Macroeconomics and ARCH

NBER WORKING PAPER SERIES MACROECONOMICS AND ARCH James D. Hamilton Working Paper 14151 http://www.nber.org/papers/w14151 NATIONAL BUREAU OF ECONOMIC RESEARCH 1050 Massachusetts Avenue Cambridge, MA 02138 June 2008 Prepared for the Festschrift in Honor of Robert F. Engle (eds. Tim Bollerslev, Jeffrey R. Russell, and Mark Watson). The views expressed herein are those of the author(s) and do not necessarily reflect the views of the National Bureau of Economic Research. NBER working papers are circulated for discussion and comment purposes. They have not been peerreviewed or been subject to the review by the NBER Board of Directors that accompanies official NBER publications. © 2008 by James D. Hamilton. All rights reserved. Short sections of text, not to exceed two paragraphs, may be quoted without explicit permission provided that full credit, including © notice, is given to the source. Macroeconomics and ARCH James D. Hamilton NBER Working Paper No. 14151 June 2008 JEL No. E52 ABSTRACT Although ARCH-related models have proven quite popular in finance, they are less frequently used in macroeconomic applications. In part this may be because macroeconomists are usually more concerned about characterizing the conditional mean rather than the conditional variance of a time series. This paper argues that even if one's interest is in the conditional mean, correctly modeling the conditional variance can still be quite important, for two reasons. First, OLS standard errors can be quite misleading, with a "spurious regression" possibility in which a true null hypothesis is asymptotically rejected with probability one. Second, the inference about the conditional mean can be inappropriately influenced by outliers and high-variance episodes if one has not incorporated the conditional variance directly into the estimation of the mean, and infinite relative efficiency gains may be possible. The practical relevance of these concerns is illustrated with two empirical examples from the macroeconomics literature, the first looking at market expectations of future changes in Federal Reserve policy, and the second looking at changes over time in the Fed's adherence to a Taylor Rule. James D. Hamilton Department of Economics, 0508 University of California, San Diego 9500 Gilman Drive La Jolla, CA 92093-0508 and NBER jhamilton@ucsd.edu 1 Introduction. One of the most influential econometric papers of the last generation was Engle’s (1982a) introduction of autoregressive conditional heteroskedasticity (ARCH) as a tool for describing how the conditional variance of a time series evolves over time. The ISI Web of Science lists over 2000 academic studies that have cited this article, and simply reciting the acronyms for the various extensions of Engle’s theme involves a not insignificant commitment of paper (see Table 1, or the more detailed glossary in Bollerslev, 2008). The vast majority of empirical applications of ARCH models have studied financial time series such as stock prices, interest rates, or exchange rates. To be sure, there have also been a number of interesting applications of ARCH to macroeconomic questions. Lee, Ni, and Ratti (1995) noted that the conditional volatility of oil prices, as captured by a GARCH model, seems to matter for the magnitude of the effect on GDP of a given movement in oil prices, and Elder and Serletis (2006) use a vector autoregression with GARCH-in-mean elements to describe the direct consequences of oil-price volatility for GDP. Grier and Perry (2000) and Fountas and Karanasos (2007) use such models to conclude that inflation and output volatility also can depress real GDP growth, while Servén (2003) studied the effects of uncertainty on investment spending. However, despite these interesting applications, studying volatility has traditionally been a much lower priority for macroeconomists than for researchers in financial markets because the former’s interest is primarily in describing the first moments. There seems to be an assumption among many macroeconomists that, if your primary interest is in the first moment, 2 ARCH has little relevance apart from possible GARCH-M effects. The purpose of this paper is to suggest that even if our primary interest is in estimating the conditional mean, having a correct description of the conditional variance can still be quite important, for two reasons. First, hypothesis tests about the mean in a model in which the variance is misspecified will be invalid. Second, by incorporating the observed features of the heteroskedasticity into the estimation of the conditional mean, substantially more efficient estimates of the conditional mean can be obtained. Section 2 develops the theoretical basis for these claims, illustrating the potential magnitude of the problem with a small Monte Carlo study and explaining why the popular White (1980) or Newey-West (1987) corrections may not fully correct for the inference problems introduced by ARCH. The subsequent sections illustrate the practical relevance of these concerns using two examples from the macroeconomics literature. The first application concerns measures of