The FED model and expected asset returns

Transcription

1 The FED model and expected asset returns Paulo Maio 1 First draft: March 2005 This version: November Bilkent University. Corresponding address: Faculty of Business Administration, Bilkent University, Ankara 06800, Turkey. paulofmaio@gmail.com or maio@bilkent.edu.tr. This paper is a substantial revision from Chapter three of my Ph.D. dissertation at Universidade Nova de Lisboa, and part of it was written when I was visiting Anderson School of Management-UCLA in I thank my advisors Pedro Santa-Clara and João Amaro de Matos for their guidance and support. I have bene ted from comments by Jan Seifert, Haim Shalit, Robert Carver, John Campbell, Jean Pierre Chateau and seminar participants at the German Finance Association Meeting (Oestrich-Winkel), the Eastern Finance Association Meeting (New Orleans), the 2007 Financial Management Association Meeting (Orlando), MAN Investments, and Bilkent University. I have also bene ted from detailed comments by João Duque, Ana Paula Serra and Richard Roll. I also thank Kenneth French and Robert Shiller for making available return data on their web pages, and Luís Catela Nunes for the help provided with the GAUSS code. I acknowledge the nancial support from Fundação para a Ciência e Tecnologia (Portuguese Government). All errors are mine.

2 Abstract The focus of this paper is on the predictive role of the stock-bond yield gap - the di erence between the market earnings yield and the ten-year Treasury bond yield - also know as the FED model, and which can be interpreted as a long-term yield or return spread of equities relative to bonds. Conditional on other forecasting variables, the yield gap forecasts positive excess market returns, both at short and long forecasting horizons, and for both value and equal-weighted indexes. On the other hand, the yield gap forecasts negative excess returns for bonds, at both short and long horizons. These ndings go in line with the predictions from a dynamic accounting identity. By performing an out-of-sample analysis, the results show that the yield gap has reasonable out-of-sample predictability over both stock market and long-term bond returns, when the comparison is made against both a simple historical average and an autoregressive speci cation. Furthermore, the yield gap proxies have greater out-of-sample predictability power than alternative forecasting variables. An investment strategy based on the forecasting ability of the yield gap produces higher Sharpe ratios than passive strategies in both the market index and long-term bond, and in addition, the certainty equivalent estimates are in general positive. Keywords: Asset pricing; FED model; Earnings yield; Bond yield; Predictability of returns; Stock and bond returns; Out-of-sample predictability; Yield gap JEL classi cation: G11; G12; G14; E44

3 1 Introduction "The Fed s 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 Fed s model asks, why would anyone buy stocks with a 4.75% earnings return, when they could get a bond with a 5.60% yield?" Barron s 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 (1988b), Campbell and Shiller (1998), Campbell and Vuolteenaho (2004), Campbell and Thompson (2008), Maio (2007a,b), among others). In addition, yield spreads related to Treasury and corporate bond yields have also been used for some time to forecast asset returns (Keim and Stambaugh (1986), Campbell (1987), Fama and French (1989), among others). This paper focus instead on the yield gap, which corresponds to the di erence 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. This variable is widely referred in the nancial press, it is used by practitioners to forecast returns, and it is even referred in FED publications and was used in o cial testimonies by FED s chairman Alan Greenspan in the late 1990s (to argue for the overvaluation of the U.S. stock market). Nevertheless, little attention has be devoted to it in academics, with Asness (2003), Koivu, Pennanen and Ziemba (2005) and Polk, Thompson and Vuolteenaho (2006) representing recent exceptions. The yield gap 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 an 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 calculated based on annual earnings) 1

4 needed for a very long-term investor who receives all the earnings in the form of dividends or other forms of cash ow 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 o 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 simpli cation, since nominal and real earnings growth over time, the earnings growth rate is time-varying, and the investor receives only a fraction of earnings as payout distributions. Nevertheless, it represents a simple straightforward measure of the long-term return on stocks. By using the de nition 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 both equity and bond returns. Several authors (Asness (2003), Koivu, Pennanen and Ziemba (2005)) have questioned the theoretical validity of the FED model, since we can not compare a real variable (earnings yield) with a nominal variable (bond yield). My model does not say that the two yields should be equal. It just postulates that the yield gap should predict both positive stock returns and negative bond returns, due to the mean reversion in stock and bond prices, and in consequence the yield gap is itself mean-reverting: If an investor buys stocks and sells bonds at a period where the yield gap is low (low earnings yield relative to bond yield), it is likely that he will enjoy negative returns from his strategy. Furthermore, if the in ation rate is not strongly time-varying, then the earnings yield (plus a constant in ation premium) should be comparable to the long-term bond yield. The results for long-horizon regressions show that conditional on other forecasting variables, the yield gap forecasts positive excess stock market returns at several horizons ahead. The yield gap has greater predictive power for the equally-weighted relative to the value-weighted excess market return, suggesting that the yield gap has greater forecasting power for the returns of small caps relative to large capitalization stocks. The yield gap has also a very signi cant e ect on bonds, forecasting negative excess returns for long-term bonds, both at short and longer horizons ahead. 2

