Interpreting OLS Cross-Sectional Macro Data

Question

I have the following model: $y=a_0+a_1(x_1/z_1)+a_2(x_2)+e $, where

$y$ = the average log of variable y,

$x_1$ = the average ratio of $ x_1/z_1 $,

$x_2$ = is the averae log of the variable $x_2$.

The values of the coefficients are :

$a_1=0.012$,

$a_2=0.049$.

The data are cross-sectional macroeconomic variables.

I wonder how can I interpret these coefficients since i am confused from some similar papers that i have seen they inerpret it with different ways. I mean in one paper says:

" 1% increase of x1 increases y by 1.2% but another paper says 0.012%" and the same with the x2 which is log " the first says 4.9% and the second 0.049%"

which of them are correct?

Answer 1

It appears you have estimated a relation

$$ \ln Y = a_0 + a_1(X_1/Z_1) + a_2 \ln X_2$$

so

$$a_2 = \frac {\partial \ln Y}{\partial \ln X_2}$$

By treating partial differentials as quantities (I won't tell if you don't)

We have

$$\frac {\partial \ln Y}{\partial \ln X_2} = \frac {\partial \ln Y}{\partial \ln X_2}\cdot \frac {1}{\partial X_2/\partial X_2} = \frac {\partial \ln Y}{\partial X_2}\cdot \frac {1}{\partial \ln X_2/\partial X_2}$$

$$=\frac {1}{Y}\frac {\partial Y}{\partial X_2} \cdot \frac {1}{1/X_2} = \frac {\partial Y}{\partial X_2} \cdot \frac {X_2}{Y} = \frac {\partial Y/\partial X_2}{Y/X_2}$$

The last expression is the definition of point elasticity (the ratio of "marginal over the average"), and for small changes in $X_2$, it approximates the more general definition of elasticity expressed as

$$\frac {\text {% Δ in Y}}{\text{% Δ in}\, X_2}$$

In your case therefore

$$\frac {\text {% Δ in Y}}{\text{% Δ in}\, X_2} = a_2 = 0.049$$

then, if $\text{% Δ in}\, X_2 =0.01$ we have

$$\text {% Δ in Y} = 0.049 \times 0.01 = 0.00049 = 0.049\%$$

So the second paper is correct (the change in $Y$ is ~"five basis points", as a banker would say).