what the market expects the U.S. Federal Reserve’s next move to be, and the second explores the extent to which U.S. monetary policy today is following a fundamentally different rule from that observed thirty years ago. I recognize that it may require more than these limited examples to persuade macroeconomists to pay more attention to ARCH. Another thing I learned from Rob Engle is that, in addition to coming up with a great idea, it doesn’t hurt if you also have a catchy acronym that people can use to describe what you’re talking about. After all, where would we be today if we all had to pronounce “autoregressive conditional heteroskedasticity” every time we wanted to discuss these issues? However, Table 1 reveals that the acronyms one 3 might logically use for “Macroeconomics and ARCH” seem already to be taken. “MARCH”, for example, is already used (twice), as is “ARCH-M”. Fortunately, Engle and Manganelli (2004) have shown us that it’s also OK to mix upperand lower-case letters, picking and choosing handy vowels or consonants so as to come up with something catchy, as in “CAViaR” (Conditional Autoregressive Value at Risk). In that spirit, I propose to designate “Macroeconomics and ARCH” as “McARCH.” Maybe not a new product so much as new packaging. Herewith, then, discussion of the relevance of McARCH. 2 GARCH and inference about the mean. We can illustrate some of the issues with the following simple model: yt = β 0 + β 1 yt−1 + ut ut = p ht vt ht = κ + αu2t−1 + δht−1 (1) (2) for t = 1, 2, ..., T h0 = κ/(1 − α − δ) vt ∼ i.i.d. N(0, 1). (3) Bollerslev (1986, pp. 312-313) showed that if 3α2 + 2αδ + δ 2 < 1, 4 (4) then the noncentral unconditional second and fourth moments of ut exist and are given by κ 1−α−δ (5) 3κ2 (1 + α + δ) . (1 − α − δ)(1 − δ 2 − 2αδ − 3α2 ) (6) µ2 = E(u2t ) = µ4 = E(u4t ) = Consider the consequences if the mean parameters β 0 and β 1 are estimated by ordinary least squares, β̂ = ³X 0 xt xt ´−1 ³X xt yt ´ β = (β 0 , β 1 )0 xt = (1, yt−1 )0 , and where all summations are for t = 1, ..., T. Suppose further that inference is based on the usual OLS formula for the variance, with no correction for heteroskedasticity: V̂ = s2 ³X 0 xt xt s2 = (T − 2)−1 ´−1 X (7) û2t 0 ût = yt − xt β̂ Consider first the consequences of this inference when the fourth-moment condition (4) is satisfied. For simplicity of exposition, consider the case when the true value of β = 0. Then from the standard consistency results (e.g., Lee and Hansen, 1994; Lumsdaine, 1996) 5 we see that ³ ´ X 0 −1 T V̂ = s2 T −1 xt xt  (8) −1 E(yt−1 )   1 p  → E(u2t )    2 ) E(yt−1 ) E(yt−1 −1   µ2 0   =    0 1 . In other words, the OLS formulas will lead us to act as if √ T β̂ 1 is approximately N(0, 1) if the true value of β 1 is zero. But notice ³ ´ ³ ´ X X √ 0 −1 T (β̂ − β) = T −1 xt xt T −1/2 xt ut . (9) Under the null hypothesis, the term inside the second summation, xt ut , is a martingale difference sequence with variance 0 E(u2t xt xt )   E(u2t ut−1 )   E(u2t ) .  =  E(ut−1 u2t ) E(u2t u2t−1 ) When the (2,2) element of this matrix is finite, it then follows from the Central Limit Theorem (e.g., Hamilton, 1994, p. 173) that T −1/2 X ¡ ¡ ¢¢ L yt−1 ut → N 0, E u2t u2t−1 . (10) To calculate the value of this variance, recall (e.g., Hamilton, 1994, p. 666) that the GARCH(1,1) structure for ut implies an ARMA(1,1) structure for u2t : u2t = κ + (δ + α)u2t−1 + wt − δwt−1 6 for wt−1 a white noise process. It follows from the first-order autocovariance for an ARMA(1,1) process (e.g., Box and Jenkins, 1976, p. 76) that E(u2t u2t−1 ) = E(u2t − µ2 )(u2t−1 − µ2 ) + µ22 = ρ(µ4 − µ22 ) + µ22 (11) for ρ= [1 − (α + δ)δ]α . 1 + δ 2 − 2(α + δ)δ (12) Substituting (11), (10) and (8) into (9), √ L T β̂ 1 → N (0, V11 ) V11 ρµ4 + (1 − ρ)µ22 = µ22 3(1 + α + δ)(1 − α − δ) + (1 − ρ). = ρ (1 − δ 2 − 2αδ − 3α2 ) with the last equality following from (5) and (6). Notice that V11 ≥ 1, with equality if and only if α = 0. Thus OLS treats √ T β̂ 1 as approximately N(0, 1), whereas the true asymptotic distribution is Normal with a variance bigger than unity, meaning that the OLS t test will systematically reject more often than it should. The probability of rejecting the null hypothesis that β 1 = 0 (even though the null hypothesis is true) gets bigger and bigger as the parameters get closer to the region at which the fourth moment becomes infinite, at which point the asymptotic rejection probability becomes unity. Figure 1 plots the rejection probability as a function of a and δ. If these parameters are in the range typically found in estimates of GARCH processes, an OLS t 7 test with no correction for heteroskedasticity would spuriously reject with arbitrarily high probability for a sufficiently large sample. The good news is that the rate of divergence is pretty slow— it may take a lot of observations before the accumulated excess kurtosis overwhelms the other factors. I simulated 10,000 samples from the above Gaussian GARCH process for samples of size T = 100, 200, and 1000 and 10,000, (and 1,000 samples of size 100,000), where the true values were specified as follows: β0 = β1 = 0 κ=2 α = 0.35 δ = 0.6. The solid line in Figure 2 