- Email: [email protected]
The predictive power of the yield curve: a theoretical assessment
ARTICLE IN PRESS
Journal of Monetary Economics 50 (2003) 1603–1622
The predictive power of the yield curve: a theoretical assessment$ Christel Rendu de Linta,*, David Stolinb b
a Pictet, PAM Fixed Income, 8 bd de la Tour, 1205 Geneva, Switzerland Toulouse Business School, 20 Boulevard Lascrosses, 31000 Toulouse, France
Accepted 7 August 2003
Abstract Although the empirical evidence about the leading indicator property of the term spread (LIPTS) is powerful, this property lacks a rigorous theoretical foundation. This paper investigates whether dynamic equilibrium asset pricing models are able to provide a theoretical underpinning for the LIPTS. We study an endowment and a production economy. The endowment economy is unable to account for the LIPTS. On the other hand, a model with endogenous production provides a reasonable theoretical justification for the LIPTS. r 2003 Elsevier B.V. All rights reserved. JEL classification: C68; E32; E43; E44 Keywords: General equilibrium; Leading indicators; Term structure of interest rates; Yield curve
1. Introduction The yield curve has long been monitored for the information it contains about future economic activity. Looking at Fig. 1, it is striking how closely the current term $ The views expressed in this paper are those of the authors and do not necessarily represent those of Pictet & Cie. This paper is based on a chapter of Christel Rendu de Lint’s dissertation at the London Business School. She is indebted to her supervisor Albert Marcet for invaluable comments and support. The paper also benefited from helpful comments by Jean-Pascal Benassy, Fabio Canova, Alexei Jiltsov, Brian Henry, Morten Ravn, Andrew Scott, Oren Sussman, participants at the 1999 SED meeting in Alghero and at the economics seminar at the Bank of Italy. Lastly we thank an anonymous referee. Remaining errors are ours. *Corresponding author. Pictet, PAM Fixed Income, 8 bd de la Tour, 1205 Geneva, Switzerland. E-mail address: [email protected] (C. Rendu de Lint).
0304-3932/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jmoneco.2003.08.007
ARTICLE IN PRESS 1604
C. Rendu de Lint, D. Stolin / Journal of Monetary Economics 50 (2003) 1603–1622
Panel A: Output 2
10 8
1 6 0
4 2
-1
0 -2 -2 -4
-3 65
70
75
80
85
90
Real GDP 1-year future g. rate
8
95
1-year spread
Panel B: Consumption
2
6
1
4
0
2
-1
0
-2
-2
-3 65
70
75
80
Real cons 1-year future g rate
85
90
95
1-year spread
Fig. 1. Predictive power of the term spread for future output and consumption growth rates. GDP and consumption growth rates are shown on the left scale, the term spread is on the right scale. All values are annualized and shown in %: The 1 year spread is defined as spreadt4 ¼ r4t r1t ; and 1 year consumption and output growth rates as c4t ¼ lnðctþ4 =ct =4Þ 400 and y4t ¼ lnðytþ4 =yt =4Þ 400; respectively.
spread (the difference between the long and short interest rates, i.e. a measure of the slope of the yield curve) tracks movements in the future output and consumption growth rates. Since in both graphs at each point in time t; the spread is known but future growth rates are not, the term spread appears to contain predictive information about future economic activity. Based on this observation, numerous
ARTICLE IN PRESS C. Rendu de Lint, D. Stolin / Journal of Monetary Economics 50 (2003) 1603–1622
1605
empirical papers studied and confirmed the leading indicator property of the term spread (henceforth LIPTS), showing that higher spreads precede periods of strong economic activity, while lower spreads announce periods of weak activity. In light of the strength of these results, the literature does not hesitate to characterize the spread’s predictive ability as ‘‘impressive’’, ‘‘robust’’ and ‘‘particularly well established’’.1 Yet, in contrast to this powerful empirical evidence stands the absence of rigorous theoretical justification for the LIPTS. Our paper is intended to bridge this gap. In an attempt to provide a theoretical underpinning for the empirically established LIPTS, this paper investigates whether dynamic equilibrium asset pricing models can explain the LIPTS. A stochastic endowment economy and a stochastic production economy are studied. Dynamic equilibrium asset pricing models appear to be a natural choice for explaining the joint behaviour of real macroeconomic variables (here consumption and output) and asset prices. Indeed, in dynamic equilibrium asset pricing models, asset prices are linked to the marginal utility of the agent, which itself depends on consumption. Under some conditions the endowment economy, in particular, implies a positive correlation between the m-period future consumption growth rate and the m-period spot interest rate.2 This property, which derives from the first-order condition for the m-period bond price, makes the endowment economy a seemingly attractive candidate for explaining the LIPTS, leading some authors to conclude that it is consistent with the LIPTS.3 Our paper questions this claim. The central findings of this paper are as follows. First, we show that, perhaps counter-intuitively, the endowment economy does not give rise to the LIPTS. Indeed, at time t an increase in the m-period spread is associated with a negative consumption growth rate over the same period. This is because in a simple endowment economy, where the endowment process has only first-order dynamics, the impact of a shock on consumption is the largest in the period it occurs and declines monotonically thereafter. As a consequence, short-term interest rates always respond more to a given shock than do long-term interest rates. Thus, after a positive endowment shock, the spread actually rises (since the short-term rate falls more than the long-term one), at odds with the expected decline in future consumption. To overcome this problem, we extend the endowment economy by introducing endogenous production. We show that term spreads generated in this economy become good predictors of future output growth, and of future consumption growth at long horizons. As opposed to the endowment model, in a production economy the 1 These are quotes, respectively, from Estrella and Hardouvelis (1991, p. 559); Kosicki (1997, p. 39), and Hamilton and Kim (2000, p. 3). See Section 2 for a survey of the empirical evidence. The predictive power of the term spread is also recognized by practioners: the Wall Street Journal and the New York Times, for instance, regularly print the yield curve as a leading indicator of the business cycle. Fortune published an article titled ‘‘A near-perfect tool for economic forecasting’’ (Clark, 1996), with the tool in question being the term spread. 2 This was noted by Harvey (1988), Donaldson et al. (1990), Estrella and Hardouvelis (1991), Plosser and Rouwenhorst (1994) and Chapman (1997). 3 Smets and Tsatsaronis (1997, p. 1), Dotsey (1998, p. 38).
ARTICLE IN PRESS 1606
C. Rendu de Lint, D. Stolin / Journal of Monetary Economics 50 (2003) 1603–1622
dynamics of output and consumption can be quite different from each other, and this turns out to be crucial in explaining the LIPTS. In particular, in this economy the impulse response function for consumption only reaches its peak a few periods after a shock (and not in the first period as in the endowment economy), due to the fact that agents try to smooth out the shock’s impact by substituting consumption intertemporally. It is precisely this ability of agents to freely substitute consumption over time which is at the root of the success of the production economy in explaining the LIPTS. Existing literature does not supply a formal theoretical model successfully justifying the LIPTS. Rouwenhorst (1995) tests for the LIPTS in a standard neoclassical growth model, modified to incorporate the endogenous choice of labour and leisure. He finds that the term spread does not forecast future real economic activity. Most papers documenting empirical evidence mention the possible role of monetary policy in explaining the LIPTS, but without formalising the argument.4 It is worth noting that our paper does not seek to explain the term structure of interest rates or to model term premia. A vast literature is already available on these issues (see e.g. Cox et al., 1985; Campbell, 1986; Backus et al., 1989; Donaldson et al., 1990; Longstaff and Schwartz, 1992; Labadie, 1994; den Haan, 1995). Nor does it study the co-movements of consumption growth and the real interest rate, as does Chapman (1997). Rather, it focuses on the predictive power of the term spread for future consumption and output growth. Donaldson et al. (1990), Labadie (1994) and Chapman (1997) all comment on how the term spread changes over the business cycle, but none studies the LIPTS in detail. The remainder of this paper is organized as follows. The next section reviews the empirical motivation for the paper. Section 3 presents results for the standard stochastic endowment economy, while Section 4 does so for the production economy. Section 5 takes a critical look at our modelling approach and discusses possible extensions. Section 6 concludes.
2. Empirical facts In this section we present empirical evidence on the LIPTS. This evidence is used to motivate the theoretical model-building of the paper and to benchmark the models’ predictions. Before turning to the various metrics existing in the empirical literature to measure the LIPTS, we present some simple evidence about the predictive power of the yield curve. Table 1 shows correlations between various term structure variables and activity growth rates calculated for the United States.5 Quantifying what can also be 4
Harvey’s (1988) derivation of the relationship between spreads and consumption growth follows Hansen and Singleton (1983) by assuming joint lognormality of consumption and returns. However, Hansen and Singleton point out that ’’by assuming that the joint distribution of consumption and returns is lognormal, we are implicitly imposing restrictions on the production technology’’ (p. 252). 5 Consumption and output data are standardized volume indices obtained for 1960:1–1998:4 from OECD Quarterly National Accounts (‘‘US final private consumption expenditure’’ and ‘‘US gross
ARTICLE IN PRESS C. Rendu de Lint, D. Stolin / Journal of Monetary Economics 50 (2003) 1603–1622
1607
Table 1 Correlations between real activity growth rates and term structure variables Real consumption growth rate Panel A: Nominal term spreads 6-month 1-year 2-year Panel B: Real term spreads 6-month 1-year 2-year Panel C: Expected inflation differential 6-month 1-year 2-year Panel D: Nominal yield 6-month 1-year 2-year Panel E: Real yield 6-month 1-year 2-year
Real output growth rate
0.36 0.46 0.46
0.34 0.52 0.54
0.29 0.39 0.36
0.04 0.15 0.17
0.06 0.07 0.10
0.13 0.14 0.05
0.40 0.37 0.00
0.44 0.46 0.24
0.22 0.31 0.44
0.02 0.04 0.09
Correlations of term structure variables with consumption and output growth rates calculated over the same period. US data for 1960–98 were used.
seen from Fig. 1, Panel A of Table 1 shows that nominal term spreads (of 2- to 8quarter durations) have exhibited a high and positive correlation with future consumption and output growth rates. In other words, nominal term spreads are good indicators of future economic activity. Based on such straightforward but striking evidence, the empirical literature has taken two main approaches to examining the LIPTS.6 The first approach relies directly on these observed correlations and proposes an OLS regression of the future consumption or output m m growth rate (cm t ; yt ) on the term spread (spreadt ): m cm t ¼ b0 þ b1 spreadt þ et ;
ð1Þ
(footnote continued) domestic product’’, respectively). US consumer price index for 1960:1–1998:4 is from IMF’s International Financial Statistics. 3-month, 6-month, 1-year, 2-year and 5-year interest rates come from Federal Reserve Bank of St. Louis’ Federal Reserve Economic Data. 3-month, 6-month and 1-year rates are obtained for 1960:1–1998:4 as auction averages on corresponding maturity T-bills. 2- and 5-year rates are constant maturity Treasury rates, obtained for 1976:1–1998:4 and 1960:1–1998:4, respectively. 6 A third approach, which we will not discuss here, consists of using vector autoregression, as in Canova and De Nicolo (1997), Estrella and Mishkin (1997) and Smets and Tsatsaronis (1997). These papers broaden the research question, but all confirm the LIPTS. However, in our setup, using a VAR would be meaningless since the models studied only allow for a single shock.
