c<iiations hold for и = 2 at sonic dale /, then the oilier equations also old lorn = 2 and the same date Л Note however lhat (10.2.9)-(10.2.11) at с not generally equivalent for particular n and I.

lematives lo Ihe Pure Expectations Hypothesis Tie expectations hypothesis (EH) is more general than the PEH in that it allows the expected returns on bonds of different maturities to differ by constants, which can depend on maturity but not on time. The differences between expected returns on bonds of different maturities are sometimes called term premia. The PEH says lhat lerm premia are zero, while die EH says that they arc constant through lime.21 Like the PEH, the EH can be formulated for one-period simple returns, for n-pcriod simple returns, or for log returns. If bond returns are lognormal and homoskedastic, as in Singleton (1990), then these formulations are consistent with one another because the Jensens Inequality effects arc constant over time. Recent empirical research typically concentrates on the log form of the EH.

Early discussions of the term structure tended to ignore the possibility that term premia might vary over time, concentrating instead on their sign. Hicks (1946) and Lutz (1940) argued that lenders prefer short maturities while borrowers prefer long maturities, so that long bonds should have higher average returns than short bonds. Modigliani and Stitch (1966) argued that different lenders and borrowers might have different preferred habitats, so thai term premia might be negative as well as positive. ЛИ these authors disputed ihe PEH but did not explicitly question the EH. More recent work has used intertemporal asset pricing theory to derive both the average sign and the time-variation of term premia; we discuss this work in Chapter 11.

10.2.2 Yield Spreads and Interest Rate Forecasts

We now consider empirical evidence on the expectations hypothesis (ELI). Since the EH allows constant differences in the expected returns on shorl-and long-term bonds, il docs not restrict constant terms so for convenience wc drop constants from all equations in ibis section.

So far wc have stated the implications of the expectations hypothesis for the levels of nominal interest rales. In post-World War II US data, nominal interest rales seem to follow a highly persistent process with a root very close to unity, so much empirical work uses yield spreads instead of yield It-Vyls.2*

Tim usage is tIn- most nnuiiiiin one in ihr literature, гаша (1УЯ4), Kama (Itl.HI), and Kanla and Bliss (1987), however, use term premia to refer to realized, rather than expet ted, cxeyss returns on long-term honds.

i-.Чее Chapters 2 and 7 for a discussion of unit roots. Ihe persistence of ihe short-rale

Recall thai the yield spread between the /i-pcriod yield and the one-period yield is s , s= у , - yu. Equation (10.1.6) implies dial

[(Tl.it/ - Vl/) + ( I I ,.n, - Tl./i/)

]jT [( - )Ду.,+/ 4- (r , i-u+. - vi./n)]

l=l

. (10.2.13)

The second equality in equation (10.2.13) replaces multiperiod interest rate changes by sums of single-period interest rate changes. The equation says that the yield spread equals a weighted average of expected future interest rate changes, plus an unweighted average of expected future excess returns on long bonds. If changes in interest rates are stationary (lhat is, if interest rates themselves have one unit root but not two), and if excess returns are stationary (as would be implied by any model in which risk aversion and bonds risk characteristics are stationary), then the yield spread is also stationary. This means that yields of different maturities arc coinlegraledP

The expectations hypothesis says that the second lerm on die right-hand side of (10.2.13) is constant. This has important implications for the relation between the yield spread and future interest-rates. Il means that the yield spread is (up to a constant) the optimal forecaster of the change in the long-bond yield over ihe life of the short bond, and the optimal forecaster of changes in short rates over the life of the long bond. Recalling thai wc have dropped all constant terms, the relations are

1-Л} -1..+ 1 - y,u\.

Sin - Ei

£(1 -i/n)Ayu

(10.2.14)

(10.2.15)

Equation (10.2.14) can be obtained by substituting the definition of r + , (10.1.5), into (10.2.9) and rearranging. It shows that when the yield spread is high, die long rale is expected lo rise. This is because a high yield spread gives the long bond a yield advantage which must be offset by an anticipated capilal loss. Such a capital loss can only come about through an increase in the long-bond yield. Equation (10.2.15) follows directly from (10.2.13) with constant expected excess returns. It shows that when the yield spread is

process is discussed tin titer in Chapter 11.