plots the fraction of samples for which an OLS t test of β 1 = 0 exceeds two in absolute value. Thinking we’re only rejecting a true null hypothesis 5% of the time, we would in fact do so 15% of the time in a sample of size T = 100 and 33% of the time when T = 1, 000. As one might imagine, for a given sample size, the OLS t-statistic is more poorly behaved if the true innovations vt in (2) are Student’s t with 5 degrees of freedom (the dashed line in Figure 2) rather than Normal. What happens if instead of the OLS formula (7) for the variance of β̂ we use White’s 8 (1980) heteroskedasticity-consistent estimate, Ṽ = ³X 0 xt xt ´−1 ³X 0 û2t xt xt ´ ³X 0 xt xt ´−1 (13) ? ARCH is not a special case of the class of heteroskedasticity for which Ṽ is intended to be robust, and indeed, unlike typical cases, T Ṽ is not a consistent estimate of a given matrix: ³ ´ ³ ´³ ´ X X X 0 −1 0 0 −1 −1 −1 2 −1 T Ṽ = T xt xt T ût xt xt T xt xt . The first and last matrices will converge as before,  T −1 but T −1 P X 0  1 0  0 p  , xt xt →    0 µ2 û2t xt xt will diverge if the fourth moment µ4 is infinite. Figure 3 plots the simu- lated value for the square root of the lower-right element of T Ṽ for the Gaussian simulations above. However, this growth in the estimated variance of the growth of the actual variance of √ T β̂ 1 is exactly right, given √ T β̂ 1 implied by the GARCH specification. And a t test based on (13) seems to perform reasonably well for all sample sizes (see the second row of Table 2). The small-sample size distortion for the White test is a little worse for Student’s t compared with Normal errors, though still acceptable. Table 2 also explores the consequences of using the Newey-West (1987) generalization of the White formula to allow for serial correlation, using a lag window of q = 5: #Ã T !−1 Ã T !−1 " T ³ 0 ´ X X X v 0 0 0 ) ût ût−v xt xt−v + xt−v xt xt xt Ṽ∗ = xt xt (1 − . q + 1 t=v+1 t=1 t=1 These results (reported in the third row of the two panels of Table 2) illustrate one potential pitfall of relying too much on “robust” statistics to solve the small-sample problems, in that 9 it has more serious size distortions than does the simple White statistic for all specifications investigated. Another reason one might not want to assume that White or Newey-West standard errors can solve all the problems is that these formulas only correct the standard error for β̂, but are still using the OLS estimate itself, which from Figure 3 was seen not to be √ T convergent. By contrast, even if the fourth moment does not exist, maximum likelihood estimation as an alternative to OLS is still √ T convergent. Hence the relative efficiency gains of MLE relative to OLS become infinite as the sample size grows for typical values of GARCH parameters. Engle (1982, p. 999) observed that it is also possible to have an infinite relative efficiency gain for some parameter values even with exogenous explanatory variables and ARCH as opposed to GARCH errors. Results here are also related to the well-known result that ARCH will render inaccurate traditional tests for serial correlation in the mean. That fact has previously been noted, for example, by Milhøj (1985, 1987), Diebold (1988), Stambaugh (1993), and Bollerslev and Mikkelsen (1996). However, none of the above seems to have commented on the fact (though it is implied by the formulas they use) that the test size goes to unity as the fourth moment approaches infinity, or noted the implications as here for OLS regression. Finally, I observe that just checking for a difference between the OLS and the White standard errors will sometimes not be sufficient to detect these problems. The difference between V̂ and Ṽ will be governed by the size of X 0 (s2 − û2t )xt xt . 10 White (1980) suggested a formal test of whether this magnitude is sufficiently small on the 0 basis of an OLS regression of û2t on the vector ψ t consisting of the unique elements of xt xt . 2 )0 . White showed that, under the null hypothesis that In the present case, ψ t = (1, yt−1 , yt−1 the OLS standard errors are correct, T R2 from a regression of û2t on ψ t would have a χ2 (2) distribution. The next-to-last row of each panel of Table 2 reports the fraction of samples for which this test would (correctly) reject the null hypothesis. It would miss about half the time in a sample as small as 100 observations but is more reliable for larger sample sizes. Alternatively, one can look at Engle’s (1982) analogous test for the null of homoskedasticity against the alternative of qth-order ARCH by looking at T R2 from a regression of û2t on (1, û2t−1 , û2t−2 , ..., û2t−q )0 , which asymptotically has a χ2 (q) distribution under the null. The last rows in Table 2 report the rejection frequency for this test using q = 3 lags. Not surprisingly, since this test is designed specifically for the ARCH class of alternatives whereas the White test is not, this test has a little more power. Its advantage over the White test for homoskedasticity is presumably greater in many macro applications in which xt includes a number of variables and their lags, in which case the vector ψ t can become unwieldy, whereas the Engle test remains a simple χ2 (q) regardless of the size of xt . The philosophy of McARCH, then, is quite simple. The Engle T R2 diagnostic should be calculated routinely in any macroeconomic analysis. If a violation of homoskedasticity is found, one should compare the OLS estimates with maximum likelihood to make sure that the inference is robust. The following sections illustrate the potential importance of doing so with two examples from applied macroeconomics. 