Durable Goods, Inflation Risk,
and Equilibrium Asset Prices
Bjørn Eraker Ivan Shaliastovich and Wenyu Wang
∗
June 2015
Abstract
High expected inflation is known to predict low future real growth. We show
that such predictability is significantly more pronounced in durable relative to
non-durable goods sectors of the economy. Consistent with this macroeconomic evidence, the equity returns of durable-goods-producing firms have a
larger negative exposure to expected inflation risks. We estimate a two-good
recursive utility model which features persistent growth fluctuations and inflation non-neutrality for durable and non-durable consumption. Our model
can quantitatively account for the levels and volatilities of bond and equity
prices, and correlations of equity returns with bond returns and with expected
inflation.
∗
Bjørn Eraker ([email protected]) is at the Wisconsin School of Business, University of
Wisconsin, Madison. Wenyu Wang ([email protected]) is at Kelley School of Business, Indiana University. Ivan Shaliastovich (corresponding author) is at Wharton School, University
of Pennsylvania, 3620 Locust Walk, Philadelphia, PA 19104. Phone: (215) 746-0005, email:
[email protected]. We thank the Journal Editor Geert Bekaert, two anonymous referees,
and Andrew Abel, Torben Andersen, Ravi Bansal, John Campbell, Anna Cieslak, Max Croce,
Michelle Harasimowicz, Urban Jermann, Stijn Van Nieuwerburgh, Mark Ready, Nick Roussanov,
Nicholas Souleles, Viktor Todorov, Max Ulrich, Pietro Veronesi, Wei Yang, and seminar and conference participants at the AFA 2013, MFA 2012, the Federal Reserve Board, Indiana University,
Kellogg School of Business, University of Warwick, Wharton, and Wisconsin School of Business for
helpful comments.
1
The impact of inflation risk on asset prices is one of the long-standing questions
in finance. A traditional view is that unlike nominal bonds, claims to real assets such
as stocks serve as a hedge against inflation and therefore should demand a negative
inflation premium. However, empirically, shocks to expected inflation depress both
bond and stock prices (see e.g., Fama and Schwert 1977 and Bekaert and Wang 2010).
A large number of studies, going back to Fama (1981), economically explain a negative
co-movement of stock prices with expected inflation building on the empirical evidence
that expected inflation has a non-neutral and negative effect on real growth. This
inflation non-neutrality channel can account for a negative exposure of stock prices
to expected inflation risks, and further leads to a positive inflation risk premium in
nominal bonds and equities.
In this paper, we provide novel evidence that the magnitude of the inflation nonneutrality varies substantially across economic sectors: it is significantly more pronounced in durable goods sectors, relative to non-durable goods sectors and the aggregate economy. Consistent with this macroeconomic evidence, the equity returns of
durable-goods-producing firms are more sensitive to expected inflation risks and have
a larger correlation with nominal bond returns than those of firms in the non-durable
goods sector. We set up and estimate an economic model which accommodates the
inflation non-neutrality for durable and non-durable consumption. Our model can
quantitatively account for the key features of the nominal bond and equity price data
and in particular, the difference in levels, volatilities, and the correlations of equity
returns in durable and non-durable goods sectors with expected inflation and bond
returns.
The key empirical motivation for our paper is that the response of future real
growth to expected inflation is negative and more pronounced in durable goods sectors
2
of the economy. We document this using direct regressions of future real growth rates
on current inflation, as well as the estimates from a macro-econometric VARMA(1,1)
model. Our macro-econometric model provides a parsimonious description for the exogenous macroeconomic factors, such as the growth rates of durable and non-durable
real consumption, inflation, and dividend growth rates for durable- and non-durable
goods producing firms. The model also includes unobservable factors that represent
the conditional expectations of future consumption and factors. We estimate the
model by Bayesian Markov Chain Monte Carlo (MCMC) methods without any of
the asset-price data, and show that it provides a very good fit to the macroeconomic
variables. We find that expected durable consumption is much more persistent than
expected non-durable consumption growth, consistent with Yogo (2006) and Yang
(2011). Durable consumption growth is also much more exposed to expected inflation news, and drops significantly more in the future in response to high expected
inflation relative to non-durable consumption.
Motivated by these empirical observations, we formulate a model of nominal stock
and bond prices based on long run risk. Our model extends the existing specifications
in Bansal and Yaron (2004), Piazzesi and Schneider (2006), Eraker (2006), Yang
(2011), Hasseltoft (2012), and Bansal and Shaliastovich (2013) to a nominal two-good
economy. In our model, the agent has a recursive utility defined over the consumption
of durable and non-durable goods. Because high expected inflation negatively impacts
future consumption and dividends, expected inflation risk is priced. As the impact of
expected inflation is much larger for durable than non-durable growth, the amount of
inflation risk significantly increases in our two-good economy relative to other models
in the literature which feature a single non-durable consumption good.
3
To assess the model performance, we fix the first-step estimates of the parameters
of the macro-econometric model and estimate the remaining preference parameters by
targeting a number of asset-pricing moments. We show that our model successfully
captures key features of the asset-price data. It closely matches the unconditional
level, slope, and curvature of the nominal interest rates in the data and it generates a
small discrepancy between the observed trajectory of nominal rates and model-implied
rates. On the equity side, we follow Gomes, Kogan, and Yogo (2009) and construct
portfolios of stocks that we classify as durable or non-durable goods producers. In the
model, durable-good producers are riskier and have higher equity premium and equity
return volatility than non-durable firms. These model implications are consistent with
the data.
Empirically we find that expected inflation is one of the key risk factors in our
model, which explains a large part of the variation in short term rates and even more
variation in long term rates. The exposure to expected inflation risks produces a large
and positive inflation premium for long term bonds, and an upward sloping nominal
yield curve in the model. The model mechanism is similar to Piazzesi and Schneider
(2006), but our results are obtained without a large risk aversion parameter, as the
amount of the inflation premium in our two-good economy is significantly larger than
in an economy with a single consumption good. We find that high expected durable
growth lowers yields in the data and in the model, and has a sizeable economic impact
on bond risk premia at long horizons.
The expected inflation channel also generates a positive correlation between stock
and bond returns. A positive shock to expected inflation, the main driver of bond
returns, increases nominal yields and therefore leads to a negative bond return. Since
expected inflation shocks are negatively correlated with future real dividends, a pos4
itive shock also leads to lower equity prices. Because durable cash flows are more
exposed to inflation risks, the magnitude of the correlation of stock returns with expected inflation, and hence of stock returns and bond returns, is larger (in absolute
value) for durable than for non-durable equities.
To the best of our knowledge, our paper is the first to consider a nominal two-good
long-run risk model to study the cross-section of equity and bond returns. In the context of the long-run risks literature, our paper is related to Yang (2011) who considers
a two-good real economy, and emphasizes the importance of durable consumption to
identify persistent economic risks for the asset markets. Gomes et al. (2009) show
the implications of durable consumption risk for the real equity returns in durable
and non-durable sectors. Eraker (2006) and Piazzesi and Schneider (2006) highlight
the role of inflation non-neutrality to account for the features of nominal bond prices,
and Kung (2015) proposes an economic mechanism for the inflation non-neutrality in
the production setting with an endogenous growth. Hasseltoft (2012) and Bansal and
Shaliastovich (2013) consider conditional movements in the bond risk premia, related
to the fluctuations in macroeconomic uncertainty, while Hasseltoft and Burkhardt
(2012) and Song (2014) relate the time-variation in the correlation of stock and bond
returns to the changes in macroeconomic regimes. In an international setting, Lustig
and Verdelhan (2007) and Colacito and Croce (2013) consider long-run risk models
with multiple goods to study currency risk premia, and Guo and Smith (2010) show
the importance of persistent durable risks in the UK data. Our paper differs from
this literature in our attention to the durable-goods channel as an important source
of inflation risk for the bond and equity markets.
In the context of the habits model, Wachter (2006) consider the role of the timevarying risk aversion for bond prices and bond risk premia, while Bekaert, Engstrom,
5
and Xing (2009) and Bekaert and Engstrom (2010) show that the fluctuations in the
risk aversion play an important role to jointly account for the bond and equity market
features. Eichenbaum and Hansen (1990) and Ogaki and Reinhart (1998) explore
asset pricing settings with multiple goods using time-additive preferences. We show
that recursive preference structure plays a significant role to match the key features
of the asset prices. In the context of the no-arbitrage literature, Koijen, Lustig, and
Nieuwerburgh (2012) develop a reduced-form asset-pricing model to jointly account
for a cross-section of bond and equity returns.
The remainder of the paper is organized as follows. In the next section, we present
empirical evidence for the inflation non-neutrality, and discuss the macro-econometric
model for the exogenous macroeconomic variables. Section 2 provides the equilibrium
asset-pricing model. Section 3 presents our empirical results, and Section 4 discusses
robustness checks. Conclusion and Appendix follow.
1.
1.1
Empirical Motivation
Data
We obtain nominal expenditures on non-durable and durable goods and services and
price levels of non-durable and durable goods from the Bureau of Economic Analysis
(BEA).1 The data are seasonally adjusted by the BEA.2 We deflate aggregate nominal
1
Bils (2009) and Bils and Klenow (2001) argue that the CPI inflation series released by the
Bureau of Labor Statistics is mis-measured because consumers shift purchases of durable goods
items from old to higher quality new models; see also the Boskin Commission Report (Boskin,
Dulberger, Gordon, Griliches, and Jorgenson (1996)) and references therein. In our implementation,
as in Piazzesi, Schneider, and Tuzel (2007), we use non-durable consumption as the numeraire and
use the inflation rate for non-durable goods.
2
We checked that our results are robust to alternative seasonality adjustment procedures.
6
series by the appropriate price levels and divide by the total population to obtain real
per-capita quantities. Since the BEA only reports the year-end durable goods stock
levels, we back out the quarterly durable goods stock level using the depreciation and
expenditure data as in Yogo (2006).
In terms of the asset price data, we collect prices of zero-coupon U.S. Treasury
bonds from CRSP. We further construct equity portfolios of durable- and non-durablegood producing firms. Following Gomes et al. (2009), we use the benchmark inputoutput accounts table in the BEA to identify industries whose final demand has
highest value added to personal consumption expenditures on non-durable goods and
services (non-durable sector portfolio) and on durable goods (durable sector portfolio).3 We collect price and cash flow data for these portfolios using CRSP database.
