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Consider two variables with levels over two time periods $\{y^i_t,x^i_t\},\{y^i_{t+1},x^i_{t+1}\}$.

For example, it could be profit and cost data of various firms over two quarters.

Suppose I take the differences of these levels over the two periods and produce new variables.

$DProfit_i=y^i_{t+1}-^iy_t$
$DCost_i=x^i_{t+1}-x^i_t$.

In other words, for each firm, I now have the difference in profit and cost for the two consecutive quarters. Here DProfit and DCost are meant to represent "D"elta.

I perform a linear regression with the following specification:

$$DProfit_i=\beta_0+\beta_iDCost_i+\varepsilon_i.$$

Is the following interpretation correct for the coefficients?:

Fix $i=$Apple Inc. Then $\hat\beta_{Apple Inc.}$ tells us for a unit increase in the change in cost between the 2 quarters result in $\hat\beta_{Apple Inc.}$ increase in the change in profit.

Is there a more sensible way of writing or interpreting the coefficients?

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Two things:

1) You should not assign a specific $i$ value to a regression coefficient. The same estimated coefficient applies to all $i$'s, so you should write it as $\beta_1$, not $\beta_i$.

2) An easier to read interpretation would be: "If a firm's cost increases by 1 unit between two consecutive periods, the firm's revenue is expected to increase by $\beta_1$ units, on average." If the sign of the estimated $\beta_1$ is negative, you can change "increase" to "decrease".

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