Munich Lecture 3

Dynamic linear and non-linear panel data models

Stefanie Schurerstefanie.schurer@rmit.edu.au

RMIT UniversitySchool of Economics, Finance, and Marketing

January 29, 2014

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Overview

1 Linear Dynamic Models

1 What’s different with linear dynamic panel data?2 First-order state dependence (SD) dynamic model;3 The Arellano-Bond estimator;4 Why is the Arellano-Bond estimator problematic if the

autoregressive parameter approaches to 1?;5 How can we test for serial correlation and over-identification?;

2 Non-linear Dynamic Models

1 What are dynamic response models and state dependence?;2 Random effects models;3 Different ways to deal with the initial conditions;4 Fixed effects (conditional logit) models.

3 An empirical application: State dependence in mental healthproblems (Roy and Schurer, 2013)

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References

1 Hsiao, C. (2003). Analysis of Panel Data. Econometric Society Monographs.Cambridge University Press: New York. pp. 69-78, 85-90, 206-224;

2 Greene, W.H. (2011). Econometric Analysis. Pearson Education Limited: pp.438-449;

3 Cameron, A.C., Trivedi, P. (2005). Microeconometrics. Methods and

Applications. Cambridge University Press. pp. 763-767;

4 Blundell, R. and S. Bond (1998). Initial conditions and moment restrictions indynamic panel data models. Journal of Econometrics 87, p. 115-143.

5 Arellano, M., and S. Bond (1991). Some tests of specification for panel data:Monte Carlo evidence and an application to employment equations. Review of

Economic Studies 58, p. 277297

6 Arulampalam, W., Stewart, M. (2011). Simplified Implementation of theHeckman Estimator of the Dynamic Probit Model and a Comparison withAlternative Estimators. Oxford Bulletin of Economics and Statistics. 17(5),659-681.

7 Roy, J., Schurer, S. (2013). Getting stuck in the blues: The persistence ofdepression in Australia. Health Economics 22(9); 1139-1157.

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1. Linear dynamic models

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1.1. What’s different with linear dynamicpanel data?

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Recap

Before we had:Yit = X ′

itβ + εit , (1)

andεit = αi + uit , (2)

for all i = 1, . . . ,N and t = 1, . . . ,T .

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Recap

• When all explanatory variables are exogenous, and theunobserved heterogeneity is potentially correlated withregressors of interest, then the fixed effects (within) estimatoris best, linear, and unbiased - this is so, because theindividual-specific heterogeneity is differenced out.

• When all explanatory variables are exogenous, and theunobserved heterogeneity is not correlated with regressors ofinterest, then the fixed effects (within) estimator is unbiasedand consistent, but it is not efficient.

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Recap, cont.

• The generalised least squares estimator will be biased underthe assumption of dependence between unobservedheterogeneity and explanatory variables.

• If the unobserved heterogeneity is linearly correlated with theexplanatory variables (or their means), then a correctlyformulated random effects model leads to the fixed effectsestimator (See Mundlak, 1978) - This is a VERY powerfulresult.

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What’s different then?

Consider the following model:

Yit = γYit−1 + X ′

itβ + αi + uit , (3)

• The OLS estimator of γ will no longer be unbiased andconsistent, even if all covariates are exogenous.

• The fixed effects (within) estimator is no longer consistent, inwhich the panel involves a large number of individuals andshort time dimension;

• The consistency of the random effects and generalized leastsquares estimator will crucially depend on the assumptions ofthe initial conditions (Yi0) (and on size of N and T ).

• However, an instrumental variables (IV) and GMM estimatorwill be consistent - that’s what we’ll cover today.

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1.2. First-order state dependence model

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First-order state dependence

Consider:Yit = γYit−1 + X ′

itβ + εit , (4)

andεit = αi + uit , (5)

for i = 1, . . . ,N and t = 1, . . . ,N, and Xit is a vector of exogenousvariables.

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First-order state dependence

The assumptions of the model and data requirements are:

• uit ∼ iid(0, σ2u);

• E (uit |Yit−1) = 0;

• |γ| < 1.

• To estimate all T-period equations we need T+1 observationson Y , i.e. Y0 is also observed. Typically, this isn’t the case, inwhich case we can only estimate equations for T-1 periods(t = 2, . . . ,T ) per i group.

