1. science
2. mathematics
3. statistics

Term Structure of Uncertainty in the Macroeconomy
Jaroslav Boroviˇcka
Lars Peter Hansen
New York University
University of Chicago and NBER
jaroslav.borovicka@nyu.edu
lhansen@uchicago.edu
April 21, 2015
Preliminary and incomplete draft.
Abstract
Impulse responses characterize how macroeconomic processes respond to alternative shocks
over different horizons. Macroeconomic models make predictions about these responses. The
impulse responses also measure how exposed macroeconomic processes and other cash flows are
to shocks. Models of financial markets assign compensations to these exposures. We describe
and apply methods for computing these compensations represented as price elasticities over
alternative payoff horizons. The outcome is a term structure of macroeconomic uncertainty.
Contents
1 Introduction
3
2 Mathematical framework
4
3 Asset pricing over alternative investment horizons
3.1 One-period pricing . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 One-period shock elasticities . . . . . . . . . . . . . . . .
3.2 Multi-period investment horizon . . . . . . . . . . . . . . . . . .
3.3 A change of measure and an impulse response for a multiplicative
3.4 Linear vector-autoregressions . . . . . . . . . . . . . . . . . . . .
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4 Discrete Markov states
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5 Examples
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6 Long-horizon pricing
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7 Beliefs and pricing measures
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8 Solution methods
8.1 Exponential-quadratic framework . . . . . . . . . . . . . . . . . . . . . .
8.1.1 Shock elasticities . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2 Perturbation methods . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.3 Approximating the state vector process . . . . . . . . . . . . . . . . . .
8.4 Approximating an additive functional and its multiplicative counterpart
8.5 Approximating shock elasticities . . . . . . . . . . . . . . . . . . . . . .
8.5.1 Approach 1: Approximation of elasticity functions . . . . . . . .
8.5.2 Approach 2: Exact elasticities under approximate dynamics . . .
8.6 Equilibrium conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.7 Related approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.8 Recursive utility investors . . . . . . . . . . . . . . . . . . . . . . . . . .
8.8.1 Recursive preferences and the robust utility interpretation . . . .
8.8.2 Expansion approaches . . . . . . . . . . . . . . . . . . . . . . . .
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9 Continuous-time framework
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10 Empirical evidence
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11 An example: Models with financial constraints
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11.1 Stochastic discount factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
11.1.1 Households . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
1
11.1.2 Financial sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
11.2 Asset pricing implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
A Exponential-quadratic framework
A.1 Definitions . . . . . . . . . . . . .
A.2 Concise notation for derivatives .
A.3 Conditional expectations . . . . .
A.4 Iterative formulas . . . . . . . . .
A.5 Coefficients Φ∗i . . . . . . . . . .
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B Shock elasticity calculations
44
B.1 Shock elasticities under the convenient functional form . . . . . . . . . . . . . . . . . 44
B.2 Approximation of the shock elasticity function . . . . . . . . . . . . . . . . . . . . . 44
C Gertler and Kiyotaki (2010) model with financial constraints
2
46
1
Introduction
Macroeconomists and financial economists have a long tradition of using impulse response functions
to characterize the quantitative importance of alternative sources of fluctuations. Frisch (1933)
initiated an important line of research when wrote on the “impulse and propagation problem”
influenced by the modeling insights of Yule (1927) and Slutsky (1927). An impulse captured
formally by a random shock tomorrow has an impact on an economic time series in all of the
subsequent time periods. The intertemporal responses are quantified by the response function.
Sims (1980) showed how to extend this approach in a tractable way to multivariate time series
with a vector of underlying shocks, and he exposed the underlying challenges for identification.
Subsequent research developed nonlinear counterparts to impulse response functions.
Macroeconomic shocks also play an important role in asset pricing. By their very nature,
macroeconomic shocks cannot be diversified. Investors exposed to those shocks require compensations. These compensations differ by the nature of the macroeconomic shocks. Since the shocks
have differential impacts on macroeconomic variables, this is reflected in the compensations. We
call the compensations uncertainty prices, and there is term structure that characterizes these prices
as a function of the investment horizon. In this chapter we study this methods for depicting this
term structure and illustrates its use by comparing pricing implications across models. Roughly
speaking we formalize the pricing counterpart to impulse response functions familiar to macroeconomist. Our calculations require either an empirical-based or model-based stochastic discount
factor process along the a representation of how microeconomic variables respond to shocks in order
to perform the calculations that interest us.