11 3 Application 1: Measuring market expectations of what the Federal Reserve is going to do next. My first example is adapted from Hamilton (forthcoming). The fed funds rate is a marketdetermined interest rates at which banks lend reserves to one another overnight. This interest rate is extremely sensitive to the supply of reserves created by the Fed, and in recent years monetary policy has been implemented in terms of a clearly announced target for the fed funds rate that the Fed intends to achieve. A critical factor that determines how Fed actions affect the economy is expectations by the public as to what the Fed is going to do next, as discussed, for example, in my (2008) paper. One natural place to look for an indication of what those expectations might be is the fed funds futures market. Let t = 1, 2, .., T index monthly observations. In the empirical results reported here, t = 1 corresponds to October, 1988 and the last observation (T = 213) is June 2006. For each month, we’re interested in what the market expects for the average effective fed funds rate over that month, denoted rt . For the empirical estimates reported in this section, rt is measured in basis points, so that for example rt = 525 corresponds to an annual interest rate of 5.25%. On any business day, one can enter into a futures contract through the Chicago Board of Trade whose settlement is based on what the value of rt+j actually turns out to be for some future month. The terms of a j-month-ahead contract traded on the last day of month t can 12 (j) be translated1 into an interest rate ft (j) such that, if rt+j turns out to be less than ft , then the seller of the contract has to compensate the buyer a certain amount (specifically, $41.67 on a standard contract) for every basis point by which ft(j) exceeds rt+j . If ft(j) < rt+j , the buyer pays the seller. (j) Since ft is known as of the end of month t but rt+j will not be known until the end of month t + j, the buyer of the contract is basically making a bet that (j) rt+j will be less than ft . If the marginal market participant were risk neutral, it would be the case that (j) ft = Et (rt+j ) (14) where Et (.) denotes the mathematical expectation on the basis of any information publicly (j) available as of the last day of month t. If (14) holds, we could just look at the value of ft to infer what market participants expect the Federal Reserve to do in the coming months. However, previous investigators such as Sack (2004) and Piazzesi and Swanson (forthcoming) have concluded that (14) does not hold. The simplest way to investigate this claim is to construct the forecast error implied by the 1-month-ahead contract, (1) (1) ut = rt − ft−1 and test whether this error indeed has mean zero, as it should if (14) were correct. For contracts at longer horizons j > 1, one can look at the monthly change in contract terms, (j) (j−1) ut = ft 1 (j) − ft−1 . Specifically, if Pt is the price of the contract agreed to by the buyer and seller on day t, then ft = 100 × (100 − Pt ). 13 (j) If (14) holds, then ut would also be a martingale difference sequence: (j) ut = Et (rt+j−1 ) − Et−1 (rt+j−1 ). One simple test is then to perform the regression (j) (j) ut = µ(j) + εt and test the null hypothesis that µ(j) = 0; this is of course just the usual t-test for a sample mean. Table 3 reports the results of this test using 1-, 2-, and 3-month-ahead futures (1) contracts. For the historical sample, the 1-month-ahead futures contract ft (j) the value of rt+1 by an average of 2.66 basis points and ft overestimated (j−1) overestimated the value of ft+1 by almost 4 basis points. One interpretation is that there is a risk premium built into these contracts. Another possibility is that the market participants failed to recognize fully the chronic decline in interest rates over this period. Before putting too much credence in such interpretations, however, recall that the theory (j) (14) implies that ut should be a martingale difference sequence but makes no claims about predictability of its variance. (j) Figure 4 reveals that each of the series ut exhibits some clustering of volatility and a significant decline in variability over time, in addition to occasional very large outliers. Engle’s T R2 test for omitted 4th-order ARCH finds very strong (1) (3) evidence of conditional heteroskedasticity at least for ut and ut ; see Table 3. Hence if we are interested in a more accurate estimate of the bias and statistical test of its significance, we might want to model these features of the data. Hamilton (forthcoming) calculated maximum likelihood estimates for parameters of the 14 following EGARCH specification (with (j) superscripts