ESTIMATION FROM CROSS-SECTIONS OF ...

C.E.P.R.E.M.A.P.

March 1998

ESTIMATION FROM CROSS-SECTIONS OF INTEGRATED TIME-SERIES Jer^ome Adda CEPREMAP, Paris and INRA Jean-Marc Robin INRA and INSEE n 9802

ABSTRACT

ESTIMATION FROM CROSS-SECTIONS OF INTEGRATED TIME-SERIES This paper studies under which conditions a cross-section regression yields unbiased estimates of the parameters of an individual dynamic model with xed eects and individual-speci c responses to macro shocks. We show that the OLS estimation of a system of non stationary variables on a cross-section yields estimates which converge to the true value when calendar time tends to in nity.

RESUME

Estimation en coupe de variables integrees Cet article etudie les conditions sous lesquelles une regression en coupe donne des estimations non biaisees des parametres d'un modele dynamique a eets xes et a chocs macroeconomiques speci ques. Nous montrons qu'une estimation en coupe par les moindres carres lineaires d'un modele ou les variables sont non stationnaires donne des estimations non biaisees lorsque l'origine des processus tend vers moins l'in ni.

JEL: F343. Key Words: liquidity constraints, debt crisis, estimation. Mots Cles: contraintes de liquidite, crise de la dette, estimation.

1 Introduction A considerable number of microeconometric studies rely on cross-section estimates. Yet, a fast growing number of other microeconometric studies use panel data and individual models involving more and more often state dependence. Were they obtained from crosssections of such data generating processes, the former cross-section estimates would generally not be consistent. This note shows that if the underlying dynamics of the regressors are integrated, the cross-section estimates do not indeed converge to the true value when the sample size tends to in nity, but the asymptotic bias becomes eventually negligible when the origin of the individual processes is far enough remote in time. This result holds for a very general speci cation of the error term, allowing xed eects, speci c macro shocks and correlations between shocks. We also show that the rate of convergence depends on whether the variables are cointegrated or not in the time dimension. These results build on other results of the time series literature, initiated by Phillips and Durlauf (1986), Park and Phillips (1988) and Park and Phillips (1989). They are more speci cally related to the the literature on cointegration in panel data (see Pedroni (1996) and Pedroni (1997)). The plan of this note is as follows. Section 2 presents the assumptions and the results. Section 3 illustrates this property by using the Summers-Heston data on annual domestic consumption and GDP for 152 countries from 1950 to 1992. By performing Monte-Carlo simulations on the calibrated model, we explore the nite sample properties of the cross section estimates. Section 4 concludes.

2 Model and Assumptions Suppose that, at the individual level, the following linear model is true:  yth = axht + ht ; (1) xht = bxht, + th; h = 1; :::; H; t = 1; :::; T; where yth and xht are two random variables, with xh = 0 for all h and jbj  1, and  ht = uh + h ut + uht; (2) th = v h +  h vt + vth; where ut and vt are two scalar macro-shocks. Notice that we allow for the presence of individual xed eects uh and vh, and of individual responses to macro-shocks h ut and  h vt : To keep things simple, we have assumed all variables scalar. But it would be easy to generalize the following results to the case of vectors. We make the following assumptions on error components. First, we assume that xed eects have asymptotically nite, non zero sample variances. 1

0

Assumption 1 Cross-section variances.

P P plim H1 Hh (uh) 2 R  ; plim H1 Hh (vh) 2 R  ; H !1 H !1 PH P 1 plim H h (h) 2 R  ; plim H1 Hh (h) 2 R  ; =1

H !1

=1

2

2

+

+

=1

H !1

=1

2

+

2

+

where R + is the set of positive real numbers and where plimH !1 is the limit in Probability.

Second, all error components have zero cross-sectional means.