Our empirical analysis is conducted for a 1963Q1 to 2006Q4 sample period. Notably, our sample excludes the Great Recession and Financial Crisis. From an economic perspective, the crisis period is quite special. It features significant movements
in risk premia and tail risk, and ascribes a potentially important role for leverage,
default, and liquidity considerations. In addition to that, the post 2008 period also
exhibits close to zero lower bound nominal rates and unconventional monetary policy
of the Federal Reserve. All these margins can play an important role for asset valuations. However, they are outside of the scope of our economic model, which instead
emphasizes small but persistent low-frequency risks in the economy. For this reason,
we restrict our sample to pre-crisis observations, and leave the analysis of the crisis
for future research.
3
Gomes et al. (2009) consider separately services, non-durables, durables, and investment industries. To be consistent with our model, we aggregate services and non-durables into a single
value-weighted non-durable portfolio. Our results are similar, and in many cases are even stronger,
using equal weights, or using only non-durable-good producing firms in the portfolio.
7
1.2
Regression evidence
In this section we present the initial empirical evidence that helped us motivate our
asset pricing model. Table 1, Panel A, shows the regression coefficients and corresponding R2 s in the projections of future average real growth rates of durable and
non-durable consumption onto the current inflation. There are two important findings: (1) The slope coefficients are negative, implying that a positive shock to inflation
today predicts lower real growth in the future. (2) The slope coefficients are more
negative for durable than for non-durable consumption goods, indicating a more pronounced response of durable consumption growth to inflation. For example, at a
three-year forecasting horizon, the slope coefficient for durable growth is more than
three times that of the non-durable (−0.31 vs. −0.075), and the R2 s are also uniformly larger for the durable growth rates (0.29 vs. 0.06). The differences remain
economically and statistically sizable for all forecasting horizons. Our findings for
the inflation non-neutrality for non-durable consumption are consistent with Piazzesi
and Schneider (2006) and Bansal and Shaliastovich (2013). The findings for durable
growth are novel, and complement the evidence in Erceg and Levin (2006) who document that an impact of monetary policy shocks on consumer durables spending is
several times larger than on other expenditures.
Our empirical results suggest that adverse effects of inflation are much more pronounced in the durable sectors of the economy. Because purchases of durable goods
are riskier and are more affected by economic conditions than those of non-durable
goods, we expect that the producers of durable goods are also more sensitive to aggregate economic risks, such as inflation, than the producers of non-durable goods.
Indeed, Panel B of Table 1 shows that future dividend growth rates of durable goods
producers are more sensitive to inflation shocks than those of non-durable goods pro8
ducers: the slope coefficients in the regressions of future average real dividend growth
rates onto inflation are two to three times larger for durable than for non-durable
firms. These findings are consistent with empirical evidence in Boudoukh, Richardson, and Whitelaw (1994), who document that the output growth in highly cyclical
industries (e.g., those involved in manufacturing of durable goods) tends to be more
negatively correlated with inflation relative to less cyclical firms (such as those that
provide necessities).
We also verify that our results are robust to alternative measures of corporate
output, such as the sectoral industrial production index, corporate earnings, and
sales. The results are omitted for brevity.
1.3
A macro-econometric model
The projection results provide supportive evidence for the inflation non-neutrality
across economic sectors. In this section, we provide a macro-econometric VARMA(1,1)
model for the joint dynamics of the exogenous macroeconomic variables. This allows
us to formally quantify the impact of expected inflation risk on aggregate consumption of durable and non-durable goods, and capture the exposures of equity dividends
to aggregate economic risks.
Denote by gt a vector of observed macroeconomic variables which includes the
logarithm of non-durable consumption growth, ∆ct = log (Ct /Ct−1 ) , non-durable
$
inflation, πt = log Pt$ /Pt−1
, and the growth rate of durable goods stock, ∆st =
log (St /St−1 ) . The dynamics of gt are specified exogenously. Following the long-run
9
risk model of Bansal and Yaron (2004), we incorporate a low-frequency time-varying
expected growth component xt , which we model as a VAR(1) process:
gt+1 = µg + xt + Σg ηt+1 ,
(1)
xt+1 = Πxt + Σx ut+1 .
The vector µg is the unconditional mean of the macroeconomic factors, Σg and Σx
are the covariance matrices, and ηt+1 and ut+1 are Gaussian shocks in realized and
expected growth factors, respectively. The matrix Π captures the persistence and
potential feedback between the expected growth states.
We assume that equity dividends in the two sectors are exposed to low-frequency
risks in the aggregate economy, and in addition are subject to idiosyncratic transitory
shocks. Specifically, the log dividend level satisfies,
di,t = d˜i,t + σitr ξi,t ,
for i = {nd, d},
(2)
where d˜i,t is a random-walk component of the dividends, and ξi,t is an i.i.d. Normal
innovation which captures transitory risks due to the dividend payout and managerial
policies of the firms. Empirically, even after seasonal adjustments quarterly dividend
growth rates are very volatile and contain substantial amounts of negative autocorrelation. Introducing transitory risk allows us to better capture these features of the
dividend dynamics in the data.
10
Following the literature, we let the random-walk component of the dividend dynamics to be exposed to the expected growth factors:4
∆d˜i,t+1 = µi + φ0i xt + σidiv ζi,t+1 ,
for i = {nd, d}.
(3)
Hence, in our specification dividend growth rates ∆di,t = ∆d˜i,t+1 + σitr ξi,t+1 − σitr ξi,t
are driven by the low-frequency aggregate economic shocks, and transitory shocks to
the dividend levels.
To estimate the model, we impose several restrictions to reduce the number of
parameters. In Section 5.1 we show that these restrictions do not materially affect
statistical fit or economic implications of the model, but allow for a cleaner interpretation of the results. First, in the persistence matrix Π, we zero out the interaction
terms between durable and non-durable consumption growth rates (Π1,3 = Π3,1 = 0)
and shut down the feedback effects from consumption growth to future inflation
(Π2,1 = Π2,3 = 0). Second, we allow dividend growth rates to load only on the
same-sector expected consumption factor. Statistically, this leads to a sharper identification of the dividend leverage parameters. Economically, such dividend dynamics
are similar to the levered consumption specification in Abel (1999), Campbell and
Cochrane (1999), and Bansal and Yaron (2004). Finally, we set the means of dividend growth rates to the means of the corresponding consumption growth rates.
While our setup does not feature co-integration of dividends and consumption, the
last restriction ensures that on average, dividends grow at the same rate as consumption in each sector.
4
Similar to the durable consumption growth, we express durable dividend growth in units of
durable goods numeraire to facilitate the interpretation of the loadings and ensure the consistency
between the units on the right- and left-hand sides of the equation.
11
We estimate the model parameters using Bayesian MCMC simulation and back
out the unobserved expected growth rates xt using standard Kalman filtering technique. The top panel of Table 2 summarizes the estimated model parameters, which
underscore three main features of the macro variables in the data.
First, the expected durable consumption growth and expected inflation are significantly more persistent than the expected non-durable consumption growth: the
implied first-order autocorrelation of expected non-durable growth is 0.4, relative to
about 0.9 for expected durable growth and expected inflation. This is consistent with
the evidence in Yang (2011) for the expected durable growth, and in Piazzesi and
Schneider (2006) and Bansal and Shaliastovich (2013) for the expected inflation. The
model also captures the autocorrelation patterns in the data at longer horizons, as
shown in Figure 1.
Second, the expected inflation shocks have a large negative impact on expected
consumption growth in both sectors, and this inflation non-neutrality is more pronounced for the durable goods sector. While the next-period feedback coefficients
of expected inflation for expected durable and non-durable consumption are not significantly different, the inflation impact on future durable consumption growth is
magnified over time due to a much larger persistence of the expected durable consumption growth. The impulse responses in Figure 2 show that expected inflation has
a similar impact on durables and non-durables initially. However, the impact of expected inflation on non-durables subsides after one year and becomes insignificant in
two years, while the inflation impact on durables peaks at about two and a half years
and remains large even five years in the future. At longer maturities, the difference
between the two responses is statistically and economically significant.
12
Third, the model estimation suggests a strong link between the expected dividend
growth rates and the expected consumption growth rates in the corresponding sectors. The dividend leverage parameter is estimated to be about 5 in the non-durable
sector and 5.5 in the durable sector. Our estimates of dividend leverage parameters
are larger than the typical values of 2.5 to 3.5 used in the literature for the aggregate market dividends (e.g., Abel 1999, Campbell and Cochrane 1999, and Bansal
and Yaron 2004). We verify that our estimates are not specific to our estimation approach: for example, running standard projections of realized dividend growth rates
on consumption growth rates, we obtain similar leverage coefficients of around 5. This
evidence suggests that decomposing real growth into sectoral components allows us
to better identify the link between corporate and macroeconomic growth rates. The
strong link between equity dividend and consumption further implies that movements
in expected inflation play a larger role for the equity of durable than non-durable producers. Our empirical evidence for the sectoral dividends and economic risk factors
can be potentially important not just for asset pricing, but also for future work on
monetary policy and economic growth.
We verify that the model provides a good fit to the macroeconomic data. First,
Figure 3 shows that the expected growth rates estimated from the model very closely
track the persistent movements in the aggregate macroeconomic variables. Second,
we use the posterior estimates of the parameters to simulate our model and compare
its implications for consumption, inflation, and dividend growth rates to the data.
Table 3 documents the model fit to the unconditional moments of the data in the
long simulation (population values) and in short samples whose size equals the length
of the data. As shown in the table, the model can capture very well the means,
standard deviations, and persistence in consumption and inflation: most model values
13
are almost identical to the data, and all the data moments contained in the 5%95% small-sample confidence interval. The bottom panel of the table shows model
implications for the dividend growth rates. At quarterly frequency, dividends are very
volatile and are subject to a large amount of transitory risk, so to better compare
low-frequency properties of dividends dynamics we time-aggregate them to an annual
horizon.
The model quantitatively captures salient features of the dividend data,
such as a higher volatility of durable dividend growth and a more negative correlation
of durable dividends growth with inflation, relative to non-durable dividends.
Overall, our estimated model provides a very good fit to the macroeconomic data.
It underscores a negative and significant impact of expected inflation on future growth
(inflation non-neutrality), and a strong exposure of the dividend cash-flows to the
consumption risk factors. The model thus provides a parsimonious description of
the exogenous macroeconomic inputs which we can use to address the asset-pricing
evidence in the bond and equity markets.
2.
