Modifying regression model equations to perform OLS?

Question

I'm not really sure why I couldn't take any of these models and perform OLS on them. Would appreciate help in understanding/finding the answer

Answer 1

Recall that one of the assumptions (actually, a constraint) of the OLS method is a linearity over the parametric set, i.e., check whether the first derivative w.r.t $\beta$ depends on the unknown $\beta$.

1) Simple linear regression. $\frac{\partial }{\partial \beta} \mathbb{E}[Y|X=x] = \ln (x) $. You can denote $\ln(x) = x^*$ if you find it more convenient.

2) Try $1/Y_i = \beta_1 + a \frac{1}{X_i}\to Y_i^* = \beta_0^* + \beta_1^*X_i^*$.

3) $\ln Y_i = \ln(a) + \beta_1\ln(X_i)$

4)Just denote $X_i^2 = X_i^*$.

5) $\ln(Y_i) = \ln (a) + \beta_1X_i. $