Simple Solutions to the Initial Conditions Problem in Dynamic ...

Simple Solutions to the Initial Conditions Problem in Dynamic, Nonlinear Panel Data Models with Unobserved Heterogeneity Author(s): Jeffrey M. Wooldridge Source: Journal of Applied Econometrics, Vol. 20, No. 1 (Jan. - Feb., 2005), pp. 39-54 Published by: Wiley Stable URL: http://www.jstor.org/stable/25146339 . Accessed: 20/04/2013 15:02 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp

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JOURNAL OF APPLIED ECONOMETRICS J. Appl Econ. 20: 39-54 Published online inWiley

(2005) InterScience

(www.interscience.wiley.com).

DOI:

10.1002/jae.770

SIMPLE SOLUTIONSTO THE INITIALCONDITIONS PROBLEM INDYNAMIC, NONLINEAR PANEL DATA MODELS WITH UNOBSERVED HETEROGENEITY JEFFREY M. WOOLDRIDGE* of Economics,

Department

Michigan

State University,

USA

SUMMARY I study a simple, widely applicable approach to handling the initial conditions problem in dynamic, nonlinear unobserved

effects

endogenous of history

variables,

Rather

models.

I propose

than finding

to obtain the attempting the distribution conditional

joint distribution on the initial

of

outcomes

all

value

(and

of

the

the observed

The approach is flexible, and results in simple estimation variables). exogenous strictly explanatory for at least three leading nonlinear models: Tobit and Poisson I treat dynamic, probit, strategies regression. the general and show that simple of estimating estimators exist for important effects, average problem partial

special cases. Copyright ? 2005 JohnWiley & Sons, Ltd.

1. INTRODUCTION In dynamic panel data models with unobserved effects, the treatment of the initial observations is an important theoretical and practical problem. Much attention has been devoted to dynamic linear models with an additive unobserved the simple AR(1) model without effect, particularly additional covariates. As is well known, the usual within estimator is inconsistent, and can be badly biased. [See, for example, Hsiao (1986, section 4.2).] For linear models with an additive unobserved effect, the problems with the within estimator can be solved by using an appropriate transformation?such as differencing?to eliminate the unobserved effects. Then, instrumental variables (IV) can usually be found for implementation in a generalized method of moments and Hsiao (1982) proposed (GMM) framework. Anderson IV estimation on a first-differenced while several authors, including Arellano and Bond equation, (1991), Arellano and Bover (1995), Ahn and Schmidt (1995), improved on the Anderson-Hsiao estimator by using additional moment restrictions in GMM estimation. More recently, Blundell and Bond (1998) and Hahn (1999) have shown that imposing restrictions on the distribution of initial conditions can greatly improve the efficiency of GMM over certain parts of the parameter space. Solving the initial conditions problem is notably more difficult in nonlinear models. Generally, there are no known transformations that eliminate the unobserved effects and result in usable moment conditions, (1992) finds although special cases have been worked out. Chamberlain moment conditions for dynamic models with a multiplicative effect in the conditional mean, and for a more general class of multiplicative models. (1997) considers transformations Wooldridge Honore for the unobserved conditions effects Tobit model with (1993) obtains orthogonality a lagged dependent variable. For the unobserved effects logit model with a lagged dependent *

to: Professor Correspondence MI 48824-1038, USA.

Lansing,

Copyright

?

2005

Jeffrey M. Wooldridge, E-mail: [email protected]

John Wiley

& Sons, Ltd.