Generally the focus is to estimate consistently γ, β, but todayfocus is mainly on γ.

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First-order state dependence

Which method can we apply to consistently estimate γ, β? Even ifαi is uncorrelated with the Xit regressors, αi is inherentlycorrelated with the lagged dependent variable Yit−1. This is so,because in period t − 1 this variable is also a dependent variable:

Yit−1 = γYit−2 + X ′

it−1β + αi + uit−1, (6)

and so (abstract from the presence of X ′

itβ for the moment):

γOLS =

∑Ni=1

∑Tt=1 YitYit−1

∑Ni=1

∑Tt=1 Y

2it−1

= γ +

∑Ni=1

∑Tt=1(αi + uit)Yit−1

∑Ni=1

∑Tt=1 Y

2it−1

.

(7)

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With a series of manipulations of continuous substitution andsumming Yit−1 over all t (see e.g. p. 74 Hsiao, 2003):

plimn→∞

1

NT

N∑

i=1

T∑

t=1

(αi + uit)Yit−1

=1

T

1− γT

1− γCov(Yi0, αi ) +

1

T

σ2α

(1− γ)2[(T − 1)− Tγ + γT ].

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First-order state dependence

Fixed effects estimation of γ, i.e. deviations from group means,may also entail some problems. The fixed effects estimator is (p.71-72 Hsiao 2003):

γFE =

∑Ni=1

∑Tt=1(Yit − Yi)(Yit−1 − Yi ,−1)

∑Ni=1

∑Tt=1(Yit−1 − Yi ,−1)2

= γ +

∑Ni=1

∑Tt=1(Yit−1 − Yi ,−1)(uit − ui )(NT )−1

∑Ni=1

∑Tt=1(Yit−1 − Yi ,−1)2(NT )−1

.

where Yi = T−1∑T

t=1 Yit and Yi ,−1 = T−1∑T

t=1 Yit−1.

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First-order state dependenceThe differences from the mean eliminates the unobservedheterogeneity αi from the model, however, the difference from themean of the error terms (uit − ui) is negatively correlated with thedifference from the mean of the lagged dependent variable(Yit−1 − Yi ,−1). Using similar continuous substitution andsumming Yit−1 over all t as before, we can show:

plimn→∞

1

NT

N∑

i=1

T∑

t=1

(Yit−1 − Yi ,−1)(uit − ui )

= −plimn→∞N−1N∑

i=1

Yi ,−1ui

= −σ2u

T 2

(T − 1)− Tγ + γT

(1− γ)2.

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Solution: Anderson and Hsiao (1982)

One solution is to find instruments for Yit−1 − Yi ,−1 OR ∆Yit−1

when applying the first difference estimator. For the latter,Anderson and Hsiao (1981, 1982) suggest using IVs that stem fromwithin the model. Consider the following first difference model:

Yit−Yit−1 = γ(Yit−1−Yit−2)+(Xit−Xit−1)′β+(uit−uit−1), (8)

or∆Yit = γ∆Yit−1 +∆X ′

itβ +∆uit , (9)

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Solution: Anderson and Hsiao (1982)

Assuming the model is correctly specified, there are naturalinstruments available for ∆Yit−1. In particular, if the first-order lagstructure is correct and exogenous variables affect Yit onlycontemporaneously, then ∆Yit−2 or the level Yit−2, and ∆Xit−1

(or Xit−k , for k > 1) will be valid instruments. E.g. consider:

Cov(Yit−2,∆uit) = Cov(Yit−2, uit − uit−1) = 0 (10)

Cov(Yit−2,∆Yit−1) = Cov(Yit−2,Yit−1 − Yit−2) 6= 0. (11)

With valid instruments for ∆Yit−1 we can use a standardinstrumental variable approach on Eq. 9 to consistently estimate γand β.

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Solution: Anderson and Hsiao (1982)

Denote Yit = ∆Yit , X′

it = (∆Yit−1,∆X ′

it) as the vector ofcovariates, and Z ′

it = (Yit−2) as the vector of instruments (Here ithas only one element for now). The IV estimator is then:

(γIV , β′

IV )′ = (Z ′X )−1Z ′Y . (12)

Under suitable regularity conditions on the instruments andcovariates, this IV estimator is consistent for the parameters(γ, β′), with

Var(γIV , β′

IV |X )′ = (Z ′X )−1Z ′E (uu′|X )Z (Z ′X )−1 (13)

where uit = ∆uit , E (uu′|X ) has a block diagonal structure.

