Adrian W. Throop*

Increasingly volatile financial markets haveput a premium on accurate forecasts of interestrates. To the extent, however, that market par­ticipants base their decisions on availableinterest-rate forecasts, the value of theseforecasts to an individual investor isdiminished because security prices wouldalready tend to reflect that information. At theextreme, the efficient-market hypothesisasserts that the market efficiently utilizes allavailable information in the pricing ofsecurities, so that market participantsgenerally can not profit from more accurateforecasts than those already incorporated insecurity prices.

This article examines the degree to whichthe Treasury-bill market efficiently utilizes allavailable information so as to incorporate thebest possible interest-rate forecast into currentmarket prices. In other words, it evaluates theapplicability of the efficient-market hypothesisto the Treasury-bill market. Two types of inde­pendent forecasts are examined: anautoregressive forecasting equation based onthe past history of the bill rate, and the forecastof a selected panel of market professionals.Statistical tests are used to determine whetherall useful information contained in these twoforecasts is efficiently incorporated into Treas­ury-bill market prices. The market's forecast isderived from the "forward rate" implied bythe term structure of yields.

If the market is not efficient, a group ofinvestors could improve their returns by alter­ing the maturity of their investments in light of

'Senior Economist. Federal Reserve Bank of San Fran­cisco. Coleman Kendall provided research assistance forthis article.

29

superior interest rate forecasts. Near a peak inthe business cycle, for example, if suchforecasts correctly foresee a larger decline ininterest rates than anticipated by the market,then investors should buy securities with amaturity longer than their expected invest­ment periods. The yields obtained by investorswould be greater than those obtainable if theyhad chosen maturities equal to their invest­ment period, due to larger capital gains thanhad been anticipated by the market. Alter­natively, if the available forecasts correctlyforesee interest rates falling more slowly thanthe market does, the investors utilizing thoseforecasts should buy maturities shorter thantheir investment periods. This is becauseinvestors would obtain a higher return from"rolling over" a series of short-term securitiesthan from buying maturities equal to theirinvestment periods.

Whether or not investors can profit by"speculating" on interest rates, through hold­ing other than their normally preferredmaturities, depends critically on whetheravailable interest-rate forecasts are moreaccurate than the market's. If the market isefficient, the information in these forecastswould already be incorporated into the pricesof securities, and therefore nothing would begained. In that case, investors would be betteroff by simply "hedging" their positions withmaturities equal to their planned investmentperiods, thereby avoiding possible risk.

This study shows that professional analysts'prediction of the Treasury-bill rate two quar­ters ahead are significantly more accurate thanmarket predictions. This indicates that themarket does not efficiently utilize all availableinformation in making bill-rate forecasts. By

making use of the information contained in theanalysts' forecast, an investor in Treasury billscould have improved his return. We show thatthe analysts' ability to "beat the market"stems from a better utilization of informationabout past movements in the bill rate, and alsofrom a more efficient utilization of other sortsof information.

Section I explains the concept of market effi­ciency and examines the forecasting accuracyof the market, compared with that of both apanel of professional analysts and a simpleautoregressive forecasting equation based onthe past history of the bill rate. In Section II,statistical tests for market inefficiency are de-

scribed and performed, on the basis of thesethree forecasts. Both the analysts' forecast andthe forecast from the autoregressive forecast­ing equation are found to contain useful infor­mation which is not fully incorporated into themarket's forecast indicating market ineffi­ciency. Section III shows that the analysts'forecast contains information similar to that inthe autoregressive forecasting equation, plusother useful information which is also not fullyincorporated into the market's forecast. Inves­tors could have traded on both types of infor­mation to improve their returns in the periodexamined. Section IV provides a summary andsome further conclusions.

I. The Concept of Market Efficiency

Efficient financial markets exist when theprices, or yields, of securities fully reflect allavailable information relevant for their valua­tion. In the case of riskless fixed-incomesecurities, such as Treasury bills, the relevantinformation consists of expectations about thefuture course of interest rates. Investors thenbid the prices of securities to the point whereexpected holding-period yields for securities ofdifferent maturities are roughly the same,given these expectations. For example, giventhe current six-month bill rate, the price andyield of a nine-month bill depends upon theexpected three-month rate for six monthsahead. New information can develop, butwhen it does it is rapidly reflected in revisedexpectations and in the prices of securities.Consequently, in an efficient Treasury-billmarket, investors would not have significantopportunities for making profits on the basis ofinformation about future interest rates l .

The hypothesis of an efficient market is anextreme one, and therefore could not beexpected to be literally true. Past tests of theefficient-market hypothesis have thusattempted to pinpoint the level of informationat which the hypothesis breaks down. In thesetests, all available information can be sepa­rated into three distinct types, or subsets, asshown in Table 1. The first subset consists of

30

the past history of rates of return, or prices. Atest of whether the market efficiently utilizesinformation in the past history of rates ofreturn, or prices, is called a test of weak-formefficiency. The second information subset con­sists of any other information publicly availa­ble at little or no cost - such as governmentstatistics on other relevant variables. Tests ofwhether the market efficiently utilizes thiskind of information in securities pricing arecalled semistrong-form tests. The third informa­tion subset consists of information that is priv­ileged or available only at significant cost.Tests of whether the market efficiently utilizesthis kind of information, so that profits cannotbe made from trading on it, are called strong­form tests.

