Monday, February 10, 2014

Models and metrics

I realized I hadn't given a lot of time to the metrics for the model fits. It turns out that they aren't really very informative -- errors are correlated so metrics like R squared tend not to show which model is worse. In fact, the adjusted R squared for the information transfer model (ITM) is 0.999 for both the monetary base including reserves (MB) and currency component (M0) versions. True it is "only" 0.9 for the quantity theory of money (QTM) where P = k  M0 and 0.75 for a constant (P = k), but therein lies the problem. However residuals tend to be a good metric and in this case tend to be a little more informative, so I'll present those.

First, here are the model fits to the price level (clockwise from top left: QTM, ITM constant κ [1], ITM M0 and ITM MB):


The residuals are given here (color coded to the graphs above):


The mean residual fractional error is 0.03 for the ITM M0 model, 0.04 for the ITM MB model, 0.10 for the ITM constant κ model and 0.37 for the QTM (horizontal lines). Basically, anyone who takes the quantity theory of money seriously should take the ITM even more seriously, but that isn't saying much except to hard core monetarists.

Here are the errors for the ITM MB and ITM M0 models blown up and put on a linear scale:


You can see that the recent QE is the only obvious reason to reject the ITM MB model -- otherwise the errors are really close to each other [2]. But then the recent QE is the major reason MB differs from M0! Fitting the recent years makes the years in 70s and 80s worse in the MB version of the ITM than the M0 version.

Still, the errors are correlated. They don't seem to have any particular relation to recessions. There are two periods of especially low error that are associated with economic booms (the late 60s and late 90s) and the period of high inflation from the 70s and 80s is associated with the largest persistent error. However, the ITM M0 version doesn't sustain 5% errors for very long.

[1] Constant κ uses the same function as the M0 and MB models P ~ (1/κ) M0^(1/κ - 1), but keeps κ constant.

[2] Trying to match the derivative (inflation rate) will give an additional reason to accept the ITM M0 version over the MB version.


2 comments:

  1. Jason you write:

    "In fact, the adjusted R squared for the information transfer model (ITM) is 0.999 for both the monetary base including reserves (MB) and currency component (M0) versions. True it is "only" 0.9 for the quantity theory of money (QTM) where P = k M0 and 0.75 for a constant (P = k), but therein lies the problem."

    Did you really mean to write "P = k" for the last one? Since you write about a "constant k" ITM later. So the four models you mention here are

    1. ITM with MB
    2. ITM with M0
    3. QTM
    4. P = k

    But then in your first figure (the four charts) we have

    1. QTM
    2. ITM M0 (constant k)
    3. ITM M0 (non-constant k)
    4. ITM MB (non-constant k)

    Those two sets seem to match up (although in different orders), except for the "P = k" part. All four match up if I ignore "P = k" and instead call that "constant k." A typo?

    When you say "the errors are correlated" do you mean in time? I.e that the error sequence, for any one particular model, has a non-delta function auto-correlation (i.e. it isn't white noise)?

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    Replies
    1. Actually the P= k (P ~ constant) was kind of a joke. The R^2 for a model that says the price level doesn't change is 0.75 ... Which means R^2 isn't a real useful metric here.

      And yes, the errors are temporally correlated so the residuals aren't white noise. If the error on the previous point is > x then it is likely the next point is > x.

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