# Does non-stationarity in logit/probit matter?

I would like to ask - I am using logit to investigate, if some variables improve the risk of currency crises. I have yearly data from 1980 for lots of countries (unbalanced panel), dummy variable is 1 if currency crises started (according to my definition), 0 otherwise. Explanatory variables are according to some theories, like current account/GDP, Net foreign assets/GDP, loans/GDP and so on... All are lagged (-1). I am using robust standard errors, which should be consistent with heteroskedasticity. However, for example loans to GDP or NFA/GDP are not stationarity (panel test). Does this matter? I have not seen any paper testing for stationarity performing logit/probit. For me it is also intuitive that it does not matter. If I am testing if a variable increases the risk of a crisis, it should not be problem, that this variable is rising permanently. On the contrary - rising variable is permanently rising the risk of the crisis and when it reach to some unsustainable level, the crisis occurs. Please could you give me an answer, whether I am right?

• Is there any reason you cannot use time fixed effects? Doing so often controls for changes in level of non stationary variables. The logit uses maximum likelihood which assumes iid latent errors. If the errors are correlated through time it could result in a biased fit regardless of how standard errors are calculated. Again using time fixed effects should account for most of this. Apr 17 '15 at 21:53
• They aren't allowing fixed effect in the logit because they are trying to avoid incidental parameter bias which on second thought makes good sense. One thing you could do is run a linear probability model, with and without random/fixed time effects. You can do residual diagnostics for these models and see if your margins of interest change. If your within group residuals are uncorrelated and time controls do not significantly change the margins than you may be fine with the logit (the average margins should for the logit should be similar to the linear model) Apr 18 '15 at 21:40
• Also there are some interesting things you can with re-sampling to reduce incidental bias in non-linear panel models which I think well apply to your problem. <a href="onlinelibrary.wiley.com/doi/10.1111/j.1468-0262.2004.00533.x/…> is a paper by Hahn and Newey that does so with a jacknife. I would not know how to implement something like this in Eveiws or STATA (Sorry), but I am sure it's do-able in STATA. Apr 18 '15 at 21:48
• I am not 100% sure. The idea with the linear model is that you can easily do residual diagnostics for non-stationarity (or auto-correlation) in the residuals. The average marginal effects for the logit should be extremely close to the linear model margins. Using this fact, if you know the linear model is unbiased (via checking the residuals), than you can say the logit is estimating average marginal effects unbiasedly as well (given you observe the average margins are pretty much the for both models). Thus, it would seem reasonable to believe the logit is unbias. Apr 19 '15 at 0:06
• When it comes to first differencing, this is sort of like "de-meaning" the time series variables which is very similar to the idea of adding fixed effects... controlling for changes in level for each individual time period. However, formal treatment of this sort can be quiet complex and there is still a chance for autocorrelation in residuals <a href="people.stern.nyu.edu/jsimonof/classes/2301/pdf/…> has some sensible suggestions using pearson residuals to check for autocorrelation. Apr 19 '15 at 0:39

Obviously in the first step you must have at list two integrated variables (in the cointegration case) or at least two variables with the same type of trend trend. If this is not the case you have a problem of omitted variables.

The altertnative to all this is that you change the scope of your analysis and transform all your regressors into a stationary ones.

I suggest looking at the results in Chang Jiang Park (2006) and Park, Phillips (2000).* According to the first paper, logit estimators are consistent even in the case of integrated series (theorem 2 at page 6-7) and usual t-statistics can be used for the parameters of interest in your case (the coefficients on the regressors). Other papers of the same authors develop econometric theory for other cases of non-stationary processes in non-linear models.

*These papers treat only theory, unfortunately I am unable to find an example of an empirical paper actually mentioning the issue of non-stationarity in this context.

I know this post is old but people do searches and often use this stuff as reference.

Let's keep it simple. Let's have a model of individual probability of defaulting on a mortgage as our Y. Now lets run level GDP on it. Lets say your data is 2002-2017, quarterly. You have millions of observations that at time T all share the same econ variables. I pick this time frame for a good reason.

What will you get as a relationship? Oh man, you will find that shazaam, lower GDP is correlated with higher defaults. Looks good right?

But now lets forecast this out, say 50 years (for the fun of it). Take the expected GDP at historical growth rate, say 2%, and extrapolate GDP. Now run the forecast. What do you find? Shazaam, like magic the probability of default will trend towards 0%.

You would get the opposite if you picked total number of unemployed (not rate). You will find that shazaam, forecast it out in the future and the probability of default trends to 100%.

Both are ridiculous. And here is the kicker. If you did a stationary test on some of these time frames you would find that they are stationary. The reason is you can dice a non stationary series into stationary parts. Particularly because Real GDP increased, decreased, and increased in the time period.

Yes, your in sample fit will look good. But your forecasts will be meaningless.

I see this frequently in risk metric modeling.

You are clearly fine from a theoretical perspective. It is a mistaken understanding of non-stationary series that they have changing means. They have no mean. The sample average is a random number because it converges to no point and so appears to change. This is also no problem for logit or probit.

Statistical models are mappings and there is no reason one cannot wrap an unbound series into a bounded series. For example, the real number line is normally thought of as having no length at all, but wrap it around a circle with the south pole being 0 and the north pole being $\infty$ and for a unit circle, the entire number line now has length $\pi$.

By mapping a non-stationary series to a well bounded set, you have created a well bounded problem as the ultimate solution has to map to the interval [0,1].

All accounting ratios must lack a variance and all financial returns must lack a variance. See the paper at https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2828744

You do not need to intrinsically worry about robust errors. It is a misunderstanding of non-stationary series that they are heteroskedastic. They are not; they are askedastic because they have no mean to form a variance about in the first place, so it is again a random number. The error terms structure has more to do with the model that maps than the lack of stationarity.

Where you could face a problem is with the concept of covariance. The distribution of equity returns is from a distribution that lacks a covariance matrix. It isn't that stocks cannot comove, but they cannot covary. The same thing is true for economies. It is a more complex concept than covariance which is a simple relationship. You will want to read the paper above and think through your model relationships carefully.