5 By performing an out-of-sample analysis, the results show that the yield gap has reasonable out-of-sample predictability over both stock market and long-term bond returns, when the comparison is made against both a simple historical average and an autoregressive speci cation. When one imposes a constraint of non-negative forecasted excess returns, the out-of-sample predictability of the yield gap generally improves for the three types of returns. Furthermore, in non-nested comparisons, the yield gap proxies have greater out-of-sample predictability power than the alternative forecasting variables earnings yield, bond yield, and dividend yield. The out-of-sample forecasting power of the yield gap is economically signi cant, as indicated by the signi cant gains in the Sharpe ratios as well as positive certainty equivalents, associated with dynamic trading strategies conditional on the predictive ability of the yield gap. Thus, the yield gap can be an important state variable to be used in dynamic portfolio optimization. The remainder of this paper is organized as follows. Section 2 presents the theoretical motivation, and Section 3 describes the data and variables. Section 4 presents the results for the long-horizon regressions, and Section 5 performs the out-of-sample predictability evaluation. Section 6 evaluates the economic signi cance associated with the out-of-sample predictive role of yield gap, and nally, Section 7 concludes. 2 Theoretical framework By representing the di erence between the market earnings yield and the long-term bond yield, the yield gap conveys information about both future expected stock market and bond returns, future aggregate dividends and future aggregate earnings. In Appendix A, I derive the following dynamic accounting identity for the log yield gap, Y G, Y G t e t p t ny nt = +E t n 1 k 1 E t 1 X j=0 j [(1 )de t+1+j + e t+1+j ] X X 1 j (r t+1+j r n;t+1+j ) + E t j r t+1+j : (1) j=0 j=n where k and are parameters of linearization de ned in Appendix A; e t p t in the rst equality represents the log earnings yield associated with the stock market index; and y nt ln (1 + Y nt ) is 3

6 the log yield at time t of a zero-coupon bond with maturity n. Equation (1) says that high values of the yield gap (log earnings yield is high relative to the log bond yield) are associated with a expected combination of higher future log stock returns (r t+1+j ), lower log dividend-to-earnings payout ratios (de t+1+j ), lower log growth rates on future equity earnings (e t+1+j ), and also lower future log bond (one-period) returns (r n;t+1+j ). 1 Thus, conditional on future dividend payout ratios and future earnings growth, Y G forecasts higher expected stock market returns and lower expected bond returns. This equation will be used to interpret the predictive regressions in the next sections, and only assumes the Log Pure Expectations Hypothesis for bond prices (Campbell, Lo and Mackinlay (1997), Chapter 10), being also based on the de nition of both stock and bond returns and a terminal condition that the log earnings-to-price ratio does not growth slower or faster than the linearization parameter,. In the actual empirical implementation, I subtract ln (Y nt ) instead of the conventional log bond yield, y nt which allows the log yield gap to represent the log of the ratio of earnings yield to the bond yield, exp (et p t ) Y G t = ln = e t p t ln (Y nt ) ; Y nt with both the numerator and denominator denoting net rates of return. This procedure allows to compare the yields on the two assets, following the prevailing idea that both stocks and longterm bonds represent competing assets, and hence, should earn approximate returns in the longrun. 2 Furthermore, due to data restrictions, in the empirical analysis I use the yield on a long-term Treasury coupon bond rather than yields associated with zero-coupon bonds. The previous dynamic identity arises from a log-linearization of both stock and bond returns, and can be seen as a generalization of the simple Gordon model, by allowing for time-varying returns, dividend payout ratios and earnings growth rates. In alternative, we can motivate the predictability of the yield gap (in levels) over simple stock and bond returns by using the Gordon model. Following Campbell and Thompson (2008), in the long-run steady state equilibrium the 1 Lowercase letters represent the logs of uppercase letters. 2 If the in ation rate is not very volatile, then the earnings yield (plus a constant in ation premium) should approximate the long-term bond yield. 4

7 simple equity market return is equal to the aggregate earnings yield (in levels), R = exp (e p) ; (2) and if we subtract both sides by the return, R n, on a long-maturity bond (equal to the bond yield, Y n ), we obtain, R R n = exp (e p) Y n ; (3) which states that positive perturbations on the yield gap (in levels) translate positively on stock returns and negatively on bond returns. 3 3 Variables and data 3.1 Data Monthly data on prices, earnings and dividends associated with the Standard & Poor s (S&P) Composite Index are obtained from Robert Shiller s website. p = ln (P ) is the log of the S&P Composite Index, e = ln (E) is the log of the annual moving average of earnings, and d = ln (D) is the log annual dividend. The smoothed log earnings yield associated with the S&P Composite Index, se p, is also from Shiller s website. Return data on both the value-weighted (R vw ) and equally-weighted (R ew ) market indexes, and on the ten-year Treasury bond (R b ) are obtained from CRSP. Interest rate data, including the ten-year and one-year Treasury bond yields, the threemonth Treasury bill rate and the Moody s seasoned AAA average corporate bond yield, are all obtained from the FRED II database, available from the St. Louis FED s website. The one-month Treasury bill rate (R f;t+1 ) is obtained from Kenneth French s website. 3 The gross simple return on a n maturity zero-coupon bond is given by R n;t+1 = (1 + Y nt ) n (1 + Y n 1;t+1 ) n 1 ; and in the steady-state equilibrium we have, R n;t+1 = R n and Y nt = Y n equation reduces to R n = 1 + Y n : 1;t+1 = Y nt, and hence, the previous 5