Table 1Types of Market Efficiency

Subset of Information Test of WhetherParticular InformationSubset Is EfficientlyUtilized by Market

I. Past History of Prices,or Rates of Return Weak-Form Test

2. All Other PubliclyAvailable Information Semistrong-Form Test

3. Privileged or CostlyInformation Strong-Form Test

The three subsets of information exhaustthe universe of all available information; thus,a market is fully efficient only if it passes allthree kinds of tests. According to previousstudies, the Treasury-bill market tends to beefficient in the weak-form sense of utilizinginformation in the past behavior of the billrate. However, the evidence has generallybeen confined to very near-term marketforecasts of only up to a few months ahead. 2

Also, little evidence is available on the billmarket's efficiency in the strong or semi­strong form sense of incorporating otheravailable information useful for forecasting. 3

In this study, we consider two specific typesof information that the Treasury-bill marketcould utilize in formulating a two-quarterahead forecast of the 3-month bill rate. Thefirst is simply an autoregressive forecast basedon the past history of the bill rate. The yield onTreasury bills may vary somewhat predictablyover time, due to the business cycle and/orpredictable patterns in monetary policy. Topass the weak-form test, the market's forecastof the future bill rate needs to take intoaccount any systematic behavior evidenced byits past history. 4

For the period 1951-IV through 1969-III, weestimated a simple autoregressive forecast thatcould have been used by the market. In thisperiod, quarterly movements in the 3-monthbill rate followed a significant autoregressivepattern, ie., past movements of the rate weresignificantly related to future movements. Anequation explaining quarterly changes in thebill rate contained significant lags at 1, 2, 3,and 6 quarters, as well as a significant constantterm indicating a positive time trend. We thenobtained autoregressive forecasts for theperiod 1970-I through 1979-III by reestimatingthis equation on a growing sample of availableobservations and computing two quarter aheadforecasts from the latest coefficient estimatesat each point in time. 5 Whether or not themarket efficiently utilized the informationcontained in this autoregressive forecast is atest of weak-form efficiency.

The second type of information is theaverage interest-rate forecast made by a panel

31

of professional analysts, and compiled by theGoldsmith-Nagan Bond and Money Market Let­ter since Septem ber 1969. 6 The forecast periodis again 1970-1 through 1979-III. This periodbegins with the first Goldsmith-Nagan sam­pling of professional forecasts and continuesthrough the quarter just prior to the FederalReserve's October 1979 shift in operating pro­cedures, which emphasized controlling bankreserves for achieving its monetary objectives.We excluded later data on the ground thatboth the market and professional forecastershad to go through a learning experience whichreduced the forecasting accuracy of each byperhaps differing amounts.

Professional analysts use, either directly orindirectly, information in the past history ofthe bill rate - but they undoubtedly use othersources of information as well. So the analysts'forecasts contain information relevant to allthree kinds of efficiency. However, it is notpossible here to distinguish between a semi­strong and strong test of efficiency. Someinterest-rate forecasts in the Goldsmith-Nagansample are not widely circulated, being privi­leged or costly to obtain, while other informa­tion may be publicly available. Since we cannotdiscriminate between these two types of infor­mation in the analysts' forecasts, we simplyuse the term "strong-form efficiency" to referto efficient market use of all types of informa­tion besides the past history of the rate.

In order to test the bill market's efficiency,we need a measure of the market's forecast ofthe 3-month bill rate for two quarters ahead.This can be obtained from the term structureof Treasury bill rates - specifically, from themarket's two quarter ahead "forward rate."This is the interest rate on a 3-month Treasurybill two quarters ahead that would be requiredto equalize expected returns on 6- and 9­month bills over a 9-month holding period. 7

The forward rate also contains a "liquidity pre­mium," which compensates investors in thelonger-term security for their sacrifice ofliquidity. So an adjustment for that premiummust be made to provide an estimate of themarket's forecast of the bill rate. A n Appendixdevelops the concept of the forward rate more

precisely, and details the technique used toestimate the liquidity premium.

For a preliminary view of market efficiency,we can compare the market's two-quarterahead forecast of the 3-month Treasury billrate (F;':2) with the professional analysts'forecast (F~: 2) and the forecast from theautoregressive equation(F:l~2)' To measureforecast accuracy, we use the mean squarederror (MSE), the arithmetic average of thesquares of the forecast errors - or better still,the root mean squared error (RMSE), since itmeasures the error in the same units as theforecasted variable (Table 2) . As a baseline forcomparison, we use the RMSE for a forecast ofno change.

