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 macroeco-

nomic 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 in-

flation 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 (beraker@bus.wisc.edu) is at the Wisconsin School of Business, University ofWisconsin, Madison. Wenyu Wang (wenywang@indiana.edu) is at Kelley School of Business, In-diana University. Ivan Shaliastovich (corresponding author) is at Wharton School, Universityof Pennsylvania, 3620 Locust Walk, Philadelphia, PA 19104. Phone: (215) 746-0005, email:ishal@wharton.upenn.edu. 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 con-ference 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 forhelpful 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 non-

neutrality varies substantially across economic sectors: it is significantly more pro-

nounced in durable goods sectors, relative to non-durable goods sectors and the ag-

gregate 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 ex-

ogenous 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 infla-

tion 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 pos-

4

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 ex-

pected 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 con-

text 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 time-

varying 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. Empirical Motivation

1.1 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

1Bils (2009) and Bils and Klenow (2001) argue that the CPI inflation series released by theBureau of Labor Statistics is mis-measured because consumers shift purchases of durable goodsitems 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 anduse the inflation rate for non-durable goods.

2We 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-durable-

good producing firms. Following Gomes et al. (2009), we use the benchmark input-

output 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 portfo-

lio).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. No-

tably, our sample excludes the Great Recession and Financial Crisis. From an eco-

nomic 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 valua-

tions. 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.

3Gomes et al. (2009) consider separately services, non-durables, durables, and investment in-dustries. To be consistent with our model, we aggregate services and non-durables into a singlevalue-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 corre-

sponding R2s in the projections of future average real growth rates of durable and

non-durable consumption onto the current inflation. There are two important find-

ings: (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 pro-

nounced 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 R2s are also uni-

formly 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 doc-

ument 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 pro-

nounced 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 ag-

gregate 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 pro-

8

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, Richard-

son, 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 consump-

tion 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(P $t /P

$t−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 + Σgηt+1,

xt+1 = Πxt + Σxut+1.

(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 = di,t + σtri ξi,t, for i = {nd, d}, (2)

where di,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 autocor-

relation. 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 dy-

namics to be exposed to the expected growth factors:4

∆di,t+1 = µi + φ′ixt + σdivi ζi,t+1, for i = {nd, d}. (3)

Hence, in our specification dividend growth rates ∆di,t = ∆di,t+1 + σtri ξi,t+1 − σtri 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 interpre-

tation 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 identi-

fication 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 div-

idend 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 consump-

tion in each sector.

4Similar to the durable consumption growth, we express durable dividend growth in units ofdurable goods numeraire to facilitate the interpretation of the loadings and ensure the consistencybetween 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 tech-

nique. 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 sig-

nificantly 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 pro-

nounced for the durable goods sector. While the next-period feedback coefficients

of expected inflation for expected durable and non-durable consumption are not sig-

nificantly different, the inflation impact on future durable consumption growth is

magnified over time due to a much larger persistence of the expected durable con-

sumption 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 ex-

pected 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 sec-

tors. 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 aggre-

gate 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 ap-

proach: 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 pro-

ducers. 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 con-

sumption are described by the Kreps-Porteus, Epstein-Zin recursive utility function

(see Kreps and Porteus 1978 and Epstein and Zin 1989):

Ut =

[(1− β)u

1− 1ψ

t + β(EtU

1−γt+1

) 1− 1ψ

1−γ

] 1

1− 1ψ

, (4)

14

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 substitu-

tion (CES) form, and thus can be expressed in the following way:

u(C, S) =[(1− α)C1− 1

ε + αS1− 1ε

] 1

1− 1ε . (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 ε captures the intratemporal elasticity of substitution between the two

goods. High values of ε indicate that the two goods can be easily substituted by the

agent, while small values for ε 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 = βθ(Zt+1

Zt

) θ

1− 1ε( 1ψ− 1ε )(Ct+1

Ct

)− θψ

Rθ−1c,t+1. (7)

The relative share of non-durable consumption of the agent Zt is defined as:

Zt =Ct

Ct +QtSt, (8)

and Qt is the user cost of durable goods given by the ratio of the marginal utilities

of durable to non-durable consumption:

Qt =ustuct

=α

1− α

(StCt

)− 1ε

. (9)

The consumption return Rc,t+1, which captures the return on the total wealth port-

folio 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 +QtSt). (10)

In a single good economy (Zt ≡ 1) we obtain a standard expression for the stochas-

tic discount factor, derived in Epstein and Zin (1991):

MNon−Durt+1 = βθ

(Ct+1

Ct

)− θψ

Rθ−1c,t+1. (11)

16

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+1Ri,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 con-

sumption as a numeraire. To change the numeraire, denote Πt 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 Mt+1,

Mt+1 = Mt+1Πt

Πt+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 = logZt+1

Zt≈ χ(∆qt+1 + ∆st+1 −∆ct+1) = χ

(1− 1

ε

)(is − ic)′gt+1, (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 = QSQS+C

. This parameter captures the

importance of durable goods in the economy. In the economy with a single non-

durable 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 + A′xxt. (15)

The price-consumption loadings satisfy:

Ax =

(1− 1

ψ

)(I − κ1Π′)

−1((1− χ)ic + χis) , (16)

where κ1 ∈ (0, 1) is the log-linearization coefficient whose value is provided in Ap-

pendix A. As in a single good economy, when the intertemporal elasticity of substi-

tution 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

5The linearization of the relative share shuts off the variation in the asset volatilities and riskpremia due to the relative share movements (see Cochrane, Longstaff, and Santa-Clara 2008 andColacito and Croce 2013). The relative share of durables is quite stable in our sample, so thesefluctuations are not likely to be important for the unconditional asset-pricing moments that weanalyze.

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 numeraire,

can be written in terms of the fundamental states and shocks in the economy:

mt+1 = m0 +m′xxt − λ′gΣgηt+1 − λ′xΣxut+1, (17)

where mx captures the loadings of the discount factor on the expected growth com-

ponents, 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)κ1Ax,

λz =

(γ(1− χ) +

1

εχ

)ic +

(γ − 1

ε

)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 non-

neutral 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:

pt,n = −B0,n −B′x,nxt, (19)

where the bond loadings depend on model parameters and are provided in the Ap-

pendix A.

The real yields are given by:

yt,n = − 1

npt,n =

1

n

(B0,n +B′x,nxt

). (20)

For a one-period bond, the vector of bond loadings satisfies,

Bx,1 =

(1

ψ(1− χ) +

1

εχ

)ic − χ

(1

ε− 1

ψ

)is. (21)

20

When χ = 0 the specification reduces to a one-good non-durable model and risk-

free 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 ε < 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 non-

durables 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 ε < 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 dis-

count factor with the excess bond return:

Etrxt+1,n +1

2V artrxt+1,n = −Covt(mt+1, rxt+1,n−1) = −B′x,n−1ΣxΣ

′xλx. (23)

The bond risk premia are determined by the bond exposure to the economic risk Bx,n,

the quantity of risk ΣxΣ′x, and the market price of risks λx.

21

The equilibrium price of nominal bonds and nominal bond risk premia are de-

rived 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:

y$t,n = yt,n + Etπt→t+n −

1

2

1

nV artπt→t+n +

1

nCovt(mt→t+n, πt→t+n), (24)

where πt→t+n and mt→t+n denote the t to t+n multi-period inflation rate and stochas-

tic discount factor, respectively. First, as nominal bonds pay in nominal dollar terms,

an increase in expected inflation directly raises nominal yields and decreases nom-

inal 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 long-

run inflation’s interaction with future non-durable growth, as discussed in Piazzesi

and Schneider (2006) and Bansal and Shaliastovich (2013). With multiple goods, in-

flation 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 non-

durable and durable sectors:

pdi,t = Hi,0 +H ′i,xxt +Hi,ξσtri ξi,t, for i = {nd, d}, (25)

22

where the expressions for the loadings are provided in Appendix A. For typical pa-

rameter values, equity prices increase during times of high expected consumption,

and decrease during times of high expected inflation. The magnitudes of equity ex-

posures 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 cur-

rent 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 non-

durable 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 pa-

rameters. We estimate the preference parameters γ, ψ, χ, and ε 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 port-

folios, 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 quan-

tities of durable and non-durable goods in the data. These macroeconomic moments

help identify χ and ε in the data. To compute the standard errors of the preference

parameters, we take into account the first-stage parameter uncertainty in the macro-

econometric 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 ε 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.