There is an alternative way to motivate the calculations that we perform. A common characterization of risk aversion looks at local certainty equivalent calculations for small variance changes in
consumption. We deviate in two ways. First when making small changes, we do not use certainty
as our benchmark but rather the equilibrium consumption from the stochastic general equilibrium
model. This leads us to make more refined adjustments in the exposure to uncertainty. Second
movements in consumption at a future dates could be induced by any of the macroeconomic shocks
with occurrences at dates between tomorrow and this future date. Thus, similar to Hansen et al.
(1999) and Alvarez and Jermann (2004), we have a differential measure depending on the specific
shock. We also find it advantageous to differentiate the dates of the impacts. This gives us a term
structure of responses analogous to what happens with impulse response functions.
We take as an alternative motivation for the methods we represent to understand better understand what the empirical findings in Hansen et al. (2008), van Binsbergen et al. (2012), van
Binsbergen et al. (2013) and Giglio et al. (2015) imply for equilibrium asset pricing models.
3
2
Mathematical framework
We introduce a framework designed to encompass a large class of macroeconomic and asset pricing
general equilibrium models. There is an underlying stationary Markov model that is used to capture
the stochastic growth of a vector of time series of economic variables. The Markov model emerges as
the “reduced form” of a solution of to dynamic stochastic equilibrium model of the macroeconomy.
Modeling stationary growth rates allows for inclusion of shocks that have permanent effects and
nontrivial long-horizon implications for risk compensations. In Section 5, we provide a range of
illustrative applications of this framework, and we devote Section 11 to a more extensive exploration
of models with financial constraints.
We start with a probability space (Ω, F, P ). On this probability space, there is an n-dimensional,
stationary and ergodic Markov process X = {Xt : t ∈ N} and a k-dimensional process W of independent and identically distributed shocks. Unless otherwise specified, we assume that each Wt is
a multivariate standard normal random variable. We will have more to say about discrete states
and shocks that are not normally distributed in Section 4.
The Markov process is initialized at X0 . Denote F = {Ft : t ∈ N} the completed filtration
generated by the histories of W and X0 . We suppose that X is a solution to a law of motion
Xt+1 = ψ (Xt , Wt+1 )
Yt+1 − Yt = φ (Xt , Wt+1 )
(1)
The state vector Xt contains both exogenously specified states and endogenous ones. We presume
full information in the sense that the shock Wt+1 can be depicted in terms of (Xt , Yt+1 − Yt ). In
more general circumstances we would be let to solve a filtering problem if we are to match an
information structure to (X, Y ).
Consistent intertemporal pricing together with the Markov property lead us to use a class of
stochastic processes called multiplicative functionals. These processes are built from the underlying
Markov process and will be used to model cash flows and stochastic discount factors. Since many
macroeconomic time series grow or decay over time, we use the state vector X to model the growth
rate of such processes. In particular, let the dynamics of a multiplicative functional M be defined
as
log Mt+1 − log Mt = κ (Xt , Wt+1 ) .
(2)
The components of Y are examples of multiplicative functionals.
Since X is stationary, the process log M has stationary increments. A revealing example is the
conditionally linear model
κ (Xt , Xt+1 ) = β (Xt ) + α (Xt ) · Wt+1
where β (x) allows for nonlinearity in the conditional mean, and α (x) introduces stochastic volatility.
4
We denote G a generic cash-flow process and S the stochastic discount factor process, both
modeled as multiplicative functionals. While we adopt a common mathematical formulation for
both, G is expected to grow and S is expected to decay over time, albeit in stochastic manners.
Equilibrium models in macroeconomics and asset pricing build on the premise of utility-maximizing
investors trading in arbitrage-free markets. Arbitrage-free pricing implies the existence of a strictly
positive stochastic discount factor process S that can be used to infer equilibrium asset prices.
Stochastic discount factors provide a convenient way to depict the observable implications of asset
pricing models.
Definition 2.1. A stochastic discount factor S is a stochastic process initialized at S0 = 1 such
that for any t, j ≥ 0 and payoff Gt+j maturing at time t + j, the time-t price is given by
Qt [Gt+j ] = E
St+j
St
Gt+j | Ft .
(3)
If markets are complete, then this stochastic discount factor is unique. Equations of the type
(3) arise from investors’ optimality conditions in the form of Euler equations. In an equilibrium
model with complete markets, the stochastic discount factor is typically equated with the marginal
rate of substitution of an unconstrained investor investor. The identity of such a person can change
over time and across states. In some models with incomplete markets, the stochastic discount
factor process ceases to unique. There are different shadow prices for untraded risk exposures
but a common pricing of the exposures with explicit compensations in financial markets. With
other forms of trading frictions, the pricing equalities can be replaced by pricing inequalities, still
expressed using a stochastic discount factor.
