Assessing Monetary Policy Effects Using Daily Federal Funds Futures Contracts

James D. Hamilton

This paper develops a generalization of the formulas proposed by Kuttner (2001) and others for purposes of measuring the effects of a change in the federal funds target on Treasury yields of different maturities. The generalization avoids the need to condition on the date of the target change and allows for deviations of the effective fed funds rate from the target as well as gradual learning by market participants about the target. The paper shows that parameters estimated solely on the basis of the behavior of the fed funds and fed funds futures can account for the broad calendar regularities in the relation between fed funds futures and Treasury yields of different maturities. Although the methods are new, the conclusion is quite similar to that reported by earlier researchers— changes in the fed funds target seem to be associated with quite large changes in Treasury yields, even for maturities of up to 10 years. (JEL: E52, E43)

Federal Reserve Bank of St. Louis Review, July/August 2008, 90(4), pp. 377-93.

conomists continue to debate how and September 1979 on which there was a change much of an effect monetary policy has in the target, Cook and Hahn estimated the follow- on the economy. But one of the more ing regression by ordinary least squares (OLS): robust empirical results is the observa- tion that changes in the target that the Federal (1) iisd,, sd1 s s d d1 u sd. E − −−=+αλξξ − + () Reserve sets for the overnight federal funds rate Their estimates of for securities of several differ- have been associated historically with large λs ent maturities are reported in the first column of changes in other interest rates, even for the Table 1. These estimates suggest that, when the longest maturities. This paper contributes to the extensive literature that tries to measure the Fed raises the overnight rate by 100 basis points, magnitude of this effect. short-term Treasury yields go up by over 50 basis One of the first efforts along these lines was by points and there is a statistically significant effect Cook and Hahn (1989), who looked at how yields even on 10-year yields. on Treasury securities of different maturities Subsequent researchers found that the mag- nitudes of the estimated coefficients for were changed on the days when the Federal Reserve λs significantly smaller when later data sets were changed its target for the fed funds rate. Let is,d denote the interest rate (in basis points) on a used. For example, column 2 of Table 1 reports Treasury bill or Treasury bond of constant matu- Kuttner’s (2001) results when the Cook-Hahn rity s months as quoted on some business day, d, regression (1) was reestimated using data from and let denote the target for the fed funds rate June 1989 to February 2000; see also Nilsen (1998). ξd as determined by the Federal Reserve for that day. However, Kuttner (2001) also identified some Using just those days between September 1974 conceptual problems with regression (1). For one

James D. Hamilton is a professor of economics at the University of California, San Diego. © 2008, The Federal Reserve Bank of St. Louis. The views expressed in this article are those of the author(s) and do not necessarily reflect the views of the Federal Reserve System, the Board of Governors, or the regional Federal Reserve Banks. Articles may be reprinted, reproduced, published, distributed, displayed, and transmitted in their entirety if copyright notice, author name(s), and full citation are included. Abstracts, synopses, and other derivative works may be made only with prior written permission of the Federal Reserve Bank of St. Louis.

FEDERALRESERVEBANKOFST. LOUIS REVIEW JULY/AUGUST 2008 377 Hamilton

Table 1 Alternative Estimates of the Response of Interest Rates to Changes in the Federal Funds Target

Study Cook-Hahn Kuttner Kuttner Poole-Rasche Specification (1) (1) (2)-(3) (4)-(3) Sample 1974-79 1989-2000 1989-2000 1988-2000 s = 3 months 0.55** 0.27** 0.79** 0.73** s = 6 months 0.54** 0.22** 0.72** — s = 1 year 0.50** 0.20** 0.72** 0.78** s = 5 years 0.21** 0.10* 0.48** — s = 10 years 0.13** 0.04* 0.32** 0.48**

NOTE: *indicates statistically significant with p-value < 0.05; **denotes p-value < 0.01. thing, the market may have anticipated much of Kuttner found that the values for were essen- γs the change in the target that occurred on day d tially zero, meaning that if target changes were ξd many days earlier, in which case those expecta- anticipated in advance, then they had no effect tions would have already been incorporated into on other interest rates. Kuttner’s estimates of , λs is,d–1. In the limiting case when the change was the effects of unanticipated target changes, are perfectly anticipated, one would not expect any reported in column 3 of Table 1 and turn out to be change in is,d to be observed on the day of the a bit larger than the original Cook-Hahn estimates. target change. To isolate the unanticipated com- Poole and Rasche (2000) proposed to side- ponent of the target change, Kuttner used fd, the step the issues associated with a mid-month target interest rate implied by the spot-month fed funds change by using not the spot-month contract on contract on day d. These contracts are settled on day d but instead the one-month-ahead contract, the basis of what the average effective fed funds that is, the interest rate implied by a contract rate turns out to be for the entire month containing purchased on day d for settlement based on the day d. Because much of the month may already average fed funds rate prevailing in the following be over by day d, a target change on day d will month, denoted f 1. They then replaced the expres- have only a fractional effect on the monthly d average. Kuttner proposed the following formula sion in (2) with to identify the unanticipated component of the %u 1 1 (4) d ffdd1. target change on day d: ξ =−−

Their estimates for s using this formulation N λ (2) %u  d  turned out to be similar to Kuttner’s and are d ffdd1 , ξ = Nt1 − − reported in column 4 of Table 1.  dd−+ () However, mid-month target changes remain where N is the number of calendar days associ- d an issue for the Poole-Rasche estimates because ated with the month in which day d occurs and there is always the possibility of a second (or even t is the calendar day of the month associated d a third) change in the target some time after day with day d. Kuttner then replaced (1) with the regression d and before the end of the following month; indeed, this turned out to be the case for about %%u u half of the target changes observed between 1988 (3) iisd,, sd1 s s d d1 d sd usd , − −−=+αγξξ() − − ξ++ λξ and 2006. Gürkaynak, Sack, and Swanson (2007) with additional modifications if d were the first developed an analog to Kuttner’s formula (2) based day or one of the last three days of a month. on the date of the next target change that followed

