# Arellano–Bond estimator

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,[1] 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.

## 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 necessarily correlated with the idiosyncratic error.

When the strict exogeneity assumption is violated, commonly used static panel data techniques such as fixed effects estimators are inconsistent, because these estimators require strict exogeneity.

Anderson and Hsiao (1981) first proposed a solution by utilising instrumental variables (IV) estimation.[2] 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.

## Formal description

Consider the static linear unobserved effects model for ${\displaystyle N}$ observations and ${\displaystyle T}$ time periods:

${\displaystyle y_{it}=X_{it}\mathbf {\beta } +\alpha _{i}+u_{it}}$ for ${\displaystyle t=1,\ldots ,T}$ and ${\displaystyle i=1,\ldots ,N}$

where ${\displaystyle y_{it}}$ is the dependent variable observed for individual ${\displaystyle i}$ at time ${\displaystyle t,}$ ${\displaystyle X_{it}}$ is the time-variant ${\displaystyle 1\times k}$ regressor matrix, ${\displaystyle \alpha _{i}}$ is the unobserved time-invariant individual effect and ${\displaystyle u_{it}}$ is the error term. Unlike ${\displaystyle X_{it}}$, ${\displaystyle \alpha _{i}}$ cannot be observed by the econometrician. Common examples for time-invariant effects ${\displaystyle \alpha _{i}}$ 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, ${\displaystyle y_{it-1}}$.

${\displaystyle y_{it}=X_{it}\mathbf {\beta } +\rho y_{it-1}+\alpha _{i}+u_{it}{\text{ for }}t=1,\ldots ,T{\text{ and }}i=1,\ldots ,N}$

Taking the first difference of this equation to eliminate the fixed effect,

${\displaystyle \Delta y_{it}=y_{it}-y_{it-1}=\Delta X_{it}\beta +\rho \Delta \,y_{it-1}+\Delta u_{it}{\text{ for }}t=1,\ldots ,T{\text{ and }}i=1,\ldots ,N.}$

This equation can be re-written as,

${\displaystyle \Delta y=\Delta R\pi +\Delta u.}$

Applying the formula for the Efficient Generalized Method of Moments Estimator, which is,

${\displaystyle \pi _{\text{EGMM}}=[\Delta R'Z(Z'\Omega Z)^{-1}Z'\,\Delta R]^{-1}\,\Delta R'Z(Z'\Omega Z)^{-1}Z'\Delta y}$

where ${\displaystyle Z}$ is the instrument matrix for ${\displaystyle \Delta R}$.

The matrix ${\displaystyle \Omega }$ can be calculated from the variance of the error terms, ${\displaystyle u_{it}}$ 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:

${\displaystyle E(y_{it-I}\,\Delta u_{it})=0{\text{ with }}I\geq 2{\text{ for each }}t\geq 3.}$

Using the single instrument ${\displaystyle y_{it-2}}$, these moment conditions form the basis for the instrument matrix ${\displaystyle Z_{di}}$:

${\displaystyle Z_{di}={\begin{bmatrix}NA&(t=2)\\y_{i1}&(t=3)\\y_{i2}&(t=4)\\\vdots &\vdots \\y_{T-2}&(t=T)\end{bmatrix}}}$

Note: The first possible observation is t = 2 due to the first difference transformation

The instrument ${\displaystyle y_{it-2}}$ enters as a single column. Since ${\displaystyle y_{it-2}}$ is unavailable at ${\displaystyle t=2}$, all observations from ${\displaystyle t=2}$ must be dropped.

Using an additional instrument ${\displaystyle y_{it-3}}$ would mean adding an additional column to ${\displaystyle Z_{di}}$. Thus, all observations from ${\displaystyle t=3}$ 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 uses the following moment conditions

${\displaystyle E(y_{it-I}\,\Delta u_{it})=0{\text{ for }}t\geq 3,\,I\geq 2.}$

Using these moment conditions, the instrument matrix ${\displaystyle Z_{di}}$ now becomes:

${\displaystyle Z_{di}={\begin{bmatrix}y_{i1}&0&0&0&0&0&\cdots \\0&y_{i2}&y_{i1}&0&0&0&\cdots \\0&0&0&y_{i3}&y_{i2}&y_{i1}&\cdots \\\vdots &\vdots &\vdots &\vdots &\vdots &\vdots &\ddots \end{bmatrix}}}$

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:

${\displaystyle \Delta u_{i}={\begin{bmatrix}\Delta u_{i3}\\\Delta u_{i4}\\\Delta u_{i5}\\\vdots \end{bmatrix}}}$

The moment conditions can be summarized as:

${\displaystyle E(Z_{di}^{T}\,\Delta u_{i})=0}$

These moment conditions are only valid when the error term ${\displaystyle u_{it}}$ 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 ${\displaystyle u_{it}}$ is correlated with all terms ${\displaystyle u_{it-s}}$ for s${\displaystyle \leq }$S (as would be the case if ${\displaystyle u_{it}}$ were a MA(S) process), it would be necessary to use only lags of ${\displaystyle y_{it}}$ of depth S + 1 or greater as instruments.

## Systems GMM

When the variance of the fixed effect term across individual observations is high, or when the stochastic process ${\displaystyle y_{it}}$ 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.[3] 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:

${\displaystyle \operatorname {E} (\Delta y_{it-1}(\alpha _{i}+u_{it}))=0{\text{ for }}t\geq 3}$

These additional moment conditions are valid under conditions provided in their paper. In this case, the full set of moment conditions can be written:

${\displaystyle \operatorname {E} (Z_{SYS,i}^{T}P_{i})=0}$

where

${\displaystyle P_{i}={\begin{pmatrix}\Delta u_{i}\\u_{i3}\\u_{i4}\\u_{i5}\\\vdots \end{pmatrix}}}$

and

${\displaystyle Z_{SYS,i}={\begin{pmatrix}Z_{di}&0&0&0\\0&\Delta y_{i2}&0&0\\0&0&\Delta y_{i3}&0\\0&0&0&\ddots \end{pmatrix}}.}$

This method is known as systems GMM.

## Implementations in statistics packages

• R: the Arellano–Bond estimator is available as part of the plm package.[4][5][6]
• Stata: the commands xtabond and xtabond2 return Arellano–Bond estimators.[7][8]