In our framework, the equilibrium stochastic discount factors will inherit the multiplicative functional structure. We include stylized examples of models of stochastic discount factors frequently
used in the literature in Section 5. Market frictions, portfolio constraints and other types of market
imperfections will then introduce distortions into formula (3). We will study such distortions in
models with financial constraints in Section 11.
Notice that the definition (3) of the stochastic discount factor involves an expectations operator.
This expectations operator in general represents investors’ beliefs about the future. Here, we
have imposed rational expectations by assuming that investors’ beliefs are identical to the datagenerating probability measure P . This measure is that implied by historical evidence or by the
fully specified model. Investors’ beliefs, however, may differ from P and there exists alternative
approaches to modeling these deviations in interesting ways . We will extend our analysis to support
the formal investigations of these approaches later in this chapter.
5
3
Asset pricing over alternative investment horizons
We are interested in pricing cash flows modeled as multiplicative processes. Consider a generic
cash flow process G, say the dividend process or aggregate consumption process. We are interested
in studying the payoffs Gt maturing in individual periods t = 0, 1, 2, . . .. In particular, we derive
measures that capture the sensitivity of expected payoff to exposure to risk, and the sensitivity of
the associated risk compensations, in units of expected return, for this exposure.
The method relies on a comparison of the pricing of the payoff Gt relative to another payoff
that is marginally more exposed to risk in a particular way. We will later show that this approach
is mathematically similar to constructing nonlinear impulse functions in an asset pricing context.
The cash flows G arising from equilibrium models will typically have the form of multiplicative
functionals (2). A special case of such cash flows are payoffs that are functions of the Markov state,
ψ (Xt ). These payoffs will be featured prominently in our subsequent analysis.
3.1
One-period pricing
We are interested in the pricing of payoffs maturing at different horizons but we start with a simple
one-period conditionally lognormal environment. This environment will provide an explicit link to
familiar calculations in asset valuation. 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 the payoff in period one
divided by the period-zero price:
.
R1 =
G1
G1
=
.
Q0 [G1 ]
E [S1 G1 | X0 ]
The logarithm of the expected return can then be calculated explicitly as:
log E [R1 | X0 = x] = log E [G1 | X0 = x] − log E [S1 G1 | X0 = x]
(4)
2
|αs (x)|
= −βs (x) −
− αs (x) · αg (x).
2 }|
|
{z
{z
}
risk-free rate
risk premium
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 these shocks. This
compensation is expressed in terms of expected returns as is typical in asset pricing.
The risk premia in this conditionally lognormal one-period model have a familiar linear structure
6
known from one-period factor models. In these models, the factor loadings αg on these individual
risk factors (components of the normally distributed shock W1 ) are multiplied by factor prices −αs .
The total compensation in terms of an expected return is thus the product of the quantity of risk
(risk exposure), and the price per unit of this risk.
In a nonlinear multiperiod environment, this calculation ceases to be straightforward. We would,
however, still like to infer measures of the quantity of risk and the associated price of the risk. We
therefore explore a related derivation that will yield the same results in this one-period lognormal
environment but will also naturally extend to a nonlinear setup and multiple-period horizons.
3.1.1
One-period shock elasticities
We paremeterize a family of random variables H1 (r) indexed by r using
log H1 (r) = rη (X0 ) · W1 −
r2
|η (X0 )|2
2
(5)
where r is an auxiliary scalar parameter. The vector of exposures αh (X0 ) is normalized to
i
h
E |η (X0 )|2 = 1.
Form a parameterized family of payoffs G1 H1 (r) given by
r2
log G1 + log H1 (r) = [αg (X0 ) + rη (X0 )] · W1 + βg (X0 ) − |η (X0 )|2 .
{z
}
2
|
new shock exposure
The new cash flow G1 H1 (r) has shock exposure αg (X0 ) + rη (X0 ) and is thus more exposed to the
vector of shocks W1 in the direction η (X0 ). By changing r we alter the magnitude of the exposure
in direction η (X0 ). By choosing different vectors η (X0 ), we alter the combinations of shocks whose
impact we want to investigate. A typical example of an η (X0 ) would be a coordinate vector ej with
a single one in j-th place. In that case, we infer the pricing implications of the j-the component of
the shock vector W1 . In some applications it may be convenient to make η (X0 ) explicitly depend
on X0 . For instance, Boroviˇcka et al. (2011) propose scaling of η with X0 in models with stochastic
volatility.
The payoffs G1 H1 (r) imply a corresponding family of logarithms of expected returns as in
equation (4):
log E [R1 (r) | X0 = x] = log E [G1 H1 (r) | X0 = x] − log E [S1 G1 H1 (r) | X0 = x] .
