A COMPANION TO Theoretical Econometrics

Partially Parametric Models

By partially parametric models we mean that we focus on modeling the data via the conditional mean and variance, and even these may not be fully specified. In Section 4.1 we consider models based on specification of the conditional mean and variance. In Section 4.2 we consider and critique the use of least squares methods that do not explicitly model the heteroskedasticity inherent in count data. In Section 4.3 we consider models that are even more partially parametric, such as incomplete specification of the conditional mean.

3.1 Quasi-ML estimation

In the econometric literature pseudo-ML (PML) or quasi-ML (QML) estimation refers to estimating by ML, under the assumption that the specified density is possibly incorrect (Gourieroux et al., 1984a). PML and QML are often used inter­changeably. The distribution of the estimator is obtained under weaker assump­tions about the data generating process than those that led to the specified likelihood function. In the statistics literature QML often refers to nonlinear generalized least squares estimation. For the Poisson regression QML in the latter sense is equivalent to standard maximum likelihood.

From (15.5), the Poisson PML estimator, PP, has first-order conditions ХПо (yi - exp(x - P))x,- = 0. As already noted in Section 2.4, the summation on the left-hand side has an expectation of zero if E [ yi|xi] = exp(x - P). Hence the Poisson PML is consistent under the weaker assumption of correct specification of the condi­tional mean - the data need not be Poisson distributed. Using standard results, the variance matrix is of the sandwich form, with

and ю, = V [ yi |xi] is the conditional variance of yi.

Given an assumption for the functional form for ю, and a consistent estimate Ю of ю,, one can consistently estimate this covariance matrix. We could use the Poisson assumption, ю, = p,, but as already noted the data are often overdispersed, with ю, > p,. Common variance functions used are ю, = (1 + api)pi, that of the NB2 model discussed in Section 3.1, and ю, = (1 + a)p,, that of the NB1 model. Note that in the latter case (15.17) simplifies to VPML[pP] = (1 + a) ( Xn=1 pixix-)-1, so with over dispersion (a > 0) the usual ML variance matrix given in (15.6) is under­stating the true variance.

If ю, = E[(yі - x-P)2 |x,] is instead unspecified, a consistent estimate of VPML[0P] can be obtained by adapting the Eicker-White robust sandwich variance estimate formula to this case. The middle sum in (15.17) needs to be estimated. If - A p,

then n1 ХП-1(y. - {i)2xix- -A lim n1 Xn-1wixix-. Thus a consistent estimate of VPML[PP] is given by (15.17) with ю, and p, replaced by (yi - p,)2 and {,.

When doubt exists about the form of the variance function, the use of the PML estimator is recommended. Computationally this is essentially the same as Poisson ML, with the qualification that the variance matrix must be recomputed. The calculation of robust variances is often an option in standard packages.

These results for Poisson PML estimation are qualitatively similar to those for PML estimation in the linear model under normality. They extend more gener­ally to PML estimation based on densities in the linear exponential family. In all cases consistency requires only correct specification of the conditional mean (Nelder and Wedderburn, 1972; Gourieroux et al., 1984a). This has led to a vast statistical literature on generalized linear models (GLM), see McCullagh and Nelder (1989), which permits valid inference providing the conditional mean is correctly specified and nests many types of data as special cases - continuous (normal), count (Poisson), discrete (binomial) and positive (gamma). Many methods for complications, such as time series and panel data models, are presented in the more general GLM framework rather than specifically for count data.

Many econometricians find it more natural to use the generalized method of moments (GMM) framework rather than GLM. Then the starting point is the conditional moment E [yi - exp(x'P)|xi] = 0. If data are independent over i and the conditional variance is a multiple of the mean it can be shown that the optimal choice of instrument is x„ leading to the estimating equations (15.5); for more detail, see Cameron and Trivedi (1998, pp. 37-44). The GMM framework has been fruitful for panel data on counts, see Section 5.3, and for endogenous regres­sors. Fully specified simultaneous equations models for counts have not yet been developed, so instrumental variables methods are used. Given instruments zi, dim(z) > dim(x), satisfying E[yi - exp(x'P)|z!] = 0, a consistent estimator of P minimizes

X(Vi - exp(x/P))zi W X(Vi - exp(x$))z,

i= 1

where W is a symmetric weighting matrix.