378 JULY/AUGUST 2008 FEDERALRESERVEBANKOFST. LOUIS REVIEW Hamilton after the one implemented on day d; see also estimates of this parameter. Although the method Gürkaynak (2005). and data set are rather different from the earlier Another potential drawback to either (2) or researchers, my estimates in fact turn out to be (4) was raised by Poole, Rasche, and Thornton quite similar to those originally found by Kuttner (2002). These authors noted that, particularly (2001) and Poole and Rasche (2000). prior to 1994, market participants may not have been perfectly aware of the target change even at the end of day d, in which case these formulas THE EFFECTIVE AND TARGET would include a measurement error that would FEDERAL FUNDS RATES bias the coefficients downward. Poole, Rasche, and Thornton developed corrections for the esti- In this paper, time is indexed in two different mates to allow for this measurement error. ways, using calendar days t for developing theo- A related issue is that the series for , the retical formulas and business days d to apply ξd actual target change, is itself subject to measure- these ideas to actual data. The theoretical formu- ment error, as indeed Kuttner (2001) and Poole, las will be developed for a typical month consist- Rasche, and Thornton (2002) used slightly differ- ing of N calendar days indexed by t = 1,2,...,N, ent series. Learning about the target change pre- whereas the data set will consist of those days sumably also began well before day d. For both d = 1,2,...,D for which there are data on both reasons, one would think that data both before Treasury interest rates and fed funds futures rates; and after day d should typically be used. In this d = 1 corresponds to October 3, 1988, and d = D = paper I develop a generalization of the Kuttner 4,552 corresponds to December 29, 2006. The (2001) and Poole, Rasche, and Thornton (2002) empirical sample for all estimates reported in adjustments for purposes of estimating the param- this paper also excludes the volatile data from eter . The basic idea is to suppose that there September 13 to September 30, 2001. λs exists some day within the month at which the The effective fed funds rate for calendar day t, target may have been changed, but to choose denoted rt, is a volume-weighted average of all deliberately not to condition on this day for pur- overnight interbank loans of Federal Reserve poses of forming an econometric estimate. The deposits for that day. All numbers in this paper paper also generalizes the earlier approaches by will be reported in basis points, so that, for exam- explicitly modeling the difference between the ple, a 5.25 percent interest rate would correspond effective fed funds rate and the actual target. to a value of rt = 525. Since October 1988, the The next section begins with an examination Chicago Board of Trade has offered futures con- of the relation between the target rate chosen by tracts whose settlement is based on the average the Fed and the actual effective fed funds rate. value for the effective fed funds rate over all the The third section develops a simple statistical calendar days of the month (with Friday rates, for description of how these deviations, along with example, also imputed to Saturday and Sunday). the process of learning by the market about what For a month that contains N calendar days, settle- the fed funds target is going to be for this month, ment of these futures contracts would be based would determine the volatility of the spot-month on the value of futures rate. The fourth section shows how the N 1 (5) SN− r. parameters estimated from the behavior of the = t ∑t 1 effective fed funds rate and the spot-month futures = rate can be used to predict calendar regularities The terms of a given fed funds futures contract in the estimated values for a generalization of the 1 can be translated into an interest rate, ft, such coefficient . The final section finds such calen- λs that, if S (which is not known at day t but will dar regularities largely borne out in the observed relation between Treasury rates and daily changes 1 Specifically, if Pt is the price of the contract agreed to by the in the spot-month futures rate and develops new buyer and seller on day t, then f = 100 100 – P . t × ͑ t͒

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could use the change in the spot-month contract Figure 1 price on day n to infer how much of the change Effective Fed Funds Rate, Target Fed Funds in the target interest rate caught the market by Rate, and Fed Funds Futures Rate, surprise according to the formula December 1990 N (7) ffnn1 . Nn1 − − Fed Funds Rate −+() 900 I will provide a formal derivation of (7) as a special case of a more general statistical inference

800 problem explored below, but would first like to comment on one potential drawback of (7), which is that it implies a huge reweighting of observa- 700 tions that come near the end of the month (n near N). Kuttner (2001, p. 529) recognized that this is Effective a potential concern here because (7) abstracts 600 Target from the deviation between the Federal Reserve’s Futures target for the effective fed funds rate and the 500 actual effective rate, and as a result magnifies 1 8 15 22 29 the measurement error for observations near the Calendar Date end of the month. Kuttner himself avoided using (7) for the last three days of the month. Other researchers like Gürkaynak (2005) avoid applying become known by the end of the month) turns it to data from the last week. Figure 1 plots the relevant variables for out to be bigger than ft, the buyer of the contract has to compensate the seller by a certain amount December 1990, which was a particularly wild month as banks adjusted to lower reserve require- for every basis point by which S exceeds ft. If the marginal market participant were risk neutral, it ments (Anderson and Rasche, 1996). Although would be the case that the Fed had lowered the target to 725 basis points on December 7, the effective fed funds rate was (6) fES, tt= () trading well above this the week after Christmas, and speculators seemed to be allowing for a pos- where E . denotes an expectation formed on the t͑ ͒ sibility of a big end-of-year spike up, such as the basis of information available to the market as of 584-basis-point increase in the effective fed funds day t. This paper will consider only spot-month rate that was seen in the last two days of 1985 or contracts, that is, contracts for which by day t we the 975-basis-point spike between December 28 already know some of the values for r (namely, r and December 30, 1986. In the event, however, for t) that will end up determining S. My forth-τ τ Յ the effective funds rate plunged 200 basis points coming paper (Hamilton, forthcoming) demon- on December 31, 1990. strates that, for futures contracts at short horizons Because the December 1990 futures contract (the spot-month, 1-month-ahead, and 2-month- was based on the effective rate rather than the ahead contracts), expression (6) appears to be target, speculators were watching these events an excellent approximation to the data, though closely. The futures rate was trending well above Piazzesi and Swanson (forthcoming) note poten- the new target of 725 basis points in the latter part tial problems with assuming that it holds for of December, partly because the month’s average longer-horizon contracts. would include the first week’s 750-basis-point Suppose that the Fed changes the target for target values, partly because the effective rate had the effective fed funds rate on calendar day n of been averaging above the new target subsequently, this month. Kuttner (2001) suggested that we and partly in anticipation of an end-of-year spike

