GMM - Basics (Generalized Method of Moments) (Applied Econometrics)

 ■ GMM

GMM is one of the generic methods to identify the parameters in the statistical models.

It uses the moment conditions that are the functions of the model parameters and the data, such that their exception is zero at the parameters' true values. The GMM is also a dynamic panel estimator.

We know that the Panel data is a Longitudinal data and generally (as discussed in the previous blogs) the T and N should be as.

● The case for GMM

Let's assume the linear regressions with the endogenous regressors,

Y = X' beta + u
               Where, Y and u are the N×1 vectors; beta is a K×1 vector of the unknown parameters.
X is a N×K matrix of explanatory variables.

Because of the assumption of the endogeneity, we assume a matrix Z that is N×L and L>K.

The Z matrix is assumed to comprise a set of variables that are highly correlated with X but orthogonal to u (i.e. a highly set of valid instruments).

● GMM specifies

a) N (Number of cross sections or groups) > T (Time Space).

b) It uses instrumental variable (IV) estimation.

c) The instruments, Z must be exogenous, E(Z',u) = 0

d) Number of instruments,  Z <= number of groups, N.

The GMM estimators are of two kinds:

i) Difference GMM
ii) System GMM

Let's say the GMM is designed to:

- Dynamic Panel Models

- The T is small and large N panels

- The independent variables that are not strictly exogenous, meaning they are correlated with past and possibly current realisation of the error term (endogeneity).

- The fixed effect are arbitrarily idistributed.

- The heteroscedasticity.

- Autocorrelation within the panel or groups.

Basically there are two instruments in GMM:

- Internal Instruments --> gmmstyle ()
- External Instruments --> ivstyle ()

i) Difference GMM

- Arellano and Bond (1991)

1. It coorects endogeneity by:

-Transforming all the regressors through differencing.
-By removing the Fixed Effects

2. The first difference transformations has weakness. It
-Subtracts the previous observations from the contemporaneous one thereby magnifies gaps in the unbalanced panel.

ii) System GMM

Two people Arellano and Bover (1995) and Blundell and Bond  (1998).

- Coorects Endogeneity by:

□ Introducing more instruments to dramatically improve efficiency.

□ Transforms the instruments to make them uncorrelated (exogenous) with the fixed effects.

- Builds a system of two equations: the original equation, the transformed one equation.

- Uses orthogonal deviations: Instead  of subtracting the previous observation from the contemporaneous one, it subtracts the average of all future available observations of a variable. No matter how many gaps, it is computable for all observations of a varable except the last for each ibdividuals so that it minimises the data loss.

To be contd...

Thank you

Aditya Raz Pokhrel 

MBA, MA Economics, MPA