-MSce Cantpheli and Shiller (11)87) for a discussion ol cointrgraiioii in the teim stiucliire ol interest rales.

1 1/. I l.utl-lllCOIIIC .VlWWW

Long bund maturities arc measured in months. The lirsl row reports the estimated regression coefficient /> 11inn (10.2. Hi), with an asvinpiolic standard error (in parentheses) raliulatcd to allow lor heteroskedasticity in the manner described in the Appendix. The second row reports the lite estimated regression coefficient y (mm (I0.2.IH). with an asymptotic standard error calculated in ihe same manner, allowing also lor residua! autocorrelation. The expectations hypothesis of the (elm stun lure implies that both j\ and y should equal one. The underlying dala are monthly /eio-соирои bond yields over the period 1952:1 to 199L2, from McCulloch and Kwon (1993).

high, short rales .ire expected lo rise so lira! ihe average short rale over ihe life of ihe long bond equals die initial long-bond yield. Near-term increases in short rates are given greater weight than further-off increases, because they alien the level of short rales during a greater pan of the life of the long bond.

Yield Spreads and Future long Rates

Kquation (10.2.1 I), which says that high yield spreads should forecast increases in long rales, fares poorly in die dala. Macaulay (1038) first noted die fact thai high yield spreads actually lend to precede decreases in long rates. I Ie wrote: Ihe yields of bonds of the highest grade should fall during a period in which short-term rates ate higher than the yields of the bonds and me during a period in which short-term rates are lower. Now experience is more nearly the opposite (Macaulay 1038, p. 33]).

Table 10.3 repoi is estimaies of the coefficient И and its standard error in the regression

.v I..H-.Y ., = a + fi (I0.2.1(i)

Ihe maturity и v.uies from 3 mouths to 120 months (10years)According -tot mainlines above one vcai ihe lahle uses the approximation y t. i V .i+- Note

HI.2. Interpreting the Ferm Structure of Interest Rates

to the expectations hypothesis, we should find Bn = 1. In fact aflthe estimates in Table 10.3 are negative; all are significandy less than one, and some are significantly less than zero. When the long-short yield spread is high the long yield tends to fall, amplifying the yield differential between long and short bonds, rather than rising to offset the yield differential as required by the expectations hypothesis.

The regression equation (10.2.16) contains the same information as a regression of the excess one-period return on an n-period bond onto the yield spread s, . Equation (10.2.5) relating excess returns to yields implies dial the excess-return regression would have a coefficient of (1 - B ). Thus the negative estimates of ftn in Table 10.3 correspond to a strong positive relationship between yield spreads and excess returns on long bonds. This is similar to the positive relationship between dividend yields and stock returns discussed in Chapter 7.25

One difficulty with the regression (10.2.16) is that it is particularly sensitive lo measurement error in the long-term interest rate (see Stambaugh [ 1988]). Since the long rate appears both in the regressor with a positive sign and in the dependent variable with a negative sign, measurement error would tend to produce the negative signs found in Table 10.3. Campbell and Shiller (1991) point out that this can be handled by using instrumental variables regression where the instruments are correlated with the yield spread but nol with the bond yield measurement error. They try a variety of instruments and find that the negative regression coefficients are quite robust.

Yield Spreads and Future Short Rates

There is much more truth in proposition (10.2.15), that high yield spreads should forecast long-term increases in short rates. This can be tested either directly or indirectly. The direct approach is to form the ex post value of the short-rate changes that appear on the right-hand side of (10.2.15) and to regress this on the yield spread. We define

/i-i

Snl = £(i -;/п)Ду,.,+1, (Ю.2. 7)

that this is nol the same as approximating p -\.,+ i by /V/+I- The numbers given differ slightly from those in Campbell (1995) because that paper uses the sample period 1951:1 to 1990:2, erroneously reported as 1952:1 to 1991:2.

