Instrumental Variables Estimation - Econometric Analysis of Panel Data - Lecture Slides, Slides of Econometrics and Mathematical Economics

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Econometric Analysis of Panel Data

8. Instrumental Variables Estimation

Structure and Regression

it it it it it it it

y E , i,t" = sibling t in family i

E ' true ' education measurable only with error

S measured 'schooling' = E w , w=measurement error

it

Earnings (structural) equation

x β

Reduced

it it it it it it it

2 it it it w

y (S w )

= S + ( w )

for least squares (OLS or GLS)

Cov[S , ( w )] 0

Consistency relies on this covariance equaling 0.

How

it it

form

x β

x β

Estimation problem

to estimate β and γ (consistently)?

Exogeneity

Structure E[ ] = Regression [ ] E[ ]= Projection "Regression of on "

y = Xβ + ε ε|X g(X) 0 y = Xβ + g(X) + ε - g(X) = E[y|X] + u, u|X 0 ε = Xθ + w y X

it it it i1 i2 iT

Exogeneity: E[ | ] 0 (current period) Strict Exogeneity: E[ | , ,..., ] 0 (all periods) (We assume no correlation a

ε = ε =

y = X(β + θ) + w The problem : X is not exogenous. x x x x cross individuals.)

The Measurement Error Problem

y x * + ν, E[x]=0, E[v|x]=

x = x* + u, E[u|x*]=0, E[v|u]=

y = x (v u) = x

E[y | x] x E[(v u) | x]

= x - E[u | x]

= x - Cov[u, x] x

Var[x]

= x -

β β ^ 

β β 0 Var[u]^ x

Var[x*] Var[u]

= Var[x*] x x, | | < | |

Var[x*] Var[u]

β ^  = γ γ β

How general is this result?

Least Squares

1 1 1 1

E[ ] E[ ]

( ) ( ) X ( )

= ( / N) ( / N)

plim = p lim( / N) plim( / N)

− − − −

= ′^ ′^ = ′^ ′ +

+ ′^ ′

+ ′^ ′

y = Xβ + ε

y|X Xβ ε|X Xβ

b X X X y X X Xβ ε

β X X X ε

b β X X X ε

β Qγ

The IV Estimator

it it it it it it it it it N i=1 i i,t it it it

[ ], [ ]

E[ | ] 0 E[ | ] E[ (y ) | ] (Using "N" to denote T ) E[(1/N) | ] E[(

= =

ε = ε = − ′ = Σ Σ ε =

1 2 K-1 K 1 2 K-

The variables

X x , x , ...x , x Z x , x , ...x , z

The Model Assumption

z z z z x β z 0

z z 1/N) (^) i,t it (yit (^) it ) | (^) it] E[(1/N) ] = E[(1/N) ] Mimic this condition (if possible) Find ˆ^ so (1/N) = (1/N) ˆ

Σ z − x^ ′ β z = 0 Z'y Z'Xβ

The Estimator :

β Z'y Z'Xβ

Consistency and Asymptotic

Normality of the IV Estimator

plim ˆ plim ( N) N

Asy.Var[ ˆ Asy.Var[ N] ( N) [plim N] (1/

-1ZX ×

-1ZX -1XZ -1ZX -1XZ

β = Z'X Z'y

= β + Z'X Z'ε (β - β ) = Z'X/ (Z'ε/ ) = Q 0 β - β] = Q Z'ε/ Q = 1/ Q Z'ΩZ/ Q =

-1 i,t^ it^ it -

i

N)

(Assuming "well behaved" data)

N ˆ ( N) N = ( N) N N

Invoke Slutsky and Lindberg-Feller for N to assert asymptotic normality.

Estimate Asy.Var[ ˆ] with

 Σ ε     

Q^ -1 ZX^ ΩZZ Q-1XZ

z (β - β ) = Z'X/ Z'X/ w

w

β

2 ,t (y^ it it ˆ)^ [ ] [-1 (^) ][ ]- N or (N-K)

x β (^) Z'X Z'Z X Z ′

Least Squares Revisited

-1 -

-1 -

plim plim ( N) plim( N)

plim ( ) ( ) = ( N) ( N) [ N]

- XX -1XX -1XX -1 -1 - XX XX XX

b = X'X X'y = β + X'X X'ε

b = β + X'X/ X'ε/ = β + Q γ

b - b = β + X'X X'ε - β + Q γ X'X/ X'ε/ - Q γ

Asy.Var[b - Q γ] = Q Asy.Var X'ε/ Q

-1 i,t it it

(1 N) [plim N] (1 N)

(Assuming "well behaved" data)

plim ) ( N) N N = ( N) N ( plim )

Invoke Slutsky and Lind

 Σ^ ε   −   

-1 -1 -1 - XX XX XX XX XX

- XX

= / Q X'ΩX/ Q = / Q Ω Q

x N(b - b = X'X/ Q γ

X'X/ w - w berg-Feller for ( plim ) to assert asymptotic normality. is also asymptotically normally distributed, but inconsistent.

