Solved – Interpreting coefficients of a dlog-dlog regression

I am running a dlog-dlog (difference of logarithm*) regression and I want to convert the coefficients into marginal effects. I know that it's different from a log-log regression, in which the coefficients directly give us the elasticities.

How can we interpret the coefficients from a dlog-dlog regression?

* For example dlog (Y)= a + b dlog(X)+ error term.

Since you have differences, this means that the data is time series and we can write

$$Y_t=Y_0+sum_{s=1}^tDelta Y_s$$

So if the true model is

$$Delta Y_t=alpha+beta Delta X_t$$

we have

$$Y_t=Y_0+sum_{s=1}^t(alpha+beta Delta X_t)=Y_0-beta X_0+alpha t+beta X_t$$

So you can say that interpretation remains the same as in model with levels.

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