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Note

Some issues are worth noting:

• We require at least three panel (time) periods to estimate thismodel consistently. More time periods are required if we seekto use t − 3 variables as instruments.

• If uit are serially correlated, then consistent estimation is moredifficult. In general, further lags of Yit (i.e. Yit−2 etc.) willnot be valid instruments, and will need to rely on exogenousvariables for IV.

• Anderson and Hsiao (1982) originally suggested using Yit−2

and/or ∆Yit−2 = Yit−2 − Yit−3 as instruments for ∆Yit−1. (Ifwe use ∆Yit−2 we lose yet another time period). We coulduse lags of Xit ’s or their differences as instruments(Z ′

it = (Yit−2,∆Xit−1)). In the case, we apply the generalised(2SLS) IV estimator.

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2SLS IV estimator

This estimator can be written as:

(γIV , βIV ) = (X ′PZ X )−1X ′PZ Y , (14)

where PZ = Z (Z ′Z )−1Z ′. In this case, we could also test forover-identification using e.g. the Sargan test. Under the nullhypothesis, we test that the instruments are all orthogonal to themodel errors.

Under the null hypothesis, the Sargan test statistic is:N(T − 2)R2 ∼ χ2

df , where R2 is the explained variation obtainedfrom regressing the errors of the IV regression on the full set ofinstruments, and df is the number of overidentifying restrictions(i.e. the number of excess instruments).

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1.3. The Arellano-Bond estimator

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Arellano and Bond (1991)

A large literature has evolved concerned with improving theefficiency of the estimation of the above model. Much of thisliterature concentrated on exploiting all available instruments foreach of the ∆Yit−k for k ≥ 1 variables. Arellano and Bond (1991)

set up the model as a system of equations, i.e. one equation pertime period, and use GMM for the estimation. Let’s consider thefollowing examples

• If T=3: Estimate one period equation and one singleinstrumental variable (Yi1) is available for ∆Yi2;

• if T=4: Estimate two period equations and two instrumentalvariables (Yi1,Yi2) are available for ∆Yi3 and Yi1 is availablefor ∆Yi2;

• if T=10: Estimate eight period equations and Yi1, . . . ,Yi8 areavailable as instruments for ∆Yi9.

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Arellano-Bond model set-up

Consider a simple AR(1) model with unobserved individual-specificeffects1:

Yit = γYit−1 + αi + uit , (15)

for all i = 1, . . . ,N and t = 1, . . . ,T . Assume E (αi ) = 0,E (uit) = 0, and E (uit , αi ) = 0, AND E (yi1uit) = 0 (Assumptionabout the initial condition).

• The auto-regressive parameter can be identified if T ≥ 3.

• There are in total .5(T-1)(T-2) orthogonality conditions.

E (Yit−s ,∆uit) = 0, (16)

for t = 3, . . . ,T , and s ≥ 2.

1Blundell and Bond (1998) illustrate this without covariates Xit . Note alsothat the notation in Blundell and Bond is different. I have adjusted thisnotation to keep the lectures consistent.

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Simple model set up

• Assume ∆uit = uit − uit−1 is a vector of first-differencederrors (∆ui3,∆ui4, . . . ,∆uit) and we assume that thedifferences are not serially correlated.

• Zi is a (T − 2)×m matrix of instrumental variables:

Zi =

Yi1 0 0 0 00 Yi1Yi2 0 0 0...

... . . . 0 00 0 0 . . . Yi1 . . .YiT−2

• The following are the moment conditions:

E (Z ′

i , uit − uit−1) = 0 (17)

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Simple model set up

• The generalised methods of moments estimator minimises thequadratic distance of:

∆u′ZANZ′∆u, (18)

where AN is some weighting matrix, Z ′ is the m × N(T − 2)matrix (Z ′

1,Z′

2, . . . ,Z′

N) and ∆u′ is the N(T − 2) vector of(∆u1,∆u2, . . . ,∆uN).

