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I was studying the linear regression in statics. When learning multiple linear regression, the textbook pointed out that $Y_i=\beta_0\exp(\beta_1X_i)+\epsilon_i $ was a nonlinear regression model. (I found some webpages about it online: https://onlinecourses.science.psu.edu/stat501/node/372 .)

I got confused by the part that, they used a $\log$ transformation to obtain the solution where $\log(Y)=\log(\beta_0)+\log(\beta_1X_i)$ like in the previous webpage and http://www.real-statistics.com/regression/exponential-regression-models/exponential-regression/ and Linear, quadratic and exponential regression .

It looked to me, and as they had pointed out, that, after the $\log$ transformation, the newly obtained variables was clearly a linear regression. Further, the term $\log(\epsilon_i)$ was missing from the expression.

My question was that:

  1. What happened to the term of $\log(\epsilon_i)$? Why we could simply dismissed it?

  2. If $\epsilon_i$ still followed a $N(0,\sigma^2) $ distribution, what's the distribution for $\log(\epsilon_i)$?

  3. Further, $\epsilon_i$ could be negative, and although we could got a value through complex analysis, it would still came with a imaginary part. What happened then?

(We had only learnt linear regression so I wan't so sure if there was some context missing.)

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First of all, don't forget that all these model are mere approximations, so don't worry so much about some logical leaps. One possible way to resolve this problem is by rewriting the original model as
$$ y(x_i) = \beta_0 e^{\beta_1 x_i}(1+\epsilon_i), $$ where $\epsilon_i \sim \text{LogNormal}(0, \sigma^2),$ hence when you take the log you have $$ \ln y_i = \ln \beta_0 + \beta_1 x_i + \ln(1+\epsilon_i), $$ as such $\ln ( 1 + \epsilon_i) \sim N (1, \sigma^2) $ or $\ln ( 1+ \epsilon_i) = 1+ \sigma \xi_i$, where $\xi_i \sim N(0,1)$. So, denote $ \ln \beta_0 + 1 = \gamma_0 $ and $\ln y_i = y_i^*$, you have a linear model $$ y_i ^* =\gamma_0 + \beta_1 x_i + \sigma \xi_i. $$

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