Dear members,

I am a student new to STATA and may need your help on the below question.

I have the level regression as below:
𝑒𝑗𝑑=𝑑𝑑 + 𝛿𝑗 + Σ𝑑β‰₯1400𝛼𝑑 WE𝑗 𝑑𝑑 + Σ𝑑β‰₯1400𝛽𝑑 𝑃𝐴𝑇𝑗 𝑑𝑑 +πœ€π‘—π‘‘

(u is log(population); WE is dummy variable if the city is in Western Europe; PAT is dummy variable if the city is involved in Atlantic trade; 𝑑𝑑 denotes time effects and 𝛿𝑗 denotes city effects)

and I am looking to transfer this to first difference regression, I wrote the equation as below (not sure if this is correct)
𝑒𝑗𝑑 βˆ’π‘’π‘—π‘‘βˆ’1 =△𝑑𝑑 +𝛼𝑑 WE𝑗 𝑑𝑑 + 𝛽𝑑 𝑃𝐴𝑇𝑗 𝑑𝑑 +πœ€π‘—π‘‘

And I tried to estimate the difference regression, and I thought that the point estimates of coefficients should be the same as those in the level regression.
I used the code:

gen yearn = 1 if year == 1300
replace yearn = 2 if year == 1400
replace yearn = 3 if year == 1500
replace yearn = 4 if year == 1600
replace yearn = 5 if year == 1700
replace yearn = 6 if year == 1750
replace yearn = 7 if year == 1800
replace yearn = 8 if year == 1850
bysort city: gen du=u-u[yearn-1]
gen du1 = du if year == 1850
reg du1 WE_y8 PAT_y8

However, when I compared the coefficients with those in level regression they are not the same. Please give me some advices on my equation and estimates.

The data sample are below
city year PAT u WE
aachen 1300 0 3.044523 1
aachen 1400 0 2.302585 1
aachen 1500 0 2.890372 1
aachen 1600 0 3.135494 1
aachen 1700 0 2.70805 1
aachen 1750 0 3.178054 1
aachen 1800 0 3.178054 1
aachen 1850 0 3.951244 1
aalst 1300 0 1.386294 1
aalst 1400 0 1.386294 1
aalst 1500 0 1.609438 1
aalst 1600 0 1.94591 1
aalst 1700 0 2.079442 1
aalst 1750 0 2.079442 1
aalst 1800 0 2.397895 1
aalst 1850 0 2.833213 1
amsterdam 1300 1 0 1
amsterdam 1400 1 1.098612 1
amsterdam 1500 1 2.70805 1
amsterdam 1600 1 3.988984 1
amsterdam 1700 1 5.298317 1
amsterdam 1750 1 5.347107 1
amsterdam 1800 1 5.379897 1
amsterdam 1850 1 5.416101 1
athens 1300 0 3.218876 0
athens 1400 0 3.555348 0
athens 1500 0 2.302585 0
athens 1600 0 3.496508 0
athens 1700 0 2.564949 0
athens 1750 0 2.302585 0
athens 1800 0 2.484907 0
athens 1850 0 3.401197 0
Thank you very much !!!