Assumption 2 Cross-section means. For all t, for all H; PH

h = PH uh = PH h = 0; u h =1 h=1 t h=1 PH h = PH v h = PH  h = 0: v h=1 h=1 t h=1

This assumption may seem restrictive. Typically, one does not expect the aggregate P P P macro-shock Hh hut to vanish. Suppose, however, that Hh uht and Hh h are not zero, then it is possible to rewrite system (1) with errors (2) satisfying assumption P P 2 if yth , H Hl ytl and xht , H Hl xlt are used in place of yth and xht. This is therefore without loss of generality that we assume all variables centered around their cross-section mean, as long as the only parameters of interest are the slope coecients a and b. Thirdly, idiosyncratic components uht and vth are mutually uncorrelated and uncorrelated with individual xed eects. =1

1

=1

1

=1

=1

Assumption 3 Cross-section covariances. For all t, t0, PH PH PH h h P h h P h h P H Ph u ut ! 0; H P h u vt ! 0; H P h v ut ! 0; H H H h h P h h P h h P h  vt ! 0; H h  vt ! 0; H Ph  ut ! 0; H H h h P h ut vt ! 0; H 1

1

1

1

=1

1

=1 =1

=1

=1

=1

1

1

=1

=1

PH 1 h h P H P h=1 v vt H 1 h h P h=1  ut H

! 0; ! 0;

0

P when H ! 1; where \! " means \convergence in probability".

Note that macro shocks ut and vt can be correlated, as well as the xed eects, uh and vh. The next assumption is on the dynamics of the idiosyncratic errors uht and vth.

Assumption 4 The idiosyncratic errors uht and vth are covariance-stationary processes with absolutely summable autocovariances and zero means.

As far as macro shocks are concerned, we assume that they can be either I(0) or I(1). Note that, if macro shocks are covariance-stationary and jbj = 1, then the rst equation of equation (1) de nes a cointegration relationship. If vt is I(1), then xht are I(2) and

whether ut is I(0) or I(1), the rst equation of equation (1) again de nes a cointegration relationship. Clearly, in these cases, one already knows that system's parameters can be consistently estimated with one in nite time-series. But if jbj = 1, vt is I(0) and ut is I(1), all three variables yth; xht and ht are I(1) and yth and xht are not cointegrated. The cross-section dimension then becomes crucial to identify the system's parameters. Let aH;t de ne the Ordinary Least Square estimator of a in the cross-section regression of yth on xht : "

aH;t =

H X h=1

#"

ythxht

H X h=1

(xht)

2

#,1

:

And let

Bt = plim [aH;t , a]

(3)

H !1

be the asymptotic bias of cross-section-t estimate aH;t . We now show the following proposition.

Proposition 5 If jbj = 1, then, under the preceding assumptions, plimt!1 Bt = 0 and the rate of convergence of Bt to 0 is as in the following table: ut is I(0) ut is I(1) vt is I(0) OP (t,1 ) OP (t,1=2 ) vt is I(1) OP (t,3=2 ) OP (t,1 )

where Bt is of order OP (t ); for all real ; if for all " > 0; there exists M" such that Prft, Bt  M"g  1 , " for all t. If xht is covariance-stationary (jbj < 1 and vt is covariance stationary) then the asymptotic bias Bt tends in general to a non null constant when t becomes large.

We refer the reader to appendix A for a proof. Provided that xht is a random walk, proposition 5 shows that a cross-section estimation yields estimates for which the asymptotic bias (when sample size H goes to in nity) tends to zero when the time origin of processes xht and yth is far remote. The rate of convergence of the asymptotic bias depends on the relative order of integration of ht and th. For example, if "ht and th are stationary, the rst equation of the system forms a cointegration relationship, and the rate of convergence of the asymptotic bias is of order t, . If the rst equation of the system is not cointegrated but variables xht and yth are I(1), then the rate of convergence is only of order t, = . When estimating a relationship on cross section data, we can expect the estimation to yield more precise results with older cohorts and if the estimated system is cointegrated. Moreover, proposition 5 shows that whatever the 1

