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If there is a linear regression model as follows:

$$y = \beta_0 + \beta_1x_1 + \beta_2x_2 + \beta_3x_3 + u$$

and we want to estimate the ratio of the slope coefficients:

$$\theta = \frac{\beta_1}{\beta_2}$$

Would the following estimator be biased for $\theta$?

$$\hat{\theta} = \frac{\hat\beta_1}{\hat\beta_2}$$

We know that given that the usual assumptions for a linear regression model are satisfied, then the slope estimators should be unbiased. Therefore, if we take the ratio of two unbiased estimators, would the resulting estimator also be unbiased?

Robin Liao
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    No, see https://en.wikipedia.org/wiki/Jensen%27s_inequality, which implies that $E(1/X)\neq 1/E(X)$. – Christoph Hanck Jan 31 '19 at 16:36
  • @Xi'an How are the slope estimates not independent? I can't see how they would be positively or negatively correlated with each other. – Robin Liao Jan 31 '19 at 23:44
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    It is well-known (you'll also find plenty of hits on this site) that $Var(\hat\beta)=\sigma^2(X'X)^{-1}$, which, in general, is not a diagonal matrix. – Christoph Hanck Feb 01 '19 at 05:36

1 Answers1

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No, it will not be unbiased (unless the estimator of the denominator have zero variance.) And it will not help if the numerator and denominator are independent. In general, if $\hat{\theta}$ is an unbiased estimator of $\theta$ and $g$ is some nonlinear function, it would be a rare case that $g(\hat{\theta})$ is unbiased estimator for $g(\theta)$.

There is more information in this related post: Test Statistic for a ratio of regression coefficients?.

kjetil b halvorsen
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