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Federal Reserve Bank of Chicago
Examining Macroeconomic Models
through the Lens of Asset Pricing
Jaroslav Borovička and Lars Peter Hansen
WP 2012-01
Examining Macroeconomic Models through the Lens of
Asset Pricing∗
Jaroslav Borovička
Lars Peter Hansen
University of Chicago
Federal Reserve Bank of Chicago
University of Chicago
National Bureau of Economic Research
borovicka@uchicago.edu
lhansen@uchicago.edu
December 9, 2011
Abstract
Dynamic stochastic equilibrium models of the macro economy are designed to match
the macro time series including impulse response functions. Since these models aim
to be structural, they also have implications for asset pricing. To assess these implications, we explore asset pricing counterparts to impulse response functions. We use
the resulting dynamic value decomposition (DVD) methods to quantify the exposures
of macroeconomic cash flows to shocks over alternative investment horizons and the
corresponding prices or compensations that investors must receive because of the exposure to such shocks. We build on the continuous-time methods developed in Hansen
and Scheinkman (2010), Borovička et al. (2011) and Hansen (2011) by constructing
discrete-time shock elasticities that measure the sensitivity of cash flows and their
prices to economic shocks including economic shocks featured in the empirical macroeconomics literature. By design, our methods are applicable to economic models that
are nonlinear, including models with stochastic volatility. We illustrate our methods
by analyzing the asset pricing model of Ai et al. (2010) with tangible and intangible
capital.
We thank John Cochrane, Jesús Fernández-Villaverde, John Heaton, Junghoon Lee and Ian Martin for
helpful comments. The views expressed herein are those of the authors and not necessarily those of the
Federal Reserve Bank of Chicago or the Federal Reserve System.
∗
1
Introduction
It is standard practice to represent implications of dynamic macroeconomic models by showing how featured time series respond to shocks. Alternative current period shocks influence
the future trajectory of macroeconomic processes such as consumption, investment or output, and these impacts are measured by impulse response functions. From an asset pricing
perspective, these functions reflect the exposures of the underlying macroeconomic processes
to shocks. These exposures depend on how much time has elapsed between the time the
shock is realized and time of its impact on the macroeconomic time series under investigation. Changing this gap of time gives a trajectory of exposure elasticities that we measure.
In this manner we build shock-exposure elasticities that are very similar to and in some cases
coincide with impulse response functions.
In a fully specified dynamic, stochastic equilibrium model, exposures to macroeconomic
shocks are priced because investors must be compensated for bearing this risk. To capture this compensation, we produce pricing counterparts to impulse response functions by
representing and computing shock-price elasticities implied by the structural model. These
prices are the risk compensations associated with the shock exposures. The shock-exposure
and shock-price elasticities provide us with dynamic value decompositions (DVD’s) to be
used in analyzing alternative structural models that have valuation implications. Quantity
dynamics reflect the impact of current shocks on future distributions of a macroeconomic
process, while pricing dynamics reflect the current period compensation for the exposure to
future shocks.
In our framework the shock-exposure and shock-price elasticities have a common underlying mathematical structure. Let M be process that grows or decays stochastically in a
geometric fashion. It captures the compounding discount and/or growth rates over time in
a stochastic fashion and is constructed from an underlying Markov process X. Let W be a
sequence of independent and identically distributed standard normal random vectors. The
common ingredient in our analysis is the ratio:
εm (x, t) = αh (x) ·
E [Mt W1 | X0 = x]
.
E [Mt | X0 = x]
(1)
where x is the current Markov state and αh selects the linear combination of the shock vector
W1 of interest. The state dependence in αh allows for analysis of stochastic volatility. We
interpret this entity as a “shock elasticity” used to quantify the date t impact on values of
exposure to the shock αh (x)W1 at date one. We justify this formula and provide ways to
compute it in practice.
While these elasticities have not been explored in the quantitative literature in macroe1
conomics, they have antecedents in the asset pricing literature. The intertemporal structure
of risk premia has been featured in the term structure of interest rates, but this literature
purposefully abstracts from the pricing of stochastic growth components in the macroeconomy. Recently Lettau and Wachter (2007) and Hansen et al. (2008) have explored the term
structure of risk premia explicitly in the context of equity claims that grow over time. Risk
premia reflect contributions from exposures and prices of those exposures. Here we build on
an analytical framework developed in Alvarez and Jermann (2005), Hansen and Scheinkman
(2009), Hansen and Scheinkman (2010) and Borovička et al. (2011) to distinguish exposure
elasticities and price elasticities. We illustrate these tools in measuring shock exposures and
model-implied prices of exposure to those shocks in a model with physical and intangible
capital constructed by Ai et al. (2010).
2
Analytical framework
In this section we describe some basic tools for valuation accounting, by which we provide
measures of shock exposures and shock prices for alternative investment horizons. We will
justify and interpret formula (1) given in the introduction. Let X be the state vector process
for a dynamic stochastic equilibrium model.
We consider dynamic systems of the form
Xt+1 = ψ(Xt , Wt+1 )
(2)
where W is a sequence of independent shocks distributed as a multivariate standard normal.
Moreover, Wt+1 is independent of the date-t state vector Xt . In much of what follows we
will focus on stationary solutions for this system.
By imposing appropriate balanced growth restrictions, we imagine that the logarithms
of many macroeconomic processes that interest us grow or decay over time and can be
represented as:
t−1
X
Yt = Y0 +
κ(Xs , Ws+1 )
(3)
s=0
where Y0 is an initial condition, which we will set conveniently to zero in much of our
discussion. A typical example of the increment to this process is
κ(Xs , Ws+1 ) = β(Xs ) + α(Xs ) · Ws+1
where the function β allows for nonlinearity in the conditional mean and the function α
introduces stochastic volatility. We call such a process Y an additive functional since it
2
accumulates additively over time, and can be built from the underlying Markov process X
provided that Wt+1 can be inferred from Xt+1 and Xt . By a suitable construction of the
state vector, this restriction can always be met. The state vector X thus determines the
dynamics of the increments in Y . When X is stationary Y has stationary increments.
While the additive specification of Y is convenient for modeling logarithms of economic
processes, to represent values of uncertain cash flows it is necessary to study levels instead
of logarithms. We therefore use the exponential of an additive functional, M = exp (Y ),
to capture growth or decay in levels. We will refer to M as a multiplicative functional
represented by κ or sometimes the more restrictive specification (α, β).
In what follows we will consider two types of multiplicative functionals, one that captures
macroeconomic growth, denoted by G, and another that captures stochastic discounting, denoted by S. The stochastic nature of discounting is needed to adjust consumption processes
or cash flows for risk. Thus S, and sometimes G as well, are computed from the underlying
economic model to reflect equilibrium price dynamics. For instance, G might be a consumption process or some other endogenously determined cash flow, or it might be an exogenously
specified technology shock process that grows through time. The interplay between S and
G will dictate valuation over multi-period investment horizons.
2.1
One-period asset pricing
It is common practice in the asset pricing literature to represent prices of risk in terms of
expected return on an investment per unit of exposure to risk. For instance, the familiar
Sharpe ratio measures the difference between the expected return on a risky and a risk-free
cash flow scaled by the volatility of the risky cash flow. We are interested in using this
approach to assign prices to shock exposures.
As a warm up for subsequent analysis, consider one-period asset pricing for conditionally
normal models. Suppose that
log G1 = βg (X0 ) + αg (X0 ) · W1
log S1 = βs (X0 ) + αs (X0 ) · W1
where G1 is the payoff to which we assign values and S1 is the one-period stochastic discount
factor used to compute these values. The one-period return on this investment is:
R1 =
G1
E [S1 G1 | X0 ]
Applying standard formulas for lognormally distributed random variables, the logarithm
3
of the expected return is:
|αs (x)|2
log E [G1 | X0 = x] − log E [S1 G1 | X0 = x] = −βs (x) − αg (x) · αs (x) −
.
2
Imagine applying this to a family of such payoffs parameterized in part by αg . The vector
αg defines a vector of exposures to the components of the normally distributed shock W1 .
Then −αs is the vector of shock “prices” representing the compensation for exposure to the
shocks. This compensation is expressed in terms of expected returns as is typical in asset
pricing.
While this calculation is straightforward, we now explore a related derivation that will
extend directly to multiple horizons. We parameterize a family of payoffs using:
r2
log H1 (r) = rαh (X0 ) · W1 − |αh (X0 )|2
2
(4)
where r is a scalar parameter and impose
E[|αh (X0 )|2 ] = 1.
In what follows we use the vector αh as an exposure direction. We have built H1 (r) so that
it has conditional expectation equal to one, but other constructions are also possible.
Form a parameterized family of payoffs G1 H1 (r) where by design:
log G1 + log H1 (r) = [αg (X0 ) + rαh (X0 )] · W1 + βg (X0 ) −
r2
|αh (X0 )|2 .
2
By changing r we alter the exposure in direction αh . These payoffs imply a corresponding
parameterized family of logarithms of expected returns:
log E[G1 H1 (r) | X0 = x] − log E[S1 G1 H1 (r) | X0 = x].
Since we are using the logarithms of the expected returns measure and our exposure direction
αh (X0 ) · W1 has a unit standard deviation, by differentiating with respect to r we compute
an elasticity:
d
log E[G1 H1 (r) | X0 = x]
dr
r=0
−
d
log E[S1 G1 H1 (r) | X0 = x]
dr
.
r=0
This calculation leads us to define counterparts to quantity and price elasticities from microeconomics:
4
1. shock-exposure elasticity:
εg (x, 1) =
d
log E[G1 H1 (r) | X0 = x]
dr
r=0
= αg (x) · αh (x)
2. shock-price elasticity:
εp (x, 1) =
d
log E[G1 H1 (r) | X0 = x]
dr
r=0
−
d
log E[S1 G1 H1 (r) | X0 = x]
dr
r=0
= −αs (x) · αh (x).
For this conditional log-normal specification, αg measures the exposure vector, −αs measures
the price vector and αh captures which combination of shocks is being targeted. In this
setting the shock price elasticity can be thought of as the “conditional covariance” between
− log S1 and αh · W1 .
Since exposure to risk requires compensation, notice that a “value elasticity” is the
difference between an exposure elasticity and a price elasticity:
d
log E[S1 G1 H1 (r) | X0 = x]
dr
r=0
= εg (x, 1) − εp (x, 1)
The value of an asset responds to changes in exposure of the associated cash flow to a shock
(a quantity effect), and to changes in the compensation resulting from the change in exposure
(a price effect). The shock elasticity of the asset value is then obtained by taking into account
both effects operating in opposite directions. Specifically, the shock price elasticity enters
with a negative sign because exposure to risk requires compensation reflected in a decline in
the asset value.
Our formulas for the shock elasticities exploit conditional log-normality of the payoffs to
be priced and of the stochastic discount factor. In this formulation we are using the possibility
of conditioning variables to fatten tails of distributions as in models with stochastic volatility.
This conditioning is captured by the Markov state x in our elasticity formulas. We use one as
the second argument for the elasticities to denote that we are pricing a one-period payoff. We
extend the analysis to multi-period cash flows in the next subsection. While the one-period
price elasticity does not depend on our specification of αg , the dependence on αg emerges
when we consider longer investment horizons.
5
2.2
Multiple-period investment horizons
Next we analyze how our analysis extends to longer investment horizons. Consider the
parameterized payoff Gt H1 (r) with a date-zero price E [St Gt H1 (r) | X0 = x]. Notice that
we are changing the exposure at date one and looking at the consequences on a t-period
investment. The logarithm of the expected return is:
log E [Gt H1 (r) | X0 = x] − log E [St Gt H1 (r) | X0 = x] .
Following our previous analysis, we construct two elasticities:
1. shock-exposure elasticity:
εg (x, t) =
d
log E[Gt H1 (r) | X0 = x]
dr
r=0
2. shock-price elasticity:
εp (x, t) =
d
log E[Gt H1 (r) | X0 = x]
dr
r=0
−
d
log E[St Gt H1 (r) | X0 = x]
dr
.
r=0
These two elasticities are functions of the investment horizon t, and thus we obtain a term
structure of elasticities. The components of these elasticities have a common mathematical
form. This is revealed by using a multiplicative functional M to represent either G or
the product SG. Taking the derivative with respect to r yields equation (1) given in the
introduction and reproduced here:
εm (x, t) = αh (x) ·
E [Mt W1 | X0 = x]
.
E [Mt | X0 = x]
This formula provides a target for computation and interpretation. Consider the pricing
of a vector of payoffs Gt W1 in comparison to the scalar payoff Gt . The shock-exposure
elasticity is constructed from the ratio of expected payoffs E [Gt W1 | X0 = x] relative to
E [Gt | X0 = x]. The shock-price elasticity includes an adjustment for the values of the
payoffs E [St Gt W1 | X0 = x] relative to E [St Gt | X0 = x]. Our interest in elasticities leads
us to the use of ratios in these computations.
Notice that
E [Mt | W1 , X0 ]
E [Mt W1 | X0 = x]
=E
W1 X0 = x .
E [Mt | X0 = x]
E [Mt | X0 ]
6
t |W1 ,X0 ]
and the
Thus a major ingredient in the computation is the covariance between E[M
E[Mt |X0 ]
shock vector W1 , which shows how the shock elasticity measures the impact of the shock W1
on the conditional conditional expectation of Mt .
Prior to our more general discussion, consider the case in which M is lognormal,
E [log Mt | W1 , X0 ] − E [log Mt | X0 ] = φt · W1
where φt is the (state-independent) vector of “impulse responses” or moving-average coefficients of M for horizon t. Then
1
E [Mt | W1 , X0 ]
2
(5)
= exp φt · W1 − |φt | ,
E [Mt | X0 ]
2
and its covariance with W1 is:
E [Mt W1 | X0 = x]
= φt .
E [Mt | X0 = x]
Thus when M is constructed as a lognormal process and αh is state-independent, our elasticities coincide with the impulse response functions typically computed in empirical macroeconomics.1 The shock-exposure elasticities are the responses for log G and the shock-price
elasticities are the impulse response functions for − log S.
Our interest is in calculating elasticities for nonlinear models and in particular for models
with stochastic volatility in which αg and possibly αh are state-dependent. Our methods
extend directly to such models provided the underlying Markov structure that we presume
is germane.
2.3
Alternative representation
To contrast transitory and long-term implications of structural shocks for the exposure and
price dynamics, we isolate growth rate and martingale components of multiplicative functionals. Hansen and Scheinkman (2009) justify the following factorization of the multiplicative
functional:
e(X0 )
Mt = exp(ηt)M̂t
(6)
e(Xt )
1
Our dating is shifted by one period vis-à-vis an impulse response function. In macroeconomic modeling
what we denote as φt is the vector of responses of log Mt−1 to the components of the shock vector W0 . The
responses are indexed by the gap of time t − 1 between the shock date and the outcome date.
7
where M̂ is multiplicative martingale and η is the growth or decay rate. Associated with
the martingale is a change of probability measure given by
h
i
Ê [Z | X0 ] = E M̂t Z | X0
for a random variable Z that is a (Borel measureable) function of the Markov process between
dates zero and t. This change of measure preserves the Markov structure for X although it
changes the transition probabilities. To study long-horizon limits, we consider only measure
changes that preserve stochastic stability in the sense that
lim Ê [f (Xt ) | X0 = x] →
t→∞
Z
f (x)dQ̂(x)
where Q̂ is a stationary distribution under the change of measure.2
Using factorization (6),
E [Mt W1 | X0 = x]
Ê [ê(Xt )W1 | X0 = x]
=
E [Mt | X0 = x]
Ê [ê(Xt ) | X0 = x]
where ê = 1e . In the large t limit, the right-hand side converges to the conditional mean of
W1 under the altered distribution:
Ê [W1 | X0 = x] .
