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Linear dynamic panel data models

Laura Magazzini

University of Verona

L. Magazzini (UniVR) Dynamic PD 1 / 67


Dynamic panel data modelsNotation & Assumptions

One of the advantage of panel data is that allows the study ofdynamics

For a randomly drawn cross section observation, the basic linearmultivariate dynamic panel data model can be written as(t = 2, 3, ...,T )

yit = ρyit−1 + x′itβ + ci + uit

or, alternatively asyit = w′

itγ + ci + uit

where wit = (yit−1, x′

it)′ and γ = (ρ, β′)′

⊲ As usual uit denotes the idiosyncratic error term⊲ ci captures individual heterogeneity⊲ More complicated dynamic structures which include additional lags of

the dependent variables and/or a distributed lag structure for thevariables in x can be accomodted in this framework



Dynamic modeling

Dynamic models are of interest in a wide range of economicapplications including

⊲ Euler equations for household consumption⊲ Adjustment cost models for firms factor demands (e.g. labor demand,

investment);⊲ Empirical models of economic growth⊲ Consistent estimates of other parameters of interest (e.g. estimation of

“dynamic” production functions)

Panel data is now widely used to estimate dynamic models

⊲ Obvious advantage over cross-section data in this context: we cannotestimate dynamic models from observations at a single point in time

⊲ Advantages over aggregate time series: (i) possibility that underlyingmicroeconomic dynamics may be obscured by aggregation biases; (ii)the scope to investigate heterogeneity in adjustment dynamics betweendifferent types of individuals



FE or RE?

Is it possible to apply FE or RE?

Obviouly no: the model violates both the orthogonality condition andthe strict exogeneity assumption

The regressor yit−1 is correlated both with ci and uit−1:

yit−1 = ρyit−2 + x′it−1β + ci + uit−1



The idea of “internal” instruments

Suppose that uit is uncorrelated over time and consider the firstdifference of our model

yit − yit−1 = ρ(yit−1− yit−2)+ (xit −xit−1)′β+(ci − ci )+ (uit −uit−1)

∆yit = ρ∆yit−1 +∆x′itβ +∆uit

OLS is inconsistent: ∆yit−1 is correlated with ∆uit⊲ You need to apply IV estimation

Can you think of a valid instrument?

Values of yit−2 are valid instruments in the equation estimated in first

differences (Anderson & Hsiao, 1981, 1982)



Sequential exogeneity

E (uit |wit ,wit−1, ...,wi2, ci ) = 0

wit (t = 1, 2, ...T ) are sequentially exogenous conditional onunobserved heterogeneity ci

One implication of sequential exogeneity is

E (wisuit) = 0

with t = 2, 3, ...,T and s ≤ t

In words, the idiosyncratic error in each time period is assumed to beuncorrelated only with past and present values of the explanatoryvariables but is allowed to be correlated with future values of theexplanatory variables



Another useful implication of sequential exogeneity is

E (uituit−j) = 0 for j > 0

In words, under sequential exogeneity, the idiosyncratic error termsare serially uncorrelated

In fact, by the Law of Total Expectations

E (uituit−j) = E [E (uituit−j |wit ,wit−1,wi2, ci )]

Since uit−j is a linear combination of the variables in the conditioningset

E [E (uituit−j |wit ,wit−1,wi2, ci )] = E [uit−jE (uit |wit ,wit−1,wi2, ci )] = 0


The first order AR panel data model


yit = ρyit−1 + vit

vit = ci + uit

|ρ| < 1 t = 2, ...,T

Standard panel data estimation methods such as within group (WG)and first differences (FD) allow for unrestricted correlation betweenunobserved heterogeneity and past, present and future values of theright hand side variables

However, in order to achieve consistency they require that allregressors are strictly exogenous

This assumption is clearly violated in the case of the laggeddependent variable since uit cannot be orthogonal (and therefore notcorrelated) to yit , yit−1, ..., yiT