After extraction of the estimated liquiditypremium, the RMSE of the market's forecast,1.24 percentage points, is only slightly lowerthan that for the forecast of no change - butsubstantially higher than the RMSEs of theanalysts' and autoregressive forecasts.Moreover, the approach used to estimate themarket's liquidity premium is more likely tohave understated rather than overstated thetrue difference between forecast errors, asshown in the Appendix. These differencesthus suggest the possibility of market ineffi­ciency. If the estimated difference between theRMSEs of the autoregressive forecast and themarket's forecast were statistically significant,then the condition of weak-form efficiencywould not be met. An autoregressive forecastbased on available past information on the

Table 2Accuracy of Forecasts

Root Mean Squared Error (RMSE)19701 - 1979111

(percentage points)Forecast of No Change (it) 1.25

Market's Forecast (F:"t-2) 1.24(Adjusted for LiquidityPremium as Estimated fromAppendix Table 1, Eq. 3)

Analysts' Forecast (Fgn ) 1.10t+2Autoregressive Forecast (Ff't) .94

Treasury- bill yields would have provided ameans for investors to "beat the market". Byaltering the maturity of their holdings in lightof a superior forecast they could haveimproved their returns.

The RMSE of the analysts' forecast is lowerthan the market's, but higher than that of theautoregressive forecast. Thus, the analystsmay not have made full use of the availableautoregressive information. Still, the analysts'forecast could contain other informationbesides that embodied in the past history ofthe rate. If an investor could have profitedfrom trading on such other information, themarket would also be inefficient in a strong­form sense. 8 The next two sections providetests of forecast accuracy that distinguish be­tween strong-form and weak-form ineffi­ciency.

II. Evidence of Market Inefficiency

We first test for weak-form inefficiency byexamining whether the autoregressive forecastcould have been systematically used to reducethe error in the market's forecast. This couldhave been done if the market's forecast error isat least partially explained by the difference be­tween the autoregressive forecast and themarket's forecast. Symbolically, this relation­ship is:

i1+ 2- F:: 2 = B I (F;"t2 - F;':2)

+ e'+2' (1)

32

where i'+2 - FIl1

2= the market's forecasterror for two quartersahead;

F;t~2 = the autoregressiveforecast for quarter t+ 2made at time t;

F;': 2= the market's forecast forquarter t+ 2 made at timet;

e t+2= a random error term.

The value of B b at .664, is more than fourtimes its estimated standard error, given in theparentheses, indicating that it is indeed signifi­cantly different from zero. Thus, theautoregressive forecast could have been usedto reduce the market's forecast error, confirm­ing the existence of weak-form inefficiency. Infact, in an optimal forecast combining the two,the autoregressive forecast would be given alarger weight (.664) than the market's forecast(.336). Also note that the unexplained forecasterror has been reduced to .90 percentagepoints (equals the equation's standard error,S .EJ from 1.24 percentage points (equalsRMSE in Table 2).

Note that this equation can be rearranged togive the optimal forecast'of the interest rate asa weighted average of the two forecasts. Thatis, it implies:

i t +2= BIF:~2 + (l - B I ) F:: 2+ e t +2 (2)

In the case of weak-form inefficiency, theautoregressive forecast would receive a signifi­cant weight. On the other hand, if there wereno weak-form inefficiency, the weight of theautoregressive forecast would be insignifi­cantly different from zero, and the weight ofthe market's forecast would be close to one.Thus, the test of weak-form inefficiency is thatthe estimated value of B I be significantlydifferent from zero, so that the autoregressiveforecast could have been used to improveupon the accuracy of the market's forecast.

F:~2 and F~:2 are apt to be highly correlatedin equation (2), tending to increase the stan­dard errors of the estimated coefficients. So itis preferable to test for weak-form inefficiencyby estimating equation (1); in addition, thisequation constrains the weights to add up tounity. We estimated this and other equationsusing ordinary least squares, with a correctionfor a moving-average pattern of serial correla­tion in the error term. 9 The estimate of equa­tion (l) is:

S.E. = .862

S.E. = .893

i'+2 - F:~2 = .685 (F~:2 - F;1~2) (5)

(.187)

"R 2= .155

A second test for confirming the existenceof strong-form inefficiency involves generaliz-

The estimated value of the B I coefficient, at1.15, is over six times its estimated standarderror. It is clearly significantly different fromzero and also not significantly different fromone. Thus, only the analysts' forecast shouldbe used in an optimal forecast combining both,because the market's forecast provides little orno additional information.

This test also supports a finding of marketinefficiency. Whether it is of the weak orstrong form, or both, depends upon the type ofinformation embodied in the analysts'forecast. If the analysts' forecast were basedonly on the historical behavior of the bill rate,only weak-form inefficiency would be con­firmed. But if it contains other information aswell, both weak and strong-form inefficiencywould be indicated.

Two additional tests can be made for thepresence of strong-form inefficiency. The firstexplains the forecast error of the autoregres­sive equation by the difference between theanalysts' forecast and the autoregressiveforecast. The significant coefficient estimatedin this equation, given below, indicates thatthe analysts' forecast can be used to explain animportant part of the error in theautoregressive forecast. Therefore, thatforecast contains other useful informationbesides the past history of rates.