ARTICLE IN PRESS C. Rendu de Lint, D. Stolin / Journal of Monetary Economics 50 (2003) 1603–1622
1608
where cm t
lnðctþm =ct Þ ; m
ym t
lnðytþm =yt Þ m
1 and spreadtm rm t rt
ð2Þ
and rm t is the m-period interest rate at time t: The second approach applies a probit model to assess the ability of the term spread to predict the probability of a future recession (or upturn). Results are reported by Estrella and Hardouvelis (1991), Estrella and Mishkin (1997), Dotsey (1998) and Bernard and Gerlach (1998), all of whom find that the term spread is a good predictor of the probability of future recessions. The last paper shows that this property holds in all G-7 countries apart from Japan. The probit approach is particularly illuminating for real data, as the obtained probabilities can then be compared with the realized state of the economy. However, to test the LIPTS in simulated economies, Eq. (1) provides a more useful benchmark, which allows us to pin down precisely the nature of the co-movements between the term spread and future consumption or output growth, and which is easily comparable across models. Papers documenting a positive and significant b1 include Estrella and Hardouvelis (1991), Plosser and Rouwenhorst (1994), Estrella and Mishkin (1997) and Dotsey (1998). This result holds for the US, the UK, France, Germany and Italy for forecast intervals ranging from 3 months to 5 years. These papers also estimate some variations of Eq. (1) introducing, for example, the short rate or some known economic leading indicators, but the underlying idea remains similar and b1 is found to be positive and significant throughout. Table 2 shows our estimates for the US economy, where as elsewhere in this paper we follow Plosser and Rouwenhorst (1994) by matching the span of the spread (m) with the length of the forecast interval for activity growth. For forecast horizons ranging from 6 months to 2 years, b1 is positive and significant. This confirms evidence from the literature that nominal term spreads forecast future real consumption and output growth rates. While the studies above use nominal interest rates, the models we present below are couched in real terms. The evidence on the predictive power of real term spreads in the US is established by Harvey (1988). In Table 1, Panel B we show our estimates of correlations between real spreads and activity growth in the US.7 The correlation coefficients are lower than for nominal spreads, but are consistent with the notion of real spreads containing information about future economic activity. Outside of the US, only the nominal spread has been used to study the LIPTS in most countries. One reason for this is the fact that ex ante real rates are not observable, and computing them requires choosing a measure of expected inflation and hence a degree of arbitrariness. Another reason may be an expectation of 7
Real rates are computed as in Harvey (1988), i.e. as the difference between nominal interest rates and expected inflation rates. Expected inflation rates are obtained in the following way. We estimate an ARIMA model (using the Ljung-Box Q-statistic to select the best specification) for each m-period inflation rate for the sample 1960:1 until 1969:1 and then compute m-step-ahead forecasts for the m-period expected inflation rates. As new data are added to the sample, parameters are re-estimated and new forecasts are obtained. We repeat this until the end of the sample, i.e. 1998:4.
ARTICLE IN PRESS C. Rendu de Lint, D. Stolin / Journal of Monetary Economics 50 (2003) 1603–1622
1609
Table 2 Regressions of m-quarter consumption and output growth on m-quarter nominal term spread, see Eq. (1) m
Coefficient
Std. error
t-Statistic
p-Value
Consumption regressions 2 1960:1–1998:2 4 1964:1–1997:4 8 1976:2–1996:4 20 1960:1–1993:4
4.296 1.942 1.230 0.118
1.100 0.305 0.266 0.124
3.91 6.37 4.63 0.95
0.001 0.000 0.000 0.340
Output regressions 2 1960:1–1998:2 4 1964:1–1997:4 8 1976:2–1996:4 20 1960:1–1993:4
4.404 2.284 1.449 0.044
1.360 0.383 0.310 0.109
3.24 5.96 4.68 0.41
0.001 0.000 0.000 0.685
Period
The number of quarters used for term spread and consumption/output growth calculations, the estimation period (quarterly data are used), the estimated coefficient on the term spread, the Newey-West (1987) autocovariance-consistent standard error, the corresponding t-statistic and p-value. Five lags were used in the Newey-West procedure (the t-statistics change little if a different number of lags is used).
comparable predictive ability for real and nominal spreads.8 A recent paper examines the ability of the real spread to predict real consumption growth in Australia (Fisher and Felmingham, 1998) and finds it to be significant. Moreover, following Donaldson et al. (1990), the use of a real model can further be justified based on two observations: (1) Nominal and real rates exhibit very similar properties in artificial economies (see Backus et al., 1989; Danthine et al., 1987), and (2) in reality nominal and real short-term rates tend to move in tandem. The latter is consistent with the observation that short-term nominal rates contain no information about future inflation (Mishkin, 1990). As further justification, Table 1, Panel C shows correlations between the differential in expected inflation rates (the ‘‘spread’’ of inflation expectations) and activity growth to be low, especially at the 2year horizon. This implies a limited role for inflation in explaining the predictive power of nominal spreads and hence reasonableness of abstracting from inflation in seeking to explain the LIPTS.
3. The standard model: a stochastic endowment economy This section outlines the standard model, whose features are common to both models studied in this paper. The production economy extension is studied in the next section. 8
Thus, in studying the predictability of economic growth for G-7 countries, Harvey (1989, Endnote 2) suggests that predictions made with nominal spreads are close to those made with real spreads: ‘‘If inflation follows a first-order integrated moving-average process, the nominal yield spread equals the expected real yield spread.’’ Similarly, ‘‘I assume that the spread between nominal rates is approximately equal to the spread between real rates. [...] Incorporating the term structure of inflation could lead to improved forecasts’’ (Harvey, 1993, p. 7–8).
ARTICLE IN PRESS 1610
C. Rendu de Lint, D. Stolin / Journal of Monetary Economics 50 (2003) 1603–1622
Agents are rational and maximize expected future discounted utility. Their utility depends on consumption (ct ) and is described by a time separable power utility function. The agents are subject to a budget constraint. The endowment is determined exogenously and is subject to a random shock (ut ). Additionally, every period agents can decide to buy or sell m-period real zero-coupon bonds (bm t ; m ¼ 1; y; M) at the prevailing price (pm ). Agents keep the bonds until maturity, t when they pay off in units of endowment (the model is unchanged if agents are allowed to sell or buy back bonds before maturity). Note that this model is a special case of Lucas (1978). Agents solve the following maximization problem: N X c1f 1 max E0 ð3Þ bt t fct g 1f t¼0 subject to ct þ
M X
m pm t bt ¼ yt þ
m¼1
M X
bm tm :
ð4Þ
m¼1
We will also assume the following law of motion for the endowment: lnðytþ1 Þ ¼ r lnðyt Þ þ utþ1 ;
ð5Þ
ut BNð0; s2u Þ
and i:i:d:: In equilibrium agents consume all of the endowment where (ct ¼ yt ) and bonds are in zero net supply (bm t ¼ 0; 8m). Solving (3) subject to (4) and the equilibrium conditions yields the following expression for the price of a bond with maturity m: ctþm f m pm ¼ b E : ð6Þ t t ct 9 m The m-period interest rate rm t is defined as lnðpt Þ=m and can be written as ! 1 ctþm f ln E rm ¼ ln b : t t m ct
The spread becomes: spreadtm
rm t
r1t
! ! 1 ctþm f ctþ1 f ¼ ln Et þ ln Et : m ct ct
ð7Þ
ð8Þ
From the first-order condition (FOC) (6) it can be seen that the bond price is negatively correlated with expected consumption growth, hence the interest rate is positively correlated with the expected consumption growth rate. This well-known property of the endowment economy follows directly from agents trying to smooth out their consumption paths. If agents expect low consumption at t þ m; they buy bonds today which mature at t þ m and provide them with additional endowment. 9
This is the continuously compounded interest rate. Under m-period discrete compounding, the rate 1=m would be given by ð1=pm 1: Differences between these two definitions are small, as long as interest t Þ rates are not too large. The continuous time definition is used to simplify the analytical expressions, as is done for example in den Haan (1995).
ARTICLE IN PRESS C. Rendu de Lint, D. Stolin / Journal of Monetary Economics 50 (2003) 1603–1622
1611
m This drives today’s price of the m-period bond (pm t ) up, and its interest rate (rt ) down. The converse holds when agents expect high consumption, thus explaining the positive correlation between the interest rate and the future consumption growth m rate, Corrðrm t ; ct Þ > 0: Some authors appear to conclude that this observation provides a justification for the LIPTS.10 In other words, they argue that this standard model is able to reproduce the positive correlation illustrated in Fig. 1B between the spread and the consumption growth rate. We can check their assertion by studying the sign of the covariance between the term spread and the consumption growth rate. Analytical expressions can be obtained for the spread, the consumption growth rate and their covariance. The derivations are reported in full in Appendix A. The spread can be expressed as
f spreadtm ¼ ½ðrm 1Þ mðr 1Þ ln ðct Þ m f2 s2u þ ½mð1 r2 Þ ð1 r2m Þ : 2mð1 r2 Þ and the consumption growth rate as " # m1 X 1 cm rj utþmj : ðrm 1Þ lnðct Þ þ t ¼ m j¼0
ð9Þ
ð10Þ
Their covariance is Covðspreadtm ; cm t Þ ¼
m1 s2u f rm 1 X ð1 rj Þ: m2 1 þ r j¼0
ð11Þ
From (11) we see that for m > 1 and a plausible calibration of the model, i.e. ro1; the covariance is negative, contradicting the above claim. In other words, the standard endowment economy fails to explain the LIPTS. This result shows that in an endowment economy, even though real interest rates are positively correlated with consumption growth rates, this property does not necessarily imply that term spreads are positively correlated with consumption growth rates, and cannot be used as a theoretical justification for the LIPTS. To illustrate this result, the model is simulated numerically, with impulse responses of the consumption growth and spread variables to an endowment shock graphed in Fig. 2, while measures of comovement between variables of interest (for different time spans m) are given in Table 3 (Appendix B contains details about the simulations and the calibration of the model). As can be seen from that table, the b1 coefficients obtained from Eq. (1) are negative and significantly different from zero, confirming that the LIPTS is rejected in this model. Further, Corrðspreadtm ; cm t Þ is never positive in any of the 500 simulations. The reason for the rejection of the LIPTS is the following. In an endowment economy, if the autoregressive coefficient (r) for the endowment shock is less than one, then a shock ut has the biggest impact on consumption in the period it occurs. 10
Smets and Tsatsaronis (1997, p. 1), Dotsey (1998, p. 38).
ARTICLE IN PRESS 1612
C. Rendu de Lint, D. Stolin / Journal of Monetary Economics 50 (2003) 1603–1622 Panel: A Impulse response for consumption 0.3
Panel: B Impulse response for spread spread2 spread4 0.3 spread24
spread40
0.2
0.2
0.1
0.1 1
6
11
16
1
6
11
16
Fig. 2. Impulse response functions for the endowment economy. The number of quarters is shown on the x-axis.
Table 3 Features of the stochastic endowment economy. This table summarizes 500 simulations, as described in Appendix B m¼2
m ¼ 20
m ¼ 40
Measures of co-movement between spread and consumption growth Corrðspreadtm ; cm 0.222 0.306 0.412 t Þ Percent Corrðspreadtm ; cm 0.0 0.0 0.0 t Þ>0 b1 18.380 6.179 2.423 7.803 2.397 0.875 SD ðb1 Þ
0.569 0.0 0.773 0.214
0.661 0.0 0.312 0.071
Additional correlations Corrðspreadtm ; ut Þ Corrðcm t ; ut Þ
0.313 0.177
0.313 0.207
0.313 0.069
m¼4
0.313 0.096
m¼8
0.313 0.127
The variables m; spreadtm ; cm t ; and ut are defined in the text. The variable b1 is the coefficient in the regression of consumption growth on the spread (Eq. (1)), and SD ðb1 Þ is its standard deviation across the 500 simulations. The row labelled ‘‘Percent’’ shows the percentage of time the variable studied was positive in the 500 simulations.
This implies that the impulse response function for consumption is at its peak when the positive shock occurs, and then declines following an AR(1) process (Fig. 2A). Consequently, after a one-off positive shock ut (and no further shocks), future consumption growth rates, as proxied by the slope of the impulse response function for consumption, are negative and Corrðcm t ; ut Þo0: Recall from (6) that bond prices are determined by expected consumption growth rates, represented by the slope of the impulse response function for consumption. In Fig. 2A, since this function is downward-sloping, consumption growth rates are negative and agents have an incentive to buy bonds. Moreover, since the impulse response function is convex, this implies that the 1-period consumption growth rate is more negative than the mperiod growth rate. Agents therefore need to buy more short-term than long-term bonds to smooth their consumption stream, thus r1t falls more than rm t and the spread rises. Hence, contrary to the LIPTS, the rise in the spread after a one-off positive shock is associated with a negative future consumption growth rate.
