# Handling error with spatially lagged variable that is related to but distinct from outcome variable. SAR, CAR, Durbin, or lagged X?

I am estimating a diffusion model in which I am using spatial lags to predict an outcome variable, but I'm having trouble specifying the distribution of the error term because the lagged variable is related to, but not the same as, my outcome variable.

Specifically, my outcome is the emergence of an attribute in observation i,t given the presence of that attribute in j,t-3 < t* < t. Or in other words, my dependent variable is the emergence of an attribute in a particular month, given that the attribute has been present at all in the previous three months in a neighboring region, regardless of when it emerged in that region.

I am pretty sure there is no spatial weights matrix that would allow me to use the outcome variable in a SAR model. In the three months after the property emerged, the weights matrix for the neighboring regions in a given month could point to the month in which the property emerged. But after 3 months, there is no information in the outcome variable to indicate whether the property continues to be present. So I can't use a standard SAR of the form: $$y=ρWy+Zδ+ϵ$$ because there is no W which could be multiplied by y to create the predictor of interest.

One way to characterize the relationship is: x implies a y in a neighboring region but not when, and y implies some of the x's, but not all.

But because the outcome variable is so closely related to the main explanatory variable, I'm not sure how I should model the error term.

One possible specification I've thought of is to make it an interaction between the lagged outcome variable (the property has emerged in the past) and a new spatially lagged predictor (the property was present in the previous three months), so the term would be: ρWyx. But the y contributes no information in this case, and the added x confuses the y-isolated form of the equation: $$y=(I-ρWx)^{-1}Xβ+(I-ρWx)^{-1}u$$ Part of the issue is that I don't feel like I have an intuitive enough understanding of the distribution of the error term in a SAR model to be able to adapt the specification to other circumstances. Most descriptions of the model show the equation above and pay little attention to the error term and why/how it differs from a linear regression.

So my questions are:

1. Can you help me to get a more intuitive understanding of the distribution on the error term in SAR in general, why epsilon can't be distributed normally with mean 0 and constant variance when spatial lags have been controlled for, and how the proper distribution relates to the autoregression?
2. How do I handle a lagged predictor that is closely related to the outcome variable?
• Actually it looks like your points 1 and 2 are closely related since the point 2 is the reason why your error term may not be normally distrubuted: the assumption of exogeneity of your explanatory variables would not be respected if one of those is explained by the outcome variable. That being said, why not use $Wx_{t*}$ as (spatially and temporaly lagged) predictor? The fact that time flows in only one direction would ensure $Wx$'s exogeneity regarding other non time-lagged variables in the model. Incidentally, "lagged X" models are often referred to as "SLX" in the litterature. – keepAlive Jan 28 '18 at 0:44
• I am using \$Wx_{t*} . So the question really is, if I do that, is iid violated and if so, how do I estimate the error term. Question 1 is really me hoping that if I understood the distribution over the error term in SAR models a little better maybe I'd be able to figure this particular problem out more easily, and apply it to more complex specifications. – Reen Jan 28 '18 at 0:47
• Your point about time flowing in one direction does help though. I was thinking in spatial lag terms without thinking about them as time lags as well. But you have to make error corrections in VARs too, so I'm still uncertain about what to do about the error. – Reen Jan 28 '18 at 0:49