Y= Intercept+ Beta1X1+ Beta2X2+ Beta3(X1X2)**

I want to determine the economic significance of Beta3 when Y, X1, and X2 are continuous variables, preferably in terms of standard deviation units. Is there any common source or article which I can refer to regarding the topic. Please help.

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    $\begingroup$ I’m voting to close this question because it belongs on the economics or quant Stack. $\endgroup$ – Dave Jun 25 at 12:07

This isn’t quite economic significance, but if this is about what you’re looking for, then we can keep the question on Cross Validated.

When you fit an OLS regression, there’s often an assumption that the response variable is $N(x_i^T\beta, \sigma^2)$. That is, you’re sliding a bell curve up and down the regression line, and that bell curve has constant variance. You don’t need these assumptions, but they simplify life, and if you’re not making those assumptions, you’d probably know it.

(Whether or not those are good assumptions is a different story and warrants a separate question that I’m sure has been addressed if you do a search.)

We can get an unbiased estimator for $\sigma^2$ by $\hat{\sigma}^2_{OLS}=\dfrac{\sum_{i=1}^n(y_i-\hat{y}_i)^2}{n-p}$.

The intercept counts toward $p$, so for you, $p=4$.

By Jensen’s inequality, the square root of that unbiased estimator is biased, but people tend to be content to use that as an estimate of standard deviation. (Biased estimators aren’t bad and can be quite useful.)

So we have the effect size from $\beta_3$ and the standard deviation from $\sqrt{\hat{\sigma}^2_{OLS}}$. Divide the effect size by the standard deviation.

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