Department

of Economics,

Michigan

Received Revised

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State

University,

East

12 July 2002 11 September 2003

40

J.M. WOOLDRIDGE

variable, Honore and Kyriazidou (2000) find an objective function that identifies the parameters under certain assumptions on the strictly exogenous covariates. The strength of semiparametric approaches is that they allow estimation of parameters without for the unobserved effect. Unfortunately, identification specifying a distribution semiparametric on some strong assumptions concerning the strictly exogenous covariates; for example, time hinges dummies are not allowed in the Honore and Kyriazidou (2000) approach. Honore and Kyriazidou also reduce the sample to cross-sectional units with no change in any discrete covariates over the last two time periods. Another limitation of the Honore and Kyriazidou (1993) and Honore practical (2000) one that often goes unnoticed?is estimators?and that partial effects on the response proba the absolute importance of covariates, or bility or conditional mean are not identified. Therefore, the amount of state dependence, cannot be determined from semiparametric approaches. In this paper I reconsider the initial conditions problem in a parametric framework for nonlinear models. A parametric approach has its usual drawbacks because I specify an auxiliary conditional distribution for the unobserved heterogeneity; misspecification of this distribution generally results in inconsistent parameter estimates. Nevertheless, in some leading cases the approach I take leads to some remarkably simple maximum likelihood estimators. Further, I show that the assumptions are sufficient for uncovering the quantities that are usually of interest in nonlinear applications: partial effects on the mean response, heterogeneity. Previous research in parametric,

averaged

across

the population

distribution

of the unobserved

has focused on three different dynamic nonlinear models are of these initial summarized Hsiao conditions; ways (1986, section 7.4). The first handling by to treat as nonrandom variables. is the initial for conditions each cross-sectional unit approach nonrandomness the initial of that conditions, y/o, implies y,o is independent of Unfortunately, unobserved heterogeneity, c,-. Even when we observe the entire history of the process {yit}, the of independence between C/ and y/o is very strong. For example, suppose we are assumption once interested in modelling of and leave y/o is earnings earnings college graduates they college, in the first post-school year. That we observe the start of this process is logically distinct from the that unobserved strong assumption 'ability' and 'motivation' are independent of initial earnings. A better approach is to allow the initial condition to be random, and then to use the joint distribution of all outcomes on the response?including that in the initial time period?conditional on unobserved and observed strictly exogenous heterogeneity explanatory variables. The main with this approach is specifying the distribution of the initial condition complication given unobserved heterogeneity. Some authors insist that the distribution of the initial condition represent a steady-state distribution. While the steady-state distribution can be found in special cases?such as the first-order linear model without exogenous variables [see Bhargava and Sargan (1983) and Hsiao (1986, section 4.3)] and in the unobserved probit model without additional conditioning cannot be done for even modest extensions. variables [see Hsiao (1986, section 7.4)]?it For the dynamic probit model with covariates, Heckman the (1981) proposed approximating conditional distribution of the initial condition. This avoids the practical problem of not being able to find the conditional distribution of the initial value. But, as we will see, it is computationally more difficult than necessary for obtaining both parameter estimates and estimates of averaged effects in nonlinear models. The approach I suggest in this paper is to model the distribution of the unobserved effect conditional on the initial value and any exogenous explanatory variables. This suggestion has been made before for particular models. For example, Chamberlain this possibility for (1980) mentions Copyright

?

2005

John Wiley

& Sons,

Ltd.

/.

Appl

Econ.

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20:

39-54

(2005)

DYNAMIC, NONLINEAR PANEL DATA MODELS 41

the linear AR(1) model without covariates, and Blundell and Smith (1991) study the conditional see also Blundell and Bond (1998). For the maximum likelihood estimator of the same model; a and Carrasco Arellano with model variable, (2003) study lagged dependent binary response a maximum likelihood estimator conditional on the initial condition, where the distribution of to the binary effect given the initial is taken to be discrete. When specialized more and is much here the flexible, response computationally simpler: the approach are or can the variables have form, exogenous response probability strictly explanatory logit probit a and standard effects software with random variable, lagged dependent easily incorporated along can be used to estimate the parameters and averaged effects. on the initial condition has several a distribution conditional of heterogeneity Specifying we to be flexible, and view it as an can choose distribution the First, auxiliary advantages. the unobserved

model,

to Heckman's ordered alternative approximation (1981). Second, in several leading cases?probit, can to distribution be chosen that leads Tobit and Poisson auxiliary regression?an probit, on mean software. effects standard estimation Third, responses, partial using straightforward are identified and can be estimated without averaged across the distribution of unobservables, much difficulty. I show how to obtain covers the probit and Tobit models.

