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The FED model and expected asset returns

AbstractThe earnings yield and long-term bond yields have been widely used to predictasset returns. In this paper, I focus on the predictive role of the stock-bond"yield gap" - the difference between the earnings yield and the 10 year Treasurybond yield also know as the FED model, and which can be interpreted as along term yield spread of stocks relative to bonds. Conditional on otherforecasting variables, the yield gap forecasts positive excess stock returns,both at short and long forecasting horizons, although the forecasting power isgreater at the near horizons. On the other hand, the yield gap forecastsnegative excess returns for bonds, at both short and long horizons. A VARvariance decomposition for stock market returns, shows that shocks in the yield

gap are highly positively correlated with innovations in both future discount-rateand cash flow news, confirming that the spread conveys information aboutfuture earnings and returns. An investment strategy based on the forecastingability of the Yield gap produces higher Sharpe ratios than passive strategies inboth the market index and longterm bond. In the context of an equilibriummultifactor ICAPM, the yield gap has some explanatory power over the crosssection of stock returns.Keywords: Asset pricing; FED model; Earnings yield; Predictability of returns;Stock and bond returns;JEL classification: G11;G12; G14; E44

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"The Feds model arrives at its conclusions by comparing the yield on the 10-year Treasury

note to the price-to-earnings ratio of the S&P 500 based on expected operating earnings in

the coming 12 months. To put stock and bonds on the same footing, the model uses the

"earnings yield" on stocks, which is the inverse of the P/E ratio. So while the yield on the

10-year Treasury is now 5.60%, the earnings yield on the S&P 500, based on a P/E ratio of

21, is 4.75%. In essence, the Feds model asks, why would anyone buy stocks with a 4.75%

earnings return, when they could get a bond with a 5.60% yield? "

Barrons online

The earnings yield and smoothed earnings yield have been widely used as predictors of

future stock market excess returns (Fama and French (1988), Campbell and Shiller (1988a,

2001)). In addition, yield spreads related to Treasury and corporate bond yields have also

been used for some time, to forecast asset returns (Fama and French (1989)).

In this paper, I focus instead on the yield gap, which corresponds to the difference between

the earnings yield on a stock market index, and the long term yield on Treasury bonds, which

is also known as the FED model. Despite the fact that this variable is widely referred in the

financial press, it is used by practioneers to forecast returns, and it is even referred in FED

publications, and has been used in official testimonies by FEDs chairman Alan Greenspan in

the recent years (to argue for the overvaluation of the stock market), little attention has be

devoted to it in academics, with the exception of Asness (2003).

This variable might be viewed as a simple measure of the yield spread of stocks versus

bonds, or a relative long-term rate of return of stocks against bonds. In fact, the earnings

yield, being the inverse of the price-earnings ratio can be interpreted as a equivalent long term

yield on stocks, analogous to the yield on long term bonds: If the earnings level were

constant, the price-earnings ratio would represent the number of years (earnings yield is

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calculated based on annual earnings) needed for a very long term investor - who receives all

the earnings in the form of dividends or other forms of cash flow distributions - to recover its

investment (which corresponds to the price paid for the stock or the index level), by

accumulating annual earnings. Thus, the greater the price-earnings ratio is, the worst off the

investor is, since he will recover its investment (in terms of earnings), in a longer period of

time, and this corresponds to a lower earnings yield - the average long term yield for investing

in stocks. Naturally, this constitutes a simplification, since nominal and real earnings growth

over time, and the earnings growth rate is time-varying, and the investor receives only a

fraction of earnings as pay-out distributions. Nevertheless, it represents a simple

straightforward measure of the long term return on stocks.

By using the definition of returns, I derive a dynamic accounting decomposition for the yield

gap, as a function of future stock and bond returns, future dividend to earnings payout ratios

and future earnings growth, which provides the rationale for the predictive role of the yield gap

over asset returns.

The reminder of this paper is organized as follows. Section I presents the theoretical

motivation. Section II describes the data and variables. Section III presents the results for the

long-horizon regressions. Section IV produces an alternative short-term VAR estimation.

Section V analyzes non-linearities in the predictive role of the yield gap. Section VI evaluates

the economic significance associated with the predictive role of yield gap, and Section VII

explores the explanatory power for the cross section of average stock returns. Finally, Section

VIII concludes.

I. Theoretical framework

The Yield gap (YG) by representing the difference between the earnings yield and the long

term bond yield, it conveys information about both future expected stock and bond returns,

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future dividends and future earnings. In the Appendix, I derive the following dynamic

accounting identity for YG,

YGt ln1 EtPt

ynt k k

11

1 Etj0 jrt1j 1 det1j et1j

1b

1bnEt

j0

n1bjrb,nj,t1j 1

where k,k,, and b are parameters of linearization defined in the Appendix, Et and Pt in

the first equality, represent the earnings and price level associated with the stock market

index, respectively, and ynt is the log yield at time t of a bond with maturity n. Equation 1

says that high values of the Yield gap (earnings yield is high relative to the bond yield), are

associated with a expected combination of higher future stock returns rt1j, lower dividend to

earnings payout ratios det1j, lower growth rate on future equity earnings et1j and lower

future bond returns rb,nj,t1j. Thus, YG forecasts higher expected stock market returns and

lower expected bond returns, although with different weights as shown by equation 1. This

equation will be used to interpret the predictive regressions in the next sections, and does not

assume any behavior for asset prices, rather being based on the definition of returns and a

terminal condition that the log earnings to price ratio does not growth slower or faster than the

linearization parameter .

II. Variables and data

A. Data

Monthly data on prices, earnings and dividends associated with the Standard & Poors

(S&P) Composite Index is obtained from the website of Professor Robert Shiller. p is the log of

the S&P Composite Index, e is the log of the annual moving average of earnings, and d is the

log annual dividend. Return data on both the value-weighted Rvw and equally-weighted Rew

market index, and the 10 year Treasury bond Rb is obtained from CRSP. Macroeconomic

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and interest rate data, including the Federal funds rate, 10 year and 1 year Treasury bond

yields and the 3 month Treasury bill rate, are all obtained from the FRED II database,

available from the website of the St. Louis FED. The 1 month Treasury bill rate Rf,t1 is

obtained from the website of Prof. Kenneth French.

B. Construction of variables and summary statistics

The khorizon continuously compounded excess return is calculated as rt1,tk rt1 . . .rtk,

where rtj lnRtj lnRf,tj is the 1 month log excess return, between dates t j 1 and

t j, Rtj is the simple (not log) gross market return, and Rf,tj is the gross risk-free rate (1

month treasury bill) at the beginning of period t j. The "Yield gap" is calculated as shown in

the last section, YGt EYt yt, with EYt ln1 EtP t

, representing the earnings yield, and

yt ln1 Yt is the log 10 year Treasury bond yield. The other forecasting state variables

known in period t, used to predict future excess returns, are the FED funds premium

(FFPREM), the term-structure spread (TERM), and the log market dividend yield (DY).

FFPREM is calculated as the difference between the FED funds rate and the 3-month

Treasury bill rate. TERM is the difference between the 10-year and 1-year Treasury bond

yields. The log dividend yield is calculated as DYt dt p t.

Table I reports descriptive statistics for excess returns, the forecasting state variables, and

the two components of YG, EY and y. By analyzing the correlation coefficients, one can see

that the two proxies for stock market excess returns are highly contemporaneously correlated

among themselves, but not significantly correlated with bond returns. The contemporaneous

correlation between YG and excess returns is negligible, and YG is positively correlated with

the earnings yield and negatively correlated with the bond yield, at similar magnitudes. The

first order autocorrelation coefficients show that the yield gap being a difference of two highly

persistent variables, has a slightly lower autocorrelation coefficient than both EY and y, but

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nevertheless it is a very persistent variable. Nevertheless, all three variables - YG, EY and y -

are less persistent than the log dividend yield.

III. Long-horizon regressions

In this section I use Fama and French (1989) multivariate long-horizon regressions, to

access the explanatory power of the Yield gap spread (YG) over future excess returns. The

typical specification used is,

rt1,tk ak bk

xt ut1,tk 2

where rt1,tk is the continuously compounded market excess return over kperiods, and xt is

a column vector of forecasting state variables known in time t. I use forecasting horizons of 1,

3, 12, 24, 36 and 48 months ahead. The compounded return rt1,tk is multiplied by 12/k,

where k is the forecasting horizon, in order for the slope coefficients bk to measure the

annualized effect of the state variable on excess returns.

For each regression, I conduct statistical inference based on both Newey-West and Hansen

and Hodrick (1980) asymptotic t-statistics, and also on a Bootstrap experiment. The

Newey-West standard errors are calculated using 5 lags. Whenever the variance-covariance

matrix associated with Hansen and Hodrick (1980) standard errors is not positive definite, I

substitute by the Newey-West standard errors calculated under the number of lags employed

by the Hansen-Hodrick estimator. The Bootstrap consists of 10,000 simulations for each

equation. Following Goyal and Santa-Clara (2003), I bootstrap the t-statistics instead of the

regression coefficients, applying the procedure for the Newey-West t-statistics. I bootstrap the

original regression residuals 10.000 times, and in each simulation, I draw the endogenous

variable (compounded excess returns) imposing the null of no predictability, i.e. the slope

coefficients in the regression are constrained to zero, and then run the regression and obtain

the associated Newey-West t-statistics. By this process, a empirical distribution of

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Newey-West t-statistics is generated - as opposed to an asymptotic theoretical distribution -

which is then compared with the Newey-West t-statistic obtained from the data, calculating the

proportion of distribution t-statistics larger (in absolute value) than the original t-statistic, to

finally obtain the corresponding p-value for the null of no predictability of returns.