ARTICLE IN PRESS C. Rendu de Lint, D. Stolin / Journal of Monetary Economics 50 (2003) 1603–1622
1613
Put differently, in an endowment economy, shocks have the biggest impact on the consumption growth rates, or marginal rates of consumption substitution, in the first period. This means that after a shock, c1t is always larger in absolute value than cm t ; 1 m and cm c always moves in opposite direction from c : This finding is not specific t t t to the chosen law of motion (5). Even when allowing for growth in the consumption process, i.e. assuming that consumption is integrated of order one, the LIPTS is still rejected. Assuming the law of motion where lnðytþ1 =yt Þ ¼ b lnðyt =yt1 Þ þ utþ1 ; also leads to a rejection of the LIPTS (see Section 5 for a more detailed discussion). It appears that in a model where the impulse response function for consumption peaks in the first period, the term spread fails to predict future economic activity in the required direction, suggesting the need to extend our model.
4. Stochastic production economy Motivated by the failure of the endowment economy to account for the LIPTS, we extend the standard model to include endogenous production, with capital as the unique production factor. In this economy, agents can decide in every period whether to consume all the available output, or to save part of it and invest it into capital (i.e. future production). Adding endogenous production is a natural and straightforward extension, and one which does not introduce additional degrees of freedom (the capital depreciation rate and the capital share are pinned down from the data in a way similar to the literature on real business cycles).11 Formally, the economy of Section 3 is augmented in the following way. Agents can now consume (ct ) and invest (it ) in the production factor capital (kt ). Production (yt ) is subject to an autocorrelated productivity shock (yt ). Agents maximize the utility function (3) but are now subject to the following budget constraint: ct þ i t þ
M X
m pm t bt ¼ yt þ
m¼1
M X
bm tm ;
ð12Þ
m¼1
where it ¼ kt ð1 dÞkt1 ;
ð13Þ
a yt ¼ yt kt1 :
ð14Þ
and Eq. (13) describes the capital accumulation process, with d being the constant depreciation rate of capital, while (14) is a Cobb-Douglas production function where the labor input is held constant and normalized to one, and a is the share of capital in total output. The productivity shock is driven by lnðyt Þ ¼ r lnðyt1 Þ þ ut ;
ð15Þ
11 den Haan (1995) compares the performance of this model with models including additionally variable labour supply or money, and shows that this model is best able to replicate the empirical bond price behaviour.
ARTICLE IN PRESS 1614
C. Rendu de Lint, D. Stolin / Journal of Monetary Economics 50 (2003) 1603–1622
where ut BNð0; s2u Þ and i:i:d:: In equilibrium, bonds are in zero net supply ðbm t ¼ 0; 8mÞ: Solving (3) subject to (12), (13), (14) and (15) yields equation (6) for the price of a k-period bond, and an additional first-order condition for consumption: a1 cf ¼ bEt ðcf þ 1 dÞÞ: t tþ1 ðytþ1 akt
ð16Þ
Since bond prices still depend on the marginal rate of consumption substitution, it follows that a necessary condition for this model to explain the LIPTS is that its impulse response function for consumption should not be monotone decreasing as it was in the endowment model. Exact analytical solutions cannot be obtained in this case, and the model is solved numerically. The parameterized expectations algorithm of den Haan and Marcet (1990) is used, whereby the conditional expectations in (6) and (16) are approximated using exponentiated polynomials; the solution is obtained by iterating on the polynomials’ coefficients until the expectations become good predictors.12 Five hundred simulations are conducted as before, with the parameters chosen to match US quarterly data and based on den Haan (1995): b ¼ 0:99;
f ¼ 3;
d ¼ 0:025;
r ¼ 0:95;
a ¼ 0:33;
su ¼ 0:018
The results are reported in Table 4, which summarises co-movement among the spread, activity growth variables, and the productivity shock. Three features of the model stand out: (1) the term spread is a significant predictor of future output growth at all forecast horizons, (2) the term spread fails to predict consumption growth (in the right direction) in the short run, but (3) the term spread predicts it for longer forecast horizons (over 10 years). Hence, the production economy is able to match the LIPTS for output growth for all forecast horizons, and for consumption growth for long-term horizons. This is a key finding since it demonstrates the ability of a simple model to match an important property of real data, and to provide it with a rigorous theoretical foundation. Note again that this ability does not follow from the FOC (6) as the literature sometimes suggests. Both the endowment and the production economy have the same FOC for the bond price, yet only the latter reproduces the LIPTS. The main mechanism triggering this positive result is the ability of agents to substitute consumption intertemporally. In the production economy when a positive shock occurs, agents increase their contemporaneous consumption, but not by the full amount of the shock. Rather, in an effort to smooth their consumption path, they save part of the shock and invest it in capital. The higher capital stock then allows higher production and higher consumption in the future. Therefore, contrary to the endowment economy case, a shock does not exert its biggest impact on consumption in the period it occurs, but rather sees its effect extended and magnified through the investment channel. Thus, the impulse response function for consumption is now hump-shaped, explaining the better performance of this model. Fig. 3 shows the impulse response functions for consumption, output and the spread. 12
Marcet and Lorenzoni (1999) further discuss applications of this algorithm.
ARTICLE IN PRESS C. Rendu de Lint, D. Stolin / Journal of Monetary Economics 50 (2003) 1603–1622
1615
Table 4 Features of the stochastic production economy. This table summarises 500 simulations, as described in Appendix B m¼2
m ¼ 20
m ¼ 40
m ¼ 44
m ¼ 48
Measures of co-movement between spread and consumption growth Corrðspreadtm ; cm 0.076 0.117 0.122 0.073 t Þ Percent Corrðspreadtm ; cm 0.0 0.0 0.0 1.0 t Þ>0 1.010 0.523 0.182 0.085 b1 ðcm t Þ SD ðb1 ðcm 6.874 1.447 0.645 0.287 t ÞÞ
0.001 65.2 0.078 0.161
0.011 60.2 0.076 0.110
0.020 63.2 0.076 0.108
Measures of co-movement between spread and output growth Corrðspreadtm ; ym 0.126 0.156 0.209 t Þ Percent Corrðspreadtm ; ym 100.0 100.0 100.0 t Þ>0 b1 ðym 27.840 5.599 2.477 t Þ SD ðb1 ðym 17.140 3.681 1.548 t ÞÞ
0.287 100.0 0.896 0.519
0.322 96.4 0.515 0.277
0.325 100.0 0.473 0.248
0.326 100.0 0.458 0.224
Additional correlations m Corrðcm t ; yt Þ ; u Corrðcm tÞ t Corrðym t ; ut Þ Corrðspreadtm ; ut Þ
0.918 0.038 0.096 0.367
0.921 0.001 0.119 0.368
0.923 0.007 0.123 0.368
0.924 0.013 0.126 0.368
0.977 0.045 0.034 0.349
m¼4
0.960 0.056 0.047 0.362
m¼8
0.938 0.060 0.064 0.364
m m The variables spreadtm ; cm t ; yt and ut are defined in the text. The variable b1 ðct Þ is the coefficient in the regression of consumption growth on the spread (Eq. (1)), and SD ðb1 ðcm t ÞÞ is its standard deviation across m the 500 simulations. The variables b1 ðym t Þ and SD ðb1 ðyt ÞÞ denote the corresponding quantities for output growth. The rows labelled ‘‘Percent’’ show the percentage of time the variable studied was positive in the 500 simulations.
Before discussing how this model relates to the empirical evidence on the LIPTS, we explain each of the three results mentioned above. First, why is the LIPTS rejected in the short-term for consumption? In Fig. 3A, the impulse response function for consumption is rising for the first 4 years; implying that the consumption growth rate is positive. Yet, since the function is concave, marginal rates of substitution are declining. In other words, c1t is larger than cm t ; and due to (7), r1t rises more than rm t ; hence the spread falls and fails to predict the future rise in consumption. Why, then, does the LIPTS hold in the long-term? Notice in Fig. 3A, that after 9 years (point B), the impulse response function falls to below its original level: the value of the function at t þ 37 is smaller than at t; so that for horizons above 9 years the consumption growth rate is negative. Hence, the fall in the long-term spread correctly predicts the future long-term decline in consumption, and the LIPTS holds for horizons over 10 years.13 13
Based on Table 4 it may seem that only long-term spreads contain some predictive information about future consumption. This is not entirely correct. Following a positive shock, all term spreads fall, independently of their maturity. In fact, the impulse response function for spreadtm and Corrðspreadtm ; ut Þ are almost invariant with respect to m: This implies that long-term and short-term spreads have nearidentical predictive power with respect to long-term consumption growth. Similarly, both long-term and short-term spreads fail badly to forecast the initial rise in consumption.
ARTICLE IN PRESS 1616
C. Rendu de Lint, D. Stolin / Journal of Monetary Economics 50 (2003) 1603–1622
Panel A: Impulse response for consumption Panel B: Impulse response for output 0.14
0.22
A
0.13
0.2
0.12 B 0.11
0.18 0.16
0.1 0.14
0.09
0.12
0.08 1 6 11 16 21 26 31 36 41 46
1
6
11
16
Panel C: Impulse response for spread 1 11 16 6 -0.1
-0.2 spread2 -0.3
spread4 spread24
-0.4
spread48
Fig. 3. Impulse response functions for the production economy. The number of quarters is shown on the x-axis.
Lastly, why does the LIPTS hold for output? Unlike for consumption, where agents are able to smooth out the benefits of a positive productivity shock by saving part of it and investing it into capital, the impact of an isolated productivity shock on output is greatest in the first period, and then decreases following an AR(1) process. In Fig. 3B this is reflected by the impulse response function for output which is at its a peak in the first period and then declining. Hence the output growth rate, as captured by the slope of the impulse response function, is negative. The declining spreads correctly predict this future fall in output. Table 4 shows that, except in the short-term, the coefficients for output, b1 ðym t Þ; are positive and significantly different from zero. Output growth rates are less volatile the longer the forecast horizon, explaining the fall in magnitude observed in the b1 coefficients. To gain further insight into the empirical LIPTS, consider its origin in the production economy. A positive productivity shock implies a positive wealth effect for the agents and increases consumption at all future dates. As we saw in Fig. 3A, the impulse response function for consumption rises until 4 years after the shock because agents are smoothing out their consumption by saving part of the output and investing into capital. This intertemporal consumption substitution is induced by the high interest rates, which follow immediately after the positive shock. However, this expansionary phase is expected to end once the accumulated capital,
ARTICLE IN PRESS C. Rendu de Lint, D. Stolin / Journal of Monetary Economics 50 (2003) 1603–1622
1617
together with a lower productivity level, bring about a fall in the marginal product. Agents then start dissaving, interest rates fall below their long-term level and consumption growth rates return to their steady-state level.14 Since interest rates are expected to fall in the future, this implies today (under the expectations hypothesis) that short-term rates rise more than long-term rates. As a result, the spread goes down today and if agents’ expectations turn out to be true, and the future productivity level is lower than today’s level, then today’s fall in the spread will have correctly predicted this future slowdown. This suggests that for the LIPTS to hold in an economy, agents must be able to substitute consumption intertemporally in reaction to productivity changes.
5. Discussion The aim of this paper is to demonstrate that while an endowment economy is unable to explain the LIPTS, a model with production goes a long way in matching its empirical properties. In this section, we take a close look at the chosen approach and how it affected the findings. Specifically, we discuss the extent to which the failure of the endowment economy followed from the chosen law of motion and outline possible future research directions. We start with the failure of the endowment economy to reproduce the LIPTS. An obvious objection to this result is to question whether it is only driven by the chosen law of motion and in particular, whether it is due to the empirically counterfactual property that after a positive disturbance, the conditional mean of all future consumption growth rates is negative. We find that even with a higher-order process for consumption, the endowment economy still fails to produce the LIPTS. Let us assume, for instance, as is done in the real business cycle literature, that consumption follows a process that is integrated of order one: Dct ¼ a þ z Dct1 þ et ;
ð17Þ
where et BNð0; s2e Þ and i:i:d:: This model still fails. This time, the impulse response for consumption is increasing, so that expected consumption growth rates are positive after a positive shock. However, the essence of the problem remains unchanged. The function is concave, implying, as seen throughout this paper, that the one-period interest rate will rise more than the m-period interest rate and therefore that spreadtm will fall. As a result, the spread will be negatively correlated with the corresponding consumption growth rate. Higher-order dynamics alone do not necessarily resolve this problem for the endowment economy. Clearly, the exogenous consumption process in the endowment economy can be fixed to be identical to the one resulting in the production economy, allowing the endowment economy to produce the LIPTS for consumption growth at long-term horizons. It is also clear that this would not be a satisfying reconciliation of the endowment economy with the LIPTS. 14
See Rouwenhorst (1995, pp. 308–309) for a detailed description.