• The GMM estimator for α is then:

γ = (∆Y ′

−1ZANZ′∆Y−1)

−1∆Y ′

−1ZANZ′∆Y , (19)

where ∆Y is the (T-2) vector (∆Yi3,∆Yi4, . . . ,∆YiT ), and∆Yi ,−1 is the (T-2) vector (∆Yi2,∆Yi3, . . . ,∆YiT−1).

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Simple model set upThere are two ways to set up the optimal weights for AN : bothmethods yield estimates of γ that are asymptotically equivalent ifthe errors are iid.

1 Two-step GMM: use residuals from an initial consistentestimator γ (Optimal choice for AN).

AN = (N−1N∑

i=1

Z ′

i∆ui∆u′iZi)−1 (20)

2 One-step GMM: Instead of using the residuals from a firststep, replace the squared residuals by a matrix H that has 2son its main diagonal, -1s on its second next diagonals, and 0severywhere else ((T − 2)× (T − 2)).

AN = (N−1N∑

i=1

Z ′

iHZi )−1, (21)

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Simple model set up

The estimated asymptotic variance is for γ is:

Var (γ) = N∆Y ′

−1ZANVNANZ′∆Y−1 × [∆Y ′

−1ZANZ′∆Y−1]

−2

(22)There is a reason why you cannot simply use the followingexpression:

Var(γ) = σ2∆u × [∆Y ′

−1Z (Z′Z )−1Z ′∆Y−1]

−2, (23)

where

σ2∆u =

∑Ni=1

∑Tt=3[(Yit − Yit−1)− γ(Yit−1 − Yit−2)]

2

N(T − 2). (24)

Eq. 24 would estimate 2× σ2∆u because of the autocorrelation over

one time-period.

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Arellano and Bond (1991)

A possible weakness with the Arellano-Bond approach is thatlagged levels of the dependent variable may be poor instrumentsfor ∆Yit−1 if the dynamic process is close to a random walk.

∆Yit = γ∆Yit−1 +∆X ′

itβ +∆uit , (25)

That would be the case if |γ| → 1. Let’s see why

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1.4. Why is the Arellano-Bond estimatorproblematic if the autoregressive parameter

approaches to 1?

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Problems

There are two situations in which using lagged levels of thedependent variable can become problematic:

1 When γ increases towards unity; of course it should still beless than 1, but imagine that it comes arbitrarily close to 1.

2 When the relative variance of αi , i.e. the variance of the fixedeffect relative to the idiosyncratic error, increases.

These arguments have been laid out in Blundell and Bond (1998).You can find the article on the Blackboard.

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Problems

Blundell and Bond (1998) consider the following special case:

• Consider T=3, so instead of 0.5(T-1)(T-2) moment conditions(See 16), we have one single orthogonality condition

E (Yi1,∆ui3) (26)

• In this case, γ reduces to an instrumental variable estimator.

• The reduced form instrumental variable equation is:

∆Yi2 = πYi1 + ri2, (27)

for all i = 1, . . . ,N.

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Problems

For sufficiently high γ or variance of αi , the least squares estimateof the reduced form π can be made arbitrarily close to 0 and thusYi1 is only weakly correlated with with ∆Yi2. WHY?

• The model with a lagged dependent variable can be re-writtenas (simply subtract Yi1 from both sides):

∆Yi2 = (γ − 1)Yi1 + αi + ui2 (28)

• The expression (γ − 1) is generally expected to be biasedupwards because E (Yi1αi ) > 0. This is an assumption aboutthe relationship between the initial condition and theunobserved heterogeneity.

• Assume stationarity and let σ2α = Var(αi ) = σ2

u.

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Situations when instruments are weak

Under these assumptions, the probability limit of π is:

plim(π) = (γ − 1)k

(σ2α/σ

2u + 1)

, (29)

where k = (1−γ)2

(1−γ2).

Hence, plim(π) → 0 if γ → 1 and/or σ2α/σ

2u → ∞.

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Graphical depiction of the bias

Fig. 1. plimnL and a!1,p2g

fixed.

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Effects on statistical inference with weak

instruments

• Generally, one would use a F-statistic, testing for jointirrelevance of the IVs in the reduced form equation in Eq. 27(Rule of thumb: F > 10 to obtain reliable IV results).