A similar test can be performed with theforecast of the Goldsmith-Nagan panel ofanalysts, F~:2' as the alternative to themarket's forecast. When the difference be­tween the analysts' and the market's forecast isused to explain the market's forecast error, weobtain the following estimated equation:

(3)

S.E. = .900

(.163)

"R 2 = .467

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Both of the estimated coefficients are larger

(.337)

+ .321 (F:~2 - F:: 2)

(.201)

R2= .498 S.E. = .873

ing the form of the original test. That is, theerror in the market's forecast might be ex­plained by both the difference between theautoregressive and the market's forecast andthe difference between the analysts' and themarket's forecast. In symbols,

than their standard errors and significantlydifferent from zero at the 5-percent level,indicating both weak- and strong-form ineffi­ciency.

To summarize the results so far, we havefound that two different types of informationcould have been used to improve upon themarket's forecasts of the 3-month Treasurybill rate, as embodied in the forward rate. Thesignificance of an autoregressive equation inexplaining the market's forecast error indi­cates weak-form inefficiency, and the addedsignificance of the analysts' forecast confirmsstrong-form inefficiency. By incorporating theinformation contained in an autoregressivemodel and the analysts' forecast, the RMSEfor a two-quarter ahead forecast of the 3­month Treasury bill rate could have beenreduced from the market's 1.24 percentagepoints to .87 percentage points (equal to S.E.of equation (8). In fact, when all threeforecasts are combined into an optimalforecast, as in equation (7), the weightattached to the market's forecast is not signifi­cantly different from zero, at .022 (equals 1 ­.657 - .321). Thus, the autoregressiveforecast and the analysts' forecast contain allthe useful information embodied in themarket's forecast, plus other useful informa­tion as well.

(6)

(8)

i'+2 - F~:2 = B I (F;"1-2 - F~+2)-I- Q {pgn _ \ -'- ~

I U2 \.1·1+2 l+2JT C;t+2

or in the alternative form,

i'+2 = BIF;I~2 + B2F~:2 (7)

+ (l - B I - B2) F:: 2+ e'+2

If both B I and B2 were significantly differentfrom zero, then both weak and strong-forminefficiency would be confirmed. We againchoose the first form of the equation forestimation, in order to reduce the extent ofmulticollinearity among the independentvariables. The estimate of the equation in thisform is:

III. Information in the Analysts' Forecast

The previous section tested whether infor­mation from a simple autoregressive modelthat could have been used to predict the Treas­ury-bill rate was fully incorporated into themarket's forecast, as embodied in the forwardrate. However, some judgment enters into theconstruction of even such a simpleau toregressi ve forecast. Specifically, if thetime pattern of rate movements is unstable,the forecasting power of the estimatedautoregressive equation could depend impor­tantly upon the period of estimation chosen.Therefore, we should also test whether theautoregressive information that was actually

used by the professional analysts was efficientlyincorporated into the market's forecast. Infact, we shall see that the autoregressive infor­mation contained in the analysts' forecast issubstantially the same as that in ourautoregressive equation.

To approach this question, we decomposedboth the professional analysts' forecast and themarket's forecast into the portion related tocurrent and past bill rates (extrapolative com­ponent) and the remainder that is not sorelated (autonomous component) .10 Thedifference between the two forecasts can thenbe decomposed into the difference between

34

the extrapolative components of the twoforecasts plus the difference between theautonomous components. If the difference be­tween the extrapolative components werestatistically significant in explaining themarket's forecast error, this would indicatethat autoregressive information actually incor­porated into the analysts' forecasts was usefulin predicting the interest rate, but neverthelesswas not fully incorporated into the market'sforecast - an instance of weak-form ineffi­ciency. Similarly, if the difference between thetwo autonomous components were significantin explaining the market's forecast error, thiswould mean that the market did not incorpor­ate other useful information available to theanalysts - an instance of strong-form ineffi­ciency.

The extrapolative components of the twoforecasts were estimated by regressing thedifference between each forecast and the cur­rent rate on lagged quarterly differences in the

bill rate. Such components can be divided con­ceptually into three parts (Figure 1). The firstis simply the current interest rate, or a predic­tion of no change corresponding to a random­walk hypothesis. If there were no significantcoefficients in the above regression, the esti­mated extrapolative component would containonly the current interest rate. The secondcomponent is a time trend, or drift factor, indi­cated by a significant constant term in theabove regression. The third component is thepart of the forecast related to past changes inthe interest rate, indicated by any significantcoefficients on past changes in the rate.

The best equation explaining the differencebetween the market's two-quarter aheadforecast of the bill rate and the current rate,chosen on the basis of a minimum standarderror, contained no constant term and no lag­ged changes in the bill rate. Thus theextrapolative component of the market'sforecast, E:: 2, is estimated to be simply a

Figure 1

Elements of a Forecast

Realized interest rate ------....."'.

Forecast error

Autonomouscomponentof forecast

j'"Related to !

past changes." '"

Time-Trend [_ .1$!

