Directory UMM :Data Elmu:jurnal:J-a:Journal of Econometrics:Vol98.Issue2.Oct2000:

A quasi-di!erencing approach to dynamic

modelling from a time series of

independent cross-sections

Sourafel Girma

*

School of Economics, University of Nottingham, Nottingham NG7 2RD, UK

Received 4 May 1998; received in revised form 10 March 2000; accepted 13 March 2000

Abstract

We motivate and describe a GMM method of estimating linear dynamic models from a time series of independent cross-sections. This involves subjecting the model to a quasi-di!erencing transformation across pairs of individuals that belong to the same group. Desirable features of the model include the fact that: (i) no aggregation is involved, (ii) dynamic response parameters can vary across groups, (iii) the presence of unobserved individual speci"c heterogeneity is explicitly allowed for, and (iv) the GMM estimators are derived by realistically assuming group size asymptotics. The Monte Carlo

experi-ments conducted to assess the"nite sample performances of the proposed estimators

provide us with encouragement. ( 2000 Elsevier Science S.A. All rights reserved.

JEL classixcation: C23

Keywords: Time series of cross sections; Panel data; GMM

1. Introduction

In this paper we propose a new method of using data from a time series of independent cross-sections (abbreviated to TISICS in what follows) in "tting

*Corresponding author. Tel.:#44-115-951-4733; fax:#44-115-951-4159. E-mail address:sourafel.girma@nottingham.ac.uk (S. Girma).

linear dynamic models such as

y

it"ayit~1#bxit#fi#eit, (1)

where i and t index individual units and time periods, respectively, f is an individual e!ect andeis a disturbance term. The main problem is the fact that such data sets do not track the same group of individuals over time, whereas the econometric models of interest require some information on intra-individual di!erences. Thus, it is not possible to consistently estimate the parameters of interest unless additional information in the shape of instrumental variables is available or further identifying restrictions are imposed on the model.

Deaton (1985) proposes a pseudo-panel approach to circumvent the

identi-"cation problem caused by the absence of repeated observations on individual units. This involves placing individuals into distinct groups or cohorts, sayG

c,

forc"1,2,Cand working with the expectation of model (1) at each point in time conditional on cohort membership. Thus taking EMy

itDi(t)3GcN, we have yH_ct"ayH_ct~1#bxH_ctit#f_ctH#eH

ct (2)

where superscripts asterisk denote the conditional expectations.

Eq. (2) is a model of the average behaviour of a cohort, much in the spirit of the&representative agent'construction. Since cohort averages are e!ectively the units of analysis and the number of cross-sections is often small in practice, one might have to assume asymptotics on C to derive large-sample properties of the ensuing econometric estimators. On the other hand as yH

ct~1 and fctH are

correlated by construction, it is desirable to di!erence the latter out of the model. This requires that f_ctH be invariant over time. One way to force this condition is to assume that all individuals within a cohort are homogeneous with respect to the unobserved heterogeneity factors. It will then be possible to apply standard dynamic panel data techniques by treating sample cohort averages as error-ridden observations on typical individuals (Collado, 1997).

response parameters are characterised by cohort-wise heterogeneity. This follows from Pesaran and Smith (1995) who showed that dynamic modelling from pooled heterogeneous panels is not trivial. Given that cohorts should be chosen so that they are as distinct from each other as possible (cf. Deaton, 1985), it is important to have a framework which allows for some sort of response heterogeneity.

The above discussion demonstrates that there is clearly scope for studying alternative estimators for dynamic models from TISICS. In a contribution to the literature Mo$t (1993) suggests a two-stage least-squares approach where the regressory

it~1in (1) is instrumented by the predicted valuey(it~1from a linear

projection of the dependent variable on a vector of varying and time-invariant variables, sayS

it andZi, respectively. To make this operational, one

needs knowledge ofS

it~1. But it is not always clear where this information is

going to come from as TISICS data sets do not typically contain the history of the variables for the cross-sectional units. Moreover two-stage least squares is potentially problematic in that the"tted instruments (which need to be corre-lated withy

it~1) will also have a non-zero correlation with the individual e!ect f

i, since the two quantities are statistically related. The only obvious exception is

ifS

it is a function of time alone, hence uncorrelated with the time invariantfi.