on all variables and parameters suppressed for ease of readability): ut = µ + p ht εt log ht − γ 0 zt = α(|εt−1 | − k2 ) + δ(log ht−1 − γ 0 zt−1 ) (15) (16) zt = (1, t/1000)0 √ 2 νΓ[(ν + 1)/2] √ k2 = E|εt | = (ν − 1) πΓ(ν/2) for εt a Student’s t variable with ν degrees of freedom and Γ(.) the gamma function: Γ(s) = Z ∞ xs−1 e−x dx. 0 The log likelihood is then found from T X log f (ut |Ut−1 ; θ) (17) t=1 ³ p ´ f(ut |Ut−1 , θ) = k1 / ht [1 + (ε2t /ν)]−(ν+1)/2 √ k1 = Γ[(ν + 1)/2]/[Γ(ν/2) νπ]. Given numerical values for the parameter vector θ = (µ, γ 0 , α, δ, ν)0 and observed data UT = (u1 , u2 , ..., uT )0 we can then begin the iteration (16) for t = 1 by setting h1 = exp(γ 0 z0 ). Plugging this into (15) gives us a value for ε1 , which from (16) gives us the number for h2 . Iterating in this fashion gives the sequence {ht , εt }Tt=1 from which the log likelihood (17) can be evaluated for the specified numerical value of θ. One then tries another guess for θ in order to numerically maximize the likelihood function. Asymptotic standard errors can 15 be obtained from numerical second derivatives of the log likelihood as in Hamilton (1994, equation [5.8.3]). Maximum likelihood parameter estimates are reported in Table 4. Adding these features provides an overwhelming improvement in fit, with a likelihood ratio test statistic well in excess of 100 when adding just 4 parameters to a simple Gaussian specification with constant variance. The very low estimated degrees of freedom results from the big outliers in the data, and both the serial dependence (δ) and trend parameter (γ 2 ) for the variance are extremely significant. A very remarkable result is that the estimates for the mean of the forecast error µ actually switch signs, shrink by an order of magnitude, and become far from statistically significant. Evidently the sample means of u(j) t are more influenced by negative outliers and observations early in the sample than they should be. Note that for this example, the problem is not adequately addressed by simply replacing OLS standard errors with White standard errors, since when the regressors consist only of a constant term, the two would be identical. Moreover, whenever, as here, there is an affirmative objective of obtaining accurate estimates of a parameter (the possible risk premium incorporated in these prices) as opposed solely to testing a hypothesis, the concern is with the quality of the coefficient estimate itself rather than the correct size of a hypothesis test. 16 4 Application 2: Using the Taylor Rule to summarize changes in Federal Reserve policy. One of the most influential papers for both macroeconomic research and policy over the last decade has been John Taylor’s (1993) proposal of a simple rule that the central bank should follow in setting an interest rate like the fed funds rate rt . Taylor’s proposal called for the Fed to raise the interest rate by an amount governed by a parameter ψ 1 when the observed inflation rate π t is higher than it wishes (so as to bring inflation back down), and to raise the interest rate by an amount governed by ψ 2 when yt , the gap between real GDP and its potential value, is positive: rt = ψ 0 + ψ 1 π t + ψ 2 yt In this equation, the value of ψ 0 reflects factors such as the Fed’s long-run inflation target and the equilibrium real interest rate. There are a variety of ways such an expression has been formulated in practice, such as “forward-looking” specifications, in which the Fed is responding to what it expects to happen next to inflation and output, and “backwardlooking” specifications, in which lags are included to capture expectations formation and adjustment dynamics. A number of studies have looked at the way that the coefficients in such a relation may have changed over time, including Judd and Rudebusch (1998), Clarida, Galí and Gertler (2000), Jalil (2004), and Boivin and Giannoni (2006). Of particular interest has been that the claim that the coefficient on inflation ψ 1 has increased relative to the 1970s, and that this increased willingness on the part of the Fed to fight inflation has been a factor helping 17 to make the U.S. economy become more stable. In this paper, I will explore the variant investigated by Judd and Rudebusch, whose reduced-form representation is ∆rt = γ 0 + γ 1 π t + γ 2 yt + γ 3 yt−1 + γ 4 rt−1 + γ 5 ∆rt−1 + vt . (18) Here t = 1, 2, ..., T now will index quarterly data, with t = 1 in my sample corresponding to 1956:Q1 and T = 205 corresponding to 2007:Q1. The value of rt for a given quarter is the average of the three monthly series for the effective fed funds rate, with ∆rt = rt − rt−1 , and for empirical results here is reported as percent rather than basis points, e.g., rt = 5.25 when the average fed funds rate over the three months of the quarter is 5.25%. Inflation π t is measured as 100 times the natural logarithm of the difference between the level of the implicit GDP deflator for quarter t and its value for the corresponding quarter of the preceding year, with data taken from Bureau of Economic Analysis Table 1.1.9. As in Judd and Rudebusch, the output gap yt was calculated as yt = 100(Yt − Yt∗ ) Yt∗ for Yt the level of real GDP (in billions of chained 2000 dollars, from BEA Table 1.1.6) and Yt∗ the series for potential GDP from the Congressional Budget Office (obtained from the St. Louis FRED database). Judd and Rudebusch focused on certain rearrangements of the parameters in (18), though here I will simply report results in terms of the reduced-form estimates themselves. The term vt in (18) is the regression error. Table 5 presents results from OLS estimation of (18) using the full sample of data. Of particular interest are γ 1 and γ 2 , the contemporary responses to inflation and output, respectively. Table 6 then re-estimates the relation, allowing for separate coefficients since 18 1979:Q3, when Paul Volcker became Chair of the Federal Reserve. The OLS results reproduce the findings of the many researchers noted above that monetary policy seems to have responded much more vigorously to disturbances since 1979, with the inflation coefficient γ 1 increasing by 0.26 and the output coefficient γ 2 increasing by 0.64. However, the White standard errors for the coefficients on dt π t and dt yt are almost twice as large as the OLS standard errors, and suggest that the increased response to inflation is in fact not statistically significant and the increased response to output is measured very imprecisely. Moreover, Engle’s LM test for the null of Gaussian errors with no heteroskedasticity against the alternative of 4th-order ARCH leads to overwhelming rejection of the null hypothesis.2 All of which suggests that, if we are indeed interested in measuring the magnitudes by which these coefficients have changed, it is preferable to adjust not just the standard errors but the parameter estimates themselves in light of the dramatic ARCH displayed in the data. I therefore estimated the following GARCH-t generalization of (18): 0 yt = xt β + vt vt = p ht εt ht = κ + h̃t 2 h̃t = α(vt−1 − κ) + δ h̃t−1 with εt a Student’s t random variable with ν degrees of freedom. 2 Siklos and Wohar (2005) also make this point. 19 (19) Iteration on (19) is initialized with h̃1 = 0. The log likelihood is then evaluated exactly as in (17). Maximum likelihood estimates are reported in Table 7. Once again generalizing a homoskedastic Gaussian specification is overwhelmingly favored by the data, with a comparison of the specifications in Tables 6 and 7 producing a likelihood ratio χ(4) statistic of 183.34. The degrees of freedom for the Student’s t distribution are only 2.29, and the implied GARCH process is highly persistent (α̂ + δ̂ = 0.82). Of particular interest is the fact that the changes in the Fed’s response to inflation and output are now considerably smaller than suggested by the OLS estimates. The change in γ 1 is now estimated to be only 0.09 and the change in γ 2 has dropped to 0.05 and no longer appears to be statistically significant. Figure 5 offers some insight into what produces these results. The top panel illustrates the tendency for interest rates to exhibit much more volatility at some times than others, with the 1979:Q2 to 1982:Q3 episode particularly dramatic. The bottom panel plots observations on the pairs (yt , ∆rt ) in the second half of the sample. The apparent positive slope in that scatter plot is strongly influenced by the observations in the 1979-82 period. If one allowed the possibility of serial dependence in the squared residuals, one would give less weight to the 1979-82 observations, resulting in a flatter slope estimate over 1979-2007 relative to OLS. This is not to attempt to overturn the conclusion of earlier researchers that there has been a change in Fed policy in the direction of a more active policy. A comparison of the changing-parameter specification of Table 7 with a fixed-parameter GARCH specification produces a χ(4) likelihood ratio statistic of 18.22, which is statistically significant with a 20 p-value of 0.001. Nevertheless, the magnitude of this change appears to be substantially smaller than one would infer on the basis of OLS estimates of the parameters. Nor is this discussion meant to displace the large and thoughtful literature on possible changes in the Taylor Rule, which has raised a number of other substantive issues not explored here. These include whether one wants to use real-time or subsequent revised data (Orphanides (2001)), the distinction between the “backward-looking” Taylor Rule explored here and “forward-looking” specifications (Clarida, Galí, and Gertler, 2000), and continuous evolution of parameters rather than a sudden break (Jalil, 2004; Boivin, 2006). The simple exercise undertaken nevertheless does in my mind establish the potential importance for macroeconomists to check for the presence of ARCH even when their primary interest is in the conditional mean. 5 Conclusions. The reader may note that both of the examples I have used to illustrate the potential relevance of McARCH use the fed funds rate as the dependent variable. This is not entirely an accident. Although Kilian and Gonçalves (2004) concluded that most macro series exhibit some ARCH, the fed funds rate may be the macro series for which one is most likely to observe wild outliers and persistent volatility clustering, regardless of the data frequency or subsample. It is nevertheless, as the examples used here illustrate, a series that features very importantly for some of the most fundamental questions in macroeconomics. The rather dramatic way in which accounting for outliers and ARCH can change one’s 21 inference that was seen in these examples presumably would not be repeated for every macroeconomic relation estimated. However, routinely checking something like a T R2 statistic, or the difference between OLS and White standard errors, seems a relatively costless and potentially quite beneficial habit. 