these partial effects

1

Example

(Dynamic and

population

t=

Probit Model

4, and Section

5

issues; we

the important

with Unobserved

return to these

For a random draw

Effect):

i from

the

T:

1, 2,...,

?(yit

to highlight

in this section

introduce three examples examples in Section 5.

in Section

EXAMPLES

2. We

generally

=

>yto, z/9 q) =

i\yi,t-i,

(z;,y+ pyi,t-\

+ ct) (l)

This equation contains several assumptions. First, the dynamics are first order, once zit and c, are also conditioned on. Second, effect is additive inside the standard normal the unobserved cumulative

distribution

.Third,

function,

the

z/r

a strict

satisfy

exogeneity

assumption:

only

z?

= (z/i,..., set on ziT) appears in the conditioning appears on the right-hand side, even though z,the left. Naturally, zit can contain lags, and even leads, of exogenous variables. As we will see in Sections 3 and 4, the parameters in equation (1), as well as average partial normal effects, can be estimated by specifying a density for c, given (y;o, z,). A homoscedastic as we will see in linear in parameters is very convenient, distribution with conditional mean Section 5. 2

Example

(Dynamic Tobit Model yu

= max[0,

uit\yij-U

for t =

with Unobserved

...,

T. This model 1, 2,..., applies that response equals zero with positive

Copyright

?

2005

John Wiley

& Sons,

Effect):

zity + g().,.-i v/o,

z,-, ct

~

Consider

)p + c,-+ uit] Normal(0,

a2)

(2) (3)

to corner

solution outcomes, where yit is an observed distributed over strictly probability but is continuously Ltd.

/.

Appl.

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Econ.

20: 39-54

(2005)

J.M. WOOLDRIDGE

42

as in that case we would positive values. It is not well suited to true data censoring applications, want a lagged value of the latent variable underlying equation (2) to appear. The function g() allows the lagged value of the observed response to appear in a variety of ways. For instance, we = = 0], which allows the effect of lagged y to l[y_i > 0]log(y_i)}, {l[y-i might have g(y_i) be different depending on whether the previous response was a corner solution (zero) or strictly positive. In this case, p is 2 x 1. Honore (1993) proposes orthogonality conditions that identify the parameters, but partial effects are unidentified. We will show how to obtain \/7V-consistent estimates of the parameters and effects in Section 5. average partial 3

Example

(Dynamic Z/, Ci) has

y/o,

(yi,t-i,...,

Unobserved

Effects

a Poisson

distribution

Poisson with

, yn, z/, ct) =

E(y/r \yi,t-\,

For

Model):

t?

each

1,...,

T, yit given

mean

c( exp[z/ry + g(y/,,_i)p\

(4)

Again, we allow for the lagged dependent variable to appear in a flexible fashion, perhaps as a set of dummy variables for specific outcomes on yij-\. The null hypothesis of no state dependence is Ho: p = 0. Chamberlain conditions (1992) and Wooldridge (1997) have proposed orthogonality are needed for yn or based only on equation (4), where no conditional distributional assumptions because the moment conditions have features similar to using first differences in the resulting GMM estimators can be very imprecise (even though the parameters would be generally identified). In Section 5 we show how a particular model for a conditional distribution for c/ leads to a simple maximum likelihood analysis. D c/. Unfortunately, a linear equation,

3. GENERAL FRAMEWORK Let

/

index

Initially,

we

a

random

assume

draw that we

from

the

observe

and

cross-section, =

(z/r, yit),t

1,...,

let T,

a

t denote along

time

particular

y/o- We

with

are

period.

