lecture 7: dynamic panel models 2 - folk.uio.nofolk.uio.no/rnymoen/e5103_lect7v10.pdf · lecture 7:...

Lecture 7: Dynamic panel models 2Ragnar Nymoen

Department of Economics, UiO

25 February 2010

ECON 5103/9103: Lecture 7

Main issues and references

The Arellano and Bond method for GMM estimation ofdynamic panel data models

A stepwise procedure for dynamic models with RE effects.

Unit roots and cointegration in panel data.

EB (book/Kompendium): Ch 9.6 and Ch 10.

EB (lecture notes) to No 8.and 12.

Baltagi, Ch 8 and 12


Arellano and Bond estimation– generalized IV I

We consider again

yit = k + αi + βxit + λyit−1 + uit , (1)

uit ∼ IID(0, σ 2),

which is a relevant model if the parameters of interest is theimpact coeffi cient of xit , β, the long-run multiplier β/(1− λ) andthe associated dynamic multipliers.After differencing, we get:

1yit = β1xit + λ1yit−1 +1uit , t = 3, ...T (2)

for observations (t = 1, 2, ...,T ).


Arellano and Bond estimation– generalized IV IIInstead of regarding (2) as one equation, regard it a system ofT − 2 equations.

IVt = 3 1yi3 = β1xi3 + λ1yi2 +1ui3 zi2 = (yi1,1xi3)t = 4 1yi4 = β1xi4 + λ1yi3 +1ui4 zi3 = (zi3, yi2,1xi4)t = T 1yiT = β1xiT + λ1yiT−1 +1uiT ziT−1 = (ziT−2, yiT−1,1xiT )

It is the use of different instruments for equations of differenttime periods that defines the A&B method relative toconventional IV estimation, which uses the same instrumentset for all endogenous variables.

Conventional instruments, wt , can be used. In both STATAand PcGIVE, there is a distinction between GMM instruments(yi1, ...yT−2) and other, conventional instruments, wt .

Note how the GMM instruments are cumulated. Number ofinstruments is increasing with T .


Arellano and Bond, matrix formulation I

From the table we see that we formally have a system of T − 2equations with different IV’s for each equation.

Zi =

zi2 0 0 00 zi3 · · · · · ·...

.... . .

...0 0 ziT−1

(3)

Vector formulation:

1yi︸︷︷︸qi

= 1yi ,−1λ+1xiβ+1ui =(1yi ,−1

... 1xi)

︸︷︷︸Wi

(λ

β

)︸︷︷︸

δ

+1ui︸︷︷︸εi

(4)which can be stacked for all individuals into one matrix equation:

q =Wδ+ ε (5)


Arellano and Bond, matrix formulation II

Similar stacking gives Z from the N matrices Zi , see Kompendiumsec 10.2. Write the orthogonality conditions for GMM estimationof (4) as

E[Z′

iεi

]=

z′

i2(1yi3 − β1xi3 − λ1yi2)z′

i3(1yi3 − β1xi3 − λ1yi2)...

z′

iT−1(1yiT − β1xiT − λ1yiT−1)

︸︷︷︸

Z′i (qi−Wiδ)

=

00...0

with sample counterpart:

gN (δ) =1N

N∑i=1

Z′i (qi −Wiδ)


Arellano and Bond 1-step and 2-step estimation I

With reference to the principles of GMM, we choose δ so thatthe quadratic form based on a weighting matrix SN isminimized:

δ= argminδgN (δ)′SNgN (δ)

When the estimated equation is in differences, as in this casewith (4), the matrix SN reflects the induced first orderMoving Average (MA(1)) in the disturbances. εi = 1ui . Thisis as with weighted least squares (GLS), but of course use IVto form the “raw moments”here.

The resulting IV estimator is usually denoted δ1 for“one-step”A&B estimator, or preliminary A&B estimator.


Arellano and Bond 1-step and 2-step estimation II

The two-step estimator uses δ1 to construct uit . And then doa second GMM based on SN using those residuals:

SN = [N−2N∑i=1

Z′i (1ui )(1ui )′Zi ]−1

This estimator is referred to as the two-step A&B estimator,and is denoted δ2 .

In principle the difference between δ1 and δ2 is that δ1 isbased on a known SN , i.e. on the assumptions of the modelnamely that uit is IID(0, σ 2), while δ2 is the GMM estimatorwhich is consistent despite those assumptions not holding inthe data.

In practice the most cited rationale for considering δ2 is thatthis estimator is more effi cient when uit is heteroscedastic.


Arellano and Bond, modifications, extensions and practicalissues I

Above the equation of interest is in terms of differencesvariables, and lagged levels is used as instruments. But asnoted in Lecture 6 can use lagged differences as instrumentsinstead.

If we are explicit about the fixed effects model, then we can ofcourse keep the equation of interest in its original levels form,and estimate that equation with GMM, since the mainproblem we are solving with IV is the finite T bias caused byyt−1 being pre-determined.δ1 will then be based on uit ∼ IID(0, σ 2), no weighting of theobservation, equivalently:

SN = [N−2σ 2N∑j=1

z′jzj ]−1


Arellano and Bond, modifications, extensions and practicalissues II

in the GMM formula.

In general the coeffi cient standard errors for 2-step estimationare downward biased, in particular in small samples. There isa small sample correction due to Windmeijer (2000) that mayor may not be standard in the software you are using.

In Stata 12: Offi cial xtabond does not contain this smallsample correction, but xtbaond2 doesIn PcGive: The correction is provided when the option “robuststandard errors” is chosen.

Additional output from the software.

STATA and PcGive both give test statistics for 1. and 2. orderresidual autocorrelation in the estimated equation. Why is thisrelevant information?