N w - w b

Cornwell and Rupert Data

Cornwell and Rupert Returns to Schooling Data, 595 Individuals, 7 Years Variables in the file are

EXP = work experience, EXPSQ = EXP^2 WKS = weeks worked OCC = occupation, 1 if blue collar, IND = 1 if manufacturing industry SOUTH = 1 if resides in south SMSA = 1 if resides in a city (SMSA) MS = 1 if married FEM = 1 if female UNION = 1 if wage set by unioin contract ED = years of education BLK = 1 if individual is black LWAGE = log of wage = dependent variable in regressions

These data were analyzed in Cornwell, C. and Rupert, P., "Efficient Estimation with Panel Data: An Empirical Comparison of Instrumental Variable Estimators," Journal of Applied Econometrics, 3, 1988, pp. 149-155. See Baltagi, page 122 for further analysis. The data were downloaded from the website for Baltagi's text.

Wage Equation with

Endogenous Weeks

logWage=β 1 + β 2 Exp + β 3 ExpSq + β 4 OCC + β 5 South + β 6 SMSA + β 7 WKS + ε

Weeks worked is believed to be endogenous in this equation.

We use the Marital Status dummy variable MS as an exogenous variable.

Condition (5.3) Cov[MS, ε] is assumed.

Auxiliary regression:

In the regression of WKS on [1,EXP,EXPSQ,OCC,South,SMSA,MS, ]

MS significantly “explains” WKS.

A projection interpretation: In the projection

X itK =θ 1 x 1it + θ 2 x 2it + … + θ K-1 x K-1,it + θ K z it , θ K ≠ 0.

(One normally doesn’t “check” the variables in this fashion.

Application: IV for WKS in Rupert

+----------------------------------------------------+ | Ordinary least squares regression | | Residuals Sum of squares = 678.5643 | | Fit R-squared = .2349075 | | Adjusted R-squared = .2338035 | +----------------------------------------------------+ +---------+--------------+----------------+--------+---------+ |Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | +---------+--------------+----------------+--------+---------+ Constant 6.07199231 .06252087 97.119. EXP .04177020 .00247262 16.893. EXPSQ -.00073626 .546183D-04 -13.480. OCC -.27443035 .01285266 -21.352. SOUTH -.14260124 .01394215 -10.228. SMSA .13383636 .01358872 9.849. WKS .00529710 .00122315 4.331.

Application: IV for wks in Rupert

+----------------------------------------------------+ | LHS=LWAGE Mean = 6.676346 | | Standard deviation = .4615122 | | Residuals Sum of squares = 13853.55 | | Standard error of e = 1.825317 | | Fit R-squared = -14.64641 | | Adjusted R-squared = -14.66899 | | Not using OLS or no constant. Rsqd & F may be < 0. | +----------------------------------------------------+ +---------+--------------+----------------+--------+---------+ |Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | +---------+--------------+----------------+--------+---------+ Constant -9.97734299 3.59921463 -2.772. EXP .01833440 .01233989 1.486. EXPSQ -.799491D-04 .00028711 -.278. OCC -.28885529 .05816301 -4.966. SOUTH -.26279891 .06848831 -3.837. SMSA .03616514 .06516665 .555. WKS .35314170 .07796292 4.530. OLS WKS .00529710 .00122315 4.331.

Generalizing the IV Estimator - 2

1 1 2 2

2 2 1 2 1 2 1 2 2

Define the set of instrumental variables

K linear combination of the M s

= an MxK matrix. is NxK

= [ , ]= [ , ]= [ , ]

Why must M be K? So can have full column

Z

Z X

Z W

WP

P WP

Z Z Z X Z X WP

Z rank.

Generalizing the IV Estimator

2

By the definitions, is a set of instrumental variables.

ˆ [ ]

is consistent and asymptotically normally distributed.

( ˆ^ )'( ˆ)

N or (N-K)

Assuming homoscedasticity and no autocorelatio

ε

Z

β = Z'X Z'y

y Xβ y Xβ

2 -1 -

n,

Est.Asy.Var[ ˆ] ˆ [ ] [ ]

β = σε Z'X Z'Z X'Z