8 3.2 Construction of variables and summary statistics The k horizon continuously compounded excess return is calculated as r t+1;t+k = r t+1 + ::: + r t+k, where r t+j = ln(r t+j ) ln(r f;t+j ) is the one-month excess log return between dates t+j 1 and t+j; R t+j is the simple gross return; and R f;t+j is the gross risk-free rate (one-month Treasury bill) at the beginning of period t+j. The log yield gap is calculated as shown in the last section, Y G t = EY t y t, with EY t e t p t representing the log earnings yield, and y t = ln(y t ) is the log of the ten-year (net) Treasury bond yield. Similarly, the smoothed log yield gap is given by SY G t = se t p t y t. The yield gap based in levels instead of logs, is calculated as Y G t = exp (e t p t ) Y t, and similarly for the smoothed yield gap, SY G t = exp (se t p t ) Y t. The other forecasting state variables known in period t, used to predict excess market returns are the relative Treasury-bill rate (RREL), the term-structure spread (T ERM), and the log market dividend yield (d p). RREL represents the di erence between the three-month Treasury-bill rate (T B3M) and a moving average of T B3M over the previous twelve months, RREL t = T B3M t P 12 j=1 T B3M t j. T ERM is the di erence between the ten-year and one-year Treasury bond yields. Table 1 reports descriptive statistics for excess returns, Y G, SY G, their corresponding components (e p, se p, ln (Y )), and the other forecasting state variables, RREL, T ERM, d p. By analyzing the correlation coe cients in Panel B, we can see that the two proxies for stock market excess returns are highly correlated among themselves, but not signi cantly correlated with excess bond returns. On the other hand, the two proxies for the yield gap are strongly correlated (0.906), and the contemporaneous correlation between both Y G and SY G with excess returns is negligible. Y G (SY G) is positively correlated with e p (se p), and both Y G and SY G are negatively correlated with the log bond yield, at similar magnitudes. The two measures of the yield gap are not signi cantly correlated with either RREL or T ERM, although they covary positively with d p. The rst order autocorrelation coe cients show that the yield gap being a di erence of two highly persistent variables, has a slightly lower autocorrelation coe cient than both e p (se p) and y, but it is still a very persistent variable. Furthermore, both Y G and SY G are also slightly less persistent than the log market dividend yield, d p. Figure 1 presents the time-series evolution of the yield gap measures and the corresponding 6

9 components. The sample is 1954:07 to 2003:12. From the picture, it looks like both Y G and SY G assumed positive values (on average) during the 1960s and 1970s, and negative values during the 1980s and 1990s, which is in part related to the long "bull" stock market in the 1990s that depressed the aggregate earnings yield. 4 Long-horizon regressions 4.1 Methodology In this section, I use Fama and French (1989) multivariate long-horizon regressions to assess the explanatory power of the log yield gap spread (Y G, SY G) over future excess returns at several horizons, in accordance to the theoretical background from Section 2. The typical speci cation used is r t+1;t+k = a k + b 0 kx t + u t+1;t+k ; (4) where r t+1;t+k is the continuously compounded excess return over k periods, and x t is a column vector of forecasting state variables known at time t. I use forecasting horizons of 1, 3, 12, 24, 36 and 48 months ahead. The compounded return r t+1;t+k is multiplied by (12=k), where k is the forecasting horizon, in order for the slope coe cients, b k to measure the annualized e ect of the state variable on future returns. For each regression, I conduct statistical inference based on Newey and West (1987) asymptotic t-statistics, and also on two di erent Bootstrap experiments. The Newey-West standard errors are calculated using four lags. The bootstrap allow us to obtain an empirical distribution that better approximates the nite sample distribution for the coe cient estimates in the above regression. Following Goyal and Santa-Clara (2003) and Santa-Clara and Valkanov (2003), I simulate the Newey-West t-statistics instead of the regression coe cients. 4 I bootstrap the original regression residuals 10,000 times, and in each simulation the dependent variable (compounded excess returns) is drawn by imposing the null of no predictability of returns, i.e. the slope coe cients in the 4 The bootstrapped p-values associated with the regression coe cients are very similar to those associated with the t-statistics. 7

10 predictive regression are constrained to zero, r m t+1;t+k = a k + u m t+1;t+k; m = 1; :::; 10; 000; where r m t+1;t+k is the simulated excess return for the mth simulation, um t+1;t+k stands for the bootstrapped residuals, and a k is the sample estimate of the intercept in the original regression (4). In the rst bootstrap experiment (type I), for each simulation the regression (4) is run with the simulated returns, r m t+1;t+k = a m k + b m0 k x t + v m t+1;t+k; m = 1; :::; 10; 000; and then one obtains the associated Newey-West t-statistics. In the second bootstrap experiment (type II), the forecasting variables are also simulated independently from the residuals, fx m t g m=1, and in each simulation the regression performed is given by r m t+1;t+k = a m k + b m0 k x m t + v m t+1;t+k; m = 1; :::; 10; 000: By simulating both the residuals and forecasting state variables in an independent way, one might control for a spurious relation between the state variables and future returns. By iterating this process forward, an empirical distribution of Newey-West t-statistics is generated for each experiment, ft m g m=1 (as opposed to the asymptotic theoretical distribution), which is then compared to the Newey-West t-statistic obtained from the real data. Finally, one computes the proportion of empirical t-statistics greater (in absolute value) than the original t-statistic to obtain the corresponding p-value associated with the null hypothesis of no predictability of returns. 5 5 I conduct two-sided bootstrap p-values instead of one-sided p-values. We could argue for the use of one-sided tests since in the predictability literature the sign of the forecasting variable over returns is usually robust and has a theoretical motivation: d p and T ERM predict positive stock market excess returns, while short term interest rates (RREL) tend to forecast negative excess market returns. Hence, the two-sided p-value should be more conservative. The p-value is calculated as, [# ft P boot = m tg + # ft m tg] =10000; if t 0 [# ft m tg + # ft m tg] =10000; if t < 0 ; where # ft m tg denotes the number of bootstrapped t-statistics that are higher than the sample t-statistic. 8

