“Arellano–Bond estimator”的意思、由来-开放百科全书

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.^[1]

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

observations and

time periods:

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,

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

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

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.

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.

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.

Systems GMM

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.^[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:

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

Implementations in statistics packages

See also

References

1. ^{{cite journal | last1 = Arellano | first1 = Manuel | last2 = Bond | first2 = Stephen | doi = 10.2307/2297968 | issue = 2 | journal = Review of Economic Studies | jstor = 2297968 | page = 277 | title = Some tests of specification for panel data: Monte Carlo evidence and an application to employment equations | url = | volume = 58 | year = 1991}}
2. ^{{cite journal | last1 = Anderson | first1 = T. W. | last2 = Hsiao | first2 = Cheng | doi = 10.1080/01621459.1981.10477691 | issue = 375 | journal = Journal of the American Statistical Association | pages = 598–606 | title = Estimation of dynamic models with error components | volume = 76 | year = 1981}}.
3. ^{{Cite journal |last1=Blundell |last2=Bond |first1=Richard |first2=Stephen |date=1998|title=Initial conditions and moment restrictions in dynamic panel data models|doi=10.1016/S0304-4076(98)00009-8|journal=Journal of Econometrics|volume=87|issue=1|pages=115–143 }}
4. ^{{cite book |first=Christian |last=Kleiber |first2=Achim |last2=Zeileis |chapter=Linear Regression with Panel Data |title=Applied Econometrics with R |location= |publisher=Springer |year=2008 |isbn=978-0-387-77316-2 |pages=84–89 |chapterurl=https://books.google.com/books?id=86rWI7WzFScC&pg=PA84 }}
5. ^{{cite journal |first=Yves |last=Croissant |first2=Giovanni |last2=Millo |year=2008 |title=Panel Data Econometrics in R: The plm Package |journal=Journal of Statistical Software |volume=27 |issue=2 |pages=1–43 |url=http://www.jstatsoft.org/v27/i02 }}
6. ^{{cite web |title=plm: Linear Models for Panel Data |work=R Project |date= |url=https://cran.r-project.org/web/packages/plm/index.html }}
7. ^{{cite web |title=xtabond — Arellano–Bond linear dynamic panel-data estimation |work=Stata Manual |date= |url=https://www.stata.com/manuals13/xtxtabond.pdf }}
8. ^{{cite web |title=Roodman, R: How to do xtabond2: An introduction to difference and system GMM in Stata |work=Stata Journal, vol. 9(1), pp. 86-136|date= |url=http://www.stata-journal.com/article.html?article=st0159 }}