Econometrics: How to Derive the OLS Estimators of a log-log model

Question

Consider the following two simple linear model specifications:

$log(y) = \beta_0 + \beta_1log(x) + u$ $(1)$

$log(y/x) = \alpha_0 + \alpha_1log(x) + v$ $(2)$

where y and x are two random variables for which we have a random sample size n and both u and v are error terms.

(a) Show that $\hat\alpha_1 = \hat\beta_1 - 1$ where both $\hat\alpha_1$ and $\hat\beta_1$ are the OLS estimates of the population parameters.

(b) Show that $se(\hat\beta_1) = se(\hat\alpha_1)$

This is what I have completed thus far but now I am having issues with Part (b) of this problem

$$ln(y) = \beta_0 + \beta_1ln(x) +u$$

$$ln(\frac{y}{x}) = ln(y) - ln(x) = \alpha_0 + \alpha_1ln(x) +v$$ $$\Rightarrow $$ $$ln(y) = \alpha_0 + \alpha_1 ln(x) +v \Rightarrow ln(y) = \alpha_0 +(x_1+1)ln(x)$$

And since we know that the ln(x) must equal we can conclude:

$$1 + \hat\alpha_1 = \hat\beta_1\Rightarrow \hat\alpha_1=\hat\beta_1 - 1$$

I will now work on Part (b) which I will add later on today hopefully. (Also, studying for an exam tomorrow) If anyone catches an error that be appreciated if you can bring it to my attention.

(b)

$$se(\hat\alpha_1)=\frac{\frac{\hat\alpha_1-\alpha_1}{sd(\hat\beta_1)}}{\sqrt(\frac{\hat\sigma^2}{\sigma^2})} = \frac{\hat\alpha_1-\alpha_1}{sd(\hat\alpha_1)} \sim N(0,1)$$

we know that:

$$tratio = \frac{\hat\alpha_1-\alpha_1}{se(\hat\alpha)}\sim t_{n-k-1}$$

If anyone can help me proceed with this question and put me on the right track that would be appreciated.

Answer 1

For (a) I would have thought $\log(y/x)= \log(y)-\log(x)$ made the answer almost immediate in terms of minimising the sum of the squares of the residuals

In the first model you want to find $\beta_0,\beta_1$ to minimise $\sum (\log(y_i)-\beta_0-\beta_1 \log(x_i))^2$ while in the second you want to find $\alpha_0,\alpha_1$ to minimise $\sum (\log(y_i/x_i)-\alpha_0-\alpha_1\log(x_i))^2$ which is $\sum (\log(y_i)-\alpha_0-(\alpha_1+1)\log(x_i))^2$.

So optimal $\hat\beta_0,\hat\beta_1$ in the first is equivalent to optimal $\hat\alpha_0=\hat\beta_0$ and $\hat\alpha_1+1=\hat\beta_1$ in the second