Interpreting log coefficients in differences model (time series/panel)

I am struggling with the interpretation of coefficients in
$$\delta(\log y)=\alpha+\beta\delta(\log x)$$

Usually in a $$y=\alpha+\beta x$$ model, $$\beta$$ represents change in $$y$$ corresponding to unit change in $$x$$, i.e. a $$\beta=0.8$$ says for an increase of 1 in $$x$$, we expect to see a corresponding change of 0.8 in $$y$$.

Logs are often interpreted as percent.

$$\log y=\alpha+\beta x$$
$$\beta=0.8 \Rightarrow$$ For an increase of 1 in $$x$$ we expect to see a corresponding change of 80% in $$y$$.

$$y=\alpha+\beta \log x$$
$$\beta=0.8 \Rightarrow$$ For an increase of 100% in $$x$$ we expect to see a corresponding change of 0.8 in $$y$$.

$$\log y=\alpha+\beta \log x$$
$$\beta=0.8 \Rightarrow$$ For an increase of 10% in $$x$$ we expect to see a corresponding change of 8% in $$y$$.

But I struggle with interpreting coefficients in time series/panel data when differences are being compared, e.g.
$$y=\alpha+\beta\delta(x)$$
$$\delta(y)=\alpha+\beta x$$
$$\delta(y)=\alpha+\beta\delta(x)$$

The reason for my confusion is that in normal (level) circumstances, we already interpret $$\beta$$ as change. If $$LHS$$ is $$\delta y$$, i.e. change in $$y$$, then what are we talking about? Change in change? How do you say that in English?

And, of course, as mentioned in the title of the post, the log differences models completely bamboozle me.
I am struggling with the interpretation of coefficients in
$$\delta(\log y)=\alpha+\beta\delta(\log x)$$
So $$\beta$$ means percent change in change? ::facepalm::

• "Change in slope" might be better understood that "change in change". – James Phillips Mar 18 at 20:13