Help: a fixed effect first order autoregressive model

[Jessica Xie]

Dear all,
I am replicating a paper, and need use panel data(firm, year) to estimate a fixed-effect first-order autoregressive model as follows:
Y_it = a_i + b_t + c*Y_it-1 + d*M_it-1+e_it

Then, I need use the time-series residuals for every firm and every firm-year residuals from the model estimation for future calculations.
My questions are: 1) what steps and which stata code can be used to estimate the fixed-effect first-order autogressive model?
I am thinking use " xtest gvkey year; xtreg Y l1.Y l1.M, fe" ?

2) how can I get the time-series residuals for every firm and every firm-year residuals after the model estimation?

Your guidance and assistance is greatly appreciated!
Best,
Jessica


BTW: my data looks like

gvkey	year	Y	M
1	2000	16	14
1	2001	17	15
1	2002	18	16
1	2003	22	20
1	2004	20	18
1	2005	10	16
1	2006	22	20
2	2000	16	14
2	2001	17	15
2	2002	18	16
2	2003	22	20
2	2004	20	18
2	2005	10	16
2	2006	22	20

[Clyde Schechter]

I assume "xtest" is a typo for "xtset." Following that -xtreg Y l1.Y l1.M, fe- is not quite what you want, because it does not contain a term corresponding to b_t in your equation. Also, while it implicitly captures the a_i, it does so implicitly and does not give any estimates for those terms. Often they are not considered of interest.

So I think you need -xtreg Y i.year l1.Y l1.M, fe- so that you incorporate the time effects. ( If you also want the a_i terms explicitly, then it would be -regress Y i.gvkey i.year l1.Y l1.M-. That will give you the same results as the -xtreg- version as far as all other coefficients are concerned, the only difference will be the explicit appearance of firm effects.)

Finally, after estimation with -xtreg-, you can get residuals with -predict firm_level, u- and firim-year level residuals with -predict observation_level, ue-.

[Jessica Xie]

Thank you for your prompt assistance, Clyde!
Thank you for your suggestions. I understand it now.

But an issue is that when I tried to get a_i terms by using code "-regress Y i.gvkey i.year l1.Y l1.M-", the stata shows error since there are too many dummy variables for gvkey (representing firm). I have more than 5000+ firms in each year in the sample. Please advice. Thanks!

Best!

[Clyde Schechter]

So use -xtreg Y i.year l1.Y l1.M, fe-. Then -predict firm_level_effect, u- will give you the firm-level intercepts.