interested

in

the conditional distribution of yit RG given (zit, y,-,r_i, C/), where zit is a vector of conditioning of z,> could be variables at time t and c, 1RJ is unobserved (The dimension heterogeneity. our in is We the conditional with but its distribution dimension denote t, fixed.) examples increasing is the of The with number time T, fixed, with c,-). |z,-f, D(y,-f y,,r-i, asymptotic analysis periods, by to the cross-section size, N, going infinity. sample on the conditional distribution of interest. First, we assume two key assumptions We make are correctly specified. This means that at most one lag of yit appears in that the dynamics = to is outcomes time back the the distribution initial {z,i, ...,ziT] period. Second, z, given on can as follows. Both of these be conditional C/. exogenous, expressed appropriately strictly Assumption

For t =

1:

1, 2,...,

DQtolz/,,

We

y/,r_i,

c/)

=

D(y/,|z/,

y/0,

y/,,_i,...,

C/)

(5)

that we have a correctly specified parametric model for the density (5) which, for lack of a better name, we call the 'structural' density.

next assume

equation Copyright

T:

?

2005

John Wiley

& Sons,

Ltd.

J.

Appl.

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20:

representing

39-54

(2005)

DYNAMIC, NONLINEAR PANEL DATA MODELS

43

2: For t = 1, 2,..., T, /"r(y.|z,, yf_1? c;$) is a correctly specified density for the Assumption conditional distribution on the left-hand side of equation (5), with respect to a cr-finite measure the true value of Oby 0o e 0. y(dyr). The parameter space, 0, is a subset of _RP.Denote is not restrictive in The requirement that we have a density with respect to a cr-finite measure a measure. v is a Lebesgue v is is If If is continuous, yt yt practice. counting purely discrete, measure. An appropriate cr-finite measure can be found for all of the possible response variables of interest in economics. Most specific analyses of dynamic, nonlinear unobserved effects models begin with assumptions 1 and 2 imply that the density of (yn,..., similar to 1 and 2. Together, Assumptions yir) given (y/o

=

yo,

Zi

=

z, cf-

=

c)

is T

(6)

n/'(yr|z,,yr_i,c;0o) t=\

where we drop the / subscript to indicate dummy arguments of the density. In using equation (6) to c. One possibility estimate B0, we must confront the fact that it depends on the unobservables, is to construct the log-likelihood function that treats the _V unobserved effects, C/, as (vectors of) the function parameters to be estimated. This leads to maximizing N

T

(1)

^^logft&ulz^yij-ucr.e) i=\ t=\

over 0 and (ci,..., this approach avoids having to restrict the distribution of c,-, it c_v). While suffers from an incidental parameters problem with fixed T: except in very special cases, the estimator of 0o is inconsistent. The alternative is to 'integrate out' the unobserved effect. As we discussed in the Introduction, there have been several suggestions for doing this. A popular approach is to find the density of ,ytr) given zt. If we specify /(y0|z, c) then (y/o, y/i, /(yo, Next, we

specify

to obtain

/(y0,

y_,

yi,.

a density f(c\z). ...,

., yrlz, We

c)

=

f(yu

..., yr|y0,

z, c)

can then integrate equation

/(y0|z,

c)

(8) with

respect

(8) to this density

yr|z).

Rather than trying to find the density of (y,-0, yn, ..., v,t) given z,-, my suggestion is to use the density of (yn, ..., yiT) conditional on (y.0, z,-). Because we already have the density of (ya, ,yn) conditional on (y,o, z,-, c,)?given need only specify the by equation (6)?we this density is not restricted by the specification density of c,- conditional on (y/o,z_). Because in Assumption or flexibility it for convenience, both. As 2, we can choose or, hopefully, in Chamberlain's effects probit models with strictly exogenous (1980) analysis of unobserved z) as a way of obtaining relatively explanatory variables, we view the device of specifying /(c|yn, a seems estimates of no worse than having to specify model for 0O. Specifying z) simple /(c|y0, must as models for /(yn|z, which themselves be viewed c), approximations, except in special cases where steady-state distributions can be derived. 3: Assumption ft(c|yo, z;8) is a correctly a to cr-finite measure n(dc). Let A c respect value of 8. Copyright

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2005

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& Sons,

specified model for the density of D(c,|y/0, z/) with 1RMbe the parameter space and let 8Q denote the true

Ltd.