• Some authors (e.g. Hansen et al., 2011) suggest to use aconcentration parameter instead, which measures the relativevariation in the instruments (relative to the variation in theerror term in the reduced form equation). The smaller thisratio, the worse the performance of the IV estimator.

• Blundell and Bond (1998) derive the concentration parameterreferring to Eq. 27 as:

τ =(σ2

uk)2

σ2α + σ2

uk, (30)

where k = (1−γ)2

(1−γ2).

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Concentration parameter and γ

Fig. 3. Concentration parameter q for ¹"3.

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Asymptotic standard error of γ

It can be also shown that the estimated asymptotic standard errorof γ will blow up in the case of γ → 1.

asyse(γ) =σu

√

2τσ2r

, (31)

where σ2r is the reduced form error variance in Eq. 27. This

standard error will explode if τ → 0.

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Extensions: System GMM

Arellano and Bover (1995) and Blundell and Bond (1998) suggestadditional moment conditions in which lagged differences of thedependent variable are orthogonal to levels of the disturbances. Toget these additional moment conditions, they assumed thatindividual effect is unrelated to the first observable first-differenceof the dependent variable:

E (uit ,∆Yit−1) = 0, t=4,5,..., T (32)

andE (ui3,∆Yi2) = 0. (33)

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Extensions: System GMM

Hence, one gets the following instrumental variable matrix:

Z+i =

Zi 0 0 0 00 ∆Yi2 0 0 0...

... ∆Yi3 0 0...

... . . .... 0

0 0 0 0 ∆YiT−1

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Extensions: System GMM

And that means in practice the following:

N∑

i=1

Z+′

i

(

∆uiui

)

=(

N∑

i=1

Yi1∆ui3,

N∑

i=1

Yi1∆ui4, . . . ,

N∑

i=1

Yi2∆ui4, . . .

N∑

i=1

Yi1∆uiT ,

N∑

i=1

Yi2∆uiT , . . . ,

N∑

i=1

YiT−2∆uiT

N∑

i=1

T∑

t=3

Xit∆uit

N∑

i=1

∆Yi2ui3,N∑

i=1

∆Yi3ui4, . . . ,N∑

i=1

∆YiT−1uiT)

.

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1.5. How can we test for serial correlation andover-identification?

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Testing: Serial correlation

The consistency of the GMM estimator hinges on the assumptionthat E (∆uit∆uit−2) = 0.2 Arellano and Bond (1991) proposed atest of the null hypothesis that all second-order auto-covariancesfor all periods are zero. More formally:

E (φi ) = E (∆ui ,−2∆ui ,∗) = 0 (34)

The test statistic for second-order serial correlation based onresiduals from the first-difference question takes the form:

SC = ∆u−2∆u∗(∆u)−1/2 ∼ N(0, 1), (35)

under the null hypothesis: E (∆uit ,∆uit−2) = 0.

2Note, however that it does not require E(∆uit∆uit−1) = 0.43 / 75

Testing: Serial correlation

You can obtain ∆u from:

∆u =

N∑

i=1

∆u′i ,−2∆ui∗∆u′i ,∗∆ui ,−2

− 2∆u′i ,−2∆Y−1,∗(∆Y ′

−1ZANZ′∆Y−1)

−1 ×

× ∆Y ′

−1ZAN(N∑

i=1

Z ′

i∆ui∆u′i∗∆ui ,−2)

+ ∆u′−2∆Y−1,∗V (γ)∆Y ′

−1,∗u−2.

This statistic is only defined if T ≥ 5. WHY?.

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Testing: Serial correlation

Here is why: If T=4, then we impose the following orthogonalityconditions:

E (∆ui3Yi1) = 0

E (∆ui4Yi1) = 0

E (∆ui4Yi2) = 0

We cannot construct differenced residuals which are two periodsapart.

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Testing: Over-identification restrictions

We also want to test whether the instruments are valid. This canbe done with a Sargan over-identification test (this is sometimescalled a Hansen test). The test statistic is computed as follows:

oid = ∆u′Z (N∑

i=1

Z ′

i∆ui∆u′iZ )−1Z ′∆u ∼ χ2

p−k , (36)

where ∆u =∑N

i=1

∑Tt=3(Yit − Yit−1)− γ(Yit−1 − Yit−2), γ stems

from the two-step estimator, and p > k (p: number ofinstruments, k: number of RHS variables).