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Rudebusch. 1998. “Taylor’s Rule and the Fed: 1970-1997,” Federal Reserve Bank of San Francisco Review 3, pp. 3-16. Kilian, Lutz, and Sílvia Gonçalves. 2004. “Bootstrapping Autoregressions with Conditional Heteroskedasticity of Unknown Form,” Journal of Econometrics 123, pp. 89-120. Lee, Kiseok, Shawn Ni, and Ronald A. Ratti. 1995. “Oil Shocks and the Macroeconomy: 26 The Role of Price Variability,” Energy Journal, 16, pp. 39-56. Lee, Sang-Won, and Bruce E. Hansen. 1994. “Asymptotic Theory for the Garch (1,1) Quasi-Maximum Likelihood Estimator,” Econometric Theory 10, pp. 29-52. Lumsdaine, Robin L. 1996. “Consistency and Asymptotic Normality of the QuasiMaximum Likelihood Estimator in IGARCH(1,1) and Covariance Stationary GARCH(1,1) Models,” Econometrica 64, pp. 575-596. Milhøj,Anders. 1985. “The Moment Structure of ARCH Processes,” Scandinavian Journal of Statistics 12, pp. 281-292. Milhøj, Aers. 1987. “A Conditional Variance Model for Daily Observations of an Exchange Rate,” Journal of Business and Economic Statistics 5, pp. 99-103. Nelson Daniel B. 1991. “Conditional Heteroscedasticity in Asset Returns: A New Approach,” Econometrica 59, pp. 347—370. Newey, Whitney K., and Kenneth D. West. 1987. “A Simple Positive Semi-Definite, Heteroskedasticity and Autocorrelation Consistent Covariance Matrix,” Econometrica 55, pp. 703-708. Orphanides, Athanasios. 2001. “Monetary Policy Rules Based on Real-Time Data.” American Economic Review 91. [[/ 964-985. Pelloni, Gianluigi and Wolfgang Polasek. 2003. “Macroeconomic Effects of Sectoral Shocks in Germany, the U.K. and the U.S.: A VAR-GARCH-M Approach,” Computational Economics 21, pp. 65-85. Piazzesi, Monika, and Eric Swanson. Forthcoming. “Futures Prices as Risk-Adjusted 27 Forecasts of Monetary Policy.” Journal of Monetary Economics. Sack, Brian. (2004) “Extracting the Expected Path of Monetary Policy from Futures Rates.” Journal of Futures Markets, 24, 733-754. Sentana, Enrique. 1995. “Quadratic ARCH Models,” Review of Economic Studies 62, pp. 639-661. Servén, Luis. 2003. “Real-Exchange-Rate Uncertainty and Private Investment in LDCS,” Review of Economics and Statistics 85, pp. 212-218. Shields, Kalvinder, Nilss Olekalns, Ólan T. Henry, and Chris Brooks. 2005. “Measuring the Response of Macroeconomic Uncertainty to Shocks,” Review of Economics and Statistics 87, pp. 362-370. Siklos, Pierre L., and Mark E. Wohar. 2005. “Estimating Taylor-Type Rules: An Unbalanced Regression?”, in Advances in Econometrics, vol. 20, edited by Thomas B. Fomby and Dek Terrell. Amsterdam: Elsevier. Stambaugh, Robert F. 1993. “Estimating Conditional Expectations when Volatility Fluctuates,” NBER Working Paper 140. Taylor, John B. 1993. “Discretion Versus Policy Rules in Practice,” Carnegie-Rochester Conference Series on Public Policy 39, pp. 195-214. White, Halbert. 1980. “A Heteroskedasticity-Consistent Covariance Matrix Estimator and a Direct Test for Heteroskedasticity,” Econometrica 48, pp. 817-38. Zakoian, J. M. 1994. “Threshold Heteroskedastic Models,” Journal of Economic Dynamics and Control 18, pp. 931—955. 28 Table 1. How many ways can you spell “ARCH”? (A partial lexicography). ________________________________________________________________________ AARCH APARCH ARCH-M FIGARCH GARCH GARCH-t GJR-ARCH EGARCH HGARCH IGARCH MARCH MARCH NARCH PNP-ARCH QARCH QTARCH SPARCH STARCH SWARCH TARCH VGARCH Augmented ARCH Asymmetric power ARCH ARCH in mean Fractionally integrated GARCH Generalized ARCH Student’s t GARCH Glosten-Jagannathan-Runkle ARCH Exponential generalized ARCH Hentschel GARCH Integrated GARCH Modified ARCH Multiplicative ARCH Nonlinear ARCH Partially Non-parametric ARCH Quadratic ARCH Qualitative Threshold ARCH Semiparametric ARCH Structural ARCH Switching ARCH Threshold ARCH Vector GARCH Bera, Higgins and Lee (1992) Ding, Engle, and Granger (1993) Engle, Lilien and Robins (1987) Baillie, Bollerslev, Mikkelsen (1996) Bollerslev (1986) Bollerslev (1987) Glosten, Jagannathan, and Runkle (1993) Nelson (1991) Hentschel (1995) Bollerslev and Engle (1986) Friedman, Laibson, and Minsky (1989) Milhøj (1987) Higgins and Bera (1992) Engle and Ng (1993) Sentana (1995) Gourieroux and Monfort (1992) Engle and Gonzalez-Rivera (1991) Harvey, Ruiz, and Sentana (1992) Hamilton and Susmel (1994) Zakoian (1994) Bollerslev, Engle, and Wooldrige (1988) 29 Table 2. Fraction of samples for which indicated hypothesis is rejected by test of nominal size 