dynamicslidesEcon507.pdf

Dynamic Panels and GMM

Walter Sosa-Escudero

Econ 507. Econometric Analysis. Spring 2012

April 23, 2012

Walter Sosa-Escudero Dynamic Panels and GMM

Preliminary issues: the GIVE estimator

Y = X + u, Z valid instruments, p K, V (u|X) = 2I

Premultiply by Z

Z Y = Z X + Z uY = X + u

It is easy to see that E(u|X) = 0, and that

V (u|X) = E(uu) = E(Z uuZ) = 2(Z Z) ,

non-spherical.


Hence, the BLUE is GLS:

GLS = (X1X)1X1Y

= (X Z(Z Z)1Z X)1X Z(Z Z)1Z Y= (X PZX)1X PZY = IV

IV is a GLS of a transformed model.


Generalization: V (u|X) =

In this case: V (u|X) = E(uu) = E(Z uuZ) = Z Z Replacing:

GLS = (X1X)1X1Y

= (X Z(Z Z)1Z X)1X Z(Z Z)1Z Y GIVThis is an efficient IV estimate under heteroskedasticity or serialcorrelation. We will label it the generalized IV.

Q? What happens if p = K?


A simple dynamic model

The goal is to estimate a dynamic model like:

yit = yi,t1 + xit + uit

with uit = i + vit

There are two sources of persistence: i y yi,t1.

Examples:

Unemployment (Galiani, et al. 2003).

Growth convergence (Islam, 1995).


Standard estimators are inconsistent

yit = yi,t1 + xit + uit

with uit = i + vit

By construction yit and yi,t1 depend on i. yi,t1 is correlatedwith uit = i + vit.

OLS is inconsistent (v/Hsiao (1986, pp.77)

Random effects is inconsistent


The fixed effects estimator is inconsistent as well. FE is based on:

yit = yi,t1 + x

it + u

it

where variables with * are deviations from individual means.Note:

yi,t1 = yi,t1 1

T 1Tt=2

yi,t1, uit = vit 1

T

Tt=1

vit

It is easy to show that yi,t1 y uit are correlated. For example,

both depend on vi,t1


If xit is exogenoous, Nickell (1981):

Cov(yi,t1, uit)

p 2v

T 2(T 1) T + T

(1 )2when n, for fixed T . Then, FE is inconsistent.

If T , Cov(yi,t1, uit)p 0, hence the inconsistency of

FE is related to T being small.

If > 0 the bias is negative.

How large is the bias?


Bias of the FE estimator

= 0.5, 0.8, 0.3

Bias diminishes with T and increases with


The Anderson/Hsiao estimator

A transformation that elliminates the individual effect consists insubstracting yi,t1:

yi,t = yi,t1 + xit + vit

OLS is inconsistent: yi,t1 are trivially correlated with vit.Both depend on vi,t1.

Note yi,t2 = yi,t2 yi,t3 though correlated with yi,t1(both depend on yi,t2) it is not correlated with vit.

Anderson and Hsiao (1981): IV using yi,t2 o yi,t2 asinstrument.

Arellano (1989): yi,t2 performs better.


The Arrellano/Bond estimator

For simplicity, consider = 0:

yit = yi,t1 + i + vit

Substracting yi,t1 we have:

yi,t = yi,t1 + vit

(i is gone!). The first period for which we observe thisrelationship is t = 3

yi,3 = yi,2 + vi3

In this case yi1 is a valid instrument: correlated withyi,2 yi,2 yi,1, but not with vi3 vi,3 vi,2.Q? How many observations do we have to estimate this?


Consider now period, t = 4:

yi,4 = yi,3 + vi4

but in this case yi2 and yi1 are valid instruments.

Using this logic, the valid instruments for period T areyi1, yi2, . . . , yi,T2.

In general, the previous argument implies the following(T 2)(T 1)/2 moment conditions:

E[vit yi,tj ] = 0, j = 2, . . . , T 1; t = 3, 4, . . . , T

From here, we will construct a GMM estimator.


Instruments in Anderson-Hsiao and Arellano-Bond

AH ABi t yit yi,t1 Dy Dyi,t1 yi,t2 1 2 3 4 5 61 1 a1 2 b a1 3 c b c-b a a 0 0 0 0 01 4 d c d-c c-b b 0 a b 0 0 01 5 e d e-d d-c c 0 0 0 a b c2 1 f2 2 g f2 3 h g h-g f f 0 0 0 0 02 4 i h i-h h-g g 0 f g 0 0 02 5 j i j-i i-h h 0 0 0 f g h


For each individual, define Wi as follows:

Wi =

[yi1] 0

[yi1, yi2]. . .

0 [yi1, . . . , yi,T2]

is (T 2) (T 2)(T 1)/2 matrix. Moment conditions can beexpressed in vector form as:

E(W ivi) = 0, i = 1, . . . , N

The sample counterpart is(1/N)N

i=1Wivi = (1/N)W

v,with W a matrix that stacks Wi vertically, for all individuals, viis defined similarly.


The GMM estimator of will be:

= argmin (Wv)AN (W v)

where AN is any sequence of symmetric pd matrices. The optimalGMM estimator can be based on:

AN = (1/N)i

W i viviWi

where vi are residuals based on any preliminary consistentestimation of . Note that, based on Hansens result, we are usinga consistent estimate of the variance of W ivi.

Now the problem is to find a preliminary estimator for .


Preliminary estimation: In matrix terms:

y = y1 + v

W is a matrix of valid instruments for this model. It is easy tocheck:

V (v) = E(vv) = 2v (IN G) = 2v (See Baltagi (2001, pp. 132) for a definition of G).Arellano and Bond (91): use the GIV estimator. Premultiply to get

W y = W y1 +W v

and replacing in the formula for the GIV estimator V IG:

1 = (y1W (W

W )1W y1)1y1W (WW )1W y

is the Arellano-Bond preliminary estimator.


Consistency depends crucially on E[vit yi,tj ] = 0, implyingno serial correlation. It is important to check this hypothesis.

6= 0. Depends on the exogeneity status of x.Finite samples: preliminary estimator works fine.


Small sample performance

Judson and Owen (1999): empirical comparisson.OLS, LSDV(FE), GMM1(AB preliminary), AH (Anderson-Hsiao),LSDVC (Kiviet)

OLS and LSDV tend to be very biased, even when T = 20 (12%and when = 0.8).

When T = 30 the bias is small.

In terms of performance, LSDVC seems to be best.

LSDVC no avabilable for unbalanded panels, not easy to compute.

T 10: GMM1, T 20 GMM1 o AH, T 30: FE


Empirical Illustration


dynamicslidesEcon507.pdf

Documents

Transcript of dynamicslidesEcon507.pdf