11 4.2 Predicting stock market returns I conduct long horizon regressions for the value-weighted market return, with Y G as the sole forecasting variable (results displayed in Table 2, Panel A). The slope estimates for Y G are positive at all horizons, and both the Newey-West t-statistics and bootstrap p-values (types I and II) indicate strong statistical signi cance (at 1%), although for very long horizons (k = 48) Y G is signi cant only at the 5% level. Thus, it seems that on a preliminary analysis, the yield gap forecasts positive equity excess returns as predicted by the dynamic identity (1). On the other hand, the forecasting power of the yield gap on excess returns declines gradually with the forecasting horizon: Y G has a coe cient estimate of at the one-month horizon compared to just at the four-year horizon. The R 2 achieves the maximum value at the twelve-month horizon regression (6.19%), declining thereafter. These estimates show that the forecasting power of the yield gap is greater and with greater statistical signi cance on the near horizons, being less relevant for forecasting more distant ahead returns, although still statistically signi cant. In Panels B and C, I add three forecasting state variables usually used in the return predictability literature: The term structure spread (T ERM, Campbell (1987), Fama and French (1989)), the relative Treasury bill rate (RREL, Campbell (1991), Hodrick (1992)), and the log market dividend yield (d p, Fama and French (1988, 1989)). Panel C includes those three additional variables, and Panel B includes only RREL and T ERM along with Y G. It is important to control for these variables since Y G is correlated with them. In particular, both T ERM and Y G depend on the ten-year bond yield, and on the other hand, both the log dividend yield and log earnings yield are strongly correlated. Hence, one needs to check if the predictive power of Y G is maintained after the inclusion of those competing forecasting variables. Thus, the benchmark regression throughout the section is given by, r t+1;t+k = a k + b k1 RREL t + b k2 T ERM t + b k3 (d t p t ) + b k4 Y G t + u t+1;t+k : (5) The results in Panel B show that when one controls for both RREL and T ERM the magnitude of the yield gap slope estimates (and the associated t-statistics) actually increases relative to the single variable forecasting regression, especially on the short-term horizons (until one-year): At 9

12 k = 1, the Y G coe cient is 0.202, and at k = 48, the corresponding estimate is In addition, the Y G slope estimates are strongly signi cant (1% level) for all horizons, as indicated by the Newey-West t-statistics and p-values from the two bootstrap experiments. The pattern of coe cients 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 market dividend yield (d p), the Y G coe cient estimates decline in magnitude relative to Panel B. Nevertheless, Y G is still signi cant at the 1% level for all horizons (with the sole exception of k = 24, where the slope is signi cant at the 5% level). The log dividend yield being a very persistent variable, has a forecasting power over returns that improves with horizon, due in part to the mean reversion of stock prices (the denominator) on the medium and long-term horizons. Hence, the higher persistence of d p makes it a better forecaster for long-horizon returns relative to short-horizon returns. Therefore, for the near horizons (one to twelve months) the slopes associated with Y G are greater than those associated with d p, and we obtain the reverse relation afterwards. Nevertheless, it is remarkable that Y G signi cantly forecasts returns at long horizons, given the fact that d p is correlated with the earnings yield (and hence Y G), and thus, the predictive power of the two variables should be overlapped. In Table 3, I replicate the long-horizon regressions for the equally-weighted excess market return (r ew ) as the variable to be forecasted. The motivation for using r ew is that the predictability over r vw might be associated with a restricted number of individual large stocks, and thus, might not provide a good picture of the return predictability in the whole market. Hence, we might be interested in evaluating return predictability for the average stock in the market. The results for the univariate case in Panel A, show that compared to the corresponding regression for the valueweighted market index (r vw ), the forecasting power of Y G is greater at all horizons: At k = 1 the slope estimate is 0.214, and at k = 48 we have an estimate of The t-statistics and bootstrapped p-values indicate statistical signi cance at the 1% level for all forecasting horizons. The R 2 s are also greater than the corresponding values in the regressions for r vw, at all horizons: At k = 1 the R 2 is 1.63% (1.36% for r vw ), and at k = 48 we obtain an estimate of 12.54% compared to 4.01% for r vw. Similarly to the case of value-weighted returns, the forecasting ability of Y G (as measured by its slope) is stronger for short horizons, with the coe cient estimates declining 10

