Could someone please help me with this exercise and tell me if I am on the right track?
Assume that based on a data set with a large number of observations of an independent variable $x$ and a dependent variable $y$ we have estimated the coefficients $\alpha$ and $\beta$ in $\ln(y) = \alpha + \beta \cdot x$.
- Compute the predicted change in $y$ (denoted $\Delta y$) that results from increasing $x$ by $\Delta x$ (from $x$ to $x + \Delta x$).
I thought maybe to put $e$ in the equation. $e(\ln(y))=e^\alpha + \beta (x+\Delta x)$ so $e(\ln)$ cancel and I have $y=e^\alpha + \beta(x+\Delta x)$ where I have substituted $x+\Delta x$ into $x$. Would this equation show the change in $y$?
- When is $\Delta y$ positive/negative/zero?
If it is an exponential function, then it just approaches zero no?
Help appreciated! Thank you!