1 2

nature of the error term in (1), a cross section regression on levels always identi es the structural parameters. At the macro level, the identi cation relies on either a regression on levels if the system is cointegrated, or a regression on rst-dierences in the alternate case. This proposition will be easily generalized to the multivariate case where yth, xht and the macro shocks are vectors. If b has all its eigenvalues strictly inside the unit circle, then the cross section estimates are generally biased. And when b is the identity matrix, the bias becomes negligible when t tends to in nity. The apparently intermediate case where b , Id, where Id is the identity matrix, is not of full rank and non null is a particular case of the case b = Id after integrating a subset of the components of xt into yt . Moreover, it is straightforward to show that exactly the same asymptotics apply to the case of pooled cross-sections, i.e. when the OLS estimator aH;t is the one obtained by regressing yth,i on xht,i in the sample f(xht,i ; yth,i); i = 0; :::; t , 1; h = 1; :::; H g. For example, Bt = OP (t, ) when both ut and vt are I(0). There is a growing number of evidence that economic series are integrated even at the micro level (see Deaton and Paxson (1994) for instance). The next section assesses the empirical relevance of these results, by using real data to calibrate Monte-Carlo simulations. 1

1

3 Application and Simulations In this section we illustrate the results established above by some Monte Carlo estimations. The motivation is to analyze the asymptotic results with real data. We use the database provided by Summers and Heston (1991) which reports annual domestic consumption and GDP for 152 countries from 1950 to 1992 to provide an empirically sensible calibration for the Monte Carlo simulations. We denote as yth the logarithm of consumption, and as xht the logarithm of GDP. We postulate the following statistical model for yth and xht :  yth = a + axht + uh + h ut + uht; (4) xht = b + bxht, + v h +  h vt + vth ; h = 1; :::; H; t = 1; :::; T; with  ut = u ut, + "ut ; (5) vt = v vt, + "vt ; 0

0

1

1

1

This is due to the fact that the numerator of Bt is OP (t) and the denominator is OP (t2 ). If all cross-sections from the origin to t are pooled together, then the numerator of the bias of the OLS estimator of a is still OP (t) and the denominator OP (t2 ); because one can easily show that, in the present context, averaging over time produces a time process which has the same order in probability than the averaged process. 1

and where uh; h; uht; "ut; vh; h; vth; "vt are independent normal random variables with means, respectively, 0, , 0, 0, 0, , 0, 0, and non zero variances. Moreover, we x u = v = xh = 0. Consistent estimates of the parameters of the error-component model (4) are reported in table 1. We then use these parameters to simulate paths of log GDP and log of consumption for a large number of periods (years), for dierent values of b; u and v corresponding to the dierent alternative cases of proposition 5. For each set of values, we constructed 100 panel data sets with 152 \countries" and 4000 periods. Table 2 displays the dierent experiments. Figure 1 displays a graph of the average absolute bias (absolute percentage deviation from the true value), as a function of the distance from the origin for the 5 cases. Coordinates are log-coordinates so that the bias paths asymptotically become straight lines. Consistent with the theory developed above, the bias is a decreasing function of time, except in the stationary case. In the stationary case, the average bias is always bigger even for small t. It appears that there exists a middle range set of values for t for which the bias is converging to zero much faster when the variables are I(2). For the \true" case, which corresponds to the estimated parameters, it takes approximately 50 years to reduce the bias by one half, and approximately 160 years to reduce it by 5. Finally, the slopes of bias paths, when t is large, are as predicted by the theory. 0

0

0

2

Table 1: Estimated Parameter Values a b s.d.(uh ) s.d.(ut ) u s.d.(uht) s.d.(vh ) s.d.(vt ) v s.d.(vth ) 0.85 0.99 0.16 0.009 0.9 0.086 0.024 0.046 0.2 0.05 Note: \s.d." : standard deviation. (1)

Table 2: Parameter Values Used in Experiment True conver-   Case u v gence rate ut  I (0); vt  I (0); b = 0 (0) 0 0 ut  I (0); vt  I (0); b = 1 (1=2) 1 0 ut  I (1); vt  I (0); b = 1 (1) 0.9 0.2 ut  I (0); vt  I (1); b = 1 (1) 1 1 ut  I (1); vt  I (1); b = 1 (3=2) 0 1 Note: Estimation results obtained on 200 simulated panel data sets of 152 "countries" and 800 periods. Est. refer to the average estimated convergence rate, and s.d. to its standard deviation. Note that h and h do not necessarilly average to 0. Yet the OLS estimate of a implies centering of the yth and xht variables and thus brings this case back to the previous set-up. 2

