Econometricians spend their lives trying coming up with new and better estimation techniques. Some of the ideas are excellent but impractical ("Just find a suitable instrumental variable"), and some complicate matters for minimal benefit (some argue that using logit or probit rather ordinary least squares estimation falls into this category).

So when I stumble across some well-intentioned econometric advice, one of first questions I ask myself is: does it matter?

Yesterday I was working on a paper where we estimate a model like this:

ln(income)=a+bX+cD+error

where X are continuous variables and D are dummy variables (variables that take on a value of either zero or one, e.g. male=0, female=1.)

People usually interpret the coefficients in such models as percentages. For example, if X was age and b was 0.02, I would write something like, "Income increases by 2 percent with each additional year of age."

Dave Giles - in a post entitled Dummies for Dummies -argues that, while this approach is fine for continuous variables like age, it is inappropriate for dummy variables like gender. As he puts it:

*The interpretation of the coefficient, b, is that it is the partial derivative of ln(Y) with respect to X. So, 100b is the percentage change in Y for a small change in X (up or down), other things held equal.*

*Unfortunately, lots of people (who really should know better) then apply the same "reasoning" to the interpretation of c. The trouble is, of course that D is not continuous, so we can't differentiate ln(Y) with respect to D. The way to get the percentage effect of D on Y is pretty obvious. Curiously enough those same people who go about this the correct way when computing marginal effects in the case of Logit and Probit models just don't seem to do it right in the present context. All we have to do is take the exponential of both sides of equation (1), then evaluateY when D = 0 and when D = 1. The difference between these two values, divided by the expression for Y based on the starting value of D gives you the correct interpretation immediately:*

*If D switches from 0 to 1, the % impact of D on Y is*

*100[exp(c) - 1]. (2)*

*If D switches from 1 to 0, the % impact of D on Y is*

*100[exp(-c) - 1]. (3)*

So, according to Giles, if I have a variable that takes on a value of 0 for male and 1 for female, and the coefficient in my regression is something like -0.1, I shouldn't say "females earn ten percent less than males, all else being equal." I should get out my spreadsheet, type the formula = exp(-.1)-1, and let it tell me that women actually earn 9.5 percent less than males.

It's a hassle to do this calculation for every variable in the model. So, honestly, is it worth it? (After all, what are the odds of getting a referee who will catch you up on this?)

I did a few calculations to work it out.

(this was generated by going to www.wolframalpha.com and typing in plot x from x=-1 to 1 and plot exp(x)-1 from x=-1 to 1) and then using the screen capture macro command-shift-4 on my mac).

That picture shows that, for coefficients between -0.2 and 0.2, it really doesn't make much difference if one takes the extra step and calculates the percentage changes directly. For coefficient values above 0.5, it makes quite a difference.

Note, too, that interpreting coefficients on dummy variables as percentages causes asymmetric errors. If the coefficient is positive, and we're talking about a swtich in the dummy variable from 0 to 1, then the coefficient-as-percentage-change approach cases one one to underestimate the true impact of a change in the dummy variable. On the other hand, if the coefficient is negative, or we're talking about a switch from 1 to 0, the coefficient-as-percentage-change approach overestimates the true effect.

Here's another picture, this one showing how much the coefficient-as-percentage approach under- or over-estimates the true impact of a change in a dummy variable:

So am I a reformed economist, who will never again interpret a coefficient on a dummy variable as a percentage, and will always take time to work out the marginal effects?

No. But if the coefficient is big enough - if it's above 0.3 or 0.5, say - I'll at least think about doing so.

(The hat tip goes to my co-author Casey Warman, who insists on calculating the marginal effects properly, even for coefficients of 0.06).

Yup. After several diagnostic tests are executed and reported and perhaps a graph of residuals is added to the paper, I'm all in favour of getting the full distribution of marginal effects with dummy variables right.

But I would prefer not to see sophisticated dummy variable coefficient estimations as an excuse to hand wave and ignore the fundamental statistical modelling issues where applied inference implicitly and almost always assumes normal distribution.

But then the reputation of applied econometricians appears best explained by an open access model (no rules) that inevitably converges to a social dilemma of untrustworthy output. The incentives to cheat and hand wave are simply overwhelming.

