# Making predictions with a distributed lag model

I have a question regarding making predictions from distributed lag models. Suppose, I have a simple linear distributed lag model of the form:

$$y\left(t\right)= \sum^{k}_{i=0}\beta_{i}x\left(t-i\right) + \epsilon\left(t\right)$$

Assuming co-linearity is not an issue. Suppose, I have many data points along the temporal and spatial domain for the explanatory variable $x$. However, I only have data for one time instance for the response variable but I have many data points in a spatial domain. Assume that there is homogeneity in the spatial domain.

The variables have a spatial domain; however, I am ignoring this and assuming homogeneity int the spatial domain for both $x$ and $y$. Also, many values of $y$ can correspond to a single value of $x$.

Is it possible to use the model for prediction? In the sense that one could take the lag coefficients and say at the initial time the only effect of the explanatory variable on the response variable is $\beta_{o}$ and at some time in the future, $t=n$, the only effect of the explanatory variable on the response variable is $\beta_{n}$.

Thanks in advance for you time and consideration.

• How is the single available value for $y$ per time period, related to the spatial values of $x$ for the same period? Is there a theoretic model behind it? By the way no, you cannot reverse as is, because the reverse is a correspondence, not a function (one $y$ many $x$'s) Feb 23, 2016 at 20:11
• @AlecosPapadopoulos Thanks for the reply! Actually, sorry I misspoke. There are many spatial values for y, say consumption expenditure for various households across a region and many spatial values for x which would be a geophysical variable $f\left(x,t\right)$. Feb 23, 2016 at 22:00
• Hi @AlecosPapadopoulos. Sorry, I should not have said reversal. I meant to say (updated question) if it's possible to use the model for prediction. Which I believe is not the same as reversing the model $f:X\rightarrowY$. Thanks! Feb 23, 2016 at 22:06
• You should include all these in the main body of the question, they alter the question so significantly that they cannot remain in the comments. And please also clarify the following: does many values of $y$ relate to the same value of $x$? It is important to be clear about these matters, so that a reader can understand the model. Feb 23, 2016 at 23:29
• @AlecosPapadopoulos. Thanks for the suggestions. I updated the original question. And yes many values of $y$ can relate to the same value of $x$. Thanks again for all your help. Feb 23, 2016 at 23:47