ARTICLE IN PRESS 1618
C. Rendu de Lint, D. Stolin / Journal of Monetary Economics 50 (2003) 1603–1622
It is important to stress that although the production economy is the more successful model, it still does not match the features of the empirical LIPTS perfectly. Indeed, for consumption growth, the correlation with the spread is positive only for horizons longer than 10 years, while in reality the predictive power of the term spread holds for much shorter durations (up to 5 years). Note that this particular shortcoming could be resolved if the impulse response function for consumption peaked earlier. Thus, while showing a possible direction for replicating the LIPTS, the production economy we studied here clearly provides only a partial answer. Further, a researcher seeking to match the empirical LIPTS by finding the right output and consumption dynamics will need to address the following issue: the two dynamics will have to be different (as suggested by the better results generated in a production economy), yet still be highly correlated with each other to match the empirical evidence, where correlations are on the order of 0.90. As can be seen from Table 4, the production economy reproduces this feature rather well. Finally, it is well known that FOC (6) implies a positive correlation between real interest rates and the corresponding consumption growth rates. In our model calibration, these correlations vary from 0.1 to 0.6 depending on the duration. This observation is consistent with empirical evidence, as reported in Table 1, Panel E and in Harvey (1988). What is counterfactual, however, is that in the production model, the correlation of real yields with consumption growth rates (and thus the predictive power of the real yields) is much higher than that of real spreads. In other words, while spreads are generally better predictors of future activity growth than yields in reality (cf. Table 1), in the stochastic production economy yields are the better predictors. This probably underlines that, unlike in the model, there is more than one source of uncertainty in the real world.
6. Concluding remarks The aim of this paper was to provide a theoretical basis for the empirically supported leading indicator property of the term spread (LIPTS). Dynamic equilibrium asset pricing models appeared natural candidates to study the joint behaviour of real macroeconomic variables (consumption and output) and asset prices. We studied two models, a stochastic endowment economy and a production economy. The first contribution of this paper is to show that not all dynamic equilibrium asset pricing models can explain the LIPTS. Contrary to what is sometimes assumed and perhaps counter-intuitively, an endowment economy does not generate the LIPTS. This is because after any shock, short-term interest rates always react more than long-term rates, resulting in a negative correlation between the term spread and future consumption growth rates. The second contribution is to show that a simple production economy, on the other hand, goes a long way in explaining the LIPTS. This positive result permits greater understanding of what may be driving the LIPTS in actuality. Indeed, one
ARTICLE IN PRESS C. Rendu de Lint, D. Stolin / Journal of Monetary Economics 50 (2003) 1603–1622
1619
mechanism appears to be crucial: agents must be able to substitute consumption intertemporally in reaction to productivity changes.
Appendix A. Analytical derivations This appendix contains the derivation of Eqs. (9), (10) and (11) presented in Section 3. The interest rate is given by Eq. (7): ! 1 ctþm f m rt ¼ ln b ln Et : ðA:1Þ m ct We need to find an expression for Et ðctþm =ct Þf : In equilibrium ct ¼ yt : Substituting backwards for lnðctþm Þ (m ¼ 1; 2; y; M) in the law of motion (5), we obtain the following expression for lnðctþm Þ: lnðctþm Þ ¼ rm lnðct Þ þ
m1 X
ri utþmi :
ðA:2Þ
i¼0
Taking the exponential of Eq. (A.2) yields ! m1 X rm i ctþm ¼ ct exp r utþmi : i¼0
We then divide by ct ; raise the resulting expression to the power ðfÞ and take expectations: " " !# # m1 X ctþm f fðrm 1Þ i Et Et exp f r utþmi : ¼ ct ðA:3Þ ct i¼0 P i Since ut is distributed normally with mean 0 and variance s2u ; f m1 i¼0 r utþmi is 2 2 Pm1 2i distributed normally with mean 0 and variance f su i¼0 r ; and therefore P Pm1 2i
i 1 2 2 expðf m1 : i¼0 r utþmi Þ is distributed lognormally with mean exp 2 f su i¼0 r Hence, Eq. (A.3) becomes " ! # m1 ctþm f 1 2 2X fðrm 1Þ exp f su r2i ¼ ct Et 2 ct i¼0 which can be rewritten as " # ctþm f 1 2 1 r2m 2 fðrm 1Þ Et exp f s : ¼ ct 2 ct 1 r2 u Substituting (A.4) into (A.1) yields rm t ¼ ln b þ
1 1 2 1 r2m 2 fðrm 1Þ lnðct Þ f s m 2m 1 r2 u
ðA:4Þ
ARTICLE IN PRESS C. Rendu de Lint, D. Stolin / Journal of Monetary Economics 50 (2003) 1603–1622
1620
and for the spread, f spreadtm ¼ ½ðrm 1Þ mðr 1Þ lnðct Þ m f2 s2u þ ½mð1 r2 Þ ð1 r2m Þ : 2mð1 r2 Þ which corresponds to Eq. (9). Consumption growth is defined as 1 ½lnðctþm Þ lnðct Þ : cm t ¼ m Inserting (A.2) into (A.6) gives " # m1 X 1 m m i ðr 1Þ lnðct Þ þ ct ¼ r utþmi m i¼0
ðA:5Þ
ðA:6Þ
ðA:7Þ
which corresponds to Eq. (10). We now turn to the covariance between the spread and the consumption growth m rate (Covðspreadtm ; cm t Þ). First of all we show that Eðct Þ ¼ 0: From the law of motion for the endowment (5), and since in equilibrium ct ¼ yt ; we can write t X lnðct Þ ¼ ri uti ; i¼0
so that E½lnðct Þ ¼ E
"
N X
# i
r uti ¼ 0:
ðA:8Þ
i¼0
From (A.7) and (A.8), we see that Eðcm t Þ ¼ 0: Therefore, m m Covðspreadtm ; cm t Þ ¼ Eðspreadt ct Þ:
Using (A.5), (A.7), (A.8) and the distribution properties of the shock ut ; a few algebraic steps are sufficient to show that
m f m ðr 1Þ s2u m m ðr 1Þ ðr 1Þ : ðA:9Þ Covðspreadt ; ct Þ ¼ m m ð1 r2 Þ This expression can be rewritten as m1 s2u f rm 1 X m m Covðspreadt ; ct Þ ¼ 2 ð1 r j Þ: m 1 þ r j¼0 which corresponds to Eq. (11) in the text.
Appendix B. Numerical simulations For the numerical simulations, our model is calibrated to match US quarterly data, and except for the risk aversion coefficient (f) the parameters are chosen as in
ARTICLE IN PRESS C. Rendu de Lint, D. Stolin / Journal of Monetary Economics 50 (2003) 1603–1622
1621
den Haan (1995): b ¼ 0:99;
f ¼ 3;
r ¼ 0:95;
su ¼ 0:007:
In the asset pricing literature, authors tend to choose relatively high risk aversion coefficients to obtain more significant risk premia. Since the purpose of this paper is to investigate whether a standard model is able to explain the LIPTS, a lower f is selected. Rouwenhorst (1995) suggests ½1; 5 as a range of plausible values for f: Throughout the paper, we work with plausible parameters, rather than selecting ones for which the model fits the LIPTS. The economy is simulated 500 times over 5000 periods, and the reported values are average values from the 500 repetitions. This approach is applied to all models studied in this paper.
References Backus, D., Gregory, A., Zin, S., 1989. Risk premiums in the term structure: evidence from artificial economies. Journal of Monetary Economics 24, 371–399. Bernard, H., Gerlach, S., 1998. Does the term structure predict recessions? The international evidence, CEPR Discussion Paper 1892. Campbell, J., 1986. Bond and stock returns in a simple exchange model. Quarterly Journal of Economics 101, 785–803. Canova, F., De Nicolo, G., 1997. Stock returns, term structure, inflation and real activity: an international perspective, CEPR Discussion Paper 1614. Chapman, D., 1997. The cyclical properties of consumption growth and the real term structure. Journal of Monetary Economics 39, 145–172. Clark, K., 1996. A near-perfect tool for economic forecasting. Fortune, July 22, 24–26. Cox, J., Ingersoll, J., Ross, S., 1985. A theory of the term structure of interest rates. Econometrica 53, 385–407. Danthine, J.P., Donaldson, J.B., Smith, L., 1987. On the superneutrality of money in a stochastic dynamic macroeconomic model. Journal of Monetary Economics 20, 475–499. den Haan, W.J., 1995. The term structure of interest rates in real and monetary economies. Journal of Economic Dynamics and Control 19, 909–940. den Haan, W.J., Marcet, A., 1990. Solving the stochastic growth model with parameterised expectations. Journal of Business and Economic Statistics 8, 31–34. Donaldson, J., Johnsen, T., Mehra, R., 1990. On the term structure of interest rates. Journal of Economic Dynamics and Control 14, 571–596. Dotsey, M., 1998. The predictive content of the interest rate term spread for future economic growth. Federal Reserve Bank of Richmond. Economic Quarterly 84, 31–51. Estrella, A., Hardouvelis, G., 1991. The term structure as a predictor of real economic activity. Journal of Finance 46, 555–576. Estrella, A., Mishkin, F., 1997. The predictive power of the term structure of interest rates in Europe and the United States: implications for the European Central Bank. European Economic Review 41, 1375–1401. Fisher, C., Felmingham, B., 1998. The Australian yield curve as a leading indicator of consumption growth. Applied Financial Economics 8, 627–635. Hamilton, J.D., Kim, D.H., 2000. A re-examination of the predictability of economic activity using the yield spread. NBER Working Paper 7954. Hansen, L.P., Singleton, K.J., 1983. Stochastic consumption, risk aversion, and the temporal behavior of asset returns. Journal of Political Economy 91, 249–265.
ARTICLE IN PRESS 1622
C. Rendu de Lint, D. Stolin / Journal of Monetary Economics 50 (2003) 1603–1622
Harvey, C.R., 1988. The real term structure and consumption growth. Journal of Financial Economics 22, 305–333. Harvey, C.R., 1989. The term structure and world economic growth. Journal of Fixed Income 1, 7–19. Harvey, C.R., 1993. Term structure forecasts economic growth, Financial Analysts Journal, May-June, 6–8. Kosicki, S., 1997. Predicting real growth and inflation with the yield spread. Federal Reserve Bank of Kansas City. Economic Review 82, 39–57. Labadie, P., 1994. The term structure of interest rates over the business cycle. Journal of Economic Dynamics and Control 18, 671–697. Longstaff, F., Schwartz, E., 1992. Interest rate volatility and the term structure: a two-factor general equilibrium model. The Journal of Finance 47, 1259–1282. Lucas, R., 1978. Asset prices in an exchange economy. Econometrica 46, 1429–1445. Marcet, A., Lorenzoni, G., 1999. The parameterized expectations approach: some practical issues. In: Marimon, R., Scott, A. (Eds.), Computational Methods for the Study of Dynamic Economies, Oxford University Press, Oxford. Mishkin, F., 1990. What does the term structure tell us about future inflation? Journal of Monetary Economics 25, 77–95. Newey, W., West, K., 1987. A simple positive semi-definite heteroskedasticity and autocorrelation consistent matrix. Econometrica 55, 703–708. Plosser, C.I., Rouwenhorst, K.G., 1994. International term structures and real economic growth. Journal of Monetary Economics 33, 133–155. Rouwenhorst, K.G., 1995. Asset pricing implications of equilibrium business cycle models. In: Cooley, T. (Ed.), Frontiers of Business Cycle Research (Princeton University Press, Princeton, NJ). Smets, F., Tsatsaronis, K., 1997. Why does the yield curve predict economic activity? Dissecting the evidence for Germany and the United States, CEPR Discussion Paper 1758.