But this will result in instruments that do not have much variation across individuals in the likely case where the number of cross-sections is much smaller than the cross-sectionals units. As a result the ensuing instrumental variables estimators will be highly inaccurate. So, although Mo$t's (1993) paper general-ises Deaton (1985) in the sense that the possibility of instrumentation procedures other than grouping is discussed, it does not seem to be too promising due to its rather strong informational requirement (Verbeek, 1992). In this paper we explore the possibility of making inference on model (1) from individual level data, by performing pair-wise quasi-di!erences across di!erent observed indi-viduals within the same group. This produces the following regression function at each time period and for all i and j: EMy

itDyjt~1,xitN. This is potentially

estimable since CovMy

it,yjt~1Ncan be identi"ed from the data as long there are

group-wise cross-sectional correlations.

can easily be allowed since the method can be applied on a single group. In Section 4 we conduct some Monte Carlo experiments to assess the"nite sample performances of the proposed estimators. Section 5 concludes.

2. Model description

2.1. Basic identifying assumptions

We focus on the estimation of the following class of models form TISICS:

y

i(t)t"ai(t)yHi(t)t~1#b@i(t)xi(t)t#fi(t)#ei(t)t

t"2,¹;i(t)"1,2,N_t. (3)

Here xis a k-dimensional vector of control variables, frepresents individual-speci"c e!ects and the e's are time-varying disturbances. The time index in parentheses is used to emphasise the non-panel nature of the data. However, when no ambiguity is apparent, we will sometimes drop this index for the sake of notational elegance. It is assumed that we have cross-sectional observations for t"1,2,¹, with¹*2. Notice that foryH_i(t)t~1the time index on the individual does not match to the (proper) time index. Such variables are not observed and they will often be denoted by a superscript asterisk. We will next discuss some of the basic identifying assumptions of the model.

H.1: The population is partitioned into a"nite set of mutually exclusive and exhaustive groups G

c, c"1,2,C. Groups may be based on one or more

underlying observed variables that are constant over time for all individuals. Without loss of generality, we assume that at each point in timenmembers are drawn at random fromG

c, ∀c.

H.2:a

i"aj"ac and bi"bj"bcfor∀(i,j)3Gc.

This assumption restricts the individual level heterogeneity in response para-meters by stating that those parapara-meters are identical within a group, but we do the next best thing by allowing for the possibility of group-wise heterogeneity.

H.3: Da

cD(1,∀c.

H.4:y

i(t)0"k1fc#k2ei(t)0, where the k's are constants and fc is a random

group speci"c e!ect.

The above assumption implies that the initial (unobserved) conditions are random draws from possibly di!erent populations, but share a common com-ponent in the speci"cation of a conditional mean. Assumption H.4 has an important implication: CovMy

i(t)t,yj(s)s)NO0 for∀(i(t),j(s))3Gcand∀t,s. That is,

1As the estimator proposed in this section deals with a single group, the index onaandbis dropped.

cross-sectional data sets (e.g., Frees, 1995). Researchers working with TISICS data often assume that each member of the group has some information about the average group value (Blundell et al., 1994).

H.5: x

i(t)t"ti(t)#j@xi(t)t~1#hct#wi(t)twhere the eigenvectors ofjlie within

the unit circle,t

i(t)is ak-dimensional vector of individual speci"c drift andhctis

group-time speci"c error component which can come about due to some shared characteristics between individuals within a group-time cell. It can be seen that

h

ct is a further source of within group correlations. From the above two

assumptions, it is clear that intra-group correlation is directly proportional to the variances off

c andhct.Further assumptions are:

H.6: CovMf

i,etiN"0,∀iandt.

H.7:f

i,fc,fct,hct andeitare all mutually, independently distributed with mean

zero and nonzero and"nite variances.

The above assumption allows for purely individual-speci"c heterogeneity, even if individuals within a group share a common e!ect. This is in contrast to Deaton's (1985) approach where within-group heterogeneity is not explicitly allowed for.

H.8: CovMx

i(t)t,fj(s)N"0.

This is an innocuous assumption sincef

j(s) is speci"c to individualj(s).