(7)
The dependence of ê(Xt ) on W1 governs the dependence of the shock elasticities on the
investment horizon and eventually decays as t → ∞.
2.4
Multi-period risk elasticities and a decomposition result
To build assets with differential exposures to risk over multiple investment horizons, consider a multi-period parameterization of an underlying cash flow GH (r), constructed as a
generalization of the family of payoffs from equation (4):
t−1
X
1 2
2
log Ht (r) =
− r |αh (Xs )| + rαh (Xs ) · Ws+1 .
2
s=0
The perturbed cash flow GH (r) is now more exposed to the shock vector W in the
direction αh at all times between the current period and the maturity date. We capture the
2
Notice that we did not specify the initial distribution for X0 in our use of M̂ . The convergence is
presumed to hold at least for almost all x under the Q̂ distribution.
8
sensitivity of the expected return to such a multi-period perturbation using the risk-price
elasticity ̺p (x, t)
̺p (x, t) =
1 d
log E [Gt Ht (r) | X0 = x]
t dr
r=0
−
1 d
log E [St Gt Ht (r) | X0 = x]
t dr
.
(8)
r=0
The risk-price elasticity measures the marginal increase in the expected return on a cash flow
in response to a marginal increase in exposure of the cash flow functional in the direction
αh in every period. Scaling by t annualizes the elasticity.
The risk-price elasticity again consists of two terms, reflecting the contribution of the exposure of the expected cash flow, and the contribution of the valuation of this cash flow. Both
terms have a common mathematical structure. Using a general multiplicative functional M
that substitutes either for S or SG, the derivative in (8) can be expressed as
̺ (x, t) =
1 d
log E [Mt Ht (r) | X0 = x]
t dr
=
r=0
1 E [Mt Dt | X0 = x]
t E [Mt | X0 = x]
where D is an additive functional
Dt =
t−1
X
s=0
αh (Xs ) · Ws+1 .
By interchanging summation and integration in the conditional expectation, and utilizing
the martingale decomposition from Section 2.3, we write the risk elasticity as3
t−1
t−1
1 X E [Mt ε (Xs , t − s) | X0 = x]
1 X Ê [ê (Xt ) ε (Xs , t − s) | X0 = x]
̺ (x, t) =
=
.
t s=0
E [Mt | X0 = x]
t s=0
Ê [ê (Xt ) | X0 = x]
This formula reveals how a risk elasticity is constructed by averaging across time the
contributions of the shock elasticities in different periods. The contributions of future shocks
are weighted by the term
ê (Xt )
(9)
Ê [ê (Xt ) | X0 = x]
which represents the contribution of the nonlinear dynamics of the model arising from both
the stationary component captured by ê, and by the martingale component incorporated in
the change of probability measure ˆ·. The shock elasticities are essential inputs into this computation because of the recursive construction of valuation as reflected by the multiplicative
functional M.
3
While we are being casual about this interchange, Hansen and Scheinkman (2010) provide a rigorous
analysis of such formulas.
9
The resulting elasticity of a payoff maturing in period t + τ to a shock that occurs in
period τ + 1 then is
Ê [ê (Xt+τ ) ε (Xτ , t) | X0 = x]
.
ε (x, t; τ ) =
Ê [ê (Xt+τ ) | X0 = x]
By construction, ε (x, t; 0) = ε (x, t).
The impact of ê in the weighting (9) is transient in two particular senses. First, fix the
time of the shock τ and extend the maturity of the cash flow by t → ∞. Then the limiting
elasticity generalizes result (7):
ε (x, ∞; τ ) = Ê [ε (Xτ , ∞) | X0 = x] = Ê [αh (Xτ ) · Wτ +1 | X0 = x]
The impact of proximate shocks on cash flows far in the future remains state-dependent but
is only determined by the change in probability measure constructed from the contribution
of permanent shocks.
Second, fix the distance between the time of the shock and the maturity date, t, but
extend the date of the shock by τ → ∞. The resulting elasticity
ε (x, t; ∞) =
Ê [ê (Xt ) ε (X0 , t)]
Ê [ê (Xt )]
=
Ê [ê(Xt )αh (X0 ) · W1 ]
Ê [ê(X0 )]
is independent of the current state, and depends on the transient term ê only through
its dynamics between the date of the shock and the maturity of the cash flow. Transient
dynamics preceding the date of the shock become irrelevant.
2.5
Partial shock elasticities
In our application in Section 7, we explore how shock elasticities are altered when we change
the shock configuration. We are interested in measuring the approximate impact of introducing new shocks. Among other things, this will allow us to quantify the contribution of
different propagation channels of the dynamics (2)–(3) to the shock elasticity. In a dynamical
system a given shock may operate through multiple channels as is the case in the example
economy we investigate. To feature a specific channel, we introduce a new shock and study
the sensitivity of the elasticities. Because of the potential nonlinear nature of the model, we
do not calculate this sensitivity by zeroing out the existing shocks. Instead we perturb the
system by exposing it to new hypothetical shocks.
10
We motivate and compute the following object:
h
i
f
d E Mt (q)W1 | X0 = x
εem (x, t) = α
eh (x) ·
dq E [Mt (q) | X0 = x]
.
(10)
q=0
f1 is a new shock vector and q as a way parameterize equilibrium outcomes when
where W
the economic model is exposed to this random vector. The vector α
e(x) determines which
combination of α
e(x) is the target of the computation. We refer to this entity as a “partial
shock elasticity”.
Formally, we consider the following perturbed model:
ft+1 , q
Xt+1 (q) = ψe Xt (q), Wt+1 , qW
for t ≥ 0
f is independent of W and X0 . Changing the
where we assume that the shock vector W
real number q changes the stochastic dynamics for the Markov process X(q). We nest our
original construction by imposing that
e w, 0, 0).
ψ(x, w) = ψ(x,
Similarly, we let
where
ft+1 , q
Yt+1 (q) − Yt (q) = κ
e Xt (q), Wt+1 , qW
for t ≥ 0,
κ(x, w) = κ
e(x, w, 0, 0).
We consider the multiplicative functional M(q) = exp[Y (q)], which depends implicitly on q.
The functions ψe and e
κ are assumed to be smooth in what follows in order that we may
compute derivatives needed to characterize sensitivity.
f to characterize a specific transmission
We measure the sensitivity to the new shock W
mechanism within the model. As in our construction of shock elasticities, we specify a
e 1 (r) analogous to (4):
parameterized perturbation H
We restrict α
eh so that
2
e 1 (r) = re
f1 − r |e
log H
αh (X0 ) · W
αh (X0 )|2 .
2
E|e
αh (Xt )|2 = 1
11
f1 is independent of X0 and W ,
analogous to our previous elasticity computation. Since W
f1 is degenerate:
the shock elasticity for W
lim α
eh (x) ·
q→0
h
i
f1 | X0 = x
E Mt (q)W
E [Mt (q) | X0 = x]
=α
eh (x) ·
h
i
f1 | X0 = x
E Mt W
E [Mt | X0 = x]
= 0.
where M is M(q) evaluated at q = 0. In what follows we compute a partial elasticity by
differentiating with respect to q:
εem (x, t) =
d
α
eh (x) ·
dq
h
i
f
E Mt (q)W1 | X0 = x
E [Mt (q) | X0 = x]
.
q=0
We use this derivative to quantify the impact of the shock elasticity when we introduce a
f1 , we will
new shock into the dynamical system. When there are multiple components to W
be able to conduct relative comparisons of their importance by evaluating the derivative
vector:
h
i
f
E
M
(q)
W
|
X
=
x
t
1
0
d
.
dq E [Mt (q) | X0 = x]
q=0
2.5.1
Construction
Let X1,· and Y1,· denote the “first derivative processes” obtained by differentiating the functions ψe and κ
e and evaluated at q = 0. These processes are represented using the recursion
ft+1 + ψeq (Xt , Wt+1 , 0, 0)
X1,t+1 = ψex (Xt , Wt+1 , 0, 0)X1,t + ψewe (Xt , Wt+1 , 0, 0)W
ft+1 + κ
Y1,t+1 − Y1,t = κ
ex (Xt , Wt+1 , 0)X1,t + κ
ewe (Xt , Wt+1 , 0, 0)W
eq (Xt , Wt+1 , 0, 0) (11)
To implement these recursions, we include X1,t as an additional state vector but we have
initialized it to be zero at date zero. The process X used in this recursion is the one associated
with the original (q = 0) dynamics.
By imitating our previous analysis, we compute:
εem (x, t) = α
eh (x) ·
h
i
f
E Mt Y1,t W1 | X0 = x
E [Mt | X0 = x]
i
h
E MW
f
|
X
=
x
t 1
0
E [Mt Y1,t | X0 = x]
−α
eh (x) ·
E [Mt | X0 = x]
E [Mt | X0 = x]
12
f1 is independent of X0 and W , the second term
where M is evaluated at q = 0. Since W
on the right-hand side is zero but the first term is not. Thus formula (10) for the partial
elasticity is valid.
f1 is independent of X and W and
We compute this expectation in two steps. Since W
h
i
′
f
e
f
future Wt ’s, in the first step we compute expectations X1,t = E X1,t (W1 ) | Ft and Ye1,t =
h
i
f1 )′ | Ft recursively using
E Y1,t (W
e1,t+1 = ψex (Xt , Wt+1 , 0, 0)X
e1,t
X
e1,t
Ye1,t+1 − Ye1,t = κ
ex (Xt , Wt+1 , 0, 0)X
for t ≥ 1 and with initial conditions:
h
i
′
e
e
f
f
X1,1 = ψwe (x, W1 , 0, 0)E W1 (W1 ) | F1 = ψewe (x, W1 , 0, 0)
h
i
f1 (W
f1 )′ | F1 = e
Ye1,1 = κ
ewe (x, W1 , 0, 0)E W
κwe (x, W1 , 0, 0).
(12)
For the recursions in (11), notice that
ψex (Xt , Wt+1 , 0, 0) = ψx (Xt , Wt+1 )
κx (Xt , Wt+1 , 0, 0) = κx (Xt , Wt+1 ).
e
With this construction, we may view Ye1,t as the approximate vector of “impulse responses”
f1 . For a nonlinear model, the date t response
of Yt to unit “impulses” of the components of W
will be a random variable. In the second step we use Ye1,t to represent the partial elasticity:
2.5.2
εem (x, t) = α
eh (x) ·
′
e
E Mt Y1,t | X0 = x
E [Mt | X0 = x]
.
An interesting special case
The following special case will be of interest in our application. Suppose that we construct
the perturbed model so that
and similarly,
ψewe (x, w, 0, 0)Υ = ψw (x, w),
(13)
κwe (x, w, 0, 0)Υ = κw (x, w)
e
(14)
13
ft+1 and the same
for some matrix Υ with the same number of rows as in the shock vector W
number of columns as in the vector Wt+1 . In this construction, Υ has at least as many rows
as columns and Υ′ Υ = I.
Given a random vector αh (x) used to model state dependence in the exposure to Wt+1 ,
form:
α
eh (x) = Υαh (x)
In light of equalities (13) and (14), and our initialization in (12),
εem (x, t) = α
eh (x) ·
E Mt Ye1,t
′
| X0 = x
E [Mt | X0 = x]
≈ αh (x) ·
E [Mt W1 | X0 = x]
,
E [Mt | X0 = x]
(15)
where the right-hand side is a shock elasticity and the left-hand side is a partial shock
elasticity. The approximation becomes arbitrarily good in a continuous-time limit. See
Borovička et al. (2011) for a continuous-time characterization of the right-hand side of this
equation. In Appendix B.3, we analyze the discrete-time approximation (15) in more detail
and provide an alternative way to characterize this approximation.
f has twice as many entries as W . We construct the
In our application in Section 7, W
f in order to explore implications of alternative transmission mechamodel perturbed by W
nisms when individual shocks have multiple impacts on the dynamic economic system. When
a component of Wt+1 influences the economic system through two channels, we design the
ft+1 are independent inputs into
perturbed system in which two distinct components of W
each of the channels. In this manner the partial elasticities in conjunction with formula (15)
allow us to unbundle the impacts of the original set of shocks.
3
Entropy decomposition
Our shock-price elasticities target particular shocks. It is also of interest to have measures
of the overall magnitude across shocks. In the construction that follows we build on ideas
from Bansal and Lehmann (1997), Alvarez and Jermann (2005), and especially Backus et al.
(2011). The relative entropy of a multiplicative functional M for horizon t is given by:
1
[log E (Mt |X0 = x) − E (log Mt |X0 = x)] ,
t
which is nonnegative as an implication of Jensen’s Inequality. When Mt is log-normal, this
notion of entropy yields one-half the conditional variance of log Mt conditioned on date zero
information, and Alvarez and Jermann (2005) propose using this measure as a “generalized
14
notion of variation.” Our primary task is to construct a decomposition that provides a
more refined quantification of how entropy depends on the investment horizon t. While our
approach in this section is similar to the construction of shock elasticities, the analysis of
entropy is global in nature and does not require localizing the risk exposure. On the other
hand, it necessarily bundles the pricing implications of alternative shocks.
For a multiplicative functional M, form:
E[Mt | W1 , X0 ]
E[Mt | X0 ]
(16)
which has conditional expectation one conditioned on X0 . By Jensen’s inequality we know
that the expected logarithm of this random variable conditioned on X0 must be less than or
equal to zero, which leads us to construct:
ζm (x, t) = log E [Mt | X0 = x] − E [log E (Mt | W1 , X0 ) | X0 = x] ≥ 0
which is a measure of “entropy” of the random variable in (16). It measures the magnitude
of new information that arrives between date zero and date one for the process M. This is
the building block for a variety of computations. We think of these measures as the entropy
counterparts to our shock elasticity measures considered previously. These measures do not
feature specific shocks but they also do not require that we localize the exposures.
Consider the case in which M is lognormal. As we showed in (5),
1
E [Mt | W1 , X0 ]
2
= exp φt · W1 − |φt | ,
E [Mt | X0 ]
2
where φt is the (state-independent) vector of “impulse responses” or moving-average coefficients of M for horizon t. Then
1
ζm (x, t) = |φt |2 .
2
which is one-half the variance of the contribution of the random vector W1 to log Mt .
Returning to our more general analysis, a straightforward calculation justifies:
h
i
lim ζm (x, t) = −E log M̂1 | X0 = x
t→∞
where M̂ is the martingale component of M in factorization (6) of the multiplicative functional.
To see why ζm (x, t) are valuable building blocks, we use the multiplicative Markov struc-
15
ture of M to obtain:
E [Mt | Fj+1 ]
=
E [Mt | Fj ]
h
h
i
i
Mt
E M
| Fj+1
|
W
,
X
j+1
j
j
h
h
i =
i ,
Mt
Mt
E M
|
F
E
|
X
j
j
Mj
j
E
Mt
Mj
and thus
log E [Mt | Fj ] − E [log E (Mt | Fj+1) | Fj ] = ζm (Xj , t − j)
for j = 0, 1, ..., t − 1. Taking expectations as of date zero,
E [log E (Mt | Fj ) | F0 ] − E [log E (Mt | Fj+1 ) | F0 ] = E [ζm (Xj , t − j) | X0 ] .