380 JULY/AUGUST 2008 FEDERALRESERVEBANKOFST. LOUIS REVIEW Hamilton

Figure 2 Squared Residuals by Day of Month

Average Squared Residual 3,000

2,500

2,000

1,500

1,000

500

–500 2468 10 12 14 16 18 20 22 24 26 28 30 Day of Month

NOTE: The figure plots the average squared residuals from a regression of the deviation of the fed funds rate from the target on its own lagged value, by day of the month (in basis points). 95 percent confidence intervals are indicated by the upper and lower box lines, and predicted values from regression (9) are indicated by the dashed line. up. When it became clear on December 31 that has in place for business day d (denoted ) and ξd the last day of the year generated a big move down the actual fed funds rate. The effective fed funds rather than up, the December futures contract fell rate, rd, was taken from the FRED database of the by 23 basis points on a single day. Formula (7) Federal Reserve Bank of St. Louis (which in turn would call for us to multiply this number by 31, is based on Board of Governors release H.15), and to deduce that the interest rate surprise on this the target prior to 1994 is from the FRED series ξd day was some 713 basis points, plausible perhaps that comes from Thornton (2005) and since 1994 if the market was anticipating a spike up to 1,250 is from Federal Open Market Committee (FOMC) rather than the plunge down to 550 that actually transcripts. I first estimated the following regres- transpired. Although this is an extreme example, sion (similar to the models in Taylor, 2001, and it drives home the lesson that one really wants Sarno, Thornton, and Valente, 2005) by OLS to downweight the end-of-month observations (standard errors are in parentheses): rather than magnify them in the manner suggested ˆ by the expression in (7). (8) rredd245.. 0300 d11 d dd . −=ξξ029.. + 0014 −− − + The next section proposes a more formal state- ()( )() ment of this problem and its solution. A necessary This regression establishes that there is mod- first step is to document some of the properties est serial correlation in deviations from the target. of the deviation between the target that the Fed Of particular interest in the next section will be

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ˆ the calendar variation in the variability of ed. Let The effective fed funds rate for each day is the = 1 if day d occurs on the jth calendar day of sum of the target for that day plus the deviation ωjd ˆ 2 the month and zero otherwise. A regression of ed from the target, denoted ut: on { }31 then gives the average squared resid- ωjd j=1 ru. ual as a function of the calendar day of the ttt=+ξ month: It follows from (5) and (6) that 31 ê2 ˆ ˆ . N dj=+βωjd ν d  1  ∑j 1 fENtt− u = =+ξττ  ∑1() ˆ  τ =  The estimated values j are plotted as a function n 1 Nn1 β − − ++ of the calendar day j in Figure 2 along with the (10)   0   Etn = N  ξ +  N  ()ξ 95 percent confidence intervals for each coeffi-     t N cient. A big outlier on January 23, 1991, (when the 1 1 NuNEu− − t . ++τ τ funds rate spiked up nearly 300 basis points on ∑1 ∑∑t 1 () τ = τ =+ a settlement Wednesday) is enough to skew the On the day before the target change, I presume results for day 23. Apart from this, the most notice- that market participants had some expectation of able feature is an increased volatility of the devi- what the target was going to be, denoted E . ation of the funds rate from the target toward the n–1͑ξn͒ The actual target would deviate from this by some end of a month. One can represent this tendency magnitude h : parametrically through the following restricted n regression: nnnnEh1 . ξξ= − ()+ 31 t ˆ 2 − d ˆ If the equilibrium fed funds price is determined (9) ed 283 1746,0.5,()d = 40 +×232 +ν by risk-neutral rational speculators, the forecast () () error h would be a martingale difference sequence where t is the calendar day of the month associ- n d that represents the content of the news about ated with business day d. The predicted values ξn that arrived on the day of the target change itself. from (9) are also plotted in Figure 2. In the next Similarly, section, a simple theoretical formulation based on (8) and (9) will be used to characterize the nnnnnEhh2 1, modest predictability of deviations from the target ξξ= −−()++ and their tendency to become more pronounced where hn–1 is the news that arrived on day n–1 at the end of the month. of the Fed’s intentions on day n, and

nnnhh1 h n210L hEn. ξξ=+−− + +++()

ACCOUNTING FOR THE Under rational expectations, {ht} should be a VOLATILITY OF SPOT-MONTH sequence of zero-mean, serially uncorrelated variables, whose unconditional variance is FUTURES PRICES denoted 2. Notice that h represents the infor- σh 1 Suppose that market participants know that, mation that the market receives on day 1 about if the Fed is going to change the target within a the value for the target that the Fed will adopt on given month consisting of N calendar days, it day n, h2 represents the new information received would do so on calendar day n, so that its target on day 2, and so on, with is a step function: Ehhhtn012ntL forÄ forÄ Åtn12 , ,..., 1 (11) E  ()ξ ++++ ≤. ξξt ==−0 tn()ξ =  Å forÄ t nn Ä forÅ Ätnn ,1,..., , N . ξn > ξξtn==+ 