Forecasted interest rate ---,-----_. ",'

Currentinterest rate

Forecast ofno change

Extrapol ativecomponentof forecast

1 1+2 Time

Note: For simplicity, this illustration assumes that elements of the forecast other than the current interest rate allcontribute to reducing the forecast error. Of course, this need not be true in general. For example, the time trend,the part of the extrapolative component related to past changes, and the autonomous component could all benegative, instead of positive as assumed here. In that case, and given the same realized interest rate, theforecasted interest rate would be below, rather than above, a forecast of no change; and the forecast error wouldbe increased, rather than reduced, by these three elements.

35

forecast of no change, equal to the currentinterest rate (i t). Symbolically,

The significant coefficient on the differencebetween the two extrapolative components

+ .323 (F:I~2- F~:2)

(.215)

R 2 = .484 S.E. = .885

+ .766 (E~:2 - E:2) + .323(.584)

The estimated standard error of the equationis reduced somewhat by the addition of thedifference between these two forecasts.However, neither the coefficient on this varia­ble nor the coefficient on the difference be­tween the extrapolative elements is signifi­cantly different from zero at the 5-percentlevel. Thus, neither the extrapolative compo­nent of the analysts' forecast nor theautoregressive equation's forecast significantlyreduces the market's forecasting error oncethe other is already being utilized. Both con­tain a positive time trend and extrapolate frompast changes in the bill rate, in contrast to theprediction of no change in the extrapolativeelement of the market's forecast. On the basisof this evidence, we conclude that the analystshave utilized the available information on pastmovements in the bill rate about as efficientlyas possible in their forecast.

shows that the analysts made better use ofinformation in the bill rate's past history thandid the market. In addition, the significantcoefficient on the difference between the twoautonomous elements indicates that theanalysts made superior use of other informa­tion besides past rates. The first finding con­firms weak-form inefficiency, and the secondsubstantiates a strong form of inefficiency.

A final point of interest is whether theextrapolative component of the analysts'forecast utilizes available information on pastbill-rate movements as efficiently as possible.To examine this question, we simply added tothe right-hand side of equation (12) thedifference between the forecast from theautoregressive equation and the market'sforecast, and then reestimated the equation.This gives

i t+2- F:: 2= .650 (A~:2 - A:) (13)

(.348)

(9)

(1 I)

S.E. = .902

(,377)

R2= .464

(.200)

F~:2 - 1+2 (A~:2+ E~:2)

- (A: 2 + E:: 2)

= (A~:2 - A::)

+ (E~:2 - E:':2)

To test for both weak and strong-form ineffi­ciency in terms of the autoregressive informa­tion actually used by analysts in theGoldsmith-Nagan survey, we substitute equa­tion (1 I) into equation (4) and reestimate thelatter. The resulting estimated equation is:

i t +2- F: 2= 1.10 (A~n~2 - A:: 2) (12)

tive element. It follows that the difference be­tween the analysts' forecast and the market'sforecast equals the sum of the differences be­tween the autonomous and extrapolative com­ponents; or

In contrast, the extrapolative component ofthe analysts' two quarter-ahead forecast, E~:2'

estimated in the same fashion, was found to bemore complex. It contains a constant term,indicating a positive time trend in the interestrate of 54 basis points per year, and significantlags at 1, 3, 4, and 5 quarters on past changesin the bill rate. Specifically, the extrapolativecomponent of the analysts' forecast is esti­mated to be:

E~:2 = it + .270 - .196(it - it-I) (10)

+ .186 (it-J - i H )

+ .306(iH - i t - s) + .200(i t •s - i H )

The autonomous components, denoted byA: 2 and A~:), are simply the difference be-tUloan tho t"ocn.a.r"ttUA fArAf"''lC'f -:lnrl ite' p-vtr<.:lnf\lo:l_L YV \,.1\,;1.1 Ll1'-' .l \,.IJP,",,"""-1 V ...... J. VI. ",,-,,"-,U'-'\. Ul.1. ..... l .. oJ .... ,.,,\.:1. ""'"p ....... ~~

36

These analysts also efficiently incorporatedother publicly available information into theautonomous component of their forecast. AsFriedman (1980) has shown, the forecastingerrors of the Goldsmith-Nagan panel for short­term interest rates have not been significantly

related to costlessly available information onpast values of macroeconomic series affectingthe interest-rate outlook. These series includethe unemployment rate, industrial production,price inflation, the money stock, and theFederal government's deficit.

IV. Summary and Conclusions

We have investigated whether the Treasury­bill market has efficiently utilized all availableinformation, so as to incorporate the bestpossible two-quarter ahead forecast of the 3­month interest rate into its pricing of Treasurybills. A forecast by a panel of professionalanalysts was used as a measure of all availableinformation, and the subset of informationrelating to the bill rate's past history was esti­mated by a simple autoregressive forecastingequation. We found that the analysts' forecastcontained a similar extrapolative component,based on past movements in the bill rate.Either this extrapolative component of theanalysts' forecast or a forecast from theautoregressive equation could have been usedto reduce the error in the market's forecast sig­nificantly; but both contained substantially thesame information. In addition, the analysts'forecast contained other useful information forexplaining a portion of the market's forecasterror. Altogether, the tests performed indicatethe existence of both weak-form and strong­form inefficiency.