We are interested in comparing the expected return of the payoff G1 H1 (r) relative to G1 ≡
G1 H1 (0). Since we are using logarithms of expected returns and our exposure direction η (X0 )
7
has a unit standard deviation, by differentiating with respect to r we compute an elasticity
d
d
log E [G1 H1 (r) | X0 = x]
log E [S1 G1 H1 (r) | X0 = x] .
−
dr
dr
r=0
r=0
This elasticity measures the sensitivity of the expected return on the payoff G1 to an increase in
exposure to the shock in the direction η (X0 ). The calculation leads us to define the counterparts
of quantity and price elasticities from microeconomics:
1. The one-period shock-exposure elasticity
d
εg (x, 1) =
log E [G1 H1 (r) | X0 = x]
= αg (x) · η (x)
dr
r=0
measures the sensitivity of the expected payoff G1 to an increase in exposure in the direction
αh (x).
2. The one-period shock-price elasticity
d
d
εp (x, 1) =
log E [G1 H1 (r) | X0 = x]
log E [S1 G1 H1 (r) | X0 = x]
−
dr
dr
r=0
r=0
= −αs (x) · η (x)
measures the sensitivity of the compensation, in units of expected return, for this exposure.
Notice that the shock-exposure elasticity recovers the exposure vector αg (x), and individual components of this vector can be obtained by varying the choice of the direction of the perturbation
η (x). Similarly, the shock-price elasticity recovers the vector of prices −αs (x) associated with the
risks embedded in the shock W1 .
In this one-period case, we replicated a straightforward decomposition of the expected return
(4) into quantities and prices of risk. Now we move to the characterization of the asset pricing
implications over longer horizons.
3.2
Multi-period investment horizon
Consider the parameterized payoff Gt H1 (r) with a date-zero price E [St Gt H1 (r)]. This is a payoff
maturing at time t that has the same growth rate as payoff Gt except period one when the growth
rate is stochastically perturbed by H1 (r). The logarithm of the expected return (yield to maturity)
is
.
log E [R0,t (r) | X0 = x] = 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
d
εg (x, t) =
log E [Gt H1 (r) | X0 = x]
dr
r=0
8
(6)
2. shock-price elasticity
d
d
log E [Gt H1 (r) | X0 = x]
log E [St G1 H1 (r) | X0 = x]
εp (x, t) =
−
dr
dr
r=0
r=0
(7)
These elasticities are functions of the investment horizon t, and thus we obtain a term structure
of elasticities. The dependence on the current state X0 = x incorporates possible time-variation in
the sensitivity of expected returns to exposure to shocks.
3.3
A change of measure and an impulse response for a multiplicative functional
Notice that the shock elasticities defined in the previous subsection have a common mathematical
structure expressed using the multiplicative functionals M = S and M = SG. Given a multiplicative functional M , we can define
d
log E [Mt H1 (r) | X0 = x]
(8)
ε (x, t) =
dr
r=0
Taking the derivative in (8), we obtain
ε (x, t) = η (x) ·
E [Mt W1 | X0 = x]
E [Mt | X0 = x]
(9)
Thus a major ingredient in the computation is the covariance between Mt and W1 conditioned on
X0 .
The random variable H1 (r) given by (5) is positive and has expectation equal to unity conditioned on X0 . Multiplication by the this random variable has the interpretation of changing the
probability distribution of W1 from having mean zero to having a mean given by rη(X0 ) instead
zero. Thus given a multiplicative functional M
E [Mt H1 (r) | X0 = x] = E [H1 (r)E [Mt | X0 , W1 | X0 ]]
e (E [Mt | X0 , W1 ] | X0 = x)
=E
e presumes that the random vector W1 is distributed as a multivariate normal with mean
where E
rη(x) consistent with our multiplication by H1 (r).
Impulse responses to specific structural shocks are a common way of representing the dynamic
properties of macroeconomic models. This idea goes back at least to Frisch (1933) who noted:
There are several alternative ways in which one may approach the impulse problem. . .
One way which I believe is particularly fruitful and promising is to study what would
become of the solution of a determinate dynamic system if it were exposed to a stream
of erratic shocks that constantly upsets the continuous evolution, and by so doing
introduces into the system the energy necessary to maintain the swings.
9
To relate our elasticity calculation to an impulse response function, consider E [Mt | X0 , W1 = w]
for alternative choices of w. Changing the value of w as gives rise to the impulse response Mt
to a shock at date one. Instead of conditioning on alternative realized value of the shock at date
one, we change the date zero distribution by targeting a shift in the mean. Another difference
is that macroeconomists typically use the logarithms of macro time series. For asset pricing it
is important that we work with levels. This leads us to study the impact on logarithm of the
conditional expectation as appears in (8). When M is a lognormal process, the impulse response
functions for log M matches exactly our elasticity as we will now see.