A. Predicting stock market returns

I conduct long horizon regressions with Yield gap as the sole forecasting variable (results

displayed in Table II, Panel A). The coefficient estimates with YG are positive at all horizons,

and the t-statistics indicate statistical significance, although for long horizons only

Newey-West t-statistics and bootstrapped p-values show significance, whereas at the 48

month horizon there is no significance. Thus, it seems that on a preliminary analysis, the yield

gap forecasts positive equity excess returns. On the other hand, the forecasting power of the

yield gap on excess returns declines gradually with the forecasting horizon: YG has a

coefficient estimate of 2.965 at the 1 month horizon comparing to 0.363 at the 4 year horizon.

The adjusted R2 achieve the maximum values at the 12 month horizon regression, declining

thereafter. This estimates show that forecasting power of the yield gap is greater, and with

higher statistical significance, on the near horizons, being less relevant for forecasting more

distant ahead returns.

In Panels B and C, I add three forecasting state variables usually used in the predictability

of returns literature: The term structure spread (TERM), the spread between the Fed funds

rate and the treasury bill rate (FFPREM), which is a measure for both monetary policy actions

and short term interest rates, and the log market dividend yield (DY). Panel C includes the 3

variables, whereas Panel B includes only FFPREM and TERM, along with YG. It is important

to control for these variables, since YG is correlated with them, in particular both TERM and

YG depend on the 10 year bond yield, and both the log dividend yield and log earnings yield

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are correlated. Hence, one needs to check if the predictive power of YG is maintained after

the inclusion of those variables.

This will be my benchmark regression throughout the paper,

rt1,tk ak bk1FFPREMt bk2TERMt bk3DYt bk4YGt ut1,tk 3

The results in Panel B show that when one controls for both FFPREM and TERM, the

magnitude of the yield gap estimates (and the associated t-statistics), actually increases

relative to the single variable forecasting regression, especially on the short-term horizons

(until 1 year): At k1, the YG coefficient is 3.889 and at k48, the corresponding estimate is

0.778. In addition the YG estimates are strongly significant (1% level) for all horizons, as

indicated by the Newey-West t-statistics and p-values from the bootstrap experiment. The

pattern of coefficients magnitudes is the same as in Panel A, with the estimates declining in a

monotonic way with the forecasting horizon.

In Panel C, by adding the log dividend yield (DY), the YG coefficient estimates decline in

magnitude relative to Panels A and B. Nevertheless YG is significant at the 5% level for

horizons until 12 months. In fact, the dividend yield being a very persistent variable, has a

forecasting power over returns which increases with horizon, which is in part due to the mean

reversion of stock prices (the denominator) in the medium and long term. Since in addition DY

is correlated with the earnings yield (EY) and hence YG, the predictive power of the variables

is overlapped, and the higher persistence of DY makes it a better forecaster for long-horizon

returns, than YG.

In Table III, I replicate the long-horizon regressions for the equally-weighted market return

rew as the variable to be forecasted. The results for the univariate case in Panel A, show that

compared to the corresponding regression for the value-weighted market index rvw, the

forecasting power of YG is greater at all horizons: At k1 the estimate is 4.763, and at k48

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we have an estimate of 1.647. The t-statistics are also higher than the corresponding ones for

rvw, showing statistical significance at the 1% level, at all forecasting horizons. The adjusted

R2s are also higher than the corresponding values in the equation for rvw, at all horizons: At

k1, the R2 is 0.022 (0.013 for rvw), and at k48, we get a value of 0.163 compared with only

0.013 for rvw. Similarly, to the case of value-weighted returns, the forecasting power of YG is

stronger on the short horizon, with the coefficient estimates declining monotonically with

horizon. The results in Panels B and C, show that by adding the control variables, YG remains

strongly significant (1% level) at all forecasting horizons, whereas the log dividend to price

ratio (DY) has no predictive power over the return on the equally-weighted index.

These results showing that the yield gap has greater forecasting power for the

equally-weighted relative to the value-weighted market excess returns, combined with the fact

that the equally weighted index is more tilted towards small caps relative to big caps, suggests

that YG has greater forecasting power for small caps excess returns, relative to large

capitalization stocks.

B. Predicting bond returns

Equation 1 above suggests that current values of Yield gap are negatively correlated with

expected future bond returns. I investigate this hypothesis, by running the long-horizon

regressions with the 10 year Treasury bond excess return as the variable to be forecasted,

which results are presented in table IV. The results in Panel A, confirm that the yield gap is

negatively correlated with future bond excess returns, at all forecasting horizons. The

coefficient estimates exhibit a hump-shaped pattern, with the magnitudes peaking at k12,

and then declining gradually. The forecasting power as measured by the coefficient

magnitudes associated with YG, is lower when compared with the regressions for the

value-weighted market return rvw for horizons until 1 year, being nevertheless higher for

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longer horizons. In terms of statistical significance the coefficients are very significant (at the

1% significance level), even for long horizons, which didnt happen in the long-horizon

regressions for rvw. The adjusted R2s increase with the horizon and are higher than the

corresponding values in the regressions for the value-weighted market return. Panels B and

C, present the results for the regressions including the control forecasting variables. In Panel

C, DY is significant for the longer horizons, which confirms Fama and French (1988) that the

dividend yield helps to predict long-horizon returns on both stocks and bonds. On the other

hand, the effect of the term structure spread (TERM) on future bond returns is not significant

in both Panels B and C, a fact that should be related with the presence of YG, which overlaps

the forecasting role of TERM, since the two variables are correlated. After accounting for the

control variables, the YG coefficients maintain the statistically significance, and even increase

in magnitudes at all horizons, relative to the estimates in Panel A. Overall these results

suggest that conditional on other forecasting variables, YG is negatively correlated with future

bond excess returns.

Common to the predictive regressions of both stock market and bond returns is that the

forecasting power of Yield Gap is greater in the near horizons (horizons until 1 year), than for

long horizons. In order to analyze this issue, I estimate long horizon regressions for the

components of the Yield Gap - the earnings yield (EY) and the log bond yield (y) - which

results presented in Table V. In the case of the value-weighted market return (Panels A and

B) , the estimates associated with the Earnings yield, have a hump-shaped pattern, peaking at

k12, and declining thereafter, which leads to a similar pattern in the forecasting ability of the

Yield gap. In the case of the bond return (Panels C and D), the responsible for the lower

forecasting power of YG at long horizons, is the slightly lower coefficient estimates associated

with the log bond yield, at more distant horizons, compared with the short-term forecasts.

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IV. A short-term VAR analysis

A. A short-horizon VAR alternative to long-horizon regressions

Following the work of Campbell and Shiller (1988a,b), Campbell (1991) and Hodrick (1992),

an alternative to long-horizon regressions is to estimate a short-horizon VAR, and obtain

implied long-horizon coefficient estimates and implied statistics which are a non-linear

function of the VAR parameters. This approach is likely to have better finite sample properties

than the long-horizon regressions, given the more data used in the estimation. I estimate the

VAR-analogue to the benchmark long-horizon regression in specification 3.

Let Zt represents a vector of the variables contained in the VAR, with

Zt FFPREMt,TERMt,DYt,YGt,rm,t where rm,t is the 1 period log excess return for the market

index. All the variables in the VAR are demeaned. I estimate a first order VAR, nevertheless

irrespective of the number of lags in the VAR, one can always write it in companion form as a

first-order VAR,

Zt1 AZt t1 4

where A is the VAR coefficient matrix 5X5, and t1 is the vector of errors 5X1.

Following Hodrick (1992), an alternative to the slope coefficients in the long horizon

regression are the estimators implied by the VAR,

blk e1C1...Ckel

elC0el5

where the unconditional variance of Zt is given by

C0 j0 AjVAj

whith V E t1 t1 representing the variance-covariance of the VAR errors. Cj, the

jth-order autocovariance of Zt, is given by Cj AjC0. e1 and el are indicator vectors that

take a value of one in the cell corresponding to the position in the VAR, of the excess returns

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and any of the forecasting variables we are analyzing, respectively, e.g., e1 0,0,0,0,1.

The implied long horizon R2 from the VAR is given

R2k 1 e1

Wke1e1Vke1

6

where

Vk kC0 j1

k1k jCj Cj

represents the variance of the sum of k consecutive Zts, and e1 Wke1 is the innovation

variance (variance not explained by the VAR variables) of the sum of k consecutive returns,

with

Wk j1

kI A1I AjVI Aj I A1

The implied variance ratio statistic which compares with Lo and Mackinlay (1988) and

Poterba and Summers (1988) variance ratio statistics, is derived as

VRk e1Vke1

ke1C0e17

The variance ratio should be equal to 1 if returns were i.i.d., since in that case, with zero

covariances, the variance of the sum of kconsecutive returns is equal to k times the variance

of one return. If the ratio is above 1, this is a sign of positive autocorrelation in returns

(momentum), whereas a ratio below 1, is evidence of negative autocorrelation in realized

returns (mean reversion in returns).

In order to compute the asymptotic standard errors associated with the long-horizon

coefficients b lk, I use the non-linear GMM distribution theory, as in Hodrick (1992),

TT0~N0,

THTH0 ~N0,HT

T

HT

T

8

where H0 and HT represent b lk as a function of the population and sample

estimation values of the parameters from the VAR, respectively. includes the slope

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coefficients A and the variance-covariance of the VAR errors V.H

T

Tdenotes the gradient of

H evaluated at T

, which is estimated numerically. The first 25 moment orthogonality

conditions from the vector of moments, deliver the OLS estimates of A, and the remaining

ones correspond to the distinct elements of E t1 t1 V 0. I compute Newey-West

standard errors associated with this GMM system, thus allowing for serial correlation between

the different moments.