Model Setup
2.1
Preferences and Stochastic Discount Factor
We specify an infinite-horizon, discrete-time, endowment economy where investors
consume durable and non-durable goods. The investors’ preferences over future consumption are described by the Kreps-Porteus, Epstein-Zin recursive utility function
(see Kreps and Porteus 1978 and Epstein and Zin 1989):
"
Ut = (1 −
1− 1
β)ut ψ
1
1− ψ
1−γ 1−γ
+ β Et Ut+1
14
#
1
1− 1
ψ
,
(4)
where Ut is the lifetime utility function, ut is the intra-period consumption aggregator,
β is the subjective discount factor, ψ is the elasticity of intertemporal substitution
(IES), and γ is the relative risk aversion coefficient. For ease of notations, we define
θ = (1 − γ)/(1 − 1/ψ). Note that when θ = 1, that is, when γ = 1/ψ, the recursive
preferences collapse to a standard Constant Relative Risk Aversion (CRRA) expected
utility.
In our economy, the agent derives utility from non-durable consumption Ct and
a service flow from durable goods, which we assume is proportional to the stock of
durables St (see e.g., Ogaki and Reinhart 1998, Yogo 2006, and Yang 2011). The
stock of durable goods accumulates over time through the purchases of new durable
goods Et net of the depreciation at the rate δ :
St = (1 − δ)St−1 + Et .
(5)
The intra-period consumption aggregator takes a constant elasticity of substitution (CES) form, and thus can be expressed in the following way:
i 1
h
1− 1 1− 1
1− 1
+ αS
.
u(C, S) = (1 − α)C
(6)
Parameter α ∈ [0, 1] determines the relative importance of durable consumption:
when α = 0 the economy collapses to a specification with a single perishable good.
Parameter captures the intratemporal elasticity of substitution between the two
goods. High values of indicate that the two goods can be easily substituted by the
agent, while small values for capture complementarity between the two goods.
15
As described in Yogo (2006), the equilibrium stochastic discount factor, valued
in the units of non-durable consumption, is driven by the fluctuations in the relative
share of non-durable goods Zt+1 , consumption growth of non-durables Ct+1 /Ct and
the return on the total wealth Rc,t+1 :
Mt+1 = β
θ
Zt+1
Zt
θ
1− 1
( ψ1 − 1 ) C
t+1
− ψθ
Ct
θ−1
Rc,t+1
.
(7)
The relative share of non-durable consumption of the agent Zt is defined as:
Zt =
Ct
,
Ct + Qt St
(8)
and Qt is the user cost of durable goods given by the ratio of the marginal utilities
of durable to non-durable consumption:
α
ust
=
Qt =
uct
1−α
St
Ct
− 1
.
(9)
The consumption return Rc,t+1 , which captures the return on the total wealth portfolio of the investor, pays off the basket of non-durable consumption and durable
consumption valued by its user cost Qt :
Rc,t+1 =
Wt+1
.
Wt − (Ct + Qt St )
(10)
In a single good economy (Zt ≡ 1) we obtain a standard expression for the stochastic discount factor, derived in Epstein and Zin (1991):
N on−Dur
Mt+1
=β
θ
Ct+1
Ct
16
− ψθ
θ−1
Rc,t+1
.
(11)
Thus, relative to a one-good economy, the specification with two goods incorporates
fluctuations in the relative share of the goods, and further, the wealth portfolio of the
agent is now composed of both non-durable and durable consumption.
To solve for the equilibrium bond and equity prices, we use a standard Euler
condition:
Et [Mt+1 Ri,t+1 ] = 1,
(12)
which holds for any return Ri,t+1 including the endogenous wealth portfolio of the
agent. The above pricing equation is specified in real terms using non-durable consumption as a numeraire. To change the numeraire, denote Π̃t the value of one unit
of non-durable consumption in units of a new numeraire. Then, we can price payoffs
expressed in units of a new numeraire using a standard Euler equation (12) under the
numeraire-adjusted stochastic discount factor M̃t+1 ,
M̃t+1 = Mt+1
Π̃t
.
Π̃t+1
(13)
When Πt denotes the dollar price of non-durables, the equation above defines the
nominal discount factor, which allows us to derive the nominal bond and equity
prices.
2.2
Equilibrium model solution
We solve for the equilibrium asset prices given the the stochastic discount factor in
(7), and the exogenous dynamics for non-durable and durable consumption growth
rates, inflation, and dividend cash flows in the macro-econometric model (1)-(3). To
obtain closed-form analytical solutions to the asset prices, we rely on a standard
17
log-linearization of the return on the wealth portfolio. In Section 4.3 we show that
the log-linearization error does not have a material impact on quantitative model
implications. We further log-linearize the relative share process:5
∆zt+1
Zt+1
1
= log
≈ χ(∆qt+1 + ∆st+1 − ∆ct+1 ) = χ 1 −
(is − ic )0 gt+1 ,
Zt
(14)
where ic and is pick out the non-durable and durable consumption growth from gt ,
and the parameter χ ∈ (0, 1) is an approximating constant equal to the average
expenditure on durables in the economy, χ =
Q̄S̄
.
Q̄S̄+C̄
This parameter captures the
importance of durable goods in the economy. In the economy with a single nondurable good, χ = 0.
Under the considered log-linearizations, our asset-pricing model is affine. The
equilibrium price-consumption ratio is a linear function of the economic states xt :
pct = A0 + A0x xt .
(15)
The price-consumption loadings satisfy:
Ax =
1
1−
ψ
−1
(I − κ1 Π0 )
((1 − χ)ic + χis ) ,
(16)
where κ1 ∈ (0, 1) is the log-linearization coefficient whose value is provided in Appendix A. As in a single good economy, when the intertemporal elasticity of substitution ψ is above one, the substitution effect dominates the wealth effect. Hence, the
equilibrium price of the consumption claim increases in good times when expected
5
The linearization of the relative share shuts off the variation in the asset volatilities and risk
premia due to the relative share movements (see Cochrane, Longstaff, and Santa-Clara 2008 and
Colacito and Croce 2013). The relative share of durables is quite stable in our sample, so these
fluctuations are not likely to be important for the unconditional asset-pricing moments that we
analyze.
18
non-durable or durable consumption is high. Under inflation non-neutrality, asset
prices drop in bad times of high expected inflation
The real stochastic discount factor, expressed in units of non-durable numéraire,
can be written in terms of the fundamental states and shocks in the economy:
mt+1 = m0 + m0x xt − λ0g Σg ηt+1 − λ0x Σx ut+1 ,
(17)
where mx captures the loadings of the discount factor on the expected growth components, and λg and λx are the market prices of immediate and expected growth
risks.
The market prices of the expected and realized growth risks are given by:
λx = (1 − θ)κ1 Ax ,
1
1
λz = γ(1 − χ) + χ ic + γ −
χis .
(18)
When the intertemporal elasticity of substitution and the risk aversion coefficient are
above one, investors’ wealth drops in bad times of low expected durable or non-durable
growth (see equation 16). Hence, the shocks in expected durable and non-durable
consumption are risky and have positive market prices of risk. The magnitudes of the
risk prices are magnified by the persistence of the shocks as fluctuations in expected
growth are perceived to be long-lasting by the investors. Similarly, due to a nonneutral and negative effect of shocks to expected inflation on future growth, the price
of the expected inflation risks is negative. Notably, the amount of expected inflation
risk can be much larger in a two-good relative to a one-good economy, because the
expected inflation is bad news both for future non-durable and durable consumption.
19
Under the expected utility (γ = 1/ψ), the market prices of expected durable and
non-durable consumption and expected inflation risks are all equal to zero: λx = 0. In
this case, only the short-run innovations in consumption are priced. The market prices
of short-run consumption risks, λz , depend on preferences. For typical parameter
values these market prices of risk are positive.
2.3
Equilibrium asset prices
Using the solution for the stochastic discount factor in (17), we can characterize the
equilibrium prices of bond and equity claims in the model. We show main results and
intuition below, and present the computational details in the Appendix A.
In our model, the equilibrium log price of a real bond with maturity n, pt,n , is
linear in the economic states:
0
pt,n = −B0,n − Bx,n
xt ,
(19)
where the bond loadings depend on model parameters and are provided in the Appendix A.
The real yields are given by:
1
1
0
B0,n + Bx,n
xt .
yt,n = − pt,n =
n
n
(20)
For a one-period bond, the vector of bond loadings satisfies,
Bx,1 =
1
1
1
1
(1 − χ) + χ ic − χ
−
is .
ψ
ψ
20
(21)
When χ = 0 the specification reduces to a one-good non-durable model and riskfree rate loading is equal to the reciprocal of the IES. In a two-good economy, the
risk-free rate loading on the expected non-durable consumption is also positive, and
is determined by the weighted average of the inter- and intratemporal elasticities of
substitution. On the other hand, when < ψ the interest rate loading on the expected
durable consumption is negative. An expected increase in durable consumption for
a given expected consumption of non-durables means that future non-durable goods
are relatively scarce. When the two goods are hard to substitute, the value of nondurables increases, and thus the interest rates drop. Hence, real risk-free rates respond
positively to news about long-run expected non-durable consumption, and negatively
to news of long-run expected durable consumption if < ψ, and similar effects persist
at longer maturities.
The sensitivities of bond yields to economic states determine the magnitudes of
the bond risk premia and the shape of the yield curve. Consider the excess log return
on buying an n-month bond at time t and selling it next period as an n − 1 period
bond:
rxt+1,n = −pt,n + pt+1,n−1 + pt,1 .
(22)
The expected excess return on n-period bonds is given by the covariance of the discount factor with the excess bond return:
1
0
Σx Σ0x λx .
Et rxt+1,n + V art rxt+1,n = −Covt (mt+1 , rxt+1,n−1 ) = −Bx,n−1
2
(23)
The bond risk premia are determined by the bond exposure to the economic risk Bx,n ,
the quantity of risk Σx Σ0x , and the market price of risks λx .
