What are the regression equation of the fixed effect estimator and the regression equation of the first difference estimator?

[Lucas Bordure]

I wrote that the regression equation of the fixed effect estimator was:

And i wrote that the regression equation of the first difference estimator was:

I think I have errors in subscripts.

Thank you for your help.

[Clyde Schechter]

In both equations, you omitted i = 1, 2, ..., N.

In addition, the error term in the fixed effects equation should not be u_it. It should be u_i + e_it. Here u_i is the panel level fixed effect--by definition, it does not vary with time. e_it is the observation level error.

And in the first-difference estimator, there should be no term involving u at all: because it is the same for all observations in the same panel, it cancels out when the first difference is calculated. Instead, the error term should be Δe_it. Also, in the first-difference estimator, depending on how you label them, t should either run from 2 through T, or from 1 through T-1, not 1 through T.

[Lucas Bordure]

Thank you very much for your answer. Could you right me these formula on Word and capture the picture of the 2 formulas? I don't want make an error of interpretation of your explanation.

[Clyde Schechter]

Sorry, no. It's very slow to do that sort of thing in Word, and I don't have time for it today. If you want to re-write the equations and then post back what you've done to confirm whether they look correct, I'll respond to that.

[Lucas Bordure]

Like this?

[Jeff Wooldridge]

I'll respectfully disagree with Clyde. I don't see why you should be tied to Stata's notation. It looks like your original equations were correct -- and might even be the notation from one of my books. There's reason you must use u(i) for the heterogenenity and e(i,t) for the idiosyncratic error. I've always used u(i,t) as the idiosyncratic error and c(i) [sometimes a(i)] for the unobserved effect. I think Clyde isn't familiar with the double dot notation, which indicates removing time averages. That's why there is no error term that depends only on i. Remember, the demeaned equation is an estimating equation, as is the FD equation.

The only problem with the original formulations -- which Clyde pointed out -- is you need to start at t = 2 in the FD equation. Your latest equation for the time-demeaned data is now incorrect, as you still have both a u term and an e term. I'd go back to the original ones.

[Clyde Schechter]

Jeff is right--I am not familiar with the double-dot notation. Thanks to him for introducing me to it. As for the meaning of u as the unobserved fixed effect and e as the idiosyncratic error, I was referring to the notation that I am familiar with, which is also that used by StataCorp in its manuals, and is the one that is conventional in my discipline, epidemiology. But with the double-dot notation explained, then Jeff is correct.

Within the system of notation that I normally use, your revised first differences equation in #5 is correct. But the fixed-effects equation (which shouldn't have the double-dots in this system) is still incorrect because it has u_it where it should have u_i.

FWIW, Greene's textbook on econometric analysis uses α_i for the unobserved fixed effect, and Є_it for the idiosyncratic error, which is structurally similar to the scheme I use, but with different letters (and from the Greek alphabet).

Just be careful about one thing. If you are going to use the notation where you have u_it as the idosyncratic error, bear in mind that that isn't what Stata is talking about when it gives you output for sigma_u, the fraction of variance due to u_i, nor a message about an F test that all u_i = 0. Don't get them confused.

[Lucas Bordure]

Thank your for your replies.

So the fixed-effect estimating equation is:

And the first-difference estimating equation is:

With

the idiosynchratic error and

also the idiosynchratic error?

[Joro Kolev]

Yes but with Uit double dot the fixed effects transformed idiosyncratic error, and dUit the first differenced idiosyncratic error.

And again when you map these equations to Stata output, do not get confused like Clyde told you: this is your and Professor Wooldridge's notation.

Also Clyde painted a very optimistic picture of a "consistent" Stata notation on which we can rely. I do not see it quite this way.

The notation Stata uses is not consistent at all.

In the methods and formulas of -xtreg- the time constant unit specific effect is called with the Greek letter nu, and the idiosyncratic error is called with the Greek letter epsilon.

In the regression output, nu becomes the Latin letter u, and epsilon becomes the Latin letter e.

While this is not totally crazy mapping, I do not quite see why whoever typed up the formulas did not specify the structural model in terms of u and e.

Thank your for your replies.

So the fixed-effect estimating equation is:
[ATTACH=CONFIG]n1616104[/ATTACH]
And the first-difference estimating equation is:
[ATTACH=CONFIG]n1616105[/ATTACH]

With [ATTACH=CONFIG]n1616106[/ATTACH] the idiosynchratic error and [ATTACH=CONFIG]n1616107[/ATTACH] also the idiosynchratic error?

[Jeff Wooldridge]

I actually like the convention that Stata uses in reporting results for xtmixed because it is neutral with respect to notation used in writing equations. It simply shows the variance estimates for the different identifiers that the user specifies.

Gary Chamberlain was using c(i) + u(i,t) as the notation for the composite error term back in 1980. That's where I got it from.