0.05. ----------------------------------------------------------------------------------------------------Errors Normally distributed H0 ------------------β1 =0 (H0 is true) β1 =0 (H0 is true) β1 =0 (H0 is true) εt homoskedastic (H0 is false) εt homoskedastic (H0 is false) Test based on --------------------OLS standard error White standard error Newey-West standard error White TR2 Engle TR2 T = 100 T = 200 T = 1000 --------- --------- ----------0.152 0.200 0.327 0.072 0.063 0.054 0.119 0.092 0.062 0.570 0.874 1.000 0.692 0.958 1.000 Errors Student’s t with 5 degrees of freedom H0 ------------------β1 =0 (H0 is true) β1 =0 (H0 is true) β1 =0 (H0 is true) εt homoskedastic (H0 is false) εt homoskedastic (H0 is false) Test based on --------------------OLS standard error White standard error Newey-West standard error White TR2 Engle TR2 T = 100 T = 200 T = 1000 --------- --------- ----------0.174 0.229 0.389 0.081 0.070 0.065 0.137 0.106 0.079 0.427 0.691 0.991 0.536 0.822 0.998 30 Table 3. OLS estimates of bias in monthly fed funds futures forecast errors. --------------------------------------------------------------------------dependent estimated standard OLS ARCH(4) Log like( j) ( j) ˆ p-value LM p-value lihood variable (ut ) mean ( µ ) error ----------------------------- -----------------------j = 1 month -2.66 0.75 0.001 0.006 -812.61 j = 2 months -3.17 1.06 0.003 0.204 -884.70 j = 3 months -3.74 1.27 0.003 0.001 -922.80 31 Table 4. Maximum likelihood estimates (asymptotic standard errors in parentheses) for EGARCH model of fed funds futures forecast errors. ------------------------------------------------------------------------------------------------------horizon (j) ut(1) ut( 2 ) ut( 3) mean (µ) 0.12 (0.24) 0.43 (0.34) 0.27 (0.67) log average variance (γ1) 5.73 (0.42) 6.47 (0.51) 7.01 (0.54) trend in variance (γ2) -22.7 (3.1) -23.6 (3.3) -17.1 (3.8) | ut-1| (α) 0.18 (0.07) 0.15 (0.07) 0.30 (0.12) log ht-1 (δ) 0.63 (0.16) 0.74 (0.22) 0.84 (0.11) Student t degrees of freedom (υ) 2.1 (0.4) 2.2 (0.4) 4.1 (1.2) -731.08 -793.38 -860.16 log likelihood 32 Table 5. Fixed-coefficient Taylor Rule as estimated from full sample OLS regression. Regressor constant πt yt yt-1 rt-1 ∆rt-1 TR2 for ARCH(4) (p-value) Log likelihood Coefficient 0.06 0.13 0.37 -0.27 -0.08 0.14 Std error (OLS) 0.13 0.04 0.07 0.07 0.03 0.07 23.94 (0.000) -252.26 33 Std error (White) 0.18 0.06 0.11 0.10 0.03 0.15 Table 6. Taylor Rule with separate pre- and post-Volcker parameters as estimated by OLS regression (dt = 1 for t > 1979:Q2). Regressor constant πt yt yt-1 rt-1 ∆rt-1 dt dtπt dtyt dtyt-1 dtrt-1 dt∆rt-1 TR2 for ARCH(4) (p-value) Log likelihood Coefficient 0.37 0.17 0.18 -0.07 -0.21 0.42 -0.50 0.26 0.64 -0.55 0.05 -0.53 Std error (OLS) 0.19 0.07 0.08 0.08 0.07 0.11 0.24 0.09 0.14 0.14 0.08 0.13 45.45 (0.000) -226.80 34 Std error (White) 0.19 0.04 0.07 0.07 0.06 0.13 0.30 0.16 0.24 0.21 0.08 0.24 Table 7. Taylor Rule with separate pre- and post-Volcker parameters as estimated by GARCH-t maximum likelihood (dt = 1 for t > 1979:Q2). Regressor constant πt yt yt-1 rt-1 ∆rt-1 dt dtπt dtyt dtyt-1 dtrt-1 dt∆rt-1 GARCH parameters constant α δ ν Log likelihood Coefficient 0.13 0.06 0.14 -0.12 -0.07 0.47 -0.03 0.09 0.05 0.02 -0.01 -0.01 Asymptotic std error 0.08 0.03 0.03 0.03 0.03 0.09 0.12 0.04 0.07 0.07 0.03 0.11 0.015 0.11 0.71 2.29 0.010 0.05 0.07 0.48 -135.13 35 Figure 1. Asymptotic rejection probability for OLS t-test that autoregressive coefficient is zero as a function of GARCH(1,1) parameters α and δ. Note: null hypothesis is actually true and test has nominal size of 5%. 36 Figure 2. Fraction of samples in which OLS t-test leads to rejection of the null hypothesis that autoregressive coefficient is zero as a function of the sample size for regression with Gaussian errors (solid line) and Student’s t errors (dashed line). Note: null hypothesis is actually true and test has nominal size of 5%. 1 Normal Student t 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 2 10 3 10 10 Sample size (T) 37 4 10 5 Figure 3. Average value of T times estimated standard error of estimated autoregressive coefficient as a function of the sample size for White standard error (solid line) and OLS standard error (dashed line). 6 White OLS 5 4 3 2 1 0 2 10 3 10 10 Sample size (T) 38 4 10 5 Figure 4. Plots of 1-month-ahead forecast errors (ut( j ) ) as a function of month t based on j = 1-, 2-, or 3-month ahead futures contracts. 1 month 30 20 10 0 -10 -20 -30 -40 -50 -60 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2 month 75 50 25 0 -25 -50 -75 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 3 month 80 60 40 20 0 -20 -40 -60 -80 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 39 Figure 5. Change in fed funds rate for the full sample (1956:Q2-2007:Q1), and scatter plot for later subsample (1979:Q2-2007:Q1) of change in fed funds rate against deviation of GDP from potential. Change in fed funds rate, 1956:Q2-2007:Q1 change in funds rate 8 6 4 2 0 -2 -4 1956 1959 1962 1965 1968 1971 1974 1977 1980 1983 1986 1989 1992 1995 1998 2001 2004 2007 date Scatter diagram, 1979:Q2-2007:Q1 8 change in funds rate 6 4 2 0 -2 -4 -8 -6 -4 -2 GDP deviation 40 0 2 4