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**Dynamic panels and unit roots**

Panel Data Models Dynamic panels and unit roots

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Introduction To describe the dynamic panel and motivate its use (This is mostly a practical guide to its use). To differentiate between the Arellano-Bond and Arellano-Bovver approaches. To discuss the problems of unit roots in panel data. To introduce the concept of cointegration in panel data.

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**Dynamic Panel Data Models**

This approach to panel data models involves the use of a dynamic effect, in this case adding a lagged dependent variable to the explanatory variables. In addition the model is estimated using Generalised Method of Moments (GMM), which works in a similar way to Two Stage least squares, overcoming problems of endogeneity. This approach requires that N>T, i.e. the cross section observations exceed the time series.

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**Theoretical Reason for the Dynamic Panel**

The main theoretical reason for the dynamic panel is that it is modelling a partial adjustment based approach. If it is a partial adjustment process, the coefficient on the lagged dependent variable measures the speed of adjustment (i.e. 1 – coefficient is speed of adjustment) In addition the lagged dependent variable can remove any autocorrelation.

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Individual Effects The dynamic panel approach accounts for the individual effects, as with other panel data models. In the main approach of Arellano-Bond, this entails differencing the data. This means it is difficult to include dummy variables in these models. Although the individual effects applies to the cross section, two way individual effects can also be included, using time dummy variables.

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**Generalised Method of Moments**

This technique is basically a method that chooses parameter estimates, such that the theoretical model is satisfied as ‘closely’ as possible. The estimates are chosen to minimise the weighted distance between the theoretical and actual values. This method requires that the theoretical relations between the parameters satisfies so called ‘orthogonality conditions’, which mean that the sample correlations between the explanatory variables and instruments is as close to zero as possible. OLS is a special case of GMM, where we assume no correlation between the explanatory variables and error term. (GMM is similar to 2SLS, in that we need to specify the instrument list)

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**Dynamic Panel Approaches**

There are basically 2 approaches, the Arellano-Bond and Arellano-Bovver approach. They differ in terms of the way that the individual effects are included in the model, with the Arellano-Bond method using differencing and the Arellano-Bovver approach using orthogonal deviations. Although the Arellano-Bond approach has proven most popular, the Arellano-Bovver approach has better small sample properties and also is better at modelling non-stationary data.

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Dynamic Panel Models Criticisms of both approaches centre around the basic dynamic model, as dynamics are usually more complicated than a single lagged dependent variable. Also when accounting for unobserved heterogeneity, the methods for modelling this are limited. The stationarity of the variables tends to be ignored, although given that these models are restricted to short time series, this may not be too much of a problem.

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Example The most well known example is the modelling of dividends (div) and earnings (earn):

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**Steps for estimating a dynamic panel data model**

Specify model. Choose whether to use differencing or orthogonal deviations to account for fixed effects. Specify instruments (often lagged values of all variables in model) Choose method for adjusting standard errors, to overcome heteroskedasticity. This is usually Whites adjustment. Use the Sargan test to determine if the instruments are suitable (Test for overidentifying restrictions)

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Panel Unit Root Tests Panel unit root tests can have the usual benefits of using a panel, in so far as increasing the number of observations In addition Levin and Lin (1992) have shown that the panel approach substantially increases the power of the test relative to the time series ADF tests.

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**Levin and Lin approach (LL)**

In the 1993 test, they adopt a similar approach to the ADF test for a unit root, where the null hypothesis is that there is a unit root. In effect ADF tests are carried out for each individual in the panel, then adjusted to account for any heteroskedasticity, a pooled t-test is then produced to test the null, which are asymptotically distributed under the normal distribution. Different lags are allowed across different cross sections.

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Levin and Lin The 1993 model takes the following form (to remove autocorrelation lagged dependent variables included):

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Levin Lin The error terms across the cross sections are assumed to be independent. It is assumed the ρ is the same across all the cross sections. The lag length for the lagged dependent variables is chosen in the usual way. As with ADF tests, a trend can also be included in the test.

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**Im Pesaran and Shin Test (IPS)**

The IPS test is an example of an alternative to the LL test, as instead of assuming a common unit root process, where all the ρ are identical, it tests for individual unit root processes. This in effect tests for all i cross sections to be stationary. The IPS test averages all the individual ADF test statistics. The null hypothesis in this case is that each series contains a unit root for all i cross sections.

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IPS Test The IPS test in effect follows the model below:

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**IPS and LL tests Compared**

The main difference between the tests, is that one assumes a common unit root, the other individual unit root, also the IPS has an alternative hypothesis stating that at least one of the I cross section series is stationary, so LL requires all to be stationary, IPS only some. Both suffer from the assumption that the error terms across the cross sections are independent, which rules out any cointegration between them. This may not always be the case, particularly where the cross sections are financial markets or banks. Depending on different values of the N and T components, the two test statistics can give different results

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Panel Unit Root Test There are a variety of different tests with panel data, which differ in terms of the assumptions regarding the null hypothesis and how the autocorrrelation is removed. For instance the Fisher PP test removes the autocorrelation using an adjustment to the standard errors, as with the usual Phillips-Perron (PP) test.

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Panel Cointegration The main approaches to cointegration have the same advantages as the panel unit root tests, in that they increase the power of the test. There are essentially two approaches, one based on the Engle-Granger approach and the other using a Johansen ML type methodology. There are in turn variations of both approaches, for example in the Engle –Granger approach, there is the Kao test, which assumes the same values across all cross sections, whereas Pedroni assumes they can vary across the cross sections, in effect allowing considerable differences in the dynamics across the cross sections.

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Conclusion The dynamic panel allows dynamic effects to be introduced into the model. There are basically two different methods, which differ in how the fixed effects are measured. Both unit root tests and tests for cointegration can be conducted with panel data, which increases the power of the tests.

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