1, x_i2, ..., x_iT, c_i) = E(yit|xit, c_i) = xitβ+ ci (10.12)

• When assumption (10.12) holds, we say that the x_it: t = 1,2,..., T are strictly exogenous conditional on the unobserved effect c_i.

• Example10.1(Program Evaluation):

log(wa ge_it) =θt+ zitγ+δ1pro g_it+ ci+ uit (10.16)

• Example 10.2 (Distributed Lag Model):

patents_it=θt+ zitγ+δ0RD_it+δ1RD_i,t−1+ · · · +δ5RD_i,t−5+ ci+ uit (10.17)

• Example 10.3 (Lagged Dependent Variable):

log(wa ge_it) =β1log(wa ge_i,t−1) + ci+ uit, t = 1,2,..., T (10.18)

10.3 Estimating Unobserved Effects Models by Pooled OLS

Under certain assumptions, the pooled OLS estimator can be used to obtain a consistent esti- mator of_βin model (10.11). Write the model as

y_it= xitβ+υit, t = 1,2,..., T (10.21)

where_υit≡ ci+ uit, t = 1,2,...T are the composite errors.

10.4 Random Effects Methods

10.4.1 Estimation and Inference under the Basic Random Effects Assumptions

• As with pooled OLS, a random effects analysis puts c_i into the error term.

• Assumption RE.1:

1. E(u_it|xi, c_i) = 0, t = 1,..., T.

2. E(c_i|xi) = E(ci) = 0 where x_i≡ (xi1, x_i2, . . . , x_iT).

• Assumption RE.2: rank E(X⁰_iΩ⁻¹X_i) = K.

• WhenΩ has the form (10.31), we say it has the random effects structure.

• Assumption RE.3:

1. E(u_iu⁰_i|xi, c_i) =σ²uI_T. 2. E(c²_i|xi) =σ²_c

• In a panel data context, the FGLS estimator that uses the variance matrix (10.33) is what is known as the random effects estimator:

βˆRE=³X^N

i=1

X⁰_iΩˆ⁻¹X_i´−1³X^N

i=1

X⁰_iΩˆ⁻¹y_i´

(10.34)

• Example 10.4(RE Estimation of the Effects of Job Training Grants):

log( ˆscra p) = .415 − .093 d88 − .270 d89 + .548 union − .215 grant − .377 grant−1

(.243) (.109) (.132) (.411) (.148) (.205)

10.4.2 Robust Variance Matrix Estimator

10.4.3 A General FGLS Analysis

If the idiosyncratic errors u_it: t = 1,2,..., T are generally heteroskedastic and serially corre- lated across t, a more general estimator ofΩ can be used in FGLS:

Ω = Nˆ ⁻¹X^N

i=1

ˆˆv_iˆˆv⁰_i (10.38)

10.4.4 Testing for the Presence of an Unobserved Effect

From equation (10.37), we base a test of H₀:_σ²_c= 0 on the null asymptotic distribution of

N^−1/2

N

X

i=1 T−1X

t=1 T

X

s=t+1

υˆit_υˆ_is (10.39)

which is essentially the estimator ˆ_σ²_c scaled up byp N.

10.5 Fixed Effects Methods

10.5.1 Consistency of the Fixed Effects Estimator

Again consider the linear unobserved effects model for T time periods:

y_it= xitβ+ ci+ uit, t = 1,..., T (10.41)

• Assumption FE.1: E(u_it|xi, c_i) = 0, t = 1,2,..., T.

• In this section we study the fixed effects transformation, also called the within transfor- mation.

• The fixed effects (FE) estimator, denoted by ˆ_{βF E},is the pooled OLS estimator from the regression

¨y_iton ¨x_it, t = 1,2,..., T; i = 1,2,..., N (10.48)

• This set of equations can be obtained by premultiplying equation (10.42) by a time- demeaning matrix.

• Assumption FE.2: rank³ PT

t=1E( ¨x⁰_it¨x_it)´

= rankh

E( ¨X⁰_iX¨_i)i

= K.

• It is also called the within estimator because it uses the time variation within each cross section.

• The between estimator, which uses only variation between the cross section observations, is the OLS estimator applied to the time-averaged equation (10.45).

10.5.2 Asymptotic Inference with Fixed Effects

• Assumption FE.3: E(u_iu⁰_i|xi, c_i) =σ²_uI_T.

• To see how to estimate_σ²_u, we use equation (10.51) summed across t :PT

t=1E( ¨u²_it) = (T − 1)_σ²_u, and so [N(T − 1)]⁻¹PN

i=1

PT

t=1E( ¨u²_it) =σ²_u.Now, define the fixed effects residuals as

ˆ

u_it= ¨yit− ¨xit_βˆ_{F E}, t = 1,2,..., T; i = 1,2,..., N (10.55)

which are simply the OLS residuals from the pooled regression (10.48).

• Example 10.5(FE Estimation of the Effects of Job Training Grants):

log( ˆscra p) = −.080 d88 − .247 d89 − .252 grant − .422 grant−1

(.109) (.133) (.151) (.210)

10.5.3 The Dummy Variable Regression

• The estimator of_βobtained from regression (10.57) is, in fact, the fixed effects estimator.