13 monotonically with horizon. Nevertheless, and in contrast to the case of r vw, the coe cients of determination rise monotonically with the horizon. The results in Panels B and C, show that by adding the control variables Y G remains strongly signi cant (1% level) at all forecasting horizons, while the log dividend-to-price ratio (d p) has no predictive power over the equally-weighted excess market return. These results showing that the yield gap has greater forecasting power for the equally-weighted relative to the value-weighted excess market return, combined to the fact that the equally weighted index is more tilted toward small capitalization stocks relative to large caps, suggest that Y G has greater forecasting power for the returns of small caps in comparison to large caps. 4.3 Predicting bond returns Equation (1) above suggests that the current values of the yield gap are negatively correlated with expected future excess bond returns. I investigate this hypothesis by running long-horizon regressions with the ten-year Treasury bond excess return as the variable to be forecasted, whose results are presented in Table 4. The results for the univariate case reported in Panel A con rm that the yield gap is negatively correlated with future excess bond returns at all forecasting horizons. The coe cient estimates exhibit a hump-shaped pattern, with the magnitudes peaking at k = 12, and then declining gradually. The forecasting power, as measured by the coe cient magnitudes associated with Y G, is lower when compared to the regressions for the value-weighted market return (r vw ) for horizons until one-year, nevertheless being greater for longer horizons (k = 24; 36; 48). In terms of statistical signi cance the coe cients are very signi cant (at the 1% signi cance level) for all horizons, with the exception of the one-month horizon, where the slope is signi cant at the 5% level. The R 2 s increase with the horizon and are greater than the corresponding values in the regressions for the value-weighted market return, with the exception of the near horizons, k = 1; 3. Panels B and C present the results for the regressions including the control forecasting variables. In Panel C, d p is signi cant for the longer horizons (k = 36; 48), which con rms the ndings in Fama and French (1989) that the dividend yield helps to predict long-horizon returns on both stocks and bonds. On the other hand, the e ect of the term structure spread (T ERM) on future bond returns is not signi cant in both Panels B and C, a fact that should be related to the presence 11

14 of Y G, which overlaps the forecasting role of T ERM since the two variables are correlated. After accounting for all the control variables, the Y G coe cients generally maintain the statistically signi cance and even have increased magnitudes (as it is the case in Panel C) at all horizons, in comparison to the estimates in Panel A. Overall these results suggest that conditional on other forecasting variables, Y G is negatively correlated with future excess bond returns, as suggested by the dynamic identity (1). 4.4 Robustness checks As a robustness check, I conduct long-horizon regressions using the alternative yield gap measure based on the log smoothed earnings yield, SY G, whose results are presented in Tables A.1 (r vw ), A.2 (r ew ), and A.3 (r b ). In the case of r vw, the slopes associated with SY G in the univariate regression have greater magnitudes than the corresponding estimates for Y G, while in the "reduced" multiple regression (including RREL and T ERM), we obtain a reversed relation, although the estimates for SY G are signi cant at the 1% level. On the other hand, in the benchmark multiple regression (including d p), SY G is not signi cant at forecasting long-horizon returns (beyond one year). In the case of r ew, the SY G slope estimates are statistically signi cant at 1% for most horizons, both at the univariate and multivariate speci cations, despite the magnitudes being slightly lower than in the Y G case. Regarding r b, the SY G coe cients have somehow lower magnitudes in comparison to Y G in both the univariate and "reduced" multiple regressions and greater magnitudes in the case of the benchmark regression. The slope estimates are signi cant in most cases, the exceptions being at the one-month horizon (Panels A and B). 5 Out-of-sample predictability 5.1 Unconstrained out-of-sample regressions The results in the last section show that the yield gap forecasts (in-sample, IS) the excess returns on both the market index (value and equal-weighted) and the long-maturity bond. In this section, and following the work of Bossaerts and Hillion (1999), Lettau and Ludvigson (2001), Goyal and Welch (2008), Campbell and Thompson (2008), among others, I investigate the out-of-sample (OS) 12

15 predictability associated with both the yield gap and its components over both stock market and bond returns. The OS regressions can be seen as complimentary to the IS regressions and try to evaluate the parameter instability in those regressions. To be consistent with Goyal and Welch (2008) and Campbell and Thompson (2008), the focus of the analysis is on the one-period ahead OS predictability, and later on some robustness checks associated with other forecasting horizons are provided. In order to assess the OS predictability of the yield gap and other competing state variables, the rst null (or restricted) model considered is the constant model, that is a regression containing just a constant in which the best forecast of future excess returns is the corresponding historical average (Lettau and Ludvigson (2001), Goyal and Welch (2008), Campbell and Thompson (2008)), H 0 : r t+1 = a + u t+1 ; H a : r t+1 = a + b 0 x t + u t+1 ; where H a corresponds to the alternative or unrestricted model, and x t represents a vector of forecasting state variables having a slope vector, b. The second null model is the AR(1) speci cation (as in Lettau and Ludvigson (2001)), H 0 : r t+1 = a + br t + u t+1 ; H a : r t+1 = a + br t + c 0 x t + u t+1 ; where b is the autoregressive coe cient, and c denotes the vector of slopes associated with the forecasting variables. The rst major measure of OS performance analyzed is the OS coe cient of determination, calculated as R 2 OS = 1 MSE U MSE R ; (6) with MSE U = 1 T OS X TOS t=1 bu2 Ut denoting the mean-squared (forecasting) error associated with the unrestricted model, and MSE R representing the same for the restricted model. T OS represents the number of observations on the evaluation (or out-of-sample) period. The OS R 2 is positive 13