Table VI presents the results for the implied long-horizon coefficients, implied long horizon

R2 and variance-ratio statistics derived from the VAR, corresponding to the excess returns on

the value weighted market index (Panel A) and equally weighted market index (Panel B).

Comparing the results in panel As VAR concerning rvw, with the corresponding

long-horizon regressions in Table II, Panel C, the implied YG estimates are of lower

magnitude in the very short term horizons (1 and 3 months), but higher in the remaining

forecasting horizons, and both the t-statistics and bootstrapped p-values indicate higher

statistical significance for horizons beyond 12 months. The implied t-statistics associated with

YG are higher than the t-statistics for DY at all horizons, contrary to what happened in the

long-horizon regressions, where for horizons bigger than 24 months, DY was more significant

than YG. The implied slopes on lagged excess returns are positive at all horizons, indicating

the existence of momentum in stock prices. The implied R2 associated with the excess return

equation in the VAR, are also higher than the corresponding values in the long-horizon

regressions, at all horizons. The implied variance ratio statistics are higher than 1 for horizons

beyond 3 months, thereby indicating the existence of momentum in value-weighted market

excess returns, as implied by the VAR, and confirming the coefficient estimates associated

with lagged market returns.

The VAR estimation results for rew presented in Panel B, show that the implied long-horizon

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coefficients for YG are lower than the corresponding long-horizon coefficients, at the near

forecasting horizon (1 and 3 months), but for the remaining horizons, the implied VAR

estimates have higher magnitudes, similarly to the VAR for rvw. In terms of statistical

significance, the YG estimates are strongly significant as shown by both t-statistics and

p-values from Bootstrap simulation, whereas the estimates associated with DY are not

significant in what concerns the bootstrap experiment. Following the results obtained for the

long-horizon regressions, the implied estimates for YG confirm that this variable has bigger

forecasting power for equally-weighted compared to value-weighted market returns. The

implied R2 are also higher than the corresponding values for the long-horizon regressions, as

for the case of rvw. The implied variance ratio statistic achieves 1.828 at k48, indicating a

higher degree of momentum in rew as compared to rvw, which suggests that small caps have

higher momentum than big caps.

Overall, the implied estimates from the first-order VAR confirm and even strengthen the

forecasting power of YG over returns associated with the long-horizon regressions.

B. Variance decomposition: The impact of discount rate news and cash-flow news

Following the work of Campbell and Shiller (1988a,b) and Campbell (1991), I employ a

log-linear approximate decomposition for unexpected excess returns:

rt1 Etrt1 Et1 Et j0

jdt1j Et1 Et

j1

jrt1j NCF,t1 NDR,t1 9

where NCF,t1 and NDR,t1 represent news about future cash flows, and news about future

discount rates, respectively. This dynamic accounting identity which results from the definition

of market return, states that positive innovations in current returns are associated with

expectations of rising future dividends/cash flows and/or expectations of declining future stock

market returns. Thus, this identity makes clear the negative correlation between realized

returns and expected returns.

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The unobserved components of market returns, NCF,t1 and NDR,t1 can be derived in the

context of the VAR in equation 4, as functions of the VAR residuals,

NCF,t1 Et1 Et j

0

jdt1j e1 e1AI A1 t1 10

NDR,t1 Et1 Et j1

jrt1j e1AI A1t1 11

Using equation 9, the variance of unexpected excess returns can be decomposed as

varrt1 Etrt1 varNCF,t1 varNDR,t1 2covNDR,t1,NCF,t1 12

and we can compute each of the three components in proportion of varrt1 Etrt1, to

produce a variance decomposition for stock market returns.

The results for this variance decomposition analysis are presented in Table VII. The results

for the value-weighted index (Panel A), show that the variance of discount-rate news

represents 0.986 of the total market variance, whereas cash-flow news have a weight of

0.378, thus confirming previous evidence that discount rate news is the main driver of stock

market volatility (Campbell (1991), Campbell and Ammer (1993), Campbell and Vuolteenaho

(2004)). The two variances sum up to more than 1, since the covariance between the two

components has a negative contribution (-0.364) for the overall market variance.

By analyzing the correlations of shocks in the individual VAR state variables with both

discount rate and cash flow news, we can see that innovations on the market return are

strongly negatively correlated with discount-rate news and weakly positively correlated with

cash-flow news, thus confirming the predictions of the dynamic identity in equation 9.

Shocks in the log dividend yield are highly positively correlated with discount rate news and

almost uncorrelated with cash flow news, which is consistent with the following decomposition

for innovations in the log dividend to price ratio, derived in the Appendix,

Et1 Etdt1 pt1 1 Et1 Et

j1

jdt1j

1 Et1 Et

j1

jrt1j 13

Innovations in the Yield gap are highly positively correlated with discount rate news

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(correlation coefficient is 0.776), and on the other hand, they are also positively correlated

with cash flow news. The fact that YG is positively correlated with discount rate news arises

from the positive correlation of the earnings yield (similarly to the dividend yield) to future

returns, and reflects mean reversion in stock prices. The fact that YG is also positively

correlated with cash flow news might be attributable to a negative correlation between current

shocks in log bond yields and future cash flows.

The variance decomposition for the equally-weighted market return, presented in Panel B,

shows that both discount rate news and cash flow news account for more than 100% of the

total market variance, which is possible since the covariance between the two news

components represents -1.819 of the overall variance. Furthermore, the variance of discount

rate news is only marginally higher than the variance of cash flow news. Since the equally

weighted index assigns an equal weight for all stocks, this result is consistent with

Vuolteenaho (2002), who found that for individual stocks, the variance of cash flow news is

higher than the variance of individual expected return news. In what concerns the correlations

between shocks in the VAR state variables and the news components, we have that

innovations in the dividend yield (market return) are only weakly positively (negatively)

correlated with discount rate news. This confirms the results for the long-horizon regressions,

in that the dividend yield does not help in forecasting the equally-weighted market return, and

it is a consequence of less mean reversion for the equally weighted index. On the other hand,

shocks in market return are weakly positive correlated with cash-flow news, and thus, part of

the rise in current market returns are due to an improvement in future earnings. Finally, YG is

strongly positively correlated with discount rate news (correlation of 0.965) and also correlated

with future cash flows.

The results of this subsection seem to confirm that innovations in the Yield gap are

important to explain the two unobserved components of market returns - Cash flow and

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discount rate news.

V. Non-linear effect of the Yield gap in expected excess returns

One interesting issue to analyze, is to check whether there is an asymmetric relation in the

predictive role of the yield gap over future excess returns, namely if the magnitude of the

correlation between YG and future excess returns changes with the sign of Yield gap: Is the

forecasting power of future returns greater in periods where the yield gap is positive (earnings

yield higher than the long term bond yield), as opposed to periods where YG is negative (long

term bond yield higher than the earnings yield), or vice-versa?

To answer this question, I construct one indicator variable, DYG, that assumes the value 1

when YGt 0, and 0 otherwise. Then, I perform the following long-horizon regression, which

corresponds to an augmented version of specification 3:

rt1,tk ak bk1FFPREMt bk2TERMt bk3DYt bk4YGtDYGtbk5YGt1 DYGt ut1,tk 14

I estimate this regression for both the value-weighted market return and 10 year bond

return, and the results are presented in table VIII. In what concerns the value weighted market

excess returns (panel A), we can see that at short horizons (1 and 3 months), the effect of YG

on future excess returns is higher when YG is negative: At k1, the estimate associated with

YGt1 DYGt is 6.056 versus 1.232 for YGtDYGt and 3.131 for YG in the benchmark

regression 3, although only the p-values from the bootstrap simulation indicate statistical

significance at the 5% level. For horizons beyond 12 months, we get the reverse relation, i.e.,

the coefficients in YGtDYGt are higher than the ones on YGt1 DYGt, and also higher than

the corresponding coefficients associated with YG in the benchmark regression 3: At k12,

the estimate on YGtDYGt is 2.011, and at k48 the coefficient is 1.106, compared to 0.501 and

-1.181 for YGt1 DYGt. The t-statistics associated with YGtDYGt, indicate statistical

significance at the 5% level, for horizons between 12 and 48 months ahead. These results

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constitute evidence in favor of an asymmetric relation between the yield gap and future

market excess returns: At short horizons, the forecasting power is greater when the ratio

assumes negative values, while at longer horizons, the forecasting power is bigger when the

yield gap is positive. In addition, the negative estimates associated with YGt1 DYGt for

longer horizons, help to interpret the low forecasting power of YG at longer horizons, since the

sum of positive and negative influences zero out, and the overall predictive power of YG is

small.

The estimation for bond excess returns presented in panel B, show somehow different

results. I found that, when the yield gap is negative, the forecasting power over future excess

bond returns, is greater than when YG is positive, at all horizons, as indicated by the

coefficients on YGt1 DYGt which are of higher magnitude (absolute value) compared with

YGtDYGt estimates, at all horizons. The higher forecasting power of YGt1 DYGt is

especially relevant at longer horizons. Thus for bond returns, contrary to stock market returns,

negative values of the yield gap, have greater forecasting power than when the variable is

positive.

