Wednesday, February 16, 2011

Macroeconomics and ARCH

That's the topic of my most recent research paper. Reader warning: this is a bit more technical than the standard Econbrowser post, so if you're not a user of regression analysis, this may not be up your alley.

One of the contributions for which my colleague Rob Engle received the Nobel Memorial Prize in Economics was development of ARCH, a class of models for predicting the volatility of a variable. One's first priority might be to predict the level of the variable, such as asking what the price of oil will be next month. With ARCH models, we instead try to predict the absolute or squared value of the change-- are oil prices likely to change more this month than usual?

ARCH models have become popular in finance, where measuring the volatility is extremely important for purposes of characterizing the riskiness of portfolios. They have been less used by macroeconomists, who are usually interested in predicting how the levels of variables might change under different circumstances.

In my latest research paper, I argue that even if one's primary interest is in measuring consequences for the levels, it can be very important to use ARCH to model any changes in the volatility, for two reasons. First, correcting for outliers can give you much more accurate estimates of the parameters you're interested in. Second, if you make no corrections, there is a possibility of a kind of spurious regression. The expression spurious regression is known to economists as a common finding when you regress two variables that have nothing in common except a tendency to drift from their starting values; (Jesus Gonzalo has some amusing examples). The result is high t statistics that would lead you to reject the null hypothesis of no relation, even though the null hypothesis is surely true.

I found there's a possibility of something similar arising if you rely on the usual OLS test of a hypothesis about a lagged dependent variable in a regression that is characterized by ARCH. If the sample size is large enough, you are certain to reject the null hypothesis that the coefficient is zero, even if the null hypothesis is true. For example, the diagram below shows the asymptotic probability you will reject a true null hypothesis of zero serial correlation as a function of the parameters α and δ of a GARCH(1,1) process for the residuals. This would be a flat plane at height 0.05 if the t test were doing what you expected, rejecting only 5% of the time when the null hypothesis is true. In fact, for α and δ in the range often found for macroeconomic series, you'd end up wrongly rejecting 100% of the time with the standard t statistic.


My paper also illustrates these issues with a couple of examples taken from the macroeconomics literature. The one I'll discuss here involves estimation of the Taylor Rule, which is a description of how the Federal Reserve changes its target for the fed funds rate in response to variables such as inflation and GDP. The conventional understanding by most macroeconomists is that since 1979, the Fed has responded more aggressively to deviations of inflation or GDP from their desired levels, and that this change in policy has helped to stabilize the economy.

The first row in the table below reproduces that finding and its apparent statistical significance using OLS estimates of the coefficients and their standard errors. However, there is strong evidence of ARCH dynamics in the residuals of this regression. When one takes those into account in the estimation, the change in the responsiveness to inflation is 1/3 the OLS estimate, while the changed responsiveness to output is less than 1/10 of the magnitude one would have inferred by OLS.

Post-1979 change in estimated coefficients (and standard errors) relating change in fed funds rate to change in inflation rate and deviation of output from potential as estimated by OLS and GARCH

inflation(std err)output(std err)
OLS0.26 (0.09) 0.64 (0.14)
GARCH0.09 (0.04) 0.05 (0.07)

The diagram below displays the features of the data that are responsible for this result. The top panel is the monthly change in the fed funds rate, in which the ARCH features are quite apparent, with increased volatility particularly over the 1979-82 period. The bottom panel is the scatter diagram relating the change in the fed funds rate (vertical axis) to the output gap (horizontal axis) over the 1979-2007 subperiod. The apparent positive slope is strongly influenced by those observations for which the variability of interest rates is highest. Because GARCH downweights these observations for purposes of estimating the slope, the post-1979 response of the Federal Reserve to the output gap is significantly smaller than that estimated by OLS.


The recommendation that the paper offers for macroeconomic researchers is quite simple. It is extremely straightforward to test for the presence of ARCH effects-- just look at the R2 of a regression of the squared residuals on their own lagged values. Macroeconomists might want to glance at this diagnostic statistic even if their primary interest is not the volatility but some other feature of the data.

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