# how to interpret/report estimated spatial lag coefficients

I estimated a spatial lag model with pysal. I want to know how to correctly interpret and report the resulting parameter estimates (given that spatial spillover exists). R's spdep has an impacts() function to correctly determine the direct, indirect, and total effects but pysal does not seem to have an equivalent.

This is a spatial lag model of the form: y = rho * W * y + intercept + beta * X

where y is the response, rho is the spatial-autoregressive coefficient, W is a queen contiguity spatial weights matrix, beta is a k-length vector of parameters to be estimated, and X is a matrix of n observations across k predictors. Everything enters the model linearly and it is estimated via maximum likelihood.

For example, if my spatial lag model estimation results in rho = 0.3222 and beta_k = -0.0359, would it be correct to report the result as: "a 1 unit increase in predictor k is associated with a 0.0359 decrease in the response variable"?

Or is it more accurate to say: "a 1 unit increase in predictor k is associated with a 0.0359 decrease in the response variable plus a spillover effect"?

On page 184 of this paper, the authors seem to indicate that the total effect = (beta_k / (1 - rho ** 2)) * (1 + rho). Given this formula and my estimated rho and beta_k, should I instead report the result as: "a 1 unit increase in predictor k is associated with a 0.0529 decrease in the response"? Assuming I'm reading those authors' paper correctly, this would report the total effect (including indirect/spillover effects) rather than just the direct effect (the beta_k parameter estimate) itself.

tldr; I'm working in python/pysal and want to know how to correctly interpret/report coefficients from a spatial lag model.

• I think it would be helpful if you actually specify which model you are using and how the parameters you refer to enter the model. Oct 4, 2019 at 17:57
• @JesperHybel I've edit the question accordingly.
– eos
Oct 4, 2019 at 18:22

You need to report what is relevant for your analysis. You know what is relevant for your analysis only when you understand the model. With respect to understanding the marginal effects it is important to notice that they depend on the choice of weight matrix through the matrix

$$D := (I - \rho W)^{-1}$$

once you have the partial derivatives there is most likely a lot of interesting quantitative measures you could calculate.

The text you refer to in your link: How to Interpret the Coefﬁcients of Spatial Models: Spillovers, Direct and Indirect Effects

by André Braz Golgher and Paul R. Voss

refer to the text

LeSage, J. P., & Pace, R. K. (2009). Introduction to spatial econometrics. Boca Raton: Taylor & Francis Group.

as a source of the terms direct effect, indirect effect and total effect and they define these as

The direct effect is represented by the mean of the diagonal terms of the partial derivatives matrix. The indirect effect is deﬁned as the mean of the off-diagonal elements in each row (or column). The total effect is represented as the sum of the direct and indirect effects.

Considering a model of 3 spatial units - areas in a city perhaps - the matrix of partial derivatives with respect to the covariate $$x_{k}$$ are found to be

$$\begin{bmatrix} \frac{\partial y_1}{\partial x_{1k}} & \frac{\partial y_1}{\partial x_{2k}} & \frac{\partial y_1}{\partial x_{3k}} \\ \frac{\partial y_2}{\partial x_{1k}} & \frac{\partial y_2}{\partial x_{2k}} & \frac{\partial y_2}{\partial x_{3k}} \\ \frac{\partial y_3}{\partial x_{1k}} & \frac{\partial y_3}{\partial x_{2k}} & \frac{\partial y_3}{\partial x_{3k}} \end{bmatrix} = \beta_k(I - \rho W)^{-1}$$

Reading the first row the first entry is $$\frac{\partial y_1}{\partial x_{1k}}$$ which is the marginal effect in area 1 of a change in the value of the independent variable in area 1. Because the place where the effect occurs (area 1) is the same as where it originates (area 1) the effect is referred to as direct.

The next entry $$\frac{\partial y_1}{\partial x_{2k}}$$ is the marginal effect in area 1 of a change in the value of the independent variable in area 2. Because the place where the effect occurs (area 1) is not the same as where it originates (area 2) the effect is referred to as indirect.

All other entries in the matrix is categorized in a similar fashion into direct and indirect such that

1. Diagonal terms are direct
2. Off-diagonal terms are indirect

Since the matrix has a lot of terms (too many to report if $$n$$ is large, which is usually the case ... otherwise one should not apply statistics based on asymptotic results) it is necessary to find some way to summarize them. The standard - but by non means the only way - to summarize a lot of terms in with a single number is to calculate the mean.

Apparently the text you are reading suggests taking the mean of the diagonal and referring to it as the direct effect (a more precise term could perhaps be mean direct effect) and similarly the mean of the off-diagonal terms referring to it as the indirect. It is then suggested to sum these and call it the total effect.

Given estimates of $$\rho$$ and $$\beta_k$$ these are easy to calculate in python or any other program that can invert a matrix $$(I - \rho W)$$ then multiply it by $$\beta_k$$ and take mean of diagonal and off-diagonal terms respectively (If I understand the text you cite correctly).