Similar to other forecasting state variables like the dividend yield or the default spread, the

Yield gap is negatively correlated with the business cycle. By using a business cycle dummy

(CYCLE), which takes the value 1 in an economic expansion as defined by the NBER, and

takes value 0 in recessions, and performing a monthly regression of YG on CYCLE, I get the

following results (OLS t-statistics in parenthesis),

YGt 0.008 0.010CYCLE

tAdj.R2 0.022

3.268 3.825

Given the countercyclical nature of YG, it is possible that its forecasting role over asset

returns, changes with the state of the business cycle, i.e. differs between economic

expansions and recessions. In order to evaluate this hypothesis, I specify the following

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regression,

rt1,tk ak bk1FFPREMt bk2TERMt bk3DYt bk4YGtCYCLEtbk5YGt1 CYCLEt ut1,tk 15

and the estimation results are presented in Table IX. In what concerns the value-weighted

market return (Panel A), the coefficient estimates associated with YGt1 CYCLEt are higher

relative to the estimates associated with YGtCYCLEt, at all forecasting horizons. Hence, the

predictive power of YG over stock market returns is greater in recessions than in economic

expansions. The results for the bond return (Panel B), show that the coefficient estimates

associated with YGtCYCLEt have higher magnitudes compared to YGt1 CYCLEt, thus in the

case of bond return, the forecasting power of YG is greater in expansions compared to

recessions, in opposition with the result for stock market returns.

Another potentially relevant question to address concerning the forecasting role of the yield

gap, is to measure how much of the forecasting power of YG over returns, comes from

predicting positive versus negative excess returns. In order to analyze this issue, I conduct a

regression similar to specification 14, using a dummy variable DRt that assumes the value 1

when future excess returns are positive, and zero otherwise. Thus, the interaction terms

YGtDRt (YGt1 DRt) measure the impact of YG on future returns, when future returns are

positive (negative). The regression to be estimated is,

rt1,tk ak bk1FFPREMt bk2TERMt bk3DYt bk4YGtDRtbk5YGt1 DRt ut1,tk 16

The results for the value-weighted market rvw and bond returns rb are presented in Table

X, panels A and B respectively. The regression for rvw show that the coefficients on

YGt1 DRt are higher than the corresponding estimates on YGtDRt, at all horizons, being

also higher than the comparable YG estimates in Table II, Panel C. The t-statistics associated

with YGt1 DRt indicate statistical significance at the 5% level for all the horizons, while YG

in the benchmark regression 2 was not significant for longer horizons. The adjusted R2s also

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compare favorably with the ones in the benchmark regression at all horizons. The estimates

associated with YGtDRt are positive at all horizons, but not statistically significant. Thus, it

seems that the forecasting power of the yield gap over stock returns is more attributable to

predicting negative excess returns into the future, relative to positive excess returns. Another

feature from the results in Panel A, is that the dividend yield is no longer significant at longer

horizons (36 and 48 months).

The results for bond returns, reported in panel B, show a different picture. In terms of

magnitudes, YGtDRt have higher coefficients and associated t-statistics than YGt1 DRt at

horizons between 1 and 24 months, and similar estimates for horizons, k36,48.

Nevertheless, YGt1 DRt is statistically significant for horizons beyond 12 months.

As was the case with the results for the value weighted market return, YGtDRt coefficients

are higher than the YG estimates for the benchmark regression, at all horizons. Thus, most of

the predictability of the yield gap over future bond returns comes from predicting positive

excess returns. Nevertheless, it must be pointed out that YGt1 DRt coefficients although

exhibit lower magnitude, are still negative at all horizons, and highly significant for horizons

beyond 12 months. Therefore, the Yield gap also forecasts negative bond returns at longer

horizons.

The combination of these results for both stock index and bond returns, suggest that the

biggest forecasting power of the yield gap is over the return of a portfolio long in bonds and

short in the market index.

VI. Economic Significance

In this section, I try to evaluate the economic significance of the predictive power of Yield

Gap over both stock market and bond returns. Following Goyal and Santa-Clara (2003), I

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define an "active" trading strategy, based on the out-of-sample 1 month ahead predicting

power of the Yield gap and other state variables, over returns. At each time t, I conduct the 1

month predictive regression,

rt1 ak bk

xt ut1 17

and the forecasted excess returns are calculated as r t1 k bk

xt, where k and bk are the

estimated coefficients from the above regression, and rt1 denotes excess returns on the risky

asset (stock index or bond). Then, the strategy allocates 100% in the risky asset if the

forecasted excess returns r t1 are positive, and otherwise, it invests 100% in the risk-free

rate. In symbols, the strategy can be written as

St 1 ifr t1 0

0 ifr t1 018

At time t 1 the realized return associated with the trading strategy, is given by

rp,t1 rt1 1 rf,t1 19

By iterating this process forward and using an expanding sample for the predictive

regressions, I generate a time-series of realized returns on the trading strategy, which are

compared to a "passive" investment strategy ("buy-and-hold") that invests either in the stock

index or long term bond. I calculate this strategy for the value-weighted market index,

equally-weighted index and bond returns, as proxies for the risky asset. In order to have an

initial sample of 60 months to conduct the first predictive regression, the active strategy starts

at 1959:07.

In addition, I consider the case of an investor who invests simultaneously in stocks and

bonds. His trading strategy is given by

St

m b 0 ifr m,t1 0,r b,t1 0

m 1,b 0 ifr b,t1 r m,t1 0

m 0,b 1 ifr m,t1 r b,t1 0

20

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and the associated realized returns are given by

rp,t1 mrm,t1 brb,t1 1 m brf,t1 21

If the excess returns on both stocks and bonds are negative, the investor allocates 100% in

the risk-free rate; if the excess returns on the stock index are higher than the bond excess

return, he invests 100% in the stock index, and otherwise he allocates 100% to bonds. This

trading strategy is compared with a passive "buy-and-hold" investment strategy that allocates

equal weights to both the stock index and long term Treasury bond.

The results for the trading strategy simulation are presented in Table XI. Panels A, B and C

offer the results for the "one risky-asset strategy", where the investor allocates either in the

value-weighted index, equally-weighted index or the long term bond, respectively, in addition

to the risk-free rate. Panels D and E contain the results for the "two risky-asset strategy",

where the investor chooses among both the value-weighted index and long maturity bond

(Panel D), or both the equally-weighted index and long maturity bond (Panel E), in addition to

the risk-free rate. The passive strategy "buy-hold" is compared with a trading strategy based

on three sets of conditioning state variables: our benchmark regression which contains

FFPREM, TERM, DY and YG; the "reduced" benchmark regression which excludes DY, and

the single predictive regression, where YG is the sole forecaster of asset returns. In each

panel, are presented the average return, standard deviation and Sharpe ratio (ratio of average

return to standard deviation) for each strategy.

For the value-weighted index, the trading strategies based on the benchmark regression

and reduced benchmark regression, produce higher average returns and lower volatilities

than the passive strategy, and therefore, have higher Sharpe ratios associated. In the case

where the conditioning set is restricted to the Yield Gap, the active strategy has similar

average returns, but lower standard deviation than the passive strategy, and this leads to a

rise in the Sharpe ratio of nearly 0.1% on a monthly basis, which translates to a 1.2% gain

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annually. For the equally-weighted index, the Sharpe ratios are slightly higher than in Panel A,

as a result of the higher average returns despite the higher volatility associated with the

trading strategies. The trading strategy based on YG produces a larger gain in the Sharpe

ratio relative to the passive strategy, than in Panel A, in consequence of both an increase in

average return and lower volatility. For the long maturity bond, the active strategy exhibits

both higher average returns and lower volatilities - and hence higher Sharpe ratios - when

compared to the passive strategy. Notice that the trading strategy based on YG has a higher

Sharpe ratio than the strategy associated with the bigger conditioning set - FFPREM, TERM,

YG.

In the case where investment opportunities depend of 2 risky assets (Stock index and long

maturity bond), the "active" trading strategies produce higher Sharpe ratios than the passive

"long" strategy which allocates half weights to both the stock index and long maturity bond.

This result is robust for both measures of stock market return, value-weighted and

equally-weighted index. The rise in the Sharpe ratio, arises from higher average returns,

which more than compensates the higher volatility associated with the trading strategies. It is

remarkable, to note that the trading strategy conditioning only on YG has higher Sharpe ratios

than the strategies based on the larger information sets. This is a confirmation, that YG has a

higher forecasting power over both stock and bond returns - both in and out of sample - at

short forecasting horizons, when compared with the other state variables.

The results in the last section show that there is evidence in favor of asymmetries in the

forecasting role of YG over the returns of both the market index and long term bond. More

specifically, the biggest forecasting power of YG is over positive excess bond returns and

negative excess stock returns. Thus, I extend the above strategies to take into account the

possibility of short-selling both stocks and bonds. In the case where the investor chooses

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among one risky asset and the risk-free rate, the active trading strategy is given by

St 1 ifr t1 0

1 ifr t1 022

where the investor allocates 100% to the risky asset if its forecasted excess return is

positive, and otherwise, he sells short the risky asset and invests the proceedings in the

risk-free asset.

In the case where the investor chooses among stocks, bond and the risk-free rate, the

strategy is as follows,

St

m 1,b 0 ifr m,t1 r b,t1 0

m 0,b 1 ifr b,t1 r m,t1 0

m 1,b 0 if 0 r b,t1 r m,t1

m 0,b 1 if 0 r m,t1 r b,t1

m 1,b 1 ifr m,t1 0,r b,t1 0

m 1,b 1 ifr b,t1 0,r m,t1 0

23

In this strategy, if the predicted excess returns on both stocks and bond are positive, the

investor allocates 100% for the asset with the highest forecasted return and if the forecasted

excess returns are both negative, the strategy consists of selling short the asset with the

lowest predicted return and invest the proceedings in the risk-free rate. Finally, if one asset

has positive - and the other asset negative - forecasted returns, the investor allocates 100% to

the former one, sell-short the last one, and invest the proceedings at the risk-free rate.

The results associated with the strategies involving short-sales are presented in Table XII.

