M acroeconom ics and A R C H*James D. Ham iltonDepartm ent of EconomicsUniversity of California, San Diegojhamilton @ u csd.eduSeptember 24, 2007Revised: May 27, 2008*Prepared for the Festschrift in Honor of Robert F. Engle (eds. Tim Bollerslev, JeffreyR. Russell and Mark Watson).AbstractAlthough ARCH-related models ha v e pro v en quite popular in finance, they are less fre-quently used in macroeconomic applications. In part this may be because macroeconomistsare usually more concerned about chara cterizing the conditional mean rather than the con-ditional variance of a time series. This pape r argues that ev en if one’s intere st is in theconditiona l mean, correctly mode ling the conditional variance can still be quite importan t,for t wo reasons. First, OLS standard errors can be quite misleading, with a “spurio us regres-sion” possib ility in which a true n u ll h y pothesis is asym p totic ally rejected with probabilityone. Second, the inference about the conditional mean can be inappropriately influenced b youtliers and high-variance episodes if one has not incorporated the conditional variance di-rectly into the estima tion of the mean, and infinite relativ e efficie ncy gain s may be possible.The practical relevan ce of t h ese concern s is illustrated with two empir ical examples fromthe macroeconomics literature, the first looking at market expectations of future chan ges inFederal Reserve policy, and the second looking at changes o ver time in the Fed’s adherencetoaTaylorRule.11 Introduction.One of the most influen tia l econometric papers of the last generation w as Engle’s (1982a)in t r oduction of autoregressive conditional heteroske da sticity (ARCH) as a tool for describinghow the conditional variance of a time series ev olves ove r time. The ISI Web of Scien c e listsover 2000 academic stud ies that have cited this article, an d simply reciting the acronym sfor the various extensions of Engle’s theme inv olv es a not insignifica nt commitment of pa per(see Table 1, or the more deta iled glossary in Bollerslev, 2008).The vast majority of empirical applic ations of ARCH models hav e studied financial timeseries such as stock prices, interest rates, or exchange rates. To be sure, there ha ve alsobeen a num ber of interesting applications of AR CH to macroeconomic questions. Lee, Ni,and Ratti (1995) noted that the conditional volatilit y of oil prices, as captured b y a GARCHmodel, seems to matter for the magnitud e of the effect on GDP of a given mov ement inoil prices, and Elder and Serletis (2006) use a vector autoregression with GARCH-in-meaneleme nts to describe the direct consequenc es of oil-price volatility for GDP. Grier and P e rry(2000) and Fountas and Karanasos (2007) use such models to conclude that inflation andoutput v olatility also can depress real GDP gro wth, while Servén (2003) studied the effectsof uncertain t y on investment spending.Howev er, despite these in te re stin g application s, studying v olatility has traditionally beena muc h lower priority for macroeconomists than for researchers in financial markets becausethe former’s interest is primarily in describing the first moment s. There seems to be an as-sumption among many macroeco no mists that, if y our primary in ter est is in the first moment,2ARCH has little relevance apart from possible GARCH-M effects.The pur pose of this paper is to sug gest that even if our primary intere st is in estimatin gthe conditio na l mean, having a correct descrip tion of the conditional variance can still bequite important, for t wo reasons. First, hypothesis tests about the mean in a model inwhich the varianc e is misspecified will be invalid. Second , b y incorporating the observedfeatures of the heteroskedasticit y into the estimation of the conditional mean, substantiallymore efficien t estimates of the conditional mean can be obtained.Section 2 develop s the theoretical basis for these claims, illustrating the potential magni-tude of the problem with a small Monte Carlo study and explaining why the popular White(1980) or New ey-West (1987) corrections may not fully correct for the inference problemsin troduced by ARC H . The subsequ e nt sections illustrate the practic al relevanc e of theseconcerns using two examples from the macroeconomics literature. The first applicationconcerns measures of what the mark et expects the U.S. Federal Reserve’s next mo ve tobe, and the second explores the exten t to which U.S. monetary policy toda y is follow ing afundamentally different rule from that observ ed thirty years ago.I recognize that it may require mor e than these limited examples to per suade macro-economists to pa y more atten tion to AR CH. Another thing I learned from Rob Engle isthat, in addition to coming up with a great idea, it doesn’t hu rt if yo u also ha ve a catc hyacronym that people can use to describe what you’re talking about. After all, where wouldw e be toda y if we all had to pronounce “autoregressive conditional heterosk edasticit y” ev erytime w e wanted to discuss these issues? H owever, Table 1 rev eals that the acron y ms one3migh t logically use for “Macroeconomics and AR CH” seem already to be taken. “M AR CH”,for example, is already used (t wice), as is “ARCH-M”.Fortunately, Engle and Man ganelli (2004) hav e shown us that it’s also OK to mix upper-and lo wer-case letters, pick ing and c hoosing handy v owels or consonants so as to come upwith something catch y, as in “CAViaR” (Conditional Autoregres sive Value at Risk). Inthat spirit, I propose to designate “Macroeconomics and ARCH” as “McA RCH.” Maybenot a new product so m uc h as new packaging.Herewith, then, discussion of the relevance of McARCH.2 GA RCH and inference about the mean.We can illustrate some of the iss ues with the follo wing simple mode l:yt= β0+ β1yt−1+ ut(1)ut=phtvt(2)ht= κ + αu2t−1+ δht−1for t =1, 2, ..., Th0= κ/(1 − α − δ)vt∼ i.i.d. N(0, 1). (3)Bollerslev (1986, pp. 312-313) sho wed that if3α2+2αδ + δ2< 1, (4)4then the noncen tral unconditional second and fourth moments of utexist and are given byµ2= E(u2t)=κ1 − α − δ(5)µ4= E(u4t)=3κ2(1 + α + δ)(1 − α − δ)(1 − δ2− 2αδ − 3α2). (6)Consider the consequences if the mean parameters β0and β1are estim ated by ordinary leastsquares,ˆβ =³Xxtx0t´−1³Xxtyt´β =(β0, β1)0xt=(1,yt−1)0,and where all summ ation s are for t =1, ..., T. Suppose