3.4
Linear vector-autoregressions
Linear vector-autoregressions (VARs) are models that are a special case of the framework (1)–(2)
with parameters that satisfy
ψ (x, w) = µ
¯x + σ
¯w
κ (x, w) = β¯ · x + α
¯·w
X is a linear vector-autoregression with autoregression coefficient µ
¯ and shock exposure matrix
σ
¯ . While the VAR literature often focuses on the responses of the individual components of the
stationary state vector X, we use the additional flexibility embedded in the construction of the
multiplicative functional M , and study the responses of Y = log M .
Let η(x) = η¯ where η¯ is a vector with norm one. In typical applications, η¯ would be a coordinate
vector, η¯ = ei . The impulse response function of Yt for the linear combination of shocks chosen by
the vector η¯ is given by
E [Yt | X0 = x, W1 = η¯] − E [Yt | X0 = x, W1 = 0] = η¯ · ε¯t .
(10)
where the coefficients satisfy the recursions
′
ε¯j+1 = ε¯j + ζ¯j β¯
ζ¯j+1 = µ
¯ζ¯j
with initial conditions ζ¯1 = σ
¯ and ε¯1 = α
¯ . Thus
ζ¯j
ε¯t
= µ
¯j−1 σ
¯
′
= α
¯ + (I − µ
¯)−1 I − µ
¯t−1 σ
¯ β¯
The impulse response function in the linear model is thus a sequence of determinstic coefficients
ε¯t . The first term, α
¯ · η¯, represents the “instantaneous” effect arising from the current shock
that directly impacts the multiplicative functional through the function κ (x, w) in period 1, while
the second term captures the subsequent propagation of the shock through the dynamics of the
10
stationary state X, which perturbs the growth rate of Y in the future.
Now consider our elasticity calculation. Notice that
log E [Mt | X0 = x, W1 = w] − log E [Mt | X0 = x, W1 = v]
= E [Yt | X0 = x, W1 = w] − E [Yt | X0 = x, W1 = v] .
Use a normal density with mean r¯
η with an identity as the covariance matrix for integrating over
w and a normal density with mean 0 and an identity matrix for v and divide by r. This gives
η¯ · ε¯t , which coincides with a horizon t impulse response. Thus for a linear model, our elasticity for
M coincides with the linear impulse response function for Y = log M , with the direction vector η¯
selecting a particular combination of shocks in W1 .
Consider in particular the shock-price elasticity (7). Notice that this shock-price elasticity
consists of the difference of shock elasticities for G and SG, and thus we are lead to compute
impulse response functions for log G and log S + log G. The additivity of the construction implies
that the impulse response function coefficients for the latter are ε¯s,t + ε¯g,t , and thus the resulting
shock-price elasticity corresponds to the impulse response function of − log S, with coefficients
−¯
εs,t .
11
4
Discrete Markov states
Many problems in macroeconomics can be characterized using models in which the underlying
dynamics transits between ‘regimes’ with qualitatively or quantitatively distinct behavior. These
switches can be exogenous (e.g., exogenously modeled periods of low or high growth and volatility)
or endogenous (e.g., interest rate at the zero lower bound, financial sector in a period of binding
financial constraints, or regime changes in government policies). Even when the dynamics are
not exactly described by switches between discrete regimes, their approximation using a regimeswitching model can be appealing from the perspective of interpretation of results (e.g., high- and
low- growth regimes interpretable as recessions and expansions). Finally, approximating continuousstate dynamics with a small number of discrete states with properly modeled transition probabilities
can be attractive for computational reasons, too.
A large class of models uses Markov chain models to capture the dynamics of exogenous shocks
(see David (2008), Chen (2010) or Bianchi (2015) for some recent examples in the asset pricing
literature and Liu et al. (2011) and Bianchi et al. (2013) in macroeconomic modeling). Regime
switches are also utilized to model time-variation in government policies, see Sims and Zha (2006),
Liu et al. (2009) and Bianchi (2012) for regime switching in monetary policy rules, Davig et al.
(2010, 2011) and Bianchi and Melosi (2015) for fiscal policy applications and Chung et al. (2007)
and Bianchi and Ilut (2015) for a combination of both. Farmer et al. (2011) and Foerster et al.
(2014) analyze solution and estimation techniques in Markov chain models in conjunction with
perturbation approximation methods that we discuss in Section 8.
