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Primer on statistical separation in Poisson models

(This guide is an abridged version of Verifying the existence of maximum likelihood estimates for generalized linear models; please see the paper for a more detailed explanation)

As noted in Santos Silva and Tenreyo (2010, 2011), maximum likelihood estimates of Poisson models might not exist, due to the problem known as "statistical separation". In practice, this is seen in regression estimates that do not converge or, even worse, converge to incorrect estimates.

Why might ML estimates not exist?

The log-likelihood function of Poisson models is log L = Σ[-exp(Xb) + (Xb) y - log(y!)]. If we denote the residuals as e = y-exp(Xb), then its first order condition is X'e=0 (same as with least squares, but with a different definition of residuals).

Now suppose we have a regressor x that is always zero when y>0, and non-negative when y=0. For instance, this simple dataset:

y x
0 1
0 1
0 0
1 0
2 0
3 0

Question: what value of b minimizes the residuals e = y-exp(a+xb)?

Note that the value of b can only affect the residuals of the first two observations (because x=0 elsewhere). Moreover, you can actually set e=0 for these first two observations if you set b=-∞ (minus infinity)!

In terms of the log likelihood, it is straightforward to verify that the log-likelihood contributions of the first two observations take their maximum possible value when b=-∞: LL_i = -exp(-∞) + (-∞)(0) - 0 = 0.

Since -∞ is not part of the real values (only of the extended reals), this means that maximum likelihood estimates will not exist. Moreover, as our own benchmarks show, trying to estimate this in any modern statistical package (Stata, R, Python, Julia, etc) can result in all kinds of issues. For instance, Stata's poisson command does converge, but to an arbitrary low negative sign (-18.8 in this case):

clear
set obs 6
gen y = max(_n - 3, 0)
gen x = _n < 3
poisson y x

screenshot-poisson

(Trivia: since mean(y)=1.5 for observations 3-6, the estimate for the constant is log(1.5)=0.405)

When does this issue occur?

As explained in the paper, separation occurs when we can find a linear combination of the regressors "z" (z = Xγ)such that:

  1. z=0 if y>0
  2. z≥0 if y=0, with at least one strict inequality

If you can find a z where this occurs, then the observations where z>0 are separated and there will be at least one estimate with infinite values that makes these observations have a perfect fit.

Moreover, z acts as a "certificate of separation", because we can regress it through least-squares against the regressors X, and if we observe a perfect fit (R2=1.0), then we can verify that the z>0 observations are indeed separated.

Notice also that this is a significantly stronger result than the one shown in Santos Silva and Tenreyo (2010), where only condition #1 is presented. Indeed, by combining these two conditions, we actually arrive at a "sharp criterion" for detecting separation.

(Note: it is equivalent to state condition #2 in terms of z≥0 or z≤0 inequalities)

The example below shows one example involving two regressors:

y x1 x2
0 2 -1
0 -1 2
0 0 0
1 0 0
2 5 -10
3 6 -12

Here, neither x1 nor x2 are equal to zero when y>0, but we can create a combination z = 2 x1 + x2 that will satisfy both conditions:

y x1 x2 z
0 2 -1 3
0 -1 2 0
0 0 0 0
1 0 0 0
2 5 -10 0
3 6 -12 0

Thus, the first observation is separated.

Now, even this simple example will be difficult for standard statistical packages. For instance, the code below creates the data in Stata and runs the poisson command:

* Create data
clear
set obs 6
gen y = max(0, _n-3)
gen x1 = 2*(_n==1) - (_n==2) + cond(_n>4, _n, 0)
gen x2 = 2 * (_n==2) - 2 * cond(_n>4, _n, 0) - (_n==1)
* Try to run -poisson-
poisson y x1 x2

(If you can, run it own your own and see what happens)

However, note that ppmlhdfe does detect and drop the separated observation. Moreover, note that by dropping the observation, the separation issue gets reduced to a collinearity problem, which in standard Stata fashion is solved by dropping one of the two collinear regressors (as in the case of perfect collinearity, this is something that you might not want to do, as it is often better to understand if there are any issues with the underlying specification):

screenshot-poisson

Even further, you can use ppmlhdfe to discover the exact linear combination of variables that causes the separation problem (i.e., the z). The command below thus generates an indicator variable sep listing the separated observations, and creates a certificate of separation z, which is then regressed against the Xs to verify that R2=1.0 and the first observation is indeed separated:

ppmlhdfe y x*, tagsep(sep) zvar(z) r2

screenshot-poisson

Now that we have seen how this issue arise, we will briefly discuss how ppmlhdfe actually detects separated observations. Also, note that the separation problem is particularly pernicious in specifications with many fixed effects (because there are many more possible linear combinations that can lead to separation), so a lot of the extra care is to ensure that all separated observations are detected.

How does ppmlhdfe actually find the separated observations?