16 whenever MSE U < MSE R, i.e., the forecasting squared errors associated with the unrestricted model have lower magnitude than those corresponding to the restricted model. The second OS evaluation measure is the F test from McCracken (1999, 2007), MSE R MSE U MSE-F = T OS ; (7) MSE U which tests the null hypothesis that the MSE associated with the restricted model is less than or equal to the corresponding value for the unrestricted model. The alternative hypothesis is that the M SE associated with the unrestricted model is lower in comparison to the restricted model. The third OS test statistic is the one proposed by Clark and McCracken (2001), ENC-NEW = X TOS t=1 (bu2 Rt bu Rt bu Ut ) MSE U ; (8) in which the null hypothesis is that the restricted model encompasses the unrestricted model, that is, the unrestricted model can not improve the forecast associated with the restricted model. The alternative hypothesis is that the unrestricted model has additional information that can improve the forecast obtained from the restricted model. In the cases of both the MSE-F and ENC-NEW tests, the statistical inference is based on asymptotic critical values obtained from McCracken (2007) and Clark and McCracken (2001), respectively. In addition, I compute bootstrapped critical values for both M SE-F and EN C- NEW, and also for ROS 2. The bootstrap experiments are similar to those performed in Kilian (1999), Lettau and Ludvigson (2001), and Goyal and Welch (2008), with the data generating process being associated with the null or restricted model (constant or AR(1) models). The OS test statistics associated with forecasting the value-weighted excess market return are provide in Table 5. Panels A and C show the results for log returns, and Panels B and D report the results associated with simple returns. In Panels A and B, the null model is the constant model, and in Panels C and D, the restricted model is the AR(1) speci cation. The forecasting variables are the log yield gap (Y G), log smoothed yield gap (SY G), yield gap in levels (Y G ), smoothed yield gap (SY G ), and the corresponding components, e p, se p, ln (Y ), exp (e p), exp (se p), and Y. The columns labeled (e p; ln (Y )), (se p; ln (Y )), (exp (e p) ; Y ), (exp (se p) ; Y ) 14

17 correspond to "unrestricted" versions of the yield gap, where the slopes of both the earnings yield and bond yield are unrestricted instead of being xed at 1 and -1, respectively. For comparison purposes, the log dividend yield (d p) and simple dividend yield (exp (d p)) are also employed as forecasting variables. The initial estimation period is 10 years (120 observations). In general, the models including the forecasting variables perform worst than the constant model, with negative estimates for both R 2 OS and MSE-F. The sole exception is the model containing Y G, in which case we have positive estimates for both ROS 2 (0.23%) and MSE-F (1.112), although they are not statistically signi cant at the 5% level. When the null model is AR(1), we have somehow di erent results. Both Y G and SY G (in forecasting log returns) and Y G (in forecasting simple returns) have positive and statistically signi cant (at 5%) estimates of ROS 2. Moreover, we reject the null (at 5%) that the MSE from the the restricted and unrestricted models are equal in the case of Y G (both asymptotic and bootstrapped critical values), and the same for both SY G and Y G (bootstrapped critical values). On the other hand, all the components of yield gap and the dividend yield have negative estimates for both R 2 OS and MSE-F. The models with the unrestricted versions of the yield gap are rejected against both the constant and AR models. Hence, the restrictions that the slopes associated with earnings yield and bond yield are xed at 1 and -1, respectively, are important to drive the OS predictability for value-weighted market returns. Regarding the EN C-N EW statistic, in general, one tends to reject the null that the restricted model (both constant and AR) encompasses the unrestricted model. The exceptions are for e p, se p, exp (e p), and exp (se p), in which cases we accept the null that each of these variables do not add forecasting power over the null model. Thus, in this application the ENC-NEW statistic tends to reject the null much more often than the MSE-F statistic. The OS results associated with the equal-weighted excess market return are provided in Table 6. In contrast to the results associated with the value-weighted market return, all the four measures of the yield gap (Y G, SY G, Y G, SY G ) exhibit statistically signi cant positive estimates of ROS 2 when the restricted model is the historical average. Moreover, we reject the null that the M SE associated with the restricted model is less or equal than that of the unrestricted model. When the null model is the AR speci cation, we have slightly higher estimates of R 2 OS compared to the constant model case for the four measures of the yield gap, and the M SE-F statistic clearly rejects 15

