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Dynamic Panel Data Model

Econometric Methodology

7.4 Dynamic Panel Data Model

155 𝐻𝐻 =𝑣𝑣𝑣𝑣𝑣𝑣(𝛽𝛽[π›½π›½π‘ˆπ‘ˆπ‘…π‘…βˆ’π›½π›½πΉπΉπ‘…π‘…]

𝐹𝐹𝑅𝑅)βˆ’π‘£π‘£π‘£π‘£π‘£π‘£(π›½π›½π‘ˆπ‘ˆπ‘…π‘…) ~ πœ’πœ’12 (7.8) Housman tests the hypothesis, H0 that 𝑒𝑒𝑖𝑖 and 𝑋𝑋𝑖𝑖𝑖𝑖’s are uncorrelated, that is REM is appropriate against that FEM is appropriate. A statistically significant value of πœ’πœ’12 indicates the rejections of the null hypothesis of REM as an appropriate model and accept that FEM is appropriate.

156 𝛽𝛽0𝑖𝑖 is the individual specific fixed effects

We make the following assumptions to estimate the model:

i) The error term is orthogonal to the exogenous variables, i.e., πΈπΈοΏ½π‘‹π‘‹πš€πš€π‘–π‘–Μ πœ€πœ€, 𝑖𝑖𝑖𝑖�= 0 ii) The exogenous variables might be correlated with the individual effects, i.e.,

πΈπΈοΏ½π‘‹π‘‹πš€πš€π‘–π‘–Μ 𝛽𝛽, 0𝑖𝑖� β‰  0

The error term is uncorrelated with the lagged endogenous variable: πΈπΈοΏ½π‘Œπ‘Œπ‘–π‘–,π‘–π‘–βˆ’1,πœ€πœ€π‘–π‘–π‘–π‘–οΏ½= 0 The dynamic structure of equation (7.9) suggests that the OLS estimator will be upward biased and inconsistent, this is because that the lagged level of dependent variable is correlated with the error term. The problem will not be solved even if the within transformation is applied owing to a downward bias (Nickell, 1981) and inconsistency.

If we estimate model (7.9) by applying simple FEM, then we see the assumption of uncorrelated error term with lagged dependent variables violates and that gives biased estimators. Therefore, we need to find some other methods to estimate the panel data model. In dynamic panel econometric analysis, there are three methods, namely, the Anderson-Hsiao method, the Areellano-Bond method and the Blundell-Bond method.

The Anderson- Hsiao method tries to solve the problem of FEM model by instrumenting the lagged endogenous variable. The idea behind instrumental variables is to find a set of variables, termed instruments that are both correlated with the explanatory variables in the equation but uncorrelated with the disturbances. These instruments are used to eliminate the correlation between regressors and the disturbances. The instrumental variable estimators are consistent when N or T or both tend to infinity. But they are inconsistent if N is fixed and T tends to infinity. This is the basic problem of the Anderson- Hsiao method.

7.4.1 The Arellano-Bond Estimation Technique

Arellano and Bond (1991) propose a method that exploits all possible instruments. Using the Generalized Method of Moments (GMM), they obtain estimators using the moment conditions generated by lagged levels of the dependent variable, (π‘Œπ‘Œπ‘–π‘–,π‘–π‘–βˆ’2 ,π‘Œπ‘Œπ‘–π‘–,π‘–π‘–βˆ’3… ) with

βˆ†π‘Œπ‘Œπ‘–π‘–π‘–π‘–. These methods are called difference GMM methods. Similar to all instrumental variables regressions, GMM estimators are unbiased. Arellano and Bond (1991) compared

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the performance of difference GMM, OLS, and GLS estimators. Using simulations, they find that GMM estimators exhibit the smallest bias and variance. The Arellano-Bond estimator is similar to the estimator suggested by Anderson and Hsiao but exploits additional moment restrictions, which enlarges the set of instruments. The dynamic equation to be estimated in levels is given as:

π‘Œπ‘Œπ‘–π‘–π‘–π‘– =πœŒπœŒπ‘Œπ‘Œπ‘–π‘–,π‘–π‘–βˆ’1+𝛽𝛽0𝑖𝑖+𝑋𝑋́ π›½π›½πš€πš€π‘–π‘– +πœ€πœ€π‘–π‘–π‘–π‘– (7.9)

where differencing eliminates the individual effects of 𝛽𝛽0𝑖𝑖 and we get the following equation:

π‘Œπ‘Œπ‘–π‘–π‘–π‘–βˆ’ π‘Œπ‘Œπ‘–π‘–,π‘–π‘–βˆ’1 =πœŒπœŒοΏ½π‘Œπ‘Œπ‘–π‘–,π‘–π‘–βˆ’1βˆ’ π‘Œπ‘Œπ‘–π‘–,π‘–π‘–βˆ’2οΏ½+�𝑋𝑋́ βˆ’ π‘‹π‘‹πš€πš€π‘–π‘– πš€πš€π‘–π‘–βˆ’1́ �𝛽𝛽+πœ€πœ€π‘–π‘–π‘–π‘–βˆ’ πœ€πœ€π‘–π‘–π‘–π‘–βˆ’1 (7.10) For each year, we now look for the instruments available for instrumenting the difference equation. For t= 3 the equation to be estimated is:

π‘Œπ‘Œπ‘–π‘–3βˆ’ π‘Œπ‘Œπ‘–π‘–,2 =𝜌𝜌(π‘Œπ‘Œπ‘–π‘–2βˆ’ π‘Œπ‘Œπ‘–π‘–1) +�𝑋𝑋́ βˆ’ π‘‹π‘‹πš€πš€3 ́ οΏ½π›½π›½πš€πš€2 +πœ€πœ€π‘–π‘–3βˆ’ πœ€πœ€π‘–π‘–2 (7.11) Where it is assumed the instruments are available. Similarly we can enlarge the instrumentation and for the equation in the final period T we get:

π‘Œπ‘Œπ‘–π‘–π‘–π‘–βˆ’ π‘Œπ‘Œπ‘–π‘–,π‘–π‘–βˆ’1 =πœŒπœŒοΏ½π‘Œπ‘Œπ‘–π‘–,π‘–π‘–βˆ’1βˆ’ π‘Œπ‘Œπ‘–π‘–,π‘–π‘–βˆ’2οΏ½+�𝑋𝑋́ βˆ’ π‘‹π‘‹πš€πš€π‘–π‘– πš€πš€π‘–π‘–βˆ’1́ �𝛽𝛽+πœ€πœ€π‘–π‘–π‘–π‘–βˆ’ πœ€πœ€π‘–π‘–π‘–π‘–βˆ’1 (7.12) Assuming the instruments π‘Œπ‘Œπ‘–π‘–,1 ,π‘Œπ‘Œπ‘–π‘–,2, β€¦π‘Œπ‘Œπ‘–π‘–,π‘–π‘–βˆ’2 and π‘‹π‘‹Μπš€πš€1,π‘‹π‘‹πš€πš€2β€¦Μπ‘‹π‘‹πš€πš€π‘–π‘–βˆ’1́ are available, finally, we get the following instrumented equation that is given in matrix form as:

π‘Šπ‘ŠΜπΉπΉπ‘¦π‘¦ =π‘Šπ‘ŠΜπΉπΉ 𝑋𝑋𝑖𝑖 +π‘Šπ‘ŠΜπΉπΉπœ€πœ€ (7.13)

Although the differencing procedure eliminates the specific country effect, it introduces a new way by construction of new error term, which is correlated to delayed dependent variable. According to the suppositions that the error term (Ξ΅) is not serially correlated, and the explanatory variables (X) are weakly exogenous, Arellano and Bond (1991) propose the following moment conditions:

πΈπΈοΏ½π‘Œπ‘Œπ‘–π‘–,π‘–π‘–βˆ’π‘ π‘ οΏ½πœ€πœ€π‘–π‘–,π‘–π‘–βˆ’ πœ€πœ€π‘–π‘–,π‘–π‘–βˆ’1οΏ½οΏ½= 0, for 𝑠𝑠 β‰₯2;𝑑𝑑= 3 …𝑇𝑇 (7.14) 𝐸𝐸�𝑋𝑋𝑖𝑖,π‘–π‘–βˆ’π‘ π‘ οΏ½πœ€πœ€π‘–π‘–,π‘–π‘–βˆ’ πœ€πœ€π‘–π‘–,π‘–π‘–βˆ’1οΏ½οΏ½= 0, for 𝑠𝑠 β‰₯2;𝑑𝑑 = 3 …𝑇𝑇 (7.15)

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By using these conditions of moment, they propose a two step GMM estimator. In the first stage, the error terms are assumed to be independent and homoscedastic through countries and time. In the second stage, residuals obtained in the first stage are used to build a coherent estimation of variance-covariance matrix, so relaxing suppositions of independence and homoscedasticity. Following simple case of instrumental variable estimation, the Arellano-Bond estimation can be seen as two-step estimation. First, a cross-section auxiliary equation is estimated and in the second step the resulting estimates are used as explanatory variables in the equation of original interest. The two step estimator is so asymptotically more efficient than that of obtained in the first step.

7.4.2 The Blundell-Bond Estimation Technique

The GMM estimator which is suggested by Arellano-Bond (1991) is known to be rather inefficient when instruments are weak because of making use of the information contained in differences only. Blundell and Bond (1998) show that in case of persistent explanatory variables, delayed explained variable in level form becomes weak instruments for difference equation regression. Asymptotically, there will have an increase in the variance of coefficients. In short samples, simulations of Monte Carlo show that weaknesses of instruments can produce biased coefficients. To reduce the potential of the way and the indistinctness associated with the GMM difference estimator, Arellano and Bover, (1995), and Blundell and Bond, (1998) suggest using a GMM system method which combines difference regression with level regression. So the combination of moment restrictions for the differences and levels results in a method which is called GMM-system method by Blundell-Bond method.

Instruments for difference regression are smae as above presented in section 7.4.1.

Instruments for level regression are the delays of corresponding variables differentiated.

These are instruments suited under the additional suppositions below: although it can have a correlation between the levels of the explanatory variables and the specific effect country in the equation (7.9), there is no correlation between the differences of these variables and the country specific effect. Given that the delayed levels are used as instruments in difference regression, the most recent difference is used as an instrument in level regression. The use of delays of additional differences would succeed in the conditions of

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moment superfluous, (Arellano and Bover, 1995). So, additional conditions of moment for level regression are:

𝐸𝐸[οΏ½π‘Œπ‘Œπ‘–π‘–,π‘–π‘–βˆ’π‘ π‘ βˆ’ π‘Œπ‘Œπ‘–π‘–,π‘–π‘–βˆ’π‘ π‘ βˆ’1οΏ½(𝛽𝛽0𝑖𝑖+πœ€πœ€π‘–π‘–π‘–π‘–] = 0 𝑓𝑓𝐢𝐢𝑓𝑓 𝑠𝑠 = 1 (7.16) 𝐸𝐸[�𝑋𝑋𝑖𝑖,π‘–π‘–βˆ’π‘ π‘ βˆ’ 𝑋𝑋𝑖𝑖,π‘–π‘–βˆ’π‘ π‘ βˆ’1οΏ½(𝛽𝛽0𝑖𝑖+πœ€πœ€π‘–π‘–π‘–π‘–] = 0 𝑓𝑓𝐢𝐢𝑓𝑓 𝑠𝑠 = 1 (7.17) Hence, we use the moment conditions presented in equations (7.16) to (7.17) and employ the system GMM method to generate consistent and efficient parameter estimates.

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