Making use of the above assumptions, we can show that CovMy

i(t)t,yj(s)sNO0

and equal∀(i,j)3G

c. In other words, our assumptions imply an equicorrelation

structure within a group-time cell. Intuitively, without equicorrelation there would not be enough independent information to estimate O(n2) covariances per cell from justnobservations. We like to note that in one form or another, the assumption of equicorrelation amongst the units of analysis is prevalent in econometric modelling. For example, the two-way panel estimation relies on equicorrelation between individuals observed during the same time period (e.g., Hsiao, 1986). The covariance stationarity assumption employed in time series models is also a case in point.

2.2. A quasi-diwerencing approach

We begin by writing (3) in its reduced-form representation as1

y

i(t)t"nt1xHi(t)1#2#nt(t~1)xHi(t)t~1#nttxi(t)t#nt0yHi(t)0#vi(t)t (4)

with

v i(t)t"

t~1

+

s/0

ase_i(t)(t~s)#1!at 1!_a fi(t)nts

2Althoughx

j(t~1)t~1is observed, it is left in the disturbances for reasons that will shortly become

apparent.

and

n_t0"_at.

Quasi-di!erencing the model asMy

i(t)t!ayj(t~1)t~1;∀[i(t),j(t!1)] and∀t*2N,

we obtain the following potentially estimable model:

y

i(t)t"ayj(t~1)t~1#b@xi(t)t#gijt (5)

where

g_ijt"_at[y

i(t)0!yj(t~1)0]#Dfijt#Dxijt#Deijt

with

*f ijt"

1!_at 1!a fi(t)!

a!_at 1!a fj(t~1),

*x ijt"b@

t~2

+

r/0

ar`1MxH

i(t)(t~1~r)!xHj(t~1)(t~1~r)N2

and

*e ijt"

t~1

+

r/1

arMe_i(t)t~r!e

j(t~1)t~rN#ei(t)t.

Let D

1"[0(T~1)C1:IT~1];D2"[IT~1: 0(T~1)C1] and A2"D2?ln?In,

where 0

a;Iaand laare the null matrix, the identity matrix and the vector of ones

of dimensiona, respectively. When all observations within a group are arranged

"rst by individuals and then by time periods, and the transformation matrix B(a)"A

1!aA2is applied to the resultingn¹]1 vector, it can be checked that

do so we"rst consider the following set of quasi-di!erences within each group between individuals with the same ordering in their respective time domains: ¸

1_{Note that}"Myi(t)t!_i(a_t)"yi(t~1)t~1_{i(t!1) is to be understood that as the case where individual};∀i(t),i(t!1) andt*2N.

iat timetand individualiat time (t!1) correspond to the same ordering within their respective time domains. It can be veri"ed that ¸

1 consists of n(¹!1)

linearly independent quasi-di!erenced equations. By augmenting ¸

1 by the

followingn!1 quasi-di!erences between the"rst observation at timet"2 and all observations bar the "rst one at time t"1, ¸

2"My1(2)2!ayj(1)1;

∀j*23G

c1N, it can be checked that¸"¸1X¸2will produce the desired full set

ofn¹!1 linearly independent quasi-di!erences. As a result¸will exhaust all information contained in the pairwise quasi-di!erenced model. Of course this is not the only set of linearly independent quasi-di!erences. However, since any other such set is a nonsingular transformation of¸, the same information will be preserved.

Let=

ijt"[yj(t~1)t~1,x@i(t)t]@ and de"ne c0"[a,b@]@ as the true parameter

vector. When the above transformations are applied to Eq. (3), the following model will then be available for estimation for

∀[i(t),j(t!1)]3S,S 1XS2 y

i(t)t"ayj(t~1)t~1#b@xi(t)t#gijt,c@0=ijt#gijt, (6)

whereS

1"M[i(t),i(t!1)] andt*2N;S2"M∀[1(2),j(1)],j(1)*2)N.

