I am building a price elasticity model using linear regression:
log(demand) ~ 1 + log(price) + ...
Does it make sense to use L1 and/or L2 regularization to prevent over-fitting, or would that lead to inaccurate elasticity estimates?
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$\begingroup$ The issue comes particularly if some of your other variables are correlated with price, which could lead to very different elasticity estimates. Regularization is often justified by better out-of-sample forecasting rather than its impact on inference $\endgroup$– HenryCommented Aug 23, 2022 at 12:51
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1 Answer
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Statistically, it is completely reasonable. The OLS $\hat\beta_{ols}=(X^TX)^{-1}X^Ty$ estimator is just one way of guessing the values of $\beta$. $\hat\beta_{ridge,\lambda}$ and $\hat\beta_{LASSO,\lambda}$ are other ways of guessing the values of $\beta$.
All three of these have desirable properties. For instance, the OLS estimator is unbiased and has the minimum variance among all linear unbiased estimators (assuming the Gauss-Markov theorem conditions apply). The ridge and LASSO estimators allow for some bias but potentially reduce the variance enough to make up for being biased.
All three have something going for them and can, in some sense, be defended. If you can show why an alternative to the usual OLS estimator is desirable in your situation, go for it! OLS isn’t magic, just the common approach.
Biased estimators can be reasonable. For instance, logistic regression, estimated the usual way, results in biased coefficients, and the usual $S=\sqrt{\dfrac{1}{n-1}\sum_{i=1}^n\left(X_i-\bar X\right)^2}$ estimator of sample standard deviation is biased.
