where θi = −β<sup>0</sup> i<sup>/φ</sup> i<sup>isthek× 1vectorofthelong-runcoefficients,andη</sup> it<sup>’sarestationarywith</sup> possibly non-zero means (including fixed effects). This allows (8) to be written as: where ηi,t−1 is the error correction term given by (9), and thus φi is the error correction coefficient measuring the speed of adjustment towards the long-run equilibrium.

This general framework allows the formulation of the PMG estimator, which allows the intercepts, short-run coefficients and error variances to differ freely across groups, but the long-run coefficients to be homogenous; i.e. θi = θ ∀ i. Group-specific short-run coefficients and the common long-run coefficients are computed by the pooled maximum likelihood estimation. Denoting these estimators by φ<sup> ̃</sup> i, β<sup> ̃</sup> i, λ<sup> ̃</sup> ij,<sup> ̃</sup> δij and θ<sup> ̃</sup> , we obtain the PMG estimators by φ<sup>ˆ</sup> PMG = <u>�Ni=1N</u><sup><u>φ ̃</u></sup> <u>i</u> , β<sup>ˆ</sup> P MG = <u>�Ni=1N</u><sup><u>β ̃</u></sup> <u>i</u> , λˆjPMG = <u>�Ni=1N</u><sup>λ ̃ij</sup> , j = 1, ..., p − 1, and<sup>ˆ</sup> δjPMG = <u>�Ni=1N</u><sup> ̃δij</sup> , j = 0, ..., q − 1, θ<sup>ˆ</sup> PMG = θ<sup> ̃</sup> . PMG estimation provides an intermediate case between the dynamic fixed effects (DFE) estimator which imposes the homogeneity assumption for all parameters except for the fixed effects, and the mean group (MG) estimator proposed by Pesaran and Smith (1995), which allows for heterogeneity of all parameters. It exploits the statistical power offered by the panel through long-run homogeneity, while still admitting short-run heterogeneity. As long as sector-homogeneity is assured, the PMG estimator offers efficiency gains over the MG estimator, while granting the possibility of dynamic heterogeneity across sectors unlike the DFE estimator. In the presence of long-run homogeneity, therefore, our preference is for the use of the PMG estimator.

The crucial question is whether the assumption of long-run homogeneity is justified, given the threat of inefficiency and inconsistency noted by Pesaran and Smith (1995). We employ a Hausman (1978) test (hereafter h test) on the difference between MG and PMG estimates of long-run coefficients to test for long-run heterogeneity.<sup>31</sup> Note that as long as the homogeneity Hausman test is passed in our estimations, we report only PMG estimation results.<sup>32</sup> Finally, it is worth pointing out that a crucial advantage of the estimation approach of the present paper, is that the dynamics generally argued to be inherent in growth processes are explicitly modelled, while recognizing the presence of a long-run equilibrium relationship underlying the dynamics. This is particularly important given the recurrent debate in the context of growth studies concerning the appropriate length of the time window used in averaging data for cross country studies. Justification for averaging rests on the need to remove short-run fluctuations in growth studies. The choice of any window is in the final instance arbitrary.<sup>33</sup> Thus the justification for the use of the PMG estimator is that it is consistent both with the underlying theory of an homogenous long-run relationship, while allowing for the explicit modelling of short-run dynamics around the long-run relationship, and the possibly heterogeneous dynamic time series nature of the data in the dynamics of adjustment.

# 4.3 The Econometric Methodology: The Time Series Analysis

Our estimation is of structural systems by standard time series techniques, with variables that are first-difference stationary. Johansen<sup>34</sup> techniques of estimation are now standard, so that discussion of estimation methodology here can be brief. We employ a vector error-correction mechanism

> 31 An alternative is offered by Log-Likelihood Ratio tests. However, the finite sample performance of such tests are generally unknown and thus unreliable. We therfore employ the h-test instead.

> 32 The author thanks Yongcheol Shin for the provision of the appropriate GAUSS code for estimation purposes.

> 33 Indeed, some panel studies do not average at all. Unfortunately the estimators used in turn are generally not dynamic, so that the results obtained may also be driven by short-term fluctuations.

> 34 See Johansen (1991) and Johansen and Juselius (1990).

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