Esman M Nyamongo Central Bank of Kenya Econometrics Course organized by the COMESA Monetary Institute CMI on 211 June 2014 KSMS Nairobi Kenya 1 Dynamic panel estimation DAY 9 2 Dynamics ID: 585032
Download Presentation The PPT/PDF document "Panel data analysis" is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
Slide1
Panel data analysis
Esman M. NyamongoCentral Bank of Kenya
Econometrics Course organized by the COMESA Monetary Institute (CMI) on 2-11 June 2014, KSMS Nairobi Kenya
1Slide2
Dynamic panel estimation
DAY 9
2Slide3
Dynamics….
…. Economic issues are dynamic in nature and use the panel data structure to understand adjustmentDemand (present demand depends on past demand)Dynamic wage equationEmployment models
Investment of firms3Slide4
Dynamic panel estimation
A dynamic panel model contains at least a lagged variable. Consider the following: with: if i=j and s=t
Here the choice between FE and RE formulation has implications for estimations that are of a different nature than those associated with the static panels.
4Slide5
If the lagged dependent variables also appear as explanatory , strict
exogeneity of the regressors no longer holds.The lagged variable introduces endogeneity problem
The LSDV is no longer consistent when N tends to infinity and T is fixed.In addition, the initial values of a dynamic process raise another problem.It turns out that with a RE formulation, the interpretation of a model depends on the assumptions of initial conditions
The consistency property of MLE and the GLS estimator also depends on this assumption and on the way in which T and N tend to infinity.
5Slide6
The problem with LSDV in DP
The LSDV estimator is consistent for the static model whether the effects are fixed or random.therefore need to show that the LSDV is inconsistent for a dynamic panel data with individual effects, whether the effects are fixed or randomThe bias of the LSDV estimator in a dynamic model is generally known as dynamic bias or
Nickell’s bias (1981)Nickell, S. 1981’ Biases in Dynamic Models with Fixed Effects, Econometrica, 49, 1399-1416.
Proof needed if possible
6Slide7
The LSDV for dynamic individual-effects model remains biased with the introduction of exogenous variables if T is small;
In this case, both estimators and are biased.What is the way out?ML or FIMLFeasible GLS
LSDV bias corrected (Kiviet, 1995)IV approach (Anderson and Hsiao, 1982)GMM approach (Arellano and Bond, 1985)
7Slide8
The GMM
The GMM framework provides a computationally convenient method of performing inferences without the need to specify the likelihood functionSome hints on momentsBackground
materialThe moment conditions
8Slide9
definition of moments
A population moment , v, as the expectation of some continuous function m(.) of a (discrete) random variable z describing the population of interest:The first momentThe population mean, v
1, or first moment about the origin, measures central tendency and is given by:In which case m(.) is the identity function
9Slide10
The second moment
The second population moment is given by:The population variance of z is a measure of spread in a distributionIt is defined as the second moment of z centred about its mean and can be expressed as a function of the first two population moments, v1 and v2:
10Slide11
The above are population moments
However, we rarely obtain information on the entire populationWe therefore use a sample {zi: i=1, 2….T}
In estimation, we therefore need to define the sample momentSample moments is a sample version of the population moments
Expectation replaced by the sample average
11Slide12
Orthogonality /moment conditions
Seeks to set expectations of functions of data, z, and unknown parameter, , to particular valuesUsually zeroMean of the data is:
Giving the population orthogonality or moment condition:Where and contains one unknown parameter, such that
12Slide13
Moment restriction for mean
Stated as sample moment as:This is the method of moments
estimator of Under random sampling, this estimator will be unbiased and consistent for regardless of the other features of the underlying population.
13Slide14
Moment restriction for variance
Given as:This contains 2 unknowns, such that This means that 2 moment conditions are needed. Here we need to include the moment condition for the mean to estimate the variance
Resulting in a system with 2 unknowns and
14Slide15
Moment restriction for covarianceStated as:
It involves 3 unknowns, , resulting in the following system:
15Slide16
….GMM
The GMM also uses the moment or orthogonality conditions and a key ingredient of GMM is the specification of the appropriate moment, or orthogonality condition,
An important approach to model specification testing is to base tests directly on certain conditions that the error terms of a model should satisfy.The moment conditions in GMM are therefore commonly based on the error terms from an economic model.
16Slide17
The intuition:
The basic idea is that if a model is correctly specified, many random quantities which are functions of the error terms should have an expectation of zero.The specification of a model sometimes allows a stronger conclusion, according to which such functions of the error terms have zero expectation conditional on some information set.
17Slide18
Back to our dynamic model
A dynamic panel model contains at least a lagged variable.
with: if i=j and s=t
The dynamic relationship is
characterised
by the presence of lagged dependent variable (Y
it-1
) among the
regressors
Including the lagged var. introduces
endogeneity
problem
Recall in FE, Y is a function of individual effects therefore it lag is also a function of these effects
18Slide19
Therefore Yit-1 is correlated with the error term => OLS cannot solve our problems.
FE cannot manage cos Yit-1 is correlated with individual effectsTo overcome this problem we use GMM.
Arellano and Bond estimatorArellano and Bover estimator
19Slide20
Arellano and bond estimator
To get consistent estimates in GMM for a dynamic panel model, Arellano and Bond appeals to orthogonality condition that exists between Y
it-1 and vit to choose the instrumentsConsider the following simple AR(10 model:
To get a consistent estimate of as N-> infinity with fixed T, we need to
difference
this equation to eliminate individual effects.
20Slide21
Consider t=3
In this case yi1 is a valid instrument of (Yi2-y
i1), since it is highly correlated with (yi2-yi1) and not correlated with (vi3-vi2)Consider t=4
What are the instruments?
21Slide22
For period T, set of instrument (w) will be:
The combination of the instruments could be defined as:Because the instruments are not correlated with the remaining error term, then our moment condition is stated as:
22Slide23
Pre-multiplying our difference equation by w
i yields:Estimating this equation by GLS yields the preliminary Arellano and Bond one-step consistent estimatorIncase there are other regressors
we have:Some practical exercise!
23