3. Inference

3.1. Identixcation

OLS on model (6) would produce inconsistent estimators because g ijt is

correlated with both y

j(t~1)t~1 andxi(t)t. This can easily be checked using the

underlying assumptions of the model. Fortunately, these assumptions also imply that the composite error termg_tcij satis"es the following two sets of linear moment conditions:

EMy

g(t~s,t~s)gijtN"0;t"2,2,¹;s"1,2,t!1;∀g(t!1)Oj(t!1)

and

EMx

g(t~s)t~sgijtN"0;t"2,2,¹;s"0,2,t!1;

∀g(t!s)Dg(t!s)Oi(t) org(t!1)"j(t!1).

3We adopt the convention of puttingg"nforj"1

correlations in thex's between the di!erent individuals comes from the h ct's,

leaving the observablex

j(t~1)t~1in the composite disturbance term ensures that

the latter does not contain a h term. Also notice that since g

ijt contains y

i(t)0!yj(t~1)0, any yg(t~s)(t~s) will not be correlated with it because of the

equicorrelation assumption imposed on the model.

A problem that we face is the possibility of an in"nite number of instruments, given that the number of observations per group-time cell is allowed to grow. For example, X

i(t~2)t~2;i"1,2,ncan be used as instruments for the

regre-ssors in the quasi-di!erenced model. Since standard optimality results for GMM estimators (e.g., Hansen, 1982) are not trivially applicable when the vector of conditioning variables is in"nite dimensional, it pays to look for some ways to economise on the number of instruments to be used. One apparent solution might be to average the instruments over cells. In this case there would not be enough internal variation in the instruments set, unless the number of groups grows withnat a suitable rate. This is bound to adversely a!ect the precision of the resulting estimators. In this paper we restrict ourselves to a set of "nite orthogonality conditions by the simple, albeit ad hoc, device of using just one orthogonality condition within each subset of the moment restrictions. For example in the Monte Carlo experiments conducted in the next section, the following moment restrictions are chosen to be the basis for estimating c: EMZ

ijtgijtN"0 with Zijt"Myg(t~1)t~1,x@g(t~1)t~1,x@g(t)tN, where g is made to

correspond to j!1.3 We have to emphasise, however, that this is entirely arbitrary. All that is needed is to assign distinct instruments to each observation.

3.2. GMM inference

GMM estimation proceeds by forming sample analogs of those population moment conditions. The sample analogs are usually sample averages ofZ

ijigijt

laws of large numbers typically place some restrictions on the dependence, heterogeneity and moments of the data process.

The main problem in the present context is the contemporaneous within group dependence introduced by the transformation function¸

2. One practical

solution may be to average all quasi-di!erences produced by¸

2 and assume

that asngrows the dependence will become negligible (and accept the risk of any

"nite sample bias). In the sequel we work with the model produced by¸

1alone,

because the large sample argument with a "xed number of groups and time periods is neater and the computations involved are simpler. Thus we have, for i"1,2,nandt"2,2,¹

y

i(t)"ayi(t~1)(t~1)#b@xi(t)t#giit,c@0=iit#giit

and the resulting orthogonality conditions can be expressed as EMZ

iitgiitN,

EMh

itN"0, with Ziit"Myi~1(t~1)t~1,x@i~1(t~1)t~1,x@i~1(t)tN In the Appendix A

we prove that the following proposition holds:

Proposition 1.

nT to be a sequence of weight matrices, the GMM estimator of c₀can thus be obtained upon solving

argmin_c

C

1

The optimal GMM estimator ofcis obtained by setting the weight matrix in the GMM minimand function to the inverse of

X,Cov

C

1

instruments matrices are constructed in an obvious way. In Appendix B we show that the following proposition holds under the assumptions of the model:

4As an anonymous referee noted, the pairwise quasi-di!erencing estimators proposed in this paper can be seen as the limiting case of this simpler estimator asMapproaches the sample size.

Thus under some regularity conditions (White, 1984),c(

c will be asymptotically

normally distributed as

J¸(c(!_c

0)PN(0,<), (7)

with

<"

CA

1

n(¹!1)Z@=

B

X~1

A

1

n(¹!1)Z@=

BD

~1

.

Obviouslyc( is not feasible sinceXis unknown, but the latter can be replaced by a consistent estimator without a!ecting the asymptotic properties of c(

(Hansen, 1982). A Newey}West (1987)-type estimator forXcan be constructed nonparametrically as

XK" 1 n(¹!1)

C

+

i,t h

ith@it#

A

+ i,t

h

ith@i(t~1)#hi(t~1)h@it

BD

.