We now have the ingredients for representing entropy over longer investment horizons. Notice
that
t
Y
E [Mt | Fj ]
Mt
=
.
E [Mt | F0 ] j=1 E [Mt | Fj−1]
Taking logarithms and expectations conditioned on date zero information, the entropy over
investment horizon-t is
t
1X
1
E [ζm (Xt−j , j) | X0 ] .
[log E (Mt | X0 ) − E (log Mt | X0 )] =
t
t j=1
(17)
The left-hand side is a conditional version of the entropy measure for alternative prospective
horizons t. The right-hand side represents the horizon t entropy in terms of averages of the
building blocks ζm (x, t).
The structure of the entropy is similar to that of the risk elasticity function ̺(x, t) from
Section 2.4. Both are constructed as averages over the investment horizon of the expected
one-period contributions captured by our fundamental building blocks.
Recall the multiplicative martingale decomposition of M constructed in Section 2.3.
Hansen (2011) compares this to an additive decomposition of log M:
log Mt = ρt + log M̃t + g(X0 ) − g(Xt )
where log M̃ is an additive martingale. Backus et al. (2011) propose the average entropy
over a t period investment horizon as a measure of horizon dependence. The large t limit of
equation (17) then is
1
[log E (Mt | X0 ) − E (log Mt | X0 )] = η − ρ.
t→∞ t
lim
16
The asymptotic entropy measure is state-independent and is expressed as the difference of
two asymptotic growth rates, one arising from the multiplicative martingale decomposion
and the other from the additive martingale decompositions in logarithms.
We now suggest some applications of our entropy decomposition. First, to relate our
calculations to the work of Backus et al. (2011), let M = S. Backus et al. (2011) study the
left-hand side of (17) averaged over the initial state X0 . They view this entropy measure for
different investment horizons as an attractive alternative to the volatility of stochastic discount factors featured by Hansen and Jagannathan (1991). To relate these entropy measures
to asset pricing models and data, Backus et al. (2011) note that
1
− E [log E (St | X0 )]
t
is the average yield on a t-period discount bond where we use the stationary distribution for
X0 . Following Bansal and Lehmann (1997),
1
− E [log St ] = −E [log S1 ] ,
t
is the average one-period return on the maximal growth portfolio under the same distribution. The right-hand side of (17) extends this analysis by featuring the role of condition
information captured by the state vector X0 and the entropy-building blocks ζ(x, t). Notice
that we may write
St
| X1 | X0 = x − E [log S1 | X0 = x] .
ζs (x, t) = log E [St | X0 = x] − E log E
S1
The first two terms compare the logarithm of a t-period bond price to the conditional average
of the logarithm of a t − 1-period bond price. The third term is the conditional growth rate
of the maximal growth-rate return. By featuring S only, these calculations by design feature
the term structure of interest rates but not the term structure of exposures of stochastic
growth factors.
As an alternative application, following Rubinstein (1976), Lettau and Wachter (2007),
Hansen et al. (2008), Hansen and Scheinkman (2009), and Hansen (2011) we consider the
interaction between stochastic growth and stochastic discounting. For instance, as in Sec-
17
tion 2.4 the logarithm of the risk premium for a t-period investment in a cash flow Gt is:
1
1
1
log E [Gt | X0 = x] − log E [St Gt | X0 = x] + log E [St | X0 = x] =
t
t
t
1
= (log E [Gt | X0 = x] − E [log Gt | X0 = x])
t
1
+ (log E [St | X0 = x] − E [log St | X0 = x])
t
1
− (log E [St Gt | X0 = x] − E [log St + log Gt | X0 = x]) .
t
The formula relates the t-period risk premium on a stochastically growing cash flow on the
left-hand side to the entropy measures for three multiplicative functionals on the right-hand
side: G, S and SG.4 Our decompositions can be applied to all three components to measure
how important one-period ahead exposures are to t-period risk premia.
4
Convenient functional form
In the preceding sections, we have developed formulas for shock-price and shock-exposure
elasticities for a wide class of models driven by a state vector with Markov dynamics (2).
While the level of generality is of advantage, it is nevertheless imperative that we find
tractable implementations. Our interest lies in providing tools for valuation analysis in
structural macroeconomic models, and we feature here a special dynamic structure for which
we can obtain closed-form solutions for the shock elasticities. Moreover, we will show in
Section 5 that this dynamic structure embeds a special class of approximate solutions to
dynamic macroeconomic models constructed using perturbation methods.
Consider the following triangular state vector system:
X1,t+1 = Θ10 + Θ11 X1,t + Λ10 Wt+1
X2,t+1 = Θ20 + Θ21 X1,t + Θ22 X2,t + Θ23 (X1,t ⊗ X1,t )
+Λ20 Wt+1 + Λ21 (X1,t ⊗ Wt+1 ) + Λ22 (Wt+1 ⊗ Wt+1 ) .
(18)
Such a system allows for stochastic volatility, and we restrict the matrices Θ11 and Θ22 to
have stable eigenvalues. The additive functionals that interest us satisfy
Yt+1 − Yt = Γ0 + Γ1 X1,t + Γ2 X2,t + Γ3 (X1,t ⊗ X1,t )
+ Ψ0 Wt+1 + Ψ1 (X1,t ⊗ Wt+1 ) + Ψ2 (Wt+1 ⊗ Wt+1 ) .
4
We thank Ian Martin for suggesting this link to entropy.
18
(19)
In what follows we use a 1×k 2 vector Ψ to construct a k×k symmetric matrix sym [matk,k (Ψ)]
such that5
w ′ (sym [matk,k (Ψ)]) w = Ψ (w ⊗ w) .
This representation will be valuable in some of the computations that follow. We use additive functionals to represent stochastic growth via a technology shock process or aggregate
consumption, and to represent stochastic discounting used in representing asset values. This
setup is rich enough to accommodate stochastic volatility, which has been featured in the
asset pricing literature and to a lesser extent in the macroeconomics literature.
A virtue of parameterization (18)–(19) is that it gives quasi-analytical formulas for our
dynamic elasticities. The implied model of the stochastic discount factor has been used in
a variety of reduced-form asset pricing models. Such calculations are free of any approximation errors to the dynamic system (18)–(19) and, as a consequence, ignore the possibility
that approximation errors compound and might become more prominent as we extend the
investment or forecast horizon t. On the other hand, we will use an approximation to deduce
this dynamical system, and we have research in progress that explores the implications of
approximation errors in the computations that interest us.
We illustrate the convenience of this functional form by calculating the logarithms of
conditional expectations of multiplicative functionals of the form (19). Consider a function
that is linear/quadratic in x = (x′1 , x′2 )′ :
log f (x) = Φ0 + Φ1 x1 + Φ2 x2 + Φ3 (x1 ⊗ x1 )
Then conditional expectations are of the form:
log E
Mt+1
Mt
f (Xt+1 ) | Xt = x = log E [exp (Yt+1 − Yt ) f (Xt+1 ) | Xt = x]
= Φ∗0 + Φ∗1 x1 + Φ∗2 x2 + Φ∗3 (x1 ⊗ x1 )
= log f ∗ (x)
(20)
where the formulas for Φ∗i , i = 0, . . . , 3 are given in Appendix A. This calculation maps a
function f into another function f ∗ with the same functional form. Our multi-period calculations exploit this link. For instance, repeating these calculations compounds stochastic
growth or discounting. Moreover, we may exploit the recursive Markov construction in (20)
5
In this formula matk,k (Ψ) converts a vector into a k × k matrix and the sym operator transforms this
square matrix into a symmetric matrix by averaging the matrix and its transpose. Appendix A introduces
convenient notation for the algebra underlying the calculations in this and subsequent sections.
19
initiated with f (x) = 1 to obtain:
log E [Mt | X0 = x] = Φ∗0,t + Φ∗1,t x1 + Φ∗2,t x2 + Φ∗3,t (x1 ⊗ x1 )
for appropriate choices of Φ∗i,t .
4.1
Shock elasticities
To compute shock elasticities given in (1) under the convenient functional form, we construct:
h
i
Mt
E
M
E
|
X
W
|
X
=
x
1
1
1
0
M1
E [Mt W1 | X0 = x]
i .
h
=
Mt
E [Mt | X0 = x]
E M1 E M1 | X1 | X0 = x
Notice that the random variable:
L1,t
Mt
M1 E M
|
X
1
1
i
= h
Mt
E M1 E M
|
X
|
X
=
x
1
0
1
has conditional expectation one. Multiplying this positive random variable by W1 and taking
expectations is equivalent to changing the conditional probability distribution and evaluating
the conditional expectation of W1 under this change of measure. Then under the transformed
measure, using a complete-the-squares argument we may show that W1 remains normally
distributed with a covariance matrix:
−1
e t = Ik − 2 sym matk,k Ψ2 + Φ∗ Λ22 + Φ∗
Σ
.
2,t−1
3,t−1 (Λ10 ⊗ Λ10 )
where Ik is the identity matrix of dimension k.6 We suppose that this matrix is positive
definite. The conditional mean vector for W1 under the change of measure is:
e t [µt,0 + µt,1 x1 ] ,
Ẽ [W1 |X0 = x] = Σ
where Ẽ is the expectation under the change of measure and the coefficients µt,0 and µt,1 are
given in Appendix B.
6
This formula uses the result that (Λ10 W1 ) ⊗ (Λ10 W1 ) = (Λ10 ⊗ Λ10 ) (W1 ⊗ W1 ).
20
Thus the shock elasticity is given by:
ε (x, t) = αh (x) · E [L1,t W1 | X0 = x]
e t [µt,0 + µt,1 x1 ]
= αh (x)′ Σ
The shock elasticity function in this environment depends on the first component, x1 , of
the state vector. Recall from (18) that this component has linear dynamics. The coefficient
matrices for the evolution of the second component, x2 , nevertheless matter for the shock
elasticities even though these elasticities do not depend on this component of the state vector.
4.2
Entropy increments
The convenient functional form (18)–(19) also provides a tractable formula for the entropy
components. Observe that
ζ (x, t) = −E [log L1,t |X0 = x] .
Consistent with our previous calculations, L1,t is the likelihood ratio built from two normal
densities for the shock vector: a multivariate normal density for the altered distribution
and a multivariate standard normal density. A consequence of this construction is that the
negative of the resulting expected log-likelihood satisfies:
′ −1
1 e
−1
e
e
e
e
ζ (x, t) =
E [W1 |X0 = x]
Σt
E [W1 |X0 = x] + log |Σt | + trace Σt
−k
2
e [W1 |X0 = x] is a critical input into both the shock elasticities
Thus the mean distortion E
and the entropy increments.7
5
Perturbation methods
In many applications it is convenient to view the functional form of the type we considered in
Section 4 as an approximation to dynamic stochastic equilibrium. Consider a parameterized
family of the dynamic systems specified in (2):
Xt+1 (q) = ψ(Xt (q), qWt+1 , q)
7
(21)
In a continuous-time limit, the only term that will remain is the counterpart to the quadratic form in
the conditional mean distortion for the shock.
21
where we let q parameterize the sensitivity of the system to shocks. We will entertain a limit
in which q = 0 and first- and second-order approximations around this limit system. Specifically, following Holmes (1995) and Lombardo (2010), we form an approximating system by
deducing the dynamic evolution for the pathwise derivatives with respect to q and evaluated
at q = 0. To build a link to the parameterization in Section 4, we feature a second-order
expansion:
q2
X2,t
2
where Xm,t is the m-th order, date t component of the stochastic process. We abstract from
the dependence on initial conditions by restricting each component process to be stationary.
Xt ≈ X0,t + qX1,t +
Our approximating process will similarly be stationary.8
5.1
Approximating the state vector process
While Xt serves as a state vector in the dynamic system (21), the state vector itself depends
on the parameter q. Let Ft be the σ-algebra generated by the infinite history of shocks
{Wj : j ≤ t}. For each dynamic system, we presume that the state vector Xt is Ft measurable
and that in forecasting future values of the state vector conditioned on Ft it suffices to
condition on Xt . Although Xt depends on q, the construction of Ft does not. As we will
see, the approximating dynamic system will require a higher-dimensional state vector for a
Markov representation, but the construction of this state vector will not depend on the value
of q. We now construct the dynamics for each of the component processes. The result will
be a recursive system that has the same structure as the triangular system (18).
Define x̄ to be the solution to the equation:
x̄ = ψ(x̄, 0, 0),
which gives the fixed point for the deterministic dynamic system. We assume that this fixed
point is locally stable. That is ψx (x̄, 0, 0) is a matrix with stable eigenvalues, eigenvalues
with absolute values that are strictly less than one. Then set
X0,t = x̄
for all t. This is the zeroth-order contribution to the solution constructed to be timeinvariant.
In computing pathwise derivatives, we consider the state vector process viewed as a func8
As argued by Lombardo (2010), this approach is computationally very similar to the pruning approach
described by Kim et al. (2008) or Andreasen et al. (2010).
22
tion of the shock history. Each shock in this history is scaled by the parameter q, which
results in a parameterized family of stochastic processes. We compute derivatives with respect to this parameter where the derivatives themselves are stochastic processes. Given the
Markov representation of the family of stochastic processes, the derivative processes will also
have convenient recursive representations. In what follows we derive these representations.9
Using the Markov representation, we compute the derivative of the state vector process
with respect to q, which we evaluate at q = 0. This derivative has the recursive representation:
X1,t+1 = ψq + ψx X1,t + ψw Wt+1
where ψq , ψx and ψw are the partial derivative matrices:
. ∂ψ
ψq =
(x̄, 0, 0),
∂q
. ∂ψ
ψx = ′ (x̄, 0, 0),
∂x
. ∂ψ
ψw =
(x̄, 0, 0).
∂w ′
In particular, the term ψw Wt+1 reveals the role of the shock vector in this recursive representation. Recall that we have presumed that x̄ has been chosen so that ψx has stable
eigenvalues. Thus the first derivative evolves as a Gaussian vector autoregression. It can be
expressed as an infinite moving average of the history of shocks, which restricts the process
to be stationary. The first-order approximation to the original process is:
Xt ≈ x̄ + qX1,t .
In particular, the approximating process on the right-hand side has x̄ + q(I − ψx )−1 ψq as its
unconditional mean.
In many applications, the first-derivative process X1,· will have unconditional mean zero,
ψq = 0. This includes a large class of models solved using the familiar log approximation
techniques, widely used in macroeconomic modeling. This applies to the example economy
we consider in Section 7. In Section 6 we suggest an alternative approach motivated by
models in which economic agents have a concern for model misspecification. This approach,
when applied to economies with production, results in a ψq 6= 0.
9
Conceptually, this approach is distinct from the approach often taken in solving dynamic stochastic
general equilibrium models. The common practice is to a compute a joint expansion in q and state vector x
around zero and x̄ respectively in approximating the one-period state dynamics. This approach often results
in approximating processes that are not globally stable, which is problematic for our calculations. We avoid
this problem by computing an expansion of the stochastic process solutions in q alone, which allows us to
impose stationarity on the approximating solution. In conjunction with the more common approach, the
method of “pruning” has been suggested as an ad hoc way to induce stochastic stability, and we suspect that
it will give similar answers for many applications. See Lombardo (2010) for further discussion.