Journal of Monetary Economics 50 (2003) 1603–1622
The predictive power of the yield curve: a theoretical assessment$ Christel Rendu de Linta,*, David Stolinb b
a Pictet, PAM Fixed Income, 8 bd de la Tour, 1205 Geneva, Switzerland Toulouse Business School, 20 Boulevard Lascrosses, 31000 Toulouse, France
Accepted 7 August 2003
Abstract Although the empirical evidence about the leading indicator property of the term spread (LIPTS) is powerful, this property lacks a rigorous theoretical foundation. This paper investigates whether dynamic equilibrium asset pricing models are able to provide a theoretical underpinning for the LIPTS. We study an endowment and a production economy. The endowment economy is unable to account for the LIPTS. On the other hand, a model with endogenous production provides a reasonable theoretical justification for the LIPTS. r 2003 Elsevier B.V. All rights reserved. JEL classification: C68; E32; E43; E44 Keywords: General equilibrium; Leading indicators; Term structure of interest rates; Yield curve
1. Introduction The yield curve has long been monitored for the information it contains about future economic activity. Looking at Fig. 1, it is striking how closely the current term $ The views expressed in this paper are those of the authors and do not necessarily represent those of Pictet & Cie. This paper is based on a chapter of Christel Rendu de Lint’s dissertation at the London Business School. She is indebted to her supervisor Albert Marcet for invaluable comments and support. The paper also benefited from helpful comments by Jean-Pascal Benassy, Fabio Canova, Alexei Jiltsov, Brian Henry, Morten Ravn, Andrew Scott, Oren Sussman, participants at the 1999 SED meeting in Alghero and at the economics seminar at the Bank of Italy. Lastly we thank an anonymous referee. Remaining errors are ours. *Corresponding author. Pictet, PAM Fixed Income, 8 bd de la Tour, 1205 Geneva, Switzerland. E-mail address: [email protected] (C. Rendu de Lint).
0304-3932/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jmoneco.2003.08.007
ARTICLE IN PRESS 1604
C. Rendu de Lint, D. Stolin / Journal of Monetary Economics 50 (2003) 1603–1622
Panel A: Output 2
10 8
1 6 0
4 2
-1
0 -2 -2 -4
-3 65
70
75
80
85
90
Real GDP 1-year future g. rate
8
95
1-year spread
Panel B: Consumption
2
6
1
4
0
2
-1
0
-2
-2
-3 65
70
75
80
Real cons 1-year future g rate
85
90
95
1-year spread
Fig. 1. Predictive power of the term spread for future output and consumption growth rates. GDP and consumption growth rates are shown on the left scale, the term spread is on the right scale. All values are annualized and shown in %: The 1 year spread is defined as spreadt4 ¼ r4t r1t ; and 1 year consumption and output growth rates as c4t ¼ lnðctþ4 =ct =4Þ 400 and y4t ¼ lnðytþ4 =yt =4Þ 400; respectively.
spread (the difference between the long and short interest rates, i.e. a measure of the slope of the yield curve) tracks movements in the future output and consumption growth rates. Since in both graphs at each point in time t; the spread is known but future growth rates are not, the term spread appears to contain predictive information about future economic activity. Based on this observation, numerous
ARTICLE IN PRESS C. Rendu de Lint, D. Stolin / Journal of Monetary Economics 50 (2003) 1603–1622
1605
empirical papers studied and confirmed the leading indicator property of the term spread (henceforth LIPTS), showing that higher spreads precede periods of strong economic activity, while lower spreads announce periods of weak activity. In light of the strength of these results, the literature does not hesitate to characterize the spread’s predictive ability as ‘‘impressive’’, ‘‘robust’’ and ‘‘particularly well established’’.1 Yet, in contrast to this powerful empirical evidence stands the absence of rigorous theoretical justification for the LIPTS. Our paper is intended to bridge this gap. In an attempt to provide a theoretical underpinning for the empirically established LIPTS, this paper investigates whether dynamic equilibrium asset pricing models can explain the LIPTS. A stochastic endowment economy and a stochastic production economy are studied. Dynamic equilibrium asset pricing models appear to be a natural choice for explaining the joint behaviour of real macroeconomic variables (here consumption and output) and asset prices. Indeed, in dynamic equilibrium asset pricing models, asset prices are linked to the marginal utility of the agent, which itself depends on consumption. Under some conditions the endowment economy, in particular, implies a positive correlation between the m-period future consumption growth rate and the m-period spot interest rate.2 This property, which derives from the first-order condition for the m-period bond price, makes the endowment economy a seemingly attractive candidate for explaining the LIPTS, leading some authors to conclude that it is consistent with the LIPTS.3 Our paper questions this claim. The central findings of this paper are as follows. First, we show that, perhaps counter-intuitively, the endowment economy does not give rise to the LIPTS. Indeed, at time t an increase in the m-period spread is associated with a negative consumption growth rate over the same period. This is because in a simple endowment economy, where the endowment process has only first-order dynamics, the impact of a shock on consumption is the largest in the period it occurs and declines monotonically thereafter. As a consequence, short-term interest rates always respond more to a given shock than do long-term interest rates. Thus, after a positive endowment shock, the spread actually rises (since the short-term rate falls more than the long-term one), at odds with the expected decline in future consumption. To overcome this problem, we extend the endowment economy by introducing endogenous production. We show that term spreads generated in this economy become good predictors of future output growth, and of future consumption growth at long horizons. As opposed to the endowment model, in a production economy the 1 These are quotes, respectively, from Estrella and Hardouvelis (1991, p. 559); Kosicki (1997, p. 39), and Hamilton and Kim (2000, p. 3). See Section 2 for a survey of the empirical evidence. The predictive power of the term spread is also recognized by practioners: the Wall Street Journal and the New York Times, for instance, regularly print the yield curve as a leading indicator of the business cycle. Fortune published an article titled ‘‘A near-perfect tool for economic forecasting’’ (Clark, 1996), with the tool in question being the term spread. 2 This was noted by Harvey (1988), Donaldson et al. (1990), Estrella and Hardouvelis (1991), Plosser and Rouwenhorst (1994) and Chapman (1997). 3 Smets and Tsatsaronis (1997, p. 1), Dotsey (1998, p. 38).
ARTICLE IN PRESS 1606
C. Rendu de Lint, D. Stolin / Journal of Monetary Economics 50 (2003) 1603–1622
dynamics of output and consumption can be quite different from each other, and this turns out to be crucial in explaining the LIPTS. In particular, in this economy the impulse response function for consumption only reaches its peak a few periods after a shock (and not in the first period as in the endowment economy), due to the fact that agents try to smooth out the shock’s impact by substituting consumption intertemporally. It is precisely this ability of agents to freely substitute consumption over time which is at the root of the success of the production economy in explaining the LIPTS. Existing literature does not supply a formal theoretical model successfully justifying the LIPTS. Rouwenhorst (1995) tests for the LIPTS in a standard neoclassical growth model, modified to incorporate the endogenous choice of labour and leisure. He finds that the term spread does not forecast future real economic activity. Most papers documenting empirical evidence mention the possible role of monetary policy in explaining the LIPTS, but without formalising the argument.4 It is worth noting that our paper does not seek to explain the term structure of interest rates or to model term premia. A vast literature is already available on these issues (see e.g. Cox et al., 1985; Campbell, 1986; Backus et al., 1989; Donaldson et al., 1990; Longstaff and Schwartz, 1992; Labadie, 1994; den Haan, 1995). Nor does it study the co-movements of consumption growth and the real interest rate, as does Chapman (1997). Rather, it focuses on the predictive power of the term spread for future consumption and output growth. Donaldson et al. (1990), Labadie (1994) and Chapman (1997) all comment on how the term spread changes over the business cycle, but none studies the LIPTS in detail. The remainder of this paper is organized as follows. The next section reviews the empirical motivation for the paper. Section 3 presents results for the standard stochastic endowment economy, while Section 4 does so for the production economy. Section 5 takes a critical look at our modelling approach and discusses possible extensions. Section 6 concludes.
2. Empirical facts In this section we present empirical evidence on the LIPTS. This evidence is used to motivate the theoretical model-building of the paper and to benchmark the models’ predictions. Before turning to the various metrics existing in the empirical literature to measure the LIPTS, we present some simple evidence about the predictive power of the yield curve. Table 1 shows correlations between various term structure variables and activity growth rates calculated for the United States.5 Quantifying what can also be 4
Harvey’s (1988) derivation of the relationship between spreads and consumption growth follows Hansen and Singleton (1983) by assuming joint lognormality of consumption and returns. However, Hansen and Singleton point out that ’’by assuming that the joint distribution of consumption and returns is lognormal, we are implicitly imposing restrictions on the production technology’’ (p. 252). 5 Consumption and output data are standardized volume indices obtained for 1960:1–1998:4 from OECD Quarterly National Accounts (‘‘US final private consumption expenditure’’ and ‘‘US gross
ARTICLE IN PRESS C. Rendu de Lint, D. Stolin / Journal of Monetary Economics 50 (2003) 1603–1622
1607
Table 1 Correlations between real activity growth rates and term structure variables Real consumption growth rate Panel A: Nominal term spreads 6-month 1-year 2-year Panel B: Real term spreads 6-month 1-year 2-year Panel C: Expected inflation differential 6-month 1-year 2-year Panel D: Nominal yield 6-month 1-year 2-year Panel E: Real yield 6-month 1-year 2-year
Real output growth rate
0.36 0.46 0.46
0.34 0.52 0.54
0.29 0.39 0.36
0.04 0.15 0.17
0.06 0.07 0.10
0.13 0.14 0.05
0.40 0.37 0.00
0.44 0.46 0.24
0.22 0.31 0.44
0.02 0.04 0.09
Correlations of term structure variables with consumption and output growth rates calculated over the same period. US data for 1960–98 were used.
seen from Fig. 1, Panel A of Table 1 shows that nominal term spreads (of 2- to 8quarter durations) have exhibited a high and positive correlation with future consumption and output growth rates. In other words, nominal term spreads are good indicators of future economic activity. Based on such straightforward but striking evidence, the empirical literature has taken two main approaches to examining the LIPTS.6 The first approach relies directly on these observed correlations and proposes an OLS regression of the future consumption or output m m growth rate (cm t ; yt ) on the term spread (spreadt ): m cm t ¼ b0 þ b1 spreadt þ et ;
ð1Þ
(footnote continued) domestic product’’, respectively). US consumer price index for 1960:1–1998:4 is from IMF’s International Financial Statistics. 3-month, 6-month, 1-year, 2-year and 5-year interest rates come from Federal Reserve Bank of St. Louis’ Federal Reserve Economic Data. 3-month, 6-month and 1-year rates are obtained for 1960:1–1998:4 as auction averages on corresponding maturity T-bills. 2- and 5-year rates are constant maturity Treasury rates, obtained for 1976:1–1998:4 and 1960:1–1998:4, respectively. 6 A third approach, which we will not discuss here, consists of using vector autoregression, as in Canova and De Nicolo (1997), Estrella and Mishkin (1997) and Smets and Tsatsaronis (1997). These papers broaden the research question, but all confirm the LIPTS. However, in our setup, using a VAR would be meaningless since the models studied only allow for a single shock.