J

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44

J.M. WOOLDRIDGE

3 is much more controversial than Assumptions 1 and 2. Ideally, Assumption have to specify anything about the relationship between c,- and (y/0, z), whereas assumes we have a complete conditional density correctly specified. In some specific and exponential models, logit models, Tobit models regression models?consistent are are complicated available But without 3. these and estimators 0o Assumption good precision. Under Assumptions

1, 2 and 3, the density

ji which

of (yn,...,

J

(n/r(yr|z,,y,-i,c;0o)

leads to the log-likelihood

li(6, 8) = log

function

m

ylT) given

we would Assumption cases?linear

= = (y,-0 yo, Z/

z) is

ft(c|yo,z;?o)??(dc) (9)

/,(y/,|z/? y/,,_!, c;0)

jj

3

estimators of need not have

on (y/0, z,-) for each observation

conditional

not

J

i:

h(c\yi0, Z/;^(dc) (10)

in equation (10) across / = 1,..., N and maxi 0o and 8Q,we sum the log-likelihoods to 0 and 8. The resulting conditional MLE is VN-consistent and asymptotically respect In dynamic unobserved normal under standard regularity conditions. effects models, the log likelihoods are typically very smooth functions, and we usually assume that the needed moments if exist and are finite. From a practical perspective, identification is the key issue. Generally, Zj) is allowed to depend on all elements of z,- then the way in which any time-constant D(c/|y/o, variables can appear in the structural density is restricted. To increase explanatory exogenous To estimate

mize with

power, we

can include time-constant explanatory variables in zit, but we will not be able to sep arately identify the partial effect of the time-constant variable from its partial correlation with c,-. in equation (10) assumes that we observe data on all cross-sectional units in The log-likelihood for unbalanced panels under certain sample selection mechanisms, all time periods. Nevertheless, a we can use the same conditional for the subset of observations log-likelihood constituting = a we 1 if selection indicator: si balanced panel. Let 57 be observe data in all time periods yiT) and st are independent conditional on (including y,o), and zero otherwise. Then, if (y/i,..., standard (y*o>Z/), the MLE using the balanced panel will be consistent, and the usual asymptotic errors and test statistics are asymptotically follows from the general argument valid. Consistency inWooldridge (2002, section 17.2.2). attrition is an issue, obtaining the density conditional on (y,o,z/) has some advantages When over the more traditional approach, where the density would be conditional only on Z/. In particular, the current approach allows attrition to depend on the initial condition, y/o, in an arbitrary way. For if y/o is annual hours worked, an MLE analysis based on equation (10) allows attrition to differ across initial hours worked. In the traditional approach, one would have to probabilities analysis. explicitly model attrition as a function of y/o and figure out the appropriate Heckit-type Of course, reducing the data set to a balanced panel can discard useful information. But available the objective function in Honore have the same feature. For example, semiparametric methods in the strictly exogenous covariates for 7 = 3. Any and Kyriazidou (2000) includes differences = 2 or 3 cannot contribute to the analysis. for t observation where AzJf is missing example,

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45

4. ESTIMATINGAVERAGE PARTIALEFFECTS we often need to go beyond estimation of the parameters, 0O, and obtain effects. Typically, we would like the effect on a mean response after averaging partial across the population. I now show how to construct consistent, the unobserved heterogeneity In nonlinear models estimated

normal estimators of these average V^-asymptotically Let q(yt) be a scalar function of yt whose conditional = yt when v. is a scalar, but q(-) leading case is q(yt) interested in probabilities. Generally, we are interested m(zt,

yr_i,

c;0o)

=

= where

zt,

and

y,_i

c

are

values

=

E[q(yit)\zit

yM_i

=

c,-

y,_i,

=

c)]

^(yr)/r(yr|Zr,y.-l,C;*oMdyr) (11)

/

that

z?