Earlier studies of the efficiency of the Treas­ury-bill market, focusing on shorter terminterest-rate forecasts, have usually indicatedweak-form efficiency. In contrast, our resultsfor a two-quarter ahead forecast of the bill rateshow the existence of weak-form inefficiency.This reflects the positive time trend found inboth the extrapolative component of the

37

analysts' forecast and the autoregressiveforecasting equation, but not in the market'sforecast. The market's forecast of the 3-monthbill rate failed to incorporate the upward driftin the bill rate attributable to rising inflation inthe forecast period, even though this driftcould have been extrapolated from past data.

In testing for a stronger form of efficiency inthe bill market, earlier studies have generallyused current and past values of relevantmacroeconomic variables to measure otheravailable information besides the past historyof the bill rate. This study used the non­extrapolative component of the professionalanalysts' forecast for this purpose. We foundthat this component also contributed signifi­cantly to reducing the market's forecast error,indicating strong-form inefficiency as well.

The difference between the overall forecasterror of the analysts and the market in the1970's was a modest 14 basis points, asmeasured by the root mean square error.Nevertheless, our results indicate that theabove types of market inefficiency would haveallowed an investor to trade on the informa­tion contained in the analysts' forecast. Aninvestor could have improved his overallreturns with a strategy of shortening maturitieswhen the analysts' forecast was above themarket's and lengthening them when theopposite was true.

AppendixEstimation of the Market's Forecast

The market's two-quarter ahead forecast ofthe 3-month Treasury bill rate is embodied inthe differential between yields on 6- and 9­month bills. To see this, first consider thatmarket arbitrage makes the yield on a 9-monthTreasury bill equal to the expected return on a3-month Treasury bill that is rolled over twice,plus premiums for the sacrifice of liquidity. Inalgebraic terms, the yield on the 9-monthTreasury bill is:

are generally unbiased, so that the realizedinterest rate is approximately equal to the anti­cipated interest rate plus a random error:

(4)

The market's forward rate (f) is composed ofan anticipated interest rate and a liquidity pre­mium (p), or:

(5)

(6)

If the liquidity premium were constant, itcould be estimated simply from the averagedifference between the forward rate and subse­quently realized interest rate. But previousresearch indicates that liquidity premiums arein fact somewhat variable. Three not mutuallyexclusive hypotheses have received support inthe literature.

The first hypothesis views short-termsecurities as better substitutes for money bal­ances than longer-term securities. If this is so,the liquidity premium would tend to beaffected by the level of interest rates. Wheninterest rates are high, the foregone interestincome in holding money is larger; and so thepublic desires to exchange part of its moneybalances for securities. However, if the publicprefers short to long securities in thisexchange, prices of short-term securitieswould be bid up relative to those on longer­term ones. This would increase long-terminterest rates relative to returns on short-termsecurities, or in other words increase theliquidity premium. Thus, if short-termsecurities are better substitutes for money thanlonger-term ones, liquidity premiums wouldvary positively with the level of interest rates. II

Subtracting (4) from (5), we obtain theliquidity premium as equal to the differencebetween the market's forward rate and therealized interest rate plus the random (expec­tationaO error:

0)

Dividing (0 by (2) and rearranging terms, weobtain equation (3) for the market's two­quarter ahead "forward rate." This is theexpected return (including a liquidity pre­mium) on a 3-month Treasury bill two quar­ters ahead that is required to bring aboutequality between the expected returns on 6­and 9-month bills over a 9-month holdingperiod.

o + 3i ) 3

To evaluate the accuracy and informationcontent of the market's two-quarter aheadforecast, it is necessary to separate the liquiditypremium from the forward rate. The approachused here assumes that market expectations

where i = yield at a quarterly rate,p = liquidity premium,left subscript = maturity of security inquarters,right subscript = time at whichinvestment in security begins in quar­ters,superscript "e" = interest rateexpected by the market as of time t.

Similarly, for a 6-month Treasury bill:

0+ 2it)2= 0 + lit) 0 + li et+ l + IPt+l) (2)

38

Two other hypotheses relate to the fact thatliquidity premiums on longer-term securitiescompensate investors for exposure to the riskof capital losses. The greater the probablevariation in interest rates, the larger would besuch risk. If the variation in interest rates isexpected to approximate its recent variance,liquidity premiums ought to vary positivelywith that variance. Moreover, at times wheninterest rates are abnormally low (high), thereis a large likelihood of unanticipated increases(decreases) producing unanticipated capitallosses (gains) for holders of longer-termsecurities. Therefore, liquidity premiumsmight also be expected to vary inversely withthe height of interest rates relative to recentlyexperienced levels. 12

These three hypotheses were tested withavailable data for the period 1963-IV through1979-III (Table A.1). Following equation (6),we regressed the difference between the two­quarter ahead forward rate and the realizedTreasury-bill rate on the current level of theTreasury-bill rate (j), the difference betweenits current level and a weighted average of its

value over the previous 2 years (i - n, and thestandard deviation of the bill rate over thesame period (SD).13 When no explanatoryvariables other than a constant term areincluded, the constant term is significantlypQ$itive - indicating a positive averageliquidity premium of 54 basis points.

