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1 The Review of Economic Studies, Ltd. Some Tests of Specification for Panel Data: Monte Carlo Evidence and an Application to Employment Equations Author(s): Manuel Arellano and Stephen Bond Reviewed work(s): Source: The Review of Economic Studies, Vol. 8, No. (Apr., 99), pp Published by: Oxford University Press Stable URL: Accessed: 6// 7:44 Your use of JSTOR archive indicates your acceptance of Terms & Conditions of Use, available at. JSTOR is a notforprofit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact Oxford University Press and The Review of Economic Studies, Ltd. are collaborating with JSTOR to digitize, preserve and extend access to The Review of Economic Studies.
2 Review of Economic Studies (99) 8, /9/877$.? 99 The Review of Economic Studies Limited Some Tests of Specification for Panel Data: Monte Carlo Evidence and an Application to Equations MANUEL ARELLANO London School of Economics and STEPHEN BOND University of Oxford Employment First version received May 988; final version accepted July 99 (Eds.) This paper presents specification tests that are applicable after estimating a dynamic model from panel data by generalized method of moments (GMM), and studies practical performance of se procedures using both generated and real data. Our GMM estimator optimally exploits all linear moment restrictions that follow from assumption of no serial correlation in errors, in an equation which contains individual effects, lagged dependent variables and no strictly exogenous variables. We propose a test of serial correlation based on GMM residuals and compare this with Sargan tests of overidentifying restrictions and Hausman specification tests.. INTRODUCTION The purpose of this paper is to present specification tests that are applicable after estimating a dynamic model from panel data by generalized method of moments (GMM) and to study practical performance of se procedures using both generated and real data. Previous work concerning dynamic equations from panel data (e.g. Chamberlain (984), Bhargava and Sargan (983)) has emphasized case where model with an arbitrary intertemporal covariance matrix of errors is identified. The fundamental identification condition for this model is strict exogeneity of some of explanatory variables (or availability of strictly exogenous instrumental variables) conditional on unobservable individual effects. In practice, this allows one to use past, present and future values of strictly exogenous variables to construct instruments for lagged dependent variables and or nonexogenous variables once permanent effects have been differenced out. Bhargava and Sargan (983) and Arellano (99) considered estimation and inference imposing restrictions on autocovariances, but assumption that model with unrestricted covariance matrix is identified was never removed. However, sometimes one is less willing to assume strict exogeneity of an explanatory variable than to restrict serial correlation structure of errors, in which case 77
3 78 REVIEW OF ECONOMIC STUDIES different identification arrangements become available. Uncorrelated errors arise in a number of environments. These include rational expectations models where disturbance is a surprise term, errorcorrection models and vector autoregressions. Moreover, if re are a priori reasons to expect autoregressive errors in a regression model, se can be represented as a dynamic regression with nonlinear common factor restrictions and uncorrelated disturbances (e.g. Sargan (98)). In se cases and also in models with movingaverage errors, lagged values of dependent variable itself become valid instruments in differenced equations corresponding to later periods. Simple estimators of this type were first proposed by Anderson and Hsiao (98, 98). Griliches and Hausman (986) have developed estimators for errorsinvariables models whose identification relies on assumptions of lack of (or limited) serial correlation in measurement errors. HoltzEakin, Newey and Rosen (988) have also considered estimators of this type for vector autoregressions which are similar to ones we employ in this paper. An estimator that uses lags as instruments under assumption of white noise errors would lose its consistency if in fact errors were serially correlated. It is refore essential to satisfy oneself that this is not case by reporting test statistics of validity of instrumental variables (i.e. tests of lack of serial correlation) toger with parameter estimates. In this paper we consider three such tests: a direct test on tne secondorder residual serial correlation coefficient, a Sargan test of overidentifying restrictions and a Hausman specification test. The operating characteristics of se tests are different as well as ir number of degrees of freedom. In addition, depending on alternative auxiliary distributional assumptions concerning stationarity and heterogeneity, different forms of each of tests are available. These alternative versions of a given test are asymptotically equivalent under less general set of auxiliary assumptions but y still may perform quite differently in finite samples. We have refore produced a number of Monte Carlo experiments to investigate relative performance of various tests. Finally, as an empirical illustration we report some estimated employment equations using Datastream panel of quoted U.K. companies. The paper is organized as follows. Section presents model and estimators. For a fixed number of time periods in sample, model specifies a finite number of moment restrictions and refore an asymptotically efficient GMM estimator is readily available. The discussion is kept as simple as possible by concentrating initially on a firstorder autoregression with a fixed effect. Exogenous variables and unbalanced panel considerations are subsequently introduced. Section 3 presents various tests of serial correlation and ir asymptotic distributions. Section 4 reports simulation results. Section contains application to employment equations and Section 6 concludes.. ESTIMATION The simplest model without strictly exogenous variables is an autoregressive specification of form Yi, ayi(tl)+ +,i + Vit, lal <. () We assume that a random sample of N individual time series (yi,...., YiT) is available. T is small and N is large. The vi, are assumed to have finite moments and in particular E(vit) E(vitvij) for t $ s. That is, we assume lack of serial correlation but not necessarily independence over time. With se assumptions, values of y lagged two periods or more are valid instruments in equations in first differences. Namely, for
