Flip Flop

This is a week filled with great angst. And, while I don't think a good electoral outcome "saves the republic," it does help stem the tide. And avert more unnecessary deaths. At least for a while. So, here's hoping the will of the people leads to a flip flop in our government. 

Our government needs a flip; not necessarily from one party to the other. It needs a flip from those in leadership positions that lack the ability or desire to lead to those that do. If politicians who are willing to and able to lead are concentrated in one party, so be it. If individuals on both sides of the aisle can stand up and lead, so be it. We need leaders in charge, not cowards. Or morons.

Alas, this is an econometrics blog, not a political one. So, I was thinking about flip flops for another reason. No, not because I have only failed to wear flip flops on only about six occasions since the pandemic began. No, not because I am typing this in my office at SMU in bare feet. In hopes of distracting myself some this week, I decided to take a break from being department chair and obsessing over the news to tell you about why I have flip flops on my mind.


In microeconometric applications there is widespread concern over selection on unobserved variables. As there should be. The presence of correlation between covariates included in the model and unobserved variables relegated to the error term must be addressed to identify causal effects. 


Historically, Instrumental Variable (IV) estimation has been the means to recover consistent estimates in the face of this problem in microeconometrics. However, since the mid-1990s, IV has lost its place atop the pantheon of selection on unobserved variables estimation techniques. In part, this is due to difficulty in interpreting IV when parameters are heterogeneous. But, more so this is due to a recognition of the potential poor properties of IV in the face of questionable instruments, small sample sizes, and weak instruments (e.g., here and here).


Confronted with a bruised and battered estimator, microeconometricians have sought other techniques to recover consistent estimates in the presence of selection on unobserved variables. Many of these alternatives rely on assumptions regarding higher order moments of the data, or other equally obtuse assumptions.


In a shameless plug, in two of my papers -- my job market paper, Millimet (2000), and the other chapter of my dissertation, Pitt et al. (2003) -- I circumvented the lack of instruments to overcome endogeneity via covariance restrictions. The idea, from my advisor, Mark Pitt, was to build on a literature from the 1980s (I think) and place restrictions on the covariance matrix of the reduced form errors in order to identify the parameters of interest. I like both papers to this day. However, the method has not caught on and probably should not. The identifying restrictions are hard to rationalize. Which I will return to below.


Perhaps the techniques that have garnered the most attention among applied researchers are estimators that exploit heteroskedasticity to achieve identification. Klein & Vella (2010) propose one such estimator (see here). Lewbel (1997, 2012) proposes a related estimator. Another shameless plug: with a former Ph.D. student of mine, Millimet & Roy (2016) use both estimators to assess the effects of environmental regulations. Learn more about Jayjit here! And his wife, Manan, who was also my student here! I am trying to stick around long enough to advise their kiddo.


So, let's now turn to the other side of our profession, macroeconometrics. 
You heard me. With my limited understanding of macroeconometrics, here is what I (think I) know. A workhorse model in macroeconometrics is the Vector Autorgression (VAR). Confronting issues of selection on unobserved variables is very, very difficult with time series data. Thus, VARs specify a system of equations for a vector of time series processes. Each process is modelled as a function of a finite number of lags of itself and the other variables in the system. Thus, all covariates in every equation in the system are lagged. If a sufficient number of lags are included such that the error terms are not serially correlated, then it is argued that Ordinary Least Squares (OLS), one equation at a time, produces consistent estimates.


Not content to leave well enough alone, a Structural VAR (SVAR) augments the set of covariates with the contemporaneous value of each of the other variables in the system. If the contemporaneous value of each variable affects the contemporaneous values of all other variables in the system, then all the contemporaneous values must be endogenous. Not good.


To overcome this problem, macroeconometricians have turned to a host of possibilities. Historically, these possibilities entail solving for the reduced form of the SVAR system, estimating this using OLS, and then backing out the structural parameters. To do so, however, requires restrictions as there are more parameters (in the structural model) than equations (parameters in the reduced form model). As discussed in Stock (2008), Fry & Pagan (2011), and in time series textbooks, macroeconomists have relied on parametric restrictions and, more recently, sign restrictions for identification. In a twist, frankly I think this is similar to what I did in my dissertation.


However, these restrictions (as in most things in macroeconomics! and as in my dissertation papers!) are often obtuse and difficult to rationalize. So, what are macroeconometricians to do? Well, it seems now they are turning to ... IV. 

Montiel Olea et al. (2020) discuss identification in SVARs using traditional, microeconometric-like, external instruments. Good old, IV!

Do you see it now? Flip flops. 

Microeconometrics has relied on IV for nearly a century. Only recently, however, it has come intense scrutiny and has fallen out of favor among many applied researchers. Instead, researchers turn to alternative estimators, some of which circumvent the need for a traditional external instrument by instead invoking restrictions on higher moments of the data or making other equally obtuse assumptions. 

Macroeconometrics has eschewed traditional external instruments for decades, instead opting for restrictions on higher moments of the data or making other equally obtuse assumptions. Only recently, however, these approaches have come under intense scrutiny and have fallen out of favor among many applied researchers. Instead, prominent macroeconometricians are touting a (re-)turn to traditional external instruments.

Flip flop. 

In econometrics, as in life, everything comes (and goes) in waves. In vogue. Out of vogue. In. Out. It's true in life and true for methods of achieving identification. 

Let's hope it's also true for our democratic principles.


Good mental health to you all!

References

Fry, R. and A. Pagan (2011), "Sign Restrictions in Structural Vector Autoregressions: A Critical Review," Journal of Economic Literature, 49, 938-960

Klein, R. and F. Vella (2010), "Estimating a Class of Triangular Simultaneous Equations Models Without Exclusion Restrictions," Journal of Econometrics, 154, 154-164

Lewbel A. (1997), "Constructing Instruments for Regressions with Measurement Error When No Additional Data are Available, with an Application to Patents and R&D," Econometrica, 65, 1201- 1213

Lewbel A. (2012), "Using Heteroskedasticity to Identify and Estimate Mismeasured and Endogenous Variables," Journal of Business and Economic Statistics, 30, 67-80

Millimet, D.L. (2000), "The Impact of Children on Wages, Job Tenure, and the Division of Household Labor," Economic Journal, 110, C139-C157

Millimet, D.L. and J. Roy (2016), "Empirical Tests of the Pollution Haven Hypothesis When Environmental Regulation is Endogenous," Journal of Applied Econometrics, 31, 652-677

Montiel Olea, J.L., J.H. Stock, and M.W. Watson (2020), "Inference in Structural Vector Autoregressions Identified With an External Instrument," Journal of Econometrics, forthcoming

Pitt, M.M., S. Khandker, O.H. Chowdhury, and D.L. Millimet (2003), "Credit Programs for the Poor and theHealth Status of Children in Rural Bangladesh," International Economic Review, 44, 87-118

Stock, J.H. (2008), "What is New in Econometrics: Time Series, Lecture 7," Short course lectures, 26 NBER Summer Institute

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