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January 23, 2014

'On R-Squared and Economic Prediction'

Jim Hamilton:
On R-squared and economic prediction: Recently I've heard a number of otherwise intelligent people assess an economic hypothesis based on the R2 of an estimated regression. I'd like to point out why that can often be very misleading. ...
Here's what you'd find if you calculated a regression of this month's stock price (pt) on last month's stock price (pt-1). Standard errors of the regression coefficients are in parentheses.
The adjusted R-squared for this relation is 0.997. ... On the other hand, another way you could summarize the same relation is by using the change in the stock price (Δpt = pt - pt-1) as the left-hand variable in the regression:
This is in fact the identical model of stock prices as the first regression. The standard errors of the regression coefficients are identical for the two regressions, and the standard error of the estimate ... is identical for the two regressions because indeed the residuals are identical for every observation. ...
Whatever you do, don't say that the first model is good given its high R-squared and the second model is bad given its low R-squared, because equations (1) and (2) represent the identical model. ...
That's not a bad empirical description of stock prices-- nobody can really predict them. ... This is actually a feature of a broad class of dynamic economic models, which posit that ... the deviation between what actually happens and what the decision-maker intended ... should be impossible to predict if the decision-maker is behaving rationally. For example, if everybody knew that a recession is coming 6 months down the road, the Fed should be more expansionary today... The implication is that when recessions do occur, they should catch the Fed and everyone else by surprise.
It's very helpful to look critically at which magnitudes we can predict and which we can't, and at whether that predictability or lack of predictability is consistent with our economic understanding of what is going on. But if what you think you learned in your statistics class was that you should always judge how good a model is by looking at the R-squared of a regression, then I hope that today you learned something new.
[There's an additional example and more explanation in the original post.]

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