interpretation of year dummies in first-difference (FD) estimation

[ana cociorva]

Hi all!

So I have a panel-data setting for which previous research has been trying to interpret year indicators (which, for this dataset, are very correlated with a linear time trend) as a measure of conservatism (more specifically, claiming that for a company with exactly the same credit risk, the rating today would be lower than 20 years ago, for reasons unrelated to company financials - it doesn't sound plausible, right?). I'm trying to formally show that one cannot attribute any kind of causality to this trending year indicators, since many financial and macroeconomic variables, for example, exhibit the same time trend, and hence this is some kind of spurious regression, where the common trend captures the variation in both ratings and many of the explanatory variables (and shows up in the year indicator coefficients).

Hence what i am trying to do is show that, once removing this secular trend, the year indicators would mostly reflect cyclical influences, which is what the theory and previous empirical evidence on ratings would suggest, and makes more sense from an economic point of view. Therefore I am trying to find ways to show this in an econometrically correct way.

One idea I've been thinking about was to use first-differencing and show that the differenced year indicators have a cyclical pattern (or are completely insignificant). I've been reading other forum posts and the relevant parts in Wooldridge (2012), but i still couldn't completely understand how to interpret the differenced coefficients. One particular part caught my attention (page 469): " Unless the time intercepts in the original model (13:28) are of direct interest, it is better to estimate the FD equation with an intercept and a single-period dummy, usually for the third period (this example was for only three periods)." But what happens if we ARE interested in the time intercepts?

Ideally, I would like a FD model with the first year as a base for the time dummies, since I need to make it comparable to the levels model.

I have found this suggestion for the FD model (1):

xi: reg D.(y x1 ... xK z1 ... zK i.year), cluster(countryid)

However, when I run this for my model, the first and last year are being dropped, therefore it doesn't seem to have the same reference as the level regression (2):

 xtreg y x1 ... xK z1 ... zK i.year, fe cluster(countryid)

Again, my ultimate goal is to show that the coefficients for i.year in (1) are extremely different from (2), but for this I need to have the same reference year. In model (2), the base year is the first one. I do have an intercept in both models.

On another hand, I first tried the FD model by keeping the original year indicators (thus, not differencing those), and the coefficients showed the cyclical patterns I wanted to have - but I guess it makes no sense to have non-differenced year effects in a FD estimation right?

And, as a more general question, the bigger the difference between all the coefficients in (1) and (2), the more I can conclude that there are various misspecification
issues ? (and here, of course, I would need to find good arguments as to why the FD model in (1) is better)?

Thank you so much for your help!

Best, Ana

Ref: Wooldridge, Jeffrey M. Introductory econometrics: A modern approach. Nelson Education, 2012.

[Phil Bromiley]

You'll increase your chances of a useful answer by following the FAQ on asking questions - provide Stata code in code delimiters, readable Stata output, and sample data using dataex.

"a company with exactly the same credit risk, the rating today would be lower than 20 years ago, for reasons unrelated to company financials - it doesn't sound plausible, right" - Much of the credit risk assessment in commercial lending is done qualitatively. And, the raters and regulatory environment have changed. So, it could easily be that there has been a change in standards over time. Obviously, we would hope that the folks who rated subprime portfolios extremely low risk have learned something.

"where the common trend captures the variation in both ratings and many of the explanatory variables" - If you have appropriate controls for firm risk, then just because you have correlated trends does not invalidate the risk assessment. Unlike many exogenous economic variables, firm risk is a judgement. The whole spurious correlation problem is very real but very hard to know what is right.

I'm also not sold on year dummies as the right way to model this. Why aren't you looking at potential changes in the parameters on the financial indicators?

Before you go too far on the econometrics, is your dv really continuous? I would think credit ratings are ordinal (higher rating is better but you can't say one rating is twice another). So, I'd think you should be looking at ordinal estimators.

While fd estimators can give different results than xtreg,fe, they are really estimating a very similar model. I'm not sure what it means to first difference an indicator variable like i.year. But, I would worry about differencing some variables and not others. First differencing also throws out continuous trends, but you description does not tell us that the changes could not have been similar year to year.

I don't think you can arbitrarily claim the fd estimator is better than the xtreg. If you have serial correlation, then you can make an argument for a serial correlation correction (xtregar or xtreg with robust standard errors).

[Dimitriy V. Masterov]

The other issue is that having an intercept in the FD specification means that you have a linear time trend in the levels equations since b*t- b*(t-1) = b. Having both time dummies and a linear trend complicated the interpretation substantially. You may want to add the nocons option to reg.