- Campbell and Ammcr( 1993), Fama and French (1989), and Keim and Stambaugh (19fifi) show that yield spreads help to forecast excess returns on bonds as well as on other long-l assets. Campbell and Shiller (1991) and Shiller, Campbell, and Schoenholtz (I98H) show that yield spreads lend ю forecast declines in long-bond yields.

ajiid run the regression

(1(1.2.18)

The expectations hypothesis implies lhat y - 1 (or all n.21

Table 10.3 reports estimated y coefficients with standard errors, correcting for heteroskedasticity and overlap in the equation errors in the manner discussed in the Appendix. The estimated coefficients have a U shape: For small n they arc smaller than one but significantly positive; up to a year or so they decline with n, becoming insignificantly different from zero; beyond one year the coefficients increase and at ten years the coefficient is even significantly greater than one. Thus Table 10.3 shows that yield spreads have forecasting power for short-rate movements over a horizon of two or three months, and again over horizons of several years. Around one-year, however, yield-spread variation seems almost unrelated to subsequent movements in short rales.

The regression equation (10.2.18) contains the same information as a regression of (I/и) times the excess м-pcriod return on an н-pcriod bond onto the yield spread s, . The relation between excess returns ami yields implies that the excess-return regression would have a coefficient of (1 - y ). Table 10.3 implies that yield spreads forecast excess returns out to horizons of several years, but ihe forecasting power diminishes towards len years.

There are several econometric difficulties with the direct approach just described. First, one loses n periods of data at die end of ihe sample period. This can be quite serious: For example, the ten-year regression in Table 10.3 ends in 1981, whereas the three-month regression ends in 1991. This makes a substantial difference to the results, as discussed by Campbell and Shiller (1991). Second, the error term e, is a moving average of order (n - 1), so standard errors must be corrected in the manner described in the Appendix. This can lead lo finite-sample problems when (и - 1) is not small relative to the sample size. Third, the rcgrcssor is serially correlated ai-(l correlated with lags of the dependent variable, and this too can cause finite-sample problems (sec Mankiw and Shapiro [1986], Richardson and Stick [1990], and Stambaugh 11980]).

J Although these econometric problems are important, they do not seem lo-account for the U-shaped pattern of coefficients. Campbell and Shiller (1091) find similar results using a vector autoregressive (VAR) methodology likj? that described in Section 7.2.3 of Chapter 7. They find that (he long-tetjm yield spread is highly correlated with an unrestricted VAR forecast of Itiliiire short-rale movements, while the intermediate-term yield spread is milch more weakly correlated with the VAR forecast.

Kama (1УЙ4) and Shillcr, Campbell, and Srhocnhuliz (1<Ж1) шс this approach at the short end of the term structure, while Kama ami Bliss (1987) extend it to (he long end. Campbell aiulShiller (1991) provide a comprehensive review.

To interpret fable 10.3, it is helpful lo return to equation ( 10.2.13) and rewrite il as

.. , = .vy , 4-.W (10.2.19)

where

*V i =

КЛ.С1 = (-F., £(н-/)Ду,.(

(r,i+i-i.n i - Yi.i+i

.i=l

In general the yield spread is the sum of two components, one that forecasts interest rate changes (vy ,) and one lhat forecasts excess returns on long bonds (.vr, ). This means that the regression coefficient y in equation (10.2.18) is

Covf.v* , S,

VarU ]

Var[yy ,] 4- Cov[.vy ), .vr, ] Var.vy,(/] 4- Vaij.vr ,] + 2Cov.vy, , .vr

(10.2.20)

For any given variance of excess-return forecasts w as the variance of interest rale forecasts sy , goes lo zero die coefficient y goes to zero, but as the variance of sy , increases die coefficient y goes to one. The U-shaped pattern of regression coefficients in fable 10.3 may be explained by reduced forecastabilily of interest rale movements at horizons around one year. There may be some short-run forecastabilily arising from Federal Reserve operating procedures, and some long-run forecastabilily arising from business-cycle effects on interest rates, but al a one-year horizon the Federal Reserve may smooth interest rates so that ihe variability of sy , is small. Ual-duzzi, Beriola, and Forest (1993), Rudebusch (199Г)). and Roberds, Runklc, and VVhiteman (1990) argue for this interpretation of the evidence. Consistent with this explanation, Mankiw and Miron (1980) show that the predictions of the expectations hypothesis lit the data belter in periods when interest rate movements have been highly forecaslable, such as the period immediately before the founding of the Federal Reserve System.

10.3 Conclusion

The results in Table 10.3 imply that naive investors, who judge bonds by their yields to maturity and buy long bonds when their yields are relatively high, have tended to earn superior returns in (he postwar period in the United