By default, ppmlhdfe uses four methods to identify separated observations. However, after reading the discussion below and seeing their pros and cons, you can choose to only include some of the methods, thus slightly increasing the speed of the command.

ppmlhdfe, separation(fe)

You can easily find some separated observations if you find categories of the fixed effects that only exist when y=0. For instance, if we have a regression with individual fixed effects, then the individuals that have always had y=0 will have their observations separated, because the indicator variables underlying their fixed effect already satisfy the requirements to be a certificate of separation z.

You can see the method in practice in the example below:

y id
0 1
0 1
0 2
1 2
2 3
3 3

Here, notice how the observations for the first individual (in the first two obs.) are separated. In Stata:

clear
set obs 6
gen y = max(0, _n - 3)
gen id = ceil(_n / 2)
li, sepby(id)

ppmlhdfe y, a(id) sep(fe)

screenshot-poisson

As you can see in the line (dropped 2 observations ...), the separated observations were indeed dropped.

ppmlhdfe, separation(simplex)

This method implements the modified simplex solver described by Clarkson and Jennrich (1991), with some twists. For instance, there is no need to run the simplex if there are no perfectly collinear regressors on the y>0 sample, in which case we stop.

This method would be sufficient except for one large drawback, that it does not handle separation arising from fixed effects, or from linear combinations of fixed effects and other regressors.

For instance, in the example below, the combination of the fe and simplex methods fails to detect separation:

clear
input byte(y id1 id2)
0 1 1
1 1 1
0 2 1
0 2 2
1 2 2
end

ppmlhdfe y, a(id1 id2) sep(fe simplex)

That said, if you are not using fixed effects, then sep(simplex) should be enough.

ppmlhdfe, separation(ir)

(Also known as ppmlhdfe, separation(relu))

This is the method described by Correia, Guimarães, Zylkin. It is easy to code and more general than the simplex method, but this comes at the cost of some speed.

To understand this method, we will first use it to solve the example above, and then actually implement it by hand.

clear
input byte(y id1 id2)
0 1 1
1 1 1
0 2 1
0 2 2
1 2 2
end

ppmlhdfe y, a(id1 id2) sep(ir)

screenshot-relu

Now, if we were to implement the algorithm by hand, we could do so in less than 20 lines of standard Stata code (!):

* Create data
clear
input byte(y id1 id2)
0 1 1
1 1 1
0 2 1
0 2 2
1 2 2
end

* Run IR (iterative rectifier) algorithm
loc tol = 1e-5
gen u =  !y
su u, mean
loc K = ceil(r(sum) / `tol' ^ 2)
gen w = cond(y, `K', 1) 

while 1 {
	qui reghdfe u [fw=w], absorb(id1 id2) resid(e)
	predict double xb, xbd
	qui replace xb = 0 if abs(xb) < `tol'

	* Stop once all predicted values become non-negative
	qui cou if xb < 0
	if !r(N) {
		continue, break
	}

	replace u = max(xb, 0)
	drop xb w
}

rename xb z
gen is_sep = z > 0
list y id1 id2 is_sep

The separation paper contains a detailed description and proof of the method, but there are only a few steps involved:

pseudo-code

A few notes:

  • We can choose the weights K equal to N0 / ϵ² (where N0 is the number of observations where y=0)
  • Running a regression with very high weights when y>0 just ensures that on those observations Xb=0 within some tolerance. This is known as the "method of weigthing".
  • The update u = max(u, 0) is known as a rectifier (ReLU) in computer science and machine learning, and is the key trick that makes the algorithm work.
  • Note that by combining the method of weighting with the rectifier, we ensure that Xb can be used as a valid certificate of separation z, once we achieve convergence.

ppmlhdfe, separation(mu)

This method, first mentioned by Clarkson and Jennrich (1991), does a simple heuristic to detect separated observations. If at any given point there are observations with y=0 where the predicted values μ=exp(xb) are also very close to zero, then it is likely that these observations are indeed separated.

However, "very close to zero" is an arbitrary number, and thus a) if set too high then it might lead to false positives, and b) if set too low it might fail to detect some separated observations. Further, if there are separated observations then the IRLS iteration used by ppmlhdfe might converge extremely slowly, so it is not ideal to exclusively rely on this method.

Thus, we agree with Clarkson and Jennrich in that this method is not very useful on its own. That said, if combined with a conservative tolerance (which we do), it can be useful as a back-stop method. Checking if μ is taking very low values after each iteration has almost no speed cost and is trivial to implement, and thus it can be used to complement the existing methods.

Using the previous example, here we can see sep(mu) in action:

clear
input byte(y id1 id2)
0 1 1
1 1 1
0 2 1
0 2 2
1 2 2
end

ppmlhdfe y, a(id1 id2) sep(mu) mu_tol(1e-5)

screenshot-mu

The iteration takes a while to run (18 iterations, compared to 6 for the IR method), but the separated observation is indeed detected, in iteration 15.

