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In econometrics, the Arellano–Bond estimator is a generalized method of moments estimator used to estimate dynamic panel data models. It was first proposed by Manuel Arellano and Stephen Bond in 1991 to solve the endogeneity, heteroscedasticity and serial correlation problems in static panel data problem. 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.
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 necessarily correlated with the idiosyncratic error.
Anderson and Hsiao (1981) first proposed a solution by utilising instrumental variables (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 variables estimation.
In the Arellano–Bond method, first difference of the regression equation are taken to eliminate the fixed 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.
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. 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 fixed 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.
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 sS (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.
When the variance of the fixed effect term across individual observations is high, or when the stochastic process is close to being a random walk, 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:
These additional moment conditions are valid under conditions provided in their paper. In this case, the full set of moment conditions can be written:
This method is known as systems GMM.
Implementations in statistics packages
- R: the Arellano–Bond estimator is available as part of the
- Stata: the commands
xtabond2return Arellano–Bond estimators.
- Arellano, Manuel; Bond, Stephen (1991). "Some tests of specification for panel data: Monte Carlo evidence and an application to employment equations". Review of Economic Studies. 58 (2): 277. doi:10.2307/2297968. JSTOR 2297968.
- Anderson, T. W.; Hsiao, Cheng (1981). "Estimation of dynamic models with error components". Journal of the American Statistical Association. 76 (375): 598–606. doi:10.1080/01621459.1981.10477691..
- Blundell, Richard; Bond, Stephen (1998). "Initial conditions and moment restrictions in dynamic panel data models". Journal of Econometrics. 87 (1): 115–143. doi:10.1016/S0304-4076(98)00009-8.
- Kleiber, Christian; Zeileis, Achim (2008). "Linear Regression with Panel Data". Applied Econometrics with R. Springer. pp. 84–89. ISBN 978-0-387-77316-2.
- Croissant, Yves; Millo, Giovanni (2008). "Panel Data Econometrics in R: The plm Package". Journal of Statistical Software. 27 (2): 1–43.
- "plm: Linear Models for Panel Data". R Project.
- "xtabond — Arellano–Bond linear dynamic panel-data estimation" (PDF). Stata Manual.
- Roodman, David (2009). "How to do xtabond2: An introduction to difference and system GMM in Stata". Stata Journal. 9 (1): 86–136.
- Baltagi, Badi H. (2008). "Dynamic Panel Data Models". Econometric Analysis of Panel Data. John Wiley & Sons. pp. 147–180. ISBN 978-0-470-51886-1.
- Cameron, A. Colin; Trivedi, Pravin K. (2005). "Dynamic Models". Microeconometrics: Methods and Applications. New York: Cambridge University Press. pp. 763–768. ISBN 0-521-84805-9.
- Hsiao, Cheng (2014). "Dynamic Simultaneous Equations Models". Analysis of Panel Data. New York: Cambridge University Press. pp. 397–402. ISBN 978-1-107-65763-2.