Posted by: westslope | July 15, 2012 at 11:33 AM

westslope - the thing is, there often isn't an easy test for fundamental modelling issues - statistical or otherwise - something that tells you whether you've done it wrong or right. Things like sample selection and treatment of missing observations often make a huge difference to the results of an analysis, but because they're more of an art than a precise science, they receive little treatment in econometrics courses.

There is a solution to untrustworthy output, though, isn't there? By which I mean requirements that people make their data available to other researchers, more replication studies, more academic points for replication studies, presentation of results for multiple different specifications/samples - with referees appendices and on-line appendices if necessary.

Posted by: Frances Woolley | July 15, 2012 at 12:09 PM

Frances - Nice post; and the second para. of your response to westslope is right on target!

My only gripe relates to your comment: "After all, what are the odds of getting a referee who will catch you up on this?" I'm sure you wouldn't dream of driving under the influence! :-)

Dave

Posted by: Dave Giles | July 15, 2012 at 12:21 PM

Isn't there the same problem with continuous variables as with dummy variables? The percentage change interpretation only works when considering infinitesimal changes so a one year increase in age leads to x% change isn't technically correct either. The only difference is that it's more practical to conduct the proper analysis for two value dummy variables then with infinitely valued continuous variables. Technically we should calculate the marginal effects at different values but we never do, so if we don't practice the proper care with continuous variables then why should we necessarily do so with dummy variables.

Posted by: Joseph | July 16, 2012 at 12:04 PM

Joseph, you're right. Suppose the 'continuous' variable is years of education, and one is interested in the impact of getting an undergraduate degree on earnings - dx is not necessarily a good approximation of the impact of a change from 12 to 16 years of education.

Posted by: Frances Woolley | July 16, 2012 at 12:25 PM

The problem I see with calling it a probability in some cases is that it suggests that all individuals have that lower probability. Say you're looking at employment. If AD is lower you could do a bunch of math and say "the probability of employment (or of being employed) decreases by x% with for each percentage decline in AD" when the effects are at the margin. Some people are almost guaranteed to be laid off if they are in marginal positions/companies in relevant sectors, but other workers can count on keeping their jobs. Often, it's best just to say "the level of employment is x% lower." I think selection models are an interesting way around this, but labour markets aren't perfect in that way either. Of course if the entire market is said to pay labour at its productivity level, then things can be made more complicated, with both wage and labour effects in various sectors and groups of workers... Similarly, you could try to say that a policy increases the probability of employment, but the effects are targeted rather than being an average probability across the population (a fact that can be exploited). Not sure if it's the best example, but sometimes I read stuff that makes me wonder about the way the causal effects are viewed as expressing themselves (or sometimes even the implied direction of causality) in the actual economy.

As for the difference between the 10% and 9.5%, being aware of that, and also knowing that things are not perfect in other ways, you could say "a quick calculation shows that..." or perhaps "for a relatively low effect we can interpret this as a ..." but then where's the cutoff where it's important? 5%? 25%? I'd say if you plan on using decimal places of any sort, and you know a method will give a more precise result then you should always use the more precise method. Imagine this: you say the effect is 6.1% but you know it's 5.7% (guessing) using the other approach. Saying "around 6%" is still fine, but saying "6.1%" is both unprecise and misleading about the precision of your results due to the use of decimal places. Also important, is whether it's being used simply to show the direction of change or whether the results are fed into other parts of other models to reach some other conclusions. It sometimes seems as though folks will carry on numerous such estimations then presenting their results to three or four decimal places, with no apparent awareness of how "significant digits" communicate precision. Fine enough for a pedagogical example to show the direction of change, or as a side note about a direction of change and a loose indication of magnitude, but not as an input for further procedures...

Lots of the time, probably not a big deal.

Posted by: Nathan W | July 16, 2012 at 04:32 PM

Surely what matters is the size of the uncertainty in the co-efficient. If the co-efficient is 10% +/- 8% then getting it correct at 10.5% is meaningless. OTH, if the data nail the co-efficient precisely, then you have to calculate it at least as precisely. Far more important is the error estimation on the parameter.

Posted by: Chris J | July 17, 2012 at 10:20 AM

Chris J, Nathan: Your general point about reporting an appropriate number of significant digits is a good one. One of my pet peeves is people who are estimating a dependent variable that is in dollars, and then report coefficients to three decimal places (you're talking 1/10th of a cent here - no, it doesn't matter).

