Abstract
Up to this point in the book, the empirical context for the various estimators and tests has been limited to a purely cross—sectional situation. In this chapter, I consider models for which observations are available in two dimensions. Typically, one dimension pertains to space and the other to time, although other combinations, such as cross-sections of cross-sections and time series of time series can be encompassed as well. This situation has become increasingly relevant in a wide range of empirical contexts. It is referred to in the literature as panel data, longitudinal data, or pooled cross-section and time series data.
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Notes on Chapter 10
For ease of notation, in this chapter covariances and variances will be expressed as cr, instead of a2.
The Kronecker product A 0 B of the m by m matrix A and the n by n matrix B is an mn by mn matrix with as elements a…B. In other words, each element in A is multiplied by the matrix B. Properties of the ’Kronecker product that will be used in the remainder of this section are: (A ® B)(C ® D) = AC ® BD; (A ® B)-1 = A-1 ® B-1; det(A O B) = [det(A)]a.[det(B))m; tr (A ® B) = trA.trB.
For details, see, e.g., Phillips (1985). See also the general comments in Section 8.2.1. for a discussion of the properties of EGLS estimators.
A specialized approach to the two—equation case is outlined in Kariya (1981).
For ease of exposition, the same weight matrix W is assumed for all time periods. The extension to the case of a different Wt for each t can be carried out in a straightforward way.
Note that here B equals the inverse of (I—kW), rather than (I—kW) itself, as in the previous chapters.
The formal derivation of this general case is given in Magnus (1978). See also Breusch (1980) for some important invariance results. The derivation of the ML etimator for the spatial SUR. model was introduced in Anselin (1980, Chapter 7).
In order for the resulting estimates to be acceptable, the X should satisfy stability requirements (e.g., lktl1) to avoid the indication of explosive spatial processes. In addition, the estimates for should result in a positive definite matrix.
More specifically, the crucial transformation matrix W(I—ktW)-1 fails to yield a zero trace in the spatial case, due to the structure of W. In time series, this matrix has zero diagonal elements, and thus yields a trace of zero. This ensures block diagonality between the information matrix partitions that correspond to the different parameters.
The resulting estimates for y should satisfy the usual stability conditions, e.g.
See also Anselin (1980), and Hooper and Hewings (1981) for some concerns about the interpretation of these space—time Moran coefficients.
The Hadamard or direct product of a matrix A and a matrix B (of the same dimensions) is obtained by multiplying each element of A with the corresponding element of B: [A*B].. = a…b…
Provided that each cluster has the same number of elements. If the clusters are of unequal size, the simplifying Kronecker product results of the SUR model no longer apply, although this situation can still be considered as a case of a nonspherical error variance. For a discussion of this issue in a time series context, see Schmidt (1977).
For a detailed discussion, see Hsiao (1988), as well as the references cited in the introduction to Section 10.2.
In a standard econometric approach, this dependence would be in the time domain, as in Lillard and Willis (1978).
Magnus (1982, p. 245) shows that for a matrix with special structure U = ivy ® C + IT ® D, the inverse and determinant can be written in a simpler form. With M = (1/T)i.iT’, and Q = D + T.C, the matrix is equivalent to: U = (M 0 Q) + (IT — M) 0 D. The determinant then follows as lUI = IQI.IDIT-1 and the inverse as U-1 = M ®Q-1 + (I—M) ® D-1.
For example, see the extensive discussion and derivations in Hsiao (1988).
For notational simplicity, only spatial equations are considered here. The simultaneous framework can easily be extended to include other non—spatial equations (and variables as well).
The special structure considered by Magnus (1978) is of the form ft=Q(2 ® A) Q’.
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© 1988 Springer Science+Business Media Dordrecht
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Anselin, L. (1988). Models in Space and Time. 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_10
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DOI: https://doi.org/10.1007/978-94-015-7799-1_10
Publisher Name: Springer, Dordrecht
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