ARTICLE IN PRESS C. Rendu de Lint, D. Stolin / Journal of Monetary Economics 50 (2003) 1603–1622
1608
where cm t
lnðctþm =ct Þ ; m
ym t
lnðytþm =yt Þ m
1 and spreadtm rm t rt
ð2Þ
and rm t is the m-period interest rate at time t: The second approach applies a probit model to assess the ability of the term spread to predict the probability of a future recession (or upturn). Results are reported by Estrella and Hardouvelis (1991), Estrella and Mishkin (1997), Dotsey (1998) and Bernard and Gerlach (1998), all of whom find that the term spread is a good predictor of the probability of future recessions. The last paper shows that this property holds in all G-7 countries apart from Japan. The probit approach is particularly illuminating for real data, as the obtained probabilities can then be compared with the realized state of the economy. However, to test the LIPTS in simulated economies, Eq. (1) provides a more useful benchmark, which allows us to pin down precisely the nature of the co-movements between the term spread and future consumption or output growth, and which is easily comparable across models. Papers documenting a positive and significant b1 include Estrella and Hardouvelis (1991), Plosser and Rouwenhorst (1994), Estrella and Mishkin (1997) and Dotsey (1998). This result holds for the US, the UK, France, Germany and Italy for forecast intervals ranging from 3 months to 5 years. These papers also estimate some variations of Eq. (1) introducing, for example, the short rate or some known economic leading indicators, but the underlying idea remains similar and b1 is found to be positive and significant throughout. Table 2 shows our estimates for the US economy, where as elsewhere in this paper we follow Plosser and Rouwenhorst (1994) by matching the span of the spread (m) with the length of the forecast interval for activity growth. For forecast horizons ranging from 6 months to 2 years, b1 is positive and significant. This confirms evidence from the literature that nominal term spreads forecast future real consumption and output growth rates. While the studies above use nominal interest rates, the models we present below are couched in real terms. The evidence on the predictive power of real term spreads in the US is established by Harvey (1988). In Table 1, Panel B we show our estimates of correlations between real spreads and activity growth in the US.7 The correlation coefficients are lower than for nominal spreads, but are consistent with the notion of real spreads containing information about future economic activity. Outside of the US, only the nominal spread has been used to study the LIPTS in most countries. One reason for this is the fact that ex ante real rates are not observable, and computing them requires choosing a measure of expected inflation and hence a degree of arbitrariness. Another reason may be an expectation of 7
Real rates are computed as in Harvey (1988), i.e. as the difference between nominal interest rates and expected inflation rates. Expected inflation rates are obtained in the following way. We estimate an ARIMA model (using the Ljung-Box Q-statistic to select the best specification) for each m-period inflation rate for the sample 1960:1 until 1969:1 and then compute m-step-ahead forecasts for the m-period expected inflation rates. As new data are added to the sample, parameters are re-estimated and new forecasts are obtained. We repeat this until the end of the sample, i.e. 1998:4.
ARTICLE IN PRESS C. Rendu de Lint, D. Stolin / Journal of Monetary Economics 50 (2003) 1603–1622
1609
Table 2 Regressions of m-quarter consumption and output growth on m-quarter nominal term spread, see Eq. (1) m
Coefficient
Std. error
t-Statistic
p-Value
Consumption regressions 2 1960:1–1998:2 4 1964:1–1997:4 8 1976:2–1996:4 20 1960:1–1993:4
4.296 1.942 1.230 0.118
1.100 0.305 0.266 0.124
3.91 6.37 4.63 0.95
0.001 0.000 0.000 0.340
Output regressions 2 1960:1–1998:2 4 1964:1–1997:4 8 1976:2–1996:4 20 1960:1–1993:4
4.404 2.284 1.449 0.044
1.360 0.383 0.310 0.109
3.24 5.96 4.68 0.41
0.001 0.000 0.000 0.685
Period
The number of quarters used for term spread and consumption/output growth calculations, the estimation period (quarterly data are used), the estimated coefficient on the term spread, the Newey-West (1987) autocovariance-consistent standard error, the corresponding t-statistic and p-value. Five lags were used in the Newey-West procedure (the t-statistics change little if a different number of lags is used).
comparable predictive ability for real and nominal spreads.8 A recent paper examines the ability of the real spread to predict real consumption growth in Australia (Fisher and Felmingham, 1998) and finds it to be significant. Moreover, following Donaldson et al. (1990), the use of a real model can further be justified based on two observations: (1) Nominal and real rates exhibit very similar properties in artificial economies (see Backus et al., 1989; Danthine et al., 1987), and (2) in reality nominal and real short-term rates tend to move in tandem. The latter is consistent with the observation that short-term nominal rates contain no information about future inflation (Mishkin, 1990). As further justification, Table 1, Panel C shows correlations between the differential in expected inflation rates (the ‘‘spread’’ of inflation expectations) and activity growth to be low, especially at the 2year horizon. This implies a limited role for inflation in explaining the predictive power of nominal spreads and hence reasonableness of abstracting from inflation in seeking to explain the LIPTS.
3. The standard model: a stochastic endowment economy This section outlines the standard model, whose features are common to both models studied in this paper. The production economy extension is studied in the next section. 8
Thus, in studying the predictability of economic growth for G-7 countries, Harvey (1989, Endnote 2) suggests that predictions made with nominal spreads are close to those made with real spreads: ‘‘If inflation follows a first-order integrated moving-average process, the nominal yield spread equals the expected real yield spread.’’ Similarly, ‘‘I assume that the spread between nominal rates is approximately equal to the spread between real rates. [...] Incorporating the term structure of inflation could lead to improved forecasts’’ (Harvey, 1993, p. 7–8).
ARTICLE IN PRESS 1610
C. Rendu de Lint, D. Stolin / Journal of Monetary Economics 50 (2003) 1603–1622
Agents are rational and maximize expected future discounted utility. Their utility depends on consumption (ct ) and is described by a time separable power utility function. The agents are subject to a budget constraint. The endowment is determined exogenously and is subject to a random shock (ut ). Additionally, every period agents can decide to buy or sell m-period real zero-coupon bonds (bm t ; m ¼ 1; y; M) at the prevailing price (pm ). Agents keep the bonds until maturity, t when they pay off in units of endowment (the model is unchanged if agents are allowed to sell or buy back bonds before maturity). Note that this model is a special case of Lucas (1978). Agents solve the following maximization problem: N X c1f 1 max E0 ð3Þ bt t fct g 1f t¼0 subject to ct þ
M X
m pm t bt ¼ yt þ
m¼1
M X
bm tm :
ð4Þ
m¼1
We will also assume the following law of motion for the endowment: lnðytþ1 Þ ¼ r lnðyt Þ þ utþ1 ;
ð5Þ
ut BNð0; s2u Þ
and i:i:d:: In equilibrium agents consume all of the endowment where (ct ¼ yt ) and bonds are in zero net supply (bm t ¼ 0; 8m). Solving (3) subject to (4) and the equilibrium conditions yields the following expression for the price of a bond with maturity m: ctþm f m pm ¼ b E : ð6Þ t t ct 9 m The m-period interest rate rm t is defined as lnðpt Þ=m and can be written as ! 1 ctþm f ln E rm ¼ ln b : t t m ct
The spread becomes: spreadtm
rm t
r1t
! ! 1 ctþm f ctþ1 f ¼ ln Et þ ln Et : m ct ct
ð7Þ
ð8Þ
From the first-order condition (FOC) (6) it can be seen that the bond price is negatively correlated with expected consumption growth, hence the interest rate is positively correlated with the expected consumption growth rate. This well-known property of the endowment economy follows directly from agents trying to smooth out their consumption paths. If agents expect low consumption at t þ m; they buy bonds today which mature at t þ m and provide them with additional endowment. 9
This is the continuously compounded interest rate. Under m-period discrete compounding, the rate 1=m would be given by ð1=pm 1: Differences between these two definitions are small, as long as interest t Þ rates are not too large. The continuous time definition is used to simplify the analytical expressions, as is done for example in den Haan (1995).
ARTICLE IN PRESS C. Rendu de Lint, D. Stolin / Journal of Monetary Economics 50 (2003) 1603–1622
1611
m This drives today’s price of the m-period bond (pm t ) up, and its interest rate (rt ) down. The converse holds when agents expect high consumption, thus explaining the positive correlation between the interest rate and the future consumption growth m rate, Corrðrm t ; ct Þ > 0: Some authors appear to conclude that this observation provides a justification for the LIPTS.10 In other words, they argue that this standard model is able to reproduce the positive correlation illustrated in Fig. 1B between the spread and the consumption growth rate. We can check their assertion by studying the sign of the covariance between the term spread and the consumption growth rate. Analytical expressions can be obtained for the spread, the consumption growth rate and their covariance. The derivations are reported in full in Appendix A. The spread can be expressed as
f spreadtm ¼ ½ðrm 1Þ mðr 1Þ ln ðct Þ m f2 s2u þ ½mð1 r2 Þ ð1 r2m Þ : 2mð1 r2 Þ and the consumption growth rate as " # m1 X 1 cm rj utþmj : ðrm 1Þ lnðct Þ þ t ¼ m j¼0
ð9Þ
ð10Þ
Their covariance is Covðspreadtm ; cm t Þ ¼
m1 s2u f rm 1 X ð1 rj Þ: m2 1 þ r j¼0
ð11Þ
From (11) we see that for m > 1 and a plausible calibration of the model, i.e. ro1; the covariance is negative, contradicting the above claim. In other words, the standard endowment economy fails to explain the LIPTS. This result shows that in an endowment economy, even though real interest rates are positively correlated with consumption growth rates, this property does not necessarily imply that term spreads are positively correlated with consumption growth rates, and cannot be used as a theoretical justification for the LIPTS. To illustrate this result, the model is simulated numerically, with impulse responses of the consumption growth and spread variables to an endowment shock graphed in Fig. 2, while measures of comovement between variables of interest (for different time spans m) are given in Table 3 (Appendix B contains details about the simulations and the calibration of the model). As can be seen from that table, the b1 coefficients obtained from Eq. (1) are negative and significantly different from zero, confirming that the LIPTS is rejected in this model. Further, Corrðspreadtm ; cm t Þ is never positive in any of the 500 simulations. The reason for the rejection of the LIPTS is the following. In an endowment economy, if the autoregressive coefficient (r) for the endowment shock is less than one, then a shock ut has the biggest impact on consumption in the period it occurs. 10
Smets and Tsatsaronis (1997, p. 1), Dotsey (1998, p. 38).
ARTICLE IN PRESS 1612
C. Rendu de Lint, D. Stolin / Journal of Monetary Economics 50 (2003) 1603–1622 Panel: A Impulse response for consumption 0.3
Panel: B Impulse response for spread spread2 spread4 0.3 spread24
spread40
0.2
0.2
0.1
0.1 1
6
11
16
1
6
11
16
Fig. 2. Impulse response functions for the endowment economy. The number of quarters is shown on the x-axis.
Table 3 Features of the stochastic endowment economy. This table summarizes 500 simulations, as described in Appendix B m¼2
m ¼ 20
m ¼ 40
Measures of co-movement between spread and consumption growth Corrðspreadtm ; cm 0.222 0.306 0.412 t Þ Percent Corrðspreadtm ; cm 0.0 0.0 0.0 t Þ>0 b1 18.380 6.179 2.423 7.803 2.397 0.875 SD ðb1 Þ
0.569 0.0 0.773 0.214
0.661 0.0 0.312 0.071
Additional correlations Corrðspreadtm ; ut Þ Corrðcm t ; ut Þ
0.313 0.177
0.313 0.207
0.313 0.069
m¼4
0.313 0.096
m¼8
0.313 0.127
The variables m; spreadtm ; cm t ; and ut are defined in the text. The variable b1 is the coefficient in the regression of consumption growth on the spread (Eq. (1)), and SD ðb1 Þ is its standard deviation across the 500 simulations. The row labelled ‘‘Percent’’ shows the percentage of time the variable studied was positive in the 500 simulations.
This implies that the impulse response function for consumption is at its peak when the positive shock occurs, and then declines following an AR(1) process (Fig. 2A). Consequently, after a one-off positive shock ut (and no further shocks), future consumption growth rates, as proxied by the slope of the impulse response function for consumption, are negative and Corrðcm t ; ut Þo0: Recall from (6) that bond prices are determined by expected consumption growth rates, represented by the slope of the impulse response function for consumption. In Fig. 2A, since this function is downward-sloping, consumption growth rates are negative and agents have an incentive to buy bonds. Moreover, since the impulse response function is convex, this implies that the 1-period consumption growth rate is more negative than the mperiod growth rate. Agents therefore need to buy more short-term than long-term bonds to smooth their consumption stream, thus r1t falls more than rm t and the spread rises. Hence, contrary to the LIPTS, the rise in the spread after a one-off positive shock is associated with a negative future consumption growth rate.