Arellano and Bond, modifications, extensions and practicalissues III

Both programmes also gives test for instrument validity".STATA reports Hansen’s test , which is also known as theJ-test. PcGive reports a test due to Sargan. Heuristically, thisstatistic tests whether the information in the set of IV set isoptimally used by treating them as instruments! The Sargantest may for example become significant if the structuralequation has too simple dynamics, or if an extraneousinstrument is an omitted variable in the equation.

Dangers of “over-instrumenting”.

It is important to keep in mind that A&B was invented toimprove estimation properties when T is small (absolute andrelative to N).The number of available A&B instruments is increasing in T ,so choosing all available instruments will inflate the moments“in” SN .


Arellano and Bond, modifications, extensions and practicalissues IV

Known of problems of overfitting.

Because of overfitting 1yit−1 ≈ 1yit−1 and the 1-stepestimator is driven towards the within group estimator, in GLSversion if MA implication is used.The second step estimator may become impractical, both interms of computation and numerical results.With enough instruments, the tests of instruments validityhave very little power.Hides weak instrument problem

Practical guidelines:

Keep the maximum number of lags used as instruments to areasonable level: 4 or 5 is still generous instrumentation inmost cases.“Judicious choice based on theory, existing studies andcomputer capacity”, (A&B in their PcGive manual)


Arellano and Bond, modifications, extensions and practicalissues V

Report results for more than one estimator.Monte Carlo studies (e.g., Judson and Owen (1999)) showthat already with T = 30, the LSDV estimator performs aswell as the IV estimators (in terms of bias).


Dynamic model with RE I

Consider now a dynamic version of the model discussed in Lecture4 (see K 4.5 and 10.4):

yit = k + αi + βxit + λyit−1 + δzi + uit ,−1 < λ < 1 (6)

uit ∼ IID(0, σ 2), αi ∼ IID(0, σ 2a), uit ⊥ αi ⊥ (xit , zi )

In addition to αi being uncorrelated with xit we include anindividual specific explanatory variable.In Lecture 4 we learned that cannot be estimated δ under fixedeffect.Kompendium Sec 10.4 motivates a 3-step procedure for estimationof β, λ and δ.


Dynamic model with RE II

1 GMM: Difference (6) and estimate β, λ by GMM.2 Between: Treating β, λ as known from step 1, estimate kand δ by OLS on

y i · − βx i · − λy i , −1 = k + δzi + ui · + αi︸︷︷︸εi

Note: εi = ui · + αi has approximate IID properties since itonly varies across i .

3 Standard errors. Define the composite residuals ε it by usingthe estimator from 1., and use residuals to estimate σ 2 andσ 2a, as in the static version of this model, Kompendium, Ch4.5.


Unit root and cointegration I

In time series, the H0 of λ = 1 is diffi cult to reject formally,for example for yt and xt .

Given this, and if we want to test H0 of β = 0 in

yt = k + βxt + ut , (7)

cannot use the usual F and t critical value, because thenreject H0 in 95 out of 100 tests, instead of in 5 out of 100 test(with 5% nominal size). This pitfall is known as spuriousregression (which is different from spurious correlationalthough the statistician Yule worked with both problems),

To do correct inference, need non-standard distribution theory,which is a central theme in a course in advanced time serieseconometrics.


Unit root and cointegration II

Interesting to ponder why we should wish focus on λ = 1,when the stationary framework lets us use the whole line from−1 < λ < 1 (and more generally the whole unit circle)....

But here only review the panel data counterparts to some ofthe popular unit-root and cointegration tests


Panel unit root test I

Tests are with reference to

yit = γ zit + λiyit−1 + uit (8)

or1yit = γ zit + (λi − 1)︸︷︷︸

ρ i

yit−1 + uit (9)

where γ zit represents deterministic terms. It can be a constant, ora constant and a trend.As with time series data, different distributions apply for thedifferent specifications of the deterministic part.


Panel unit root test II

Homogenous unit-root testsuit ∼ IID(0, σ 2) and λi = λ, so homogenous DGP assumed inthis test.Use “t-statistics” for

H0: ρ = 0 against ρ < 0

based on the relevant estimation of (8) or (9), for example toaccount of fixed individual effects.These “t-statistics” are asymptotically N(0, 1) when T →∞,and N →∞. See Baltagi Ch 12.Entails that these tests have larger power than thecorresponding Dickey-Fuller test for pure time series.When T is fixed, the “t-statistics” are still asymptoticallynormal but their variance depend on T .


Panel unit root test III

Tests for individual unit roots.

H0: ρ i = 0 for all i

against

H1 :{ρ i = 0, for i = 1, 2, 3, ..,N1.ρ i < 0, for i = N1 + 1, ...,N

N Dickey-Fuller (DF) statistics are calculated, and thenaveraged for all N.A computer program like EViews gives these statistics withapproximate critical values, also for the case when (9) isaugmented by 1yt−j (j > 0) terms.In the simplest case, they are N(0, 1) under H0.


Panel cointegration tests I

Like the first generation of cointegration test, the H0 of nocointegration in the static panel model

yit = k + αi + βxit + uit

where both yit and xit have unit roots, can be based on theresiduals uit .The tests are based on

uit = λi uit−1 + vit (10)

where the H0 of no cointegration translates to H0: λi = 1 (orρ i ≡ λi − 1 = 0).Like in the unit root-test there are two branches of tests:


Panel cointegration tests II

Homogenous λi = λGives Dickey-Fuller type tests, for example:

DFt =√1.25tp +

√1.875N ∼

ass, NTN(0, 1)

Heterogenous λi

Averaging from N individual tests, see e.g. Baltagi ch 12.4.2 fordetails.


lecture 7: dynamic panel models 2 - folk.uio.nofolk.uio.no/rnymoen/e5103_lect7v10.pdf · lecture 7:...

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