Markov chain models can be accommodated within the multiplicative functional framework,
and we will use them for some particularly revealing examples. In this paper, we pay particular
attention to a class of discrete-time models. In Boroviˇcka et al. (2011), we constructed shock
elasticities for shocks associated with state transitions in continuous-time Markov chains. See also
the discussion on the continuous-time framework in Section 9.
An n-state discrete-time Markov chain is represented by an n-dimensional process X where Xt
is a coordinate vector determining the state of the chain at time t. The transition probabilities are
defined by an n × n matrix P such that
Pij = P (Xt+1 = ej | Xt = ei )
where ei is the i-th coordinate vector.
Define the innovation Wt+1 = Xt+1 − E [Xt+1 | Xt ]. A multiplicative functional is determined
by n × n matrices L0 , L1 such that
log Mt+1 − log Mt = (Xt )′ L1 Wt+1 + (Xt )′ L0 Xt .
(11)
By exponentiating the right-hand side and evaluating it for all possible coordinate vector com-
12
binations for Xt and Xt+1 , it may be shown that
Mt+1
= (Xt )′ MXt+1
Mt
(12)
where M is a matrix with positive entries.
Typically, in time series models we also include continuous states and normal shocks. This
framework can also be easily extended to such environments in which the discrete states are used
to shift the conditional normal distributions in the Markov evolution.
13
5
Examples
This section is still under construction.
14
6
Long-horizon pricing
Shock elasticities constructed in the previous section depict the term structure of risk as we change
the maturity of priced payoffs. In this section, we focus on the characterization of long-horizon limits
of the shock elasticities. These limits will be associated with martingale components embedded in
the stochastic discount factor and cash flow processes.
Hansen and Scheinkman (2009) used Perron–Frobenius theory to provide a factorization of
multiplicative functionals. Given a multiplicative functional M , consider solving the equation
E [Mt e (Xt ) | X0 = x] = exp (ηt) e (x)
(13)
for an unknown function e (x) that is strictly positive and an unknown number η. The solution is
independent of the choice of the horizon t.
Example 6.1. In a finite-state Markov chain environment, equation (13) is a standard eigenvalue
problem. As in Section 4, write
Mt+1
.
= κ (Xt , Wt+1 ) = (Xt )′ MXt+1
Mt
where M is an n × n matrix. In the same way, the probability measure P defines a one-period
transition matrix P. For t = 1, equation (13) becomes a vector equation
(P ∗ M) e = exp (η) e
where the operator ∗ depicts elementwise multiplication, (P ∗ M)ij = Pij Mij . When P ∗ M is a
matrix with strictly positive elements, then the Perron–Frobenius theorem implies the existence of
a unique normalized strictly positive eigenvector e associated with the largest eigenvalue exp (η) of
the matrix P ∗ M.
In continuous state spaces, this factorization may not yield a unique strictly positive solution
e (x). Hansen and Scheinkman (2009) provide a selection criterion based on stochastic stability
that guarantees uniqueness. For now, we will operate with this unique solution, and revisit this
question in Section 7.
Consider the pair (e, η) that solves (13) and form
e (Xt )
.
ft =
Mt .
M
exp (−ηt)
e (X0 )
f is a martingale under P , since
The stochastic process M
exp (−η (t + 1))
Mt+1
Mt E
e (Xt+1 ) | Ft =
e (X0 )
Mt
e (Xt )
ft .
Mt = M
= exp (−ηt)
e (X0 )
h
i
ft+1 | Ft =
E M
15
(14)
Consequently, expression (14) can be reorganized as
Mt = exp (ηt)
e (X0 ) f
Mt
e (Xt )
(15)
This formula provides a multiplicative decomposition of the multiplicative functional M into a
f.
deterministic drift exp (ηt), a stationary function of the Markov state e (x) and a martingale M
This martingale component will be critical in characterizing long-term pricing implications.
f is a probability measure Pe such that for every measurable
Associated with the martingale M
function Z of the Markov process between dates zero and t,
h
i
ft Z | X0 = x = E
e [Z | X0 = x]
E M
e [· | X0 = x] is the conditional expectation operator under the probability measure Pe.1
where E
This factorization shares some features of additive decompositions of linear time series. Beveridge and Nelson (1981) and Blanchard and Quah (1989) extracted a martingale component in
linear models and used it to characterize the impact of permanent shocks. Hansen (2012) constructs
an additive decomposition of log M in a continuous-time version of our nonlinear framework.