23
We compute the pathwise second derivative with respect to q recursively by differentiating
the recursion for the first derivative. As a consequence, the second derivative has the recursive
representation:
X2,t+1 = ψqq + 2 (ψxq X1,t + ψwq Wt+1 ) +
+ψx X2,t + ψxx (X1,t ⊗ X1,t ) + 2ψxw (X1,t ⊗ Wt+1 ) + ψww (Wt+1 ⊗ Wt+1 )
where matrices ψij denote the second-order derivatives of ψ evaluated at (x̄, 0, 0) and formed
using the construction of the derivative matrices described in Appendix A.2. As noted by
Schmitt-Grohé and Uribe (2004), the mixed second-order derivatives ψxq and ψwq are often
zero using second-order refinements to the familiar log approximation methods.
The second-derivative process X2,· evolves as a stable recursion that feeds back on itself
and depends on the first derivative process. We have already argued that the first derivative
process X1,t can be constructed as a linear function of the infinite history of the shocks. Since
the matrix ψx has stable eigenvalues, X2,t can be expressed as a linear-quadratic function
of this same shock history. Since there are no feedback effects from X2,t to X1,t+1 , the joint
process (X1,· , X2,· ) constructed in this manner is necessarily stationary.
With this second-order adjustment, we approximate Xt as
Xt ≈ x̄ + qX1,t +
q2
X2,t .
2
When using this approach we replace Xt with these three components, thus increasing the
number of state variables. Since X0,t is invariant to t, we essentially double the number of
state variables by using X1,t and X2,t in place of Xt .
Further, the dynamic evolution for (X1,· , X2,· ) becomes a special case of the the triangular
system (18) given in Section 4. When the shock vector Wt is a multivariate standard normal,
we can utilize results from Section 4 to produce exact formulas for conditional expectations
of exponentials of linear-quadratic functions in (X1,t , X2,t ). We exploit this construction in
the subsequent subsection. For details on the derivation of the approximating formulas see
Appendix A.
5.2
Approximating an additive functional and its multiplicative
counterpart
Consider the approximation of a parameterized family of additive functionals with increments
given by:
Yt+1 (q) − Yt (q) = κ(Xt (q), qWt+1 , q)
24
and an initial condition Y0 (q) = 0. We use the function κ in conjunction with q to parameterize implicitly a family of additive functionals. We approximate the resulting additive
functionals by
q2
Yt ≈ Y0,t + qY1,t + Y2,t
(22)
2
where each additive functional is initialized at zero and has stationary increments.
Following the steps of our approximation of X, the recursive representation of the zerothorder contribution to Y is
.
Y0,t+1 − Y0,t = κ(x̄, 0, 0) = κ̄;
the first-order contribution is
Y1,t+1 − Y1,t = κq + κx X1,t + κw Wt+1
where κx and κw are the respective first derivatives of κ evaluated at (x̄, 0, 0); and the
second-order contribution is
Y2,t+1 − Y2,t = κqq + 2 (κxq X1,t + κwq Wt+1 ) +
+κx X2,t + κxx (X1,t ⊗ X1,t ) + 2κxw (X1,t ⊗ Wt+1 ) + κww (Wt+1 ⊗ Wt+1 )
where the κij ’s are the second derivative matrices constructed as in Appendix A.2. The
resulting component additive functionals are special cases of the additive functional given in
(19) that we introduced in Section 4.
Consider next the approximation of a multiplicative functional:
Mt = exp (Yt ) .
The corresponding components in the second-order expansion of Mt are
M0,t = exp (tκ̄)
M1,t = M0,t Y1,t
M2,t = M0,t (Y1,t )2 + M0,t Y2,t .
Since Y has stationary increments constructed from Xt and Wt+1 , errors in approximating
X and κ may accumulate when we extend the horizon t. Thus caution is required for this
and other approximations to additive functionals and their multiplicative counterparts. In
what follows we will be approximating elasticities computed as conditional expectations
of multiplicative functionals that scale the shock vector or functions of the state vector.
25
Previously, we have argued that the nonstationary martingale component of multiplicative
functionals can be absorbed conveniently into a change of measure. Thus for our purposes,
this problem of approximation of a multiplicative functional is essentially equivalent to the
problem of approximating a change in measure. Since our elasticities are measured per unit
of time, the potential accumulation of errors is at least partly offset by this scaling. In our
applications we will perform some ad hoc checks, but such approximation issues warrant
further investigation.
5.3
Approximating shock elasticities
We consider two alternative approaches to approximating shock elasticities of the form:
ε(x, t) = αh (x) ·
E [Mt W1 | X0 = x]
.
E [Mt | X0 = x]
(23)
Recall that we produced this formula by localizing the risk exposure and computing a (logarithmic) derivative.
5.3.1
Approach 1: Approximation of elasticity functions
Our first approach is a direct extension of the perturbation method just applied. We will
show how to construct a second-order approximation to the shock elasticity function of the
form
q2
ε2 (X1,0 , X2,0 , t)
2
where only the second-order component is state-dependent. First, observe that the zerothε(X0 , t) ≈ ε0 (t) + qε1 (t) +
order approximation is
ε0 (t) = 0
because the zeroth-order contribution in the numerator of (23) is
E [exp(tκ̄)W1 |X0 = x] = 0.
This result replicates the well-known fact that first-order perturbations of a smooth deterministic system do not lead to any compensation for risk exposure.
The first-order approximation is:
ε1 (t) = αh (x̄) · E [Y1,t W1 | F0 ] = αh (x̄) ·
26
" t−1
X
j=1
j−1
κx (ψx )
ψw + κw
#′
which is state-independent. This approximation shows the explicit link between the impulse
response function for a log-linear approximation and the shock elasticity function.
The second-order adjustment to the approximation is:
ε2 (X1,0 , X2,0 , t) = αh (x̄) · E (Y1,t )2 W1 + Y2,t W1 | F0 − 2E [Y1,t W1 | F0 ] E [Y1,t | F0 ] +
∂αh
(x̄) X1,0 · E [Y1,t W1 | F0 ] .
+2
∂x′
This adjustment can be expressed as a function of X1,0 and X2,0 since (X1,· , X2,· ) is Markov.
Notice that the second-order approximation can induce state dependence in the shock
elasticities. Often it is argued that higher than second-order approximations are required to
capture state dependence in risk premia. Since we have already performed a differentiation
to construct an elasticity, the second-order approximation of an elasticity implicitly include
third-order terms. Relatedly, in approximating elasticities using representation (23), we have
normalized the exposure to have a unit standard deviation and this magnitude is held fixed
even when q declines to zero. By fixing the exposure we reduce the order of differentiation
required for state dependence to be exposed.
To illustrate these calculations, consider a special case in which
Yt+1 − Yt = κ(Xt , qWt+1 , q) = β(Xt ) + qα(Xt ) · Wt+1 .
Then
ε(x, 1) = αh (x) ·
E [M1 W1 | X0 = x]
= qαh (x) · α(x).
E [M1 | X0 = x]
We may use our previous formulas or perform a direct calculation to show that
ε1 (1) = αh (x̄) · α(x̄)
′
′
′ ∂αh
′ ∂α
ε2 (X1,0 , X2,0 , 1) = 2(X1,0 )
(x̄) α(x̄) + 2(X1,0 )
(x̄) αh (x̄)
∂x′
∂x′
In comparison, suppose that we compute a risk premium for the one-period cash flow
G1 = exp [βg (X0 ) + qαg (X0 ) · W1 ]
priced using the one-period stochastic discount factor:
S1 = exp [βs (X0 ) + qαs (X0 ) · W1 ]
27
The one-period risk premium (in logarithms) is:
log E [G1 | X0 = x] − log E [S1 G1 | X0 = x] + log E [S1 | X0 = x] = (q)2 αg (x) · αs (x).
The first two terms on the left when taken together give the logarithm of the expected one
period return, and the negative of the third term is an adjustment for the risk-free rate. Since
we scaled the cash flow exposure by q, the risk premium scales in q2 and the second-order
approximation to this premium will be constant in contrast to our shock elasticities.
5.3.2
Approach 2: Exact elasticities under approximate dynamics
As an alternative approach, we exploit the fact that the second-order approximation is a
special case of the convenient functional form that we discussed in Section 4. This allows
us to compute elasticities using the quasi-analytical formulas we described in that section.
With this second approach, we calculate approximating stochastic growth and discounting
functionals and then use these to represent arbitrage-free pricing. This second approach
leads us to include some (but not all) third-order terms in q as we now illustrate.
Recall that in the example just considered, we approximated the one-period shock elasticity as
ε(x, 1) = qαh (x) · α(x).
With this second approach, we obtain
∂αh
∂α
ε(x, 1) ≈ q αh (x̄) + q ′ (x̄)X1,0 · α(x̄) + q ′ (x̄)X1,0 .
∂x
∂x
The q and q2 terms agree with the outcome of our first approach, but we now include an
additional third-order term in q. Both approaches are straightforward to implement and can
be compared.
There are applications where it is natural to make the perturbation vector αh (x) depend
on x, for example, when calculating shock elasticities in models with stochastic volatility.
However, in line with the literature on impulse response functions, αh (x) will often be
chosen to be a constant vector of zeros with a single one. In this case, both notions of the
second-order approximation of a shock elasticity function coincide.
5.4
Approximating partial shock elasticities
In Section 2.5 we defined the partial shock elasticity function as a way to explore alternative
transmission mechanisms and the impact of introducing new shocks. We may either compute
28
direct expansions or we may use the second-order expansion in q as a starting point. The
formulas in Section 2.5 are directly applicable to these, except that we must compute the
initializations:
e1,1 = ψewe (x, W1 , 0, 0)
X
Ye1,1 = κ
ewe (x, W1 , 0, 0).
We may approximate these initial conditions by constructing a joint expansion based on
scaling Wt+1 by q and qW̃ and including first-order terms in q. This allows us to exploit the
analytical tractability of the convenient functional form in Section 4.
In Appendix B.3, we show that the first-order expansion in r of the partial elasticity
function
εe(X0 , t) ≈ εe0 (t) + qe
ε1 (X1,0 , t).
corresponds to the second-order expansion of the shock elasticity function for appropriately
chosen shock configurations. The differentiation in q that we used to construct the partial
elasticity (10) implies that the partial elasticity function is nonzero already in its zerothorder:
εe0 (t) = α
e(x̄) ·
" t−1
X
j=1
j−1
κx ψex
e
ψewe + κ
ewe
#′
where the derivative matrices are evaluated at the deterministic steady state (x̄, 0, 0, 0).
f evaluated at the
Observe that εe0 (t) is linear in the partial derivatives with respect to W
deterministic steady state, which is also true for the higher-order terms in the expansion of
εe(x, t). This illustrates why partial elasticities decompose additively in shock configurations,
as we documented in the ‘interesting special case’ in Section 2.5. We utilize this additive decomposition in Section 7 to quantify the contribution of different shock propagation channels
to shock elasticities in an example economy.
5.5
Equilibrium conditions
In our discussion for pedagogical simplicity we took as a starting point the Markov representation for the law of motion (21). In economic applications, this law of motion is
expressed in terms equilibrium conditions that involve conditional expectations of state and
co-state variables. Using the perturbation methods described in Judd (1998), we may compute the necessary derivatives at the deterministic steady state without explicitly computing
the function ψ in advance. As in our calculations there is a convenient recursive structure
29
to the derivatives in which higher-order derivatives can be built easily from the lower-order
counterparts. The requisite derivatives can be constructed sequentially, order by order.
5.6
Related approaches
There also exist ad-hoc approaches which mix orders of approximation for different components of the model or state vector. The aim of these methods is to improve the precision of
the approximation along specific dimensions of interest, while retaining tractability in the
computation of the derivatives of the function ψ. Justiniano and Primiceri (2008) use a
first-order approximations but augment the solution with heteroskedastic innovations. Benigno et al. (2010) study second-order approximations for the endogenous state variables in
which exogenous state variables follow a conditionally linear Markov process. Malkhozov
and Shamloo (2011) combine a first-order perturbation with heteroskedasticity in the shocks
to the exogenous process and corrections for the variance of future shocks. These solution
methods are designed to produce nontrivial roles for stochastic volatility in the solution of
the model and in the pricing of exposure to risk. The approach of Benigno et al. (2010) or
Malkhozov and Shamloo (2011) give alternative ways to construct the functional form used
in Section 4.
6
Recursive and robust utility investors
In this section we contrast two preference specifications which share some common features
but can lead to different approaches for local approximation. The first preference specification is the recursive utility of Kreps and Porteus (1978). By design, this specification avoids
presuming that investors reduce intertemporal, compound consumption lotteries. Instead
investors may care about the intertemporal composition of risk. As an alternative, we consider an investor whose preferences are influenced by his concern for robustness, which leads
him to evaluate his utility under alternative distributions and checking for sensitivity.
While the two preference specifications may be observationally equivalent in particular
model economies, here we briefly explore conceptual differences in the construction of their
expansions. We have ongoing work that studies the comparisons in more detail. Here, we
focus on the second-order expansion to illustrate the impact of such preferences on valuation and pricing and, in particular, on the construction of the approximations of the shock
elasticity functions.
30
6.1
Recursive preferences and the robust utility interpretation
We follow Epstein and Zin (1989) and others by using a homogeneous aggregator in modeling
recursive preferences in the study of asset pricing implications. For simplicity we focus on
the special case in which investors’ preferences exhibit a unitary elasticity of intertemporal
substitution. In this case the continuation value process satisfies the forward recursion:
log Vt = [1 − exp(−δ)] log Ct +
exp(−δ)
log E (Vt+1 )1−γ |Ft .
1−γ
(24)
where Vt is the date t continuation value associated with the consumption process {Ct+j :
j = 0, 1, ...}. The parameter δ is the subjective rate of discount and γ is used for making
a risk adjustment in the continuation value. The limiting γ = 1 version gives the separable
logarithmic utility. We focus on the case in which γ > 1. As we will see, the forward-looking
nature of the continuation value process can amplify the role of beliefs and uncertainty about
the future in asset valuation.
We suppose that the equilibrium consumption process from an economic model is a multiplicative functional of the type described previously. For numerical convenience, subtract
log Ct from both sides of this equation:
exp(−δ)
log E
log Vt − log Ct =
1−γ
or
log Ut =
"
Vt+1
Ct
1−γ
#
| Ft ,
exp(−δ)
log E (exp [(1 − γ) log Ut+1 + (1 − γ)(log Ct+1 − log Ct )] | Ft )
1−γ
where log Ut = log Vt −log Ct . The stochastic discount factor process is given by the recursion:
(Vt+1 )1−γ
E (Vt+1 )1−γ | Ft
1−γ
1−γ Ct+1
(U
)
t+1
Ct
Ct
,
= exp(−δ)
1−γ
Ct+1
1−γ Ct+1
E (Ut+1 )
| Ft
Ct
St+1
= exp(−δ)
St
Ct
Ct+1
(25)
which gives the one-period intertemporal marginal rate of substitution for a recursive utility
investor. When γ = 1 the expression for the stochastic discount factor simplifies and reveals
the intertemporal marginal rate of substitution for discounted logarithmic utility. When
γ > 1, there is a potentially important contribution from the forward-looking continuation
value process reflected in Vt+1 or Ut+1 .