ARTICLE IN PRESS C. Rendu de Lint, D. Stolin / Journal of Monetary Economics 50 (2003) 1603–1622
1613
Put differently, in an endowment economy, shocks have the biggest impact on the consumption growth rates, or marginal rates of consumption substitution, in the first period. This means that after a shock, c1t is always larger in absolute value than cm t ; 1 m and cm c always moves in opposite direction from c : This finding is not specific t t t to the chosen law of motion (5). Even when allowing for growth in the consumption process, i.e. assuming that consumption is integrated of order one, the LIPTS is still rejected. Assuming the law of motion where lnðytþ1 =yt Þ ¼ b lnðyt =yt1 Þ þ utþ1 ; also leads to a rejection of the LIPTS (see Section 5 for a more detailed discussion). It appears that in a model where the impulse response function for consumption peaks in the first period, the term spread fails to predict future economic activity in the required direction, suggesting the need to extend our model.
4. Stochastic production economy Motivated by the failure of the endowment economy to account for the LIPTS, we extend the standard model to include endogenous production, with capital as the unique production factor. In this economy, agents can decide in every period whether to consume all the available output, or to save part of it and invest it into capital (i.e. future production). Adding endogenous production is a natural and straightforward extension, and one which does not introduce additional degrees of freedom (the capital depreciation rate and the capital share are pinned down from the data in a way similar to the literature on real business cycles).11 Formally, the economy of Section 3 is augmented in the following way. Agents can now consume (ct ) and invest (it ) in the production factor capital (kt ). Production (yt ) is subject to an autocorrelated productivity shock (yt ). Agents maximize the utility function (3) but are now subject to the following budget constraint: ct þ i t þ
M X
m pm t bt ¼ yt þ
m¼1
M X
bm tm ;
ð12Þ
m¼1
where it ¼ kt ð1 dÞkt1 ;
ð13Þ
a yt ¼ yt kt1 :
ð14Þ
and Eq. (13) describes the capital accumulation process, with d being the constant depreciation rate of capital, while (14) is a Cobb-Douglas production function where the labor input is held constant and normalized to one, and a is the share of capital in total output. The productivity shock is driven by lnðyt Þ ¼ r lnðyt1 Þ þ ut ;
ð15Þ
11 den Haan (1995) compares the performance of this model with models including additionally variable labour supply or money, and shows that this model is best able to replicate the empirical bond price behaviour.
ARTICLE IN PRESS 1614
C. Rendu de Lint, D. Stolin / Journal of Monetary Economics 50 (2003) 1603–1622
where ut BNð0; s2u Þ and i:i:d:: In equilibrium, bonds are in zero net supply ðbm t ¼ 0; 8mÞ: Solving (3) subject to (12), (13), (14) and (15) yields equation (6) for the price of a k-period bond, and an additional first-order condition for consumption: a1 cf ¼ bEt ðcf þ 1 dÞÞ: t tþ1 ðytþ1 akt
ð16Þ
Since bond prices still depend on the marginal rate of consumption substitution, it follows that a necessary condition for this model to explain the LIPTS is that its impulse response function for consumption should not be monotone decreasing as it was in the endowment model. Exact analytical solutions cannot be obtained in this case, and the model is solved numerically. The parameterized expectations algorithm of den Haan and Marcet (1990) is used, whereby the conditional expectations in (6) and (16) are approximated using exponentiated polynomials; the solution is obtained by iterating on the polynomials’ coefficients until the expectations become good predictors.12 Five hundred simulations are conducted as before, with the parameters chosen to match US quarterly data and based on den Haan (1995): b ¼ 0:99;
f ¼ 3;
d ¼ 0:025;
r ¼ 0:95;
a ¼ 0:33;
su ¼ 0:018
The results are reported in Table 4, which summarises co-movement among the spread, activity growth variables, and the productivity shock. Three features of the model stand out: (1) the term spread is a significant predictor of future output growth at all forecast horizons, (2) the term spread fails to predict consumption growth (in the right direction) in the short run, but (3) the term spread predicts it for longer forecast horizons (over 10 years). Hence, the production economy is able to match the LIPTS for output growth for all forecast horizons, and for consumption growth for long-term horizons. This is a key finding since it demonstrates the ability of a simple model to match an important property of real data, and to provide it with a rigorous theoretical foundation. Note again that this ability does not follow from the FOC (6) as the literature sometimes suggests. Both the endowment and the production economy have the same FOC for the bond price, yet only the latter reproduces the LIPTS. The main mechanism triggering this positive result is the ability of agents to substitute consumption intertemporally. In the production economy when a positive shock occurs, agents increase their contemporaneous consumption, but not by the full amount of the shock. Rather, in an effort to smooth their consumption path, they save part of the shock and invest it in capital. The higher capital stock then allows higher production and higher consumption in the future. Therefore, contrary to the endowment economy case, a shock does not exert its biggest impact on consumption in the period it occurs, but rather sees its effect extended and magnified through the investment channel. Thus, the impulse response function for consumption is now hump-shaped, explaining the better performance of this model. Fig. 3 shows the impulse response functions for consumption, output and the spread. 12
Marcet and Lorenzoni (1999) further discuss applications of this algorithm.
ARTICLE IN PRESS C. Rendu de Lint, D. Stolin / Journal of Monetary Economics 50 (2003) 1603–1622
1615
Table 4 Features of the stochastic production economy. This table summarises 500 simulations, as described in Appendix B m¼2
m ¼ 20
m ¼ 40
m ¼ 44
m ¼ 48
Measures of co-movement between spread and consumption growth Corrðspreadtm ; cm 0.076 0.117 0.122 0.073 t Þ Percent Corrðspreadtm ; cm 0.0 0.0 0.0 1.0 t Þ>0 1.010 0.523 0.182 0.085 b1 ðcm t Þ SD ðb1 ðcm 6.874 1.447 0.645 0.287 t ÞÞ
0.001 65.2 0.078 0.161
0.011 60.2 0.076 0.110
0.020 63.2 0.076 0.108
Measures of co-movement between spread and output growth Corrðspreadtm ; ym 0.126 0.156 0.209 t Þ Percent Corrðspreadtm ; ym 100.0 100.0 100.0 t Þ>0 b1 ðym 27.840 5.599 2.477 t Þ SD ðb1 ðym 17.140 3.681 1.548 t ÞÞ
0.287 100.0 0.896 0.519
0.322 96.4 0.515 0.277
0.325 100.0 0.473 0.248
0.326 100.0 0.458 0.224
Additional correlations m Corrðcm t ; yt Þ ; u Corrðcm tÞ t Corrðym t ; ut Þ Corrðspreadtm ; ut Þ
0.918 0.038 0.096 0.367
0.921 0.001 0.119 0.368
0.923 0.007 0.123 0.368
0.924 0.013 0.126 0.368
0.977 0.045 0.034 0.349
m¼4
0.960 0.056 0.047 0.362
m¼8
0.938 0.060 0.064 0.364
m m The variables spreadtm ; cm t ; yt and ut are defined in the text. The variable b1 ðct Þ is the coefficient in the regression of consumption growth on the spread (Eq. (1)), and SD ðb1 ðcm t ÞÞ is its standard deviation across m the 500 simulations. The variables b1 ðym t Þ and SD ðb1 ðyt ÞÞ denote the corresponding quantities for output growth. The rows labelled ‘‘Percent’’ show the percentage of time the variable studied was positive in the 500 simulations.
Before discussing how this model relates to the empirical evidence on the LIPTS, we explain each of the three results mentioned above. First, why is the LIPTS rejected in the short-term for consumption? In Fig. 3A, the impulse response function for consumption is rising for the first 4 years; implying that the consumption growth rate is positive. Yet, since the function is concave, marginal rates of substitution are declining. In other words, c1t is larger than cm t ; and due to (7), r1t rises more than rm t ; hence the spread falls and fails to predict the future rise in consumption. Why, then, does the LIPTS hold in the long-term? Notice in Fig. 3A, that after 9 years (point B), the impulse response function falls to below its original level: the value of the function at t þ 37 is smaller than at t; so that for horizons above 9 years the consumption growth rate is negative. Hence, the fall in the long-term spread correctly predicts the future long-term decline in consumption, and the LIPTS holds for horizons over 10 years.13 13
Based on Table 4 it may seem that only long-term spreads contain some predictive information about future consumption. This is not entirely correct. Following a positive shock, all term spreads fall, independently of their maturity. In fact, the impulse response function for spreadtm and Corrðspreadtm ; ut Þ are almost invariant with respect to m: This implies that long-term and short-term spreads have nearidentical predictive power with respect to long-term consumption growth. Similarly, both long-term and short-term spreads fail badly to forecast the initial rise in consumption.
ARTICLE IN PRESS 1616
C. Rendu de Lint, D. Stolin / Journal of Monetary Economics 50 (2003) 1603–1622
Panel A: Impulse response for consumption Panel B: Impulse response for output 0.14
0.22
A
0.13
0.2
0.12 B 0.11
0.18 0.16
0.1 0.14
0.09
0.12
0.08 1 6 11 16 21 26 31 36 41 46
1
6
11
16
Panel C: Impulse response for spread 1 11 16 6 -0.1
-0.2 spread2 -0.3
spread4 spread24
-0.4
spread48
Fig. 3. Impulse response functions for the production economy. The number of quarters is shown on the x-axis.
Lastly, why does the LIPTS hold for output? Unlike for consumption, where agents are able to smooth out the benefits of a positive productivity shock by saving part of it and investing it into capital, the impact of an isolated productivity shock on output is greatest in the first period, and then decreases following an AR(1) process. In Fig. 3B this is reflected by the impulse response function for output which is at its a peak in the first period and then declining. Hence the output growth rate, as captured by the slope of the impulse response function, is negative. The declining spreads correctly predict this future fall in output. Table 4 shows that, except in the short-term, the coefficients for output, b1 ðym t Þ; are positive and significantly different from zero. Output growth rates are less volatile the longer the forecast horizon, explaining the fall in magnitude observed in the b1 coefficients. To gain further insight into the empirical LIPTS, consider its origin in the production economy. A positive productivity shock implies a positive wealth effect for the agents and increases consumption at all future dates. As we saw in Fig. 3A, the impulse response function for consumption rises until 4 years after the shock because agents are smoothing out their consumption by saving part of the output and investing into capital. This intertemporal consumption substitution is induced by the high interest rates, which follow immediately after the positive shock. However, this expansionary phase is expected to end once the accumulated capital,
ARTICLE IN PRESS C. Rendu de Lint, D. Stolin / Journal of Monetary Economics 50 (2003) 1603–1622
1617
together with a lower productivity level, bring about a fall in the marginal product. Agents then start dissaving, interest rates fall below their long-term level and consumption growth rates return to their steady-state level.14 Since interest rates are expected to fall in the future, this implies today (under the expectations hypothesis) that short-term rates rise more than long-term rates. As a result, the spread goes down today and if agents’ expectations turn out to be true, and the future productivity level is lower than today’s level, then today’s fall in the spread will have correctly predicted this future slowdown. This suggests that for the LIPTS to hold in an economy, agents must be able to substitute consumption intertemporally in reaction to productivity changes.
5. Discussion The aim of this paper is to demonstrate that while an endowment economy is unable to explain the LIPTS, a model with production goes a long way in matching its empirical properties. In this section, we take a close look at the chosen approach and how it affected the findings. Specifically, we discuss the extent to which the failure of the endowment economy followed from the chosen law of motion and outline possible future research directions. We start with the failure of the endowment economy to reproduce the LIPTS. An obvious objection to this result is to question whether it is only driven by the chosen law of motion and in particular, whether it is due to the empirically counterfactual property that after a positive disturbance, the conditional mean of all future consumption growth rates is negative. We find that even with a higher-order process for consumption, the endowment economy still fails to produce the LIPTS. Let us assume, for instance, as is done in the real business cycle literature, that consumption follows a process that is integrated of order one: Dct ¼ a þ z Dct1 þ et ;
ð17Þ
where et BNð0; s2e Þ and i:i:d:: This model still fails. This time, the impulse response for consumption is increasing, so that expected consumption growth rates are positive after a positive shock. However, the essence of the problem remains unchanged. The function is concave, implying, as seen throughout this paper, that the one-period interest rate will rise more than the m-period interest rate and therefore that spreadtm will fall. As a result, the spread will be negatively correlated with the corresponding consumption growth rate. Higher-order dynamics alone do not necessarily resolve this problem for the endowment economy. Clearly, the exogenous consumption process in the endowment economy can be fixed to be identical to the one resulting in the production economy, allowing the endowment economy to produce the LIPTS for consumption growth at long-term horizons. It is also clear that this would not be a satisfying reconciliation of the endowment economy with the LIPTS. 14
See Rouwenhorst (1995, pp. 308–309) for a detailed description.