The level of the bill rate (reflecting theeffect of better substitutability between short­term securities and money) and the deviationof the bill rate from its recent trend (proxyingfor the probability of unanticipated capitalgains on long-term securities) show significanteffects when entered together. So also does themoving standard deviation of the bill ratewhen entered alone. But when all three varia­bles are entered together, only the standarddeviation retains at least marginal significance.Also the equation containing only the movingstandard deviation of the bill rate explainsmore of the variation in the liquidity premiumthan does the equation containing the othertwo variables. Mainly on statistical grounds,we chose this equation ((3) in Table A.1) forseparating out the liquidity premium from the

Table A.1.Estimation of liquidity Premium in the Two-Quarter Ahead

Three-Month Forward Rate (percentage points)19631V - 1979111

Constant i-i SO R2D.W. S.E.

(I) .540 .000 1.28* 117(.134)**

(2) -1.30 3.54 -.443 .148 1.67** 1.08(.565)** (I.03)** (.153)**

(3) -1.56 .162 .175 1.53** 1.07(.565)** (.0424)**

(4) -1.49 -.0909 .195 .169 1.62** 1.07(.578)** (.120) (.0611)**

(5) -1.49 -.0746 .158 .166 1.62** 1.07(.582)** (.126) (.0431)**

(6) -1.58 -.602 .545 .409 .165 1.51 ** 1.07(.560)*- (.645) (.675) (.273)

Note: Standard errors are in the parentheses.•• indicates a regression coefficient that is significantly differentfrom zero at the 1- percent level on the basis of a single-tailed test; and' indicates significance at the 5-percentlevel. The same symbols on the Durbin-Watson statistic (OW.) indicate the absence of significant serial correla­tion in the residuals at the 5- and 1-percent levels, respectively.

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forward rate to arrive at an estimate of themarket's anticipated 3-month bill rate.Because this equation has a minimum standarderror, the resulting estimate of the market'sforecast has a minimum forecast error.However, equations incorporating the othervariables did about as well; and our results arenot particularly sensitive to this choice.

The procedure used to estimate the liquiditypremium assumes that the market's trueexpectational error, v, is uncorrelated with thevariables used to model the liquidity premium.To the extent that this condition is not met,the estimated liquidity premium captures aportion of the market's true expectationalerror. This portion of the true expectationalerror would then be removed from the esti­mate of the market's forecast when the esti­mated liquidity premium is subtracted fromthe forward rate. As a simple example of theproblem, if the true expectational error ispositively biased, the procedure would over­estimate the liquidity premium and correspon­dingly underestimate the size of the market'sforecast errors.

We investigated the seriousness of thisproblem by regressing the difference betweenthe forecast of the Goldsmith-Nagan panel andthe realized interest rate on the variables usedin Table A.l to explain the liquidity premium.The estimated coefficients from these regres­sions carried the same signs as those in TableA.l and were frequently statistically signifi­cant, indicating that these variables are capableof proxying for a portion of the analysts'forecast errors. However, the size of thesecoefficients was generally considerably smallerthan those in Table A.I., as would be true ifthese variables were also related to themarket's liquidity premium.

To minimize inclusion of expectationalerrors, we estimated the liquidity-premiummodels on all available data going back to1963, rather than just on the forecast period of1971-1 thru 1979-111. This reduced contamina­tion of the estimated liquidity premium with

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expectational error, because the additionaldata allowed more of an ex ante estimation. Apurely ex post fit of the forward rate's forecast­ing error against the liquidity-premium varia­bles would have generated a larger bias.

Even so, our estimate of the liquidity pre­mium appears to have picked up some of thetrue expectational error in the market'sforecast. This was tested for by removing fromboth the analysts' forecast and the market's for­ward rate the portion of the respectiveforecast error associated with the moving stan­dard deviation of the bill rate (and a constantterm) during 1971-1 thru 1979-IIl. Theseadjusted forecasts were obtained by regressingeach forecast error on the moving standarddeviation of the bill rate, and subtracting thevalues predicted by these regression equationsfrom each respective forecast. The procedureremoves a similar amount of true expecta­tional error from both forecasts, but also anestimated liquidity premium from themarket's forward rate. The RMSE's of theseadjusted forecasts are 1.01 and 1.18 percentagepoints for the analysts' and the forward rate,respectively. The difference of 17 basis pointsexceeds the 14- basis-point difference betweenthe RMSE's of the analysts' forecast and ourestimate of the market's forecast, suggestingthat our estimate of the latter tends to under­estimate the size of its forecasting error.

The procedure used for estimating theliquidity premium has thus tended to under­state the difference in true expectational errorsbetween the market's forecast and the twoalternative forecasts. A possible bias in theopposite direction could result from the omis­sion of an important variable for explaining theliquidity premium. However, we have alreadyconsidered all such possibilities. On balance, itis more likely that our estimate of the market'sliquidity premium has understated, rather thanoverstated, the true difference between theforecast errors of the market and those of alter­native forecasts.