Note, that this test tends to over-reject in the presence ofheteroskedasticity.

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2. Non-linear dynamic models

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Dynamic binary response models

We consider alternative methods for estimating dynamic binaryresponse models and discuss the issues that may arise. Consider thefirst-order lagged dependent variable (”state dependence”) model:

Yit = 1(γYit−1 + X ′

itβ + αi + uit > 0), (37)

for i = 1, . . . ,N, and t = 1, . . . ,T − 1.

P(Yi0|Xi , αi ) = p0(Xi , αi ) (38)

P(Yit |Yi0, . . . ,Yit−1,Xi , αi ) = F (γYit−1 + X ′

itβ + αi ) (39)

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Dynamic response models

• Yit is an outcome state of interest for individual i in period t;

• 1 is an indicator function that takes the value 1 if thestatement is true, and 0 otherwise;

• Xit is a vector of observable characteristics;

• αi is an individual-specific effect, assumed to betime-invariant;

• uit is an idiosyncratic term that is assumed to be iid over timeand across individuals with distribution function F(.), and souit ∼ iid(0, σ2

u);

• γ and β are parameters of interest to be estimated.

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2.1. What are dynamic response models andstate dependence?

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Dynamic response models

• γ measures the degree to which last period’s state directlyaffects the probability of being in the same state in thecurrent time period, and often is referred to as a measure oftrue state dependence;

• In contrast, the effects of observed (Xit) and/or unobserved(αi ) factors capture heterogeneity across individuals thataffect the propensity to experience outcome Y in period t,and represent sources of spurious state dependence;

• Often the challenge is to identify these two sources ofheterogeneity separately from each other;

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What is state dependence?

State dependence often has an economic importance and/orinterpretation, e.g.:

• Welfare or poverty traps;

• Search costs in labour force participation context;

• Habit formation in a consumer choice context;

• Narcotic effects in an arbitration or dispute resolution context;

• Hysteresis in a trade context;

• Mental health problems.

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What is unobserved heterogeneity?

Unobserved heterogeneity confounds the identification of suchbehaviours, e.g.:

• Different individuals may have inherently different propensitiesto experience some event;

• Possibly due to differences in taste, abilities, soft-skills;

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2.2. Random effects models

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Random effects models

The usual approach to estimating the dynamic binary responsemodel expressed in Eq. (1) is to specify the distribution of αi

conditional on Xi , e.g. αi ∼ N(0, σ2αi), and express the likelihood

contribution for individual i as:

Li(θ|Yi ,Xi ) = P(Yi0, . . . ,YiT−1|Xi ) (40)

=

∫

P(Yi1, . . . ,YiT−1|Yi0,Xi , αi )×

×P(Yi0|Xi , αi )dFα(αi |Xi ),

where Fαiis the distribution function for αi conditional on Xi , and

θ is a vector of parameters that parameterises the model (e.g.θ = (γ, β′, σ2

α, σ2u)

′).

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The initial conditions problem and

solutions

The main econometric issue that arises in this case relates to thesample initial conditions for the dynamic process (Yi0); inparticular the relationship between Yi0 and αi .

Intuitively, the problem arises because we do not observe Yi ,−1,which belongs on the RHS of Eq. 37, as an explanatory variable forYi0, and so we are not able to estimate the model for this period.

There are four approaches to the initial conditions commonlyadopted. These are discussed in Heckman (1981a,b,c) andWooldridge (2005).

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2.3. Different ways to deal with the initialconditions

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Exogenous initial conditions

The simplest approach assumes that Yi0 is independent of αi . Inthe context of Eq. 40, this assumption involves ignoring thecomponent P(Yi0|Xi , αi ):

Li (θ|Yi ,Xi ) =

∫

P(Yi1, . . . ,YiT−1|Yi0,Xi , αi )P(Yi0|Xi , αi )dFα(αi |Xi),

(41)and, if uit ∼ iid

Li (θ|Yi ,Xi) =

∫

{

T−1∏

t=1

P(Yit |Yit−1,Xit , αi )}

dFα(αi |Xi ) (42)

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Exogenous initial conditions

• The exogeneity assumption may be reasonable if t = 0corresponds to the true beginning of the dynamic process thatdetermines Yit . This is, for instance the case for Card andHyslop (2005) who study the dynamic effects of a welfareexperiment and condition on being on welfare in the initialperiod.