This is why ˆ_{βF E} is sometimes referred to as the dummy variable estimator.

10.5.4 Serial Correlation and the Robust Variance Matrix Estimator

Applying equation (7.26), the robust variance matrix estimator of ˆ_{βF E}is

Ava ˆr( ˆ_{βF E}) = ( ¨X⁰X )¨ ⁻¹¡

N

X

i=1

X¨⁰_iuˆ_iuˆ⁰_iX¨_i¢( ¨X⁰X )¨ ⁻¹ (10.59)

• Example 10.5 (continued):

log( ˆscra p) = −.080 d88 − .247 d89 − .252 grant − .422 grant−1

(.109) (.133) (.151) (.210) [.096] [.193] [.140] [.276]

10.5.5 Fixed Effects GLS

• Assumption FEGLS.3: E(u_iu⁰_i|xi, c_i) =Λ, a T × T positive definite matrix.

• The fixed effects GLS (FEGLS) estimator is defined by

βˆF EGLS=¡

N

X

i=1

X¨_i⁰Ωˆ⁻¹X¨_i¢₋₁¡

N

X

i=1

X¨_i⁰Ωˆ⁻¹¨y_i¢

where ¨X_iand ¨y_iare defined with the last time period dropped.

• Assumption FEGLS.2: rankE( ¨X⁰_iΩˆ⁻¹X¨_i) = K.

10.5.6 Using Fixed Effects Estimation for Policy Analysis

10.6 First Differencing Methods

10.6.1 Inference

• Assumption FD.1: Same as Assumption FE.1.

Lagging the model (10.41) one period and subtracting gives

∆yit=∆xitβ+∆uit, t = 2,3,..., T (10.63)

where∆yit= yit− y_i,t−1,∆xit= xit− x_i,t−1, and∆uit= uit− u_i,t−1.

• As with the FE transformation, this first-differencing transformation eliminates the un- observed effect c_i.

• The first-difference (FD) estimator, ˆ_βFD, is the pooled OLS estimator from the regression

∆yiton∆xit, t = 2,..., T; i = 1,2,..., N (10.65)

• Assumption FD.2: rank³ PT

t=2E(∆x⁰_it∆xit)´

= K.

• Assumption FD.3: E(e_ie⁰_i|xi1, . . . , x_iT, c_i) =σ²eI_T−1, where e_i is the (T − 1) × 1 vector con- taining e_it,t = 2,..., T.

10.6.2 Robust Variance Matrix

If Assumption FD.3 is violated, then, as usual, we can compute a robust variance matrix. The estimator in equation (7.26) applied in this context is

A ˆvar( ˆ_{βF D}) = (∆X⁰∆X)⁻¹³X^N

i=1

∆X⁰_iˆe_iˆe⁰_i∆Xi

´

(∆X⁰∆X)⁻¹ (10.70)

where∆X denotes the N(T −1)× K matrix of stacked first differences of xit.

• Example 10.6 (FD Estimation of the Effects of Job Training Grants):

∆log(scrapit) =δ1+δ2d89_t+β1∆grantit+β2∆grant_i,t−1+∆uit

10.6.3 Testing for Serial Correlation

The regression is based on T − 2 time periods:

ˆe_it= ˆp1ˆe_i,t−1+ errorit, t = 3,4,..., T; i = 1,2,..., N (10.71)

• Example10.6 (continued):

10.6.4 Policy Analysis Using First Differencing

In this one case,∆progi=∆progi2,and the first-differenced equation can be written as

∆yi2=θ2+∆zi2γ+δ1 pro g_i2+∆ui2 (10.72)

• When ∆zi2 is omitted, the estimate of _δ1 from equation (10.72) is the difference-in- differences (DID) estimator (see Problem 10.4): ˆ∆1=∆ytreat −∆ycortrol.

10.7 Comparison of Estimators

10.7.1 Fixed Effects versus First Differencing

With more than two time periods, a test of strict exogeneity is a test of H₀:_γ= 0 in the expanded equation

∆yt=∆xtβ+ wtγ+∆ut, t = 2,..., T

where w_tis a subset of x_t(that would exclude time dummies).

10.7.2 The Relationship between the Random Effects and Fixed Effects Estimators

Using the fact that j⁰_Tj_T= T, we can writeΩ under the random effects structure as

Ω =σ²uI_T+σ²cj_Tj⁰_T=σ²uI_T+ Tσ²cj_T( j⁰_Tj_T)⁻¹j⁰_T=σ²uI_T+ Tσ²cP_T= (σ²u+σ²c)(P_T+ηQ_T)

where P_T≡ IT− QT= jT( j⁰_Tj_T)⁻¹j⁰_T and_η≡σ²u/(_σ²_u+ Tσ²c).

• Equation (10.76) shows that the random effects estimator is obtained by a quasitime demeaning: rather than removing the time average from the explanatory and dependent variables at each t, random effects removes a fraction of the time average.

• Example 10.7 (Job Training Grants):

10.7.3 The Hausman Test Comparing the RE and FE Estimators