Log linear model interpretation - % Contributions?

Question

I know that for log-lin models the interpretation for the coefficiente is this one, that is: Coefficients⋅100 have a semi-elasticity interpretation: for a 1 unit change in x, you get b*100% change in y.

The questions is: Could it also be interpreted as a % Contribution of variable x over y? For example,

t: 1,2.
Xt = TV Grps aired during period t.
Yt = \$ sales for a product during t.

x1 = 50; x2 = 20; y1 = 100; y2 = 70... Model => ln(yt) = a + b * xt + error

So, being b = 0.001 for t = 1--> 50 * 0.0001 *100 = 5% and for t = 2--> 20 * 0.0001 *100 = 2%

Makes sense to say that television contribution to \$ sales in the first period was 5% while in the next was 2%?

Answer 1

No. I mean, you have the following setup:

• $t = \{1,2\}$ i.e. two observations;
• $X_{t} = \{50, 20\}$ TV Grps aired in $t$;
• $Y_{t} = \{100, 70\}$ sales (\$) for a product in $t$.

You build the following log-linear model: $$ ln(Y_{t}) = \beta_{0} + \beta_{1} X_{t} + \epsilon, $$

where $\beta_{0}$ is the intercept, $\beta_{1}$ is the slope, and $\epsilon$ is an error on which you make some assumptions.

Now, if you run OLS to estimate the coefficients of the model, you find that $\hat{\beta}_{0} = 4.01$ and $\hat{\beta}_{1} = 0.012$, so that you can say that:

for a 1 unit change in $X$, you get $1.2\%$ change in $Y$

The $\%$ change in $Y$ due to the variable $X$ is fixed once you have estimated the coefficient $\beta_{1}$. Of course, since $\hat{\beta}_{1} > 0$, to a greater number of TV Grps aired will correspond higher sales.