An advantage of this approach is that it can be implemented with only two cross-sections. And when we have more than one group and the parameters of the model are heterogeneous across these groups and the equation errors are not related, it can be implemented separately for each group without loss of e$ciency. If the groups share a common parameter, however, a joint estimation can enhance e$ciency. For example, with C(Rgroups and only avarying across groups, the model can be estimated within a system of instrumental variables method by stacking the observation by groups.

3.3. A simpler estimator

For some M in each time period, randomly divide observations within a group into M groups. Let y6

mt be the sample mean across all observations

in subgroup m,m"1,M. Now consider the quasi-di!erenced equations:

My

it!ay6mt~1N which gives rise to the following estimable model: yit" ay6

mt~1#b@xit#g6imt. One can then implement the GMM estimation strategy of

this paper by using [y6

(m~1)t~1x6(m~1)tx6(m~1)t~1] as instrumental variables.4The

relative merits of these alternative estimators by conducting some simulation experiments.

3.4. Some extensions to the basic model

The study of the e!ects of observed time-invariant variables such as sex, region and other socio-economic background variables (assuming that they are not used to de"ne groups) could be important in practice. To see how such variables"t into the quasi-di!erencing framework, de"ne by =_{the vector of}

observed time-invariant regressors and consider the following model:

y

i(t)t"ayHi(t)t~1#u=i(t)#ei(t)t. (8)

Pairwise quasi-di!erencing Eq. (8) would yield

y

i(t)t"ayj(t~1)t~1#o1=i(t)#o2=j(t~1)#qtij (9)

where

q_tij"atxy

0i!y0jy#*ftij#*ftij; o₁"1!at

1!au and o2"! a!_at 1!au.

The identi"cation ofufrom (9) would be greatly simpli"ed if=

i(t)"=j(t~1),

that is if individualsi(t) andj(t!1) exhibit the same time-invariant character-istics. Eq. (9) then simpli"es to y

ti"ayj(t~1)t~1#o=i(t)#qtcij.

Thus a suitable choice of pairs for di!erencing would ease the estimation ofu, a parameter which is not identi"able in"rst-di!erenced dynamic (pseudo) panel models.

The model we have considered in this paper can also be easily extended to include lagged regressors of higher orders. For example, to estimate the pure AR(2) modely

i(t)t"a1yi(t)t~1#a2yi(t)t~2#eit, we need to perform the

follow-ing transformation to the data:

My

i(t)t!a1yj(t~1)(t~1)!a2yk(t~2)(t~2);∀i(t),j(t!1) andk(t!2)3GcN.

4. Monte Carlo simulations

Since an important factor a!ecting the behaviour of the estimators is the magnitude of IV-regressor correlation, we consider alternative data generating mechanism in which 25%, 50% and 75% of the total variability in the (scalar) exogenous variable and the initial values is attributed to a common group component. In each of these cases, the individual-speci"c variance is taken to be unity. Fixing the number of groups and cross sections to 1 and 5 respectively, we experimented with three di!erent group-time cell sizes, 100, 200 and 400, which we think are not unrealistic. For each of the resulting 18 con"gurations, one hundred Monte Carlo experiments are conducted. For the AQD estimators the observations are randomly assigned to 10 subgroups.

Tables 1 and 2 summarise the results from the experiments. The means and standard deviations are computed over the 100 replications and the"gures in parentheses represent the percentage biases from the true values. The sample means of the asymptotic standard errors calculated from the GMM formula are also reported, with the percentage deviation of these asymptotic values from the respective across replication standard deviations of estimated values given in parentheses.

A noteworthy result from the experiments is that the"nite sample bias of the GMM estimator is more pronounced when the relative variance of the group e!ect (sayf) is small. For example in Table 1, the bias in the estimator fora(b) decreases from 3.6% (3.8%) to 1.5% (1.8%) whenn"100 asfincreases from 25% to 50%. But, it is encouraging to note that this bias seems to be tolerable for all sample sizes.