382 JULY/AUGUST 2008 FEDERALRESERVEBANKOFST. LOUIS REVIEW Hamilton

Given (8) and (9), I assume that deviations follow whereas for t > n, changes in futures prices are an AR(1) process with an innovation variance driven solely by the deviation of the effective fed that increases at the end of the month: funds rate from the target: uu t tt1 Nt1 =+φε− 1 −+ 2 Nt 1 − ff N− − φ forÅtn . E t 0 1 (), t t 1 ()t εγγδ=+ −=− 1 ε > () − φ where the empirical results suggest values of = φ It follows that the variance of daily changes in the 0.30, 0 = 283, 1 = 1,746, and = 0.5. Then γ γ δ spot-month futures rate would be given by N Eut ()τ = 2 ∑t 1 Ef tt f1 targetÄ changeÄ onÄ dayÄ n (12) τ = + − − Nt () 1 −   2 N t φφ− 2 22Nt 12 uuttL − ut ut .  Nn1 //N 1 −+ φφ+++ φ =()1  −+  σσht+−, φ − φφ (() ε () (15) 2  N 2 1 forÄ Å tn Substituting (11) and (12) into (10) gives =  − φ  ≤  ()  21Nt 2 2 2 n 1  ,t 11−+ / N  foor t >n − σφε − ()− φ ft   0  ()  =  N  ξ 2 Nt −   ,t 01(). Nn1 σγγδε =+  −+ Ehhhh +  012ξn ++++L t   N   ()  Prior to 1992, the day of a target change would Nt t 1 − often (but not always) occur the day after an FOMC 1 1 φφ− NuN− − ut for tn (13) ++τ () ≤≤ meeting. Since 1994, it usually has occurred on ∑1 1 τ = −φ the day of an FOMC meeting, but there are excep- tions: Three times in 2001 (January 3, April 18, t n 11Nn 1 and September 17) the Fed changed the target ftn −  0  −+ Nu− = ξξ+ + τ  N   N  ∑∑1 without a meeting, and in August and September     τ = Nt 1 − of 2007 there was active speculation that the Fed 1 φφ− Nutn− for . was considering or possibly had even already + ()1 t > − φ implemented an intermeeting rate cut. Rather From (13) we can then calculate the change than treat day n as if always known to the econo- in the spot-month futures rate for t Յ n to be metrician, I have followed a different philosophy, which is to ask, How would the data look if they Nn1 ff  −+ h were generated by (15) but the econometrician tt−=1 t −  N  does not condition on knowledge of the particular Nt1 Nt1 1 −+ 1 −+ value of n? Suppose that the day of the target 1 − φ 1 φφ− NuN− ()t −− ()ut 1 change (which the formula assumed was known + 1 − 1 − − φ − φ to market participants as of the start of the month) NNt1 Nn1 1 −+ −+ 1 − φ could have occurred with equal probability on any (14)   hNt − ()utt1 =  N  + 1 ()φε− + one of the calendar days n = 1,2,...,N. If we let   − φ η Nt1 1 −+ denote the unknown day of the target change, then 1 φφ− Nu− ()t 1 the unconditional data would exhibit a calendar − 1 − − φ regularity in the variance that is described by Nt1 1 −+ Nn1 1 −+ − − φ   hNt ()t fortn , ==  N  + 1 ε ≤   − φ

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Figure 3 Squared Spot-Month Change by Day of Month

Average Squared Change 20

–5 24 68 10 12 14 16 18 20 22 24 26 28 30 Day of Month

NOTE: The figure plots the average squared change in the spot-month futures rate, by day of the month (in basis points). 95 percent confidence intervals are indicated by upper and lower box lines, and predicted values from regression (19) are indicated by the dashed line.

N Expression (16) describes the variance of 2 1 2 Eftt f1 N− E ftt f1 n changes in the spot-month rate as the sum of two − − =− − η =  () n∑1 () = terms. The first term ( 1͑t͒) represents solely the N t 1 2 κ 1 −−+ N Nn1 2 contribution of deviations of the effective funds − φ 21 −+ 2 ,t N − ()h rate from the target. For days near the beginning = ()2 2 σσε + 2 (16) N 1 nt∑ N ()− φ = of the month (N – t large), this is essentially equal Nt1 2 2 1 −+ Nt1 2 to / 1 – (the unconditional variance of u ) Nt −+ 2 γ0 ͑ φ͒ t − φ ()− τ 2 == ()2 2 γγδ01+  + 3 σ h divided by N (because each ut contributes with N 1   ∑11 N ()− φ τ = weight 1/N to the monthly average). This declines tt2 , = κκσ12()+ () h gradually during the month (because there are fewer days remaining for which the serial corre- where lation in ut contributes to the variance) but then Nt1 2 rises quickly at the end of the month because of 1 −+ Nt t − φ − the large value of , reflecting the increased volatil- 1 ()2  01() 1 κ ()= N 2 1 γγδ+ γ ()− φ   ity of the deviations from the target at month end. Nt12223 Nt N t The second term ( 2͑t͒) represents the contribution t ()−+ ()−+ ()−+. κ κ2 (()= 6N 3 of target changes to the volatility of the spot-month rate. This contribution declines monotonically as