21
The equilibrium price of nominal bonds and nominal bond risk premia are derived in an analogous way using the solution to the nominal discount factor. The
nominal yields are affine functions of the expected consumption growth rates and
the expected inflation factors. Relative to the real bonds, a significant component
of the nominal bond yields is related to the expected inflation. Indeed, consider a
Fisher-type equation for nominal bonds:
$
yt,n
= yt,n + Et πt→t+n −
1
11
V art πt→t+n + Covt (mt→t+n , πt→t+n ),
2n
n
(24)
where πt→t+n and mt→t+n denote the t to t+n multi-period inflation rate and stochastic discount factor, respectively. First, as nominal bonds pay in nominal dollar terms,
an increase in expected inflation directly raises nominal yields and decreases nominal bond prices. Second, under inflation non-neutrality, long-term inflation rates
are high in bad times, so the the last term in the above equation which captures
the inflation premium is positive and increasing with maturity. This leads to a a
positive risk premium and a positive slope of the term structure for the nominal
bonds. In a single-good economy, the inflation premium arises only because of longrun inflation’s interaction with future non-durable growth, as discussed in Piazzesi
and Schneider (2006) and Bansal and Shaliastovich (2013). With multiple goods, inflation non-neutrality also arises through a persistent negative covariation of inflation
with durable consumption, which magnifies the amount of inflation premium in the
economy.
Finally, we derive approximate solutions to the price-dividend ratios in the nondurable and durable sectors:
0
xt + Hi,ξ σitr ξi,t ,
pdi,t = Hi,0 + Hi,x
22
for i = {nd, d},
(25)
where the expressions for the loadings are provided in Appendix A. For typical parameter values, equity prices increase during times of high expected consumption,
and decrease during times of high expected inflation. The magnitudes of equity exposures are governed by the sensitivity of dividend cash flows to the aggregate risks.
The asset valuations also respond to the transitory dividend risk ξi,t . A positive shock
ξt raises current dividend level but depresses expected dividend growth by the same
amount, so asset prices fall (Hi,ξ = −1). Because of the two offsetting effects on current dividends and current prices, transitory dividend risks have virtually no impact
on equity returns. Further, transitory risks do not impact aggregate consumption
and the stochastic discount factor, so they do not contribute to the level of the equity
premium.
3.
Empirical Model Evaluation
3.1
Asset-price data
Table 4 presents the key moments of the nominal bond and equity prices in the data.
As seen in the first panel, nominal term structure is upward-sloping with a five-year
term spread of 58 bps. The volatilities of bond yields decrease with maturity from
2.76% for a one-year bond to 2.53% for a five-year bond. The yields are very persistent
with AR(1) coefficients ranging from 0.93 to 0.96 in our sample.
The bottom panel of Table 4 presents evidence on the equity returns in nondurable and durable sectors. Consistent with Gomes et al. (2009), we find that
durable-goods-producing firms are riskier than non-durable ones: the risk premium
on the durable portfolio is 7.11% in the data, compared to 5.62% for non-durables.
23
Equity returns of durable goods producers also have a substantially higher volatility
(21%) relative to non-durables (about 15%). Finally, as shown in Table 4, durable
equity returns have a higher correlation with nominal bond returns, and are more
exposed to the expected inflation risks. The correlation of stock and bond returns is
larger for durables relative to non-durables for the whole sample and in the various
subsamples, as can be seen from on a 10-year rolling window plot in Figure 4.
3.2
Preference parameter estimation
Our empirical evaluation of the economic model takes the joint dynamics of the
consumption, dividends, and inflation as exogenous, adopting the macro-econometric
specification in Section 1.3. This estimation only uses macroeconomic data, so the
estimated dynamics directly reflect the economic fundamentals in the data, and are
not hardwired to match asset prices.
Naturally, using macro data alone does not allow us to identify preference parameters. We estimate the preference parameters γ, ψ, χ, and in the second stage
by targeting selected asset-price moments, such as the average nominal yields over
maturities from one to five years, average return and volatility of the two equity portfolios, and the sensitivities of nominal yields to expected consumption growth rates
and expected inflation. In addition to that, we include the moment condition for the
relative share of non-durable consumption, as well as for the slope coefficient in the
regression of log real user cost of durable goods on log difference between the quantities of durable and non-durable goods in the data. These macroeconomic moments
help identify χ and in the data. To compute the standard errors of the preference
parameters, we take into account the first-stage parameter uncertainty in the macroeconometric specification. That is, we repeat the preference parameter estimation
24
for each draw of the first-stage parameter estimates obtained in the Bayesian MCMC
chain. This approach generates a chain of preference parameters, and the posterior
means and standard deviations of the preference parameters are computed directly
from this chain.
The estimates of preference parameters are reported in the bottom panel of Table
2. For identification purposes, we fix the subjective discount factor at 0.994.6 At
the quarterly frequency, the estimated intertemporal elasticity of substitution is 1.42,
and the risk aversion coefficient is 16.21. Our estimate of the IES is similar to the
literature (see Bansal and Shaliastovich 2013 and Doh 2013), while our estimate of
the risk aversion is lower than the values entertained in single good economies, such as
Piazzesi and Schneider (2006) and Bansal and Shaliastovich (2013). The amount of
inflation risk increases in a two-good relative to a single-good economy, which puts less
stress on the preference parameters to match asset prices. The intertemporal elasticity
of substitution is estimated at 0.32 (0.22), which suggests the complementarity
between the two goods. There is a wide range of estimates in the literature, from
0.13 in Flavin and Nakagawa (2008) to 1.75 in McGrattan, Rogerson, and Wright
(1997), which could be attributed to considerable measurement errors in estimating
this parameter from the data. Yogo (2006) and Gomes et al. (2009) also emphasize the
importance of the complementarity of non-durable and durable consumption goods,
and estimate this coefficient to be below one, around 0.7 to 0.8. Finally, the parameter
which governs the share of durable goods χ is estimated at 0.35 with a standard error
of 0.37.
6
This is related to the discussion of identification issues in Kocherlakota (1990) and is similar to
the approach in Marshall (1992) and Bansal and Shaliastovich (2013).
25
3.3
Model implications for bond prices
Table 5 presents the benchmark model implications for the bond yields and bond
returns. The table reports model-implied moments obtained from a long simulation
(population values), and also in short samples whose size is equal to the length of the
data. As shown in the top and middle panels of Table 5, our model matches very
well the unconditional level, slope, and curvature of the nominal yield curve. The
model-implied one-year nominal yield is 6.15% on average, compared to 6.26% in
data. The nominal yield curve is upward sloping, with a five-year term spread equal
to 84 bps in the model and 58 bps in the data. The model can further account for
the volatility and persistence of nominal yields and term spreads. Yield volatilities
decline with maturities both in the model and in the data, though the model-implied
volatilities are somewhat lower than in the data, especially for long maturities. Both
nominal yields and term spread are highly persistent in the model and in the data.
The bottom panel of Table 5 confirms that the model also produces satisfactory fit
to the short-term (two-year) and long-term (five-year) bond excess returns. Longterm bonds earn higher average excess returns and exhibit higher volatilities than
short-term bonds in data, consistent with our model predictions.
We also check model implications for the real bonds, defined as the assets which
pay one unit of non-durable consumption in the future. In the model, the real term
structure is nearly flat with a slight U-shape. The one-quarter real yield yield is
1.97%. It declines to 1.73% at a one-year horizon, and increases to about 1.87% at
five years. Our real yield implications are broadly consistent with Ang, Bekaert, and
Wei (2008) who estimate an alternative no-arbitrage term-structure model and find
that the real term spread is close to zero.
26
Figure 5 shows the conditional fit of the model to short rates, term spreads, and
bond returns in the data. Generally, the model tracks the observed data quite well.
Some of the noticeable deviations of the model predictions to the data include the
mid-1980s, where interest rates peaked significantly above what is predicted by our
model, as well as the recent episode in early- and late-2000s, when the yields in the
data were below the model predictions. We formally evaluate the conditional fit of the
model through the annualized mean-squared error (MSE). The MSEs for the bond
yields, reported in Table 5, are about 2%. The MSE for the term spread is 1.1%, and
the MSEs for the bond returns range from 1.8% for a two-year bond to 5.9% for a
five-year bond.7
Figure 6 examines the sensitivity of nominal yields to the expected durable and
non-durable consumption growth and expected inflation risks. In the model, nominal yields increase at times of high expected non-durable consumption growth and
high expected inflation, and decrease at times of high expected durable growth. In
terms of the relative contribution of consumption risks, we find that short-term yields
load similarly in magnitude on the expected non-durable and durable consumption
growth rates (2.4 for non-durable vs. -2.8 for durable), but long-term yields load more
significantly on the expected durable consumption growth than on the expected nondurable consumption growth (0.5 for non-durable vs. -1.3 for durable). The expected
durable growth is also more volatile than expected non-durable growth. This implies
7
Note that our model abstracts from other economic channels, such as movements in risk and
term premia, interest rate shocks, changes in the monetary policy regimes, or fluctuations in risk
aversion and uncertainty which can improve the fit of the model to the bond prices. Bekaert et al.
(2009), Wachter (2006), and Bansal and Shaliastovich (2013) highlight the economic importance of
time-variation in risk premia in the bond markets, while Hasseltoft and Burkhardt (2012), Gallmeyer,
Hollifeld, Palomino, and Zin (2009), Bansal and Zhou (2002), Baele, Bekaert, Cho, Inghelbrecht,
and Moreno (2015), and Bikbov and Chernov (2013) discuss the role of other channels for bond
markets.
27
that durable consumption growth contributes more than the non-durable consumption growth in determining long-term yields and term spreads.
We measure bond sensitivities in the data by regressing nominal yields on our
estimates of expected growth state variables, and compare these data loadings to the
model. Figure 6 shows that the signs and magnitudes of bond exposures are similar
in the data and in the model, and the estimates are not statistically different from
each other.
Notably, the largest deviations between the model and the data are for the loadings
on expected durable growth. While they are all negative, the model bond loadings
decline in absolute value with maturity, while in the data they are nearly flat. Recall
that in our model, bond risk premia are constant, so the fluctuations in long term
yields reflect just the variations in the expected future short rates, the so-called
“Expectation Hypothesis” component of the yields. To examine the importance of
the time-varying risk premia for our bond loading results, we adopt a reduced-form
approach to filter out the time-varying risk premia from long-term yields in the data,
and assess whether the remaining ”Expectations Hypothesis” components better line
up with the model. Specifically, we estimate an unrestricted VAR(1) specification
which includes bond yields with one quarter, one-, three-, and five-year to maturity,
as well as our three expected growth states.8 We can use this VAR(1) to flexibly
estimate the expected future short rates in the data, without making any modelspecific assumptions:
EH
=
yt,n
8
1 X
Et
yt+j−1,1 ,
n
Our results are similar if we use only yield variables or only macroeconomic variables.
28
(26)
where the conditional expectations are based on the VAR(1) representation of bond
yields and macroeconomic factors. These expected short rates correspond to the
“Expectation Hypothesis” component of the long-term yield which would prevail
under the constant risk premia assumption.
