Abstract
Many phenomena studied in regional science lead to structural instability over space, in the form of different response functions or systematically varying parameters. In addition, the measurement errors that result from the use of ad hoc spatial units of observation are likely to be non—homogeneous and can be expected to vary with location, area or other characteristics of the spatial units.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Notes on Chapter 9
Overviews of econometric issues related to the instability of parameters are given in, e.g., Swamy (1971, 1974), Belsley and Kuh (1973), Cooley and Prescott (1973, 1976), Rosenberg (1973), Pagan (1980), Raj and Ullah (1981), and Chow (1984). Instability of the functional form is discussed in, e.g., Quandt (1958, 1972, 1982), Goldfeld and Quandt (1973, 1976), Kiefer (1978), Quandt and Ramsey (1978), and Maddala (1983).
The tests in question are discussed in Glejser (1969), and Goldfeld and Quandt (1972). Other, more recent tests for heteroskedasticity are treated in, e.g., Harvey (1978), Godfrey (1978), Breusch and Pagan (1979), White (1980), Koenker and Bassett (1982), Cragg (1983), and MacKinnon and White (1985).
Early discussion of the Chow test and related issues of coefficient stability can be found in, e.g., Chow (1960), Zellner (1962), Fisher (1970), and Brown, Durbin and Evans (1975). The effect of heteroskedasticity on Chow-type tests has received considerably more attention in the literature than that of serial correlation, e.g., in Toyoda (1974), Jayatissa (1977), Schmidt and Sickles (1977), Watt (1979), and Honda (1982). See also Dufour (1982) for a recent review.
The simulation experiments consider a wide range of interactions between the spatial autocorrelation and extent of heteroskedasticity, with the latter varying from a relative error variance ratio of 1 (no heteroskedasticity) to 4. Based on the invariance results of Breusch (1980), they may therefore be considered quite general, since the effect of the regression coefficients can be ignored. The only other factor which needs to be further considered is the structure of the weight matrix. For additional details, see Anselin (1987b), and also Anselin and Griffith (1988).
See, e.g., Savin (1980) for a more extensive discussion.
Asymptotically, this is equivalent to a Wald or Likelihood Ratio test. However, since these necessitate the derivation of the model under the alternative hypothesis, they are considerably more complex computationally. As pointed out before, the Lagrange Multiplier is based on the model estimated under the null hypothesis, i.e., without any heteroskedasticity.
The block-diagonality of the information matrix pertains to the 3 coefficients and the parameters of the error covariance as a group, but not to the latter individually.
For a detailed derivation, see Anselin (1987b).
This is not a rigorous statement, since the residuals and error variance will differ between the two cases.
The LM, W and LR statistics are of this same form and differ only in whether the estimates for 11 and cr2 are computed under the null hypothesis, the alternative or both.
Other, more complex situations for the error covariance structure could be considered as well. For example, the error variance could be heteroskedastic, or a different weight matrix may drive the spatial dependence in each subset of the data. The results for these more complex cases can be found as direct extensions of the simple model considered here.
See also Ansein (1987a), and Chapters 13 and 14. Recent overviews of the econometric problems associated with ad hoc specification searches are given in, e.g., Learner (1974, 1978, 1983), Zellner (1979), Hendry (1980), Mayer (1980), Sims (1980), Frisch (1981), Malinvaud (1981), Lovell (1983), Ziemer (1984), and Cooley and LeRoy ( 1985, 1986 ).
The orthogonal expansion approach, as used in, e.g., Casetti and Jones (1987, 1988) provides a means to avoid some of the multicolinearity problems. The original trend surface polynomial is replaced by a small number of principal components, which significantly reduces the number of expansion variables in the terminal model.
For a similar approach in a time series context, see Singh et al. (1976).
For example, if the trend is quadratic, two of the heteroskedastic components would be fourth powers of the coordinates. Since the coordinates in an infinitely large lattice approach infinity themselves, this is a situation which does not fit within the regularity conditions for an asymptotic approach, such as a bounded error variance. A similar problem is the potential infinity of the cross products of the explanatory variables. Although this does not fit within these regularity conditions either, most asymptotic results still hold if the so—called Grenander conditions are satisfied. For details, see, e.g., Judge et al. (1985, pp. 161-163) and also White (1984).
For example, see the discussion in Grether and Maddala (1973), Granger and Newbold (1974), McCallum (1976), Thursby (1981), and Kiviet (1986).
By convention, the regression equation is formulated without a disturbance term, since the error associated with the random intercept fullfils this role.
For details, see, e. Swam (1971, 1974), Magnus 1978 Raj e.g., Y (B (J and Ullah J (1981), Schwallie (1982), and Hsiao (1986).
This is in contrast to the statement by Arora and Brown (1977, p. 76), that the “random coefficient regression model overcomes the problem of spatial autocorrelation by assuming that the coefficients are random.”
In most applications of the switching regression approach in urban and regional analysis, the classification of observations into one or the other regime is done in analogy to the time series case. For example, in urban density studies, the observations are ordered with respect to distance from the CBD, and the switching points correspond to given distances (or distances to be determined by a search over all possible switching points). In general however, any organization into subsets of observations would be a valid classification into regimes.
For an overview of the salient features of this technique in a time aeries context, see, e.g., Carbone and Longini (1977), Carbone and Gorr (1978), and Bretschneider and Gorr ( 1981, 1983 ).
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1988 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Anselin, L. (1988). Spatial Heterogeneity. 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_9
Download citation
DOI: https://doi.org/10.1007/978-94-015-7799-1_9
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-8311-1
Online ISBN: 978-94-015-7799-1
eBook Packages: Springer Book Archive