It's also the case that, if one is transforming the reported coefficient as suggested by Dave Giles above, it's probably a good idea to transform the error terms as well. E.g. a coefficient that's between 0.2 and .3 with 95% confidence represents a proportional change of X to Y with 95% confidence.

At the same time, it's not an either/or - it's possible to do a better job in terms of reporting marginal effects accurately *and* pay attention to reporting the appropriate number of significant digits.

Posted by: Frances Woolley | July 17, 2012 at 11:36 AM

I don't see why finding an instrumental variable is impractical. Surely we agree there is no dearth of economic data, especially financial data. How spoiled can we be? Sure, one needs to think outside the center, be creative and clever, and tell a compelling story, but that is the same criteria for good theoretical research. Finding a good instrument is akin to writing a good theory: they both require significant thought and effort.

Endogeneity abounds, so finding a suitable instrument, aside from running a barrage of regressions that increasingly isolate the partial effect of interest and that indicate the same story is being told, is really the only correct way to isolate the exogenous variation of a partial effect and to get nearer to the true coefficient. In light of today's increasingly complex microfounded macromodels, which often times use empirical micro studies parameter estimates as inputs, I think taking the time to get the coefficient right is worth it.

Posted by: Colin | July 17, 2012 at 01:47 PM

The discusion is about how close log (s1/s2) is to (s1/s2). The two are close enough when s1/s1 is within +0.01 and – 0.01, but grows apart beyond the range.

But things are more complicated than this. The estimates from the model using log earnings as the outcome will be different from the estimates in the model using earnings as the outcome because the log transformation changes the distribution of the outcome variable. Taking exponential of the both side does not give the same results as using (un-logged) earnings as the outcome. So in the end, no matter what you do with the coefficients from the logged model, we are still not certain to what extent the coefficients represent the ‘real’ group differences in earnings.

Posted by: Feng | July 17, 2012 at 03:35 PM

I think a perfectly legitimate although underused alternative interpretation could be as follows (assuming female=1 and its coefficient is -0.06): Females earn exp(-0.06) times as much as males, holding constant other independent variables (provided there are no interaction terms).

On that note, I think it is very useful to think even harder about the interpretation of a dummy variable in a model with a log dependent variable when there is an interaction term as is often the case for such models. 100*[(exp(b1+b2*X))-1]

Posted by: primedprimate | July 17, 2012 at 03:56 PM

Colin: "I don't see why finding an instrumental variable is impractical."

Sure, there are some papers out there, e.g. Nathan Nunn's work on aid and conflict in Africa, that use really smart and convincing instrumental variables. But I would say that 80%+ of the papers that I see with instrumental variable techniques use some instrument that leaves me scratching my head and saying "really?" because the connection between the instrument and the variable being instrumented is so tenuous I just can't believe the results.

Feng - absolutely. It's difficult to overstress the importance of doing exploratory analysis - just plotting the distribution of the dependent variable and asking yourself 'what does this look like? does it look normal? if I take a log does it looks vaguely normal?" Often it's worth considering doing some kind of quantile regression.

primedprimate - very true.

Posted by: Frances Woolley | July 17, 2012 at 05:02 PM

I wasn't so much going on about sig figs as I was about uncertainty estimation. Data are only so good and hearing a discussion about how to calculate parameters without a discussion of how to calculate errors is a bit odd to a physicist.

If you do a chi square estimation you have a clear.way to define both goodness of fit and get the error on the derived parameters and their covariances.

Discussion of errors in parameters is trickier. And if the coefficients have non linear transformation to the meaningful values then so do the unvertainties.

Posted by: Chris J | July 17, 2012 at 09:22 PM

frances - Making data available for replication is an excellent idea. Been talked about for decades.

But with all due respect, I don't buy your argument that simple diagnostic testing or a graphic representation of residuals is all that difficult to do. It is not just "exploratory analysis" that should go unreported in the final paper. An explicit description of the data mining process might also be helpful.

Agreed on instrumental variables.

Modeller choices between different configurations of essentially the same model is another complication.

Part of the problem is that even within the economics profession, most economists, including academics, are not steeply schooled in econometrics and so their ability to assess modelling and inference exercises is limited. Then there is the widespread cynicism that tends to reject the statistical modelling results out of hand.

Posted by: westslope | July 18, 2012 at 11:35 PM