For both the value-weighted and equally-weighted market index, the trading strategies have

higher Sharpe ratios than the passive strategy, in result of higher average returns which

overstates the rise in the associated volatility. Similar to Table XI, the equally weighted index

has higher out of sample predictability, as indicated by the Sharpe ratios, compared to the

value-weighted index. In the case of bond returns, the three trading strategies have also

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higher Sharpe ratios than the passive strategy. For the augmented scenario with two risky

assets, the trading strategies are more profitable than the passive strategy. The main

difference associated with the strategies that allow short sales, is that the Sharpe ratios are in

general lower, than the corresponding values in Table XI, which is the result of the higher

volatility associated with the trading strategies when short-sales are available, despite the

increase in average returns.

Overall, the results of this simulation indicate that YG has both in and out of sample

predictive power over stock and bond returns and this forecasting power is significantly

relevant in terms of asset allocation/portfolio choice, as showed by the gains in the Sharpe

ratio.

VII. The Yield gap and the cross-section of returns

According to Mertons (1973) ICAPM, state variables that predict future investment

opportunities or market returns, should act as risk factors that price the cross-section of

ex-post average returns. Campbell (1993) derives a discrete-time ICAPM, where the factors

are the market return and news on future market returns (discount-rate news), and Campbell

(1996) derives an equivalent k factor ICAPM, using as factors the innovations on state

variables that help to forecast market returns. Recently, and using the same basic framework

as Campbell (1993, 1996), there has been some new versions of the ICAPM, e.g., Chen

(2003), Campbell and Vuolteenaho (2004) and Maio (2005a,b). Given the evidence in the

previous sections, that the yield gap helps to forecast market returns, then in the ICAPM

context, it must be a factor that explains the cross-section of returns.

Following Campbell (1993, 1996), I use an Epstein and Zin utility function,

Ut 1 Ct1

EtUt1

1

1

1 24

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where 1

1 1, is the elasticity of intertemporal substitution, is the relative risk aversion

parameter, is a time discount factor, and Ct denotes consumption. This utility function has

the advantage of allowing to separate and , contrary to the power utility function, where

is the reciprocal of . The stochastic discount factor (SDF) associated with the objective

function 24 is equal to

Mt1 Ct1Ct

1Rm,t1

1 25

where Rm,t1 denotes the simple return on market wealth. The corresponding log SDF is

given by,

mt1 ln

ct1 1 rm,t1 26

By summing and subtracting both Etct1 and 1 Etrm,t1 yields,

mt1 Etmt1 ct1 Etct1 1 rm,t1 Etrm,t1 27

where Etmt1 ln Etct1 1 Etrm,t1, and making use of the fact that

c t1 Etct1 ct1 Etct1. Substituting ct1 Etct1 by its expression derived in the

Appendix, it follows

mt1 Etmt1 rm,t1 Etrm,t1 1 rt1H 1 rm,t1 Etrm,t1 Etmt1 rm,t1 Etrm,t1 1 rt1

H 28

where the last equality follows from substituting the expression for . If we substitute news

in future discount rates for its expression in equation 11 above,

rt1H Et1 Et

j1

jrm,t1j e1AI A1t1 t1

k1

Kkk,t1 29

we have,

mt1 Etmt1 rm,t1 Etrm,t1 1 k1K

kk,t1 30

where k is the kth element of e1AI A1 and k,t1 is the kth element of the VAR

error vector t1. By letting the market return be positioned last in the VAR state vector, i.e.,

Et1 Etrm,t1 K,t1, it follows,

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mt1 Etmt1 1 k1

K1kk,t1 1 K K,t1 31

Finally, by substituting the VAR error vector t1 FFPREM,t1,TERM,t1,DY,t1,YG,t1,m,t1

corresponding to the state vector in section IV, the log SDF is given by,

mt1 Etmt1 1 1FFPREM,t1 2TERM,t1 3DY,t1 4YG,t11 5 m,t1 32

By Making ft1 FFPREM,t1,TERM,t1,DY,t1,YG,t1,m,t1 and

b b1,b2,b3,b4,b5 1 1, 1 2, 1 3, 1 4, 1 5 , and using

Theorem 1 in the Appendix, one has the following asset pricing model,

Eri,t1 rf,t1 i

2

2 11i,FFPREM 2i,TERM 3i,DY

4i,YG 15 i,m 33

where i,m Covri,t1,m,t1 represents the covariance of the return of asset i with

innovations in the market return and similarly for the other factors. Equation 33 represents a

5 factor equilibrium asset pricing model, similar to Campbell (1996), where the factors are the

innovations in the state variables used to forecast stock market returns, in the spirit of Merton

(1973). The factor risk prices are theoretically constrained, and variables which have higher

forecasting power over market returns (as measured by the respective elements of ) have

higher risk prices. The risk prices also depend on the coefficient of relative risk aversion ,

which is the only parameter to be estimated in the cross-section of returns. More specifically,

a factor which has a large positive element of associated, will be highly positively correlated

with future investment opportunities, and will demand a high positive risk price, if investors are

risk averse 1. In the case of the innovations in market return, for a risk-averse investor,

the risk price will be below the risk aversion parameter, if 5 0 as suggested by the results

in Table VII.

Since most asset pricing models are estimated and evaluated in terms of factor betas risk

prices, we can restate equation 33 in terms of single regression betas, by multiplying and

dividing each covariance by the associated factor variance,

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Eri,t1 rf,t1 i

2

2 11FFPREM

2 i,FFPREM 2TERM2 i,TERM 3DY

2 i,DY

4YG2 i,YG 15 m

2 i,m 34

where m2 Varm,t1, and similarly for the other factors. The risk prices for betas can be

derived from 33, by FFPREM,TERM,DY,YG,m

fb, where f is a diagonal matrix

with the factor variances on its main diagonal.

A natural econometric framework to use in estimating and testing the asset pricing model

33, is first-stage GMM using as weighting matrix the identity matrix, where the N sample

moments correspond to the pricing errors for each of the N test assets at hand,

gTb 1T t1

Tri,t1 rf,t1

i2

2 11i,FFPREM 2i,TERM 3i,DY

4i,YG 15 i,m 0, i 1 , . . . ,N

The standard errors for the parameter estimates and moments are presented in the

Appendix, and the asymptotic test that the pricing errors are jointly zero, with gTb

, is

given by var 1 ~2N K, with K being the number of parameters estimated by the

system, and N K denoting the number of overidentifying conditions. For the model above,

we have K 1, since only one parameter is estimated in the cross-section.

Following Cochrane (1996), and given the fact that var is singular in most of the cases, I

perform a eigenvalue decomposition of the moments variance-covariance matrix,

var QQ, where Q is a matrix containing the eigenvectors of var on its columns, and

is a diagonal matrix of eigenvalues, and then I invert only the non-zero eigenvalues of .

In Table XIII, I present the estimation results for the ICAPM model of equations 33 and

34 above. Following Lo and Mackinlay (1990), who argue against testing asset-pricing

models by using returns on portfolios sorted on some characteristic associated with returns

themselves, I use the returns on 38 industry portfolios (IND38), and the combination of

size/book-to-market and industry portfolios (SBV25IND38), as additional groups of test

assets relative to the 25 size/book-to-market portfolios (SBV25). All the data on portfolio

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returns is obtained from Prof. Kenneth Frenchs website.

The estimated sensitivities of discount rate news to VAR shocks,

0.237,1.935,0.395,7.611, 0.372

, indicate that the yield gap is relatively important at

forecasting future market returns (coefficient of 7.611), confirming the results of Table VII. The

estimates of the relative risk aversion coefficient indicate that is clearly above 1, and it is

statistically significant for the three sets of testing portfolios. The covariance risk prices of

innovations in the market return are positive and significant, but also lower than , in result of

both 5 being negative and 1 (for SBV25 we have m 7.417 compared to 11.213).

The covariance risk price associated with innovations on the yield gap has the highest

magnitude across all factors, due to the high magnitude of 5. In terms of beta risk prices, YG

is lower compared to both DY and m, in result of the lower variance of the yield gap,

compared with these two factors. On the other hand, the magnitude of YG is higher compared

to both FFPREM and TERM. The p-values associated with the asymptotic 2 test, confirm that

the model is not rejected for the three sets of portfolios.

The factor loading estimates associated with model 34 are presented in Table XIV. The

betas associated with YG are negative, which is mainly due to the positive contemporaneous

correlation between market and individual stock returns. The average betas across quintiles

indicate that both growth and small stocks are more sensitive to innovations in YG, relative to

value and large stocks, respectively - the difference in beta magnitudes between BV1 and

BV5 is 2.327, whereas the spread between S1 and S5 is 2.323. In general, growth stocks

discount their cash flows more distant into the future (have higher "duration risk"), and in many

cases start to distribute cash flows for shareholders, only after some periods. Thus, growth

stocks should be more correlated with future discount rates, and given the fact that

innovations in yield gap are positively correlated with future investment opportunities, growth

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stocks are more correlated with YG than value stocks.

To measure the pricing ability of each factor, I calculate the factor risk premium (beta times

risk price), and compute the averages across book-to-market quintiles, which results appear

in Table XV. The results show that innovations in YG have a negative risk premium, due to

the respective negative betas, and have the third largest contribution to the total pricing errors,

after the market return and dividend yield factors. Comparing across quintiles, the magnitude

of YGs risk premium is higher for growth stocks than value stocks (-0.440 versus -0.291), in

consequence of identical pattern for the average betas, as shown in Table XIV. The presence

of YG in the factor model, is not enough to price accurately both growth and value stocks,

which have pricing errors of -0.385 and 0.267 respectively, which goes in line with the findings

of Maio (2005a,b), that an ICAPM equivalent to model 33 does not price the value premium.

Nevertheless, the results of this section suggest that innovations in the yield gap have some

explanatory power for the cross section of returns.

