Abstract
To round off the discussion in Part II, in this chapter several of the estimation methods and tests for spatial process models that were previously developed in formal terms will be illustrated empirically. Three aspects are considered in particular. In the first section, I focus on some operational issues related to the implementation of maximum likelihood estimation and the associated nonlinear optimization problem. In the second section, the analysis of cross-sectional data is considered. Using a simple model of determinants of crime for 49 contiguous neighborhoods in Columbus, Ohio, several estimation methods and tests from Chapters 6, 8 and 9 are implemented empirically. In the third section, attention shifts to space-time data sets. A Phillips curve model is estimated at two points in time for 25 contiguous counties in South-Western Ohio, to illustrate the instrumental variable methods from Chapter 7 and various diagnostics for spatial effects from Chapter 10.
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Notes on Chapter 12
Although OLS on the transformed variable is easily carried out, the iteration back and forth between the estimation of ß and the optimization for X may be tedious in an econometric package without macro facilities.
Typically, a simultaneous optimization of the likelihood for all parameters will be numerically more complex. Moreover, not all standard optimization routines will converge to coefficient values within an acceptable range, i.e., spatial parameters less than one in absolute value and positive estimates for variance terms. Also, a number of approaches are very sensitive to the proper choice of a starting value for the parameters, and may not converge at all if this is not achieved. A further discussion of these issues is given in Section 12. 1. 4.
It is important to note that a standardized weight matrix will not be symmetric, so that a more complex numerical procedure is needed to compute the eigenvalues. The most common routines are based on the so—called QR algorithm. For details, see, e.g., the algorithms discussed in Wilkinson and Reinsch (1971). These routines are present in mathematical packages such as IMSL and UNPACK, but are not always included in standard statistical packages, where often only the symmetric case is considered. Ord (1975) suggests a transformation of the nonsymmetric matrix to a symmetric one which has the same eigenvalues (but not the same eigenvectors).
As shown in Chapter 8, the partial derivative of 1n1A1 with respect to p is —trAW.
For example, all illustrations in the following sections are computed by means of routines written in the GAUSS matrix language, which is optimized for precision and speed for matrices smaller than 90 by 90. See Edlefsen and Jones (1988) for a description of the language.
It should be noted that a smaller scale factor will slow down the speed of convergence, since the parameter moves towards the local optimum by smaller increments.
See also Anselin and Griffith (1988) for an elaboration of this point.
Examples of the integration of spatial routines into SAS and MINITAB are presented in Griffith (1988b). Instrumental variables estimators for spatial process models are incorporated into SAS in the routines reported in Anselin (1985).
Some examples are the FORTRAN routines for measures of spatial association based on the Hubert—Golledge quadratic assignment approach, in Costanzo (1982) and Anselin (1988c), and a set of FORTRAN programs for the estimation of spatial process models listed in Anselin (1985). Other programming efforts are sometimes referred to in the footnotes of journal articles.
The traces needed in the computation of the variance matrix are: tr(WA-1)2=19.440 and tr(WA-1)’(WA-1)=22.338.
The values listed in Table 12.5. are rounded. The actual estimation was carried out with a convergence criterion associated with a precision of better than six decimals. The corresponding estimate is 0.431023, with a derivative of the concentrated likelihood of 0.0000016, obtained after 23 iterations.
See, e.g., the discussion in Anselin (1980).
As before, the values reported here are rounded. The final value is rounded from a more precise calculation, and is not the result of a manipulation of the rounded values.
The traces necessary in the computation of the variance matrix are: tr(WB-1)2=27.551, and tr(WB-1) (WB-1)=31.582.
The values in the table are rounded. With a more rigorous precision criterion of six decimals, the estimation takes 11 iterations. The final estimate is k=0.561790, with an associated cr2 of 95. 5745.
A further analysis of the common factor hypothesis is given in Section 13.3.1.
Estimation was carried out by means of and iterated EGLS, with an adjusted Gauss—Newton procedure to obtain estimates for the random components. Starting values were generated from a Godfeld—Quandt approach. The latter consists of regressing the squared residuals on the squared explanatory variables. Convergence, with a precision criterion of six decimals, was achieved in 25 rounds for the model with INC and HOUSE, and in 9 rounds for the model with HOUSE only.
The corresponding Moran statistics for the residuals are 1=0.256 and I=0.208, with associated z—values of 2.995 and 2.472 (under the assumption of randomization).
Critical levels for an F(3,43) variate are 2.83 for p=0.05 and 4.28 for p=0.01.
As before, the results in Table 12.14. are rounded. The actual estimation was carried out with a precision criterion of 6 decimals and converged after 15 iterations. The final values of the parameters were: k=0.84803, and for the random components: CONSTANT=67.27923, HOUSE=0.00991.
For a recent example, see, e.g., Greenwood, Hunt and McDowell (1988).
A more rigorous approach could be based on canonical correlations, as in Bowden and Turkington (1984). See also the remarks in Section 7. 1. 1.
The measure of fit in the 3SLS results is a pseudo R2, in the form of the squared correlation between observed and predicted dependent variables.
Strictly speaking, for the example used here this approach is not appropriate, since it ignores the different time lags involved in the explanatory variables. Aside from this conceptual problem, which is less important in this purely numerical illustration, there is statistical evidence that the coefficients are not stable over time. For example, for the SUR estimates in Table 12.23, a Wald test on coefficient equality rejects the null hypothesis for both UN (W=5.770, p=0.02) and NMR (W=6.289, p=0.01) individually, as well as for all parameters jointly (W=16.460, p 0. 01 ).
Estimation is carried out by iterated EGLS, where the estimates for if and to at each iteration are based on transformed residuals, in the standard way. Convergence, with a precision of more than six decimals, is reached after 7 iterations.
As before, the values in Table 12.28. are rounded, and the actual estimation was carried out with a precision of higher than 8 decimals. This converged after 22 iterations and yielded the estimates X1=-0.944042 and h2=-0.801991.
An earlier version of this algorithm is presented in Anselin (1980, pp. 50–53).
In a mixed regressive spatial autoregressive model there is no equivalent to a Moran coefficient. A suitable alternative may be to take zero, or a fraction of the OLS estimate. In the illustrations presented here, a fraction of 0.2 is used. Even though it is chosen rather arbitrarily, this value has provided a useful starting point in many other empirical implementations as well.
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© 1988 Springer Science+Business Media Dordrecht
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Anselin, L. (1988). Operational Issues and Empirical Applications. 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_12
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DOI: https://doi.org/10.1007/978-94-015-7799-1_12
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