18 the null in all four cases. On the other hand, for all remaining forecasting variables (including the unrestricted versions of the yield gap) we have negative estimates of R 2 OS and the MSE-F statistic does not reject the null for the two types of restricted models (both constant and AR(1)). The EN C-N EW statistic typically does not reject the null when the forecasting variables are e p, se p, exp (e p), exp (se p), Y, and ln (Y ), and rejects the null for all the remaining speci cations, and speci cally for the yield gap proxies. The OS results when the forecasting variable is the excess return on the long-term bond are reported in Table 7. We can see that both Y G (0.52%) and Y G (0.71%) have signi cant positive estimates of R 2 OS when the comparison is made against the historical average. In addition, the MSE-F statistic rejects the null for both Y G and Y G. In contrast, both SY G (-0.03%) and SY G (0.06%) have negligible values for ROS 2, and the null is not rejected by the MSE-F statistic. When the restricted model is AR, we obtain similar results, with both Y G and Y G having signi cant positive estimates for ROS 2, and the null hypothesis associated with the MSE-F test being rejected. On the other hand, both SY G and SY G have nearly zero estimates of ROS 2, and the MSE-F statistic does not reject the null. Similarly to the cases of value and equal-weighted market returns, the components of the yield gap, the unrestricted measures of the yield gap, and the dividend yield all have negative estimates associated with both R 2 OS and the MSE-F statistic. In contrast to the results in Tables 5 and 6, the ENC-NEW statistic does not reject the null for all forecasting variables, with the sole exception of Y G. The results presented in this subsection show that the yield gap has some OS predictability over r vw, r ew and r b, when compared against both the historical average and the AR(1) model. This evidence is generally stronger when the comparison is made relative to the AR(1) model than the constant model. The OS predictability is also stronger in the case of r ew than for either r vw or r b, which goes in line with the IS predictability evidence from the previous section. 5.2 Constrained out-of-sample regressions As pointed out by Campbell and Thompson (2008), the OS predictability evaluation can su er from some distortions. Thus, the estimation of the predictive regressions over a short sample can lead to perverse slope estimates, that is, the model has low power (for example, we might obtain 16

19 negative slope estimates for Y G in predicting the equity premium, when the theory calls for a positive slope). As a result, this potentially leads to negative forecasts of the equity premium, which are not con rmed ex-post, and in consequence a downward bias in the OS performance. To overcome this issue, I impose the prior restriction that the forecasted market equity premium (and also forecasted excess bond returns) can not be negative, that is, a investor would rule out a model that forecasts negative risk premia. Table 8 reports estimates for ROS 2 when the restriction of a non-negative forecasted excess return is imposed, i.e., whenever the unrestricted model forecasts a negative excess return, this estimate is truncated to zero. Panels A and B show the results for the value-weighted index; Panels C and D are associated with the equal-weighted index; and Panels E and F report the estimates for the long-term bond. In Panels A, C and E, the forecasted variable are log returns, and Panels B, D and F report the results for simple returns. In each panel, the unrestricted model is compared against both the constant and AR(1) models. The results for the value-weighted market return show that most yield gap proxies produce positive and signi cant R 2 OS estimates (both in forecasting log and simple returns) when the comparison is done against the constant model. Only in the case of (exp (e p) ; Y ) (0.11%) there is no statistical signi cance at the 5% level. In the case where the null model is AR the results are better than in the constant model case, with both restricted and free measures of the yield gap producing signi cant positive estimates for ROS 2. In particular, in the comparison against AR(1), most of the yield gap proxies have estimates above 1%, the exceptions being (e p; ln (Y )) and (exp (e p) ; Y ). The R 2 OS estimates for the equal-weighted market return also compare favorably to the unrestricted case, being signi cant for all the eight yield gap proxies when the comparison is done against both the constant and autoregressive models. Most of the estimates are above 1%, and in particular, SY G and (se p; ln (Y )) produce values above 2%. In the speci cations containing the earnings yield, the estimates for R 2 OS associated with the constrained regressions are still negative. On the other hand, when the forecasting variable is ln (Y ) we have signi cant positive R 2 OS estimates for both r vw and r ew. In the case of Y, the estimates for R 2 OS are only signi cant in forecasting the value-weighted market return. Overall, these results con rm the ndings in Campbell and Thompson (2008) that by imposing the constraint of positive forecasted equity 17

20 risk premium, the OS performance of the forecasting variables tends to improve. In the case of bond returns, only the restricted yield gap measures, Y G, SY G, Y G, SY G, generate positive estimates for ROS 2, and among those, only for Y G and Y G these estimates are statistically signi cant. These results are not very di erent from those associated with the unrestricted regression in Table 7, which suggests that the constraint of positive excess returns is not very important for driving the OS predictability of long-term bond returns. To o er a better sense about the evolution over time concerning the OS forecasting performance associated with the yield gap, I plot the cumulative SSE (sum squared error) di erence as in Goyal and Welch (2008). For the OS regressions the performance measure is the cumulative di erence between the sum of squared forecasting errors associated with the restricted (constant or AR(1)) and the unrestricted models, tx bu 2 Rs bu 2 Us; t = 1; :::; T OS : s=1 In the case of the IS regressions the performance measure is the cumulative di erence between the sum of squared residuals associated with the restricted and the unrestricted models, more speci cally, tx bu 2 Rs bu 2 Us; t = 1; :::; T OS ; s=1 bu R;s+1 = r s+1 ba; bu U;s+1 = r s+1 ba b bxs ; and tx bu 2 Rs bu 2 Us; t = 1; :::; T OS ; s=1 bu R;s+1 = r s+1 ba b brs ; bu U;s+1 = r s+1 ba b brs bcx s ; for the constant and AR(1) models, respectively, with the intercepts and slopes being estimated from the IS regressions. Given the results above, the cumulative SSE di erence associated with 18