4 ARELLANO & BOND SPECIFICATION TESTS 79 Ti_3 model implies following m(t)(t)/ linear moment restrictions E[(yit  ay(t_))yj(t_j) (j,..., (t ); t 3,..., T) () where for simplicity Yi Yi Yi(ti) We wish to obtain optimal estimator of a as N  co for fixed T on basis of se moment restrictions alone. That is, in absence of any or knowledge concerning initial conditions or distributions of vi, and qj. Note that our assumptions also imply quadratic moment restrictions, for example E( eiti(t)), which however we shall not exploit in order to avoid iterative procedures. This estimation problem is an example of those analyzed by Hansen (98) and White (98), and an optimal GMM or twostage instrumental variables estimator should be available. The moment equations in () can be conveniently written in vector form as E(Z'ibi) where vi (vi3... lbit)' and Zi is a (T) x m block diagonal matrix whose sth block is given by (yi *... Yis) The GMM estimator a' is based on sample moments N' Z'i3 N'Z'i where v3yajll(iu',..., I'N)' is a N( T)x vector and Z(Z,..., Z'N)' is a N(T  ) x m matrix. a is given by a argmina (ViZ)AN(Z') lzanz (3) Y_,ZANZ'Yl' Multivariate standard CLT implies that VPN"ZZ'I is asymptotically standard normal where VN N 'E E(Z'ifAiZj) is average covariance matrix of Ziui. Under our assumptions, VN can be replaced by VN N Zivi3Z, Ii where are residuals from a preliminary consistent estimator at. The onestep estimator ca is obtained by setting AN (N Ei ZHZi)l where H is a (T) square matrix which has twos in main diagonal, minus ones in first subdiagonals and zeroes orwise. A consistent estimate of avar (c) for arbitrary AN is given by aa (a) NY lzanvnanz'y avar( an  ZN,p_) 4 The optimal choice for AN is VN (e.g. see Hansen (98)) which produces a twostep A A A estimator a. a, and a will be asymptotically equivalent if vi, are independent and homoskedastic both across units and over time. It is useful to relate se estimators to AndersonHsiao (AH) estimator which is commonly used in practice. Anderson and Hsiao (98) proposed to estimate a by regressing y on Y using eir Y or Y as instruments. Since both Y and Y are linear combinations of Z resulting estimators will be inefficient. Note that under stationarity, namely when E(YitYi(tk)) Cik for all t, estimator that uses Z+ diag (yit) (t,..., T  ) is asymptotically equivalent to one based on stacked vector Y, whose computation is much simpler (since AN becomes irrelevant). However neir of two is asymptotically equivalent to a, or a, not even under stationarity. The extension of previous results to case where a limited amount of serial correlation is allowed in vit is straightforward. Suppose that vit is MA (q) in. In this paper we represent this type of matrix by Zi diag (yil,. Yis), (s... i T).. An alternative choice of AN is (N ZfAlZi)l with A N S The resulting estimator does not depend on data fourthorder moments and is asymptotically equivalent to ' provided vi, are serially independent. Note that in this case E(Z!ti3ivZi) E(Z!fliZi) and limn o N, E[Z!(fli fn)zi] O (see White (98), p. 49).
5 8 REVIEW OF ECONOMIC STUDIES sense that E(vivi(,_k)) $ O for k _ q and zero orwise. In this case a is just identified with T q + 3 and re are mq T  q  )( T  q  )/ restrictions available. Models with exogenous variables We now turn to consider an extended version of equation () where (k ) independent explanatory variables have been included Yit ayi(t,l)+ xu* + 'qi + Vit 8'xi, + 7Ri + vi, () where xit (Yi(t) x*')' is k x and vit are not serially correlated. Suppose initially that x* are all correlated with qi. In this context form of optimal matrix of instruments depends on wher x* are predetermined or strictly exogenous variables. If x* are predetermined, in sense that E(xi*vis) $ for s < t and zero orwise, n only x*,..., x*4s) are valid instruments in differenced equation for period s so that optimal Zi is a (T)x(T)[(k)(T+)+(T)]/ matrix of form Zi diag (yi,... yisx*l... x*( +)), (s,..., T  ). On or hand, if x* are strictly exogenous, i.e. E(x viis) for all t, s, n all x*'s are valid instruments for all equations and Zi takes form Zi diag (yi, * yis x*x' * X** ), (s,..., T ). Clearly, x* may also include a combination of both predetermined and strictly exogenous variables. In eir case, form of GMM estimator of k x coefficient vector 8 is (X'ZANZ'X) X'ZANZ'y (6) where X is a stacked (T ) N x k matrix of observations on xit, and y and Z are as above for appropriate choice of Zi. As before, alternative choices of AN will produce onestep or twostep estimators.3 Turning now to case where x* can be partitioned into (x*t,x*x,) and x*, is uncorrelated with qi, additional moment restrictions exploiting this lack of correlation in levels equations become available. For example, if x*, is predetermined and letting uit qi + vi,, we have T extra restrictions. Namely, E(uiXIi4) and E (ui,xi,i), (t,..., T). Note that all remaining restrictions from levels equations are redundant given those previously exploited for equations in first differences. Define ui (Ui.. UiTY), let v+ be [(T)+(T)]xl vector v+(i3u')' and let v+ (V+'... v+' Y+  X+8. The optimal matrix of instruments Z+ is block diagonal and consists of two blocks: Zi which is as in predetermined x* case above (assuming that x* t is also predetermined), and Za which is itself block diagonal with (x*lx*4) in first block and x*', s 3,..., T in remaining blocks. The twostep estimator is of same form as (6) with X+, y+ and Z+ replacing X, y and Z respectively, and AN [N Eji Z+A+A+t'ZP+]. On or hand, if x4*t is strictly exogenous, observations for all periods become valid instruments in levels equations. However, given those previously exploited in first differences we only have T extra restrictions which in this case can be conveniently expressed as E(T,i uisxit) (t,..., T). Thus, twostep estimator would just combine ( T ) first difference equations and average level equation.4 3. Note that if E(vjvt) is unrestricted (i.e. vi, is MA(q) with T< q+3) but x* is strictly exogenous, model is identified with Z, diag (*... *x), * (s,..., T ), in which case twostep estimator coincides with generalized threestage least squares estimator proposed by Chamberlain (984). 4. Note that when x*, variables are available it may be possible to identify and estimate coefficients for timeinvariant variables on lines suggested by Hausman and Taylor (98), Bhargava and Sargan (983) and Amemiya and MaCurdy (986).