6This is related to the discussion of identification issues in Kocherlakota (1990) and is similar tothe 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. Long-

term 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, nom-

inal 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 non-

durable 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

7Note that our model abstracts from other economic channels, such as movements in risk andterm premia, interest rate shocks, changes in the monetary policy regimes, or fluctuations in riskaversion 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 oftime-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 bondmarkets.

27

that durable consumption growth contributes more than the non-durable consump-

tion 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 model-

specific assumptions:

yEHt,n =1

nEt∑

yt+j−1,1, (26)

8Our results are similar if we use only yield variables or only macroeconomic variables.

28

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 pre-

dictability 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 con-

trasts 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 evi-

dence: 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%

9This 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 bondreturns.

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. 10

3.4 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

10Indeed, we also consider consumption growth projections on the risk premia adjusted compo-nents of the term spread, constructed from the Expectations Hypothesis components of yields. Atshort horizons, the loadings on the risk-premia-adjusted term spread are above those on unadjustedterm 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 pro-

nounced 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. Fur-

ther, 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 ex-

plain 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.

11Hasseltoft (2009), Bekaert, Baele, and Inghelbrecht (2010), and Campbell, Sunderam, and Vi-ceira (2012) show that the stock-bond correlation switches sign and becomes negative in the recentperiod, which can be driven by regime changes in the fundamentals, time-varying risk premia orliquidity considerations. For simplicity, our model does not feature these economic channels, so thatthe 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:

Etrxt+1,n +1

2V artrxt+1,n = −B′x,n−1ΣxΣ

′xλx,

V art(yt,n) =

(1

n

)2

B′x,nΣxΣ′xBx,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 ΣxΣ′x 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 non-

durable 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 con-

sumption, 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, non-

durable 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 in-

flation 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 es-

timates 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 in-

flation impact on non-durable consumption growth, on durable consumption growth,

and finally on both consumption growth rates at the same time. To obtain infla-

tion 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 infla-

tion 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 con-

sumption, 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 ε = 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 bench-

mark 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:

Qt = P dt − (1− δ)Et[Mt+1P

dt+1], (28)

where P dt 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 approxima-

tions 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 cor-

related 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 magni-

tude 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 fu-

ture 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 − λ′zµ, (A.1)

where the log-linearization parameter κ1 satisfies the following recursive equation:

log κ1 = log β +

(1− 1

ψ

)((1− χ)ic + χ(ia + is))

′ µ

+1

2θ

(1− 1

ψ

)2

((1− χ)ic + χ(ia + is))′ΣgΣ

′g ((1− χ)ic + χ(ia + is))

+1

2θκ2

1A′xΣxΣ′xAx.

(A.2)

The nominal discount factor is given by:

mt+1 = m$′0 +m$′

x xt − λ$′g Σgηt+1 − λ$′

x Σxut+1. (A.3)

The parameters satisfy:

m$0 = m0 − iπµz, m$

x = mx − F ′iπ, λ$x = λx, λ$

z = λz + iπ, (A.4)

where iπ =[0 1

]′.

The solution for real bond price loadings are given by,

B0,n = B0,n−1 −m0 −1

2λ′gΣgΣ

′gλg −

1

2(λx +Bx,n−1)′ΣxΣ′x(λx +Bx,n−1),

Bx,n = Π′Bx,n−1 −mx.

(A.5)

The nominal bond equations are very similar, and use the parameters of the nominal dis-

count factor.