4 Conclusion In this note we have shown that the OLS estimation of a system of non stationary variables on a cross-section yields estimates which converge to the true value when calendar time tends to in nity. We emphasize the importance of the variability of idiosyncratic responses. We provide empirical evidence on the elasticity of domestic consumption to the GDP, for 152 countries, and explore the predictions of our proposition with real data. Given the variability of the dierent shocks in our example, the convergence of the bias to values of the order of 10% takes several decades, if not centuries, depending on the order of integration of the series. This result casts serious doubts on the conclusions that can be drawn from cross-sectional estimations when the regressors are known to be dynamic processes. To refer to only one empirical case, we can mention demand system estimation where one of the conditioning variables, total expenditure if not relative prices, is known to be highly autocorrelated, if not a random walk.

References Deaton, A. and C. Paxson (1994). \Intertemporal Choice and Inequality." Journal

of Political Economy , 102(3), 437{467.

Park, J. Y. and P. C. Phillips (1988). \Statistical Inference in Regression with

Integrated Processes : Part 1." Econometric Theory , 4, 468{497.

Park, J. Y. and P. C. Phillips (1989). \Statistical Inference in Regression with

Integrated Processes : Part 2." Econometric Theory , 5, 95{131.

Pedroni, P. (1996). \Fully Modi ed OLS for Heterogeneous Cointegrated Panels and

the Case of Purchasing Power Parity." mimeo, Indiana University.

Pedroni, P. (1997). \Panel Cointegration: Asymptotic and Finite Sample Properties

of Pooled Time Series Tests with an Application to the PPP Hypothesis." mimeo, Indiana University.

Phillips, P. C. and S. Durlauf (1986). \Multiple Time Series Regression with

Integrated Processes." Review of Economic Studies , 53, 473{496.

Summers, R. and A. Heston (1991). \The Penn World Table (Mark 5): An Ex-

panded Set of International Comparisons, 1950-1988." Quarterly Journal of Economics , 106(9).

Appendix A. Proof of Theorem 5 Replacing yth by axht + ht in the formula for aH;t yields: aH;t = a +

" H X

h=1

#"

ht xht

H X h=1

(xht )

#,1

2

:

Moreover, let t (L) = 1 + bL + ::: + bt,1 Lt,1 ;

where L is the lag-operator. Then: xht = t (L)th = t (1)vh + ht (L)vt + t (L)vth:

Step 1 : computation of bias Bt = plimH !1

h

1

H

PH

h h h=1 t xt

ih

1

H

PH

h=1

(xht)

2

i,1

:

First, we have 1

H

"

H X

#

"

#

H H H X X X   1 1 1 h h h h h h t xt = t (1) uv + u  [t (L)vt ] + uh t (L)vth Hh Hh Hh " # " # H H H X   1X 1X + (1) 1 h v h u + h  h u [ (L)v ] + h  (L)v h u

h=1

=1

t

H

t

h=1 "

+t (1) 1

=1

H X

H

h=1

#

uhtv h

H

h=1

"

+ 1

H

=1

t

H X h=1

#

uht h

t

t

H

h=1 H X

[t (L)vt ] + 1

H

h=1

t



t

t



uht t (L)vth :

Hence, for all t, assumption 3 implies that: "

H X

#

"

#

H H X X 1 1 h h h h  x = t (1) uv + uh h [t (L)vt ] Hh t t Hh Hh " # " # H H X X 1 1 + (1) h v h u + h  h u [ (L)v ] + oH (1)

1

=1

t

=1

H

h=1

t

=1

H

h=1

t

t

t

P

where oHP (1) is a random variable which converges to 0 in probability when H tends to in nity.

