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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.

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    $\begingroup$ 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) $\endgroup$ – Alecos Papadopoulos Feb 23 '16 at 20:11
  • $\begingroup$ @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 $f\left(x,t\right)$. $\endgroup$ – user877334 Feb 23 '16 at 22:00
  • $\begingroup$ 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! $\endgroup$ – user877334 Feb 23 '16 at 22:06
  • $\begingroup$ 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. $\endgroup$ – Alecos Papadopoulos Feb 23 '16 at 23:29
  • $\begingroup$ @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. $\endgroup$ – user877334 Feb 23 '16 at 23:47
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So the short answer is yes, you can generate some predictions based on your model. Whether it will generate good predictions is another question. Basically you are only using the spatial variation to estimate your model and no time variation. So your predictions would most likely be very poor unless the relationship is very stable over time. There's also no way of knowing which lagged values of the explanatory variable would be useful in prediction, unless there is some spatial structure that you can use (i.e. some regions are leading indicators for other regions)

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