• However, if the process is ongoing at t = 0 (i.e., it has startedbefore t = 0) and if there is a lot of heterogeneity in thepopulation, the exogeneity assumption is not reasonable.

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Exogenous initial conditions

• For example, if there is heterogeneity in the propensity toexperience welfare in the population (indexed by αi), thenthose with a high propensity to experience welfare (high αi )will be more likely to be on welfare in period t = 0 than thosewith a low propensity (low αi ). In this case, αi will not beindependent of Yi0.

• Assuming the initial conditions are exogenous will tend toupward bias the estimated state dependence and downwardbias the estimated heterogeneity.

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Equilibrium initial conditions

• An alternative way to deal with initial conditions is to assumethat the dynamic process is in equilibrium, perhaps conditionalon the vector of covariates, at the beginning of the sampleperiod (e.g. Card and Sullivan, 1988).

• This assumption implies restrictions on the parameters of thedynamic process, and in particular, the initial periodprobability P(Yi0|Xi , αi ) in Eq. 40.

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Equilibrium initial conditions

For example:

pi0 = P(Yi0 = 1|Xi , αi ) = pi0pRi0 + (1− pi0)p

Ai0, (43)

where pRi0 = F (γ + X ′

i0β + αi) and pAi0 = F (X ′

i0β + αi ) are theretention and accession probabilities, and therefore:

pi0 =pAi0

1 + pAi0 − pRi0. (44)

This assumption is unattractive too, although its implications areless clear than the exogeneity assumption.

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Reduced form initial conditions

Another approach suggests to adopt a flexible reduced formspecification for the initial conditions of the dynamic process. Inpractice, this typically involves specifying the initial periodoutcome in the following form:

Yi0 = 1(X ′

i0β0 + εi0 > 0), (45)

where the parameters in this equation, β0, are unrelated to thosein eq. 37, β; the errors εi0 should be flexibly specified, especiallywith respect to εit = αi + uit , e.g. εit ∼ iidN(0, σ2

0) andcorr(εi0, εit) = ρt .

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Reduced form initial conditions

• Estimation of the model would then combine eq. 45 for theinitial period outcomes with the structural specification for thedynamic process of subsequent period outcomes in Eq. 37.

• This formulation of the initial conditions tends to complicatethe computational requirements of the model, e.g. withoutfurther restriction, the covariance structure loses itsequicorrelation structure that enables the T-dimensionalintegral to be expressed as a product of single integrals in Eq.40.

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Wooldridge’s (2005) approach

A much simpler approach was proposed by Wooldridge (2005),rewriting the log-likelihood contribution in Eq. 40 to form the jointprobability conditioning the unobserved heterogeneity on the initialperiod outcome Yi0 as well as the exogenous covariates Xi , asfollows:

L(θ|Yi ,Xi) =

∫

{

T−1∏

t=1

P(Yit |Yit−1,Xit , αi )}

dFα(αi |Yi0,Xi ) (46)

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Wooldridge’s (2005) approach

That is, rather than modelling the initial conditions directly,Wooldridge suggested specifying a model for the unobservedheterogeneity conditional on the initial conditions, using a flexibleform for the unobserved heterogeneity model. For example, byanalogy to the correlated random effects (CRE) approach, anatural specification may be:

αi = γ0Yi0 +

T−1∑

s=0

X ′

isλs + ηi , (47)

where ηi ∼ N(0, σ2η).

This is one of the most commonly used approaches in theliterature, as it is very simple: One needs to insert the initial timeperiod of the outcome variable as a RHS variable.

