# Log linear model interpretation - % Contributions?

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%?

• $t = \{1,2\}$ 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.