Allowing the parameter γ in the recursive utility specification to be large has become
31
common in the macro-asset pricing literature. For this reason we are led to consider motivations other than risk aversion for large values of this parameter. Anderson et al. (2003)
extend the literature on risk-sensitive control by Jacobson (1973), Whittle (1990) and others
and provide a “concern for robustness” interpretation of the utility recursion (24). Under
this interpretation the decision maker explores alternative specifications of the transition
dynamics as part of the decision-making process. This yields a substantially different interpretation of the utility recursion and the parameter γ. An outcome of this robustness
assessment is an exponentially-tilted worst case model (subject to penalization) in which the
term
Set+1
(Vt+1 )
=
≡
E (Vt+1 )1−γ | Ft
Set
1−γ
1−γ
Ct+1
Ct
1−γ
(Ut+1 )
1−γ
1−γ Ct+1
| Ft
E (Ut+1 )
Ct
in the stochastic discount factor ratio (25) induces an alternative specification of the transitional dynamics used to implement robustness. Notice that this term has conditional
expectation equal to one, and as a consequence it implies an alternative density for the
shock vector Wt+1 conditioned on date t information.
6.2
Expansion approaches
Since an essential ingredient for the evolution of the logarithm of the stochastic discount factor process is the continuation value process, as a precursor to approximating the stochastic
discount factor process we first approximate log U. As previously, we seek an approximation
of the form:
q2
log U2,t
2
where the terms on the right-hand side are themselves components of stationary processes.
We will construct the approximation of the continuation value as a function of a corresponding approximation of the logarithm of the consumption process log C given by equation (22).
log Ut ≈ log U0,t + q log U1,t +
For ease of comparison, we will hold fixed the second-order approximation for consumption
as we explore two different approaches. In a production economy the approximation of the
consumption process will itself change as we alter the specification of preferences.
6.2.1
Recursive utility approach
The conventional approach that is valid for the recursive utility specification dictates to
treat both the scaled continuation value process U as well as the consumption process C as
32
functions of the perturbation parameter q:
log Ut (q) =
exp(−δ)
log E (exp [(1 − γ) (log Ut+1 (q) + log Ct+1 (q) − log Ct (q))] | Ft ) .
1−γ
The zero-th order expansion implies a constant contribution
log U0,t ≡ ū =
exp (−δ)
(log C0,t+1 − log C0,t )
1 − exp (−δ)
(26)
and the higher-order terms can be represented recursively as
log U1,t = exp(−δ)E [log U1,t+1 + log C1,t+1 − log C1,t | Ft ]
log U2,t = exp(−δ)E [log U2,t+1 + log C2,t+1 − log C2,t | Ft ] +
+(1 − γ) exp(−δ)E (log U1,t+1 + log C1,t+1 − log C1,t )2 | Ft
−(1 − γ) exp(−δ) [E (log U1,t+1 + log C1,t+1 − log C1,t | Ft )]2
and can be solved forward. This approach assures that both log U and log C will conform functional forms introduced when constructing expansions of additive functionals in
Section 5.2. Observe that only the second-order term log U2,· in the expansion of the continuation value depends on the risk aversion parameter γ, and only scales the first-order terms.
We next consider an alternative approach motivated by a concern for robustness.
6.2.2
Robust utility approach
To obtain a lower-order contribution for a concern about robustness, control theorists explore
alternative ways to parameterize robustness as the exposure to uncertainty is altered.10 The
term (γ − 1)−1 ≡ θ in expression (24) can be viewed as a parameter that penalizes alternative
probability models for the continuation value in the search for a “worst-case” model, and
the associated martingale Se as the probability distortion used to represent this alternative
model.
While the risk aversion coefficient γ might be plausibly viewed as a preference parameter
that should be held constant as we change the riskiness of the stochastic environment by
changing q, the parameter θ does not have such an interpretation. Instead, we may be more
interested in the consequences of θ for the minimizing probability distortion as we change q.
For instance, Anderson et al. (2003) and Hansen and Sargent (2011) suggest using measures
10
For instance, Campi and James (1996) provide links between the risk-sensitive optimal control problem
and different stochastic and nonstochastic limiting counterparts. Anderson et al. (2011) apply and extend
their approach to dynamic economic models.
33
of statistical detection as aids to calibrating θ which leads them to look directly at the
e It turns out that if we fix θ as we change the amount of extrinsic
implied distortions S.
uncertainty by scaling the shocks by q, the probability distortion for the shock vector Wt+1
given by the solution of the robust decision problem vanishes as q → 0 even though the
covariance matrix of this shock remains fixed.11
This leads us to consider perturbations where the penalty parameter θ depends on q as
well. We thus define
1
= qθ
(27)
γ−1
for θ > 0, which results in the recursion
1
log Ut (q) = − exp(−δ)qθ log E exp − (log Ut+1 (q) + log Ct+1 (q) − log Ct (q)) | Ft .
qθ
This modification of the perturbation approach has profound consequences on the resulting functional form not only of the scaled continuation value log U but also of the consumption process. Define log Πt+1 (q) = log Ut+1 (q) + log Ct+1 (q) − log Ct (q) and write
log Ut (q) = exp (−δ) log Π0,t+1
1
− exp(−δ)qθ log E exp − (log Πt+1 (q) − log Π0,t+1 )
qθ
| Ft .
The term on the second line is zero for q = 0, and the zero-th order term of log U thus
coincides with that of the recursive utility expansion in expression (26). Higher-order terms
are different, though. The first-derivative process now satisfies
log U1,t
1
= − exp(−δ)θ log E exp − (log U1,t+1 + log C1,t+1 − log C1,t )
θ
| Ft ,
and the second-derivative process is
e [(log U2,t+1 + log C2,t+1 − log C2,t ) | Ft ]
log U2,t = exp(−δ)E
e represents an expectation operator under a change of measure induced by the
where E
11
Using a related perspective, Petersen et al. (2000) and Hansen et al. (2006) consider specifications for
which there is a constraint on a measure of relative entropy where θ now becomes a Lagrange multiplier used
to compute and implement robust decision making. Holding the relative entropy of the distortion constant
for alternative specifications of q results in multipliers that scale approximately linearly in q.
34
positive random variable
exp − θ1 (log U1,t+1 + log C1,t+1 − log C1,t )
E exp − 1θ (log U1,t+1 + log C1,t+1 − log C1,t ) | Ft
(28)
with a unit expectation. Under this change of measure, the shock Wt+1 will retain its unitary
covariance matrix but will have a nonzero, constant mean.
6.3
Stochastic discount factors
The robustness interpretation of the risk aversion parameter (27) will also have implications
for the expansion of the stochastic discount factor
log St ≈ log S0,t + q log S1,t +
q2
log S2,t .
2
Under the recursive expansion, the risk aversion parameter γ is held constant, and the
terms in the expansion of the stochastic discount factor are linear in continuation values and
changes in consumption:
log S0,t+1 − log S0,t = −δ + log C0,t − log C0,t+1
log S1,t+1 − log S1,t = log C1,t − log C1,t+1
+ (1 − γ) [log U1,t+1 + log C1,t+1 − log C1,t − exp (δ) log U1,t ]
log S2,t+1 − log S2,t = log C2,t − log C2,t+1
+ (1 − γ) [log U2,t+1 + log C2,t+1 − log C2,t − (1 − γ) exp (δ) log U2,t ]
The robust utility expansion, on the other hand, scales the penalization parameter θ by
q as well. This implies the following first-order expansion:
log S0,t+1 − log S0,t = −δ + log C0,t − log C0,t+1
1
− [log U1,t+1 + log C1,t+1 − log C1,t − exp (δ) log U1,t ]
θ
log S1,t+1 − log S1,t = log C1,t − log C1,t+1
1
− [log U2,t+1 + log C2,t+1 − log C2,t − exp (δ) log U2,t ]
2θ
The robust expansion moves the terms that represent the risk-adjusted continuation value
to a lower-order term in the expansion of the stochastic discount factor. Under the robust
utility expansion, there are first-order adjustments to the continuation value and zeroth-order
adjustments to the stochastic discount factor process. These are lower order modifications
35
than those implied by the recursive utility specification. On the other hand, the first terms
log Ck,t − log Ck,t+1 on the right-hand sides of the respective formulas are the same for both
expansions. These terms represent the contribution of substitution between periods t and
t + 1. With the exception of these terms, our first-order expansion for the robust utility
specification expansion is comparable to the second-order expansion under recursive utility.12
6.4
Shock elasticities
We compare the first-order expansions of the shock elasticities for the robust and recursive
utility specifications. Consider growth functionals with expansions of the type given in
Section 5. With a slight abuse of notation define a time horizon t twisted conditional
expectation via:
M0,t
E [M0,t | X0 ]
where M is G or SG. In the case of G, there is no twisting because G0,t is a deterministic
function of time, but the shock elasticity calculation for SG leads to a twist for the robust
utility model because of the contribution of S0,t . This twisting coincides with the compounding of the one-period change in probability measure implied by formula (28). Both elasticity
approximations have a common functional form:
ε (x, t) = ε0 (x, t) + qε1 (x, t)
where
e [W1 | X0 = x]
ε0 (X1,0 , X2,0 , t) = αh (x̄) · E
∂αh
e [W1 | X0 = x] + αh (x̄) · Cov
g [log M1,t , W1 | X0 = x]
(x̄) X1,0 · E
ε1 (X1,0 , X2,0 , t) =
∂x′
where thee· notation is interpreted differently for the two approximations. Under the recursive
e [W1 | X0 = x] = 0, and the first-order shock elasticity coincides with
utility expansion, E
e [W1 | X0 = x] is no
formulas derived in Section 5. For the robust utility specification the E
longer zero. In particular, the zeroth-order term contributes to the elasticity approximation.
12
The derivatives of the stochastic discount factor S under the robust expansion have a different stochastic
structure than that assumed in Section 5. In particular, under the expansion for robust utility, log S0,t is not
a linear function of time. The stochastic nature of log S0,t will also lead to first-order risk premia, reflecting
the covariance between log S0,t and the first-order term of the cash-flow process.
36
7
Application: Intangible risk
We use the model of Ai et al. (2010) to illustrate our methodology by analyzing shock
elasticities associated with consumption and capital dynamics in a model with two types
of capital. The two capital stocks face different risk exposures, which leads to differences
in their valuation. We decompose shock elasticities to understand the mechanism how risk
propagates in the model economy.
The model is motivated by an extensive literature that confronts challenges in measuring
capital. In this literature, one component of the capital stock, tangible capital, is measured
while another one, intangible capital, is not. In what follows we will refer to the tangible
component as physical capital. Intangible capital is introduced to account fully for firm
values. For instance, if firms accumulate large quantities of unmeasured productive intangible
capital, their market valuation will differ from valuation based on the replacement value of
the stock of physical capital. Hall (2000, 2001) uses this argument to understand the secular
movement in asset values relative to measures of capital. Similarly, McGrattan and Prescott
(2010a,b) argue that accounting properly for the accumulation of intangible capital explains
the heterogeneity in measured returns and the observed macroeconomic dynamics including
the period of the 1990’s.13
Following Hansen et al. (2005) we consider a related question by exploring risk-based
explanations for the heterogeneity in the returns to physical and intangible capital. Hansen
et al. (2005) use the return heterogeneity documented by Fama and French (1992, 1996) to
motivate studies of the risk exposure differences between returns on tangible and intangible
capital. Among other things, Fama and French (1992, 1996) show that firms with high bookto-market (B/M) ratios (value firms) have systematically higher expected returns compared
to their low B/M counterparts (growth firms).14 Ai et al. (2010) build a stylized model to
investigate formally the link between the value premium featured by Fama and French and
the differential contribution of intangible capital to what are classified as growth or value
firms. In the Ai et al. (2010) model growth firms are those with relatively large amounts
of intangible capital, are less exposed to aggregate risk, and therefore earn lower expected
returns.
13
This literature implicitly confronts the potential fragility in asset values because to the extent tangible
capital is used to explain increases in asset values, it must also account for large declines in these values.
14
For related empirical motivation see the cross-sectional heterogeneity in cash-flow risk exposures of
growth and value firms documented by Bansal et al. (2005) and Hansen et al. (2008).
37
7.1
The model
We use the aggregate version of the Ai et al. (2010) model inclusive of adjustment costs. Ai
et al. (2010) suggest a more primitive starting point meant to provide microfoundations for
the model. We use shock elasticities to characterize the valuation of measured and intangible
capital stocks. Parameters and specification of some of the functional forms can be found
in Appendix C. While a more explicit use of econometric methods to the estimation of this
model is a welcome extension, we find it useful to exposit properties of the model as given
in the Ai et al. (2010) paper.
7.1.1
Technology
The economy consists of two sectors. Final output is produced using physical capital K and
labor, and allocated to consumption C and investment into physical capital I and intangible
capital I ∗ :
Ct + It + It∗ = (Kt )ν (Zt )1−ν .
The model abstracts from endogenous labor supply and instead normalizes the labor input
to be one. The technology process Z is specified exogenously. To produce new capital,
investment I must be combined with the stock of intangible capital K ∗
Kt+1 = (1 − λ) Kt +
∗
Zt+1
Zt∗
G (It , Kt∗ ) .
The investment-specific technology process Z ∗ is also specified exogenously. In the process
of capital accumulation G (It , Kt∗ ) units of intangible capital are depleted in the production
of one unit of new physical capital. With this adjustment, intangible capital accumulates in
accordance with:
∗
Kt+1
= (1 − λ∗ ) [Kt∗ − G (It , Kt∗ )] + H (It∗ , Kt ) .
The functions G and H used to model adjustment costs are both concave.
7.1.2
Exogenous inputs
The technology processes Z and Z ∗ evolve according to:
log Zt+1 − log Zt = Γ0 + Γ1 Xt + ΨWt+1
∗
log Zt+1
− log Zt∗ = Γ∗0 + Γ∗1 Xt + Ψ∗ Wt+1
Xt+1 = Θ1 Xt + ΛWt+1 .
38
(29)
where Xt and Wt+1 are both two-dimensional. The first component of the shock vector W is
a direct shock to the growth rate of technology Z, while the second component represents a
long-run risk shock to the expected growth rates. The persistence in these expected growth
rates is modeled using a first-order, bivariate Markov process X. Correspondingly, Ψ and
Ψ∗ are two-dimensional row vectors with a zero in their second columns, and Λ is a twodimensional square matrix with zeros in its first column.
The matrix Θ1 is a diagonal matrix with common diagonal entries strictly less then one,
and Λ has identical entries in the second column. By design, the two components of X
remain the same when they have a common initialization. We include both components to
the state vector because we will consider perturbations of the original dynamics (29) where
the two components will have distinct roles. Observe that the first component of W impacts
both Z and Z ∗ . Moreover, we impose the restrictions
Ψ∗ = −
h
i
Γ1 = 1 0 ,
1−ν
Ψ,
ν
h
i
Γ∗1 = 0 − 1−ν
.
ν
Under the maintained restrictions,
Γ∗1 Xt
∗
+ Ψ Wt+1
1−ν
=−
ν
(Γ1 Xt + ΨWt+1 )
and shocks thus have offsetting impacts on the technology processes Z and Z ∗ . A positive
shock movement increases the growth rate in the neutral technology process Z but simultaneously decreases the investment-specific process Z ∗ . Ai et al. (2010) interpret Z ∗ as a
wedge that temporarily mitigates the risk exposure of newly installed capital. In summary,
there are two underlying shocks whose impacts we seek to characterize: a direct shock and
a long-run risk shock.