Similarly, H X

1

H

h=1

"

#

2

2

H X

"

#

H H  X X  (xht) = t (1) H (vh) + H1 (h) [t (L)vt ] + H1 t (L)vth h h h # " H H X 1 X vh  (L)vh v h h [ (L)v ] + 2 (1) + 2 (1) 1

1

2

=1

t

+2 1

H

H X

H

h=1

2

2

2

=1

t

h=1



=1

t

t

H

t

h=1

t



 h t (L)vth [t (L)vt ] :

And by assumption 3 1

H

H X h=1



v h t (L)vth

! 0 and H1

 P

H X h=1





P  h t (L)vth [t (L)vt ] ! 0

when H tends to in nity. Moreover, by assumption 3: t 2 P X h j j t (L)vt j =0

H  X

1

H

h=1

!

where 

j = E vth vth,j



H X 1 = plim H vthvth,j ; H !1

h=1

and where 



j = 2bj 1 + b2 + ::: + b2(t,j ) ; j = 1; :::; t:

It follows that

"

#

"

#

H H H X X X 1 1 1 h h (xt ) = t (1) plim H (v ) + plim H (h) [t (L)vt ] plim H !1 H !1 H !1 H h h h " # H t X X 1 h h + 2 (1) plim v  [ (L)v ] +  + oH (1): 2

2

2

=1

=1

t

H !1

H

=1

t

h=1

t

j =0

Step 2 : computation of limit bias plimt!1 Bt when b = 1: P We now show that Bt ! 0 if b = 1: When b = 1; t (1) = t and

t (L)vt =

t X j =0

vt,j ;

2

2

t (L)vth

=

t X j =0

vth,j :

j j

P

Thus 1

H

H X h=1

"

#

=1

"

+t 1

H

H X h=1

and 1

"

H H X X 1 1 h h h h t xt = t uv + uh  h Hh Hh

H

"

H X h=1

#

h v h

1

(xht) = t H 2

2

"

+ 2t H1

1 ut +

H

h=1 H X h=1

t X

=1

"

#

H X

#"

H X h=1

#

h  h

j =0

ut

j =0

2

#"

t X

v h h

j =0

#

vt,j +

t X

#

vt,j + oHP (1)

#"

H X 1 h (v ) + H (h) h=1

(A.1)

vt,j

"

"

2

#

t X j =0

t X j =0

#2

(A.2)

vt,j

j j + oHP (1)

We rst prove three lemmas which will be helpful in deriving the nal results.

Lemma A1 If vt is a covariance stationary variable, then t, = P

1 2

vt is integrated of order one, then t,3=2

t j =0 vt,j

Pt

= OP (1).

j =0 vt,j

= OP (1). If

Proof: See Park and Phillips (1988). Lemma A2 With the notation de ned above, t 1X j j = O(1):

t

j =0

Proof: The lemma is a simple consequence of t t 1 X X 1 X j j j j  2 j j j  2 j j j < 1: t+1 j j j =0

=0

=0

since (t + 1), j  2 and where the last inequality follows from the fact that vth is covariance stationary. 1

Lemma A3 1u

t! t

t X j =1

P vt,j ! 0

in one of the three cases de ned below, 1. ut and vt are I(0), with ! = 1. 2. ut is I(1) and vt is I(0), with ! = 3=2.

3. ut is I(0) and vt is I(1), with ! = 2.

Proof: Lets de ne the variables u~t and v~t such as 1. if ut and vt are I(0), u~t = ut and v~t =

1

t

Pt

j =0 vt,j .

2. if ut is I(1) and vt is I(0), u~t = ut =t = and v~t = 1 2

3. if ut is I(0) and vt is I(1), u~t = ut and v~t = t2 1

1

t

Pt

j =0 vt,j .

Pt

j =0 vt,j .

P Then from lemma A1 we have v~t ! 0 when t tends to in nity. As u~t is covariance stationary, E u~t is equal to some nite constant  independent of t. Chebyshev's inequality then applies to show that: 2

2

Prfju~t j > g   for all positive . Moreover, it is true that

2

2

ju~t j   and ju~tj jv~t j   =) jv~t j  : Hence Pr fjv~t j  g  Pr fju~tj   and ju~tj jv~t j   g for all positive numbers  and . Then remark also that Pr fju~t j > g  Pr fju~tj >  and ju~t j jv~t j   g : Consequently Pr fju~tj jv~t j   g  Pr fjv~tj  g + Pr fju~tj > g : For all positive number , the convergence of jv~tj to 0 in probability makes possible to nd such as And, choosing  =

q

Pr fjv~t j  g  =2:  

2 2

yields that Pr fju~tj jv~t j   g  :

This achieves to show that P u~t v~t ! 0

when t tends to in nity.