When it comes to the e$ciency of the GMM estimators as measured by the across-simulation standard deviations, the role of fis more crucial. For given sample size, e$ciency improves dramatically as f increases. For example, in Table 2 the standard deviations of the estimators is divided by a factor of 3 as

fdoubles to 50%, forn"200. A marked gain in e$ciency is also observed as

fincreases to 75%.

A worrying aspect of the results is the absolute magnitude of the standard deviations (and asymptotic standard errors) whenfis small, irrespective of the sample sizes. Assuming that the estimators will be deemed precise enough if the margin of error at 5% level is within 20% for the true parameter value, the standard deviation should be less than a tenth of the corresponding parameter. Using this rough guide most estimates based onn"100 and 200 are disappoint-ing whenf"25%. In general good precision is obtained whenfis set to 75%, although some reasonably precise estimates emerge withf"50%.

Mean (% bias), standard deviation and asymptotic standard error of the estimators fora"0.8;b"!0.5 Relative variance of group e!ect

25% 50% 75%

PQD AQD PQD AQD PQD AQD

n"100,a

Mean 0.771 (3.6%) 0.358 (55.2%) 0.812 (1.5%) 0.539 (32.6%). 0.810 (1.3%) 0.669 (16.4%)

Std deviation 0.294 0.078 0.209 0.087 0.071 0.046

Asym s.error 0.297 (1%) 0.034 (56.4%) 0.175 (16.3%) 0.037 (47.5%) 0.070 (1.4%) 0.026 (43.5%)

b

Mean !0.519 (3.8%) !0.796 (59.2%) !0.491 (1.8%) !0.665 (33%) !0.494 (1.2%) !0.589 (17.8%)

Std deviation 0.211 0.06 0.135 0.062 0.057 0.042

Asym s.error 0.213 (1%) 0.034 (43.3%) 0.122 (9.6%) 0.034 (45.1%) 0.054 (5.3%) 0.023 (45.2%)

n"200,a a

Mean 0.818 (2.2%) 0.462 (42.2%) 0.805 (0.6%) 0.623 (22.2%) 0.786 (1.8%) 0.717 (10.4%)

Std deviation 0.439 0.090 0.151 0.073 0.049 0.041

Asym s.error 0.421 (4.1%) 0.042 (53.3%) 0.131 (13.2%) 0.029 (61.3%) 0.050 (2%) 0.019 (53.6%)

b

Mean !0.463 (7.4%) !0.714 (42.8%) !0.496 (1%) !0.616 (23.2%) !0.506 (1%) !0.559 (11.8%)

Std deviation 0.342 0.085 0.112 0.057 0.040 0.029

Asym s.error 0.299 (12.6%) 0.039 (54.1%) 0.094 (16.1%) 0.026 (54.4%) 0.038 (5%) 0.017 (41.4%)

n"400,a a

Mean 0.764 (4.5%) 0.574 (28.2%) 0.801 (0%) 0.695 (13.1%) 0.803 (0%) 0.754 (5.7%)

Std deviation 0.247 0.084 0.099 0.043 0.038 0.028

Asym s.error 0.238 (3.6%) 0.032 (62%) 0.091 (8.1)% 0.022 (48.9%) 0.037 (2.6%) 0.014 (50%)

b

Mean !0.546 (9.2%) !0.663 (32.6%) !0.504 (1%) !0.569 (13.8%) !0.498 (0.4%) !0.532 (6.4%)

Std deviation 0.196 0.063 0.059 0.037 0.026 0.019

Asym s.error 0.182 (7.2%) 0.030 (52.4%) 0.065 (10.2%) 0.019 (48.6%) 0.029 (11.5%) 0.013 (31.6%)

S.

Girma

/

Journal

of

Econometrics

98

(2000)

365

}

383

Table 2

Mean (% bias), standard deviation and asymptotic standard error of the estimators fora"0.3;b"!0.8 Relative variance of group e!ect

25% 50% 75%

PQD AQD PQD AQD PQD AQD

n"100,a

Mean 0.340 (13%) 0.121 (60%) 0.299 (0%) 0.185 (38%) 0.304 (1.3%) 0.242 (19%)