In Figure 6 we confirm that the model-implied loadings of yields are much more
aligned with the risk-premia adjusted component of yields in the data. Specifically,
the loadings of “Expectation Hypothesis” components of bond yields in the data on
expected durables monotonically decrease, in absolute value, to zero, and are similar
to the loadings in the model. The loadings on the other two state variables also
become closer to the model. This evidence suggests an important relation between
risk premia fluctuations and the macroeconomic factors, and we leave a detailed study
of this for future research.9
Finally, we consider the equilibrium model implications for the univariate predictability of future real consumption growth rates by the term structure variables.
We simulate the model and regress the simulated consumption growth rates up to five
years in the future on the model-implied short rate and term spread. Figure 7 contrasts the projection slope coefficients obtained from the model to those in the data.
Our model can broadly account for the signs and magnitudes of the projection evidence: short rates negatively predict future consumption growth of both non-durable
and durable goods and the coefficients are much larger for durable goods. Further,
the term spread forecasts positively future consumption growth rates, and again the
slope coefficients are larger for the durable consumption. The model implications are
generally consistent with the data, and most of the estimates are within the 5%-95%
9
This evidence is related to Joslin, Priebsch, and Singleton (2010), Cooper and Priestley (2009),
and Ludvigson and Ng (2009) who show that aggregate growth variables can predict future bond
returns.
29
confidence interval in the data. Some of the noticeable deviations are for the durable
consumption predictability by the term spread, where the values in the model are
above the estimates in the data. Recall that the model does not feature movements
in the risk premia and stochastic volatilities, which can play an important role for
the fluctuations in the term spread. If these factors also negatively impact future
real growth, this can potentially account for an observed discrepancy between the
projection coefficients in the model and in the data.
3.4
10
Model implications for equity prices
Table 6 summarizes model-implied evidence for equity returns in non-durable and
durable production sectors and compares it to evidence from our data. In the model,
equity dividends are exposed to aggregate consumption and inflation risks. This
makes equity returns risky, which leads to a positive equity premium and a high
volatility of returns. In the model, as in the data, the portfolio of durable goods
producers is riskier than the portfolio of non-durable goods producers. The average
equity premium is 6.3% in the durable goods production sector and 4.2% in the
non-durable goods production sector. This is close to the data estimates of 7.1%,
and 5.6%, respectively. The model-implied volatility of excess returns in the durable
production sector is 29.7%, which is larger than that in the non-durable production
sector of 19.9%. This is consistent with the empirical evidence in the data.
We further consider model implications for the covariation of stock returns with
expected inflation and bond returns. In the model, equity valuations decrease during
10
Indeed, we also consider consumption growth projections on the risk premia adjusted components of the term spread, constructed from the Expectations Hypothesis components of yields. At
short horizons, the loadings on the risk-premia-adjusted term spread are above those on unadjusted
term spread, and are closer to the model.
30
times of high expected inflation. As a result, excess stock returns have a negative
correlation with expected inflation. The model predicts that this will be more pronounced for stocks of durable goods producers because durables have higher exposure
to expected inflation risks. These predictions are supported by the data. As shown
in Table 6, the model-implied correlation of equity returns with expected inflation is
-0.30 for the durable production sector relative to -0.26 for the non-durable sector. In
the data, the levels and the gaps between the correlations are somewhat larger (-0.54
relative to -0.44), but the estimates are in the confidence interval of the model. Further, our model implies that excess bond returns and excess equity returns co-move
positively, and the degree of the co-movement is larger for the durable portfolio. These
model implications are also supported by the data. Specifically, the model-implied
correlation between excess returns of the durable portfolio with excess bond returns
is 0.36, relative to 0.31 for the non-durable portfolio, close to the estimates in our
sample.11
3.5
Importance of model channels
In this section we assess the significance of the economic channels of the model to explain the asset-pricing evidence in bond and equity markets. Specifically, we consider
the relative importance of our macroeconomic factors for the levels and volatilities of
the asset prices, the impact of inflation non-neutrality, and the role of the preference
structure.
11
Hasseltoft (2009), Bekaert, Baele, and Inghelbrecht (2010), and Campbell, Sunderam, and Viceira (2012) show that the stock-bond correlation switches sign and becomes negative in the recent
period, which can be driven by regime changes in the fundamentals, time-varying risk premia or
liquidity considerations. For simplicity, our model does not feature these economic channels, so that
the conditional correlations are constant over time.
31
In the model, the levels of bond risk premia and the fluctuations in bond prices
are determined by three state variables which drive the movements in the expected
consumption growth of non-durable and durable goods and expected inflation:
1
0
Σx Σ0x λx ,
Et rxt+1,n + V art rxt+1,n = −Bx,n−1
2
2
1
0
V art (yt,n ) =
Bx,n
Σx Σ0x Bx,n .
n
(27)
To understand the relative importance of the factors, we decompose these quantities
into the components associated with each source of risks. As the three shocks are
correlated and the variance-covariance matrix Σx Σ0x is non-diagonal, to simplify the
presentation we distribute the covariance terms with a weight proportional to the
variance of each factor. Table 7 documents these decompositions. The key state
variables which drive the movements of bond prices are the expected inflation and
the expected durable consumption growth. Indeed, the expected inflation factor
contributes 64% to the total variance of the one-year yield. As the expected inflation
is the most persistent factor, its relative role increases with maturity, and reaches
78% for the five-year yields and 82% for the ten-year yields. The contribution of
the durable factor is 35% for the one-year yield, and it still contributes a non-trivial
amount of about 20% at five- and ten-year maturities. The contribution of nondurable expected growth factor is virtually zero: the percentages are even slightly
negative at long maturities because of the negative covariance terms.
Similarly, the expected inflation and the expected durable growth are the most
important factors in generating a positive risk premium for nominal bonds. In our
model, most of the bond risk premium at long maturities reflects compensation for
the expected inflation risk. Bonds hedge movements in expected non-durable consumption, so that the risk compensation for non-durable risks is negative. On the
32
other hand, bonds are risky with respect to the expected durable consumption. The
durable risk factor contributes almost 50% to the bond risk premia at one year, and
about 15% at five- and ten-year maturities.
We also consider similar decompositions for the equity volatility and equity risk
premia. We find that the expected inflation risks play an important role for these
quantities in the model. In terms of the aggregate consumption risks, naturally, nondurable equity returns are more exposed to expected non-durable growth risks, while
durable returns are more affected by the risks in durable consumption.
In our model, inflation non-neutrality is the key economic channel which allows us
to generate a positive inflation premium and a sizeable compensation for expected inflation risks. We quantify the importance of this channel in Table 8. In the benchmark
model, inflation has a negative effect on both non-durable and durable consumption
growth. As shown in the first column of the table, the benchmark model produces a
large term spread, sizeable equity premia, and large correlations of stock returns with
bonds and expected inflation. The model also generates a sizeable inflation premium
which is positive and increasing with maturity. It is equal to 0.24% at a one year
horizon and reaches 0.93% at five years. These magnitudes are consistent with the estimates in Ang et al. (2008), who consider an alternative no-arbitrage term structure
model, and find the inflation premia of 0.31% and 1.14%, respectively.
We next consider several restricted model specifications in which we shut down inflation impact on non-durable consumption growth, on durable consumption growth,
and finally on both consumption growth rates at the same time. To obtain inflation neutrality on future growth, we set the appropriate coefficients which govern the
co-movements of expected inflation and expected consumption to zero. For inflation
neutrality for non-durables, these coefficients are Π(1, 2) and Σx (2, 1); for durables,
33
these coefficients are Π(3, 2) and Σx (3, 2), and all four coefficients are set to zero when
inflation is neutral for both non-durable and durable growth. Because of the structure
of our model, when inflation is made neutral for non-durable growth, expected inflation has a zero effect on future non-durable consumption, while its impact on future
durable consumption is identical to the benchmark model. Similarly, under inflation
neutrality for durable growth, expected inflation has no effect on future durable consumption, and its effect on non-durable consumption is identical to the benchmark
model. Under inflation neutrality for both durable and non-durable coonsumption,
the inflation has no effect on either consumption growth rates in the future.
As shown in Table 8, removing the impact of expected inflation shocks on the
real growth rates significantly affects the model fit to the asset prices. Indeed, under
full inflation neutrality, the nominal term structure is effectively flat, the inflation
premium is zero, the equity premia are reduced by about a half, and the correlations of
stock returns with bonds and with expected inflation are nearly zero. Quantitatively,
because of the high persistence of the expected durable factor, inflation non-neutrality
for durable consumption plays a more important role for the inflation risks than
inflation non-neutrality for non-durable consumption. For example, the slope of the
term structure reduces by about 10 basis points when expected inflation has no impact
on future non-durables, while it reduces by 50 basis points when expected inflation has
no effect on future durables. Similarly, the five-year inflation premium is diminished
from 93 basis points in the full model to 53 basis points under inflation neutrality
for non-durables, and it goes down to 38 basis points when inflation is neutral for
durables.
Finally, we assess the role of the preference structure which features utility over
two goods, a low (below one) elasticity of intratemporal substitution, and a preference
34
for early uncertainty resolution. We summarize the main findings below, and provide
detailed results in the Appendix Table B.1. In a one-good restricted version of the
model, the market price of expected durable risks is zero, and the market price of
expected inflation risks reduces by about half relative to the benchmark case. As a
result, the model cannot explain the levels of the bond and equity risk premia and a
negative impact of expected durables on nominal bond yields. In the second exercise,
we entertain a high (above one) elasticity of intra-temporal substitution = 1.5. The
elasticity of intratemporal substitution significantly affects the interest rate loadings
on the factors. In particular, when goods are viewed as substitutes, an increase in
expected durables actually raises interest rates, opposite to the benchmark model
and the data. Finally, in the last specification we set the elasticity of intertemporal
substitution ψ to 0.05, so that the agents have a preference for late resolution of
uncertainty. With this specification, all the market prices of risks have opposite signs
to the benchmark specification. Counter to the data and the benchmark model, the
model-implied yields now load positively on the expected durables, and negatively on
the expected inflation. This model specification also implies implausibly high values
of the yields of above 60%.
4.
Robustness
4.1
Alternative model estimations
In our benchmark model specification, we impose economic and statistical restrictions
on the model parameters. In this section, we show that they do not materially affect
the statistical fit or economic implications of the model.
