I am analyzing ARIMA or multiple regression model. $Y= X_1 + X_2 + X_3 + X_4$

Can I have the **log on Y** and **square root on X1** and **square root on X2**, at the same time? That is,

$log Y = sqrt{X_1} + sqrt{X_2} + X_3 + X_4$

If yes, how can I interpret the results?

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#### Best Answer

Yes; you can do this and still achieve a valid model.

Interpretation, however, is complicated by this transformation. Your model then looks like

$$log(E(Y_i)) = beta_0+ beta_1sqrt{X_{i1}} + beta_2sqrt{X_{i2}} + beta_3X_{i3} + beta_4X_{i4}$$

This means that e.g. a doubling of $X_1$ is associated with an expected increase of $Y$ by a factor $exp(beta_1sqrt{2})$, all other factors kept equal. For a unit increase in $X_1$ there is no easy interpretation; that depends on the current value of $X_1$.

A unit increase of $X_3$ is associated with an expected increase of $Y$ by a factor $exp(beta_3)$. So, interpretation is easier without transforming the variables. However, if square root transformation improves the fit or is necessary in some other way, then it is mathematically perfectly correct.

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