41

Finally, the price-dividend ratio is given by,

pdi,t = Hi,0 +H ′i,xxt +Hi,ξσtri ξi,t, (A.6)

for i = nd, d. The solutions to the price coefficients are given by,

Hi,x =(I − κ1,dΠ

′)−1(m$x + φi + iπ

),

Hi,ξ = −1.(A.7)

42

B. Alternative Model Specifications

Table B.1 Role of preference parameters

Full One Good High ε Low ψModel χ = 0 ε = 1.5 ψ = 0.05

Mkt Price of Risk:Exp. Nondur 16.00 24.61 16.00 -3.18Exp. Dur 42.19 0.00 42.17 -4.83Exp. Inflation -75.20 -39.43 -75.20 4.05

1Y Yield Loadings:Exp. Nondur 2.43 1.10 1.08 22.10Exp. Dur. -2.81 0.00 0.04 19.58Exp. Infl. 3.20 3.16 3.16 -6.93

Bond Premia:1Y 0.22 0.13 0.28 0.965Y 1.82 0.60 1.09 1.7010Y 2.41 0.73 1.30 1.93

Equity Premia:Non-durable 4.20 2.12 3.05 0.82Durable 6.33 3.42 8.54 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

ε), or low intertemporal elasticity of substitution (Low ψ).

43

Table B.2 Estimates of the model with unrestricted persistence matrix

Π Σx × 1000 diag(Σg)× 1000

∆c ∆π ∆s ∆c ∆π ∆s∆c 0.312

(0.107)−0.163(0.055)

−0.060(0.086)

3.822(0.369)

1.421(0.782)

∆π 0.102(0.068)

0.969(0.035)

0.076(0.045)

−0.312(0.321)

1.923(0.331)

1.988(0.262)

∆s −0.064(0.068)

−0.112(0.035)

0.879(0.046)

1.511(0.291)

0.298(0.294)

1.202(0.278)

1.579(0.359)

φ σd × 100 σtrd × 100

∆dnd 4.96(2.28)

0 0 7.58(0.81)

7.46(0.93)

∆dnd 0 0 5.43(2.99)

9.89(1.47)

14.31(1.58)

The table shows parameter estimates of the model with an unrestricted persistence matrix. Macroe-

conomic parameters are estimated by Bayesian MCMC using the quarterly observation on consump-

tion 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

Π Σx × 1000 diag(Σg)× 1000

∆c ∆π ∆s ∆c ∆π ∆s∆c 0.395

(0.099)−0.136(0.055)

0 3.602(0.631)

1.912(0.912)

∆π 0 0.917(0.035)

0 −0.102(0.319)

2.291(0.301)

1.878(0.298)

∆s 0 −0.102(0.027)

0.878(0.031)

1.393(0.197)

0.301(0.205)

1.102(0.198)

1.598(0.164)

φ σd × 100 σtrd × 100

∆dnd 4.74(2.720)

−1.10(1.182)

−0.159(1.708)

7.45(0.821)

7.69(1.027)

∆dnd 5.78(5.604)

−2.41(2.113)

2.63(3.134)

10.35(1.401)

13.69(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

1965 1970 1975 1980 1985 1990 1995 2000 2005−0.02

−0.01

0

0.01

0.02

Non−durable

1965 1970 1975 1980 1985 1990 1995 2000 2005−0.01

0

0.01

0.02

Inflation

Baseline model

Unrestricted model

1965 1970 1975 1980 1985 1990 1995 2000 2005−0.015

−0.01

−0.005

0

0.005

0.01

Durable

The figure shows the time series of the expected non-durable consumption growth rate, expected non-

durable 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

0 5 10 15 20−1

0

1

2

3

4

5x 10

−3 nondur resp to nondur

5%−95% CI

Baseline model

Unrestricted model

0 5 10 15 20−1

0

1

2

3

4

5x 10

−3 infl resp to nondur

0 5 10 15 20−1

0

1

2

3

4

5x 10

−3 dur resp to nondur

0 5 10 15 20−5

0

5x 10

−3 nondur resp to dur

0 5 10 15 20−5

0

5x 10

−3 infl resp to dur

0 5 10 15 20−2

0

2

4

6x 10

−3 dur resp to dur

0 5 10 15 20−4

−3

−2

−1

0

1x 10

−3 nondur resp to infl

0 5 10 15 20−2

0

2

4

6x 10

−3 infl resp to infl

0 5 10 15 20−4

−3

−2

−1

0

1x 10

−3 dur resp to infl

The figure shows impulse response functions of macroeconomic variables 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 persistence matrix. Grey regions correspond

to 5%-95% confidence interval.