To understand better the shock transmission mechanisms in this model, we also consider
ft+1 that has four
a less rigid specification by introducing an independent shock vector W
components:
where
eW
ft+1
log Zt+1 (q) − log Zt (q) = Γ0 + Γ1 Xt (q) + ΨWt+1 + qΨ
∗
ft+1
log Zt+1
(q) − log Zt∗ (q) = Γ∗0 + Γ∗1 Xt (q) + Ψ∗ Wt+1 + qΨ̃∗ W
eW
ft+1
Xt+1 (q) = Θ1 Xt (q) + ΛWt+1 + qΛ
i
√ h
e= 2 Ψ 0 ,
Ψ
i
√ h
e∗ = 2 0 Ψ ,
Ψ
39
e=
Λ
√
"
#
Λ1 0
2
0 Λ1
ft+1
and Λ1 is the first row (or the second row as they are the same) of Λ. We construct W
in order to explore independent shocks that impinge directly on each technology as well as
independent shocks that shift the predictable components to these technologies. The first
f only impact the neutral technology process Z while the remaining
two components of W
two components impact the investment-specific technology process Z ∗ . We compute partial
ft+1 parameterized by q. By
elasticities by exploring small changes in the exposure to W
ft+1 satisfy:
design, the constructed impact matrices for W
e = Ψ,
ΨΥ
where
e ∗ Υ = Ψ∗ ,
Ψ
" #
1 I
.
Υ= √
2 I
e = Λ.
ΛΥ
Notice that Υ′ Υ = I. We impose these restrictions to ensure that restrictions (13) and (14)
given in Section 2.5 are satisfied.
7.1.3
Preferences
The model is closed by introducing a representative household with recursive preferences of
the Epstein and Zin (1989) type:
Vt =
[1 − exp (−δ)] (Ct )
1−ρ
1−ρ
+ exp (−δ) E (Vt+1 )1−γ | Ft 1−γ
1
1−ρ
.
(30)
This specification is more general than the recursion considered in Section 6 by allowing the
elasticity of intertemporal substitution ρ−1 to be different from one. We obtain equation (24)
by taking the limit as ρ → 1. The preference recursion (30) implies a stochastic discount
factor which is a generalization of expression (25):
St+1
=β
St
Ct+1
Ct
−ρ
Vt+1
E (Vt+1 )
1−γ
| Ft
1
1−γ
ρ−γ
.
The first-order conditions from a fictitious planner problem then lead to recursive formulas for the (shadow) prices of existing physical and intangible capital Q and Q∗ , respectively:
Qt
Q∗t
1−ν
St+1
Zt
∗
∗
HK (It , Kt ) Qt+1 + (1 − λ) Qt+1 | Ft
+E
= ν
Kt
St
∗
Zt+1
St+1
∗
∗
∗
∗
= E
GK ∗ (It , Kt ) Qt+1 + (1 − λ ) [1 − GK ∗ (It , Kt )] Qt+1 | Ft
St
Zt∗
40
This equation system can be solved forward to compute the prices of the two capital stocks.15
The resulting solution will, at least implicitly, use the multi-period stochastic discount factors
to make risk adjustments in future time periods. Dividing both equations by the right-hand
side variables gives the pricing formula for one-period returns to physical and intangible
capital. The conditional expectation of the one-period stochastic discount factor times the
one-period return is equal to one.
7.2
Dynare implementation
Following Ai et al. (2010), we solve the model using a second-order perturbation around the
deterministic steady state. We provide online the Dynare code for the model, and the toolbox
that computes shock elasticities from the solution generated by Dynare.16 The toolbox is
general and can be employed to analyze shock elasticities in conjunction with Dynare using
only minor modifications to the model files.17
We exploit Dynare to construct the equilibrium dynamics for the increments of additive
functionals that are of our interest. With the characterization of the dynamics (18)–(19),
we only need to implement the elasticity formulas developed in Section 5.
7.3
Shock price and exposure dynamics
We use elasticities and partial elasticities to obtain a more complete characterization of the
equilibrium expected return heterogeneity. We analyze the dynamics of aggregate consumption which determines the characteristics of the stochastic discount factor, and the pricing
implications for the two capital stocks.
7.3.1
Consumption price and exposure elasticities
We first consider the shock elasticities for the equilibrium consumption process. To make
comparisons to the literature on long-run consumption risk, we use consumption as the
growth functional. The resulting elasticities are reported in Figure 1.
The top left panel gives the shock-price elasticities. The flat trajectories are familiar
from our earlier analysis of consumption-based models of the type suggested by Bansal
15
Alternative formulas can be obtained by looking at the first-order conditions for investment.
See http://home.uchicago.edu/∼borovicka/software.html.
17
Dynare produces a full second-order approximation of the model solution as in Schmitt-Grohé and
Uribe (2004). This approximation is globally unstable, and does not fit the convenient triangular structure
introduced in Section 4. However, we can apply the perturbation methods from Section 5 to the second-order
solution itself. This step effectively doubles the number of state variables, generating separate vectors of
variables for the first- and second-order dynamics. This method also corresponds to the algorithms used in
Andreasen et al. (2010).
16
41
shock−price elasticity − consumption
shock−exposure elasticity − consumption
1
0.15
0.8
0.1
0.6
0.05
0.4
0
0.2
0
direct
long−run risk
5
10
15
maturity (years)
20
−0.05
25
contribution of the neutral technology channel
5
10
15
maturity (years)
20
25
contribution of the neutral technology channel
1
0.15
0.8
0.1
0.6
0.05
0.4
0
0.2
0
5
10
15
maturity (years)
20
−0.05
25
contribution of the investment−specific channel
0.15
−0.005
0.1
−0.01
0.05
−0.015
0
5
10
15
maturity (years)
20
10
15
maturity (years)
20
25
contribution of the investment−specific channel
0
−0.02
5
−0.05
25
5
10
15
maturity (years)
20
25
Figure 1: Shock elasticities for consumption. The left panels give the shock-price elasticities
and the right panels give the shock-exposure elasticities. The top row shows elasticities for
alternative investment horizons in the original model. The second and third rows show the
corresponding elasticities using the perturbed specification. The second row features the
transmission mechanism for neutral technology shocks, and the third row for investmentspecific shocks. To capture the state dependence in the elasticities, we report three quartiles.
and Yaron (2004). See Hansen (2011) and Borovička et al. (2011). As is shown in these
two papers, with large specifications of the risk aversion coefficient γ, a forward-looking
martingale component associated with the continuation value process dominates the pricing
implications. Expected future growth in consumption is an important contributor to this
42
martingale component. The magnitudes of the shock-price elasticities reported in Figure 1
are about double of those reported in our earlier work.
There is a substantive difference in the structure of the Bansal and Yaron (2004) and
the Ai et al. (2010) models. Bansal and Yaron (2004) specify directly predictability in the
growth rates in consumption whereas Ai et al. (2010) specify the predictability in technology
processes that are inputs into production. The two models in fact produce very different
implied predictability for consumption, reflected in the shock-exposure elasticities. For instance, the limiting shock-exposure elasticity for the shock to the growth rates in technology
reported in the top right panel of Figure 1 is about double that implied by the Bansal and
Yaron (2004) model. Given the forward-looking role for continuation values in pricing, the
approximate doubling of the long-run responses also doubles the entire trajectory of the
shock-price elasticity function.
The direct empirical evidence for the long-run predictability in consumption is weak,
however. For instance, see Hansen et al. (2008). This has led one of us to view long-run
risk models as models of sentiments (Hansen (2011)) and to explore related models in which
investors have skepticism about their model as in Hansen (2007) and Hansen and Sargent
(2010). Given the even more prominent role of this forward-looking channel in the Ai et al.
(2010) model, it would be valuable either to reconsider the evidence for predictability in
growth using other macroeconomic time series or to reduce the degree of the confidence that
investors have in the long-run risk model.18
Since the long-run risk shocks have a common impact on both technology processes, we
use partial elasticities to explore the two channels of influence: i) neutral technology channel
and ii) investment-specific channel. As is evident from comparing the panels in rows two and
three, the neutral technology channel is much more important for equilibrium consumption
as reflected by the larger exposure elasticities. This same channel dominates pricing again
with a flat trajectory. The investment-specific channel has only a small and transitory
impact on equilibrium consumption dynamics, reflected in elasticities that start small and
decay quickly to zero. The partial shock-price elasticities for the investment specific channel
are also very small, although they do not decay to zero due to the forward-looking channel
of the recursive preference specification.
Another difference between the model used Bansal and Yaron (2004) and that used by
Ai et al. (2010) is that Bansal and Yaron introduce stochastic volatility in consumption as
an exogenously specified process. There is no counterpart process in the Ai et al. (2010)
model, although stochastic volatility could be generated endogenously by the nonlinearity in
18
Hansen et al. (2008) feature corporate earnings but do not report findings for other macroeconomic
aggregates.
43
the equilibrium evolution. Stochastic volatility would be manifested in the state dependence
of the shock elasticities. Figure 1 shows that this endogenous source is only noticeable for
the partial elasticities associated with the investment channel and these elasticities are small
in magnitude.
7.3.2
Elasticities for capital and the associated prices
The Ai et al. (2010) model features differences in valuation of physical and intangible capital.
To understand what underlies the differences, we report exposure elasticities for quantities
and prices of capital. Figure 2 shows the differential exposures of the two capital stocks, K
and K ∗ , to the underlying shocks, and Figure 3 complements the analysis by depicting the
exposures of the corresponding prices of capital, Q and Q∗ . The prices are of direct interest,
but they are also important components to returns to holding capital over time.
The responses of physical capital (top left panel in Figure 2) start small and build up over
time, as is typically the case in business cycle models. The long-run responses of intangible
capital (top right panel) necessarily coincide with the positive responses for physical capital
but the short-run responses are very different for both shocks. The exposure of intangible
capital to the direct shock to the technology processes is initially strongly negative (beginning
after a one-period delay), while the exposure elasticity for the long-run risk shock provides
a mirror image of the direct shock elasticity in the short run. For the physical capital the
short-run exposure elasticities are slightly negative for both shocks but then both eventually
become positive and more pronounced.
The partial elasticities in the second and third row of Figure 2 show that the neutral
technology shock channel dominates the long-term responses for both capital stocks as might
be expected. The investment-specific channel is important for intangible capital for the
shorter investment horizons but not for the physical capital stock. In fact, the investmentspecific channel inhibits the accumulation of physical capital after a positive shock because
new vintages of physical capital are temporarily less productive.
Consider next the exposure elasticities for the prices of the two types of capital reported
in Figure 3. Overall these exposure elasticities are much smaller than the corresponding
quantity elasticities and are only transitory because prices of capital in this model are stationary. The important differences are in the elasticities to the long-run risk shock. They are
initially negative for the price of intangible capital but substantially positive for the physical
capital stock. Recall that intangible capital is expected to increase in response to such a
shock in contrast to the physical capital stock, but the physical capital stock becomes more
valuable. From the partial elasticity plots it is evident that the important differences are
accounted for by the investment-specific channel.
44
shock−exposure elasticity − physical capital
shock−exposure elasticity − intangible capital
0.15
0.15
0.1
0.1
0.05
0.05
0
0
direct
long−run risk
−0.05
5
10
15
maturity (years)
20
−0.05
25
contribution of the neutral technology channel
0.15
0.1
0.1
0.05
0.05
0
0
5
10
15
maturity (years)
20
−0.05
25
contribution of the investment specific channel
0.15
0.1
0.1
0.05
0.05
0
0
5
10
15
maturity (years)
20
20
25
5
10
15
maturity (years)
20
25
contribution of the investment specific channel
0.15
−0.05
10
15
maturity (years)
contribution of the neutral technology channel
0.15
−0.05
5
−0.05
25
5
10
15
maturity (years)
20
25
Figure 2: Shock-exposure elasticities for physical and intangible capital. The left panels give
the elasticities for physical capital and the right panels give the elasticities for intangible
capital. The top row shows elasticities for alternative investment horizons in the original
model. The second and third rows show the corresponding partial elasticities using the
perturbed specification. The second row features the transmission mechanism for neutral
technology shocks, and the third row for investment-specific shocks. To capture the state
dependence in the elasticities, we report three quartiles.
Overall, the partial elasticities illuminate the interaction between the quantity and price
dynamics for the two types of capital. While the neutral technology shock channel dominates
the long-term quantity responses for both capital stocks, the investment-specific channel
45
shock−exposure elasticity − price of physical capital
shock−exposure elasticity − price of intangible capital
0.02
0.02
0.015
0.015
0.01
0.01
0.005
0.005
0
0
direct
long−run risk
−0.005
−0.01
5
10
15
maturity (years)
20
−0.005
−0.01
25
contribution of the neutral technology channel
0.02
0.015
0.015
0.01
0.01
0.005
0.005
0
0
−0.005
−0.005
5
10
15
maturity (years)
20
−0.01
25
contribution of the investment−specific channel
0.02
0.015
0.015
0.01
0.01
0.005
0.005
0
0
−0.005
−0.005
5
10
15
maturity (years)
20
20
25
5
10
15
maturity (years)
20
25
contribution of the investment−specific channel
0.02
−0.01
10
15
maturity (years)
contribution of the neutral technology channel
0.02
−0.01
5
−0.01
25
5
10
15
maturity (years)
20
25
Figure 3: Shock-exposure elasticities for the prices of physical and intangible capital. The
left panels give the elasticities for the price of physical capital Q and the right panels give
the elasticities for the price of intangible capital Q∗ . The top row shows elasticities for
alternative investment horizons in the original model. The second and third rows show the
corresponding partial elasticities using the perturbed specification. The second row features
the transmission mechanism for neutral technology shocks, and the third row for investmentspecific shocks. To capture the state dependence in the elasticities, we report three quartiles.
plays a crucial role in the short-run dynamics after a long-run risk shock. This latter channel
drives both the quantity response of intangible capital, and the price response of physical
capital.
46
cumulative excess return on physical capital
cumulative excess return on intangible capital
0.02
0.02
0.015
0.015
0.01
0.01
0.005
0.005
0
0
direct
long−run risk
−0.005
−0.01
5
10
15
maturity (years)
20
−0.005
−0.01
25
contribution of the neutral technology channel
0.02
0.015
0.015
0.01
0.01
0.005
0.005
0
0
−0.005
−0.005
5
10
15
maturity (years)
20
−0.01
25
contribution of the investment−specific channel
0.02
0.015
0.015
0.01
0.01
0.005
0.005
0
0
−0.005
−0.005
5
10
15
maturity (years)
20
20
25
5
10
15
maturity (years)
20
25
contribution of the investment−specific channel
0.02
−0.01
10
15
maturity (years)
contribution of the neutral technology channel
0.02
−0.01
5
−0.01
25
5
10
15
maturity (years)
20
25
Figure 4: Shock exposure elasticities for cumulative excess returns on physical and intangible
capital in the Ai et al. (2010) model. The left column gives the elasticities and partial
elasticities for physical capital, the right column for intangible capital. The top row shows
elasticities for alternative investment horizons in the original model. The second and third
rows show the corresponding partial elasticities using the perturbed specification. The second
row features the transmission mechanism for neutral technology shocks, and the third row
for investment-specific shocks. To capture the state dependence in the elasticities, we report
three quartiles.