35
In the first robustness check, we relax the restriction on the persistence matrix,
that is, we allow Π1,3 , Π3,1 , Π2,1 , and Π2,3 to be estimated from the data, rather
than imposing them to be zero. Table B.2 shows that the zeroed-out coefficients
are all insignificant, while all the remaining parameter estimates are very similar to
the benchmark model. The unrestricted estimation further produces very similar
estimates of the filtered states (Figure B.1) and the impulse response functions for
the macroeconomic variables (Figure B.2).
In the second robustness check, we relax the restriction on the dividend growth
specification, and allow the dividend growth rates to load on all the expected states.
Table B.3 shows that all the parameter estimates remain very similar to the benchmark case except for the dividend loadings which are imprecisely identified and are
insignificant. The long-term properties of dividends are similar to the benchmark
model. Indeed, the impulse responses of dividend growth to expected states, shown
in Figure B.3, are not statistically different from the benchmark model.
4.2
Alternative numeraires and implications for CPI
In our benchmark model, we choose the non-durable consumption as the numeraire
and use the change in its price index, that is, the inflation rate for non-durable goods,
to compute the equilibrium prices for nominal assets. Alternatively, one can use
durable goods as the numeraire, or the aggregate consumption basket of non-durable
and durable goods. Because of the data and measurement issues, and consistent with
the existing literature, we chose to proceed with non-durable numeraire.
36
First, the standard estimates for the relative user cost of durable goods Q in the
literature typically rely on the Euler condition equation:
d
Qt = Ptd − (1 − δ)Et [Mt+1 Pt+1
],
(28)
where Ptd is the real purchase price of durable goods in terms of non-durable goods.
These implied user costs are very noisy, and are quite sensitive to the measurement
assumptions, specifically, to those regarding computing expectations and approximations of the stochastic discount factors. The literature typically replaces the stochastic
discount factor with one over the risk free rate, and replaces expectations by future
realized values, current values, or AR(1) forecasts. We find that changes in user costs
implied by these computations are only weakly, and sometimes even negatively correlated across the measurements, and have quite different first-order properties. For
this reason, we choose to minimize the direct use of the user cost model estimation,
and use non-durable numeraire.
We further choose not to use the Consumer Price Index (CPI) numeraire implied
by basket of goods. In the model, one can compute the ideal price index associated
with the preferences of the agent. However, the aggregate consumption and CPI
inflation series in the data which aggregate quantities and prices of both non-durable
and durable goods are based on the weights according to the National Income and
Product Account (NIPA) conventions, which, in principle, can be different from the
weights in the economic model. Hence, as in Piazzesi et al. (2007), we choose to
use dis-aggregated series for non-durable and durable consumption and non-durable
inflation to specify and estimate the model. For robustness, we verify that the inflation
rate for the ideal price index in the model is reasonably close to the CPI inflation data:
37
the means and standard deviations of the two are quite similar, and the correlation
between the two is 60%.
4.3
Log-linearization of the wealth portfolio return
To solve the model analytically, we log-linearized the return on the wealth portfolio.
To check the accuracy of the log-linearization, we also solve the model numerically and
compare the numerical solution to our analytical solution. Specifically, we discretize
each expected growth state into 10 grid points. Then we solve for the equilibrium
price of the wealth portfolio and the stochastic discount factor numerically through
the fixed point iteration of the Euler equation (12).
We find that the log-linearization of the wealth portfolio return does not have any
material impact on the model solution. Indeed, the mean, volatility, and persistence
of the wealth-consumption ratio and nominal yields of one- to ten-year to maturity,
shown in Table C.1, are nearly identical. For example, the mean price-consumption
ratio is 5.45 under both solution methods, its volatility is 1.16 in an approximate
analytical solution relative to 1.20 in a numerical approach, and the persistence is 0.93
and 0.91, respectively. For bond prices, the difference in mean yields is less then 1 basis
point for one-year yields, and is about 5 basis points at ten-year horizon. We chose to
use the analytical solution, because it provides closed-form equations for equilibrium
asset prices, reveals more intuition about the underlying model mechanism and allows
us to give clearer interpretation of the model implications, and in the end is not
materially different from the numerical results.
38
5.
Conclusions
In the data, high expected inflation predicts low future real growth, and the magnitude of this inflation non-neutrality is larger in the durable goods sector. Consistent
with these findings, the equity prices in the durable goods sector are more correlated
with expected inflation and bond returns. We set up and estimate a two-good model
which features recursive utility over consumption of durable and non-durable goods,
persistent fluctuations in expected growth rates, and inflation non-neutrality for future non-durable and durable consumption. The inflation non-neutrality for durable
consumption leads to a significant role for the expected inflation risks and a large
inflation premium in the economy, which goes a long way in accounting for the key
features of the macroeconomic data, nominal bond yields, and equity price data.
There are several extensions of our framework that could be interesting to address
in future work. First, we take the dynamics of real consumption and inflation as given
and do not endogenize inflation non-neutrality. Kung (2015) shows that a negative
correlation between low-frequency movements in aggregate growth and inflation can
arise endogenously in a stochastic endogenous growth model. It would be interesting
to explore whether these economic mechanisms can explain our novel evidence for
multiple consumption goods. Second, for parsimony in our model the variances of
the underlying shocks are constant, so the risk premia, asset price volatilities, and
correlations between asset returns do not fluctuate over time. In the data, these
conditional moments are time-varying, and it would be useful to incorporate their
dynamics into the economic framework. Finally, we also abstract from policy and
welfare considerations. It would be interesting to examine the aspects of a time-
39
varying monetary policy and its impact on sectoral growth, asset prices, and welfare
(see e.g., Diercks 2015). We leave all these model extensions for future research.
40
Appendix
A.
Model Solution
The unconditional mean of the real discount factor is given by:
m0 = θ log δ + (1 − θ) log κ1 − λ0z µ,
(A.1)
where the log-linearization parameter κ1 satisfies the following recursive equation:
1
((1 − χ)ic + χ(ia + is ))0 µ
log κ1 = log β + 1 −
ψ
1
1 2
+ θ 1−
((1 − χ)ic + χ(ia + is ))0 Σg Σ0g ((1 − χ)ic + χ(ia + is ))
2
ψ
1
+ θκ21 A0x Σx Σ0x Ax .
2
(A.2)
The nominal discount factor is given by:
0
0
0
0
mt+1 = m$0 + m$x xt − λ$g Σg ηt+1 − λ$x Σx ut+1 .
(A.3)
The parameters satisfy:
m$0 = m0 − iπ µz ,
m$x = mx − F 0 iπ ,
λ$x = λx ,
λ$z = λz + iπ ,
(A.4)
h
i0
where iπ = 0 1 .
The solution for real bond price loadings are given by,
1
1
B0,n = B0,n−1 − m0 − λ0g Σg Σ0g λg − (λx + Bx,n−1 )0 Σx Σ0x (λx + Bx,n−1 ),
2
2
(A.5)
0
Bx,n = Π Bx,n−1 − mx .
The nominal bond equations are very similar, and use the parameters of the nominal discount factor.
41
Finally, the price-dividend ratio is given by,
0
pdi,t = Hi,0 + Hi,x
xt + Hi,ξ σitr ξi,t ,
(A.6)
for i = nd, d. The solutions to the price coefficients are given by,
Hi,x = I − κ1,d Π0
−1 Hi,ξ = −1.
42
m$x + φi + iπ ,
(A.7)
B.
Alternative Model Specifications
Table B.1 Role of preference parameters
Full
Model
One Good
χ=0
High = 1.5
Low ψ
ψ = 0.05
Mkt Price of Risk:
Exp. Nondur
Exp. Dur
Exp. Inflation
16.00
42.19
-75.20
24.61
0.00
-39.43
16.00
42.17
-75.20
-3.18
-4.83
4.05
1Y Yield Loadings:
Exp. Nondur
Exp. Dur.
Exp. Infl.
2.43
-2.81
3.20
1.10
0.00
3.16
1.08
0.04
3.16
22.10
19.58
-6.93
Bond Premia:
1Y
5Y
10Y
0.22
1.82
2.41
0.13
0.60
0.73
0.28
1.09
1.30
0.96
1.70
1.93
Equity Premia:
Non-durable
Durable
4.20
6.33
2.12
3.42
3.05
8.54
0.82
0.74
This table analyzes the role of preference parameters for the market prices of risk, yield loadings,
and bond and equity risk premia. The benchmark model (Full Model) is compared to the restricted
models with one consumption good (One Good), high intratemporal elasticity of substitution (High
), or low intertemporal elasticity of substitution (Low ψ).
43
Table B.2 Estimates of the model with unrestricted persistence matrix
Σx × 1000
Π
∆c
∆π
∆s
∆c
0.312
∆π
−0.163
∆s
−0.060
∆c
3.822
(0.107)
(0.055)
(0.086)
(0.369)
∆dnd
∆s
1.421
(0.782)
0.102
0.969
0.076
−0.312
1.923
1.988
(0.068)
(0.035)
(0.045)
(0.321)
(0.331)
(0.262)
−0.064
−0.112
0.879
1.511
0.298
1.202
1.579
(0.068)
(0.035)
(0.046)
(0.291)
(0.294)
(0.278)
(0.359)
σd × 100
φ
∆dnd
∆π
diag(Σg ) × 1000
4.96
0
0
7.58
7.46
(0.81)
(0.93)
5.43
9.89
14.31
(2.99)
(1.47)
(1.58)
(2.28)
0
0
σdtr × 100
The table shows parameter estimates of the model with an unrestricted persistence matrix. Macroeconomic parameters are estimated by Bayesian MCMC using the quarterly observation on consumption growth rates, dividend growth rates, and inflation. Quarterly data from 1963Q1 to 2006Q4.
44
Table B.3 Estimates of the model with unrestricted dividend loadings
Σx × 1000
Π
∆c
∆π
∆c
0.395
∆π
−0.136
(0.099)
(0.055)
0
0.917
∆s
0
0
0
∆dnd
∆s
1.912
(0.912)
−0.102
2.291
1.878
(0.319)
(0.301)
(0.298)
−0.102
0.878
1.393
0.301
1.102
1.598
(0.027)
(0.031)
(0.197)
(0.205)
(0.198)
(0.164)
σd × 100
φ
∆dnd
∆π
(0.631)
(0.035)
∆s
∆c
3.602
diag(Σg ) × 1000
σdtr × 100
4.74
−1.10
−0.159
7.45
7.69
(2.720)
(1.182)
(1.708)
(0.821)
(1.027)
5.78
−2.41
2.63
10.35
13.69
(5.604)
(2.113)
(3.134)
(1.401)
(1.642)
The table shows parameter estimates of the model with unrestricted dividend growth loadings.
