Topic 3: The Generalized Method of Moments

 In the previous IV estimator we have considered the case where the number of instruments is equal to the number of coefficients we want to estimate  Size of Z is the same as the size

Dr Pham Thi Bich Ngoc

Hoa Sen University

ngoc.phamthibich@hoasen.edu.vn

 Simple moment conditions

0 ˆ

0 ]

, cov[

0 ˆ

0 ]

[

' 1

t T

X X

ˆ (

0 ]

0 )

ˆ (

' 0

] '

E

X X

 IV is a MM estimator

 MM estimator:

0 ]

, cov[

but ,

0 ]

,

0 )

ˆ (

' 0

] '

X Z

y

 In the previous IV estimator we have considered

the case where the number of instruments is

equal to the number of coefficients we want to

estimate

 Size of Z is the same as the size of IV

 What happens if the number of instruments is

greater than the number of coefficients?

 Essentially, the number of equations is greater

than the number of coefficients you want to

estimate: model is over-identified

 Maintain the moment condition as before

W Z

Z Z

' )

' ( )

' '

ˆ '

(

' '

1 1

 First order conditions:

 MM estimator (looks like an IV estimator

with more instruments than parameters

to estimate):

0 )

ˆ (

' )

' ( '

Index: i = 1, ,N for individuals

g = 1, ,G for equations (this would be t=1, T for a panel) Data matrices: G rows,

y X β + ε

i1,1 i1

i1,2 i1 i1 i1

i1 1 i2 2

iG G

i1 i2

iG

i1 i2

Z ε

z

1 2

iG G

L rows

L rows

0

(1) In case of TRIANGLE RELATIONSHIP:

IV is an external variable

◦ ivregress gmm depvar1 [varlist1] (depvar2 =

varlistiv)

Estat endog / estat overid

◦ ivreg2 depvar1 [varlist1] (depvar2 = varlistiv),

(1) In case of TRIANGLE RELATIONSHIP:

(2) In case IV is the lagged endogenous

Linear generalized method of moments (GMM)

Arellano-Bond (1991); Arellano-Bover (1995)/ Blundell-Bond(1998)

yit = αyi,t-1 + x’itβ + αi + εit

Reasons: True dependence, observed or unobserved heterogeneity, error correlation

yi,t-1 is correlated with αi OLS – inconsistent (Nickell, 1981)

(yi,t-1 - yi,t-2 ) is correlated with (εit - εi,t-1 )

collapsed

DIF – GMM

 STATA/panel data

◦ xtabond/xtabond2 depvar varlist [if exp] [in

range]

◦ Options:

 gmmstyle(varlist [, laglimits(# #) collapse orthogonal

equation({diff | level | both}) passthru {cmdab:sp: d:)}

 ivstyle(varlist [, equation({diff | level | both}) passthru

mz])

 Common Obtions:

◦ noleveleq

specifies that level equation should be excluded

from the estimation, yielding difference rather than system GMM

 Common Obtions:

◦ robust

For one-step estimation, robust specifies that therobust estimator of the covariance matrix of theparameter estimates be calculated The resultingstandard error estimates are consistent in thepresence of any pattern of heteroskedasticityand autocorrelation within panels

In two-step estimation, the standard covariancematrix is already robust in theory but typicallyyields standard errors that are downward biased.twostep robust requests Windmeijer’s finite-sample correction for the two-step covariancematrix

◦

tranh phuong sai thay doi va da cong tuyen

 Common Obtions:

twostep

twostep specifies that the two-step estimator is to

be calculated instead of the one-step

ivstyle()

specifies a set of variables to serve as standardinstruments, with one column in the instrumentmatrix per variable Normally, strictly exogenousregressors are included in ivstyle options, in order

to enter the instrument matrix, as well as beinglisted before the main comma of the commandline

trong ngoac la nhung bien ko b noi sinh

 Common Options:

gmmstyle : specifies a set of variables to be used

as bases for "GMM-style" instrument sets

laglimits(a b): for the transformed equation, laggedlevels dated t-a to t-b are used as instruments,while for the levels equation, the first-differencedated t-a+1 is normally used

+ a and b can each be missing ("."); a defaults to 1and b to infinity

+ E.g., gmm(w, lag(2 )), the standard treatment for

an endogenous variable, is equivalent to gmm(L.w,lag(1 )), thus gmm(L.w)

+ the lag limits are a and b, then lags of thespecified variables in differences dated t-b to t-aare used tat ca bien qua khu deu dung lam bien cong cu

 Arellano-Bond test for AR(1) in first differences.

The null hypothesis: no autocorrelation

If p-value <5%  AR(1) in first differences exist

 The Sargan-Hansen test is a test of overidentifying

restrictions (REF Topic 19a)

If p-value >5%  instruments are valid

The augmented Cobb- Doughlas model:

LnY = F (LnK, LnL, LnM, LnFDI)

INPUT ENDOGENEITY PROBLEM