384 JULY/AUGUST 2008 FEDERALRESERVEBANKOFST. LOUIS REVIEW Hamilton the day of the month t increases. This is because, Let td denote the calendar day associated with as the month progresses, it becomes increasingly business day d (in other words, if = 1, then ωjd likely that the target change for the month has td = j). I then tested whether the specific function already occurred and there is no more uncertainty derived in (17) could account for this pattern by about the value of for that month. Added estimating via OLS the following relation: ξn together, expression (16) implies that the variance of changes in the spot-month futures rate should 2 ˆ (19) ffd dd1 1 t27. 9 2 t dd. − − = κκν+ + decline over most of the month but then increase ()() ()33. () at the very end. Note that all the parameters appearing in the Expression (16) was derived under the functions t and t are known as described κ1͑ ͒ κ2͑ ͒ assumption that at the beginning of every month above on the basis of the observed behavior of market participants are certain that there will be deviations of the effective fed funds rate from its a target change on day n of the month. If instead target, so that only a single parameter—the coef- there is a fraction of months for which people ficient on t in equation (19)—was estimated ρ κ2͑ d͒ anticipate a change on some day n and a fraction directly from the behavior of the futures data. 1 – for which they are certain there will be no This parameter, , has the interpretation of being ρ γ2 change, the result would be that the last term in the variance of daily news the market receives in (16) would be multiplied by : ρ a typical month about the upcoming Fed target 2 (recall equation (17)): (17) Eftt f1 122 t t, − − = κγκ()+ () () ˆˆˆ2 2 h 27. 9. γρσ= = 33. where 2 . () γρσ2 = h Note also that (19) imposes 30 separate restric- This model can be tested using daily data on tions on the 31 parameters of the unrestricted fed funds futures contracts.2 Figure 3 plots regres- regression (18). The F(30, 4,521) = 0.59 test sta- sion coefficients along with 95 percent confidence tistic leads to ready acceptance of the null hypoth- intervals from a regression of the squared change esis that this relation is indeed described by the in the spot-month futures rate on calendar day j: function given in (16) with a p-value of 0.96 (again, treating t as known functions). The 31 κj͑ ͒ 2 ˆ ˆ model thus successfully accounts for the tendency (18) ffdd1 jjd d , − − =+βω ν ()∑j 1 of the volatility of the spot-month futures rate to = decline over most of the month but then increase where = 1 if business day d occurs on calendar ωjd the last few days. The actual volatility seems to day j and is zero otherwise. In other words, ˆ is βj increase more at the end of the month than the the average squared change for observations falling model predicts, though it is possible to attribute on the jth day of a month. These indeed exhibit this entirely to sampling error. a tendency to fall over most of the month but then rise at the end. INFERRING MARKET EXPECTATIONS OF TARGET CHANGES FROM THE SPOT- 2 Data for October 3, 1988, through June 30, 2006, were purchased MONTH FUTURES RATE from the Chicago Board of Trade; data for July 3 through January 29, 2007, were downloaded from the now-defunct web site We are now in a position to answer the pri- spotmarketplace.com. For d corresponding to the first day of the mary question of this paper, which is, What does month (say the first day of February for illustration), fd – fd–1 was calculated as the change in the February contract between an observed movement in the spot-month futures February 1 and the last business day in January. For all other days rate signal about market expectations about the of the month, it was simply the change in the spot-month contract between day d and the previous business day. target rate that is going to be set for this month?

FEDERALRESERVEBANKOFST. LOUIS REVIEW JULY/AUGUST 2008 385 Hamilton

N 1 Figure 4 EYfftt t1 N− EYfftt t1 n  − −  =− − η =   () n∑∑1  () = Plot of κ4(t) as a Function of t N NNt1 12Nn1 1− + NE−  −+ hN − τ (21) = ρ  N  t = N γ 2 κ nt∑    ∑1 4͑t͒ =   τ = 2.5

2.0 Nt12 Nt −+ −+ t . ()()2 223 1.5 = 2N γγκ= () 1.0 Substituting (21) and (17) into (20) establishes 0.5 t 0 ˆ κγ3 2 EYtt () fft t1 1 6 11 16 21 26 31 Ω = − − (22) ()1 tt22() Day of Month κκγ()+ () 4 t fft t 1 . = κ ( ))()− −

The parameters determining 4͑t͒ have all been Let denote the information set available to mar- κ Λt estimated above from the properties of the devia- ket participants as of date t, and let = {f ,f ,…} tions of the fed funds rate from the target and Ωt t t–1 be the information set that is going to be used by squared changes in the spot-month futures rate. Figure 4 plots the function t for these the econometrician to form an inference, where κ4͑ ͒ it is assumed that is a subset of , the previous parameter values. To understand the intuition Ωt Λt target is an element of both and , and the for this function, consider first the case in which ξ0 Ωt Λt day n target change is an element of but not of the fed funds rate is always identically equal to Λt the target, so that 2 and t are both zero. From . Our task is to use the observed data to form σ ,t κ1͑ ͒ Ωt Ωt (14), the expected squaredε change in the spot- an inference about how the market changed its month rate conditional on knowing that the target assessment of based on information it received ξn change will occur on day n would be given by at t, that is, to form an assessment about 2 2 Ef tt f1 n, ,t0 − − ησ==ε YEt nt E nt1 () = ξξΛΛ− −   ()() (23) 2 22 htnforÄ Å Nn1 / N forÅ Ä tn t σ h  −+  ≤ ,  ≤ = () = 0 forÄ Åtn . 0 forÄ Å tn  >  > We can calculate the linear projection of Y on whereas the covariance of the spot-month futures t Ωt rate change with the expected target rate change as follows (e.g., Hamilton, 1994, equation [4.5.27]): would for this case be

EY f f 2 tt t1 Ef f Y n, 0 (20) ÊY  ()− −  ff .  tt1 t, t  tt 2 t tt 1 ()− − ησ==ε ()Ω = Ef f ()− −    tt− 1  2 ()− h Nn1 / NforÅ Ä tn   σ ()−+  ≤ . Recalling (14), the numerator of (20) can be found = 0 forÄ Å tn  > from Thus, if we knew both the day of the target change and that there were no targeting errors, the inference would be

