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I have a regression model that looks like this:

$$y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \beta_3 X_1 X_2 + \beta_4 X_1X_2$$

What is the correct name for this regression: multiple linear regression? polynomial regression? or something else?

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    $\begingroup$ Regular old linear regression. $\endgroup$ Dec 7, 2021 at 8:05
  • $\begingroup$ b_3*X_1*X_2 + b_4*X_1*X_2 seems redundant, is that a typo? $\endgroup$ Dec 7, 2021 at 8:08
  • $\begingroup$ Maybe to add '…with interactions'. $\endgroup$
    – deschen
    Dec 7, 2021 at 8:10
  • $\begingroup$ Yes, of course, it's a typo. I just want to show that there are interactions in the equation. $\endgroup$
    – Helena Star
    Dec 7, 2021 at 8:40
  • $\begingroup$ Aside from "interaction", you can also call b_3*X_1*X_2 a feature cross $\endgroup$ Dec 7, 2021 at 8:44

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That is a multiple linear regression with two interaction terms, probalbly with a typo, because there is unnecessary redundancy of the b_3 and the b_4 term which seem to be identical. Thus the model is probably underidentified, unless you use special regression techniques (regularized like Bayesian with non-flat priors or ridge regression etc could probably compute that, though I cannot imagine a situation where that would be useful).

If this was to be put in R formula type you could denote it as

y ~ X_1 * X_2

because in the formula notation, the * implies the single terms and the intercept b automatically.

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