47
7.3.3
Exposure elasticities for cumulative returns
The Ai et al. (2010) model generates a large expected return on physical capital, much larger
than for intangible capital. To enhance our understanding of the differences in the risk premia associated with the two capital investments, we study the shock-exposure elasticities of
their associated excess returns. An n-period return is a cash flow delivered in n periods for
a unitary initial investment. Figure 4 plots the shock-exposure elasticities of the cumulative excess returns on physical and intangible capital and their decomposition into partial
elasticities.
The elasticities of the cumulative excess returns are flat. The excess return exposures
for the physical capital are essentially the same for both shocks, but they are substantially
different for the excess returns on intangible capital. The exposure elasticity for the long-run
risk shock is slightly negative for the intangible capital excess return whereas this exposure
elasticity is much bigger in magnitude and positive for the direct shock. Recall that the
shock-price elasticities are much larger for the long-run risk shock and hence investors in the
physical capital are compensated more than investors in intangible capital. The negative
exposure elasticity of intangible capital to the long-run risk shock makes intangible capital
a good hedge against such a shock and this is reflected in equilibrium expected returns.
The partial elasticities are particularly revealing for the excess return to the physical
capital asset. The primary channel for the large exposure to the direct shock is through
the impact of the neutral technology process, while the primary channel for the long-run
risk shock is through the impact of the investment-specific technology. Consider the partial
elasticities for the long-run risk shock. The impact on the expected returns via the neutral
technology process Z is very small. This same impact via the investment-specific technology
Z ∗ is large for the physical capital stock but small and actually negative for the intangible
capital stock for the reasons given in our discussion of exposure elasticities for the quantities
and prices of capital. This investment-specific channel is the critical one for generating large
expected returns for physical capital vis-à-vis intangible capital.
In summary, distinguishing price from exposure elasticities and exploring separately channels with two technological inputs reveal key features underlying the differences in risk premia
between physical and intangible capital investments. As in the earlier literature, shocks to
long-run risk are central to understanding these differences. The partial elasticities for the
shock prices are large for the neutral technology process. Exposure to the shock to long-run
risk in this technology requires compensation . At the same time, excess returns to physical
capital have large exposure elasticities to the long-run risk shocks to the investment-specific
technology process. The large premium for returns to physical capital are generated by the
high (in fact perfect) correlation between the two long-run risk shocks.
48
8
Conclusion and directions for further research
In this paper, we build on our previous work in Hansen and Scheinkman (2010), Borovička
et al. (2011), and Hansen (2011) by developing tractable ways to measure the sensitivity
of expected cash-flows with macroeconomic components and the associated expected returns to structural shocks. These shock elasticities measure prices and quantities of risk in
macro-asset pricing models. They constitute fundamental building blocks for dynamic value
decompositions within stochastic equilibrium models. We show that the same approach
can be used to deconstruct dynamic entropy measures analyzed in Alvarez and Jermann
(2005) and Backus et al. (2011) by taking account of the role of conditioning information
for alternative investment horizons.
This paper focuses on tractable implementability in contrast to Hansen and Scheinkman
(2010), who provide a more rigorous basis for some of our calculations by taking continuoustime limits. We show that a second-order perturbation approach to model solution along the
lines of Holmes (1995) and Lombardo (2010) results in tractable closed-form formulas for the
shock elasticities. To support the use of our methodology, we provide a set of Matlab codes19
that can be integrated with Dynare/Dynare++ and generate the shock elasticities for secondorder solutions to dynamic macroeconomic models. It remains to provide more rigor to some
of these approximations and to explore other more global approaches to approximation.
This paper also sketches an approach for constructing low-order expansions applicable to
economies in which either private agents or policy makers have a concern for robustness. Our
emphasis is to show how robustness can have consequences for even first-order approximations to continuation values and for initial terms in expansions for stochastic discount factors
and the resulting elasticities. We suspect this same approach will also provide additional
insights into the study and design of robust macroeconomic policy rules.
In this paper we used shock elasticities as interpretive diagnostics for comparing the asset
valuation implications of alternative macroeconomic models and for understanding better the
channels by which exogenous shocks influence equilibrium outcomes. We have not described
formally shock identification and statistical uncertainty in our measurements, but we should
be able to build on the related macroeconomic literature on identification and inference for
impulse response functions. Also methods like the ones we describe here should provide
useful complements for the recent empirical work by Binsbergen et al. (2011) and others
on the decomposition of cash flow contributions to equity returns for alternative investment
horizons.
19
See http://home.uchicago.edu/∼borovicka/software.html.
49
Appendix
A
Conditional expectations of multiplicative functionals
Let X = (X1′ , X2′ )′ be a 2n × 1 vector of states, W ∼ N (0, I) a k × 1 vector of independent Gaussian
shocks, and Ft the filtration generated by (X0 , W1 , . . . , Wt ). In this appendix, we show that given
the law of motion from equation (18)
X1,t+1 = Θ10 + Θ11 X1,t + Λ10 Wt+1
(31)
X2,t+1 = Θ20 + Θ21 X1,t + Θ22 X2,t + Θ23 (X1,t ⊗ X1,t ) +
+Λ20 Wt+1 + Λ21 (X1,t ⊗ Wt+1 ) + Λ22 (Wt+1 ⊗ Wt+1 )
and a multiplicative functional Mt = exp (Yt ) whose additive increment is given in equation (19):
Yt+1 − Yt = Γ0 + Γ1 X1,t + Γ2 X2,t + Γ3 (X1,t ⊗ X1,t ) +
(32)
+Ψ0 Wt+1 + Ψ1 (X1,t ⊗ W1,t+1 ) + Ψ2 (Wt+1 ⊗ Wt+1 )
we can write the conditional expectation of M as
log E [Mt | F0 ] = Γ̄0
where Γ̄i
t
t
+ Γ̄1 t X1,0 + Γ̄2 t X2,0 + Γ̄3 t (X0 ⊗ X0 )
(33)
are constant coefficients to be determined.
The dynamics given by (31)–(32) embeds the perturbation approximation constructed in Section 5 as a special case. The Θ and Λ matrices needed to map the perturbed model into the
above structure are constructed from the first and second derivatives of the function ψ(x, w, q) that
captures the law of motion of the model, evaluated at (x̄, 0, 0):
Θ10 = ψq
Θ20 = ψqq
Λ20 = 2ψwq
Θ11 = ψx
Λ10 = ψw
Θ21 = 2ψxq
Λ21 = 2ψxw
Θ22 = ψx
Θ23 = ψxx
Λ22 = ψww
where the notation for the derivatives is defined in Appendix A.2.
A.1
Definitions
To simplify work with Kronecker products, we define two operators vec and matm,n . For an m × n
matrix H, vec (H) produces a column vector of length mn created by stacking the columns of H:
h(j−1)m+i = [vec(H)](j−1)m+i = Hij .
50
For a vector (column or row) h of length mn, matm,n (h) produces an m × n matrix H created by
‘columnizing’ the vector:
Hij = [matm,n (h)]ij = h(j−1)m+i .
We drop the m, n subindex if the dimensions of the resulting matrix are obvious from the context.
For a square matrix A, define the sym operator as
sym (A) =
1
A + A′ .
2
Apart from the standard operations with Kronecker products, notice that the following is true. For
a row vector H1×nk and column vectors Xn×1 and Wn×1
H (X ⊗ W ) = X ′ [matk,n (H)]′ W
and for a matrix An×k , we have
′
X ′ AW = vecA′ (X ⊗ W ) .
(34)
Also, for An×n , Xn×1 , Kk×1 , we have
(AX) ⊗ K = (A ⊗ K) X
K ⊗ (AX) = (K ⊗ A) X
Finally, for column vectors Xn×1 and Wk×1 ,
(AX) ⊗ (BW ) = (A ⊗ B) (X ⊗ W )
and
(BW ) ⊗ (AX) = [B ⊗ A•j ]nj=1 (X ⊗ W )
where
[B ⊗ A•j ]nj=1 = [B ⊗ A•1 B ⊗ A•2 . . . B ⊗ A•n ] .
A.2
Concise notation for derivatives
Consider a vector function f (x, w) where x and w are column vectors of length m and n, respectively. The first-derivative matrix fi where i = x, w is constructed as follows. The k-th row [fi ]k•
corresponds to the derivative of the k-th component of f
[fi (x, w)]k• =
∂f (k)
(x, w) .
∂i′
Similarly, the second-derivative matrix is the matrix of vectorized and stacked Hessians of
51
individual components with k-th row
[fij (x, w)]k• =
∂ 2 f (k)
vec
(x, w)
∂j∂i′
!′
.
It follows from formula (34) that, for example,
x
A.3
′
!
∂ 2 f (k)
(x, w) w =
∂x∂w′
!′
∂ 2 f (k)
vec
(x, w) (x ⊗ w) = [fxw (x, w)]k• (x ⊗ w) .
∂w∂x′
Conditional expectations
Notice that a complete-the squares argument implies that, for a 1 × k vector A, a 1 × k2 vector B,
and a scalar function f (w),
E [exp (B (Wt+1 ⊗ Wt+1 ) + AWt+1 ) f (Wt+1 ) | Ft ] =
(35)
1 ′
W (matk,k (2B)) Wt+1 + AWt+1 f (Wt+1 ) | Ft
= E exp
2 t+1
1
−1/2
−1 ′
= |Ik − sym [matk,k (2B)]|
exp
A (Ik − sym [matk,k (2B)]) A Ẽ [f (Wt+1 ) | Ft ]
2
where ˜· is a measure under which
Wt+1 ∼ N (Ik − sym [matk,k (2B)])−1 A′ , (Ik − sym [matk,k (2B)])−1 .
We start by utilizing formula (35) to compute
Ȳ (Xt ) = log E [exp (Yt+1 − Yt ) | Ft ] =
= Γ0 + Γ1 X1,t + Γ2 X2,t + Γ3 (X1,t ⊗ X1,t ) +
1 ′
′
′
+ log E exp Ψ0 + X1t [matk,n (Ψ1 )] Wt+1 + Wt+1 [matk,k (Ψ2 )] Wt+1 | Ft
2
= Γ0 + Γ1 X1,t + Γ2 X2,t + Γ3 (X1,t ⊗ X1,t ) −
1
1
− log |Ik − sym [matk,k (2Ψ2 )]| + µ′ (Ik − sym [matk,k (2Ψ2 )])−1 µ
2
2
with µ defined as
µ = Ψ′0 + [matk,n (Ψ1 )] X1,t .
Reorganizing terms, we obtain
Ȳ (Xt ) = Γ̄0 + Γ̄1 X1,t + Γ̄2 X2,t + Γ̄3 (X1,t ⊗ X1,t )
52
where
1
1
log |Ik − sym [matk,k (2Ψ2 )]| + Ψ0 (Ik − sym [matk,k (2Ψ2 )])−1 Ψ′0
2
2
−1
= Γ1 + Ψ0 (Ik − sym [matk,k (2Ψ2 )]) [matk,n (Ψ1 )]
Γ̄0 = Γ0 −
Γ̄1
(36)
Γ̄2 = Γ2
Γ̄3
h
i′
1
′
−1
= Γ3 + vec [matk,n (Ψ1 )] (Ik − sym [matk,k (2Ψ2 )]) [matk,n (Ψ1 )]
2
For the set of parameters P = (Γ0 , . . . , Γ3 , Ψ0 , . . . , Ψ2 ), equations (36) define a mapping
P̄ = E¯ (P) ,
with all Ψ̄j = 0. We now substitute the law of motion for X1 and X2 to produce Ȳ (Xt ) =
Ỹ (Xt−1 , Wt ). It is just a matter of algebraic operations to determine that
Ỹ (Xt−1 , Wt ) = log E [exp (Yt+1 − Yt ) | Ft ] =
= Γ̃0 + Γ̃1 X1,t−1 + Γ̃2 X2,t−1 + Γ̃3 (X1,t−1 ⊗ X1,t−1 )
+Ψ̃0 Wt + Ψ̃1 (X1,t−1 ⊗ Wt ) + Ψ̃2 (Wt ⊗ Wt )
where
Γ̃0 = Γ̄0 + Γ̄1 Θ10 + Γ̄2 Θ20 + Γ̄3 (Θ10 ⊗ Θ10 )
(37)
Γ̃1 = Γ̄1 Θ11 + Γ̄2 Θ21 + Γ̄3 (Θ10 ⊗ Θ11 + Θ11 ⊗ Θ10 )
Γ̃2 = Γ̄2 Θ22
Γ̃3 = Γ̄2 Θ23 + Γ̄3 (Θ11 ⊗ Θ11 )
Ψ̃0 = Γ̄1 Λ10 + Γ̄2 Λ20 + Γ̄3 (Θ10 ⊗ Λ10 + Λ10 ⊗ Θ10 )
in
h
Ψ̃1 = Γ̄2 Λ21 + Γ̄3 Θ11 ⊗ Λ10 + Λ10 ⊗ (Θ11 )•j
j=1
Ψ̃2 = Γ̄2 Λ22 + Γ̄3 (Λ10 ⊗ Λ10 )
This set of equations defines the mapping
P̃ = Ẽ P̄ .
A.4
Iterative formulas
We can write the conditional expectation in (33) recursively as
Mt
log E [Mt | F0 ] = log E exp (Y1 − Y0 ) E
| F1 | F0 .
M1
53
Given the mappings Ē and Ẽ, we can therefore express the coefficients P̄ in (33) using the
recursion
P̄t = E¯ P + Ẽ P̄t−1
where the addition is by coefficients and all coefficients in P̄0 are zero matrices.
B
Shock elasticity calculations
In this appendix, we provide details on some of the calculations underlying the derived shock
elasticity formulas.
B.1
Shock elasticities under the convenient functional form
To calculate the shock elasticities in Section 4.1, utilize the formulas derived in Appendix A to
deduce the one-period change of measure
log L1,t = log M1 + log E
Mt
| X1
M1
− log E M1 E
Mt
| X1
M1
| X0 = x .
In particular, following the set of formulas (37), define
′
Ψ1 + Φ∗1,t−1 Λ1,0 + Φ∗2,t−1 Λ20 + Φ∗3,t−1 (Θ10 ⊗ Λ10 + Λ10 ⊗ Θ10 )
in
h
∗
∗
= matk,n Ψ1 + Φ2,t−1 Λ21 + Φ3,t−1 Θ11 ⊗ Λ10 + Λ10 ⊗ (Θ11 )•j
j=1
= sym matk,k Ψ2 + Γ̄2 Λ22 + Γ̄3 (Λ10 ⊗ Λ10 )
µ0,t =
µ1,t
µ2,t
Then it follows that
log L1,t = (µ0,t + µ1,t X1,0 )′ W1 + (W1 )′ µ2,t W1 −
1
− log E exp (µ0,t + µ1,t X1,0 )′ W1 + (W1 )′ µ2,t W1 | F0
2
Expression (35) then implies that
e [W1 | F0 ] =
E [L1,t W1 | F0 ] = E
= (Ik − 2µ2,t )−1 (µ0,t + µ1t X1,0 )
The variance of W1 under the ˜· measure satisfies
e t = Ik − 2sym matk,k Ψ2 + Γ̄2 Λ22 + Γ̄3 (Λ10 ⊗ Λ10 ) −1 .