Êh,, n 2 0 t f f ,  ttΩ ησ==, t = β n t− t1  ε  ()()−

386 JULY/AUGUST 2008 FEDERALRESERVEBANKOFST. LOUIS REVIEW Hamilton where In the presence of targeting errors, expression (22) adds the term t / to the denominator of NNn/ 1forÅÄ tn κ1͑ ͒ γ2 t  −+ ≤ , (24), so, as noted by Poole, Rasche, and Thornton βn = () () 0 forÄ Å tn (2002), the optimal inference in the presence of  > targeting errors always puts a smaller weight on which reproduces Kuttner’s (2001) formula (7) f – f than does (24). This explains why the for the special case considered by Kuttner, namely, t t–1 function t in Figure 4 begins at a value below t = n. If we don’t know the day of the target change, κ4͑ ͒ 1.5 for t = 1. The function t then begins to but still impose no targeting error, we’d use the κ4͑ ͒ increase monotonically in t for the same reason as unconditional moments: in (24). However, as t increases, both the numer-

2 2 ator and denominator in (24) become smaller, Ef tt f1 , t0 − − σ ε = whereas t / is approximately constant (at least ()  κ1͑ ͒ γ2 N for small t). This latter effect eventually over- 12 2 NNnN− 1 / =−σ h  + whelms the tendency of (22) to increase in t, and ∑nnt () = it begins to fall after the 20th day of the month. Ef f Y 2 0  t − ttt1 σ , =  This decline accelerates toward the very end of ()− ε  N the month as 1͑t͒ starts to spike up from the end- 1 2 κ NNn− 1 // N of-month targeting errors. =−σ h  + ∑nt () = ˆ 2 Eh tt, , t0 t f t f t1 Ω σβε = = () − −   ( )) RESPONSE OF INTEREST RATES

1 N NNnN− 1 / TO CHANGES IN FEDERAL FUNDS nt −+  (24) t ∑ = () . β = 1 N 2 FUTURES () NNnN− 1 / nnt −+  ∑ = () We’re now ready to return to the original ques- For N large and t = 1, the numerator of (24) would tion of how interest rates for Treasuries of various be approximately (1/2) and the denominator about maturities seem to respond to the spot-month fed (1/3), so that the coefficient 1 would be close funds futures rate. Deviations of the funds rate β͑ ͒ to 1.5. This is bigger than Kuttner’s expression (7), from the target should have a quite negligible effect which equals unity at n = 1, because a one-unit on maturities greater than three months, because the autocorrelation implied by (8) dies out within increase in h1 will increase the expected target on day n > 1 by one unit but increase the futures rate a matter of days. We should therefore find that, if we regress the change in Treasury yields on the on day t = 1 by only [͑N – n + 1͒/N] < 1. Kuttner’s formula assumes that, if we use the day t = 1 change in the spot-month futures rate, the value change in the futures, the target change occurs of the regression coefficient should exhibit exactly on day n = 1, whereas our formula assumes that the same pattern over the month as the function in all probability the actual change is going to be in Figure 4—the impact should rise gradually implemented on some day n > 1. through the first half of the month and fall off –1 quickly toward the end of the month. Going from t to t + 1, we drop N [͑N – t + 1͒/ N] from the numerator and drop the smaller mag- As a first step in evaluating this conjecture, –1 2 divide the calendar days of a month into j = 1,2, nitude N [͑N – t + 1͒/ N] from the denominator, ...,8 octiles and let = 1 if business day d is so that the ratio (24) monotonically increases in t ψjd until it finally reaches the same value as (7) on associated with a calendar date in the jth octile of the month. For example, = 1 if day d falls the last day of the month: ψ1d on one of the first four days of the month, whereas NN. = 1 if it falls on the 29th, 30th, or 31st. Let β()= ψ8d is,d denote the yield in basis points on day d for a Treasury bill or bond of constant maturity s

FEDERALRESERVEBANKOFST. LOUIS REVIEW JULY/AUGUST 2008 387 Hamilton

Figure 5 The Effect of Federal Funds Rate Changes on 1-Year Treasury Yields

3.0

2.5

2.0

1.5

1.0

0.5

0.0

–0.5

–1.0 2468 10 12 14 16 18 20 22 24 26 28 30 Day of Month

NOTE: The figure plots the coefficients and 95 percent confidence intervals for the OLS regression of daily change in the 1-year Treasury yield on daily change in the spot-month futures rate, with different coefficients for each octile based on the calendar day of the month (denoted by the rectangles and vertical lines) and the predicted values for the coefficients for each day of the month as implied by (26) (denoted by the dashed line).

months; for example, i12,d would be the 1-year and a much bigger effect than it would have rate. (Daily Treasury yields were taken from the toward the end of the month. The same pattern St. Louis FRED database.) Consider OLS estima- holds for shorter yields (Figure 6) and longer tion of yields (Figure 7). According to the theory, we can capture the 8 exact effect predicted for each calendar day by (25) iisd,, sd 11js jdffu d d sd. − −−=−αψ + regressing the change in interest rates on the prod- ∑j 1 () = uct between the change in fed funds futures and In Figure 5, the OLS estimates, ˆ , along with the function in (22): α j͑t͒,s their 95 percent confidence intervals, are plotted (26) iisd,, sd1 s41 tffu d d d sd, as a function of calendar day t = 1,2,...,31 for −−− = λκ ()()− + s = 12, which corresponds to a 1-year Treasury where td is the calendar day of the month associ- security. These indeed display very much the ated with business day d, is the effect of a one- λs predicted pattern—an increase in the fed funds basis-point increase in the target rate on a Treasury futures rate around the middle of the month has security of maturity s, and usd results from factors a slightly bigger effect on the 1-year Treasury rate influencing yields that are uncorrelated with than it would have at the beginning of the month, changes in the expected target rate. Note that all

388 JULY/AUGUST 2008 FEDERALRESERVEBANKOFST. LOUIS REVIEW Hamilton

Figure 6 The Effect of Federal Funds Rate Changes on 3-Month and 6-Month Treasury Yields