Inconsistent of pooled OLS, WG, FD

Under sequential exogeneity it can be shown that pooled OLS, WGand FD estimators are all inconsistent for fixed T

There is a difference however since under appropriate stabilityconditions (|ρ| < 1 in the AR(1) model) the inconsistency in the WGestimator – but not in the pooled OLS or in the FD estimator – is oforder T−1

This in turn implies that the asymptotic bias of the WG estimator –but not of the OLS and FD estimators – goes to zero as T goes toinfinity



Inconsistency of pooled OLS

For the pooled OLS estimator we can write

ρOLS =

(

N∑

i=1

T∑

t=2

y2it−1

)−1( N∑

i=1

T∑

t=2

yit−1yit

)

To study the asymptotic properties of OLS, we substituteyit = ρyit−1 + vit to get:

ρOLS = ρ+

(

N∑

i=1

T∑

t=2

y2it−1

)−1( N∑

i=1

T∑

t=2

yit−1vit

)



Inconsistency of pooled OLS

The asymptotic bias of the simple pooled OLS estimator for ρ is givenby

p lim(ρOLS − ρ) = (1− ρ)

σ2c

σ2u

σ2c

σ2u+ 1−ρ

1+ρ

> 0

where E (c2i ) = σ2c and E (u2it) = σ2

u

It follows that the pooled OLS estimator is biased upwards and itsasymptotic bias does not depend on T

The consistency of the pooled OLS estimator cannot be rescued byallowing the time dimension to grow without bounds



Inconsistency of WG

For the WG regression, we can write

ρWG =

(

N∑

i=1

T∑

t=2

y2it−1

)−1( N∑

i=1

T∑

t=2

yit−1yit

)

with yit = yit − yi and yi =∑

T yit/T

As in the previous case, to analyse the asymptotic properties of ρWG

we substitute yit = ρyit−1 + uit :

ρWG = ρ+

(

N∑

i=1

T∑

t=2

y2it−1

)−1( N∑

i=1

T∑

t=2

yit−1uit

)



Inconsistency of WG

Nickell (1981) derives the analytical expression for the asymptotic bias

p lim(ρWG − ρ) = −

1+ρ

T−1

(

1− 1T

1−ρT

1−ρ

)

1− 2ρ(1−ρ)(T−1)

(

1− 1T

1−ρT

1−ρ

)

Nickell also provides a simple approximation

p lim(ρWG − ρ) ≈ −1 + ρ

T − 1

for reasonably large values of T

This makes it clear that when ρ > 0, the WG estimator is biaseddownwards

Furthermore, even with T = 10 , which is the order of magnitude ofmost sets of panel data, if ρ = 0.5, than the asymptotic bias is −.167

If T also goes to infinity, the asymptotic bias converges to 0 so thatthe WG estimator is consistent for ρ if both N → ∞ and T → ∞



Suggestion for applied work

The fact that the pooled OLS estimator is biased upward and the WGestimator is biased downward can be used to assess the performanceof consistent estimators in applications

If the estimate produced by a consistent estimator lies outside theinterval defined by the pooled OLS (upper bound) and the WG (loeerbound) estimates, the chosen consistent estimator is likely to havefinite sample problems



Inconsistency of FD

For the FD estimator, we can write

ρFD =

(

N∑

i=1

T∑

t=2

∆y2it−1

)−1( N∑

i=1

T∑

t=2

∆yit−1∆yit

)

with ∆yit = yit − yit−1

To study the asymptotic properties of ρFD , we substitute∆yit = ρ∆yit−1 +∆uit to get:

ρFD = ρ+

(

N∑

i=1

T∑

t=2

y2it−1

)−1( N∑

i=1

T∑

t=2

yit−1∆uit

)



Inconsistency of FD

The asymptotic bias of the FD estimator is given by:

p lim(ρFD − ρ) = −1 + ρ

2< 0

and therefore the FD estimator is biased downwards and itsasymptotic bias does not depend on T



How can ρ be consistently estimated?