The additive linear decomposition of Beveridge and Nelson (1981) and Blanchard and Quah
(1989) allows to interpret the martingale and the residual component as permanent and transitory
parts of the time series because the two components can be constructed as orthogonal. This
ceases to be true for the decomposition (15). The eigenfunction e (Xt ) is often correlated with the
ft . The martingale component still determines the long-run implications
martingale component M
for valuation but the mutual correlation invalidates the interpretation of e (Xt ) as the ‘transitory’
component of the multiplicative functional.
Factorization (15) leads to a characterization of long-horizon limits for the shock elasticities.
Using this factorization in expression (8), we obtain
ε (x, t) = η (x) ·
e [b
E
e (Xt ) W1 | X0 = x]
e
E [b
e (Xt ) | X0 = x]
.
where eb (x) = 1/e (x). Under technical assumptions
e [b
lim E
e (Xt ) | Xs = x]
t→∞
converges to a finite constant, and thus the long-maturity limit for the shock elasticity is given by
e [W1 | X0 = x] .
lim ε (x, t) = αh (x) · E
t→∞
The sensitivity of long-horizon payoffs to current shocks is therefore determined by the martingale
components of the stochastic discount factor and the cash flow, and their covariance with the shock
In order to completely define the measure Pe , we also need to specify the unconditional probability distribution,
a question to which we return in Section 7.
1
16
W1 .
17
7
Beliefs and pricing measures
Still under construction.
18
8
Solution methods
In the preceding sections, we developed formulas for shock-price and shock-exposure elasticities
for a wide class of models driven by a state vector with Markov dynamics (1). While the general
analysis is revealing, we now propose a tractable implementation. Our interest lies in providing tools
for valuation analysis in structural macroeconomic models, and we now feature a special dynamic
structure for which we obtain closed-form solutions for the shock elasticities. The discussion draws
on methods developed in Boroviˇcka and Hansen (2013, 2014). We also provide Matlab software
that implements the solution methods described in this section, and a toolkit that computes shock
elasticities for models solved using Dynare.2
We start by introducing a convenient exponential-quadratic framework that we use for modeling
the state vector X and the resulting multiplicative functionals. In this framework, conditional
expectations of multiplicative functionals and the shock elasticities are available in closed form.
We then consider a special class of approximate solutions to dynamic macroeconomic models
constructed using perturbation methods. We show how to approximate the equilibrium dynamics,
additive and multiplicative functionals, and the resulting shock elasticities. The dynamics of these
approximate solutions will be nested within the exponential-quadratic framework.
8.1
Exponential-quadratic framework
We study dynamic systems for which the state vector can be partitioned as X = (X1′ , X2′ )′ where
the two components follow the laws of motion:
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 ) .
(16)
We restrict the matrices Θ11 and Θ22 to have stable eigenvalues. Notice that the restrictions
imposed by the triangular structure imply that the process X1 is linear, while the process X2 is
linear conditional on the evolution of X1 .
The class of multiplicative functionals M that interest us satisfies, for Y = log M , the restriction
Yt+1 − Yt = Γ0 + Γ1 X1,t + Γ2 X2,t + Γ3 (X1,t ⊗ X1,t )
(17)
+Ψ0 Wt+1 + Ψ1 (X1,t ⊗ Wt+1 ) + Ψ2 (Wt+1 ⊗ Wt+1 ) .
In what follows we use a 1 × k2 vector Ψ to construct a k × k symmetric matrix sym [matk,k (Ψ)]
2
Dynare is a freely available Matlab/Octave toolkit for solving and analyzing dynamic general equilibrium models.
See http://www.dynare.org. Our software is available at https://files.nyu.edu/jb4457/public/software.html.
19
such that3
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.
The system (16)–(17) 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. For
instance, the state variable X1,t can capture a linear process for conditional volatility, and X2,t the
conditional growth rate of cash flows. The coefficient Ψ1 in (17) then determines the time-variation
in the conditional volatility of the growth rate of M , while Λ21 in (16) impacts the conditional
volatility of the changes in the growth rate. In Section 8.2, we will map the solution obtained using
perturbation approximations into this framework as well.
A virtue of parameterization (16)–(17) 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 (16)–(17) 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 (17). Consider a function that is linearquadratic in x = (x′1 , x′2 )′ :
log f (x) = Φ0 + Φ1 x1 + Φ2 x2 + Φ3 (x1 ⊗ x1 )
(18)
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)
(19)
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 (19) initiated with f (x) = 1 to
3
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.
20
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 .
8.1.1
Shock elasticities
To compute shock elasticities given in (9) under the convenient functional form, we construct:
i
h
Mt
W
|
X
=
x
|
X
E
M
E
1
0
1
1
M1
E [Mt W1 | X0 = x]
i .
h
=
Mt
E [Mt | X0 = x]
|
X
=
x
|
X
E M1 E M
0
1
1
Notice that the random variable:
L1,t
Mt
M1 E M
|
X
1
1
i
= h
Mt
E M1 E M1 | X1 | X0 = x
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∗
.
