Decomposition by covariates

Question

I have a fixed effects regression with the following specification.

$$Y_{it}=Cost_{it}+Year_t+Year_t*Cost_{it}+X_{it}+State_i+e_{it}$$

where $Y$ is a continuous outcome variable, Cost is the main variable of interest, Year = 1999, 2000, 2001, other control variables represented by $X$, and state fixed effects. I had a program change in 2000 - so, 1999 represents pre-intervention period.

What I want to see is the total change in $Y$ in the years 2000,2001 vs. 1999 and also the change in $Y$ in these two years that was contributed by Cost variable. Basically, I want to see the % of total change in Y in 2000 and 2001 that was contributed by Cost. I have read through some of the decomposition techniques but couldn't figure out how to do this one.

Appreciate your help.

Answer 1

Just got some random thoughts, since no context is informed. Why not regress the change in Y on Cost (or the change in Cost), in a cross-sectional regression?

To my knowledge, this kind of regression-based method can only answer:

• what's the change in Y as a function of Cost;
• the % of variation of change in Y can be explained by Cost (i.e., $R^2$).

If you "want to see the % of change in Y that was contributed by Cost", you may need to turn to other methods. So some valuable information would be useful to attract help from the general audience, e.g., what decomposition has been explored, or what's the standard way to do this decomposition in practice, and why those would not apply to this question. Otherwise only someone has done this decomposition can answer this question.

Also it's not clear for the definition of "the total change in Y in the years 2000,2001 vs. 1999".