Macroeconomic parameters are estimated by Bayesian MCMC using the quarterly observation on
consumption growth rates, dividend growth rates, and inflation. Quarterly data from 1963Q1 to
2006Q4.
45
Figure B.1: Estimated economic states across models
Non−durable
0.02
Baseline model
Unrestricted model
0.01
0
−0.01
−0.02
1965
1970
1975
1980
1985
1990
1995
2000
2005
1990
1995
2000
2005
1990
1995
2000
2005
Inflation
0.02
0.01
0
−0.01
1965
1970
1975
1980
1985
Durable
0.01
0.005
0
−0.005
−0.01
−0.015
1965
1970
1975
1980
1985
The figure shows the time series of the expected non-durable consumption growth rate, expected nondurable inflation rate, and expected durable consumption growth rate, estimated in the benchmark
model and the model with an unrestricted persistence matrix. Quarterly data from 1963Q1 to
2006Q4.
46
Figure B.2: Impulse responses of expected growth shocks across models
−3
5
−3
nondur resp to nondur
x 10
5
5%−95% CI
Baseline model
Unrestricted model
4
−3
infl resp to nondur
x 10
5
4
4
3
3
2
2
2
1
1
1
0
0
0
−1
−1
−1
3
0
5
−3
1
10
15
20
0
5
−3
nondur resp to infl
x 10
6
0
10
15
20
0
5
−3
infl resp to infl
x 10
dur resp to nondur
x 10
1
10
15
20
15
20
15
20
dur resp to infl
x 10
0
4
−1
−1
2
−2
−2
0
−3
−4
0
5
−3
5
10
15
20
−2
0
5
−3
nondur resp to dur
x 10
−3
5
10
15
20
−4
0
5
−3
infl resp to dur
x 10
6
10
dur resp to dur
x 10
4
0
0
2
0
−5
0
5
10
15
20
−5
0
5
10
15
20
−2
0
5
10
The figure shows impulse response functions of macroeconomic variables to shocks in expected nondurable consumption, expected durable consumption, and expected non-durable inflation, based on
the benchmark model and the model with unrestricted persistence matrix. Grey regions correspond
to 5%-95% confidence interval.
47
Figure B.3: Impulse responses of dividend growth
dur div resp to nd cons
nd div resp to nd cons
0.04
0.08
5%−95% CI
Baseline model
Unrestricted
0.03
0.06
0.02
0.04
0.01
0.02
0
0
0
5
10
15
20
0
5
nd div resp to inf
10
15
20
15
20
15
20
dur div resp to infl
0.01
0.01
0
0
−0.01
−0.01
−0.02
−0.02
−0.03
−0.03
−0.04
0
5
10
15
20
0
5
nd div resp to dur cons
10
dur div resp to dur cons
0.04
0.02
0.02
0.01
0
0
−0.01
−0.02
−0.03
−0.02
0
5
10
15
20
0
5
10
The figure shows impulse response functions of dividend growth to shocks in expected non-durable
consumption, expected durable consumption, and expected non-durable inflation, based on the
benchmark model and the model with unrestricted dividend growth loadings. Grey regions correspond to 5%-95% confidence interval.
48
C.
Accuracy of Log-linearization Approach
Table C.1 Approximate analytical and numerical solutions
Analytical
Numerical
Price-Consumption Ratio:
Mean
5.453
Std Dev
1.159
AR(1)
0.932
5.449
1.199
0.911
1-year Nominal Yield:
Mean
6.151
Std Dev
2.494
AR(1)
0.912
6.148
2.564
0.917
5-year Nominal Yield:
Mean
6.983
Std Dev
1.651
AR(1)
0.947
6.994
1.722
0.951
10-year Nominal Yield:
Mean
7.536
Std Dev
1.050
AR(1)
0.944
7.588
1.107
0.947
The table reports the moments of the price-consumption ratio and nominal yields under the approximate analytical solution based on the log-linearization of the wealth portfolio, and the numerical
solution without the approximation.
49
Tables and Figures
Table 1 Inflation non-neutrality for real growth
1Y
Non-durable
R2
Durable
R2
Non-durable
R2
Durable
R2
-0.187
(0.056)
0.144
-0.305
(0.073)
0.201
-1.641
(0.551)
0.044
-4.277
(1.027)
0.100
2Y
3Y
4Y
5Y
-0.041
(0.053)
0.027
-0.029
(0.047)
0.017
-0.310
(0.091)
0.286
-0.270
(0.084)
0.260
-0.230
(0.067)
0.227
Dividend Growth:
-1.188
-0.787
(0.531)
(0.519)
0.050
0.039
-0.554
(0.411)
0.034
-0.315
(0.315)
0.017
-1.175
(0.906)
0.033
-0.461
(0.754)
0.008
Consumption Growth:
-0.119
-0.075
(0.062)
(0.061)
0.099
0.061
-0.329
(0.085)
0.272
-3.227
(0.890)
0.123
-1.915
(0.937)
0.067
The table reports the slope coefficients, the standard errors, and the R2 s in the projections of
average cumulative real economic growth rate on current inflation. The top panel reports the
evidence for aggregate consumption growth of non-durable goods and durable goods; and the bottom
panel reports the results for dividend growth in non-durable and durable goods production sectors.
Quarterly data from 1963Q1 to 2006Q4. Standard errors are Newey-West adjusted with a number
of lags equal to the horizon of the regression.
50
Table 2 Model parameter estimates
Macroeconomic Parameters:
Σx × 1000
Π
∆c
π
∆c
0.375
π
−0.143
(0.115)
(0.063)
0
0.916
∆s
0
∆c
3.83
0
0
∆dnd
4.98
2.35
1.78
(0.306)
(0.334)
(0.252)
−0.104
0.876
1.37
0.37
1.20
1.63
(0.040)
(0.043)
(0.385)
(0.254)
(0.273)
(0.369)
σd × 100
0
0
0
σdtr × 100
7.80
7.56
(0.80)
(0.91)
5.45
11.39
12.80
(2.48)
(1.49)
(1.52)
(2.12)
0
1.46
(0.979)
−0.16
φ
∆dnd
∆s
(0.552)
(0.034)
∆s
π
diag(Σg ) × 1000
Preference Parameters:
δ
γ
ψ
χ
0.994
16.21
1.42
0.32
0.35
(8.33)
(0.11)
(0.22)
(0.37)
The table reports parameter estimates of the benchmark model specification. Macroeconomic parameters are estimated by Bayesian MCMC using the quarterly observation on consumption growth
rates, dividend growth rates, and inflation. Preference parameters are estimated in the second stage
using the moments of nominal yields and equity returns data. Quarterly data from 1963Q1 to
2006Q4.
51
Table 3 Model fit to macroeconomic variables
Data
Est
SE
Pop
5%
Model
50%
95%
Consumption Growth and Inflation, Quarterly:
Mean:
Non-durable
Durable
Inflation
2.20
4.32
4.16
(0.20)
(0.36)
(0.52)
2.20
4.32
4.16
2.16
4.16
3.98
2.20
4.32
4.16
2.27
4.51
4.34
Std. Dev.:
Non-durable
Durable
Inflation
0.90
0.94
1.25
(0.07)
(0.09)
(0.19)
0.91
1.10
1.25
0.85
0.87
1.00
0.95
1.12
1.24
1.08
1.77
1.92
AR(1):
Non-durable
Durable
Inflation
0.33
0.78
0.85
(0.07)
(0.06)
(0.05)
0.39
0.84
0.85
0.25
0.76
0.74
0.39
0.85
0.84
0.53
0.94
0.93
Dividend Growth Rates, Annual:
Mean:
Non-durable
Durable
5.89
4.55
(2.17)
(4.24)
2.32
4.40
-2.16
-4.63
2.08
4.30
6.84
13.35
Std. Dev.:
Non-durable
Durable
14.41
28.12
(1.48)
(2.91)
15.29
23.42
12.66
19.05
15.10
23.08
17.97
27.45
AR(1):
Non-durable
Durable
0.27
0.10
(0.14)
(0.15)
0.16
0.24
-0.12
-0.09
0.13
0.20
0.37
0.45
Corr. with
Nondur. Cons:
Non-durable
Durable
0.42
0.38
(0.13)
(0.13)
0.34
0.27
0.08
-0.04
0.34
0.24
0.54
0.48
Corr. with
Dur. Cons:
Non-durable
Durable
0.14
0.21
(0.15)
(0.15)
0.20
0.45
-0.09
0.14
0.20
0.41
0.43
0.63
Corr. with
Inflation:
Non-durable
Durable
-0.24
-0.32
(0.15)
(0.14)
-0.16
-0.23
-0.42
-0.47
-0.16
-0.19
0.12
0.12
The table reports the data and model properties of non-durable and durable consumption growth,
inflation, and dividends growth rates in non-durable and durable production sectors. Summary
statistics for consumption and inflation are based on quarterly data, annualized, and dividend data
are annual. Population values correspond to a long simulation at the estimated parameter values.
The 5%, 50%, and 95% model values capture the model estimate distributions across the small
samples whose size equals the data. Standard errors are Newey-West adjusted with 10 lags.
52
Table 4 Summary statistics of asset-price data
1Y
2Y
3Y
4Y
5Y
Nominal Bond Yields:
Mean
Std. Dev.
AR(1)
6.26
2.76
0.93
6.47
2.71
0.94
6.63
2.62
0.95
6.76
2.58
0.95
6.84
2.53
0.96
Excess Bond Returns:
Mean
Std. Dev.
0.42
1.83
0.68
3.45
0.87
4.72
Non-durable
0.89
5.88
Durable
Excess Stock Returns:
Mean
Std. Dev.
AR(1)
5.62
14.93
0.06
7.11
21.37
-0.02
Corr. with Bond Return
Corr. with Exp. Inflation
0.28
-0.44
0.38
-0.54
The table reports summary statistics for nominal bond yields, excess nominal bond returns, and
excess equity returns in non-durable and durable production sectors. Data are from 1963Q1 to
2006Q4.
