Friday, June 12, 2015

Linear Literalism & Fundamentalist Econometrics

Your tweet stream may have included the recently trending article by Angrist and Pischke entitled: "Why econometrics teaching needs an overhaul". Here is an excerpt:

“In addition to its more up-to-date contents, our book renews the econometrics canon by abandoning the childish literalism of the legacy approach to econometric instruction. In this spirit, we eschew the notion that regression is tied to a literal linear model. Regression describes differences in averages whether or not these averages fit a linear equation. This is a universal property – one that is reliably true – and we don’t intimidate readers with descriptions of the punishments to be meted out for the failure of classical assumptions. Our regression discussion begins by challenging readers to ask themselves, first, what the target causal effect is, and, second, by asking, ‘what is the regression you want’? In other words, what would you like to hold fixed when trying to regress-out an average causal effect?”

They are referencing their text Mastering Metrics, which I highly recommend.  In their other text, Mostly Harmles Econometrics, they also state:

"In fact, the validity of linear regression as an empirical tool does not turn on linearity either...The statement that regression approximates the CEF lines up with our view of empirical work as an effort to describe the essential features of statistical relationships, without necessarily trying to pin them down exactly."  - Mostly Harmless Econometrics, p. 26 & 29

On their MHE blog a reader asks about their pedagogy which focuses on 'best linear projection' vs the traditional BLUE criteria to which they respond:

"our undergrad econometrics training (like most people’s) focused on the sampling distribution of OLS. Hence you were tortured with the Gauss-Markov Thm, which says that OLS is a Best Linear Unbiased Estimator (BLUE). MHE and MM are largely unconcerned with such things. Rather, we try to give our students a clear understanding of what regression means. To that end, we introduce regression as the best linear approximation to whatever conditional expectation fn. (CEF) motivates your empirical work – this is the BLP property you mention, which is a regression feature unrelated to samples. (MM also emphasises our interpretation of regression as a form of “automated matching”)." read more...

The notion of using regression as a means of making like comparisons has also been echoed by Andrew Gelman:

http://andrewgelman.com/2013/01/understanding-regression-models-and-regression-coefficients/

"It's all about comparisons, nothing about how a variable "responds to change." Why? Because, in its most basic form, regression tells you nothing at all about change. It's a structured way of computing average comparisons in data."

Linear literalism or fundamentalist undergraduate econometrics (being tortured with BLUE as A&P might put it) can have long term consequences for students. I think this has caused harms that I encounter from time to time even among more seasoned practitioners and even graduate degree holders. This isn't too different from what Leo Brieman described as a 'statistical straight jacket' that can arbitrarily limit fruitful empirical work. Overly clinical concerns with linearity, heteroskedasticity, and multicollinearity might crowd out more important concerns around causality and prediction.

Heteroskedasticity

We could simplify this as a notion of non-constant variance. As Angrist and Pischke note:

"Our view of regression as an approximation to the CEF makes heteroskedasticity seem natural. If the CEF is nonlinear....the residuals will be larger, on average, at values of X where the fit is poorer...as an empirical matter, heteroskedasticity may matter little" -Ch 3, p.46-47 MHE

 Of course the concern is correct standard errors and valid inference, which can be addressed via heteroskedasticity corrected standard errors. But I am afraid some students, after taking a traditional econometrics course, may terminate all thought processes after a cookbook test hints of its existence.

Multicollinearity

When covariates are highly correlated, it may be difficult to parse out the independent information about each variable and  lead to inflated standard errors. Again this is a phenomena related to inference, not prediction. Even in professional and academic settings when I have presented or attended other presentations related to forecasting or predictive analtyics you will get the occasional criticism or self aggrandizing questions about multicollinearity being a concern.

"Multicollinearity has a very different impact if your goal is prediction from when your goal is estimation. When predicting, multicollinearity is not really a problem provided the values of your predictors lie within the hyper-​​region of the predictors used when estimating the model."-  Statist. Sci.  Volume 25, Number 3 (2010), 289-310.

Undue criticisms and literalism related to multicollinearity often results from a failure to recognize the differences between goals related to explaining vs. predicting.

Paul Allison offers some additional advice on when not to worry about multicolinearity. I have highlighted a couple points of interest here.

This kind of linear fundamentalist paradigm can lead students and practitioners to adopt more complicated methods than necessary or abandon promising empirical work altogether, become overly critical and dismissive of other important work done by others, or completely miss more important questions related to selection bias and identification and unobserved heterogeneity and endogeneity.

Some of this also is a the result of the huge gap between theoretical and applied econometrics.

See also:
Marc Bellemare discusses a similar vein of literalism that is averse to linear probability models here.

Linear Probability Models

Regression as an empirical tool
Quasi-Experimental Design Roundup

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