6 ARELLANO & BOND SPECIFICATION TESTS 8 Models from unbalanced panel data By unbalanced panel data we refer to a sample in which consecutive observations on individual units are available, but number of time periods available may vary from unit to unit as well as historical points to which observations correspond. This type of sample is very common particularly with firm data which is context of application reported below. Aside from often allowing one to exploit a much larger sample or to pool more than one panel, use of unbalanced panels may lessen impact of self selection of firms in sample. In fact nothing fundamental changes in econometric methods provided a minimal number of continuous time periods are available on each unit, and one assumes that if periodspecific parameters are present number of observations on se periods tend to infinity. Of course, essential assumption is that observations in initial crosssection are independently distributed and that subsequent additions and deletions take place at random (see Hsiao (986), Chapter 8). The previous notation can accommodate unbalanced panels with minor changes. We now have Ti timeseries observations on ith unit, and re are N individual units in sample. The matrices X and Z, and vectors y and v are made of N rowblocks, ith block containing (Ti  ) rows. Note that now number of nonzero columns in each Zi may vary across units. For example, in firstorder autoregressive specification above, number of columns in Zi is p (r )(r  )/ where r is total number of periods on which observations are available for some individuals in sample, and Zi diag (yi,.., Yis), (s,..., r), only if r observations are available on i. For individuals with Ti < r, rows of diag (yi,..., yis) corresponding to missing equations are deleted and missing values of Yit in remaining rows are replaced by zeroes. The twostep GMM estimator of a for this choice of instruments: is same as in (3) using AN(N'E'Z'ivZiZl) where Zi is (Ti)xp and vi is (Ti ) x. 3. TESTING THE SPECIFICATION In order to keep notation simple we now drop bars from variables in first differences, so that firstdifference equation for unbalanced panel is now yx 8+ v (7) nxl nxk kxl nxl where n i (Ti ). We also assume that x* are all potentially correlated with mi. The n x vector of residuals is given by AyX8 vx(88) where 8 can be any estimator of form (6) for a particular choice of Z and AN. Let v be vector of residuals lagged twice, of order q i (Ti 4) and let v* be a q x vector of trimmed v to match v and similarly for X*. Since vit are first differences of serially uncorrelated errors, E(vitvi(tl)) need not be zero, but consistency of GMM estimators above hinges heavily upon assumption that E(vitvi(t)). In an. This is optimal choice amongst estimators that can be obtained by stacking equations for all periods and individuals. An alternative estimator would minimize sum of GMM criteria for each of balanced subpanels in sample. Although latter is strictly more efficient when number of units in all subsamples tend to infinity, it may have a poorer finitesample performance when various subsample sizes are not sufficiently large.
7 8 REVIEW OF ECONOMIC STUDIES unbalanced panel (r4) such covariances can be estimated in total, in principle with varying number of sample observations to estimate each of covariances. Provided one assumes that all subsamples tend to infinity, a (r4) degrees of freedom test can be constructed of hyposis that secondorder autocovariances for all periods in sample are zero. However, a considerably simpler procedure will look at average covariances Xi  ()V These averages are independent random variables across units with zero mean under null although with unequal variances in general. So a straightforward one degree of freedom test statistic can be constructed to test wher E(4i) is zero or not. The test statistic for secondorder serial correlation based on residuals from firstdifference equation takes form M 'A/* di N(, ) (8) V under E(vitvi(t)), where v is given by V Z, V ) Vi* V* V'i() 'X*(X'ZANZXY)XPZAN(N Zviv i*vi()) + v' X avar (8)X *V. (9) Note that m is only defined if min Ti'. A proof of asymptotic normality result is sketched in Appendix. It is interesting to notice that m criterion is rar flexible, in that it can be defined in terms of any consistent GMM estimator, not necessarily in terms of efficient estimators, eir in sense of using optimal Z or AN or both. However, asymptotic power of m test will depend on efficiency of estimators used. The m statistic tests for lack of secondorder serial correlation in firstdifference residuals. This will certainly be case if errors in model in levels are not serially correlated, but also if errors in levels follow a randomwalk process. One may attempt to discriminate between two situations by calculating an ml statistic, on same lines as M, to test for lack of firstorder serial correlation in differenced residuals. Alternatively, notice that if errors in levels follow a random walk, n both OLS and GMM estimates in firstdifference model are consistent which suggests a Hausman test based on difference between two estimators. We now turn to consider two or tests of specification which are applicable in same context. One is a Sargan test of overidentifying restrictions (cf. Sargan (98, 988), Hansen (98)) given by SV Z(Zl ZiviZZi) YZVa Xpk () where v y  X8, and 8 is a twostep estimator of 8 for a given Z. Notice that Z need not be optimal set of instruments; here p just refers to number of columns in Z provided p > k. Also notice that while we are able to produce a version of serial correlation test based upon a onestep estimator of 8 which remained asymptotically normal on null under more general distributional assumptions, no robust chisquare Sargan test based on onestep estimates is available. Under null a statistic of form lav Z(EI V ZHiHZi) Z where v are onestep residuals, will only have a limiting chisquare distribution if errors are indeed i.i.d. over time and individuals. In general, asymptotic distribution
