Nonlinearities and the Macroeconomic E ff ects of Oil Prices* James D. Hamilton [email protected]Department of Economics University of California, San Diego December 9, 2009 Revised: November 15, 2010 ABSTRACT This paper reviews some of the literature on the macroeconomic effects of oil price shocks with a particular focus on possible nonlinearities in the relation and recent new results obtained by Kilian and Vigfusson (2009). ∗ I thank Rob Vigfusson for graciously sharing his data and helping me follow his code. An earlier version of this paper was circulated under the title, “Yes, the Response of the U.S. Economy to Energy Prices is Nonlinear.”
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Nonlinearities and the Macroeconomic Effects of OilPrices*
then there is a range of positive realizations of x#t that are defined to be a “negative oil
shock”. More generally, insofar as an impulse-response function is intended to summarize
the revision in expectations of future variables associated with a particular realization of
(9), as Gallant, Rossi, and Tauchen (1993), Koop, Pesaran and Potter (1996), and Potter
(2000) emphasized, such an object is in principle different for every different information set
{xt−1, yt−1, xt−2, yt−2, ...} and size of the shock ut. There are an infinite number of questions
one could ask about dynamic response functions in a nonlinear system, with a potentially
different answer for each history and each size shock. Which of these is “the” impulse-
response function of interest? For small shocks, one would expect from Taylor’s Theorem
that a linear representation of the function would be a good approximation around the
point of linearization. In most of their analysis, Kilian and Vigfusson seem to assume that
the object of interest is a one-standard deviation shock averaged across the dates in the
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sample. Given this decision as to the question they propose to answer, and particularly
given the underlying weak evidence of nonlinearity for their data set and specification, Kilian
and Vigfusson find limited evidence of nonlinearity in the impulse-response function. On
the other hand, by “oil shock,” many of us may instead have in mind the consequences of
extraordinary events. I note that, even with their favored specification and data set, Kilian
and Vigfusson find statistically significant evidence of nonlinearity when they examine the
effects of two-standard-deviation shocks..
In any case, there is a much simpler and direct way to get at this question. Any answer
from the infinite set of possible impulse-response functions in a nonlinear system is nothing
more than an answer to a particular conditional forecasting question plotted as a function of
the horizon. Jordá (2005) notes that it is possible to estimate the latter directly as primitive
objects independent of the equation used to forecast oil prices themselves, by simple OLS
estimation of the equation for forecasting GDP h periods ahead directly,
yt+h−1 = α+
pXi=1
φiyt−i +pX
i=1
βixt−i +pX
i=1
γix̃t−i + εt, (10)
on which one can readily test the null hypothesis of linearity in the form of the restriction
γ1 = γ2 = · · · = γp = 0. For h > 1, the errors in (10) are serially correlated, for which one
could correct using the regression-coefficient covariance matrix proposed by either Hansen
and Hodrick (1980) with h− 1 lags or by Newey and West (1987) using L > h lags.3
The Jordá (2005) approach is also perfectly valid when xt exhibits discrete dynamics, a
3 The Hansen-Hodrick results reported in Table 2 for h = 1 differ slightly from those reported in Table 1because for h = 1, the Hansen-Hodrick formula becomes White’s (1980) heteroskasticity-consistent estimaterather than the usual OLS, and the White standard errors turn out to be smaller than OLS standard errorsfor this application. The Newey-West results used L = 5 lags.
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case for which Kilian and Vigfusson (2010) note that their impulse-response analysis could
be problematic.
Table 2 reports results of these tests using both the Kilian-Vigfusson data set and spec-
ification (that is, xt the real RAC, p = 6, and t+ h− 1 running from 1974:Q4 to 2007:Q4)
and the original Hamilton (2003) data set and specification (xt the nominal PPI, p = 4, and
t+h− 1 running from 1949:Q2 to 2001:Q3). Interestingly, for every specification and every
horizon one finds quite strong evidence of nonlinearity.
The evident reconciliation of results is that, although there is not much evidence of a
nonlinear response to small changes in the Kilian-Vigfusson data set and specification, the
results are quite consistently indicating nonlinear consequences of larger movements in oil
prices.
5 Conclusion.
The evidence is convincing that the relation between GDP growth and oil prices is nonlinear.
The recent paper by Kilian and Vigfusson (2009) does not challenge that conclusion, but
does offer a useful reminder that we need to think carefully about what question we want
to ask with an impulse-response function in such a system and cannot rely on off-the-shelf
linear methods for an answer.
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Table 1 P-values for test of null hypothesis of linearity for alternative specifications
sample oil measure price
adjustment contemporaneous # lags p-value
(1) 1974:Q4-2007:Q4 RAC real include 6 0.046 (2) 1949:Q2-2001:Q3 PPI nominal exclude 4 0.002 (3) 1974:Q4-2007:Q4 PPI nominal exclude 4 0.013 (4) 1974:Q4-2007:Q4 PPI real include 6 0.024 (5) 1974:Q4-2007:Q4 RAC nominal include 6 0.028 (6) 1974:Q4-2007:Q4 RAC real exclude 6 0.027 (7) 1974:Q4-2007:Q4 RAC real include 4 0.036 Notes to Table 1: P-values for test that 0 1 0pγ γ γ= = = = in equation (6) (for rows with “include” in contemporaneous column) or test that 1 0pγ γ= = = in equation (4) (for rows with “exclude” in contemporaneous column). Boldface entries in each row indicate those details of the specification that differ from the first row.
Table 2 P-values for test of null hypothesis of linearity of h-quarter-ahead forecasts of real GDP using alternative data sets and specifications.
Forecast horizon Kilian and Vigfusson (2009) Hamilton (2003)
Hansen_Hodrick Newey-West Hansen-Hodrick Newey-West h = 1 0.002 0.000 0.001 0.000 h = 2 0.000 0.000 0.000 0.000 h = 3 0.000 0.037 0.000 0.000 h = 4 0.000 0.000 0.001 0.001