Question on dummy variables

Dear all,

I have a question I met in my project.

I have a panel data, including 20 cities and there are 18 years OBS (x and y) for each city. I want to establish a equation with dummy variables for control time-invariant city-specific characteristics and control for time trend.

Thus my equation is:

Y = a1 + a2*X + a3*D_city + a4*D_year (1)

Where, D_city and D_year are dummiy variables for controling variant of city and time. So I have two dummy variables.

But If I do that, is it correct that I actually assumed a common time trend for each city? (question 1)

It maybe not correct since each city has their own time trend features. So I want to give a city-specific time trend.

Thus I have another function form:

Y = a1 + a2*X + a3*D_city + a4*D_city*year (2)

Others are the same with previous function except the last term which I multiple D_city with value of year.

So the equ(2) could change to

Y = a1 + a2*X + D_city(a3 + a4*year) (3)

In the form, can I say I considered city-specific time trend? (question 2)

Finally, I tried the equ(2) in R by inputting:

D_city<-as.factor(data$city)

result<-lm(Y~X + D_city + D_city*year,data)

But I don't think R gave me the result I need. How can I do this? (question 3)

Thanks!

Regards,

Tianyi