8 ARELLANO & BOND SPECIFICATION TESTS 83 of s is a quadratic form in standard normal variables. Critical values can still be calculated by numerical integration but this clearly leads to a burdensome test procedure. On or hand, re may be circumstances where serial correlation test is not defined while Sargan test can still be computed. As a simple example, take firstorder autoregressive equation at beginning of Section with T 4; in this case Sargan statistic tests two linear combinations of three moment restrictions available, namely E(Ui3yi) E(lbi4yi) E(ti4yi), but no differenced residuals two periods apart are available to construct an m test. A furr possibility is to use Sargan difference tests to discriminate between nested hyposes concerning serial correlation in a sequential way. For example, let Z, be a n x p, matrix containing columns of Z which remain valid instrumental variables when errors in levels are firstorder moving average, and let 8, be a twostep estimator of 8 based on Z, with associated residuals v^, n S VI Z,( I viiii IV a I k if errors in levels are MA () or MA (). In addition ds s_siax'p l) if errors in levels are not serially correlated. Moreover ds is asymptotically independent of s, (see Appendix). A closely related alternative is to construct a Hausman test based on difference (8, 8) (cf. Hausman (978) and Hausman and Taylor (98)). This type of test has been proposed by Griliches and Hausman (986) in context of movingaverage measurement errors. A test is based on statistic h, (?  )'[av'r (?SI)  ava'r (3)](?s ?) a x() A A where r rank avar (3, 8) and ( ) indicates a generalized inverse. The value of r will depend on number of columns of X which are maintained to be strictly exogenous. In particular, if only nonexogenous variable is lagged dependent variable n r. 4. EXPERIMENTAL EVIDENCE A limited simulation was carried out to study performance of estimation and testing procedures discussed above in samples of a size likely to be encountered in practice. In all experiments dependent variable yi, was generated from a model of form Yit ayi(,i) + oxit,q7i+ vi, (i 9...., N; t,...., T+ ) Vit 7it(eit + 4ei(t)), it + Xit (3) where q, i i.i.d. N(O, o ), {it  i.i.d. N(, ) and yio. The first ten crosssections were discarded so that actual samples contain NT observations. With regard to xi,, we considered following generating equation xit pxi(,,) + 6it (4) with Eit i.i.d. N(O, (J) independent of qi and vi, for all t, s and kept observations on xi, fixed over replications. As an alternative choice of regressors we used total sales from a sample of quoted U.K. firms where large variations across units and outliers are likely to be present. In both cases, xi, is strictly exogenous and uncorrelated with individual effects. However, since we are interested in performance of estimators that
9 84 REVIEW OF ECONOMIC STUDIES rely on lags of yi, for identification of a and,3, overidentifying restrictions arising from strict exogeneity of xi, and lack of correlation with qi were not used in simulated GMM estimator. Thus, we chose6 Zi  [diag(yi,.. **Yis)*C(X3 ** ity ] (S I.. T) which is a valid instrument set provided 4?. In base design, sample size is N and T 7, vi, are independent over time and homoskedastic:, 4?, a. (o, /, p 8 and or 9. Tables and summarize results for a ,, 8 obtained from replications. Results for or variants of this design were calculated (N, T 6, r.,,, p ], and are available from authors on request. However conclusions are same as for results reported here. Table reports sample means and standard deviations for onestep and twostep GMM estimators (GMM and GMM respectively), OLS in levels, withingroups, and TABLE Biases in estimates Robust Within Onestep Onestep Twostep GMM GMM OLS groups AHd AHI ASE ASE ASE a,, Coefficient: a Mean St. Dev Coefficient:,3 Mean St. Dev a ,,3 Coefficient: a Mean 937 d St. Dev Coefficient: /8 Mean St. Dev a 8,/3 Coefficient: a Mean St. Dev Coefficient:,3 Mean St. Dev Notes. (i) N, T 7, replications, o a. (ii) Exogenous variable is first order autoregressive with p 8 and or 9. (iii) GMMI and GMM are respectively one step and two step differenceiv estimators of type described in Section. Both GMM use Z, [diag (y,,... Yi) (Xj3 * * *,T)'] (S,. * *, T). (iv) AHd and AHl are AndersonHsiao stackediv estimators of equation in first differences that use AYi(,) and Yi(t) as an instrument for Ayi(,) respectively. (v) One Step ASE and Robust One Step ASE are estimates of asymptotic standard errors of GMM. The former are only valid for i.i.d. errors while latter are robust to general heteroskedasticity over individuals and over time. Two step ASE is a robust estimate of asymptotic standard errors of GMM. 6. The optimal instrument set for system of first difference equations would be zi diag (yi, *.* Yis, Xi,  , XiT)
10 ARELLANO & BOND SPECIFICATION TESTS 8 two AH estimators. The AHd estimator is given by ( ^) (i Et4 AZitw I)t t 4 AZit Ayit () where wi (yi(t_), xit)' and zi, (yi(t), xi,)'. The AHI estimator replaces Azi, with (yi(t), Axit)' and summation goes from t 3 to T The next two columns report sample means and standard deviations for two alternative estimates of asymptotic standard errors of one step GMM estimator. The first one is only valid for i.i.d. errors while second is robust to heteroskedasticity of arbitrary form. The last column corresponds to estimates of asymptotic standard errors of twostep GMM estimator. The tabulated results show a small downward finitesample bias in GMM estimators of a (of about to 3%). Not surprisingly, OLS and withingroup (WG) estimators of a exhibit large biases in opposite directions (i.e. upward bias in OLS whose size depends on (J; downward bias in WG whose size depends on T). The behaviour of AH estimators is more surprising. Concerning AHd, re is evidence of lack of identification for a and negligible biases, though coupled with large variances, for a  and 8. On or hand, standard deviation of AHl is small for a  and a but it more than doubles that of AHd for a 8. These results are consistent with calculations of asymptotic variance matrices for AH estimators reported elsewhere (cf. Arellano (989)). As explained in that note, in a model containing an exogenous variable in addition to lagged dependent variable, re are values of a and p between and for which re is no correlation between AYi(,t) and AYi(,), in which case AYi(t) is not a valid instrument and AHd is not identified. In our first experiment, AHd is close to such a singularity which explains result. In contrast, AHl has no singularities for stationary values of a and p but can neverless be even less precise than AHd for large values of a. An interesting result is that standard deviation of GMM estimators of a is about three times smaller than that of AHd for a  and 8 and between four and five times smaller than that of AHl for a 8 (although standard deviation