Using the lemmas A1 through A3, we are now able to calculate the limit bias p limt!1 Bt . If both ut and vt are I(0), dividing equation (A.1) by t and applying lemmas A1 through A3, gives plim 1 1

H !1

tH

H X h=1

"

#

H H 1X 1X ht xht = plim uhv h + plim h v h ut + otP (1) H H H !1 H !1 h h =1

=1

where otP (1) is a random variables which converges to 0 in probability when t tends to in nity. Dividing equation (A.2) by t gives, 2

H H X X 1 1 1 h plim t H (xt ) = plim H (vh) + otP (1): 2

H !1

2

H !1

h=1

(A.3)

2

h=1

Hence, 1 plimH !1

Bt = t

h

i

PH

h

h h h=1 u v + plimH !1

1

H

plimH !1 H !1

h

1

H

PH

h=1

i

PH

1

H

(vh)2

i

h h h=1  v ut

+ otP (t, )

(A.4)

1

If ut is I(1) and vt is I(0), dividing equation (A.1) by t = and applying lemmas A1 through A3, gives 3 2

"

#

H H X u 1X plim t 1= H1 ht xht = =t plim h v h + otP (1) t H H !1 H !1 h h 3 2

1 2

=1

=1

Dividing equation (A.2) by t and using equation (A.3) gives 2

Bt = t 1=

1 2

P

plimH !1 H !1 H h H hvh t , = + oP (t ) P plimH !1 H !1 H Hh (vh)

ut t1=2

1

=1

1

(A.5)

1 2

2

=1

If ut is I(0) and vt is I(1), dividing equation (A.1) by t = and applying lemmas A1 through A3, gives 3 2

plim t 1= H1

H !1

3 2

H X h=1

"

1

ht xht = plim

H X

H !1 H h=1

#

uh h

"

t

1

=

3 2

t X j =0

#

"

vt,j + plim

1

H X

H !1 H h=1

#

h v h

and dividing equation (A.2) by t gives, 3

plim 1 1

H X

H !1 t3 H h=1

"

(xht) = plim 1 2

H X

H !1 H h=1

(h)

# 2

1

t3

"

t X j =0

#2

vt,j

+ otP (1):

ut + otP (1) 1=2 t

Hence,

Bt = t 1=

1

t3=2

P

t j =0 vt,j



3 2

plimH !1 H 1

1

t3

P

t j =0 vt,j

h

PH 2

1 h h h=1 u  + plimH !1 H

PH

plimH !1 H

h=1 (

1

h

PH

h h h=1  v

i

)

2

ut t1=2

+otP (t, = ) 3 2

where otP (t, = ) is a random variables such that t = otP (t, = ) converges to 0 in probability when t tends to in nity. If ut and vt are I(1), dividing equation (A.1) by t and applying lemmas A1 through A3, gives 3 2

3 2

3 2

2

plim 1 1

H X

u ht xht = 2t 2 t H !1 t H h=1

" t X

j =0

#"

vt,j

plim 1

#

H X

h  h + otP (1)

H !1 H h=1

and dividing equation (A.2) by t gives, 3

"

#"

H H X X plim t1 H1 (xht) = t1 plim H1 (h) H !1 H !1 h h 2

3

2

3

=1

t X j =0

=1

#2

vt,j

+ otP (1):

Hence,

Bt = 1t

  h P i H ut P t v 1 hh plim  H !1 H j =0 t,j h=1 t2 1

t3

P

t j =0 vt,j

2

plimH !1 H !1 H

This nishes to show proposition 5.

1

PH

h=1 (

+ otP (t, ):

h)

1

2