3.0 Effect on 3-Month Treasury Yield 2.5 2.0 1.5 1.0 0.5 0.0 –0.5 –1.0 2468 10 12 14 16 18 20 22 24 26 28 30 Day of Month

3.0 Effect on 6-Month Treasury Yield 2.5 2.0 1.5 1.0 0.5 0.0 –0.5 –1.0 2468 10 12 14 16 18 20 22 24 26 28 30 Day of Month

NOTE: The figure plots the coefficients and 95 percent confidence intervals for the OLS regression of daily change in the 3-month and 6-month Treasury yields on daily change in the spot-month futures rate, with different coefficients for each octile based on the calendar day of the month (denoted by the rectangles and vertical lines) and the predicted values for the coefficients for each day of the month as implied by (26) (denoted by the dashed line).

the parameters governing t have been inferred iisd,, sd 1 κ4͑ ͒ −=− from the behavior of the fed funds rate and futures (28) cfsdd f14 s t ddd f f 1 ussd . alone. Estimates of for different maturities, s, ()− −−+ λκ ()()− + λs are reported in the first column of Table 2, and If model (26) is correct, then we should be able values of ˆ t for different maturities, s, are to accept the null hypothesis that c = 0, whereas λ sκ4͑ ͒ s plotted as a function of t in Figures 5 to 7. if (27) is correct, we should accept the null hypothesis that = 0. If neither specification is correct, The adequacy of (26) was investigated in a λs number of different ways. One obvious question then we should reject both null hypotheses. The is how important the function t is for the second and third columns of Table 2 report the κ4͑ d͒ regression. This can be explored by comparing OLS coefficient estimates and standard errors for (26) with a specification in which changes in (28). For maturities greater than two years, we accept the null hypothesis that c = 0 and strongly futures prices have the same effect on interest s reject the hypothesis that = 0. For maturities rates regardless of when within the month they λs less than two years, both hypotheses are rejected, occur: suggesting that there is more to the response of short-term interest rates to fed funds futures than (27) iisd,, sd1 cffu s d d1 sd. − −−=−()+ is captured by (26) alone. Even in these cases, The specifications (26) and (27) are non-nested, however, the term involving t makes by far κ4͑ ͒ but it is simple enough to generalize to a model the more important contribution statistically. I that includes them both as special cases: conclude that the model successfully captures a

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Figure 7 The Effect of Federal Funds Rate Changes on 2-Year, 3-Year, and 10-Year Treasury Yields

3.0 Effect on 2-year Treasury Yield 2.5 2.0 1.5 1.0 0.5 0.0 –0.5 –1.0 2 468 10 12 14 16 18 20 22 24 26 28 30 Day of Month

3.0 Effect on 3-year Treasury Yield 2.5 2.0 1.5 1.0 0.5 0.0 –0.5 –1.0 2468 10 12 14 16 18 20 22 24 26 28 30 Day of Month

3.0 Effect on 10-year Treasury Yield 2.5 2.0 1.5 1.0 0.5 0.0 –0.5 –1.0 2468 10 12 14 16 18 20 22 24 26 28 30 Day of Month

NOTE: The figure plots the coefficients and 95 percent confidence intervals for the OLS regression of daily change in 2-year, 3-year, and 10-year Treasury yields on daily change in the spot-month futures rate, with different coefficients for each octile based on the calendar day of the month (denoted by the rectangles and vertical lines) and the predicted values for the coefficients for each day of the month as implied by (26) (denoted by the dashed line).

390 JULY/AUGUST 2008 FEDERALRESERVEBANKOFST. LOUIS REVIEW Hamilton

Table 2 The Effect of Federal Funds Futures on Interest Rates

With separate Restricted octile effects added effect With constant effect added (p-value for indicated H0) … Maturity λs cs λs α1s = = α8s = 0 λs = 0 3 Months 0.658** 0.256** 0.499** (0.00)** (0.98) (0.022) (0.089) (0.060) 6 Months 0.706** 0.286** 0.529** (0.00)** (0.45) (0.021) (0.084) (0.056) 1 Year 0.748** 0.226** 0.608** (0.00)** (0.60) (0.023) (0.095) (0.063) 2 Years 0.685** 0.159 0.586** (0.00)** (0.74) (0.029) (0.112) (0.079) 3 Years 0.641** 0.143 0.552** (0.01)** (0.62) (0.030) (0.122) (0.081) 10 Years 0.426** 0.082 0.375** (0.05)* (0.45) (0.028) (0.115) (0.077)

NOTE: This table shows the regression coefficients relating change in the interest rate on securities with maturity s to change in the fed funds futures rate. *indicates statistically significant with p-value <0.05; **denotes p-value <0.01. OLS standard errors are in parentheses. clear tendency in the data for the impact to vary both weekend effects and that some business days across the month, although it seems to leave convey much more important economic news something out in the description of the response than others (on this last point, see Poole and of short-term interest rates. Rasche, 2000, and Gürkaynak, Sack, and Swanson, In the same spirit, we can nest (26) and (25): 2005). We can in fact carry that last point a step fur- iisd,, sd 1 ther and estimate a separate coefficient for −=− λjs (29) 8 every calendar day j = 1,…,31: js jdff d d14 s tff d d d 1usd . αψ − −−+ λκ − + ∑j 1 ()()(() 31 = (30) iisd,, sd 11js jdffu d d sd, − −−=−λω + ∑j 1 () The results, shown in the last two columns of = Table 2, are not as encouraging. In every case, we where = 1 if day d falls on the jth day of the strongly reject the hypothesis that = … = = 0, ωjd 1s 8s month. Figure 8 plots the OLS estimates of as α α λjs meaning that for each maturity, s, there are sta- a function of the calendar day j along with 95 per- tistically significant deviations from the broad cent confidence intervals and the predicted values monthly pattern that is predicted by (26), and in for the function js implied by (26) for 1-year every case readily accept the hypothesis that = 0, λ λs Treasuries. Again the broad pattern seems to fit meaning that the specific variation within octiles well, though again there are large deviations on that is predicted by (26) is not particularly found some days that are well beyond what could be in the data. attributed to sampling error, and formal hypoth- These last results are perhaps not too surpris- esis tests comparing (30) with (26) (which the ing given the many approximations embodied in former formally nests as a special case) lead to (26), which assumed among other things that all overwhelming rejection, with a p-value less than months have N = 31 calendar days and ignored 10–10 for each s. In addition to the details noted