The simplest linear dynamic panel data model is the first orderautoregressive panel data model:

yit = ρyit−1 + ci + uit

with |ρ| < 1 and t = 2, ...,T

In order to consistently estimate ρ:

1. Use the First Difference transformation to remove the individual effect

∆yit = ρ∆yit−1 +∆uit

2. Exploit the sequential exogeneity assumption to find instruments for∆yit−1 (and, therefore, moment conditions)

3. ρ can be consistently estimated by using the GMM estimator



Assumption: sequential exogeneity

E (uit |yit−1, yit−2, ..., yi1, ci ) = E (uit |yt−1i , ci ) = 0

Recall that the seq.exo. assumption implies lack of autocorrelation inuit

⊲ Lagged uis are linear combinations of the variables in the conditioningset (e.g. uit−1 = yit−1 − ρyit−2 − ci )

How can this assumption be exploited to find instruments for ∆yit−1?



Anderson & Hsiao (1981, 1982)

One simple possibility suggested by Anderson and Hsiao (1981, 1982)is to use yit−2 as instrument for ∆yit−1

⊲ Relevance: Is yit−2 correlated with ∆yit−1?⊲ Exogeneity: Is yit−2 uncorrelated with ∆uit−1? That is,

E [yit−2∆uit ] = 0?

They also proposed an alternative where ∆yit−2 is used instead asinstrument

Both these IV estimators thus exploit one moment condition inestimation, respectively

E [yit−2∆uit ] = E [yit−2(∆yit − ρ∆yit−1)] = 0

andE [∆yit−2∆uit ] = E [∆yit−2(∆yit − ρ∆yit−1)] = 0



Arellano & Bond (1991)

Arellano and Bond (1991) proposed to use the entire set ofinstruments (or at least a larger subset) in a GMM procedure

⊲ Under seq.exo. values of y lagged twice or more are valid instrumentsfor ∆yit−1 in the first differecing version of the equation of interest

∆yit = ρ∆yit−1 +∆uit

⊲ So, at a generic time t, we can use yi1, ..., yit−2 as potentialinstruments for ∆yit−1



Arellano & Bond (1991)

For example, when T = 5, we can write

⊲ Moment conditions for t = 3

E [yi1(∆yi3 − ρ∆yi2)] = 0


E [yi2(∆yi4 − ρ∆yi3)] = 0

E [yi1(∆yi4 − ρ∆yi3)] = 0


E [yi3(∆yi5 − ρ∆yi4)] = 0

E [yi2(∆yi5 − ρ∆yi4)] = 0

E [yi1(∆yi5 − ρ∆yi4)] = 0



Practical suggestion

The first difference GMM approach generates moment conditionsprolifically, with the instrument count quadratic in T

T 3 4 5 6 7 8 9 10 15

0.5(T − 1)(T − 2) 1 3 6 10 15 21 28 36 91

Since the number of elements in the estimated variance matrix isquadratic in the instrument count, it is quartic in T

A finite sample may lack adequate information to estimate such alarge matrix

In applications it is common (and wise) practice to use only lessdistant lags as instruments (t − 2, t − 3 or t − 2, t − 3, t − 4)



Extensions

This method extends easily to models with limited moving-averageserial correlation in uit

For instance, if uit is itself MA(1), then the first differenced errorterm ∆uit is MA(2)

As a consequence, yit−2 would not be a valid instrumental variable inthese first-differenced equations but yit−3 and longer lags remainavailable as instruments

In this case ρ is identified provided T > 4



Asymptotic Distribution

Under the assumptions

⊲ Sequential exogeneity⊲ Conditional homoschedasticity⊲ Time series homoschedasticity

the first difference one-step GMM estimator is consistent, efficientand asymptotically normal

If homoschedasticity is not satisfied, the one-step estimator is nolonger efficient (still, it remains consistent)

⊲ A heteroschedasticity robust estimator of the asymptotic variancematrix for the one-step estimator is also available

Under the assumption of sequential exogeneity, the first differencetwo-step GMM estimator is efficient and asymptotically normallydistributed