of AHl for a  and a  is of a similar magnitude as for GMM estimators). This suggests that re may be significant efficiency gains in practice by using GMM as opposed to AH, aside from overcoming potential singularities as in our first experiment. Concerning GMM, two alternative estimators of ir asymptotic standard errors behave in a similar way, although robust alternatives always have a bigger standard deviation. Their sample mean is always very close to finitesample standard deviation in column one, suggesting that asymptotic approximation is quite accurate for simulated designs. On or hand, estimator of asymptotic standard errors of GMM in last column shows a downward bias of around percent relative to finitesample standard deviations reported in second column. Table reports number of rejections toger with sample means and variances for test statistics discussed in Section 3. The first three columns contain two alternative versions of one step m statistic and two step m statistic (see notes to table). The Sargan tests are tests of overidentifying restrictions based on minimized criteria of GMM estimators of Table. The differencesargan tests are based on difference between minimized GMM criteria and restricted versions of se that remain valid when errors are MA (). The Hausman statistics test distance between GMM and restricted GMM estimates of a. With only replications we cannot hope to provide accurate estimates of tail probabilities associated with test statistics; our results can only be suggestive. The
11 86 REVIEW OF ECONOMIC STUDIES TABLE Sizes of Test Statistics, Number of rejections out of cases Onestep Twostep Robust Onestep Twostep difference difference Onestep Twostep Onestep onestep Twostep Sargan Sargan Sargan Sargan Hausman Hausman M M M (df 4) (df 4) (df ) (df ) (df ) (df ) a Mean Variance a Mean Variance a Mean Variance Notes. (i) N, T 7, replications, U U p. (ii) Exogenous variable is first order autoregressive with p 8 and o 9 (iii) All tests are based on one and twostep GMM estimates reported in Table as well as on restricted versions of those which only use columns of Zi that remain valid instruments when errors are MA (). The onestep m statistic is described in Appendix. (iv) Results on an extended set of experiments are available from authors on request.
12 ARELLANO & BOND SPECIFICATION TESTS 87 TABLE 3 Power of Test Statistics, Number of rejections out of cases Onestep Twostep Robust Onestep Twostep difference difference Onestep Twostep Onestep onestep Twostep Sargan Sargan Sargan Sargan Hausman Hausman M M M (df 4) (df 4) (df ) (df ) (df ) (df ) Serial correlation: Corr (v,v,) Mean Variance Serial correlation: Corr (v,v,l) Mean * Variance Heteroskedasticity: Var (vi,) xi,, Xit AR () data Mean Variance Heteroskedasticity: Var (Vi,) xi,, : U.K. Sales Mean * Variance * *73 Notes. (i) N, T 7, replications, a, /3, o. (ii) In serial correlation designs vi, has been generated as MA () with standard normal random errors. (iii) In first three designs xi, is AR () with p 8 and T. 9. (iv) In fourth design, xi, are total sales from a sample of quoted U.K. companies scaled to have Var (Axi,). x,, behaves as a random walk process with large differences in mean and variance across firms. (v) In heteroskedastic designs re is no serial correlation in vi,.
13 88 REVIEW OF ECONOMIC STUDIES robust m statistics, which depend on fourthorder moments of data, both tend to reject too often at % level, suggesting that y have a slower convergence to normality by comparison with or test, but y are still to be recommended when heteroskedasticity is suspected. Overall all three m tests seem to be quite well approximated by ir asymptotic distributions under null, with no obvious indications of need for systematic finitesample size corrections. The same is true for Sargan, differencesargan and one step Hausman tests. However, twostep Hausman statistic appears to overreject consistently in se experiments. Table 3 repeats exercise for two models with MA () serial correlation (4 9 and.333) and two or experiments with heteroskedastic errors. The m statistic will reject null more than half time at per cent level when correlation between vi, and vi(,i) is only. However when autocorrelation rises to 3, null will be rejected in 9% of cases. The Hausman test has considerably less power than differencesargan test or m statistics, and with increasing autocorrelation difference in power becomes wider. The last two panels of Table 3 investigate effects of departures from homoskedasticity of error distribution on probabilities of rejection of tests. Both experiments have and. In first, xi, are generated AR () data as in previous experiments, while in second xi, are U.K. sales data. This has a dramatic effect on onestep tests which are not robust to heteroskedasticity. On or hand, robust m statistics and twostep differencesargan test show no serious departures from ir nominal size. The twostep Sargan test tends to underreject and twostep Hausman test overrejects, especially in last experiment where variance of xi, is much greater. We suspect that twostep Hausman statistic is very sensitive to presence of outliers.. AN APPLICATION TO EMPLOYMENT EQUATIONS In this section we apply strategy for estimation and testing outlined earlier to a model of employment, using panel data for a sample of U.K. companies. We consider a dynamic employment equation of form nit a, ni(tl) + ani(t) + i'(l)xit + At + qi + vit. (6) Here ni, is logarithm of U.K. employment in company i at end of year t,7 vector xi, contains a set of explanatory variables and 3(L) is a vector of polynomials in lag operator. The specification also contains a time effect At that is common to all companies,8 a permanent but unobservable firmspecific effect 'qi and an error term vi,. Equation (6) will admit more than one oretical interpretation. Suppose first that, in absence of adjustment costs, a pricesetting firm facing a constant elasticity demand curve would choose to set employment according to a loglinear labour demand equation (see, for example, Layard and Nickell (986)) n* hyo+ Yl wi, + 7k + Ce, + 7(7) where YI <, Y > and Y3?. Here wi, is log of real product wage, kit is log 7. Note that time period is taken to be month period covered in company's accounts ("accounting year") and so differs across companies in sample. 8. These time effects relate to calendar years and a company's accounting year is allocated to calendar year in which it ends.