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2.4. Fixed effects (conditional logit) models

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Cox-Chamberlain model

First, consider the dynamic logit model version of Eq. 37 withoutcovariates (i.e. uit ∼ iidlogistic):

Yit = 1(γYit−1 + αi + uit > 0), (48)

for i = 1, . . . ,N and t = 1, . . . ,T − 1 and with:

P(Yi0 = 1|αi ) = p0(αi ) (49)

P(Yit = 1|Yi0, . . . ,Yit−1, αi ) =exp(γYit−1 + αi)

1 + exp(γYit−1 + αi ). (50)

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Cox-Chamberlain model

Cox (1958) and Chamberlain (1985) show thatB =

{

Yi0,YiT−1,∑T−1

t=0 Yit

}

is a set of sufficient statistics for αi

(and p0(αi )) in this model and:

P(Yi0, . . . ,YiT−1|B) =exp(γ

∑T−1t=1 YitYit−1)

∑

d∈B exp(γ∑T−1

t=1 dtdt−1). (51)

Essentially, this conditional logit approach identifies and estimatesγ by considering the likelihood of different outcome paths betweenthe same initial and final states. In the absence of statedependence (γ = 0) among the sequences within B , seriallycorrelated outcomes should be no more prevalent than uncorrelatedoutcomes.

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Cox-Chamberlain model

This estimator conditional logit estimator requires T ≥ 4, because:

• We condition on the first and last period outcomes(Yi0,YiT−1);

• and for there to be alternative outcome paths,∑T−1

t=0 Yit mustcover at least 2 other time periods;

• e.g. if T=4, consider the setB =

{

yi0 = d0, yi3 = d3, yi1 + yi2 = 1, d0 6= d3}

has twooptions for Yi1,Yi2 { = (1,0) or (0,1)}:

P(Yi0 = d0,Yi1 = 1,Yi3 = d3|B) =exp

(

γ(Yi0 − Yi3))

1 + exp(

γ(Yi0 − Yi3)) .

(52)

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Cox-Chamberlain model

This means that the CL estimator compares pairs of sequences(1100 versus 1010) and (0011 versus 0101). State dependence,however, implies that the sequences 1100 or 0011 are more likelythan the sequences 1010 and 0101. Maximising the samplelog-likelihood provides the CL estimator for γ, which can be shownto be consistent and asymptotically normal.

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Honore and Kyrizidou (2000) model

• Honore and Kyrizidou (2000) show that the Cox-Chamberlainmodel can be extended in the presence of strictly exogenouscovariates

• Although conditioning on the set B will not eliminate theunobserved αi in this case, HK show that, if Xi2 = Xi3

(T=4), then the resulting conditional probabilities areindependent of αi , and β and γ are identified.

• For the case of T=4, we have:

P(Yi0 = d0,Yi1 = 1,Yi2 = 0,Yi3 = d3|Xi , αi ,Yi0 = d0,

Yi1 + Yi2 = 1,Yi3 = d3, d0 6= d3,Xi2 = Xi3)

=exp(γ(Yi0 − Yi3) + (Xi1 − Xi2)

′β)

1 + exp(γ(Yi0 − Yi3) + (Xi1 − Xi2)′β). (53)

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Honore and Kyrizidou (2000) model

• As for the Cox-Chamberlain case, identification of γ comesfrom differences in the outcome paths, while β is identifiedfrom changes in the X ’s in the middle two time periods (t=1and t=2).

• The condition of Xi2 = Xi3 is stringent, but based on thisinsight HK derive an estimator that puts greater weight onobservations that have Xi2 closer to Xi3. For the case of T=4,their estimator is:

(β, γ) = argmax

N∑

i=1

1(Yi1 + Yi2 = 1)K(Xi2 − Xi3

σN

)

×

ln( exp(γ(Yi0 − Yi3)

Yi1 + (Xi1 − Xi2)′β)

1 + exp(γ(Yi0 − Yi3) + (Xi1 − Xi2)′β)

)

.

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Honore and Kyrizidou (2000) model

• Where K (.) is a kernel weighting function that gives greaterweight to observations with smaller differences;

• σN is the bandwidth that goes to zero as N increases;

• If P(Xi2 − Xi3) > 0 and there is sufficient variation in(Xi1 − Xi2), the K (.) function can be replaced by1(Xi2 − Xi3 = 0);

• then the estimator has the usual N−1/2 asymptoticconvergence, otherwise, the estimator has slower convergence,but is still consistent and asymptotically normal.

• The convergence rate falls as the number of covariatesincreases.

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3. An empirical application: Roy and Schurer

(2013)

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