Testing the AssumptionsAutocorrelation test by Arellano & Bond

The consistency of the GMM-FD estimator relies on the absence offirst (and higher) order serial correlation in the disturbances

E (uituit−1) = 0 t = 3, 4, ...,T

Since the model is estimated in first differences it is useful to rewritethis condition as

E (∆uit∆uit−2) = 0 t = 5, 4, ...,T

⊲ Lack of autocorrelation in uit implies that ∆uit is autocorrelated oforder 1, whereas higher order autocorrelation is zero



Testing the assumption

This assumption can be tested by using the standardized averageresidual autocovariance

m2 =

∑Ni=1

∑Tt=5∆uit∆uit−2

(∗)

The test statistic is asymptotically N(0,1) under the null of no serialcorrelation in uit

(∗): for details refer to Arellano and Bond (1991)



Sargan test

More generally, the Sargan test of overidentifying restrictions can alsobe computed when AN is chosen optimally

The test statistic s is asymptotically χ2 with as many degrees offreedom as overidentifying restrictions, under the null hypothesis ofthe validity of the instruments

⊲ # overidentifying restrictions = # moment conditions − # estimatedparameters



Finite sample resultsMonte Carlo set up by Arellano and Bond (1991)

yit = ρyit−1 + xit + ci + uit with yi0 = 0

xit = 0.8xit−1 + εit

ci ∼ iidN(0, 1); uit ∼ iidN(0, 1); and εit ∼ iidN(0, 1) and independentof ci , uis for all t, s

Sample size: N = 100; T = 7

Mean and standard deviations computed on the basis of 100replications



Monte Carlo results

GMM1 GMM2 OLS WG 1sASE R1sASE 2sASE

ρ = 0.5Mean .4884 .4920 .7216 .3954 .0683 .0677 .0604Std.Dev. .0671 .0739 .0216 .0272 .0096 .0120 .0106

β = 1Mean 1.005 .9976 .7002 1.041 .0612 .0607 .0548Std.Dev. .0631 .0668 .0484 .0480 .0031 .0055 .0052

1sASE: one-step aymptotic standard errors

R1sASE: robust one-step aymptotic standard errors

2sASE: two-step aymptotic standard errors



Comments - 2sASE

In the Monte Carlo simulations the estimator of the asymptoticstandard errors of GMM2 in the last column (2sASE) shows a sizabledownward bias relative to the finite-sample standard deviationreported in the second column

Why?

The efficient GMM estimator uses A2sN as the weighting matrix

A2sN =

(

1

N

N∑

i=1

Z′

Di∆ui∆u′iZDi

)−1

A2sN is a function of estimated fourth moments

Generally, it takes a substantially larger sample size to estimate fourthmoments reliably than to estimate first and second moments



Size of the test statisticsNumber of rejections (out of 100 cases)

LoS m1s2 m1sr

2 m2s2 Sargan

10% 12 15 14 75% 5 5 6 41% 1 1 1 0

⊲ LoS: level of significance

⊲ Experiments with ρ = 0.5



Power of the test statisticsNumber of rejections (out of 100 cases)

Experiments with different degree of serial correlation:corr(uit , uit−1) = ρu 6= 0

LoS m1s2 m1sr

2 m2s2 Sargan

ρu = 0.210% 54 53 53 335% 45 46 46 201% 24 25 25 2

ρu = 0.310% 95 96 96 715% 91 92 92 471% 78 77 78 30



Robustness of the test statisticsNumber of rejections (out of 100 cases)

Experiments with heteroschedasticity: var(uit) = x2it

LoS m1s2 m1sr

2 m2s2 Sargan

10% 23 9 10 85% 15 2 2 01% 3 0 0 0



The weak instruments problem

Blundell and Bond (1998) show that the instruments used in thestandard first difference GMM estimator become less informative intwo important cases

