In
econometrics
Econometrics is the application of Statistics, statistical methods to economic data in order to give Empirical evidence, empirical content to economic relationships.M. Hashem Pesaran (1987). "Econometrics," ''The New Palgrave: A Dictionary of ...
, the Arellano–Bond estimator is a
generalized method of moments estimator used to estimate
dynamic models of
panel data
In statistics and econometrics, panel data and longitudinal data are both multi-dimensional data set, data involving measurements over time. Panel data is a subset of longitudinal data where observations are for the same subjects each time.
Time s ...
. It was proposed in 1991 by
Manuel Arellano and
Stephen Bond, based on the earlier work by
Alok Bhargava
Alok Bhargava (born 13 July 1954) is an Indian econometrician. He studied mathematics at Delhi University and economics and econometrics at the London School of Economics. He is currently a full professor at the University of Maryland School ...
and
John Denis Sargan
John Denis Sargan, FBA (23 August 1924 – 13 April 1996) was a British econometrician who specialized in the analysis of economic time-series.
Sargan was born in Doncaster, Yorkshire in 1924, and was educated at Doncaster Grammar School and ...
in 1983, for addressing certain endogeneity problems.
The GMM-SYS estimator is a system that contains both the levels and the first difference equations. It provides an alternative to the standard first difference GMM estimator.
Qualitative description
Unlike static panel data models, dynamic panel data models include lagged levels of the dependent variable as regressors. Including a lagged dependent variable as a regressor violates strict exogeneity, because the lagged dependent variable is likely to be correlated with the
random effects
In statistics, a random effects model, also called a variance components model, is a statistical model where the model parameters are random variables. It is a kind of hierarchical linear model, which assumes that the data being analysed are ...
and/or the general errors.
The Bhargava-Sargan article developed optimal linear combinations of predetermined variables from different time periods, provided sufficient conditions for identification of model parameters using restrictions across time periods, and developed tests for exogeneity for a subset of the variables. When the exogeneity assumptions are violated and correlation pattern between time varying variables and errors may be complicated, commonly used static panel data techniques such as
fixed effects
In statistics, a fixed effects model is a statistical model in which the model parameters are fixed or non-random quantities. This is in contrast to random effects models and mixed models in which all or some of the model parameters are random va ...
estimators are likely to produce inconsistent estimators because they require certain
strict exogeneity assumptions.
Anderson
Anderson or Andersson may refer to:
Companies
* Anderson (Carriage), a company that manufactured automobiles from 1907 to 1910
* Anderson Electric, an early 20th-century electric car
* Anderson Greenwood, an industrial manufacturer
* Anderson ...
and
Hsiao (1981) first proposed a solution by utilising
instrumental variables
In statistics, econometrics, epidemiology and related disciplines, the method of instrumental variables (IV) is used to estimate causal relationships when controlled experiments are not feasible or when a treatment is not successfully delivered to ...
(IV) estimation. However, the Anderson–Hsiao estimator is asymptotically inefficient, as its asymptotic variance is higher than the Arellano–Bond estimator, which uses a similar set of instruments, but uses
generalized method of moments estimation rather than
instrumental variable
In statistics, econometrics, epidemiology and related disciplines, the method of instrumental variables (IV) is used to estimate causal relationships when controlled experiments are not feasible or when a treatment is not successfully delivered to ...
s estimation.
In the Arellano–Bond method,
first difference
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a paramete ...
of the
regression equation
In statistical modeling, regression analysis is a set of statistical processes for estimating the relationships between a dependent variable (often called the 'outcome' or 'response' variable, or a 'label' in machine learning parlance) and one o ...
are taken to eliminate the individual effects. Then, deeper lags of the dependent variable are used as instruments for differenced lags of the dependent variable (which are endogenous).
In traditional panel data techniques, adding deeper lags of the dependent variable reduces the number of observations available. For example, if observations are available at T time periods, then after first differencing, only T-1 lags are usable. Then, if K lags of the dependent variable are used as instruments, only T-K-1 observations are usable in the regression. This creates a trade-off: adding more lags provides more instruments, but reduces the sample size. The Arellano–Bond method circumvents this problem.
Formal description
Consider the static linear unobserved effects model for
observations and
time periods:
:
for
and
where
is the dependent variable observed for individual
at time
is the time-variant
regressor matrix,
is the unobserved time-invariant individual effect and
is the
error term In mathematics and statistics, an error term is an additive type of error. Common examples include:
* errors and residuals in statistics, e.g. in linear regression
* the error term in numerical integration
In analysis, numerical integration ...