However, this method is fragile, especially when the dependent variable has a skewed distribution. For instance, this method would fail to detect separation if we replace mu_tol(1e-5) with mu_tol(1e-6) (the default).

Alternatively, also depending on its tolerance, the μ method might be too aggressive and incorrectly drop observations. In the example below we we add three observations to the dataset, so the third observation is no longer separated. As a consequence, the sep(mu) method might converge extremely slowly (in 115 iterations), and to the wrong solution (incorrectly dropping one observation that is not separated), depending on tolerance for μ:

clear
input double(y id1 id2)
0 1 1
1 1 1
0 2 1
0 2 2
1 2 2
1e-6 2 1
1e-6 2 1
1e-6 2 1
end

ppmlhdfe y, a(id1 id2) sep(mu) mu_tol(1e-2) // takes a while to converge, and erroneously drops one obs.

ppmlhdfe y, a(id1 id2) sep(ir) // converges quickly and to the correct number of observations
ppmlhdfe y, a(id1 id2) sep(mu) mu_tol(1e-6) // converges quickly and to the correct number of observations

Nonetheless, we selected very conservative default values for mu_tol(), and also added some extra checks for highly skewed data, so in practical scenarios the μ method is quite unlikely to fail.

Recap

The table below summarizes our views on the pros and cons of each method.

Method Pro Con
fe Simple Only detects separation from a single category
simplex Robust Does not work for fixed effects
ir General Slower for small problems, as each iteration involves computing weighted least squares
mu Fast Works poorly with skewed data; convergence may be slow

For simple regressions without any fixed effects, the sep(simplex) method is a good choice, while for more complex regressions with many levels of fixed effects sep(fe ir) or sep(fe simplex ir) should work well. Optionally, the mu method can be added as a back-stop, and the user should also inspect the iteration log to see if there are very low values of mu.

To read more on separation, see the separation benchmarks, as well as our separation paper. For more information on ppmlhdfe, see the help file, the list of undocumented options, as well as our ppmlhdfe paper.

"To Infinity and Beyond!"

Given that the ML estimates are actually infinite, one may ask what are we reporting exactly, given that there are no infinite symbols in the regression tables. For this, let's revisit an earlier example:

y x1 x2 z
0 2 -1 3
0 -1 2 0
0 0 0 0
1 0 0 0
2 5 -10 0
3 6 -12 0
* Create data
clear
set obs 6
gen y = max(0, _n-3)
gen x1 = 2*(_n==1) - (_n==2) + cond(_n>4, _n, 0)
gen x2 = 2 * (_n==2) - 2 * cond(_n>4, _n, 0) - (_n==1)

ppmlhdfe y x1 x2

Here, ppmlhdfe drops x2 and returns b1 = 0.35. This is, however, not entirely accurate. If we allow ourselves to think in terms of infinities, as Geyer (2009) does, we can argue that the true estimates are b1 = lim 2c + 0.35 and b2 = lim c, as c goes to infinity. This is described by Geyer in terms of a "direction of recession" in a Barndorff-Nielsen completion, as otherwise one would just say that b1 = b2 = ∞.

This also raises the question of how the b1 = 0.35 reported by ppmlhdfe should be interpreted. As we discuss in the paper, it is useful to think of the issue as being similar to (though not exactly the same as) a perfect collinearity problem. That is, the reported "b1" is really an estimate for the combined parameter b1-2*b2, similar to how one would interpret estimates in models with omitted perfectly collinear regressors. Furthermore, we show in the paper that this is a consistent estimate for b1-2*b2 (and that any regressors not involved in separation - in this case, the constant - are consistently estimated as well.)

Another way of framing the problem would be to add a third variable to the regression, z = 2 x1 + x2. Then, we can do:

ppmlhdfe y z x1 x2

Here, the only estimate with "infinities" would be the one for z. However, it is a matter of interpretation whether you can add such a z variable.

If you don't know exactly the linear combination of regressors that produces z, you can also use ppmlhdfe to obtain it. For instance, below we reproduce Table 1, Example 2.3 of Geyer (2009):

import delimited using "http://www.stat.umn.edu/geyer/gdor/catrec.txt", delim(" ") clear
ppmlhdfe y i.(v*)#i.(v*)#i.(v*) , tagsep(sep) zvar(z) r2 // Get certificate of separation Z, and regress it against the Xs
* Code below is just to present a prettier output:
matrix b = e(b)
mata: vars = st_matrixcolstripe("b")
mata: directions = round(st_matrix("b"), 0.001)'
mata: idx = selectindex(directions)
mata: (vars, strofreal(directions))[idx, .]

screenshot-poisson

As we can see, we are able to recover Geyer's "direction of recession" by employing the IR algorithm, which has the added advantage of being easy to implement, and not requiring exact algebra routines.

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