Problem
Suppose that the logarithm of $Y$ is regressed on the logarithm of $X$, the fitted regression being:
$$log \hat{Y} = \hat{\beta_1} + \hat{\beta_2} log X$$ Suppose $X^* = \mu X$, where $\mu$ is a constant, and suppose that $ log Y$ is regressed on log $X^*$ . Determine how the regression coefficients are related to those of the original regression. Determine also how the t statistic for $\hat{\beta_2}$ and $R^2$ for the equation are related to those in the original regression.
My ideas:
We can substitute the $X^* = \mu X$ into the original regression and get that $log \hat{Y} = \hat{\beta_1} + \hat{\beta_2}log \mu + \hat{\beta_2}log X$. As $\mu$ is a constant the regression we stay the same, with exception of new intercept $\hat{\beta} := \hat{\beta_1} + \hat{\beta_2}log \mu $.
Also, as t-statistics is $t = \frac{\hat{\beta_2} - \beta_2}{s.e(\hat{\beta_2})}$, it seems that $t_{\text{stat}}$ will increase in $\mu$ times as $ s.e(\hat{\beta_2})$ decrease in $\mu$ times.
Can it somehow help me?
You are almost right. The only thing, yo should've substituted $X = \frac{X^*}{\mu}$. As you correctly mentioned the slope coefficient stays the same. However, new intercept is $\hat{\beta_1} - \hat{\beta_2}log\mu$.
You may check that resiaduals for the original and new models are the same. Hence, RSS and $R^2$ stay unchanged.