First, as the value of ρ increases towards unity, and second as thevariance of the unobserved heterogeneity component, σ2

c increaseswith respect to the variance of the idiosyncratic error term, σ2

u

To see this we can focus on the case with T = 3:

⊲ We are left with only one moment condition E (yi1∆ui3) = 0⊲ ρ is just identified



Weak instruments problem with T = 3

In this case we use one equation in first differences (i = 1, ...,N)

∆yi3 = ρ∆yi2 +∆ui3

where we use yi1 as an instrument for ∆yi2

The GMM estimator reduces to a simple IV estimator with thefollowing reduced form equation

∆yi2 = πyi1 + ri

For sufficiently high values of ρ or σ2c/σ

2u, the OLS estimate of the

reduced form coefficient π can be made arbitrarily close to zero

In this case yi1 is only weakly correlated with ∆yi2 and thus a weak

instruments problem arise



Weak instruments problem with T = 3

It can be shown that

p lim πLS = (ρ− 1)

(1−ρ)2

(1−ρ2)

σ2c

σ2u+ (1−ρ)2

(1−ρ2)

For instance, with σ2c

σ2u= 1 and ρ = 0.8, p lim πLS = −0.02

In words, the instruments available for the equations infirst-differences are likely to be weak when the individual series arepersistent, that is when they have near unit root properties

Blundell and Bond (1998) propose additional moment conditions thatallow to solve this problem

⊲ See also Arellano and Bover (1995)



GMM “system” estimator


Arellano and Bond (1991) GMM-DIF estimator uses lags of yit (lagtwo or older) as instruments for the equation in first differences

Blundell and Bond (1998) suggest using lags of ∆yit (lag one orolder) as instruments for the equation in levels

E (vit∆yit−1) = 0 t = 3, 4, ...,T

An additional assumption is needed for consistency: the mean

stationarity assumption



The mean stationarity assumptionRoodman (2009)


Entities in this system can evolve much like GDP per worker in theSolow growth model, converging towards mean stationarityBy itself, a positive fixed effect, for instance, provides a constant,repeated boost to y in each period, like investment does for thecapital stockBut, assuming |ρ| < 1, this increment is offset by reversion towardsthe meanThe observed entities therefore converge to steady-state levels definedby

E [yit |ci ] = E [yit+1|ci ] ⇒ yit = ρyit + ci ⇒ yit =ci

1− ρ

The fixed effects and the autocorrelation coefficient interact todetermine the long-run mean of the seriesL. Magazzini (UniVR) Dynamic PD 39 / 67



Now consider the momemt conditions proposed by Blundell and Bond(1998): E (vit∆yit−1) = 0, t = 3, 4, ...,T

We can write (t ≥ 3)

E (vit∆yit−1) = E [(ci + uit)(yit−1 − yit−2)]

= E [(ci + uit)(ρyit−2 + ci + uit−1 − yit−2)]

= E [(ci + uit)((ρ− 1)yit−2 + ci + uit−1)]

= E [ci ((ρ− 1)yit−2 + ci )] = 0

which is equivalent to E [ci ((ρ− 1)yit + ci )] = 0 for t ≥ 1




E [ci ((ρ− 1)yit + ci )] = 0 for t ≥ 1

By dividing this condition by (1− ρ) we get:

E

[

ci

(

yit −ci

1− ρ

)]

= 0

Deviations from long-run means must not be correlated with the fixedeffects

It is possible to show that if this condition holds in t then it alsoholds in all subsequent periods

Effectively, this is a condition on the initial observation




The conditions imposed by Blundell and Bond (1998) imply thatadditional 0.5(T − 1)(T − 2) moment conditions for the equation inlevels are available

However, all but T − 3 moment restrictions are redundant since theycan be expressed as linear combinations of the GMM-DIF momentrestrictions



For example, with T = 5Which restrictions are redundant?