FOOTNOTES

Li t = .118 + .239 Li t_1 - .241 Li t_2(.055) (1.19) (.120)

market prices. Any systematic pattern in expectedshort-period returns would quickly be eliminated asinvestors bid prices up or down in attempts to profitfrom them. In contrast, the returns on 3-month Treas­ury bills held to maturity cannot be affected by thiskind of speculation, because the price at the end. ofthe 3 months is fixed contractually. Therefore, even afully anticipated time pattern in 3-month Treasury billyields is not likely to be arbitraged away in an effi­cient market. However, an efficient market would takesuch a pattern into account in its bill-rate forecasts.

5. The data sample was extended back to 1951-IVusing Salomon Brothers, An Analytical Record ofYields and Yield Spreads. The best autoregressiveequation for explaining quarterly changes in the 3­month Treasury bill rate during 1951-IV thru 1969-111was found to be:

Standard errors are indicated in the parenthesis. Atwo-step ahead forecast from this autoregressivemodel of quarterly changes was found to be con­siderably more accurate than a one-step aheadforecast from past changes over six-month periods.Hence, the two-step ahead forecast (with coefficientsreestimated at each point in time) was used for Ff~2'

6. The author wishes to thank Mr. Peter Nagan of TheGoldsmith-Nagan Bond and Money Market Letter forpermitting use of this data.

7. A large body of literature exists on the term struc­ture of interest rates. Useful surveys are contained inDodds and Ford (1974), Malkiel (1966), and VanHorne (1966).

An alternative measure of the market's forecast canbe obtained from yields on Treasury-bill futures con­tracts. We did not examine the accuracy of this typeof forecast because a futures market in 3-monthTreasury bills has existed only since January 1976,reducing the number of observations by more thanhalf. For a comparison of yields on futures contractsand implied forward rates, see Lang and Rasche(1978) and Poole (1978).

8. Prell (1973) compared the accuracy of forecastsby the Goldsmith-Nagan panel of analysts with thoseof no change and an autoregressive model for 1970through 1973. In that period, the RM8E of theGoldsmith-Nagan panel for a two-quarter ahead pre­diction of the 3-month bill rate was smaller than forboth of these forecasts. However, for long-term bondyields, the panel's forecasts were no more accuratethan a forecast of no change. Prell (1973) did notcompare the accuracy of the analysts' forecast withthose of the market implied by the forward rate, aswould be necessary for a test of bill-market effi-

- .126 Li t_3 - .338 Li t_6(.124) (.122)

DW.=1.93S.E. = .449f'l2=.193

1. Tests of market efficiency were first applied to thestock market. Surveys of this literature are containedin Cootner (1964), Fama (1970), and Lorie andHamilton (1973, Ch. 4).

2. The weak form of the efficient-market hypothesishas been tested in several ways in the bill market.One approach is to determine whether the forwardrate applicable to any particular point in time follows arandom walk. If the market's adjustment to new infor­mation is virtually instantaneous, as in an efficientmarket, successive changes in the forward rateapplicable to any particular period should be random.This approach is followed in Shiller (1973) and Roll(1970).

A second approach to testing the weak form is todetermine whether the market reacts appropriately toany autocorrelation in the bill rate. While changes inthe forward rate applicable to any particular timeperiod should be random, it does not follow that forweak-form efficiency the spot rate should also followa random walk. Indeed, in an efficient market the for­ward rate should extrapolate any systematic autocor­relation that tends to occur in the spot rate. Evidenceof weak-form efficiency in market forecasts of up to afew months ahead is contained in Hamburger andPlatt (1975), Fama (1975b), and Fildes and Fitzgerald(1980).

Fama (1975a, 1977) takes a rather differentapproach by focusing on the relationship between thenominal Treasury-bill rate and the subsequentlyobserved inflation rate. For weak-form efficiency, thenominal interest rate would summarize all the infor­mation about future inflation rates contained in thetime series of past inflation rates, so long as theexpected real return is constant. Fama argues thatthe Treasury-bill market is efficient in this sense, andalso that expected real returns on Treasury bills areapproximately constant. Carlson (1977), Joines(1977), and Nelson and Schwer! (1977) present con­trary evidence on the constancy of the expected realrate, but their evidence with respect to weak-formefficiency is not conclusive.

3. Hamburger and Platt (1975) conduct a test of thesemistrong form of efficiency by investigatingwhether variables other than current and past levelsof interest rates are better than forward rates in pre­dicting future bill rates. They considered such poten­tial predictors as the current and past values of per­sonal income and three alternative monetary aggre­gates, but found that the root-mean-squared error forsuch forecasts was no smaller than from a forecastusing the forward rate, consistent with the semistrongform of efficiency.

4. For the market to be considered efficient, theTreasury-bill rate does not necessarily have to followa random walk, in which the change in the return fromone period to the next is completely random. A ran­dom walk would be consistent with weak-form effi­ciency in stock and bond markets because short­period returns on these securities are dominated bycapital gains or losses resulting from changes in

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