VIII. Conclusion

The Yield gap - the difference between the stock market earnings yield and the long term

bond yield - can be interpreted a simple measure of the yield spread of stocks versus bonds,

or a relative long-term rate of return of stocks against bonds.

By using the definition of returns, I derive a dynamic accounting decomposition for the yield

gap, where it is positively correlated with future stock market returns and negatively correlated

with future dividend to earnings payout ratios, growth rate on future equity earnings and future

bond returns. This decomposition provides the rationale for the predictive role of the yield gap

over asset returns.

Conditional on other forecasting variables, the yield gap forecasts positive stock excess

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returns, at several horizons ahead. The greatest forecasting power is achieved at near

horizons (until one year) declining gradually with the horizon, contrary to the majority of other

forecasting variables, which have forecasting power increasing with horizon. Hence, at short

horizons the yield gap has bigger forecasting power than most other state forecasting

variables. The yield gap has greater predictive power for the equally-weighted relative to the

value-weighted market excess returns, suggesting that the yield gap has greater forecasting

power for small caps excess returns, relative to large capitalization stocks. The yield gap has

also a very significant effect on bonds, forecasting negative excess returns for long-term

bonds, both at short and longer horizons ahead.

Implied estimates from a first-order VAR, confirm the forecasting power of YG over returns

associated with the long-horizon regressions, and in addition, innovations in the Yield gap are

positively correlated with the two unobserved components of market returns - cash flow and

discount rate news.

The observed correlation between the yield gap and future returns on both stocks and

bonds is subject to non-linearities: The bulk of the predictability of stock returns, comes from

predicting negative excess returns ahead, whereas for bonds, the bigger proportion of

forecasting power over bond returns, is associated with predicting positive excess returns in

the future.

The out of sample forecasting power of the yield gap, is economically significant, as

indicated by the significant gains in the Sharpe ratios, associated with dynamic trading

strategies conditional on the predictive ability of YG and other state variables. Thus, the yield

gap can be an important state variable to be used in dynamic portfolio optimization.

Furthermore, in the context of a 5 factor ICAPM, the innovations in the yield gap have some

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explanatory power for the cross section of average returns.

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References

Ang, Andrew, and Geert Bekaert, 2004, Stock return predictability: Is it there?,

unpublished manuscript, Columbia University.

Asness, Clifford, 2003, Fight the FED model: the relationship between future returns and

stock and bond market yields, Journal of Portfolio Management, 30, 11-24.

Bary, Andrew, Is Alan addled?, Barrons online.

Campbell, John Y., 1987, Stock returns and the term structure, Journal of Financial

Economics 18, 373-99.

Campbell, John Y., 1991, A variance decomposition for stock returns, Economic Journal

101, 157-179.

Campbell, John Y., 1993, Intertemporal asset pricing without consumption data, American

Economic Review, 1993, 83, 487-512.

Campbell, John Y., 1996, Understanding risk and return, Journal of Political Economy,

104, 298-345.

Campbell, John Y., and John Ammer, 1993, What moves the stock and bond markets? A

variance decomposition for long-term asset returns, Journal of Finance 48, 3-26.

Campbell, John Y., Andrew W. Lo, and A. Craig Mackinlay, 1997, The econometrics of

financial markets, Princeton University Press.

Campbell, John Y., and Robert J. Shiller, 1988a, Stock prices, earnings, and expected

dividends, Journal of Finance 43, 661-676.

33

8/2/2019 Fed Model

34/68

Campbell, John Y., and Robert J. Shiller, 1988b, The dividend price ratio and

expectations of future dividends and discount factors, Review of Financial Studies 1, 195-228.

Campbell, John Y., and Robert J. Shiller, 2001, Valuation ratios and the long-run stock

market outlook: An update, Forthcoming in Advances in behavioral finance, vol. II, Princeton

University Press.

Campbell, John Y., and Samuel B. Thompson, 2004, Predicting the equity premium out

of sample: Can anything beat the historical average?, unpublished manuscript, Harvard

University.

Campbell, John Y., and Tuomo Vuolteenaho, 2004, Bad beta, good beta, American

Economic Review, 94, 1249-1275.

Chen, Joseph, 2003, Intertemporal CAPM and the cross-section of stock returns,

unpublished manuscript, University of Southern California.

Chen, Nai-Fu, Richard Roll, and Stephen A. Ross, 1986, Economic forces and the stock

market, Journal of Business 59, 383-403.

Cochrane, John H., 1992, Explaining the variance of price-dividend ratios, Review of

Financial Studies 5, 243-280.

Cochrane, John H., 1996, A cross-sectional test of an investment-based asset pricing

model , Journal of Political Economy, 104, 572-621.

Cochrane, John H., 2001, Asset pricing, Princeton University Press.

Estrella, Arturo, and Gikas A. Hardouvelis, 1991, The term structure as a predictor of

real economic activity, Journal of Finance 46, 555-576.

34

8/2/2019 Fed Model

35/68

Fama, Eugene F., and Kenneth R. French, 1988, Dividend yields and expected stock

returns, Journal of Financial Economics 22, 3-25.

Fama, Eugene F., and Kenneth R. French, 1989, Business conditions and expected

returns on stock and bonds, Journal of Financial Economics 25, 23-49.

Goyal, Amit, and Pedro Santa-Clara, 2003, Idiosyncratic risk matters, Journal of Finance

58, 975-1006.

Hamilton, James D., 1994, Time series analysis, Princeton University Press.

Hayashi, Fumio, 2000, Econometrics, Princeton University Press.

Hodrick, Robert J., 1992, Dividend yields and expected stock returns: Alternative

procedures for inference and measurement, Review of Financial Studies 5, 357-386.

Hong, Harrison, Walter Torous and Rossen Valkanov, 2004, Do industries lead stock

markets?, unpublished manuscript, UCLA.

Jegadeesh, Narasimhan, and Sheridan Titman, 1993, Returns to buying winners and

selling losers: Implications for stock market efficiency, Journal of Finance, 48, 65-91.

Li, Lingfend, 2002, Macroeconomic factors and the correlation of stock and bond returns,

unpublished manuscript, Yale University.

Lo, Andrew W., and A. Craig Mackinlay, 1988, Stock market prices do not follow random

walks: Evidence from a simple specification test, Review of Financial Studies 1, 41-66.

Lo, Andrew, and Craig Mackinlay, 1990, Data-snooping biases in tests of financial asset

pricing models, Review of Financial Studies, 3, 431-468.

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Maio, Paulo, 2005a, ICAPM with time-varying risk aversion, unpublished manuscript,

Universidade Nova, Lisbon.

Maio, Paulo, 2005b, Bad, good and excellent: An ICAPM with bond risk premia,

unpublished manuscript, Universidade Nova, Lisbon.

Maio, Paulo, 2005c, The predictive role of monetary policy and the cross section of stock

returns, unpublished manuscript, Universidade Nova, Lisbon.

Philips, Thomas K., 1999, Why do valuation ratios forecast long-run equity returns?,

Journal of Portfolio Management, 25, 39-44.

Poterba, James M., and Lawrence Summers, 1988, Mean reversion in stock returns:

Evidence and implications, Journal of Financial Economics 22, 27-60.

Vuolteenaho, Tuomo, 2002, What drives firm-level stock returns?, Journal of Finance 57,

233-264.

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Appendices

A. A decomposition for the yield gap

Following Cochrane (2001), chapter 20, and using the definition of simple return,

1 Rt11 Rt1 Rt1

1 Pt1Dt1Pt

A. 1

Dividing by the current dividend Dt, leads to

PtDt

R t11 1

P t1Dt1

Dt1Dt

A. 2

By taking logs, one has

dt pt rt1 ln1 exppt1 dt1 dt1 A. 3

where the lowercase letters denote the logs of variables. By taking a first-order Taylor

expansion around the mean of p t1 dt1, it follows

dt pt k rt1 dt1 pt1 dt1 A. 4

where 11expEdt1pt1

and k ln1 ln 1 1are parameters of linearization. By

adding log earnings et to both sides of A. 4, and rearranging, we have

et pt k rt1 1 de t1 et1 et1 pt1 A. 5

where det1 dt1 et1 denotes the log dividend payout ratio. If we iterate forward A. 5,

impose a "transversality condition" that the log earnings to price ratio cant growth faster

(slower) than rate 1/, limjjetj p tj 0, and take conditional expectations, we obtain the

log earnings yield as a function of expected future stock returns, dividend payout ratios and

earnings growth,

et pt k

1 Et

j0

jrt1j 1 det1j et1j A. 6

By defining the earnings yield as ln1 EtPt

, we have

ln1 EtP t

ln1 expet pt k 1 et pt A. 7

where 11expEetpt

and k ln 1 ln 1

1 are parameters of

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linearization from a first-order Taylor expansion around the mean of et p t.

Following Campbell et al (1997), the log 1 period return on a n maturity coupon bond, can

be represented as

rb,n,t1 kb bpn1,t1 1 bc pnt A. 8

where c is the log coupon, and b 1

1expEcpn1,t1, kb lnb 1 b ln

1b

1define

linearization parameters from a first-order Taylor expansion around the mean of c pn1,t1.