Moment conditions when t = 3

E [∆yi2(yi3 − ρyi2)] = 0


E [∆yi3(yi4 − ρyi3)] = 0

E [∆yi2(yi4 − ρyi3)] = 0


E [∆yi4(yi5 − ρyi4)] = 0

E [∆yi3(yi5 − ρyi4)] = 0

E [∆yi2(yi5 − ρyi4)] = 0




Summarizing, the enlarged set of moment conditions is

E [yt−2i (∆yit − ρ∆yit−1)] = 0 t = 3, 4, ...,T

E [∆yit−1(yit − ρyit−1)] = 0 t = 3, 4, ...,T

The GMM method can be applied for the estimation of ρ



Practical Suggestion

The validity of the additional orthogonality conditions (spanning fromthe mean stationarity assumption) can be tested by using theso-called Sargan difference test of overidentified retstrictions

The model is estimated twice, without and with the additionalmoment conditions

The difference in the value of the Sargan statistic is found to have aχ2 distribution under the null (the validity of these additional momentconditions), where the degrees of freedom are given by the number ofadditional restrictions



The weak instrument problem is solved

To show that these additional moment restrictions remain informativewhen ρ increases towards unity and/or when the variance of theunobserved heterogeneity component, σ2

c increases with respect tothe variance of the idiosyncratic error term, σ2

u, we can again focus onthe case with T = 3

We are left with only one moment condition E (vi3∆yi2) = 0, so thatρ is just identified

Here, we can use one equation in level

yi3 = ρyi2 + ci + ui3

where we use ∆yi2 as an instrument for yi2The corresponding GMM estimator is an IV estimator with thefollowing reduced form equation

yi2 = π∆yi2 + ri



The weak instrument problem is solved

It is possible to show that, in this case

p lim πLS =1

2

1− ρ

1− ρ2

This moment condition stays informative for high values of ρ, incontrast to the moment condition available for the first differencedmethod

For instance, p lim πLS = 0.28 for ρ = 0.8



Finite sample resultsMonte Carlo set up by Blundell and Bond (1998)


The mean stationarity assumption is satisfied: yi0 =ci

1−ρ+ εi1

ci ∼ iidN(0, 1); uit ∼ iidN(0, 1)

εi1 ∼ iidN(0, σ2ε) and independent of ci , uis for all t, s

Sample size: N = 100; T = 4, 11

Mean and standard deviations computed on the basis of 1000replications



Monte Carlo resultsT = 4

GMM-DIF. GMM-SYSρ Mean Std.Dev RMSE Mean Std.Dev RMSE

0.0 -0.0044 0.1227 0.1227 0.0100 0.0990 0.09940.3 0.2865 0.1849 0.1853 0.3132 0.1215 0.12210.5 0.4641 0.2674 0.2693 0.5100 0.1330 0.13300.8 0.4844 0.8224 0.8805 0.8101 0.1618 0.16200.9 0.2264 0.8264 1.0659 0.9405 0.1564 0.1615



Monte Carlo resultsT = 11

GMM-DIF GMM-SYSρ Mean Std.Dev RMSE Mean Std.Dev RMSE

0.0 -0.0138 0.0463 0.0483 -0.0183 0.0431 0.04680.3 0.2762 0.0541 0.0591 0.2728 0.0487 0.05580.5 0.4629 0.0623 0.0725 0.4689 0.0535 0.05680.8 0.6812 0.1036 0.1576 0.7925 0.0651 0.06550.9 0.6455 0.1581 0.2996 0.9259 0.0453 0.0522



Comments

In these simulations, when T = 4 we observe both the poorperformance of the first-differenced GMM estimator at high values ofρ and the dramatic improvement in precision that results fromexploiting the additional moment conditions

The poor performance of the FD-GMM estimator improves with thenumber of time periods available

However, even with T = 11, for high values of ρ there remain sizablegains in bias, precision and RMSE



Multivariate dynamic panel data models

The multivariate dynamic panel data model is (|ρ| < 1;t = 2, 3, ...,T )

yit = ρyit−1 + x′itβ + ci + uit

Sequential exogeneity assumption

E (uit |xti , y

t−1i , ci ) = 0

⊲ As in the first order autoregressive panel data model, note thatGMM-DIF1 implies lack of serial correlation in uit



