Marginal Effect of a Rate of Decline - Logit Regression

[Jeremy Hockle]

All,

I am wondering if I may trouble you for some clarity over a marginal effect I have recently calculated.

I am using a logit model to understand the effect of Routinisation (explained below) and a number of educational and demographic variables on the probability of ending up in one of six occupational categories at the end of a period of transition, t.

Routinisation is the rate of decline of routine jobs (derived through a classification scheme I have put together) over each period of transition. For the sake of example, for the first four periods it takes the following values:

Period	1	2	3	4
Change in share or routine jobs	-5.77%	-6.08%	-3.78%	-1.40%
Initial share of routine jobs	57.99%	52.22%	46.14%	42.36%
Rate	-9.95%	-11.64%	-8.19%	-3.30%
Routinisation	0.0995017	0.116384	0.081888	0.03300123

There are various destination groups individuals can end up in at the end of the period of transition (Professional, Managerial, etc). I estimate a logit model for the probability of ending up at each of the final occupational groups, conditional on them starting in the Routine group. Of these final groups, one is Routine. In this case, the model essentially estimates the probability of remaining in the Routine group.

Being a logit model, I look at the Average Marginal Effects (AME) and Marginal Effects at the Average (MEA) in order to gauge some feel of the magnitude of the effects. In the estimation for remaining in the routine category, I notice that the marginal effect of Routinisation is very high: it is around 0.9 for both MEA and AME.

My question is this: how an earth do I interpret this? Routinisation is ultimately a continuous variable, but it is calculated as a rate of decline. Surely this cannot mean that for a 1 unit increase in Routinisation (what is this? 1%?), the probability of remaining in the Routine category decreases by 90%. Can it?

Many thanks,
Jeremy.

[Jeremy Hockle]

Ah, I just spotted something that I thought would solve the issue - I had an interaction term between rountinsation and another variable, and I know that this can cause trouble. Dropping this, and running the marginal effect on remaining in routine again on a model with no interaction on routine (and in fact no interactions at all), It's still quite high at 0.73.

Any thoughts?

[Jeremy Hockle]

One final point - all marginal effects discussed above are significant at 1%

[Clyde Schechter]

Well, maybe it does mean just that. A 1 unit increase in Routinization, the latter being, according to your description, a fraction, not a percentage, is a change from 0 to 1, or, equivalently, from 0% to 100%. In light of that, does a marginal effect of 0.9 seem plausible?

If not, please post the commands and output from both your -logit- and the -margins-. Do that between code delimiters so that it is readable. (See FAQ #12 if you are not familiar with code delimiters.)

[Jeremy Hockle]

Yes, that does make much more sense. Clearly my afternoon coffee is yet to kick in. As routinisation is essentially the fraction of routine jobs at the start of the period that decline over the period, it is between 0 and 1 (as you can only get rid of 100% of the initially routine jobs). In that sense, a marginal effect of 90% seems very plausible as routinisation goes from 0 to 1.

I just didn’t think about the ‘unit’ increase in terms of 0-1. I can now just divide this by 100 and claim that a 1% increase in routinisation decreases the probability of remaining in a routine occupation by 0.9%.

These qualms aside, am I correct in my understanding of the bad practice of evaluating marginal effects for variables that are also contained within an interaction term in the model? Aren’t I much better having a specification with zero interactions whatsoever for calculating marginal effects, and then a separate specification which may include some interaction terms, for which I will run –inteff– to get an understanding of magnitude?

Thank you for this; you have been tremendously helpful.

[Clyde Schechter]

These qualms aside, am I correct in my understanding of the bad practice of evaluating marginal effects for variables that are also contained within an interaction term in the model?

It's not a bad practice if you understand what you're doing and doing correctly. When you have an interaction, v1#v2 in your model, you are, just by using that interaction term, stipulating that there is no such thing as the marginal effect of v1, nor the marginal effect of v2. Rather, v1's marginal effect varies according to the value of v2, and vice versa. The simplest case is when v1 and v2 are discrete:

regression_command outcome_var i.v1##i.v2  // AND OTHER COVARIATES PERHAPS

margins v2, dydx(v1)
margins v1, dydx(v2)

gets you the marginal effect of v1 at each value of v2, and then the marginal effect of v2 at each value of v1.

If either variable is continuous, it's a bit more complicated and you have to specify particular values of each variable at which you would like to evaluate the marginal effect of the other, using the -at()- option of -margins-.

-inteff- is an old user-written command whose functions have been largely, if not entirely, superseded by the -margins- command. So assuming you are using a modern version of Stata, mimicking or modifying my code above will serve you, and you won't need -inteff-.

[Jeremy Hockle]

Thank you Clive - will give that a try.

Re your very first point on routinisation above, although your explanation makes sense, on what basis can you claim the unit increase in routinisation is from 0 to 1?

Isn't the marginal effect of a continuous variable measuring the effect of a very small (/instantaneous) change in that variable on the dependant? In my case, the the effect of a very small change in routinisation on Pr(routine occupation at end of period t after having started in a routine occupation). In what sense is the increase from 0-1 small if it's essentially 100%?

[Clyde Schechter]

Strictly speaking, yes, the marginal effect of x on y is defined as dy/dx, the first derivative. Because not everybody who works with these things is comfortable with calculus, people have developed a shorthand way of speaking about it and say that it is the effect of a unit change in x on y. These things would be equivalent if the marginal effect were a constant, independent of the value of x. Just like if you are driving and your speedometer shows 60 mph, it is true that if you maintained that exact speed you would travel 60 miles after driving for an hour. The true marginal effect is like the speedometer reading.

So in your case, you found that the derivative of your outcome probability is 0.9. You don't say at what value of routinisation you evaluated that. Maybe that is the "average" marginal effect. In any event, the implication is that if the marginal effect were 0.9 at all values of x (which, by the way, is not even possible in a logistic model), then a full unit increase of x would increase the outcome probability by 0.9. Now, a more careful statement is that at whatever value of x you evaluated it at, dy/dx = 0.9. So if x we look at some small variation in x, call it h, then the corresponding variation in y is 0.9h + o(h), where o(h) is negligibly small compared to h in the limit as h -> 0. (Strictly speaking o(h)/h -> 0 as h -> 0.) So the statement about a unit change is a very large extrapolation of what is, in fact, a very local result. It extrapolates to the case h = 1. But of course, o(h) will not necessarily (and, in the case of a logistic model will definitely not be) negligibly small compared to h.

This whole business of the marginal effect representing the effect of a unit change is just loose talk. But it is so entrenched that everybody I know, including people who try to be careful about things, uses it.

The one circumstance where a marginal effect is the expected difference associated with a unit change is when x is a discrete variable taking on integer values. Then the definition of marginal effect is not dy/dx but delta y/delta x. And, by the way, the marginal effects calculated by -margins- are handled accordingly.

[Phil Bromiley]

On a slightly different point, you might consider ordinal or multinomial probit. You really have multiple discrete outcomes, not a set of 0/1 choices. However, this makes the interpretation a bit trickier.