Std deviation 0.221 0.029 0.107 0.028 0.042 0.024

Asym s.error 0.251 (13.6%) 0.023 (20%) 0.105 (1.9%) 0.024 (14.3%) 0.041 (2.4%) 0.016 (33.3%)

b

Mean 0.769 (3.9%) 0.934 (16.8%) 0.800 (0%) 0.888 (11.1%) 0.798 (0%) 0.846 (5.75%)

Std deviation 0.188 0.027 0.086 0.026 0.035 0.022

Asym s.error 0.211 (12.2%) 0.022 (19.5%) 0.089 (3.48%) 0.022 (15.4%) 0.034 (2.8%) 0.014 (36.3%)

n"200,a a

Mean 0.290 (3.3%) 0.164 (45.3%) 0.308 (2.6%) 0.230 (23%) 0.301 (0%) 0.267 (11%)

Std deviation 0.242 0.046 0.66 0.026 0.031 0.019

Asym s.error 0.206 (14.9%) 0.027 (41.3%) 0.071 (7.6%) 0.018 (31.8%) 0.030 (3.3) 0.12 (36.8%)

b

Mean 0.815 (1.88%) 0.902 (12.8%) 0.791 (1.12%) 0.854 (6.8%) 0.799 (0%) 0.826 (3.25%)

Std deviation 0.177 0.033 0.063 0.023 0.025 0.016

Asym s.error 0.166 (6.2%) 0.025 (24.2%) 0.060 (4.8%) 0.016 (23.1%) 0.025 (0%) 0.011 (31.3%)

n"400,a a

Mean 0.324 (8%) 0.208 (30.6%) 0.294 (2%) 0.258 (14%) 0.297 (0%) 0.279 (7%)

Std deviation 0.164 0.032 0.049 0.020 0.029 0.013

Asym s.error 0.170 (3.7%) 0.022 (31.2%) 0.046 (6.1)% 0.014 (30%) 0.021 (27.6%) 0.010 (27.6%)

b

Mean 0.789 (1.4%) 0.871 (8.9%) 0.805 (0.6%) 0.832 (0.4%) 0.803 (0.4%) 0.817 (2.1%)

Std deviation 0.146 0.029 0.039 0.017 0.020 0.010

Asym s.error 0.139 (4.8%) 0.020 (31%) 0.040 (2.6%) 0.013 (23.5%) 0.017 (15%) 0.007 (30%)

S.

Girma

/

Journal

of

Econometrics

98

(2000)

365

}

mostly within 20% of each other. We thus "nd it encouraging that the nonparametric covariance matrix estimator performs quite well in small samples.

The above discussion has highlighted the desirability of having a reasonably high level of relative variation in the group-speci"c component (and hence a high degree of correlation between individuals within the same group) for the success of the quasi-di!erencing method proposed in this paper. This is not a surprising result in view of recent work which emphasises the pitfalls of IV-type estimators when the instrument-regressor correlation is low (Staiger and Stock, 1997). So an important implication for applied work is to routinely examine the correlations between the regressors and the instrumental variables candidates and exercise caution when these values are low.

The small sample performance of the simpler AQD proposed in Section 3.3 proves to be disappointing, except when the sample size is quite large (n"400) and most of the variation in the data comes from the group speci"c e!ect (f"75%). A large number of observation per sub-group is needed to guarantee that the AQD disturbances are not correlated; while the consequences of throwing some individual speci"c information following the aggregation pro-cedure is not serious if only a small part of the sample variation is due to individual-speci"c e!ects.

5. Concluding remarks

In this paper we motivated and described a new method of estimating linear dynamic models from a time series of independent cross-sections. Some of the desirable features of the method include the fact that no aggregation is involved; the response parameters can freely vary across groups; the presence of unobser-ved individual speci"c heterogeneity is explicitly allowed for and the large-sample results are derived by realistically assuming group size asymptotics. Moreover the Monte Carlo experiments conducted to assess the"nite sample performances of the proposed estimators provide us with some encouragement. For these reasons, we think that it can be considered as a feasible alternative in applied work.