From A. 8 it follows that the log bond yield can be represented as the discounted sum of

future 1 period log bond returns.

ynt 1b1bn Etj0

n1

bj rb,nj,t1j A. 9

By combining equations A. 6, A. 7 and A. 9, we have the decomposition for the yield

gap,

YGt ln1 EtPt

ynt k k

11 1 Et

j0

jrt1j 1 det1j

e t1j 1b

1bnEt

j0

n1bjrb,nj,t1j A. 10

B. A decomposition for innovations in the log dividend to price ratio

Following Campbell and Shiller (1988b), the log stock market return can be approximated

as,

rt1 k pt1 1 dt1 pt A. 11

where and kare parameters of linearization. By combining A. 11 with the decomposition

for innovations in market returns 9, and by noting that Et1 Etk Et1 Etp t 0, we

have,

Et1 Etdt1 pt1 Et1 Etdt1 Et1 Et j0

jdt1j

Et1 Et j1

jrt1j A. 12

In addition by noting that Et1 Etdt 0, it follows

Et1 Etdt1 pt1 1 Et1 Et

j1

jdt1j

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1 Et1 Et

j1

jrt1j A. 13

C. Theorem 1

Given the asset pricing model

1 EtMt1Ri,t1 A. 14

and with the assumption that the log SDF, mt1 lnMt1, is a linear function of K risk

factors ft1,

mt1 a bft1 A. 15

the unconditional model in discount factor form for log returns, ri,t1 lnRi,t1, can be

represented as,

Eri,t1 rf,t1 0. 5i2 b Covri,t1, ft1 A. 16

Proof:

Taking logs of A. 14, one gets the pricing equation in the log form,

0 lnEtexpmt1 ri,t1 A. 17

Since the log is a non-linear function, one can use a second-order Taylor expansion to the

right hand side of A. 17, leading to the following approximation

0 Etmt1 ri,t1 0. 5Vartmt1 ri,t1 A. 18

By expanding A. 18 and rearranging, one obtains,

Etri,t1 0. 5Vartri,t1 Etmt1 0. 5Vartmt1 Covtmt1, ri,t1 A. 19

Applying the pricing equation A. 19 to the risk-free rate, rf,t1, and noting that

Vartrf,t1 Covtmt1, rf,t1 0, since rf,t1 is known in period t, one has,

rf,t1 Etmt1 0. 5Vartmt1 A. 20

Subtracting A. 20 from A. 19, we obtain,

Etri,t1 rf,t1 0. 5Vartri,t1 Covtmt1, ri,t1 A. 21

Given the assumption that the log SDF is linear in the risk factors, mt1 a bft1, and

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substituting in A. 21, we have the following conditional pricing equation for excess returns,

Etri,t1 rf,t1 0. 5Vartri,t1 bCovtri,t1, ft1 A. 22

By applying the law of iterated expectations to equation A. 22, one has the unconditional

pricing model

Eri,t1 rf,t1 0. 5i2 bCovri,t1, ft1 k1

Kbki,k A. 23

where i2 Varri,t1, i,k Covri,t1,fk,t1,k 1 , . . . ,Kand fk,t1 denotes the kth factor.

Theorem 1 represents a straightforward generalization of the theorem in section 6.3 of

Cochrane (2001), for the case in which the SDF is nonlinear, but the log SDF is a linear

function of the factors.

D. Substituting out consumption as in Campbell (1993) model

Using an Epstein and Zin utility function,

Ut 1 Ct1

EtUt1

1

1

1 A. 24

where 1

1 1, is the elasticity of intertemporal substitution, and is the RRA coefficient.

The corresponding SDF is given by

Mt1 Ct1Ct

1Rm,t1

1 A. 25

and the corresponding log SDF is equal to

mt1 ln Etct1 1 Etrm,t1

c t1 Etct1 1 rm,t1 Etrm,t1 A. 26

Applying the conditional log pricing equation A. 19 to the market return, rm,t1, leads to

Etrm,t1 0. 5Vartrm,t1 Etmt1 0. 5Vartmt1 Covtmt1, rm,t1 A. 27

substituting the expressions for Etmt1, Vartmt1 and Covtmt1, rm,t1, and using the fact

that Covtmt1,rm,t1 Covtmt1, rm,t1 Etrm,t1 and Vartrm,t1 Vartrm,t1 Etrm,t1, we

have

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Etrm,t1 0. 5mt2 ln Etct1 1 Etrm,t1

0. 5 2ct

2 1 2mt2 2 1 c,mt

c,mt 1 mt

2 A. 28

where ct2 Vartct1 Etct1, mt

2 Vartrm,t1 Etrm,t1 and

c,mt Covtct1 Etct1, rm,t1 Etrm,t1

Solving for Etct1, and imposing joint conditional homoskedasticity for log consumption

growth and log market returns, it follows,

Etct1 ln 0. 51 ct

2 mt2 2c,mt Etrm,t1

ln 0. 5 1 c2 m

2 2c,m Etrm,t1 A. 29

where c2 Varct1 Etct1, m

2 Varrm,t1 Etrm,t1 and

c,m Covct1 Etct1, rm,t1 Etrm,t1.

Equation A. 29 can be restated as

Etct1 m Etrm,t1 A. 30

with m ln 0. 51 c

2 m

2 2c,m.

Giving a relation similar to equation A. 30, Campbell (1993) shows that innovations in log

consumption and log market returns are related by the following expression,

ct1 Etc t1 rm,t1 Etrm,t1 1 Et1 Et j1

jrm,t1j A. 31

E. GMM standard errors formulas for parameter estimates and moments

The parameter estimates b

associated with GMM system in section VII, have variance

formulas given by,

Varb

1T

dINd1dININdd

INd1 A. 32

where IN is a N order Identity matrix, d gTb

brepresents the matrix of moments

sensitivities to the parameters, and is a estimator for the spectral density matrix S, derived

under the Newey-West procedure with 5 lags. The variance-covariance matrix for the

moments is given by,

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Var 1T

INddINd

1dINININddINd

1d A. 33

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Table I

Descriptive statistics of returns and forecasting state variables

This table reports descriptive statistics for asset excess returns and forecastingstate variables. The excess returns data is on the value-weighted market index(rvw), equally-weighted market index (rew) and 10 year Treasury bond (rb). Theforecasting variables are the earnings yield (EY), the log 10 year Treasury bondyield (y), the yield gap (YG), the FED funds premium (FFPREM), the termstructure spread (TERM), and the log market dividend yield (DY). The originalsample is 1954:07- 2003:12. Autocorr designates the first order autocorrelation.For details on the variables construction refer to Section II.

Panel A

Mean Stdev. Min Max Autocorr

rvw 0.005 0.044 -0.261 0.148 0.073rew 0.007 0.055 -0.324 0.256 0.218

rb 0.001 0.022 -0.078 0.089 0.066

EY 0.064 0.024 0.025 0.133 0.995

y 0.065 0.025 0.023 0.143 0.993

YG -0.001 0.021 -0.044 0.059 0.982

FFPREM 0.005 0.008 -0.011 0.054 0.878

TERM 0.008 0.011 -0.031 0.033 0.967

DY -3.488 0.371 -4.495 -2.796 0.997

Panel Brvw rew rb EY y YG FFPREM TERM DY

rvw 1.000 0.860 0.159 -0.012 -0.075 0.075 -0.129 0.137 0.009

rew 1.000 0.065 -0.005 -0.089 0.099 -0.157 0.167 -0.003

rb 1.000 0.000 0.023 -0.027 0.028 0.115 0.005

EY 1.000 0.622 0.400 0.480 -0.364 0.840

y 1.000 -0.469 0.561 -0.082 0.433

YG 1.000 -0.115 -0.314 0.441

FFPREM 1.000 -0.441 0.269

TERM 1.000 -0.192

DY 1.000

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Table II

Yield gap predicting the value-weighted market return

Long horizon regressions for the monthly continuously compounded excessreturns on the value-weighted market index, at horizons of 1, 3, 12, 24, 36 and48 months ahead. The forecasting variables are the current values of the yieldgap (YG), FED funds premium (FFPREM), term structure spread (TERM), andlog market dividend yield (DY). The original sample is 1954:07- 2003:12, and kobservations are lost in each of the respective k-horizon regressions fork1,3,12,24,36,48. For each equation, in line 1 are reported the coefficientestimates, and in lines 2 and 3 respectively, are reported the asymptoticNewey-West (with 5 lags) and Hansen-Hodrick t-statistics. In line 4, arepresented the p-values from a bootstrap experiment, consisting of 10,000simulations to the Newey-West t-statistics, under the null of no predictability ofreturns. The t-statistics and p-values at bold denote significance at the 1%

level, while the underlined ones indicate significance at the 5% level. Adj. R2

denotes the adjusted R2.

Panel A

Const. YG Adj. R2

Const. YG Adj. R2

K=1 0.055 2.965 0.013 K=24 0.046 0.853 0.030

2.470 2.815 5.101 2.238

2.564 2.701 2.520 1.255

0.473 0.003 0.920 0.024

K=3 0.054 2.662 0.031 K=36 0.047 0.606 0.0262.673 2.961 6.658 1.963

2.449 2.679 2.583 0.907

0.603 0.005 0.972 0.041

K=12 0.049 1.730 0.053 K=48 0.050 0.363 0.013

3.685 3.170 8.394 1.564

2.286 2.084 2.780 0.601

0.821 0.003 0.997 0.077

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Panel B

Const. FFPREM TERM YG Adj. R2

Const. FFPREM TERM YG Adj. R2

K=1 0.017 -2.912 6.764 3.889 0.033 K=24 0.023 0.856 2.670 1.256 0.075

0.457 -0.628 2.908 3.336 1.467 0.802 2.829 2.842

0.437 -0.681 2.656 3.091 0.904 0.660 2.002 1.800

0.808 0.263 0.003 0.001 0.990 0.248 0.006 0.009

K=3 0.019 -2.178 5.987 3.486 0.075 K=36 0.020 1.233 2.920 1.050 0.120

0.617 -0.597 2.940 3.628 1.831 1.929 4.818 3.163

0.567 -0.555 2.712 3.298 0.898 1.111 3.035 1.645

0.844 0.279 0.005 0.001 0.998 0.050 0.000 0.004

K=12 0.014 0.157 4.577 2.416 0.126 K=48 0.027 0.680 2.623 0.778 0.136

0.714 0.081 3.313 4.299 3.108 1.177 5.240 3.235

0.486 0.074 2.356 2.803 1.551 0.947 3.114 1.630

0.969 0.474 0.002 0.000 0.999 0.151 0.000 0.003

Panel C

Const. FFPREM TERM DY YG Adj. R2

Const. FFPREM TERM DY YG Adj. R

K=1 0.311 -4.482 6.310 0.081 3.131 0.033 K=24 0.381 -1.163 1.644 0.098 0.324 0.150

1.086 -0.904 2.696 1.026 2.339 3.496 -1.006 2.167 3.120 0.815

1.043 -0.963 2.441 0.991 2.148 1.928 -0.950 1.820 1.643 0.753

0.190 0.188 0.006 0.149 0.013 0.005 0.178 0.026 0.002 0.235

K=3 0.347 -3.944 5.435 0.090 2.637 0.081 K=36 0.294 -0.368 2.012 0.075 0.350 0.185

1.405 -1.039 2.720 1.308 2.448 3.441 -0.554 3.935 3.047 1.323

1.274 -0.962 2.502 1.187 2.219 2.368 -0.601 3.603 1.975 0.9830.151 0.174 0.008 0.118 0.012 0.010 0.307 0.000 0.004 0.118