47

Figure B.3: Impulse responses of dividend growth

0 5 10 15 20

0

0.01

0.02

0.03

0.04nd div resp to nd cons

5%−95% CI

Baseline model

Unrestricted

0 5 10 15 20

0

0.02

0.04

0.06

0.08dur div resp to nd cons

0 5 10 15 20−0.03

−0.02

−0.01

0

0.01

0.02

nd div resp to dur cons

0 5 10 15 20

−0.02

0

0.02

0.04dur div resp to dur cons

0 5 10 15 20

−0.03

−0.02

−0.01

0

0.01nd div resp to inf

0 5 10 15 20

−0.04

−0.03

−0.02

−0.01

0

0.01dur div resp to infl

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 corre-

spond 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 5.449Std Dev 1.159 1.199AR(1) 0.932 0.911

1-year Nominal Yield:Mean 6.151 6.148Std Dev 2.494 2.564AR(1) 0.912 0.917

5-year Nominal Yield:Mean 6.983 6.994Std Dev 1.651 1.722AR(1) 0.947 0.951

10-year Nominal Yield:Mean 7.536 7.588Std Dev 1.050 1.107AR(1) 0.944 0.947

The table reports the moments of the price-consumption ratio and nominal yields under the approx-

imate 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 2Y 3Y 4Y 5Y

Consumption Growth:Non-durable -0.187 -0.119 -0.075 -0.041 -0.029

(0.056) (0.062) (0.061) (0.053) (0.047)R2 0.144 0.099 0.061 0.027 0.017

Durable -0.305 -0.329 -0.310 -0.270 -0.230(0.073) (0.085) (0.091) (0.084) (0.067)

R2 0.201 0.272 0.286 0.260 0.227

Dividend Growth:Non-durable -1.641 -1.188 -0.787 -0.554 -0.315

(0.551) (0.531) (0.519) (0.411) (0.315)R2 0.044 0.050 0.039 0.034 0.017

Durable -4.277 -3.227 -1.915 -1.175 -0.461(1.027) (0.890) (0.937) (0.906) (0.754)

R2 0.100 0.123 0.067 0.033 0.008

The table reports the slope coefficients, the standard errors, and the R2s 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:

Π Σx × 1000 diag(Σg)× 1000

∆c π ∆s ∆c π ∆s∆c 0.375

(0.115)−0.143(0.063)

0 3.83(0.552)

1.46(0.979)

π 0 0.916(0.034)

0 −0.16(0.306)

2.35(0.334)

1.78(0.252)

∆s 0 −0.104(0.040)

0.876(0.043)

1.37(0.385)

0.37(0.254)

1.20(0.273)

1.63(0.369)

φ σd × 100 σtrd × 100

∆dnd 4.98(2.12)

0 0 7.80(0.80)

7.56(0.91)

∆dnd 0 0 5.45(2.48)

11.39(1.49)

12.80(1.52)

Preference Parameters:

δ γ ψ ε χ

0.994 16.21(8.33)

1.42(0.11)

0.32(0.22)

0.35(0.37)

The table reports parameter estimates of the benchmark model specification. Macroeconomic pa-

rameters 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 ModelEst SE Pop 5% 50% 95%

Consumption Growth and Inflation, Quarterly:

Mean: Non-durable 2.20 (0.20) 2.20 2.16 2.20 2.27Durable 4.32 (0.36) 4.32 4.16 4.32 4.51Inflation 4.16 (0.52) 4.16 3.98 4.16 4.34

Std. Dev.: Non-durable 0.90 (0.07) 0.91 0.85 0.95 1.08Durable 0.94 (0.09) 1.10 0.87 1.12 1.77Inflation 1.25 (0.19) 1.25 1.00 1.24 1.92

AR(1): Non-durable 0.33 (0.07) 0.39 0.25 0.39 0.53Durable 0.78 (0.06) 0.84 0.76 0.85 0.94Inflation 0.85 (0.05) 0.85 0.74 0.84 0.93

Dividend Growth Rates, Annual:

Mean: Non-durable 5.89 (2.17) 2.32 -2.16 2.08 6.84Durable 4.55 (4.24) 4.40 -4.63 4.30 13.35

Std. Dev.: Non-durable 14.41 (1.48) 15.29 12.66 15.10 17.97Durable 28.12 (2.91) 23.42 19.05 23.08 27.45

AR(1): Non-durable 0.27 (0.14) 0.16 -0.12 0.13 0.37Durable 0.10 (0.15) 0.24 -0.09 0.20 0.45

Corr. with Non-durable 0.42 (0.13) 0.34 0.08 0.34 0.54Nondur. Cons: Durable 0.38 (0.13) 0.27 -0.04 0.24 0.48

Corr. with Non-durable 0.14 (0.15) 0.20 -0.09 0.20 0.43Dur. Cons: Durable 0.21 (0.15) 0.45 0.14 0.41 0.63

Corr. with Non-durable -0.24 (0.15) -0.16 -0.42 -0.16 0.12Inflation: Durable -0.32 (0.14) -0.23 -0.47 -0.19 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 6.26 6.47 6.63 6.76 6.84Std. Dev. 2.76 2.71 2.62 2.58 2.53AR(1) 0.93 0.94 0.95 0.95 0.96

Excess Bond Returns:

Mean 0.42 0.68 0.87 0.89Std. Dev. 1.83 3.45 4.72 5.88

Non-durable Durable

Excess Stock Returns:

Mean 5.62 7.11Std. Dev. 14.93 21.37AR(1) 0.06 -0.02

Corr. with Bond Return 0.28 0.38Corr. with Exp. Inflation -0.44 -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 ModelEst SE Pop 5% 50% 95%

Nominal Bond Yields:

Mean: 1Y 6.26 (0.62) 6.15 4.55 6.18 7.833Y 6.63 (0.60) 6.59 5.28 6.63 7.995Y 6.84 (0.58) 6.98 5.95 7.01 8.13

Std. Dev.: 1Y 2.75 (0.47) 2.49 1.62 2.23 3.083Y 2.62 (0.44) 2.02 1.30 1.81 2.515Y 2.53 (0.42) 1.65 1.07 1.48 2.08

AR(1): 1Y 0.93 (0.02) 0.91 0.79 0.89 0.943Y 0.95 (0.02) 0.95 0.87 0.94 0.975Y 0.96 (0.02) 0.95 0.87 0.93 0.96

MSE: 1Y 2.063Y 2.015Y 2.03

Nominal Term Spread:

Mean: 0.58 (0.16) 0.84 0.28 0.81 1.39Std. Dev.: 0.86 (0.09) 0.99 0.76 0.93 1.17AR(1): 0.82 (0.07) 0.73 0.53 0.68 0.81MSE: 1.13

Excess Bond Returns:

Mean: 2Y 0.42 (0.26) 0.39 0.06 0.39 0.725Y 0.89 (0.85) 1.33 0.34 1.29 2.27

Std. Dev.: 2Y 1.83 (0.17) 1.35 1.10 1.34 1.595Y 5.88 (0.57) 3.83 3.16 3.81 4.49

MSE: 2Y 1.835Y 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 ModelEst SE Pop 5% 50% 95%

Mean: Non-durable 5.62 (2.25) 4.20 -0.27 4.29 9.48Durable 7.11 (3.22) 6.33 0.58 6.52 13.87

Std. Dev.: Non-durable 14.73 (1.35) 19.85 16.08 19.33 23.77Durable 21.11 (1.94) 29.71 23.03 28.95 36.46

Corr. with: Non-durable -0.44 (0.13) -0.26 -0.49 -0.27 -0.04Exp. Infl.: Durable -0.54 (0.11) -0.30 -0.56 -0.31 -0.05

Corr. with: Non-durable 0.28 (0.14) 0.31 0.08 0.31 0.52Bond Ret.: Durable 0.38 (0.13) 0.36 0.11 0.35 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

Relative Contributions of Risks:Total Exp. Nondur Exp. Infl Exp. Dur

Yield Volatility:1Y 2.49 0.01 0.64 0.355Y 1.65 -0.03 0.78 0.2510Y 1.05 -0.02 0.82 0.20