14 ARELLANO & BOND SPECIFICATION TESTS 89 of gross capital, a'e is a measure of expected demand for firm's product relative to potential output, and intercept may contain a firmspecific component q'. If employment adjustment is costly n actual employment will deviate from n* in short run. This suggests a dynamic labour demand model of form of (6), where xit contains k, w, and o, and unrestricted lag structures are included to model this sluggish adjustment. We include log of industry output (ysi,) to capture industry demand shocks, and aggregate demand shocks are also included through time dummies. The resulting employment equation is a skeleton version of those estimated on U.K. time series data by Layard and Nickell (986) and on micro data by Nickell and Wadhwani (989). The shortrun dynamics will compound influences from adjustment costs, expectations formation and decision processes. Alternatively, if adjustment costs take standard additivelyseparable quadratic form (/a)(ni,  Ni(t_l)), where Nit denotes level rar than logarithm of company employment, n Dolado (987), following Nickell (984), derives a loglinear approximation to Euler equation for a firm maximising present discounted value of profits as E.l(nit) 8+ (+ r)ni(t,l)( + r)ni(t)+ a( + r)[n(l)  ni(t,l)]. (8) Here r is a real discount rate, assumed constant, and n* is given by (7). Replacing conditional expectation by its realisation and introducing an expectational error vit yields a model with form of (6), though with strong restrictions on dynamic structure in this case. In particular rational expectations hyposis suggests a oretical motivation for assumption of seriallyuncorrelated errors in this kind of model. The principal data source used is published accounts of 4 quoted companies whose main activity is manufacturing and for which we have seven or more continuous observations during period The panel is unbalanced both in sense that we have more observations on some firms than on ors, and because se observations correspond to different points in historical time. We allocate each of our companies to one of nine broad subsectors of manufacturing according to ir main product by sales, and use valueadded in that sector as our measure of industry output. Our wage variable is a measure of average remuneration per employee in company, which we deflate using a valueadded price deflator at industry level. Finally we use an inflationadjusted estimate of company's gross capital stock. More information about sample and construction of se variables is provided in Data Appendix. In Table 4 we report GMM estimates of se dynamic employment equations.9 We begin by including currentdated variables and unrestricted lag structures. Columns (al) and (a) present onestep and twostep results respectively for most general dynamic specification that we considered. Three crosssections are lost in constructing lags and taking first differences, so that estimation period is , with 6 useable observations. Here all variables or than lagged dependent variables are assumed to be strictly exogenous, although none of overidentifying restrictions that follow from this assumption are exploited. Comparing columns (al) and (a) shows that estimated coefficients are quite similar in all cases. Both models are well determined and have sensible longrun properties for a labour demand equation. However asymptotic standard errors associated with twostep estimates are generally around 3% lower than those associated with 9. Estimation was performed using DPD program written in GAUSS, described in Arellano and Bond (988a) and available from authors on request.
15 9 REVIEW OF ECONOMIC STUDIES Dependent variable: In (Employment)i, TABLE 4 Employment equations GMM estimates (all variables in first differences) Sample Period: (4 companies) Independent Instrumenting wages and capital* variables (al) (a) (b) (c) (d) ni(,_l) 686 ( 4) 69 ( 9) 474 ( 8) 8 (.48) 8 (.6) ni(t) 8 ( 6) 6 ( 7) 3 (7) 6 ()  74 (.) Wit *68 (78) 6 (4) 3 (*49) 64 (.4) wi(t) *393 (68) 3 (94)  ( 8) *64 (66) 43 (.76) ki, *37 (.9) 78 (.4) 93 ( 39)  () k,(,_l)  8 (*73) 4 ( 3) 77 ( 4) ki(t)  (33) 4 (.6) ysi, *68 (.7) 9 (6) 6 ( 9) 89 ( 98) ysi(t)  7 (.3) 66 (4) () 87 () () YSi(t) 6 ( 4)  (3) 96 ( 9) M Sargan test 6.8 () 3 4() 3 () 63() 683 () DifferenceSargan 49 (6) 4 (6)  (6) 86 () 36 () Hausman 8 () 44 () 34 ()  () 9 () Wald test 483 () 667 () 37 (7) (7) 639 (6) No. of observations * A subset of valid moment restrictions involving lagged wages and capital are exploitedsee note (vi). Additional instruments used are stacked levels and first differences of (firm real sales)(,) and (firm real stocks)(t) Notes (i) Time dummies are included in all equations. (ii) Asymptotic standard errors robust to general crosssection and time series heteroskedasticity are reported in parenses. (iii) The GMM estimates reported are all two step except column (al). (iv) The M, Sargan, differencesargan and Hausman statistics are all two step versions of se tests except in column (al). In column (al) m and Hausman statistics are asymptotically robust to general heteroskedasticity, whilst Sargan and differencesargan tests are only valid in case of i.i.d. errors. All Hausman statistics test only coefficient on ni(,t). Degrees of freedom for x statistics are reported in parenses. (v) The Wald statistic is a test of joint significance of independent variables asymptotically distributed as X under null of no relationship, where k is number of coefficients estimated (excluding time dummies). (vi) The basic instrument set used in columns (al), (a) and (b) is of form nil ni... O... O *&Xlj4] 979 l nil ni ni3 AX* 98 ZiI. O * nil... n7. &x'9] 984 where x, is vector of exogenous variables included in equation. For example, equation for 979 in first differences can be written as An,4 :alani3 + aani + Ax'4f3 + Avi4 For companies on which less than 9 observations are available, rows of Z; corresponding to missing equations are deleted and missing values of n in remaining rows are replaced by zeroes. In columns (c) and (d) Zi is modified to take form Zi [diag (ni I.. niswi(s_l)wski(sl)kis): (Ax'4... Axi9)'] (s,..., 7) where xi, is now vector of explanatory variables excluding wages and capital but including stacked lagged sales and stocks.
