Esman M. Nyamongo Central Bank of Kenya Panel data analysis Econometrics Course organized by the COMESA Monetary Institute (CMI) on 2-11 June 2014, KSMS Nairobi Kenya Esman M. Nyamongo Central Bank of Kenya

Dynamic panel estimation DAY 9 Dynamic panel estimation

Dynamics…. …. Economic issues are dynamic in nature and use the panel data structure to understand adjustment Demand (present demand depends on past demand) Dynamic wage equation Employment models Investment of firms

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.

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.

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 random The 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

In this case, both estimators and are biased. What is the way out? 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 FIML Feasible GLS LSDV bias corrected (Kiviet, 1995) IV approach (Anderson and Hsiao, 1982) GMM approach (Arellano and Bond, 1985)

The GMM The GMM framework provides a computationally convenient method of performing inferences without the need to specify the likelihood function Some hints on moments Background material The moment conditions

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 moment The population mean, v1, or first moment about the origin, measures central tendency and is given by: In which case m(.) is the identity function

The second moment The second population moment is given by: The population variance of z is a measure of spread in a distribution It 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:

The above are population moments However, we rarely obtain information on the entire population We therefore use a sample {zi: i=1, 2….T} In estimation, we therefore need to define the sample moment Sample moments is a sample version of the population moments Expectation replaced by the sample average

Orthogonality /moment conditions Seeks to set expectations of functions of data, z, and unknown parameter, , to particular values Usually zero Mean of the data is: Giving the population orthogonality or moment condition: Where and contains one unknown parameter, such that

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.

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

Moment restriction for covariance Stated as: It involves 3 unknowns, , resulting in the following system:

….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.

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.

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 (Yit-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

To overcome this problem we use GMM. Therefore Yit-1 is correlated with the error term => OLS cannot solve our problems. FE cannot manage cos Yit-1 is correlated with individual effects To overcome this problem we use GMM. Arellano and Bond estimator Arellano and Bover estimator

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 Yit-1 and vit to choose the instruments Consider 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.

Consider t=3 In this case yi1 is a valid instrument of (Yi2- yi1), since it is highly correlated with (yi2-yi1) and not correlated with (vi3-vi2) Consider t=4 What are the instruments?

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:

Pre-multiplying our difference equation by wi yields: Estimating this equation by GLS yields the preliminary Arellano and Bond one-step consistent estimator Incase there are other regressors we have: Some practical exercise!