53
Table 5 Model implications for nominal yields
Data
Est
SE
Pop
5%
Model
50%
95%
Nominal Bond Yields:
Mean:
1Y
3Y
5Y
6.26
6.63
6.84
(0.62)
(0.60)
(0.58)
6.15
6.59
6.98
4.55
5.28
5.95
6.18
6.63
7.01
7.83
7.99
8.13
Std. Dev.:
1Y
3Y
5Y
2.75
2.62
2.53
(0.47)
(0.44)
(0.42)
2.49
2.02
1.65
1.62
1.30
1.07
2.23
1.81
1.48
3.08
2.51
2.08
AR(1):
1Y
3Y
5Y
0.93
0.95
0.96
(0.02)
(0.02)
(0.02)
0.91
0.95
0.95
0.79
0.87
0.87
0.89
0.94
0.93
0.94
0.97
0.96
MSE:
1Y
3Y
5Y
2.06
2.01
2.03
0.28
0.76
0.53
0.81
0.93
0.68
1.39
1.17
0.81
Nominal Term Spread:
Mean:
Std. Dev.:
AR(1):
MSE:
0.58
0.86
0.82
1.13
(0.16)
(0.09)
(0.07)
0.84
0.99
0.73
Excess Bond Returns:
Mean:
2Y
5Y
0.42
0.89
(0.26)
(0.85)
0.39
1.33
0.06
0.34
0.39
1.29
0.72
2.27
Std. Dev.:
2Y
5Y
1.83
5.88
(0.17)
(0.57)
1.35
3.83
1.10
3.16
1.34
3.81
1.59
4.49
MSE:
2Y
5Y
1.83
5.91
The table reports the data and model output for nominal yields. Population values correspond to
a long simulation at the estimated parameter values. The 5%, 50%, and 95% model values capture
the model estimate distributions across the small samples whose size equals to the size of the data.
Quarterly data from 1963Q1 to 2006Q4. Standard errors are Newey-West adjusted with 10 lags.
54
Table 6 Model implications for excess equity returns
Data
Est
SE
Pop
5%
Model
50%
95%
Mean:
Non-durable
Durable
5.62
7.11
(2.25)
(3.22)
4.20
6.33
-0.27
0.58
4.29
6.52
9.48
13.87
Std. Dev.:
Non-durable
Durable
14.73
21.11
(1.35)
(1.94)
19.85
29.71
16.08
23.03
19.33
28.95
23.77
36.46
Corr. with:
Exp. Infl.:
Non-durable
Durable
-0.44
-0.54
(0.13)
(0.11)
-0.26
-0.30
-0.49
-0.56
-0.27
-0.31
-0.04
-0.05
Corr. with:
Bond Ret.:
Non-durable
Durable
0.28
0.38
(0.14)
(0.13)
0.31
0.36
0.08
0.11
0.31
0.35
0.52
0.55
The table reports the data and model output for equity portfolios of non-durable and durable
production sectors. Population values correspond to a long simulation at the estimated parameter
values. The 5%, 50% and 95% model values capture the model estimate distributions across the
small samples whose size equals to the size of the data. Data are from 1963 to 2006. Standard errors
are Newey-West adjusted with 10 lags.
55
Table 7 Model decomposition of yield volatility and risk premium
Total
Relative Contributions of Risks:
Exp. Nondur
Exp. Infl
Exp. Dur
Yield Volatility:
1Y
2.49
5Y
1.65
10Y
1.05
0.01
-0.03
-0.02
0.64
0.78
0.82
0.35
0.25
0.20
Bond Premia:
1Y
0.22
5Y
1.82
10Y
2.41
-1.57
-0.13
-0.09
2.09
0.98
0.97
0.48
0.15
0.12
This table reports model decomposition of yield volatility and nominal bond risk premium. The
relative contribution of risks statistics show the percentage contribution of the expected durable and
expected non-durable consumption growth and expected inflation to the variance of yields and bond
risk premium. The covariance terms are assigned to each factor proportionally to the variance of
the factor.
56
Table 8 Role of inflation non-neutrality
Full
Model
Infl. Neutrality
NonDur
Dur
All
Yield Level:
1y
3y
5y
6.15
6.59
6.98
6.19
6.55
6.84
6.16
6.33
6.47
6.11
6.14
6.19
Inflation Premium:
1y
3y
5y
0.24
0.67
0.93
0.15
0.39
0.53
0.10
0.27
0.38
0.00
0.00
0.00
Equity Premium:
Non-durable
Durable
4.20
6.33
2.32
4.52
2.29
2.91
1.66
2.61
Corr. with Exp. Infl. :
Non-durable
Durable
-0.26
-0.30
-0.13
-0.24
-0.19
-0.06
0.03
0.02
Corr. with Bond Ret. :
Non-durable
Durable
0.31
0.36
0.16
0.33
0.18
0.08
0.02
0.01
The table shows the impact of inflation non-neutrality on bond and equity moments in the model.
Inflation neutrality specifications zero out the impact of expected inflation on expected non-durable
consumption growth (NonDur), expected durable consumption growth (Dur), and both (All).
57
Figure 1: Autocorrelation of consumption growth and inflation
Non−durable
0.6
5%−95% CI
Model
Data
0.4
0.2
0
−0.2
1
2
3
4
5
6
7
8
9
10
7
8
9
10
7
8
9
10
Inflation
1
0.8
0.6
0.4
0.2
0
1
2
3
4
5
6
Durable
1
0.8
0.6
0.4
0.2
0
1
2
3
4
5
6
The figure shows the autocorrelation functions of non-durable and durable consumption growth and
non-durable inflation rate in the data and implied by the estimated model. Quarterly observations
from 1963Q1 to 2006Q4.
58
Figure 2: Impulse response of expected growth shocks
5
−3
x 10 nondur resp to nondur
−3
5
5%−95% CI
Baseline model
4
x 10
−3
infl resp to nondur
5
4
4
3
3
3
2
2
2
1
1
1
0
0
0
−1
−1
−1
0
5
−3
1
x 10
10
15
20
0
5
−3
nondur resp to infl
6
10
15
20
1
0
−2
5
10
15
20
15
20
dur resp to infl
x 10
−1
−2
2
−3
−3
0
5
−3
5
0
0
4
−1
−4
dur resp to nondur
−3
infl resp to infl
x 10
x 10
x 10
10
15
20
0
0
5
−3
nondur resp to dur
5
10
15
20
−4
x 10
0
5
−3
infl resp to dur
5
10
dur resp to dur
x 10
4
3
0
0
2
1
−5
0
5
10
15
20
−5
0
5
10
15
20
0
0
5
10
15
20
The figure shows the impulse response functions for shocks to expected non-durable consumption,
expected durable consumption, and expected non-durable inflation, based on the estimated benchmark model. Grey regions correspond to 5%-95% confidence interval. Quarterly data from 1963Q1
to 2006Q4.
59
Figure 3: Realized and filtered macroeconomic variables
Non−durable
0.02
0.01
0
5%−95% CI
Model
Data
−0.01
−0.02
1965
1970
1975
1980
1985
1990
1995
2000
2005
1990
1995
2000
2005
1990
1995
2000
2005
Inflation
0.04
0.03
0.02
0.01
0
1965
1970
1975
1980
1985
Durable
0.03
0.02
0.01
0
−0.01
1965
1970
1975
1980
1985
The figure shows the time series of realized (dashed line) and expected (solid line) non-durable
consumption growth rate, non-durable inflation rate, and durable goods growth rate. Grey regions
correspond to 5%-95% confidence interval. Quarterly data from 1963Q1 to 2006Q4.
60
Figure 4: Correlation of equity with bond returns in durable relative to non-durable sector
0.3
0.2
0.1
0
−0.1
−0.2
1975
1980
1985
1990
1995
2000
2005
The figure shows the difference in stock and bond correlations in a durable sector relative to a nondurable sector. The correlations are computed using a 10-year rolling window. Quarterly data from
1963Q1 to 2006Q4.
61
Figure 5: Nominal bond prices: data and model
One-Year Yield and Five-Year Term Spread:
0.04
0.16
5%−95% CI
0.14
model
0.03
data
0.12
0.02
0.1
0.01
0.08
0
0.06
−0.01
0.04
−0.02
0.02
−0.03
0
1965
1970
1975
1980
1985
1990
1995
2000
1965
2005
1970
1975
1980
1985
1990
1995
2000
2005
1990
1995
2000
2005
Two- and Five-Year Bond Excess Returns:
0.06
0.2
0.15
0.04
0.1
0.02
0.05
0
0
−0.05
−0.02
−0.1
−0.15
−0.04
−0.2
−0.06
−0.25
1965
1970
1975
1980
1985
1990
1995
2000
2005
1965
1970
1975
1980
1985
The figure shows data and model implications for bond yields and bond excess returns. Top panel
shows the time series of one-year nominal yield (left) and five-year term spread (right). Bottom
panel shows the time series of two- and five-year nominal bond excess returns. The data is plotted
in dashed line and the model-implied time series is plotted in solid line. Quarterly data from 1963Q1
to 2006Q4.
62
Figure 6: Yield loadings
Yield Loading on Exp. Non−durable
3.5
Data
Data EH
Model
95% CI
3
2.5
2
1.5
1
0.5
0
−0.5
1
2
3
4
5
4
5
4
5
Yield Loading on Exp. Inflation
4
3.5
3
2.5
2
1.5
1
1
2
3
Yield Loading on Exp. Durable
0
−0.5
−1
−1.5
−2
−2.5
−3
−3.5
−4
−4.5
−5
1
2
3
The figure shows the bond loadings on expected non-durable growth (top panel), expected inflation
(middle panel), and expected durable growth (bottom panel). Solid line shows bond loadings in
the model, dashed line represents bond loadings in the data, and circles show the loadings of the
Expectation-Hypothesis component of observed yields in data. 5%-95% confidence interval is based
on the standard errors in the data.
63
Figure 7: Model implications for consumption growth predictability
By Short Rate:
Non−Durable Consumption
Durable Consumption
0.1
0.1
Data
Model Pop
95% CI
0
0
−0.1
−0.1
−0.2
−0.2
−0.3
−0.3
−0.4
−0.4
−0.5
1
2
3
4
−0.5
5
1
2
3
4
5
4
5
By Term Spread:
Non−Durable Consumption
Durable Consumption
1.2
1.2
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
−0.2
−0.2
−0.4
−0.4
1
2
3
4
5
1
2
3
The figure shows the data and model-implied slope coefficients in the regressions of future consumption growth of non-durable goods (left) and durable goods (right panel) on short-term interest rate
(top panel) and the term spread (bottom panel). Solid line shows slope coefficients in the model, and
dashed line represents coefficients in the data. 5%-95% confidence interval is based on the standard
errors in the data.
64
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