16 ARELLANO & BOND SPECIFICATION TESTS 9 onestep estimates, with discrepancy being even larger in some cases. We suspect that most of this apparent gain in precision may reflect a downward finitesample bias in estimates of twostep standard errors as indicated by simulation results in Table, suggesting that caution would be advisable in making inferences based on twostep estimator alone in samples of this size. Turning to test statistics, neir of robust m statistics nor twostep Sargan test provide evidence to suggest that assumption of serially uncorrelated errors is inappropriate in this example. The onestep Sargan and differencesargan statistics do reject overidentifying restrictions but our simulation results showed a strong tendency for those tests to reject too often in presence of heteroskedasticity. The twostep differencesargan test is more marginal but does reject at per cent significance level. Both Hausman tests also reject but se too show a tendency to overreject in our simulation experiments. One possibility is that this instability across different instrument sets reflects failure of strict exogeneity assumption for wages and capital, rar than serial correlation per se. In Table we present some alternative estimates of this same model for comparison. Columns (e) and (f) report two instrumental variable estimates of differenced equation using simpler instrument sets of AH type. In column (e) we use difference Ani(t3) to instrument Ani(t_), losing one furr crosssection, whilst in column (f) we use level ni(t3) as instrument. In both cases coefficient estimates are poorly determined, indicating a massive loss in efficiency compared to eir GMM estimator in this application. Using both Afni(t3) and ni(t3) as instruments (not reported) helped a little, but estimates remained very imprecise. In column (g) we report OLS estimates of employment equation in levels. In this case 978 crosssection is available and longer estimation period has been used here. Compared to GMM estimates re is a serious upward bias on lagged dependent variable, which suggests presence of firmspecific effects. Column (h) reports withingroups estimates, which are close to GMM in this example. In fact WG estimate of firstorder autoregressive coefficient is bigger than corresponding GMM estimates, although comparison between WG and GMM in this case is obscured by likely endogeneity of wages and capital. Returning to GMM estimates in Table 4, column (b) omits insignificant dynamics with little change in longrun properties of previous model. In columns (b)(d) we report only twostep estimates though onestep coefficient estimates were invariably similar; In column (b) twostep differencesargan test now marginally accepts hyposis of no serial correlation, but twostep Hausman statistic remains an outlier. In column (c) we relax assumption that real wage and capital stock are strictly exogenous and instead treat m as being endogenous. We refore use lags of w and k dated (t ) and earlier as instruments for wit, wi(tl) and kit. We also use lagged values of company's real sales and real stocks as additional instruments. Given size of our sample, not all available moment restrictions were used. The precise form of instrument matrix is described in note (vi) to Table 4. The results in column (c) suggest that it is inappropriate to treat wages and capital as strictly exogenous in this model. In this case none of test statistics indicate presence of misspecification. The coefficient estimates for our preferred specification in column (c) suggest a longrun wage elasticity of 4 (standard error 8) and a longrun elasticity with respect to capital of 7 (S.E. 4). There is a strong suggestion that industry output enters model in changes rar than levels, which is appealing since o' in (7) measures demand shocks relative to potential output. Layard and Nickell (986) interpret
17 9 REVIEW OF ECONOMIC STUDIES Dependent variable: In (Employment)i, TABLE Employment equations Alternative estimates Independent (e) (f) (g) (h) variables AHd AHI OLS Withingroups P (.) () (*) (.8) ni(t) (*8) (7) (48) (.77) Wit (.3) (83) (7) () Wi(t,l) ( 768) (8) (69) (43) k,, () ( ) (48) (6) (.386) (*37) (64) ( 3) ki(t) (3) (4) (3) (.4) ysi, (3) (338) (76) (93) ysi(,_) (88) (.99) ( 48) ( 8) YSi(t) (46) (.4) (3) (39) M Wald test 993 () () R Number of observations Notes. (i) Time dummies are included in all equations. (ii) Asymptotic standard errors robust to general crosssection and time series heteroskedasticity are reported in parenses. (iii) The m and Wald tests are asymptotically robust to general heteroskedasticity. (iv) Columns (e) and (f) report AndersonHsiaotype estimates of equation in first differences:,&n,(,l) is treated as endogenous and additional instruments used are An,(,3) in (e), and ni(,3) in (f), so that one furr crosssection is lost in (e) and effective sample period becomes (v) Column (g) reports OLS estimates of equation in levels, where effective sample period becomes (vi) Column (h) reports withingroups estimates. These are OLS estimates of equation in deviations from time means. shortrun effect of product demand fluctuations on labour demand as reflecting practice of normal cost pricing. Finally in column (d) we report estimates of Euler equation model given in (8). Here again we treat wages and capital as endogenous variables. Although tests for serial correlation remain below ir critical values, coefficient estimates are not favourable to Euler equation interpretation. The coefficients on capital and industry output are poorly determined, whilst those on lagged dependent variable imply a real discount rate of around %. Very similar results were obtained for versions of Euler equation model allowing for an MA () error process and estimating in levels as opposed to logs. It appears that process of employment adjustment is not well described by this model.