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Figure 8 Effects of Federal Funds Rate Changes on 1-Year Treasury Yields

3.0

2.5

2.0

1.5

1.0

0.5

0.0

–0.5

–1.0 24 68 10 12 14 16 18 20 22 24 26 28 30 Day of Month

NOTE: The figure plots the coefficients and 95 percent confidence intervals for the OLS regression of daily change in the 1-year Treasury yields on daily change in the spot-month futures rate, with different coefficients for each calendar day of the month (denoted by the rectangles and vertical lines) and the predicted values for the coefficients for each day of the month as implied by (26) (denoted by the dashed lines). above, individual outliers are highly influential approach developed here gives us a plausible for the daily regression (30), and one would want interpretation of the broad regularities found in to carefully model these non-Gaussian innova- the data and a sound basis for generalizing the Kuttner (2001) and Poole, Rasche, and Thornton tions, usd, and GARCH effects before trying to build a more detailed model that could reproduce (2002) approaches. Although the methods involve more of the unrestricted pattern. This and related some new uses of the data, the conclusion I draw tasks, such as trying to use information about the is quite consistent with earlier researchers— actual date of the target change when it is unam- changes in the fed funds target seem to be associ- biguously known, using one-month or two-month ated with quite large changes in Treasury yields, futures contracts in place of the spot rate, and even for maturities of up to 10 years. exploring the consequences of a secular change in 2 (e.g., Lang, Sack, and Whitesell (2003) and σh Swanson, 2006), we leave as topics for future REFERENCES research. Anderson, Richard G. and Rasche, Robert H. Although there is much more to be done “A Revised Measure of the St. Louis Adjusted before having a completely satisfactory under- Monetary Base.” Federal Reserve Bank of St. Louis standing of these relations, I believe that the Review, March/April 1996, 78(2), pp. 3-14.

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Cook, Timothy and Hahn, Thomas. “The Effect of Piazzesi, Monika and Swanson, Eric T. “Futures Prices Changes in the Federal Funds Rate Target on Market as Risk-Adjusted Forecasts of Monetary Policy.” Interest Rates in the 1970s.” Journal of Monetary Journal of Monetary Economics (forthcoming). Economics, November 1989, 24(9), pp. 331-51. Poole, William and Rasche, Robert H. “Perfecting the Gürkaynak, Refet S. “Using Federal Funds Futures Market’s Knowledge of Monetary Policy.” Journal Contracts for Monetary Policy Analysis.” Working of Financial Services Research, December 2000, paper, Board of Governors of the Federal Reserve 18(2/3), pp. 255-98. System, 2005. Poole, William; Rasche, Robert H. and Thornton, Gürkaynak, Refet S.; Sack, Brian P. and Swanson, Daniel L. “Market Anticipations of Monetary Policy Eric T. “Do Actions Speak Louder Than Words? Actions.” Federal Reserve Bank of St. Louis Review, The Response of Asset Prices to Monetary Policy July/August 2002, 84(4), pp. 65-94. Actions and Statements.” International Journal of Central Banking, June 2005, 1(1), pp. 55-93. Sarno, Lucio; Thornton, Daniel L. and Valente, Giorgio. “Federal Funds Rate Prediction.” Journal Gürkaynak, Refet S.; Sack, Brian P. and Swanson, of Money, Credit, and Banking, June 2005, 37(3), Eric T. “Market-Based Measures of Monetary Policy pp. 449-71. Expectations.” Journal of Business and Economic Statistics, April 2007, 25(2), pp. 201-12. Swanson, Eric T. “Have Increases in Federal Reserve Transparency Improved Private Sector Interest Rate Hamilton, James D. Time Series Analysis. Princeton, Forecasts?” Journal of Money, Credit, and Banking, NJ: Princeton University Press, 1994. April 2006, 38(3), pp. 791-819.

Hamilton, James D. “Daily Changes in Fed Funds Taylor, John B. “Expectations, Open Market Futures Prices.” Journal of Money, Credit, and Operations, and Changes in the Federal Funds Banking (forthcoming). Rate.” Federal Reserve Bank of St. Louis Review, July/August 2001, 83(4), pp. 33-47. Kuttner, Kenneth N. “Monetary Policy Surprises and Interest Rates: Evidence from the Fed Funds Futures Thornton, Daniel L. “A New Federal Funds Rate Market.” Journal of Monetary Economics, June 2001, Target Series: September 27, 1982–December 31, 47(3), pp. 523-44. 1993.” Working Paper 2005-032, Federal Reserve Bank of St. Louis, 2005; Lang, Joe; Sack, Brian P. and Whitesell, William. http://research.stlouisfed.org/wp/2005/2005-032.pdf. “Anticipations of Monetary Policy in Financial Markets.” Journal of Money, Credit, and Banking, December 2003, 35(6, part 1), pp. 889-909.

Nilsen, Jeffrey H. “Borrowed Reserves, Fed Funds Rate Targets, and the Term Structure,” in Ignazio Angeloni and Riccardo Rovelli, eds., Monetary Policy and Interest Rates. London: Macmillan Press, 1998.

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