Abstract
In this chapter, I consider the application of the maximum likelihood principle to estimation and hypothesis testing for spatial process models. The models follow the taxonomy for cross-sectional situations presented in Chapter 4. Space-time formulations will be discussed in Chapter 10.
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Notes on Chapter 6
As in the time series case, this is due to the complex stochastic nature of the inverse term. This will typically contain elements which are a function of the y’s (and therefore the e) at every observation point. As a result, this term will not be uncorrelated with E. Moreover, while in time series E[yL’e]=0 if there is no serial residual correlation, this is not the case in space. Indeed, in the spatial model: E[yL’e]=E{[W.(I—pW)—le]’e) which is only zero for p=0.
This in contrast to some of the early suggestions of Hordijk (1974). A more rigorous demonstration of this point is given in Anselin (1981).
As pointed out in the introduction to this chapter, these approaches are discussed in detail in Cliff and Ord (1981), Ripley (1981), and Upton and Fingleton (1985). See also Anselin (1980) for a detailed description of issues involved in the derivation of maximum likelihood estimators for spatial autoregressive models.
Previous formal treatments of this issue can be found in, e.g., Silvey (1961), Bar—Shalom (1971), Bhat (1974), and Crowder (1976), based on a conditional probability framework. However, it is only in the more recent articles that the simultaneous case is considered. This situation is most relevant to econometric models with lagged dependent variables and general error variance structures. See also the various laws of large numbers and central limit theorems discussed in Chapter 5.
A more rigorous formulation, for the situation where a normal distribution is assumed, is given in Heijmans and Magnus (1986c). The conditions presented there can be taken as the formal structure within which ML estimation can be carried out for the spatial autoregressive models considered here.
Here, as well as in the rest of the chapter, the notation det stands for determinant of a matrix. Where appropriate for notational simplicity, the symbol I I will be used as well.
The Jacobian in the first order spatial autoregressive model, y = pW + E, is det (I—pW). Its properties have been explored in detail, and related to characteristics of the weight matrix on regular lattice structures, in Griffith (1980) and Ord (1981).
Given the usual specification of the model in terms of standardized weight matrices, the parameterization of A, B, and O will lead to well—behaved partial derivatives. However, for more general specifications, e.g., with a nonlinear distance decay function as spatial weights, the conditions needed to satisfy the existence of a continuously differentiable log—likelihood should also be checked.
It should be noted that the conditions (6.18)—(6.20) do not have to hold jointly to satisfy (6.17). They are sufficient, but too strict.
Ord (1975) has derived a simplification of determinants such as CAI and IBS in terms of their eigenvalues, more specifically as a product over i of (1—p.w.), where the tut are the eigenvalues of W. Consequently, the regularity conditions can be expressed in terms of these eigenvalues. Although this is usually taken as —1 < p < +1, in Anselin (1982, p. 1025) I showed that a more precise inequality is: — (1/6)m) < p < +1, where o1m is the largest negative eigenvalue of W (in absolute value).
In the notation below, tr stands for trace of a matrix, and a stands for the p—th element of the vector a, with p=0,1,…,P. H stands for the diagonal matrix with elements dh/da p, where h is h(za), or, explicitly, forP (dh/8s).zp, where s=za and sip is the p—th element of the s vector.
Detailed results for the special models are given in Ord (1975), Anselin (1980), Cliff and Ord (1981), and Upton and Fingleton (1985).
See in particular the literature on the estimation of random coefficient models, e.g., in Swamy (1971), Raj and Ullah (1981), and the overview in Amemiya (1985).
In the literature, there is no uniform practice for estimating the asymptotic variance matrix for the parameters of spatial process models. The variance matrix is often estimated by its sample equivalent, and not necessarily based on the explicit derivation of the expected values of the second partial derivatives. Also, sometimes invalid simplifying assumptions have been introduced for the traces of the various matrices. For example, in Bivand (1984, p. 32), it is suggested for a mixed regressive spatial autoregressive model that tr(W1A-1)2 equals tr(W1A-1)’(W A-1). Unless the weight matrices are symmetric (which they are not in Bivand’s example), this will not hold, as demonstrated in Anselin (1988a), and further illustrated in the empirical applications in Section 12.2.
See also the discussion of central limit theorems in Section 5.2., and the treatment of the properties of the ML estimator in the previous section.
However, as shown in Anselin (1988a), under the null hypothesis, the block diagonality holds between the spatially dependent and heteroskedastic components.
For an overview of the standard econometric approach, see e.g., Breusch and Godfrey (1981), Pagan and Hall (1983).
See, e.g., Cliff and Ord (1981). This coefficient has also been called R01 in Anselin (1982) and Bivand (1984).
This is in addition to the problem of the indeterminacy of the information matrix estimate associated with LM tests. In many situations this matrix can be estimated in a number of different ways, e.g., based on the expected values or on sample equivalents. For a review of this issue, see, e.g., Davidson and MacKinnon (1983).
When only a single constraint is tested, appropriate size corrections result in equal power between the tests. For several constraints, the results are not clear. For more extensive discussion, see the references cited in the text.
For overview, see, e.g., Sargan (1976), Phillips (1977, 1982), and Serfling (1980). A critical view on the relevance and usefulness of these procedures is expressed in a theoretical review by Taylor (1983), and in extensive Monte Carlo simulations by Kiviet (1985, 1988).
This is based on the asymptotic convergence in distribution of an F(q, N—K) statistic to a x2(q)/q. Also, when the null hypothesis consists of only one constraint, the significance levels could be based on a more conservative Student—t variate (with N—K degrees of freedom), instead of the standard normal.
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© 1988 Springer Science+Business Media Dordrecht
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Anselin, L. (1988). The Maximum Likelihood Approach to Spatial Process Models. In: Spatial Econometrics: Methods and Models. Studies in Operational Regional Science, vol 4. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-7799-1_6
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