ARTICLE IN PRESS 1618
C. Rendu de Lint, D. Stolin / Journal of Monetary Economics 50 (2003) 1603–1622
It is important to stress that although the production economy is the more successful model, it still does not match the features of the empirical LIPTS perfectly. Indeed, for consumption growth, the correlation with the spread is positive only for horizons longer than 10 years, while in reality the predictive power of the term spread holds for much shorter durations (up to 5 years). Note that this particular shortcoming could be resolved if the impulse response function for consumption peaked earlier. Thus, while showing a possible direction for replicating the LIPTS, the production economy we studied here clearly provides only a partial answer. Further, a researcher seeking to match the empirical LIPTS by finding the right output and consumption dynamics will need to address the following issue: the two dynamics will have to be different (as suggested by the better results generated in a production economy), yet still be highly correlated with each other to match the empirical evidence, where correlations are on the order of 0.90. As can be seen from Table 4, the production economy reproduces this feature rather well. Finally, it is well known that FOC (6) implies a positive correlation between real interest rates and the corresponding consumption growth rates. In our model calibration, these correlations vary from 0.1 to 0.6 depending on the duration. This observation is consistent with empirical evidence, as reported in Table 1, Panel E and in Harvey (1988). What is counterfactual, however, is that in the production model, the correlation of real yields with consumption growth rates (and thus the predictive power of the real yields) is much higher than that of real spreads. In other words, while spreads are generally better predictors of future activity growth than yields in reality (cf. Table 1), in the stochastic production economy yields are the better predictors. This probably underlines that, unlike in the model, there is more than one source of uncertainty in the real world.
6. Concluding remarks The aim of this paper was to provide a theoretical basis for the empirically supported leading indicator property of the term spread (LIPTS). Dynamic equilibrium asset pricing models appeared natural candidates to study the joint behaviour of real macroeconomic variables (consumption and output) and asset prices. We studied two models, a stochastic endowment economy and a production economy. The first contribution of this paper is to show that not all dynamic equilibrium asset pricing models can explain the LIPTS. Contrary to what is sometimes assumed and perhaps counter-intuitively, an endowment economy does not generate the LIPTS. This is because after any shock, short-term interest rates always react more than long-term rates, resulting in a negative correlation between the term spread and future consumption growth rates. The second contribution is to show that a simple production economy, on the other hand, goes a long way in explaining the LIPTS. This positive result permits greater understanding of what may be driving the LIPTS in actuality. Indeed, one
ARTICLE IN PRESS C. Rendu de Lint, D. Stolin / Journal of Monetary Economics 50 (2003) 1603–1622
1619
mechanism appears to be crucial: agents must be able to substitute consumption intertemporally in reaction to productivity changes.
Appendix A. Analytical derivations This appendix contains the derivation of Eqs. (9), (10) and (11) presented in Section 3. The interest rate is given by Eq. (7): ! 1 ctþm f m rt ¼ ln b ln Et : ðA:1Þ m ct We need to find an expression for Et ðctþm =ct Þf : In equilibrium ct ¼ yt : Substituting backwards for lnðctþm Þ (m ¼ 1; 2; y; M) in the law of motion (5), we obtain the following expression for lnðctþm Þ: lnðctþm Þ ¼ rm lnðct Þ þ
m1 X
ri utþmi :
ðA:2Þ
i¼0
Taking the exponential of Eq. (A.2) yields ! m1 X rm i ctþm ¼ ct exp r utþmi : i¼0
We then divide by ct ; raise the resulting expression to the power ðfÞ and take expectations: " " !# # m1 X ctþm f fðrm 1Þ i Et Et exp f r utþmi : ¼ ct ðA:3Þ ct i¼0 P i Since ut is distributed normally with mean 0 and variance s2u ; f m1 i¼0 r utþmi is 2 2 Pm1 2i distributed normally with mean 0 and variance f su i¼0 r ; and therefore P Pm1 2i
i 1 2 2 expðf m1 : i¼0 r utþmi Þ is distributed lognormally with mean exp 2 f su i¼0 r Hence, Eq. (A.3) becomes " ! # m1 ctþm f 1 2 2X fðrm 1Þ exp f su r2i ¼ ct Et 2 ct i¼0 which can be rewritten as " # ctþm f 1 2 1 r2m 2 fðrm 1Þ Et exp f s : ¼ ct 2 ct 1 r2 u Substituting (A.4) into (A.1) yields rm t ¼ ln b þ
1 1 2 1 r2m 2 fðrm 1Þ lnðct Þ f s m 2m 1 r2 u
ðA:4Þ
ARTICLE IN PRESS C. Rendu de Lint, D. Stolin / Journal of Monetary Economics 50 (2003) 1603–1622
1620
and for the spread, f spreadtm ¼ ½ðrm 1Þ mðr 1Þ lnðct Þ m f2 s2u þ ½mð1 r2 Þ ð1 r2m Þ : 2mð1 r2 Þ which corresponds to Eq. (9). Consumption growth is defined as 1 ½lnðctþm Þ lnðct Þ : cm t ¼ m Inserting (A.2) into (A.6) gives " # m1 X 1 m m i ðr 1Þ lnðct Þ þ ct ¼ r utþmi m i¼0
ðA:5Þ
ðA:6Þ
ðA:7Þ
which corresponds to Eq. (10). We now turn to the covariance between the spread and the consumption growth m rate (Covðspreadtm ; cm t Þ). First of all we show that Eðct Þ ¼ 0: From the law of motion for the endowment (5), and since in equilibrium ct ¼ yt ; we can write t X lnðct Þ ¼ ri uti ; i¼0
so that E½lnðct Þ ¼ E
"
N X
# i
r uti ¼ 0:
ðA:8Þ
i¼0
From (A.7) and (A.8), we see that Eðcm t Þ ¼ 0: Therefore, m m Covðspreadtm ; cm t Þ ¼ Eðspreadt ct Þ:
Using (A.5), (A.7), (A.8) and the distribution properties of the shock ut ; a few algebraic steps are sufficient to show that
m f m ðr 1Þ s2u m m ðr 1Þ ðr 1Þ : ðA:9Þ Covðspreadt ; ct Þ ¼ m m ð1 r2 Þ This expression can be rewritten as m1 s2u f rm 1 X m m Covðspreadt ; ct Þ ¼ 2 ð1 r j Þ: m 1 þ r j¼0 which corresponds to Eq. (11) in the text.
Appendix B. Numerical simulations For the numerical simulations, our model is calibrated to match US quarterly data, and except for the risk aversion coefficient (f) the parameters are chosen as in
ARTICLE IN PRESS C. Rendu de Lint, D. Stolin / Journal of Monetary Economics 50 (2003) 1603–1622
1621
den Haan (1995): b ¼ 0:99;
f ¼ 3;
r ¼ 0:95;
su ¼ 0:007:
In the asset pricing literature, authors tend to choose relatively high risk aversion coefficients to obtain more significant risk premia. Since the purpose of this paper is to investigate whether a standard model is able to explain the LIPTS, a lower f is selected. Rouwenhorst (1995) suggests ½1; 5 as a range of plausible values for f: Throughout the paper, we work with plausible parameters, rather than selecting ones for which the model fits the LIPTS. The economy is simulated 500 times over 5000 periods, and the reported values are average values from the 500 repetitions. This approach is applied to all models studied in this paper.
References Backus, D., Gregory, A., Zin, S., 1989. Risk premiums in the term structure: evidence from artificial economies. Journal of Monetary Economics 24, 371–399. Bernard, H., Gerlach, S., 1998. Does the term structure predict recessions? The international evidence, CEPR Discussion Paper 1892. Campbell, J., 1986. Bond and stock returns in a simple exchange model. Quarterly Journal of Economics 101, 785–803. Canova, F., De Nicolo, G., 1997. Stock returns, term structure, inflation and real activity: an international perspective, CEPR Discussion Paper 1614. Chapman, D., 1997. The cyclical properties of consumption growth and the real term structure. Journal of Monetary Economics 39, 145–172. Clark, K., 1996. A near-perfect tool for economic forecasting. Fortune, July 22, 24–26. Cox, J., Ingersoll, J., Ross, S., 1985. A theory of the term structure of interest rates. Econometrica 53, 385–407. Danthine, J.P., Donaldson, J.B., Smith, L., 1987. On the superneutrality of money in a stochastic dynamic macroeconomic model. Journal of Monetary Economics 20, 475–499. den Haan, W.J., 1995. The term structure of interest rates in real and monetary economies. Journal of Economic Dynamics and Control 19, 909–940. den Haan, W.J., Marcet, A., 1990. Solving the stochastic growth model with parameterised expectations. Journal of Business and Economic Statistics 8, 31–34. Donaldson, J., Johnsen, T., Mehra, R., 1990. On the term structure of interest rates. Journal of Economic Dynamics and Control 14, 571–596. Dotsey, M., 1998. The predictive content of the interest rate term spread for future economic growth. Federal Reserve Bank of Richmond. Economic Quarterly 84, 31–51. Estrella, A., Hardouvelis, G., 1991. The term structure as a predictor of real economic activity. Journal of Finance 46, 555–576. Estrella, A., Mishkin, F., 1997. The predictive power of the term structure of interest rates in Europe and the United States: implications for the European Central Bank. European Economic Review 41, 1375–1401. Fisher, C., Felmingham, B., 1998. The Australian yield curve as a leading indicator of consumption growth. Applied Financial Economics 8, 627–635. Hamilton, J.D., Kim, D.H., 2000. A re-examination of the predictability of economic activity using the yield spread. NBER Working Paper 7954. Hansen, L.P., Singleton, K.J., 1983. Stochastic consumption, risk aversion, and the temporal behavior of asset returns. Journal of Political Economy 91, 249–265.
ARTICLE IN PRESS 1622
C. Rendu de Lint, D. Stolin / Journal of Monetary Economics 50 (2003) 1603–1622
Harvey, C.R., 1988. The real term structure and consumption growth. Journal of Financial Economics 22, 305–333. Harvey, C.R., 1989. The term structure and world economic growth. Journal of Fixed Income 1, 7–19. Harvey, C.R., 1993. Term structure forecasts economic growth, Financial Analysts Journal, May-June, 6–8. Kosicki, S., 1997. Predicting real growth and inflation with the yield spread. Federal Reserve Bank of Kansas City. Economic Review 82, 39–57. Labadie, P., 1994. The term structure of interest rates over the business cycle. Journal of Economic Dynamics and Control 18, 671–697. Longstaff, F., Schwartz, E., 1992. Interest rate volatility and the term structure: a two-factor general equilibrium model. The Journal of Finance 47, 1259–1282. Lucas, R., 1978. Asset prices in an exchange economy. Econometrica 46, 1429–1445. Marcet, A., Lorenzoni, G., 1999. The parameterized expectations approach: some practical issues. In: Marimon, R., Scott, A. (Eds.), Computational Methods for the Study of Dynamic Economies, Oxford University Press, Oxford. Mishkin, F., 1990. What does the term structure tell us about future inflation? Journal of Monetary Economics 25, 77–95. Newey, W., West, K., 1987. A simple positive semi-definite heteroskedasticity and autocorrelation consistent matrix. Econometrica 55, 703–708. Plosser, C.I., Rouwenhorst, K.G., 1994. International term structures and real economic growth. Journal of Monetary Economics 33, 133–155. Rouwenhorst, K.G., 1995. Asset pricing implications of equilibrium business cycle models. In: Cooley, T. (Ed.), Frontiers of Business Cycle Research (Princeton University Press, Princeton, NJ). Smets, F., Tsatsaronis, K., 1997. Why does the yield curve predict economic activity? Dissecting the evidence for Germany and the United States, CEPR Discussion Paper 1758.