partial effects (APEs). mean we are interested in at time t. The could be an indicator function if we are in

we

must

choose.

since

Unfortunately,

the

it is unclear which heterogeneity rarely, if ever, has natural units of measurement, should plug in for c. Instead, we can hope to estimate the partial effects averaged distribution of c,. That is, we estimate p(zt,

y,_i)

=

E[m(z?

yf_i,

c,-;0o)]

unobserved

values we across the

(12)

where the expectation is with respect to c,-. (For emphasis, variables with an i subscript are random variables in the expectations; others are fixed values.) Under Assumptions 1, 2 and 3, we do not have a parametric model for the unconditional distribution of ct, and so it may seem that we to estimate equation (12). Instead, we can obtain a consistent need to add additional assumptions estimator of equation (12) using iterated expectations: E[m(z,,

y,_i,

C/;0O)]

=

E{E[m(z,,

= E

y,_i,

c,-;0o)lyio, Z/]}

q(yt)ft(yt\zt, y,_!, (Jg

/i(c|y/0, z,-;?oMdc)

(13)

c;0oMdy,))

where

the outside, unconditional is with respect to the distribution of (y/o, Z;). While expectation it simplifies considerably in some leading cases, as we will equation (13) is generally complicated, see in Section 5. In effect, we first compute the expectation of q(yu) conditional on (zit, y^-i, C/), which is possible because we have specified the density /.(y.lz., y._i, c;0o). Then, we (hopefully) integrate m(zt, y,_i, c) against /z(c|y/0, zt;80) to obtain E[m(z,, y,_i, C/;0o)|y/O, z,-]. One point worth emphasizing about equation (13) is that 8Q appears explicitly. In other words, as a nuisance parameter for estimating 0O, 8Q is not a nuisance while _?0 be viewed may properly literature treats A(c|yo, z) as a nuisance parameter for estimating APEs. Because the semiparametric seems little hope that semiparametric approaches will deliver consistent estimates function, there of APEs Given

in dynamic, unobserved effects panel data models. (13), a consistent estimator of p(zt, yt-\)

equation

follows

immediately:

TV p(zt,

Copyright

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y,_i)

= AT1

Yl

r(zt, y,_i,

y/0, z_; 0, 8)

(14)

i=\

Ltd.

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Appl.

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J.M. WOOLDRIDGE

46

where r(zt, y,_i, y/0, Z/;0O, S0) = E[m(zt, y,_i, C/;0o)|y/O, z,]. Under weak assumptions, ji(zt, y,_i) is aV/V-asymptotically normal estimator of p,(zt, yf_i), whose asymptotic variance can be obtained using the delta method.

5. THE EXAMPLES REVISITED We reconsider the examples from Section 2, showing how we can apply the results from Sections 3 and 4. Our focus is on choices of the density h(c\yo, z;8) that lead to computational simplicity. For notational convenience, we drop the 'o' subscript on the true values of the parameters.

5.1.

Dynamic

In addition

Probit

to equation

and Ordered

Probit Models

(1), assume

that

Ci\yto, z/

~

Normal(or0 + aiy/o 4- z,-a2, ?l)

(15)

where z,- is the row vector of all (nonredundant) explanatory variables in all time periods. If zJf contains a full set of time period dummy variables these elements would be dropped from z,-. The on time-constant that we cannot identify the coefficients presence of z,- in equation (15) means covariates in z/,, although time-constant covariates can be included in z/ in equation (15). Given

equation

(1), we

can write T

f(yu

, yr\yo, z, c; fi) =

yi,

JJ{(zty+pyt-i

(16)

+c)]1^}

=

in (y/, p)'. When we integrate equation (16) with respect to the normal distribution we of the obtain y/rly/o* z/). D(y/i,..., density equation (15), It turns out that we can specify the density in such a way that standard random effects probit If we write software can be used for estimation.

where

fi

ct = where with

~ Normal(0, a,-|(y/o, Z/) response probability

al),

o?o+ aiy/o + z/cr2 + ai (17)

then yit given