Thursday, January 26, 2012

A brief parable of over-differencing

The Grumpy Economist has sat through one too many seminars with triple differenced data, 5 fixed effects and 30 willy-nilly controls. I wrote up a little note (7 pages, but too long for a blog post), relating the experience (from a Bob Lucas paper) that made me skeptical of highly processed empirical work.

The graph here shows velocity and interest rates.  You can see the nice sensible relationship.

(The graph has an important lesson for policy debates. There is a lot of puzzling why people and companies are sitting on so much cash. Well, at zero interest rates, the opportunity cost of holding cash is zero, so it's a wonder they don't hold more. This measure of velocity is tracking interest rates with exactly the historical pattern.) 

But when you run the regression, the econometrics books tell you to use first differences, and then the whole relationship falls apart. The estimated coefficient falls by a factor of 10, and a scatterplot shows no reliable relationship.  See the the note for details, but you can see in the second graph  how differencing throws out the important variation in the data. 

The perils of over differencing, too many fixed effects, too many controls, and that GLS or maximum likelihood will jump on silly implications of necessarily simplified theories are well known in principle. But a few clear parables might make people more wary in practice.  Needed: a similarly clear panel-data example.

17 comments:

  1. From the note:

    "When interest rates rise, the opportunity cost of holding cash rather than interest-bearing assets falls."

    This is backwards, no?

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    1. pretty sure it is, but i didn't want to be the one to point it out...

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  2. >the econometrics books tell you to use first differences

    Perhaps the econometrics books are wrong. I am sure if one regress V_{i+m} - V_{i} against r_{i+m} - r_i, where m > 1 the quality of the regression will improve, but what is the "correct" value of m then?


    The whole issue - whether to over-difference or not - has no practical significance from the probabilistic viewpoint - if one has a model of the underlying process then the regression quality (t-statistics or whatever other measure one uses for that) is the quality of the model. Bayesians' talk about probability of the model or its parameters makes much more sense than whether one should take first differences or one should not.

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    1. Clearly you are not an engineer. I'm a researcher in signal processing and you don't need to go into technical detail to know simple rules of thumb like, differencing amplifies noise, the ratio of two noisy numbers is noisier, etc. And yes, we have Bayesian models of the processes. And they are useful but are almost always wrong. I find economists often place way too much faith in their models; a common sense, first-order approximation post like this is a breath of fresh air.

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    2. I am not sure if it's relevant whether I am an engineer or not. I think John's parable is directed precisely against simple rules of thumb (first differences).

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  3. The problem is, our "models" are really quantitative parables. You can't ask a formula, Bayesian or not, to tell you where a parable makes sense and where it does not.

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  4. Von Neumann said it best:

    "With four parameters I can fit an elephant, and with five I can make him wiggle his trunk."

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  5. True, almost all "models" in social sciences are quantitative parables. While real models have greater or smaller predictive power, parables have not. Are we ready to deal with the implications of this fact? For instance, can we base our policy recommendations on parables - quantitative or not?

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  6. Pardon my econometric rustiness (exacerbated by being out of Academia and in Industry), but couldn't you just cointegrate this sucker?

    More broadly, when there is a cointegrating relationship b/w the variables, is the point about the perils of first-differencing less valid?

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    1. You should read Lucas' paper :D

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    2. "when there is a cointegrating relationship b/w the variables, is the point about the perils of first-differencing less valid?"

      No, the opposite is true. When the variables are cointegrated then the level regression is correctly specified and if you difference then you introduce a unit root that destroys your regression.

      The whole point here is that money demand is cointegrated with the interest rate so you don't want to difference your data.

      You could view differencing as an attempt to find a cointegrating relationship if one isn't already there, the non-stationary variable is cointegrated with its own lagged value.

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  7. Why would the textbook say to take differences? You only do that if the processes are non sationary, but they look broadly stationary to me.

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  8. 1. Two curves you show in the upper panel are obviously not stationary ones and any unit root test would confirm that assertion.

    2. For two not stationary time series to be linked by a valid correlation relation, i.e. by a link confirmed by methods from textbooks, it should be a linear combination of those two series which has a I(0) residual error time series. It is necessary since the non stationary time series will diverge in the long run if the residual error is not a I(0).

    3. The lower panel shows that there is no linear relation between the original (not stationary) times series which provides an I(0) residual error. This means that one should not use the link between the original time series as obtained by linear regression in the long run. Coefficients of linear regresion are biased and the relation will not guarantee the convergence of the nonstationary time series. Thus, there is no link in the long run despite you may "see"it in the short run.

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    1. "The lower panel shows that there is no linear relation between the original (not stationary) times series which provides an I(0) residual error."

      The lower panel shows no such thing, in fact in the note Cochrane displays 4 year difference that have a very tight correlation and a clearly stationary residual. In the long run the regression is valid.

      Thus, there is a link in the long run despite the fact that you may not "see" it in the high frequency differenced data.

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  9. I agree with Ivan, original series are non-stationary thus the regression results may be spurious in levels. If the series are cointegrated then the levels regression correspond to the cointegration vector regression. Thus if the series are cointegrated , there is not need for differences. But if the series are not cointegrated, then we need to differentiate.

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  10. With all due respect I agree with some people above. You are missing co-integration which is a reasonable prior if you have in mind some monetary models with frictions

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