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I am running linear regression in an area where there is no theory that I am aware of to determine if various variables interact. I am predicting income at closure for customers of a social program. I am using various measures of spending and control variables such as age gender etc to analyze this.

One possibility would to specify interaction among all control variables and then drop out the ones that are not significant. But I realize there are questions about dropping out variables based just on p values (for what its worth I have the entire population so I am not sure p values matter - a disputed point I know). Any suggestions on how to deal with this?

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Predicting unknown observations implies you have not observed the entire population (of observable events). Population does not only refer to physical experimental units. The population of coin flips of even a single coin, for example, is infinite.

As for your question, there are various ways you can check for interactions. If you only have a limited number of variables, you can simply compare different models to see whether those with certain interactions outperform others and use criteria like AIC or BIC, or even better: an estimate of out-of-sample performance, such as cross-validated loss, or if your data is large enough, an independent test set.

You could also try estimating a conditional independence network between your predictors to screen for potential interactions. The graphical lasso (or ridge) comes to mind.

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    $\begingroup$ thank you. If it matters I actually do have the entire population of interest (we are analyzing the impact of various factors on income gain for our customers, we have all the customers rather than a sample). $\endgroup$ – user54285 Apr 30 at 23:44
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    $\begingroup$ Then why are you predicting income at closure? If you already know every income at closure, this seems futile. $\endgroup$ – Frans Rodenburg May 8 at 1:19

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