. Unlike
,
cannot be observed by the econometrician. Common examples for time-invariant effects
are innate ability for individuals or historical and institutional factors for countries.
Unlike a static panel data model, a dynamic panel model also contains lags of the dependent variable as regressors, accounting for concepts such as momentum and inertia. In addition to the regressors outlined above, consider a case where one lag of the dependent variable is included as a regressor,
.
:
Taking the first difference of this equation to eliminate the individual effect,
:
Note that if
had a time varying coefficient, then differencing the equation will not remove the individual effect. This equation can be re-written as,
:
Applying the formula for the Efficient Generalized Method of Moments Estimator, which is,
:
where
is the instrument matrix for
.
The matrix
can be calculated from the variance of the error terms,
for the one-step Arellano–Bond estimator or using the residual vectors of the one-step Arellano–Bond estimator for the two-step Arellano–Bond estimator, which is
consistent and asymptotically efficient in the presence of heteroskedasticity.
Instrument matrix
The original Anderson and Hsiao (1981) IV estimator uses the following moment conditions:
:
Using the single instrument
, these moment conditions form the basis for the instrument matrix
:
:
Note: ''The first possible observation is t = 2 due to the first difference transformation''
The instrument
enters as a single column. Since
is unavailable at
, all observations from
must be dropped.
Using an additional instrument
would mean adding an additional column to
. Thus, all observations from
would have to be dropped.
While adding additional instruments increases the efficiency of the IV estimator, the smaller sample size decreases efficiency. This is the efficiency - sample size trade-off.
The Arellano-bond estimator addresses this trade-off by using time-specific instruments.
The Arellano–Bond estimator uses the following moment conditions
:
Using these moment conditions, the instrument matrix
now becomes:
:
Note that the number of moments is increasing in the time period: this is how the efficiency - sample size tradeoff is avoided. Time periods further in the future have more lags available to use as instruments.
Then if one defines:
:
The moment conditions can be summarized as:
:
These moment conditions are only valid when the error term
has no serial correlation. If serial correlation is present, then the Arellano–Bond estimator can still be used under some circumstances, but deeper lags will be required. For example, if the error term
is correlated with all terms
for s
S (as would be the case if
were a MA(S) process), it would be necessary to use only lags of
of depth S + 1 or greater as instruments.
System GMM
When the variance of the individual effect term across individual observations is high, or when the stochastic process
is close to being a
random walk
In mathematics, a random walk is a random process that describes a path that consists of a succession of random steps on some mathematical space.
An elementary example of a random walk is the random walk on the integer number line \mathbb Z ...
, then the Arellano–Bond estimator may perform very poorly in finite samples. This is because the lagged dependent variables will be weak instruments in these circumstances.
Blundell and Bond (1998) derived a condition under which it is possible to use an additional set of moment conditions. These additional moment conditions can be used to improve the small sample performance of the Arellano–Bond estimator. Specifically, they advocated using the moment conditions:
:
(1)
These additional moment conditions are valid under conditions provided in their paper. In this case, the full set of moment conditions can be written:
where
:
and
:
This method is known as system GMM. Note that the consistency and efficiency of the estimator depends on validity of the assumption that the errors can be decomposed as in equation (1). This assumption can be tested in empirical applications and likelihood ratio test often reject the simple random effects decomposition.
Implementations in statistics packages
*
R: the Arellano–Bond estimator is available as part of the
plm
package.
*
Stata
Stata (, , alternatively , occasionally stylized as STATA) is a general-purpose statistical software package developed by StataCorp for data manipulation, visualization, statistics, and automated reporting. It is used by researchers in many fie ...
: the commands
xtabond
and
xtabond2
return Arellano–Bond estimators.
See also
*
Random effects model
In statistics, a random effects model, also called a variance components model, is a statistical model where the model parameters are random variables. It is a kind of hierarchical linear model, which assumes that the data being analysed are dra ...
*
Mixed model
A mixed model, mixed-effects model or mixed error-component model is a statistical model containing both fixed effects and random effects. These models are useful in a wide variety of disciplines in the physical, biological and social sciences. ...
References
Further reading
*
*
*
{{DEFAULTSORT:Arellano-Bond estimator
Estimator
Analysis of variance
Regression with time series structure