Σ
54
B.2
Approximation of the shock elasticity function
In Section 5, we constructed the approximation of the shock elasticity function ε (x, t). The firstorder approximation is constructed by differentiating the elasticity function under the perturbed
dynamics
ε1 (X1,0 , t) =
E [Mt (q) W1 | X0 = x]
d
α(X0 (q)) ·
dq
E [Mt (q) | X0 = x]
q=0
= α (x̄) · E [Y1,t W1 | X0 = x] .
The first-derivative process Y1,t can be expressed in terms of its increments, and we obtain a
state-independent function
ε1 (t) = α (x̄) · E
t−1
X
j=1
′
κx (ψx )j−1 ψw + κw
where κx , ψx , κw , ψw are derivative matrices evaluated at the steady state (x̄, 0).
Continuing with the second derivative, we have
d2
E [Mt (q) W1 | X0 = x]
=
α(X0 (q)) ·
dq2
E [Mt (q) | X0 = x] q=0
i
o
n h
= α (x̄) · E (Y1,t )2 W1 + Y2,t W1 | F0 − 2E [Y1,t W1 | F0 ] E [Y1,t | F0 ] +
∂α
+2
(x̄) X1,0 · E [Y1,t W1 | F0 ] .
∂x′
ε2 (X1,0 , X2,0 , t) =
However, notice that
h
E (Y1,t )2 W1 | F0
i
= 2
t−1
X
j=0
′
t−1
X
κx (ψx )j X1,0
κx (ψx )j−1 ψw + κw
j=1
′
t−1
X
κx (ψx )j−1 ψw + κw
E [Y1,t W1 | F0 ] =
j=1
E [Y1,t | F0 ] =
and thus
t−1
X
κx (ψx )j X1,0
j=0
i
h
E (Y1,t )2 W1 | F0 − 2E [Y1,t W1 | F0 ] E [Y1,t | F0 ] = 0.
The second-order term in the approximation of the shock elasticity function thus simplifies to
∂α
ε2 (X1,0 , X2,0 , t) = α (x̄) · E [Y2,t W1 | F0 ] + 2
(x̄) X1,0 · E [Y1,t W1 | F0 ] .
∂x′
55
(38)
The expression for the first term on the right-hand side is
t−1
X
E [Y2,t W1 | F0 ] = E
(Y2,j+1 − Y2,j ) W1 | F0 = 2matk,n (κxw ) X1,0 +
j=0
+2
t−1 h
X
′
ψw
ψx′
j=1
+2
j−1 h
t−1 X
X
j−1
′
ψw
ψx′
j=1 k=1
ii
h
matn,n (κxx ) (ψx )j + matk,n κx (ψx )j−1 ψxw X1,0
k−1
i
i
h
matn,n κx (ψx )j−k−1 ψxx (ψx )k X1,0
To obtain this result, notice that repeated substitution for Y1,j+1 − Y1,j into the above formula
yields a variety of terms but only those containing X1,0 ⊗W1 have a nonzero conditional expectation
when interacted with W1 .
B.3
Partial shock elasticities
In Section 5.4, we constructed the first-order approximation of the partial shock elasticity function,
and argued that it is equivalent to the second-order approximation of the shock elasticity function.
f that is independent of W ,
Recall that for a shock vector W
εe(x, t) = α
e(x) ·
where
Y1,t =
t−1
X
j=0
+
f1 | X0 = x
E Mt Y1,t W
E [Mt | X0 = x]
f1 +
(Y1,j+1 − Y1,j ) = κ
ewe (X0 , W1 , 0, 0) W
t−1
X
j=1
=
h
κ
ex (Xj, Wj+1 , 0, 0)
t−1
X
f1
Ye1,j+1 − Ye1,j W
j−1
Y
k=1
i
!
f1 =
ψex (Xk , Wk+1 , 0, 0) ψewe (X0 , W1 , 0, 0) W
j=0
′
e
f
where Y1,t = E Y1,t W1 | Ft , with Ft being the σ-algebra generated by (X0 , W1 , . . . , Wt ). Once
f1 is conditioned out, we proceed with the parameterization of the sensitivity to the shock W given
W
by (21), and follow the approximations from Section 5.
We construct a first-order approximation of the partial shock elasticity function
εe (x, t) ≈ εe0 (x, t) + qe
ε1 (x, t) .
The zero-th order approximation to the partial shock elasticity function evaluates Ye1,t at the de-
56
terministic steady state
εe0 (x, t) = α
e(x̄) ·
t−1
X
j=1
h
κ
ex (x̄, 0, 0, 0) ψex (x̄, 0, 0, 0)
ij−1
ψewe (x̄, 0, 0, 0) + κ
ewe (x̄, 0, 0, 0) .
Notice that derivatives κ
ex and ψex evaluated at the deterministic steady state coincide with κx and
ψx . In line with the interesting special case from Section 2.5.2, consider the following positioning
f:
of the shock vector W
ψe (x, w, qw,
e q) ≡ ψ x, w + qΥ′ w,
e q
κ
e (x, w, qw,
e q) ≡ κ x, w + qΥ′ w,
e q .
(39)
Then the derivatives evaluated at q = 0 satisfy:
ψewe (x, w, 0, 0) ≡ ψw (x, w, 0) Υ′
κ
ewe (x, w, 0, 0) ≡ κw (x, w, 0) Υ′ ,
and post-multiplying by Υ yields expressions (13)–(14). Choosing the exposure direction vector as
α
eh = Υαh , we obtain εe0 (x, t) = ε1 (x, t). By constructing alternative configurations of the shock
vector W̃ in the functions ψe and κ
e, the partial elasticity function allows us to study a richer class
of dynamic responses.
In order to construct the first-order approximation, notice that
εe1 (X1,0 , t) =
d
α
e(X0 )
dq
= α
e(x̄) · E
"
E Mt
Ye1,t
′
| X0 = x
E [Mt | X0 = x]
d e ′
Y1,t
dq
q=0
#
| F0 +
=
q=0
∂α
e
(x̄) X1,0 · εe0 (x, t) .
∂x′
The second term on the second line corresponds to one half of the second term in expression (38).
It remains to express the derivative in the first term. Recall that
Ye1,1 (q) = κ
ewe (X0 (q) , qW1 , 0, 0)
Ye1,j+1 (q) − Ye1,j (q) = κ
ex (Xj (q) , qWj+1 , 0, 0)
j−1
Y
k=1
We then have
E
"
!
e
ψx (Xk (q) , qWk+1 , 0, 0) ψewe (X0 (q) , qW1 , 0, 0) ,
d e ′
Y1,1
dq
q=0
#
κxwe ) X1,0
| F0 = matek,n (e
57
j > 0.
and, for j > 0,
E
"
#
′
d e
Y1,j+1 (q) − Ye1,j (q)
| F0 =
dq
q=0
j−1
j−1
′
′
e
e
e
e
ex ψx
ψxwe X1,0 +
matn,n (e
κxx ) E [X1,j | F0 ] + matek,n κ
= ψwe ψx
+
j−1
X
k=1
k−1
j−k−1
′
′
e
e
e
e
matn,n κ
ψwe ψx
ex ψx
ψxx E [X1,k | F0 ] .
Collecting the terms and substituting for E [X1,k | F0 ], we obtain a result that is analogous to
the first term of 12 ε2 (X1,0 , X2,0 , t) in expression (38):
E
"
d e ′
Y1,t
dq
q=0
#
| F0 =
t−1
′
d Xe
e
=E
Y1,j+1 − Y1,j
dq
+
t−1
X
j=1
+
j=0
′
ψew
e
j−1
t−1 X
X
j=1 k=1
ψex′
′
ψew
e
j−1
q=0
| F0 = matek,n (e
κxwe ) X1,0 +
j
j−1
matn,n (e
κxx ) ψex + matek,n κ
ex ψex
ψexwe X1,0 +
k−1
j−k−1
k
′
matn,n e
ψe
κx ψex
ψexx ψex X1,0
x
e and ψe with
Once again, if we construct ψe and e
κ to satisfy (39), then all partial derivatives of κ
f correspond to those of κ and ψ with respect to W multiplied by Υ′ . When we choose
respect to W
α
eh = Υαh , we obtain
1
εe1 (X1,0 , t) = ε2 (X1,0 , t)
2
and thus the approximations coincide.
Moreover, an inspection of the above expressions for εe0 (x, t) and εe1 (x1,· , t) reveals that all terms
f . Partial elasticities will thus be additive in
are linear in a single partial derivative with respect to W
shock configurations, and we can naturally additively decompose elasticities by positioning shocks
in alternative locations in the functions ψe and κ
e.
C
Parameterization of the Ai et al. (2010) model
For sake of illustration and comparability, we use the same parameters as used by Ai et al. (2010) in
their extended model with adjustment costs, H (I ∗ , K), in the accumulation of intangible capital.
The production technology for turning intangible capital into new vintages of physical capital is
58
Preferences
Time preference
Risk aversion
Intertemporal elasticity of substitution
Technology
Capital share
Depreciation rate of physical capital
Depreciation rate of intangible capital
Weight on physical investment
Elasticity of substitution in G(I, K ∗ )
Elasticity of substitution in H(I ∗ , K)
Scaling parameters H(I ∗ , K)
Exogenous shocks
Mean growth rate
β
γ
ρ−1
0.971
10
2
ν
λ
λ∗
ϕ
η
ξ
a1
a2
0.3
0.11
0.11
0.88
2.5
5
0.6645
-0.0324
Γ0
Γ∗0
Ψ
(Θ1 )1,1
Λ1
Volatility of the direct shock
Autocorrelation of the long-run risk process
Volatility of the long-run risk shock
0.02
0
[0.0508 0]
0.925
[0 0.008636]
Table 1: Parameterization of the Ai et al. (2010) model. All parameters correspond to a
calibration at the annual frequency.
specified by the CES aggregator
∗
G (I, K ) = ϕI
1−1/η
∗ 1−1/η
+ (1 − ϕ) (K )
1
1−1/η
and the adjustment cost function for the production of new intangible capital is chosen to be
"
a1
H (I , K) =
1 − 1/ξ
∗
I∗
K
1−1/ξ
#
+ a2 K
where a1 and a2 are chosen so as to assure that H I¯∗ , K̄ = HI ∗ I¯∗ , K̄ = 1 for steady state values
I¯∗ and K̄. The parameter values are summarized in Table 1.
59
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63
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Morris A. Davis, Jonas D.M. Fisher and Toni M. Whited
WP-10-02
Accounting for non-annuitization
Svetlana Pashchenko
WP-10-03
2
Working Paper Series (continued)
Robustness and Macroeconomic Policy
Gadi Barlevy
WP-10-04
Benefits of Relationship Banking: Evidence from Consumer Credit Markets
Sumit Agarwal, Souphala Chomsisengphet, Chunlin Liu, and Nicholas S. Souleles
WP-10-05
The Effect of Sales Tax Holidays on Household Consumption Patterns
Nathan Marwell and Leslie McGranahan
WP-10-06
Gathering Insights on the Forest from the Trees: A New Metric for Financial Conditions
Scott Brave and R. Andrew Butters
WP-10-07
Identification of Models of the Labor Market
Eric French and Christopher Taber
WP-10-08
Public Pensions and Labor Supply Over the Life Cycle
Eric French and John Jones
WP-10-09
Explaining Asset Pricing Puzzles Associated with the 1987 Market Crash
Luca Benzoni, Pierre Collin-Dufresne, and Robert S. Goldstein
WP-10-10
Prenatal Sex Selection and Girls’ Well‐Being: Evidence from India
Luojia Hu and Analía Schlosser
WP-10-11
Mortgage Choices and Housing Speculation
Gadi Barlevy and Jonas D.M. Fisher
WP-10-12
Did Adhering to the Gold Standard Reduce the Cost of Capital?
Ron Alquist and Benjamin Chabot
WP-10-13
Introduction to the Macroeconomic Dynamics:
Special issues on money, credit, and liquidity
Ed Nosal, Christopher Waller, and Randall Wright
WP-10-14
Summer Workshop on Money, Banking, Payments and Finance: An Overview
Ed Nosal and Randall Wright
WP-10-15
Cognitive Abilities and Household Financial Decision Making
Sumit Agarwal and Bhashkar Mazumder
WP-10-16
Complex Mortgages
Gene Amromin, Jennifer Huang, Clemens Sialm, and Edward Zhong
WP-10-17
The Role of Housing in Labor Reallocation
Morris Davis, Jonas Fisher, and Marcelo Veracierto
WP-10-18
Why Do Banks Reward their Customers to Use their Credit Cards?
Sumit Agarwal, Sujit Chakravorti, and Anna Lunn
WP-10-19
3
Working Paper Series (continued)
The impact of the originate-to-distribute model on banks
before and during the financial crisis
Richard J. Rosen
WP-10-20
Simple Markov-Perfect Industry Dynamics
Jaap H. Abbring, Jeffrey R. Campbell, and Nan Yang
WP-10-21
Commodity Money with Frequent Search
Ezra Oberfield and Nicholas Trachter
WP-10-22
Corporate Average Fuel Economy Standards and the Market for New Vehicles
Thomas Klier and Joshua Linn
WP-11-01
The Role of Securitization in Mortgage Renegotiation
Sumit Agarwal, Gene Amromin, Itzhak Ben-David, Souphala Chomsisengphet,
and Douglas D. Evanoff
WP-11-02
Market-Based Loss Mitigation Practices for Troubled Mortgages
Following the Financial Crisis
Sumit Agarwal, Gene Amromin, Itzhak Ben-David, Souphala Chomsisengphet,
and Douglas D. Evanoff
WP-11-03
Federal Reserve Policies and Financial Market Conditions During the Crisis
Scott A. Brave and Hesna Genay
WP-11-04
The Financial Labor Supply Accelerator
Jeffrey R. Campbell and Zvi Hercowitz
WP-11-05
Survival and long-run dynamics with heterogeneous beliefs under recursive preferences
Jaroslav Borovička
WP-11-06
A Leverage-based Model of Speculative Bubbles (Revised)
Gadi Barlevy
WP-11-07
Estimation of Panel Data Regression Models with Two-Sided Censoring or Truncation
Sule Alan, Bo E. Honoré, Luojia Hu, and Søren Leth–Petersen
WP-11-08
Fertility Transitions Along the Extensive and Intensive Margins
Daniel Aaronson, Fabian Lange, and Bhashkar Mazumder
WP-11-09
Black-White Differences in Intergenerational Economic Mobility in the US
Bhashkar Mazumder
WP-11-10
Can Standard Preferences Explain the Prices of Out-of-the-Money S&P 500 Put Options?
Luca Benzoni, Pierre Collin-Dufresne, and Robert S. Goldstein
WP-11-11
Business Networks, Production Chains, and Productivity:
A Theory of Input-Output Architecture
Ezra Oberfield
Equilibrium Bank Runs Revisited
Ed Nosal
WP-11-12
WP-11-13
4
Working Paper Series (continued)
Are Covered Bonds a Substitute for Mortgage-Backed Securities?
Santiago Carbó-Valverde, Richard J. Rosen, and Francisco Rodríguez-Fernández
WP-11-14
The Cost of Banking Panics in an Age before “Too Big to Fail”
Benjamin Chabot
WP-11-15
Import Protection, Business Cycles, and Exchange Rates:
Evidence from the Great Recession
Chad P. Bown and Meredith A. Crowley
Examining Macroeconomic Models through the Lens of Asset Pricing
Jaroslav Borovička and Lars Peter Hansen
WP-11-16
WP-12-01
5
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