Acknowledgements

Appendix A. Proof of consistency

For the sake of notational elegance we pretend thattruns from 1 to¹and for ease of exposition we assume thathis scalar. We want to show that the sequence of random vectorMh

it;t"1,2,¹;n"1, 2,2Nsatis"es a Weak Law of Large

Numbers (WLLN). Using the assumptions of the model and further assuming that the various fourth-order moments exist and are"nite, it can be established that the covariance of the GMM disturbances have the following general structure:

represents covariance terms between two individuals which correspond to the same ordering within adjacent time periods, andf

3(t,s) is the result of using the

stochastic y

i~1(t~1)t~1 as an instrument. Thus we have a zero-mean random

variable which is heterogeneous across time but whose covariance structure is independent of the individual index.

Now re-write

Chebyshev inequality, fore'0

Now, since E(h

a continuous function, we use Proposition 6.1.4 in Brockwell and Davis (1991) to establish that g(H

nTconverges in probability toSwhich is O(1) and uniformly

positive de"nite.Then it follows from the theorem in White (1984, p. 25) thatc(, which is the solution to establish that

argmin_c

C

1

Appendix B. Proof of asymptotic normality

Consider (1/Jn)+_ni/1H

i with E(Hi)"0, by the orthogonality conditions. The

variance ofH

i can be written as

Var(H

Similarly we establish that E(H

iHi~1)"+T~1t/1[f3(t,t#1)#f3(t#1,t)],

X

2. By assumption E(HiHi~j)"0 ifj*2, implying thatMHiN=i/1is&covariance

stationary'. Thus we can use Wold's decomposition theorem to write H

IfX is uniformly positive de"nite and O(1) and assumptions (a) and (b) of Appendix A hold, we can use the theorem in White (1984, p. 69) to establish that

where D,(Q@

nTSnTQnT)~1Q@nTSnTXSnTQnT(Q@nTSnTQnT)~1. In practice, the

asymptotic covariance matrixXis substituted by its positive semide"nite consis-tent estimator andQ

nTis replaced byZ@=/n¹, whereZand=are the matrices

of instruments and regressors, respectively. SinceMH

iN=i/1is&covariance

station-ary', the nonparametric estimators of asymptotic covariance matrices in the econometric literature (e.g., Newey and West, 1987) should work under some regularity conditions regarding the underlying error terms.

References

Anderson, T.W., 1971. The Statistical Analysis Time Series. Wiley, New York.

Brockwell, P.J., Davis, R.A., 1991. Time Series: Theory and Methods. Springer, New York. Blundell, R., Browning, M., Meghir, C., 1994. Consumer demand and the life-cycle allocation of

household expenditure. Review of Economic Studies 61, 57}80.

Crepon, B., Mairesse, J., 1996. The Chamberlain approach. In: Matyas, L., Sevestre, P. (Eds.), The Econometrics of Panel Data: Handbook of Theory and Applications. Kluwer Academic Publishers, Dordrecht, pp. 323}395.

Collado, M.D., 1997. Estimating dynamic models from time series of independent cross sections. Journal of Econometrics 82, 37}62.

Deaton, A., 1985. Panel data from time series of cross sections. Journal of Econometrics 30, 109}126. Frees, E.D., 1995. Assessing cross-sectional correlation in panel data. Journal of Econometrics 69,

393}414.

Hansen, L.P., 1982. Large sample properties of generalised method of moments estimators. Econometrica 50, 1029}1054.

Hsiao, C., 1986. Analysis of Panel Data. Cambridge University Press, Cambridge.

Judge, G., Gri$ths, W., Hill, W., Lutkepohl, R.C., Lee, T.C., 1985. The Theory and Practice of Econometrics. Wiley, New York.

Mo$tt, R., 1993. Identi"cation and estimation of dynamic models with time series of repeated cross sections. Journal of Econometrics 59, 99}123.

Newey, W.K., West, K.D., 1987. A simple, positive de"nite, heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica 55, 703}708.

Pesaran, M.H., Smith, R., 1995. Estimating long-run relationship from dynamic heterogeneous panels. Journal of Econometrics 68, 79}112.

Staiger, D., Stock, J.H., 1997. Instrumental variables regression with weak instruments. Econo-metrica 65, 557}586.

Verbeek, M., 1992. Pseudo Panel Data. In: Matyas, L., Sevestre, P. (Eds.), The Econometrics of Panel Data: Handbook of Theory and Applications. Kluwer Academic Publishers, Dordrecht, pp. 303}315.