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Empirical Economics

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06.01.2016

A J test for dynamic panel model with fixed effects, and nonparametric spatial and time dependence

verfasst von: Harry H. Kelejian, Gianfranco Piras

Erschienen in: Empirical Economics | Ausgabe 4/2016

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Abstract

In this paper, we suggest a J test in a dynamic spatial panel framework of a null model against one or more alternatives. The null model we consider has fixed effects, along with nonparametrically specified spatial and time dependence. The alternatives can have either fixed or random effects with the same complications. We implement our procedure to test the specifications of a demand for cigarette model. We find that the most appropriate specification is one that contains the average price of cigarettes in neighboring states, as well as the spatial lag of the dependent variable. Along with formal large sample results, we also give small sample Monte Carlo results. Our large sample results are based on the assumption $N\rightarrow \infty $ and T is fixed. Our Monte Carlo results suggest that our proposed J test has good power and proper size even for small to moderately sized samples.

Vorheriger Artikel Forecasting US real private residential fixed investment using a large number of predictors

Nächster Artikel Bivariate generalized Poisson regression model: applications on health care data

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There is, of course, a large literature relating to the J test. For example, see Davidson and MacKinnon (1981), MacKinnon et al. (1983), Godfrey (1983), Pesaran and Deaton (1978), Dastoor (1983), Pesaran (1974, 1982), Delgado and Stengos (1994) and the reviews given in Greene (2003, pp.153–155, 178–180) and Kmenta (1986, pp. 593–600). A nice overview of issues relating to non-nested models is given in Pesaran and Weeks (2001).

See, e.g., Anselin et al. (2008), Kapoor et al. (2007), Baltagi et al. (2003, 2007a, c, 2013), Baltagi and Liu (2008), Debarsy and Ertur (2010), Elhorst (2003, 2008, 2009, 2010), Elhorst and Freret (2009), Elhorst et al. (2010), Lee and Yu (2010a, b, c, d), Mutl and Pfaffermayr (2011), Pesaran and Tosetti (2011), Yu and Lee (2010), Yu et al. (2008), Parent and LeSage (2010), Piras (2013).

In a slightly different context, Mutl and Pfaffermayr (2011) suggest and give large sample results for a Hausman test to discriminate between such models.

This was pointed out by a referee.

For the estimation of a dynamic panel data model with random effect, see Parent and LeSage (2012).

Clearly if the model also had regressors which only varied cross-sectionally, the corresponding coefficients would not be identified. This was the case in a paper by Kelejian et al. (2013).

See, e.g., Kapoor et al. (2007).

As a simple illustration, note from (1), (3), and (4) that

$$\begin{aligned} u_{T}= & {} R_{T1}\varepsilon _{1}+R_{T2}\varepsilon _{2}+\cdots +R_{TT}\varepsilon _{T} \\ u_{1}= & {} R_{11}\varepsilon _{1} \end{aligned}$$

Thus, since $y_{1}$ depends on $u_{1},$ which in turn depends on $ \varepsilon _{1}$, as does $u_{T},$ $y_{1}$ can not be viewed as exogenous even at time period T.

Essentially, the rational of the J test is that if the null model is correct, these augmenting variables should not add to the explanation of the dependent variable in (9).

The proof for these statements is tedious. It can be obtained by writing to the authors.

Further specifications are given below.

We indicate that the value of i should be preselected in order to avoid pretest problems.

See footnote 6.

To estimate the null model, we use the following matrix of instruments:

$$\begin{aligned} H_{0}=Q_{0}[X,X_{-1},P] \end{aligned}$$

Results from the estimation of the null, alternative and augmented model are reported in the Appendix.

The weighting matrix employed in this paper is based on the six nearest neighbors.

There is an additional difference between the null and alternative model since the model under $H_{1}$ is specified in terms of a spatially lagged dependent variable. This was brought to our attention by an anonymous referee. In light of this, we decided to run an additional Monte Carlo experiment where the competing models presented two differences: both choice of W and inclusion or exclusion of the spatially lagged dependent variable. To make this additional Monte Carlo experiment more similar to our empirical application, we set $N=49$ and $T=25$. The details on this experiment are not reported in the paper but are available from the authors. In summary, while the power is very high, for all parameters combination, estimates for the size are, on average, 0.038 and 0.039 for the two predictors.

Note that the spatial lag in the price variable is one of the columns in the spatial lag of M.

In the paper, we only present the test based on the predictor corresponding to the minimum information set. The results for the other predictor are qualitatively similar and, therefore, are only available upon request from the authors.

In the empirical application as well as in all of the Monte Carlo experiments in Sect. 8, the denominator of $\tilde{\Omega }_{H^{\prime }Q_{0}RR^{\prime }Q_{0}H}$ is $N(T-1)-k$ where k is the number of regressors in the model. This degrees of freedom correction is typically done in fixed effects studies (see, e.g., Baltagi 2008, equation 2.24). Also, in order to estimate the residuals more efficiently, $Q_{0}u$ is estimated from the null model.

We generate the spatial panel data with $100+T$ periods and then take the last T as our sample, and we set T equal to 5 in all experiments. The initial values are generated as

$$\begin{aligned} y_{0}=(I_{T}\otimes (I_{N}-\lambda _{0}W_{0})^{-1})X_{0}\beta _{0} \end{aligned}$$

In a sense, this result is not very surprising given that the power of any test will depend on the extent to which the null and alternative hypotheses differ. A further discussion of this is given by LeSage and Pace (2009). Using Bayesian posterior model comparisons, they illustrate for alternative weight matrices that as the spatial dependence approaches low levels, the posterior probabilities approach the prior probabilities. In the limiting case, if the spatial dependence were zero, an empirical test would not be able to distinguish between two different weighting matrices.

However, as pointed out by Jin and Lee (2013), there might be situations where the gap in the power for the two versions of the spatial J test may be large. This is an issue that could be explored in a larger Monte Carlo study, and we leave it for further research.

Some studies have suggested to implement bootstrap testing procedure to improve the small sample performance of the test (see, e.g., Burridge and Fingleton 2010, for an example in a spatial context). We decided to leave this for future research.

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Titel: A J test for dynamic panel model with fixed effects, and nonparametric spatial and time dependence
verfasst von: Harry H. Kelejian
Gianfranco Piras
Publikationsdatum: 06.01.2016
Verlag: Springer Berlin Heidelberg
Erschienen in: Empirical Economics / Ausgabe 4/2016
Print ISSN: 0377-7332
Elektronische ISSN: 1435-8921
DOI: https://doi.org/10.1007/s00181-015-1056-2

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