Predetermined versus endogenous variables

The sequential exogeneity assumption (E (uit |xti , y

t−1i , ci ) = 0) implies

that the idiosyncratic error in each time period is assumed to beuncorrelated only with past and present values of the explanatoryvariables (predetermined variables)

This allows for feedback effects but not for simultaneity

To allow for simultaneity, the idiosyncratic error term in each timeperiod has to be assumed to be uncorrelated only with past values ofthe explanatory variables (endogenous variables)

In this case we write:

E (uit |xt−1i , yt−1

i , ci ) = 0



Consistency

The sequential exogeneity assumption also implies that the following0.5(T − 1)(T − 2) + 0.5(T + 1)(T − 2)K linear moment restrictionshold

E [yt−2i (∆yit − ρ∆yit−1 −∆xitβ)] = 0

E [xt−1ki (∆yit − ρ∆yit−1 −∆xitβ)] = 0

The estimating equation is in first differences

∆yit = ρ∆yit−1 +∆xitβ +∆uit

Under the sequential exogeneity assumption, values of y lagged twiceor more and values of x lagged once or more are valid instruments for∆yit−1 and ∆xit



Example: T = 5

1 + 2k moment conditions for t = 3

E [yi1(∆yi3 − ρ∆yi2 −∆xi3β)] = 0

E [xi1(∆yi3 − ρ∆yi2 −∆xi3β)] = 0

E [xi2(∆yi3 − ρ∆yi2 −∆xi3β)] = 0

2 + 3K moment conditions for t = 4

E [yi1(∆yi4 − ρ∆yi3 −∆xi4β)] = 0

E [yi2(∆yi4 − ρ∆yi3 −∆xi4β)] = 0

E [xi1(∆yi4 − ρ∆yi3 −∆xi4β)] = 0

E [xi2(∆yi4 − ρ∆yi3 −∆xi4β)] = 0

E [xi3(∆yi4 − ρ∆yi3 −∆xi4β)] = 0



Finally, 3 + 4K moment conditions for t = 5

E [yi1(∆yi5 − ρ∆yi4 −∆xi5β)] = 0

E [yi2(∆yi5 − ρ∆yi4 −∆xi5β)] = 0

E [yi3(∆yi5 − ρ∆yi4 −∆xi5β)] = 0

E [xi1(∆yi5 − ρ∆yi4 −∆xi5β)] = 0

E [xi2(∆yi5 − ρ∆yi4 −∆xi5β)] = 0

E [xi3(∆yi5 − ρ∆yi4 −∆xi5β)] = 0

E [xi4(∆yi5 − ρ∆yi4 −∆xi5β)] = 0

The GMM method allows you to obtain consistent estimates of theparameters



Practical suggestion

In the multivariate case, the first difference GMM approach becomeseven more prolific than in the simple autoregressive case

For instance, with K = 2

T 3 4 5 6 7 10

0.5(T − 1)(T − 2) + (T + 1)(T − 2) 5 13 24 38 55 124

In applications it is common (and wise) practice to use only lessdistant lags as instruments: t − 2, t − 3 for y and t − 1 and t − 2 forx (t − 2, t − 3 if the variables in x are endogenous)



Extensions

This method extends easily to models with limited moving-averageserial correlation in uit

For instance, if uit is itself MA(1), then the first differenced errorterm ∆uit is MA(2)

As a consequence, yit−2 would not be a valid instrumental variable inthese first-differenced equations but yit−3 and longer lags remainavailable as instruments

In this case ρ is identified provided T > 4



GMM-SYS estimator

Assumptions

⊲ Sequential exogeneity⊲ Mean stationarity also for x

These assumptions lead to additional non-redundant momentconditions

E [∆wit(yit − ρyit−1 − xitβ)] = 0

which can be exploited in the context of the GMM-SYS estimator



Variance of the 2-step GMM estimatorFinite sample correction

Montecarlo studies (including AB, 1991) have shown that estimatedasymptotic standard errors of the efficient two-step GMM estimatorare severely downward biased in small samples