K=12 0.387 -1.915 3.749 0.103 1.432 0.162 K=48 0.263 -0.562 2.024 0.066 0.257 0.201

2.275 -0.920 2.873 2.141 2.187 3.775 -0.914 4.632 3.281 1.185

1.441 -0.940 2.212 1.308 1.676 2.560 -0.892 2.923 2.167 0.734

0.048 0.211 0.006 0.028 0.031 0.008 0.214 0.000 0.003 0.150

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Table III

Yield gap predicting the equally-weighted market return

Long horizon regressions for the monthly continuously compounded excessreturns on the equally-weighted market index, at horizons of 1, 3, 12, 24, 36and 48 months ahead. The forecasting variables are the current values of theyield gap (YG), FED funds premium (FFPREM), term structure spread (TERM),and log market dividend yield (DY). The original sample is 1954:07- 2003:12,and k observations are lost in each of the respective k-horizon regressions fork1,3,12,24,36,48. For each equation, in line 1 are reported the coefficientestimates, and in lines 2 and 3 respectively, are reported the asymptoticNewey-West (with 5 lags) and Hansen-Hodrick t-statistics. In line 4, arepresented the p-values from a bootstrap experiment, consisting of 10,000

simulations to the Newey-West t-statistics, under the null of no predictability ofreturns. The t-statistics and p-values at bold denote significance at the 1%level, while the underlined ones indicate significance at the 5% level. Adj. R 2

denotes the adjusted R2.

Panel A

Const. YG Adj. R2

Const. YG Adj. R2

K=1 0.082 4.763 0.022 K=24 0.069 2.056 0.106

2.709 3.418 6.093 4.848

3.082 3.584 3.180 2.517

0.483 0.000 0.922 0.000

K=3 0.081 4.565 0.049 K=36 0.069 1.819 0.144

2.932 3.811 7.855 5.636

2.664 3.429 3.129 2.391

0.603 0.000 0.960 0.000

K=12 0.073 3.130 0.101 K=48 0.069 1.647 0.163

4.290 4.754 9.264 6.582

2.751 3.043 3.325 2.420

0.831 0.000 0.986 0.000

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Panel B

Const. FFPREM TERM YG Adj. R2

Const. FFPREM TERM YG Adj. R2

K=1 0.014 -3.481 11.102 6.333 0.055 K=24 0.051 0.940 1.902 2.358 0.117

0.270 -0.642 3.397 4.054 2.844 0.743 1.691 5.261

0.283 -0.734 3.451 4.157 1.750 0.710 0.992 2.758

0.886 0.264 0.001 0.000 0.942 0.265 0.063 0.000

K=3 0.010 -1.478 10.207 6.064 0.107 K=36 0.053 0.972 1.465 2.058 0.155

0.237 -0.322 3.579 4.677 4.046 0.999 1.717 5.965

0.217 -0.298 3.249 4.217 2.214 0.790 0.952 2.768

0.934 0.380 0.001 0.000 0.957 0.191 0.061 0.000

K=12 0.021 0.993 6.339 4.113 0.174 K=48 0.053 0.751 1.731 1.937 0.187

0.784 0.375 3.803 6.350 4.675 0.726 2.281 6.721

0.537 0.329 2.883 4.060 2.223 0.565 1.483 2.816

0.984 0.373 0.001 0.000 0.980 0.266 0.023 0.000

Panel C

Const. FFPREM TERM DY YG Adj. R2

Const. FFPREM TERM DY YG Adj.

K=1 0.148 -4.197 10.895 0.037 5.988 0.054 K=24 0.211 0.039 1.444 0.044 1.943 0.1

0.413 -0.726 3.313 0.374 3.440 1.973 0.027 1.207 1.554 3.457

0.407 -0.807 3.361 0.368 3.516 1.949 0.028 0.764 1.435 2.144

0.423 0.244 0.001 0.356 0.001 0.120 0.499 0.145 0.082 0.002

K=3 0.162 -2.297 9.951 0.042 5.670 0.106 K=36 0.082 0.802 1.368 0.008 1.984 0.1

0.519 -0.478 3.480 0.482 3.975 1.117 0.713 1.539 0.410 4.868

0.463 -0.443 3.167 0.429 3.604 0.976 0.722 0.918 0.357 2.5050.409 0.333 0.001 0.329 0.001 0.377 0.275 0.091 0.352 0.000

K=12 0.228 -0.156 5.880 0.057 3.568 0.179 K=48 0.023 0.911 1.808 -0.008 2.004 0.1

1.159 -0.056 3.444 1.051 4.405 0.314 0.783 2.314 -0.441 5.875

0.918 -0.053 2.790 0.806 3.171 0.201 0.630 1.449 -0.300 2.588

0.257 0.481 0.002 0.172 0.000 0.688 0.248 0.024 0.349 0.000

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Table IV

Yield gap predicting the long-term bond return

Long horizon regressions for the monthly continuously compounded excessreturns on the 10 year Treasury bond, at horizons of 1, 3, 12, 24, 36 and 48months ahead. The forecasting variables are the current values of the yield gap(YG), FED funds premium (FFPREM), term structure spread (TERM), and logmarket dividend yield (DY). The original sample is 1954:07- 2003:12, and kobservations are lost in each of the respective k-horizon regressions fork1,3,12,24,36,48. For each equation, in line 1 are reported the coefficientestimates, and in lines 2 and 3 respectively, are reported the asymptoticNewey-West (with 5 lags) and Hansen-Hodrick t-statistics. In line 4, arepresented the p-values from a bootstrap experiment, consisting of 10,000

simulations to the Newey-West t-statistics, under the null of no predictability ofreturns. The t-statistics and p-values at bold denote significance at the 1%level, while the underlined ones indicate significance at the 5% level. Adj. R 2

denotes the adjusted R2.

Panel A

Const. YG Adj. R2

Const. YG Adj. R2

K=1 0.009 -1.245 0.009 K=24 0.009 -1.424 0.319

0.796 -2.474 2.371 -7.353

0.819 -2.634 1.278 -3.640

0.526 0.008 0.644 0.000

K=3 0.009 -1.359 0.033 K=36 0.010 -1.201 0.370

0.875 -3.043 3.175 -8.268

0.794 -2.767 1.485 -3.920

0.580 0.003 0.617 0.000

K=12 0.009 -1.528 0.168 K=48 0.010 -1.103 0.404

1.485 -5.961 3.805 -9.922

0.931 -3.880 1.585 -4.364

0.689 0.000 0.502 0.000

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Panel B

Const. FFPREM TERM YG Adj. R2

Const. FFPREM TERM YG Adj. R2

K=1 0.008 -1.280 0.882 -1.161 0.009 K=24 0.000 0.773 0.791 -1.281 0.332

0.430 - 0.609 0.610 -2.048 -0.033 1.467 1.726 -6.536

0.423 - 0.645 0.588 -2.216 -0.017 1.141 1.189 -3.3150.524 0.276 0.271 0.027 0.060 0.092 0.065 0.000

K=3 0.008 -1.017 0.719 -1.291 0.037 K=36 0.002 1.047 0.332 -1.108 0.395

0.483 -0.568 0.557 -2.562 0.408 3.042 1.068 -7.424

0.454 -0.537 0.516 -2.359 0.182 1.678 0.692 -3.797

0.556 0.309 0.303 0.012 0.931 0.003 0.177 0.000

K=12 -0.001 -0.079 1.500 -1.309 0.200 K=48 0.005 0.974 0.001 -1.053 0.444

-0.161 -0.088 2.206 -4.959 1.174 2.700 0.004 -8.328

-0.118 -0.069 1.594 -3.782 0.555 1.629 0.002 -4.274

0.109 0.471 0.025 0.000 0.841 0.010 0.488 0.000

Panel C

Const. FFPREM TERM DY YG Adj. R2

Const. FFPREM TERM DY YG Adj

K=1 0.253 -2.591 0.504 0.068 -1.794 0.014 K=24 0.107 0.169 0.484 0.029 -1.559 0.3

1.648 -1.190 0.352 1.615 -2.426 1.900 0.269 0.998 1.935 -6.015

1.712 -1.251 0.326 1.711 -2.535 1.245 0.194 0.698 1.268 -3.656

0.063 0.127 0.367 0.060 0.011 0.072 0.402 0.181 0.048 0.00