Q:

Interpreting coefficients of first differences of logarithms

My problem is interpreting coefficients of such time series model:
\begin{equation}
\ln Y_t - \ln Y_{t-1} =b_1 \cdot \left(X_{t}-X_{t-1}\right)+b_2 \cdot Z_t.\end{equation}
I don't know how to interpret coefficients $b_1$ and $b_2$.
Hope that someone can help.

A:

For $small$ changes, you can interpret logged differences as percentage changes after multiplying by 100.
For example, $y_t=9$ and $y_{t-1}=8$. Then $\ln 9 - \ln 8=.118$ or 11.8%, which is the logarithmic approximation to the actual 12.5% increase. Note that I had to multiply by 100 here. For $y_t=9$ and $y_{t-1}=8.5$ the approximation will be much better ($5.9\% \approx 5.7\%$).
Usually, a coefficient tells you the effect on $y$ of a one unit change in that explanatory variable, holding other variables constant. A one unit change in $\Delta \ln x$ corresponds to a 100% change (using the approximation above, which is terrible since this is not a small change). This means that $b_1$ tells you the percentage change in $y$ associated with a 1% increase in x.
But your $x$ is not logged, so the coefficient needs to be interpreted differently. When $x$ grows by one unit, you get $100 \cdot b_1\%$ more $y$.
Moreover, $100 \cdot b_2$ tells you the percentage change in $y$ associated with a 1 unit increase in $z$.