18 ARELLANO & BOND SPECIFICATION TESTS 93 The results of this empirical application are generally in agreement with those of our Monte Carlo simulations. The GMM estimator offers significant efficiency gains compared to simpler IV alternatives, and produces estimates that are welldetermined in dynamic panel data models. The tendency for nonrobust test statistics to overreject is confirmed. The robust m statistics perform satisfactorily as do twostep Sargan and differencesargan tests, but twostep Hausman test must be considered suspect in samples of this size. 6. CONCLUSION In this paper we have discussed estimation of dynamic panel data models by generalized method of moments. The estimators we consider exploit optimally all linear moment restrictions that follow from particular specifications, and are extended to cover case of unbalanced panel data. We focus on models with predetermined but not strictly exogenous explanatory variables in which identification results from lack of serial correlation in errors. A test of serial correlation based on GMM residuals is proposed and compared with Sargan tests of overidentifying restrictions and Hausman specification tests. To study practical performance of se procedures we performed a Monte Carlo simulation for units, seven timeperiods and two parameters. The results indicate negligible finite sample biases in GMM estimators and substantially smaller variances than those associated with simpler IV estimators of kind introduced by Anderson and Hsiao (98). We also find that distributions of serialcorrelation tests are wellapproximated by ir asymptotic counterparts. We applied se methods to estimate employment equations using an unbalanced panel of 4 quoted U.K. companies for period The GMM estimators and serial correlation tests performed well in this application. A potentially serious problem, suggested by both experimental evidence and application, concerns estimates of standard errors for twostep GMM estimator which we find to be downward biased in our samples. Furr results on alternative estimators of se standard errors would be very useful. A. The asymptotic normality of m statistic APPENDIX Following notation of Section 3, under assumption that (X v*/ N) op () we have and also where Nd^' 6 Nvv  (v' X*/N)N(88)+op() as N*oo N/A/ N V_U* A~ N N/vLI/gNZ v_ g* gn V + is( op(), vx*(x'zanz'xv)'x'zan. Then a multivariate central limit orem for independent observations ensures
19 94 REVIEW OF ECONOMIC STUDIES where WN is average covariance matrix of (4i, ct)': and {i Zvi. Therefore where N I iei~ _'N ipn\ WN N E( lifi ifi ) N VN) N INlIVV* Lv +d N(O, ) (Al) or VN )N g'n + gnvngn N (~A VN i) N [J, E(vi()vi*vs*vl()) (VLX*)(XZANZX) X ZANLi E(Zviv*vi()) with +(v' X*) avar (X*V)] avar () N(X'ZANZ'X)(X'ZANVNANZ'X)(X'ZANZ'X)Y A consistent estimate of VN can be obtained by replacing population average expectations of errors by sample averages of residuals. Finally, noticing that under our assumptions (Al) remains valid after this replacement, result follows. For a onestep?, we can also consider an alternative m criterion ("one step M") which relies on more restrictive auxiliary distributional assumptions. Assuming that errors in model in levels are independent and identically distributed across individuals and time, we have N OJN  N a, E(V!()H,*Vi()) where Hi* is a ( Ti  4) square matrix which has twos in main diagonal, minus ones in first subdiagonals and zeros orwise. Moreover where Hit is a ( Ti) x ( Ti4) matrix with N N N tvi()) N VH,*J and Hi.. is a x (T 4) matrix with minus one in (, ) position and zeroes elsewhere. Under such conditions v can be replaced by V ff Lil Vi()Hi. AX*(X ZANZ X) X ZAN (Si z (Hit ))a + v^'x* avar (8)X *V where is an estimate of a* B. The Sargan difference test Note that s in () can be rewritten as iyz* *j 6*iN zvi A S AzI N \I (N N 'vivizi '' IN` where Z* ZH and H is a p x p linear transformation matrix such that Z*(ZiIZl) with N Z V^V^'Z * O
20 ARELLANO & BOND SPECIFICATION TESTS 9 Letting where CN is p x p nonsingular, and noting that ( zn z' IZ z> CNC'N Z*tv z*x Xz* z*x 'xlz* z*v Z/ IP Z XXZ CNC'N ] CNC'N} v we have where vpz* s CN[IP  G(G'G) IG']C'N Z ZN G CN N( Following usual argument one can show that s de'me M X k, where E  N(O, IP) and M is of form IP D(D'D) D' with rank p  k. On same lines we can write SI NI CIN[IP G(G' GI) G']C' Z'v where G C'N (Z X) and ( N z i l) CINC IN Let G* contain top p, rows of G. Notice that G*  G, P. Therefore Ml ds dsss~~deme~p' S SI  E ME E ) O)E where Ml IP DI (D'DY) D and rank (MI) p,  k with D' (D'I D'). Finally notice that is symmetric and idempotent with rank p  p, and also from which () follows. [M Q??)] /Ml Ml o\ / (a) Sample DATA APPENDIX The principal data source used is company accounts from Datastream International which provide accounts records of employment and remuneration (i.e. wage bill) for all U.K. quoted companies from 976 onwards. We have used a sample of 4 companies with operations mainly in U.K., whose main activity is manufacturing and for which we have at least 7 continuous observations during period Where more than 7 observations are available we have exploited this additional information, so that our sample has unbalanced structure described in Table Al.
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