2024年4月2日发(作者：)

Using Arellano – Bond Dynamic Panel GMM Estimators in Stata

Tutorial with Examples using Stata 9.0

(xtabond and xtabond2)

Elitza Mileva,

Economics Department

Fordham University

July 9, 2007

1. The model

The following model examines the impact of capital flows on investment in a panel dataset

of 22 countries for 10 years (1995 – 2004):

I

it

=

β

1

I

i,t−1

+

β

2

K

it

+

β

3

X

it

+u

it

. (1)

In equation (1) above I

it

is gross fixed capital formation as a percentage of GDP and I

it-1

is its

lagged value. K

it

is a matrix of the components of foreign resource flows – FDI, loans and

portfolio (equity and bonds) – as percentage shares of GDP. X

it

is a matrix of the following control

variables: lagged real GDP growth to account for the accelerator effect; the absolute value of one

step ahead growth forecast errors as a measure of uncertainty; the change in the log terms of trade

to gauge the price of imported capital goods; and, finally, the deviation of M2 from its three-year

trend as a proxy for the liquidity available to finance investment.

2. Why the Arellano – Bond GMM estimator?

Several econometric problems may arise from estimating equation (1):

1. The capital flows variables in K

it

are assumed to be endogenous. Because causality may run in

both directions – from capital inflows to investment and vice versa – these regressors may be

correlated with the error term.

2. Time-invariant country characteristics (fixed effects), such as geography and demographics,

may be correlated with the explanatory variables. The fixed effects are contained in the error term

in equation (1), which consists of the unobserved country-specific effects, v

i

, and the observation-

specific errors, e

it

:

1

u

it

=v

i

+e

it

(2).

3. The presence of the lagged dependent variable I

it-1

gives rise to autocorrelation.

4. The panel dataset has a short time dimension (T =10) and a larger country dimension (N =22).

To solve problem 1 (and problem 2) one would usually use fixed-effects instrumental

variables estimation (two-stage least squares or 2SLS), which is what I tried first. The exogenous

instruments I used were the following: the aggregate long-term capital inflows to the countries in

our sample as a group as a percentage of the sum of their cumulative GDP (I labelled these

‘regional flows’), an index of financial openness and the EBRD transition index. However, the

first-stage statistics of the 2SLS regressions showed that my instruments were weak. With weak

instruments the fixed-effects IV estimators are likely to be biased in the way of the OLS

estimators. Therefore, I decided to use the Arellano – Bond (1991) difference GMM estimator first

proposed by Holtz-Eakin, Newey and Rosen (1988). Instead of using only the exogenous

instruments listed above lagged levels of the endogenous regressors in K

it

(FDI, loans and

portfolio) are also added. This makes the endogenous variables pre-determined and, therefore, not

correlated with the error term in equation (1).

To cope with problem 2 (fixed effects) the difference GMM uses first-differences to

transform equation (1) into

ΔI

it

=

β

1

ΔI

i,t−1

+

β

2

ΔK

it

+

β

3

ΔX

it

+Δu

it

(3).

′

(In general form the transformation is given by:

Δy

it

=

α

Δy

it−1

+Δx

it

β

+Δu

it

.)

By transforming the regressors by first differencing the fixed country-specific effect is

removed, because it does not vary with time. From equation (2) we get

Δu

it

=Δv

i

+Δe

it

or

u

it

−u

i,t−1

=(v

i

−v

i

)+(e

it

−e

i,t−1

)=e

it

−e

i,t−1

.

The first-differenced lagged dependent variable (problem 3) is also instrumented with its

past levels.

2