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 R^{2} 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 (*p*_{t}) on last month's stock price (*p*_{t-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 (Δ*p*_{t} = p_{t} - *p*_{t-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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