Bond Premia:1Y 0.22 -1.57 2.09 0.485Y 1.82 -0.13 0.98 0.1510Y 2.41 -0.09 0.97 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 Infl. NeutralityModel NonDur Dur All

Yield Level:1y 6.15 6.19 6.16 6.113y 6.59 6.55 6.33 6.145y 6.98 6.84 6.47 6.19

Inflation Premium:1y 0.24 0.15 0.10 0.003y 0.67 0.39 0.27 0.005y 0.93 0.53 0.38 0.00

Equity Premium:Non-durable 4.20 2.32 2.29 1.66Durable 6.33 4.52 2.91 2.61

Corr. with Exp. Infl. :Non-durable -0.26 -0.13 -0.19 0.03Durable -0.30 -0.24 -0.06 0.02

Corr. with Bond Ret. :Non-durable 0.31 0.16 0.18 0.02Durable 0.36 0.33 0.08 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

1 2 3 4 5 6 7 8 9 10−0.2

0

0.2

0.4

0.6

Non−durable

5%−95% CI

Model

Data

1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

Inflation

1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

Durable

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

0 5 10 15 20−1

0

1

2

3

4

5x 10

−3nondur resp to nondur

5%−95% CI

Baseline model

0 5 10 15 20−1

0

1

2

3

4

5x 10

−3 infl resp to nondur

0 5 10 15 20−1

0

1

2

3

4

5x 10

−3 dur resp to nondur

0 5 10 15 20−5

0

5x 10

−3 nondur resp to dur

0 5 10 15 20−5

0

5x 10

−3 infl resp to dur

0 5 10 15 200

1

2

3

4

5x 10

−3 dur resp to dur

0 5 10 15 20−4

−3

−2

−1

0

1x 10

−3 nondur resp to infl

0 5 10 15 200

2

4

6x 10

−3 infl resp to infl

0 5 10 15 20−4

−3

−2

−1

0

1x 10

−3 dur resp to infl

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 bench-

mark model. Grey regions correspond to 5%-95% confidence interval. Quarterly data from 1963Q1

to 2006Q4.

59

Figure 3: Realized and filtered macroeconomic variables

1965 1970 1975 1980 1985 1990 1995 2000 2005−0.02

−0.01

0

0.01

0.02

Non−durable

5%−95% CI

Model

Data

1965 1970 1975 1980 1985 1990 1995 2000 2005

0

0.01

0.02

0.03

0.04

Inflation

1965 1970 1975 1980 1985 1990 1995 2000 2005−0.01

0

0.01

0.02

0.03

Durable

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

1975 1980 1985 1990 1995 2000 2005−0.2

−0.1

0

0.1

0.2

0.3

The figure shows the difference in stock and bond correlations in a durable sector relative to a non-

durable 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 modelOne-Year Yield and Five-Year Term Spread:

1965 1970 1975 1980 1985 1990 1995 2000 20050

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

5%−95% CI

model

data

1965 1970 1975 1980 1985 1990 1995 2000 2005−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

Two- and Five-Year Bond Excess Returns:

1965 1970 1975 1980 1985 1990 1995 2000 2005−0.06

−0.04

−0.02

0

0.02

0.04

0.06

1965 1970 1975 1980 1985 1990 1995 2000 2005−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

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

1 2 3 4 5−0.5

0

0.5

1

1.5

2

2.5

3

3.5Yield Loading on Exp. Non−durable

Data

Data EH

Model

95% CI

1 2 3 4 51

1.5

2

2.5

3

3.5

4Yield Loading on Exp. Inflation

1 2 3 4 5−5

−4.5

−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0Yield Loading on Exp. Durable

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:

1 2 3 4 5−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1Non−Durable Consumption

Data

Model Pop

95% CI

1 2 3 4 5−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1Durable Consumption

By Term Spread:

1 2 3 4 5

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2Non−Durable Consumption

1 2 3 4 5

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2Durable Consumption

The figure shows the data and model-implied slope coefficients in the regressions of future consump-

tion 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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