The weight matrix used in the calculation of the efficient two-stepestimator is based on initial consistent parameter estimates

Windmeijer (2005) shows that the extra variation due to the presenceof these estimated parameters in the weight matrix accounts for muchof the difference between the finite sample and the asymptoticvariance of the two step estimator

This difference can be estimated, resulting in a finite samplecorrection of the variance



Monte Carlo experimentsWindmejier (2005)

yit = βxit + ci + uit

xit = 0.5xit−1 + ci + 0.5uit−1 + εit

ci ∼ iidN(0, 1) and εit ∼ iidN(0, 1)

uit = δiτuωit with ωit ∼ χ21− 1; δi ∼ U(0.5, 1.5); τt = 0.5+0.1(t− 1)

N = 100

10000 replications

Note that

⊲ xit are correlated with ci and predetermined with respect to uit⊲ uit are skewed and heteroschedastic over time and units



Monte Carlo resultsβ = 1

CorrectedMean Std.Dev. Std.Err. Std.Err.

T = 4

β1sGMMd 0.9800 0.1538 0.1471

β2sGMMd 0.9868 0.1423 0.1244 0.1391

T = 8

β1sGMMd 0.9784 0.0832 0.0809

β2sGMMd 0.9810 0.0721 0.0477 0.0715



Comments

Non-corrected two-step standard errors are severely downward biased,especially when T = 8

Both one-step and corrected two-step standard errors are very close tostandard deviations

Additional simulations also show that Wald tests based onnon-corrected two-step estimated variances are oversized

This is no more the case when corrected two-step variances are usedinstead

Finally, the size performance of the Sargan/Hansen test is not affectedby the estimation of β1s

GMMd used to construct the weight matrix



Selected references

Anderson, T.W. and C. Hsiao (1981), Estimation of Dynamic Modelswith Error Components, Journal of the American Statistical

Association, 76, 589-606

Arellano, M. and S. Bond (1991), Some Tests of Specification forPanel Data: Montecarlo Evidence and an Application to EmploymentEquations, Review of Economic Studies, 58, 277-297

Blundell, R. and S. Bond (1998), Initial Conditions and MomentRestrictions in Dynamic Panel Data Models, Journal of Econometrics,87, 115-143

Nickell, S. (1981), Biases in Dynamic Models with Fixed Effects,Econometrica, 49, 1417-1426



Selected references

Roodman, D. (2009), A Note on the Theme of Too ManyInstruments, Oxford Bulletin of Economics and Statistics, 71, 135-157

Windmeijer, F. (2005), A Finite Sample Correction for the Variance ofLinear Two-step GMM Estimators, Journal of Econometrics, 126,26-51



Further readings

Bun, M. (2003), Bias Correction in the Dynamic Panel Data Modelwith a Nonscalar Disturbance Covariance Matrix, Econometric

Reviews, 22, 29-58

Bun, M. and M. Carree (2006), Bias Corrected Estimation inDynamic Panel Data Models with Heteroskedasticity, Economic

Letters, 92, 220-227

Bun, M. and J.F Kiviet (2006), The Effects of Dynamic Feedbacks onLS and MM Estimator Accuracy in Panel Data Models, Journal ofEconometrics, 127, 409-444

Kiviet, J. F. (1995), On Bias, Inconsistency and Efficiency of VariousEstimators in Dynamic Panel Data Models, Journal of Econometrics,68, 53-78



Further readings

Judson, R. and A. Owen (1999), Estimating Dynamic Panel DataModels: a Guide for Macroeconomists, Economic Letters, 65, 9-15

Hsiao, C., Pesaran, M.H. and Tahmiscioglu A.K.: 2002, MaximumLikelihood Estimation of Fixed Effects Dynamic Panel Data ModelsCovering Short Time Periods, Journal of Econometrics 109(1),107–150.


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