Scatterplot of change in Y against change in X with a fitted line

[Mohsin Javed]

I am working with a replication package and came across the following which I cannot seem to understand (for the ease of convenience I will refer to the dependent variable as Y and explanatory variable as X):

The paper examines the relationship between Y and X using the following equation:

Y_st = B₀ + B₁X_st + A_s + A_t + e_st

where s refers to states, t refers to years, A_s and A_t are state and time fixed effects, respectively. The data is a balanced panel with 51 states and 5 years.

In one of the graphs, the authors show a simple scatter plot of change in Y against change in X along with a fitted line. The adopted method is what I find confusing i.e. the paper plots this graph by:

1. regressing Y on state and year fixed effects

reg Y st1-st51 y1-y5, robust cluster(state)
predict dy, residual

2. regressing X on state and year fixed effects

reg X st1-st51 y1-y5, robust cluster(state)
predict dx, residual

3. Finally, regressing dy on dx

reg dy dx, robust cluster(state)
predict line1

4. The plot is produced using:

scatter dy dx || line line1 dx, sort

I tried reproducing the graph as follows but it appears they are not the same:

egen id=group(state)
egen t=group(year)
tsset id t

gen dY = Y  - l.Y
gen dX = X - l.X

reg dY dX, robust cluster(state)
twoway (scatter dY dX) ///
title("Differences") ///
xtitle("Change in X") ytitle("Change in Y") legend(off) ///
(lfit dY dX, lwidth(medthick))

Am I missing anything?