Per Wikipedia, recall that the Durbin-Wu-Hausman test (hereafter the Hausman test)
evaluates the consistency of an estimator when compared to an alternative, less efficient estimator which is already known to be consistent.
One common way in which the Hausman test is used is to compare OLS with 2SLS–that is, to perform a test of the null of exogeneity. This tests consists in estimating an OLS specification, estimating a 2SLS specification of the same equation, and then in comparing whether the two parameter vectors are statistically the same. If you fail to reject the null of exogeneity, OLS is to be preferred. If you reject the null, then 2SLS is to be preferred.
Even if you know only a little about discrete-choice models, you know that when comparing multinomial alternatives, or un-ordered categories– for instance, the choice to drive, take the bus, bike, or walk to work, or the decision to pick a major in arts and sciences, business, engineering, etc.–the go-to model is the multinomial logit (MNL), which is a version of the logit that allows comparing more than two alternatives without there being any order between those alternatives. And if you’ve studied the MNL, you know that its main drawback is the fact that it assumes that the independence of irrelevant alternative (IIA) assumption holds.
In the context of of the mode of transportation you choose to go to work, the classic example of IIA is this: Suppose you face a choice between driving or taking a red bus to work, each with equal likelihood, i.e., 0.50-0.50. Introducing a blue bus as a third alternative should not affect your likelihood of driving to work–your choice to drive should be independent from the irrelevant alternatives blue bus or red bus. If the introduction of the blue bus changes the likelihood you’ll drive in any way, then the IIA does not hold.
There are many contexts where the IIA cannot be argued to hold. Perhaps the simplest case is Condorcet’s paradox: A group of voters might have a clear preference between candidates A and B, but introducing a third candidate may well make their preferences cyclical.
Some well-known alternatives to the MNL relaxing the IIA assumption are the generalized extreme value (GEV) model and the multinomial probit (MNP) model, but both the GEV and MNP models often make undesirable assumptions or are difficult to estimate–the MNP, for example, involves integrating over an N-variate normal distribution, something which gets computationally intensive beyond the bi- or trivariate cases.
One alternative which I suspect only one or two of you have ever heard of is the dogit (pronounced “dodge it”; I’ll get to why in a minute), which offers a nice alternative to MNL in that it allows relaxing the independence of irrelevant alternative assumption fully or only partially.
This is not a post about Fox Mulder’s replacement.
The dogit model was introduced by Gaudry and Dagenais in a 1977 article in Transportation Research B (a lot of the early microeconometric models were developed by transportation economists, given the categorical nature of many transportation choices).* They called it “dogit” because, according to the first footnote of the paper (the emphasis is mine):
The model avoids or dodges the researcher’s dilemma of choosing a priori between a format which commits to IIA restrictions or one which excludes them–whence its name.
The cool thing is that the dogit model allows the data to speak for itself when it comes to the IIA.
Another cool thing is that it readily allows for a certain amount of captivity to a given choice category. For instance, consumers must often spend a certain amount on certain expenditure categories (e.g., food) irrespective of their price before they can start spending on other categories (e.g., books), which makes them “captive” to some expenditure categories. Lastly, it also allows the possibility of estimating an “income effect” in addition to the substitution effect one can estimate with the MNL, but that is less clear to me from reading Gaudry and Dagenais’s article.
If you are interested in seeing what the distribution looks like for the dogit, here it is:
Obviously, all the p_{i}s have to be between zero and one and must all sum up to one. The \theta parameters are what’s new here–if they all equal zero, the dogit reverts to an MNL.
I have never estimated a dogit model, but in the interest of paying tribute to those who set me on the path to becoming an applied econometrician, I’d like to write something estimates a dogit model one day (though for better or for worse, I don’t often encounter multinomial choices in the things I study).
* I only know about the dogit because Marc Gaudry taught me the first econometrics class I’ve ever taken, and Marcel Dagenais taught me the second one, both when I was an undergraduate, and one day I decided to look up what my instructors’ contributions to their fields had been and stumbled upon it.
Today marks the seventh anniversary of this blog, so on top of wishing Happy New Year to all, I am also wishing this site Happy Blog Anniversary!
Today also marks the close of this blog’s most successful year ever: Between 2016 and 2017, the number of page views increased by 22 percent, for a total of over 150,000 pageviews in 2017.
The most popular posts of 2017 were, starting with the most popular:
Looks like there is a common theme there… which means I will most keep writing about applied econometrics. I should be able to post more frequently in the next few months, as I don’t have to teach between January and September.
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This has been a good year for me. In terms of research, I have published three new articles–two in my discipline’s top journal, and one in a top general political science journal. Another article was just accepted and is forthcoming in early 2018, and I have placed an op-ed in the Wall Street Journal.
(Update: And in an article titled “In Praise of Quinoa,” The Economistmentioned our work on quinoa–and me by name, although I wish they would’ve acknowledged my coauthors, too–in an article in their Leaders section.)
I have traveled for work within the US to Chicago (three times), Gainesville, FL, Washington, DC (three times), West Lafayette, IN, New York, Vail, CO, Columbus, OH (twice), Madison, WI, and New Orleans, LA; and internationally to the United Kingdom, back home to Canada (twice), and to Italy.
At Food Policy, my co-editor and I have seen our impact factor go up significantly yet again this year.
I have helped conduct, for the first time in my career, the external review of one of our peer departments, learning many new things in the process.
My personal favorite has been to see three of my PhD students successfully defend, graduate, and start in tenure-track positions that they are really happy with.
The most wonderful part of all this has mixed the personal with the professional: it has been to meet and talk with so many interesting people, old and new.
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Whether you have been reading this blog since its very beginning or you have only started recently, thank you from the bottom of my heart for making time to read what I write. When I registered this domain name and installed WordPress on the site, I wasn’t sure I could keep this blog going for a year, let alone seven.*
Happy New Year! I hope 2018 brings you joy, health, and prosperity.
* At that point, I had tried and failed many times to keep a diary. This only changed recently, as I now write a few pages in a daily journal every morning as part of a more elaborate early-morning ritual.