21 the constrained OS regressions are also presented. Figures 2, 3 and 4 present the cumulative SSE di erences associated with the yield gap in forecasting value-weighted market returns, equal-weighted market returns and bond returns, respectively, with the comparison being made against the constant model. When the plotted line has a positive slope it means that the unrestricted model outperforms the restricted model (in terms of squared forecasting errors), while a negative slope indicates that the restricted model has a better performance. In the case of the value-weighted market index, the unrestricted model (containing some proxy for the yield gap) outperforms until the mid 1980s, from mid 1980s to late 1990s we have a period where the historical average has greater forecasting power, and in the early 2000s, the unrestricted model outperforms again. This pattern is robust across the three forecasting regressions in-sample (IS), out-of-sample (OS), and constrained out-of-sample (OSR). It must be pointed out that the cumulative SSE di erence associated with the OS regression becomes negative only in late 1990s, coming back to positive values in the 2000s, when the yield gap measures are Y G and Y G. In the case of the smoothed yield gap, SY G and SY G, the recovery in the 2000s is not enough to bring back the cumulative di erence to positive values. The underperformance in the 1990s is related to the very low values of the yield gap (and all the market valuation ratios, in general) in those years, which forecasted low returns that were not con rmed ex post (Campbell and Shiller (1998), Campbell and Thompson (2008)). The cumulative SSE di erence associated with the constrained OS regression is positive for most of the period in analysis, and speci cally in the 1990s, where the constraint of positive forecasted equity premium is binded, thus con rming the results in Table 8 above. The OS performance of the yield gap is signi cantly greater into forecasting the equally-weighted market return (in comparison to the value-weighted return) with the cumulative SSE di erence being positive for most of the period, apart from the beginning of the sample coincident with the 1960s. In the cases of Y G and Y G, the lines labeled OS and OSR are almost coincident, which means that the constraint of positive equity premium does not bind most of the time, in contrast to the value-weighted return case. Regarding the Treasury bond return, we can see that the forecasting performance has improved over the sample. When the forecasting variables are Y G and Y G, all three regressions produce 19

22 high positive estimates for the cumulative SSE di erence in the 1990s and early 2000s. The yield gap proxies based on the smoothed earnings yield, SY G and SY G have lower performance in forecasting bond returns, with the cumulative SSE di erence associated with the OS regressions being negative for most of the sample, although achieving positive values in the 2000s, in the case of SY G. Nevertheless, the constrained regression yields a positive performance in the second half of the sample, meaning that the constraint is binded in that period for bond returns. Figures 5, 6 and 7 present the cumulative SSE di erence in the case where the comparison is conducted against the AR(1) model. For the market returns the performance is better than in the constant model case, which con rms the results from Tables 5, 6, and 8. In the case of the bond return, the performance is very similar to that displayed in Figure Non-nested comparisons The test statistics used in the previous subsections are associated with nested models, i.e., the null (or restricted) model represents a special case of the alternative (unrestricted) speci cation. However, one might be interested in non-nested comparisons, that is in comparing the OS performance of two alternative variables, for example, a given yield gap proxy and each of its components. In the forthcoming analysis, the alternative model will be some measure of the yield gap, and the null model will be some of its components. For example, in the comparison between Y G and e p, the competing models are, H 0 : r t+1 = a + b (e t p t ) + u t+1 ; H a : r t+1 = c + dy G t + u t+1 : The rst statistic employed to compare non-nested models is the OS coe cient of determination, R 2 OSC = 1 MSE A MSE N ; (9) where MSE A and MSE N denote the mean-squared error associated with the alternative and null models, respectively. This coe cient of determination is constructed similarly to the corresponding measure for the nested case, the only di erence being that the null model is not a special case of 20

23 the alternative model. The second statistic is the encompassing test statistic proposed by Harvey, Leybourne and Newbold (1998), ENC-T = r TOS 1 T OS c 1 T OS X TOS t=1 (bu2 Nt bu Nt bu At ) r t (TOS XTOS 1); (10) t=1 (bu2 Nt bu Nt bu At ) 2 c 2 XT OS t=1 bu 2 Nt bu Nt bu At which is distributed as a t-student, thus it is valid for nite samples. This test is about the covariance between bu Nt and bu Nt bu At. The null hypothesis is that the null model (the model without the yield gap proxy) encompasses the preferred model containing the yield gap measure. The alternative hypothesis is that the preferred model contains information that can improve the forecast of the null model. The statistics associated with the comparison across non-nested models are presented in Table 9. The results for the value-weighted index are displayed in Panels A and B; Panels C and D are associated with the equal-weighted index; and Panels E and F show the estimates for the long-term bond. In Panels A, C and E, the variable to be forecasted are log returns, and Panels B, D and F report the results for simple returns. The columns labeled ENC-T and p-value show the value and associated p-value for the ENC-T statistic. We can see that the estimates for R 2 OSC are all positive when we compare the yield gap proxies against the alternative forecasting variables. This result is robust for both value and equal-weighted market returns and bond returns, and also for both log and simple returns. In many cases, the R 2 OSC estimates are above 1%, especially when one forecasts equally-weighted market returns. Furthermore, the EN C-T statistic assumes positive values in all the comparisons, and the null is rejected at 5% in all cases except for the comparison between SY G and ln (Y ) for log bond returns (in which case the null is rejected at 10%). In summary, the results in this subsection provide further evidence that the yield gap proxies have greater OS predictability power than the alternative forecasting variables earnings yield, bond yield, and dividend yield. 21