Consider the following population regression model:
$$y_{i} = \beta _{1} + \beta_{2}x_{i} + \epsilon _{i},$$
where $i=1,...,n$. Assume $\epsilon \sim iid$, with the pdf in equation: $f(\epsilon ) = \alpha \epsilon$ for $0\leq \epsilon \leq 1$.
For $n=4$, we have $f(\epsilon _{1},\epsilon _{2},\epsilon _{3},\epsilon _{4}) = f_{1}(\epsilon _{1}) f_{2}(\epsilon _{2}) f_{3}(\epsilon _{3}) f_{4}(\epsilon _{4}) = \alpha \prod _{i=1}^{4} \epsilon _{i}$. This is because of no autocorrelation in errors.
Now we substitute $f(\epsilon _{1},\epsilon _{2},\epsilon _{3},\epsilon _{4})$ in the regression model and obtain
$$y_{i} = \beta _{1} + \beta_{2}x_{i}+\prod _{i=1}^{4} \epsilon _{i}$$.
Taking the natural logs, we obtain
$$ln(y_{i}) = ln(\beta _{1} + \beta_{2}x_{i}+\prod _{i=1}^{4} \epsilon _{i})$$.
What is a common name for the resulting function?
I am guessing it is the maximum log likelihood function. Am I correct?
No, you are wrong it have nothing common with log likelihood function. Moreover the model has defects, because you do not guarantee that $y_i>0$ otherwise you cannot make a log.