lm() and glm() equivalence for log-transformed response variable [duplicate]

Model b matches the lm() model. Both of those assume that log(y) has a Gaussian distribution with mean a0 + a1*x1 + a2*x2.

Model a assumes that y has a Gaussian distribution, with mean exp(a0 + a1*x1 + a2*x2).

In both cases a0, a1, a2 are the coefficients you are estimating, and the Gaussian variance is constant.

B is equivalent; A is not.

Let’s write out the math.

A $$ \log(\mathbb E[y]) = \beta_0+\beta_1x_1+\beta_2x_2\\\iff \mathbb E[y]=e^{\beta_0+\beta_1x_1+\beta_2x_2} $$

B $$ \mathbb E[\log(y)]=\beta_0+\beta_1x_1+\beta_2x_2 $$

B certainly looks like it is equivalent, especially considering that the estimation specified by your code will be minimization of square loss (same as in OLS, equivalent to Gaussian maximum likelihood estimation).

By the strong form of Jensen’s inequality, $\log(\mathbb E[y])<\mathbb E[\log(y)]$ with strict inequality, so the two are not equivalent to each other, ruling out the possibility that A is also equivalent to the lm specification.

Overall, B is equivalent to the lm specification while A is not.