This function obtains the multiplier effects, on the explained variable, of a change in a regressor for the model that has been estimated. For reasons given below, this function only applies to models with an autoregressive structure ("slm", "sdm" and "sarar") or with spatial lags of the regressors ("slx", "sdem"). The measurement of the multiplier effects is a bit more complicated than in a pure time series context because, due to the spatial structure of the model, part of the impacts spills over non uniformly over the space. Using the notation introduced by LeSage and Pace (2009) we distinguish between:
Average Direct effects: The average over the N spatial units and Tm time periods of the effect of a unitary change in the value of a explanatory variable on the contemporaneous value of the corresponding explained variable, located in the same point of the intervened regressor. This calculus is solved for all the regressors that appear in the G equations of the model.
Average Indirect effects: The average over the N spatial units and Tm time periods of the effects of a unitary change in the value of a explanatory variable on the contemporaneous value of the corresponding explained variable, located in a different spatial unit that that of the intervened regressor. This calculus is solved for all the regressors that appear in the G equations of the model.
Average total effects: The sum of Direct and Indirect effects.
The information on the three estimated effects is supplement with an indirect measure of statistical significance obtained from the randomization approach introduced in LeSage and Pace (2009).
impacts(spsurfit, nsim = 1000)A fitted object of class spsur.
Number of simulations for the randomization procedure. Default = 1000.
Returns the Direct, Indirect and Total effects of the estimated spatial SUR model and simulted significance measures.
table_dir_eff |
Table of average direct effects. |
table_ind_eff |
Table of average indirect effects. |
table_tot_eff |
Table of average total effects. |
LeSage and Pace (2009) adapt the classical notion of 'economic multiplier' to the problem of measuring the impact that a unitary change in the value of a regressor, produced in a certain point in space, has on the explained variable. The question is interesting because, due to the spatial structure of the model, the impacts of such change spill non uniformly over the space. In fact, the reaction of the explained variable depends on its relative location in relation to the point of intervention.
To simplify matters, LeSage and Pace (2009) propose to obtain aggregated multipliers for each regressor, just averaging the \(N^{2}\) impacts that results from intervening the value of each regressor on each of the N points in Space, on the explained variable, measured also in each of the \(N\) points in space. This aggregated average is the so-called Total effect.
Part of this impact will be absorved by the explained variable located in the same point of the regressor whose value has been changed (for example, the k-th regresor in the g-th equation, in the n-th spatial unit) or, in other words, we expect that \([d y_{tgn}]/[d x_{ktgn}] ne 0\). The aggregated average for the N points in space (n=1,2,...,N) and Tm time periods is the so-called Direct effect. The difference between the Total effect and the Direct effect measures the portion of the impact on the explained variable that leakes to other points in space, \([d y_{tgn}]/[d x_{ktgm}] for n ne m\); this is the Indirect effect.
impacts obtains the three multipliers together with an indirect measure of statistical significance,
according to the randomization approach described in Lesage and Pace (2009). Briefly, they suggest to obtain
a sequence of nsim random matrices of order (NTmxG) from a multivariate normal distribution
N(0; Sigma), being Sigma the estimated covariance matrix of the G equations in the SUR
model. These random matrices, combined with the observed values of the regressors and the estimated values of
the parameters of the corresponding spatial SUR model, are used to obtain simulated values of the explained
variables. Then, for each one of the nsim experiments, the SUR model is estimated, and the effects
are evaluated. The function impacts obtains the standard deviations of the nsim estimated
effects in the randomization procedure, which are used to test the significance of the estimated effects for the
original data.
Finally, let us note that this is a SUR model where the G equations are connected only through the error
terms. This means that if we intervene a regressor in equation g, in any point is space, only the explained
variable of the same equation g should react. The impacts do not spill over equations.
Moreover, the impact of a regressor, intervened in the spatial unit n, will
cross the borders of this spatial unit only if in the right hand side of the equation there are spatial lags of the
explained variables or of the regressors. In other words, the Indirect effect is zero for the
"sim" and "sem" models. impacts produces no output for these two models.
Lastly, it is clear that all the impacts are contemporaneous because the equations in the SUR model
have no time dynamics.
LeSage, J., and Pace, R. K. (2009). Introduction to spatial econometrics. Chapman and Hall/CRC.
L<U+00F3>pez, F.A., Mur, J., and Angulo, A. (2014). Spatial model selection strategies in a SUR framework. The case of regional productivity in EU. Annals of Regional Science, 53(1), 197-220.
Mur, J., L<U+00F3>pez, F., and Herrera, M. (2010). Testing for spatial effects in seemingly unrelated regressions. Spatial Economic Analysis, 5(4), 399-440.
# NOT RUN {
####################################
### PURE CROSS SECTIONAL DATA ######
########## (G>1; Tm=1) ###########
####################################
#### Example 1: Spatial Phillips-Curve. Anselin (1988, p. 203)
rm(list = ls()) # Clean memory
data(spc)
Tformula <- WAGE83 | WAGE81 ~ UN83 + NMR83 + SMSA | UN80 + NMR80 + SMSA
## A SUR-SLM model.
spcsur.slm <-spsur3sls(Form = Tformula, data = spc, type = "slm", W = Wspc)
summary(spcsur.slm)
eff.spcsur.slm <- impacts(spcsur.slm, nsim = 30)
# }
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