Σ
2,t−1 Λ22 + Φ3,t−1 (Λ10 ⊗ Λ10 )
where Ik is the identity matrix of dimension k.4 We suppose that this matrix is positive definite.
The conditional mean vector for W1 under the change of measure is:
e [W1 | X0 = x] = Σ
e t [µt,0 + µt,1 x1 ] ,
E
e is the expectation under the change of measure and the coefficients µt,0 and µt,1 are given
where E
in Appendix B.
Thus the shock elasticity is given by:
ε (x, t) = η(x) · E [L1,t W1 | X0 = x]
e t [µt,0 + µt,1 x1 ] .
= η(x)′ Σ
The shock elasticity function in this environment depends on the first component, x1 , of the state
vector. Recall from (16) 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
This formula uses the result that (Λ10 W1 ) ⊗ (Λ10 W1 ) = (Λ10 ⊗ Λ10 ) (W1 ⊗ W1 ).
21
8.2
Perturbation methods
In macroeconomic models, the equilibrium Markov dynamics (1) is typically ex-ante unknown and
needs to be solved for from a set of equilibrium conditions. We now describe a solution method
for dynamic general equilibrium models that yields a solution in the form of an approximate law of
motion that is a special case of the exponential-quadratic functional form analyzed in Section 8.1.
This solution method, based on Holmes (1995) and Lombardo (2010), constructs a perturbation
approximation where the first- and second-order terms follow the restricted dynamics (16).
We assume that we can represent these equilibrium conditions in the form
0 = E [e
g (Xt+1 , Xt , Xt−1 , Wt+1 , Wt ) | Ft ]
(20)
where ge is an n × 1 vector function and the dynamics for Xt is implied by (1). These conditions
include, with sufficient generality, expectational and nonexpectational equations, including laws
of motion for exogenous variables. They can also involve one-period increments of multiplicative
functionals (2).
There are well-known saddle-point stability conditions on the system (20) that lead to a unique
equilibrium of the linear approximation (see Blanchard and Kahn (1980) or Sims (2002)) and we
assume that these are satisfied. While the specification of the equilibrium conditions involves expectations under the data-generating measure P , alternative belief specifications of individual agents
are possible by incorporating the corresponding distorting martingales described in Section ?? into
the specification of the vector function e
g.
Consider a parameterized family of the dynamic systems specified in (1):
Xt+1 (q) = ψ(Xt (q) , qWt+1 , q)
(21)
where we let q parameterize the sensitivity of the system to shocks. The dynamics of X(q) for q = 1
coincide with the dynamics for X in the original model as introduced in equation (1). We consider
a limit in which q = 0 and first- and second-order approximations around this limit system. For
each q, the system (21) is a solution to the set of equations
0 = E [e
g (Xt+1 (q) , Xt (q) , Xt−1 (q) , qWt+1 , qWt , q) | Ft ] .
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 8.1, we feature a second-order expansion:
Xt (q) ≈ X0,t + qX1,t +
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. Our
22
approximating process will similarly be stationary.5 The expansion leads to laws of motion for the
component processes X1,· and X2,· . The joint process (X1,· , X2,· ) will again be Markov, although
the dimension of the state vector under the approximate dynamics doubles.
The goal of the following subsections is to derive expansion approximations for the law of motion
of the state vector X and the equilibrium conditions that lead to a set of algebraic equations for
the coefficients in the approximate law of motion. We also derive expansion approximations for
multiplicative functionals of X and for the shock elasticities constructed for these functionals.
8.3
Approximating the state vector process
While Xt (q) serves as a state vector in the dynamic system (21), the state vector itself depends on
the parameter q. Recall that 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 (q) is Ft measurable and
that in forecasting future values of the state vector conditioned on Ft it suffices to condition on Xt .
Although Xt (q) 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 (16).
Define x
¯ to be the solution to the equation:
x
¯ = ψ(¯
x, 0, 0),
which gives the fixed point for the deterministic dynamic system. We assume that this fixed point
is locally stable. That is ψx (¯
x, 0, 0) is a matrix with stable eigenvalues, eigenvalues with absolute
values that are strictly less than one. Then set
X0,t = x
¯
for all t. This is the zeroth-order contribution to the solution constructed to be time-invariant.
In computing pathwise derivatives, we consider the state vector process viewed as a function
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.6
5
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).
6
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 x
¯ 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
23
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
(22)
where ψq , ψx and ψw are the partial derivative matrices:
. ∂ψ
(¯
x, 0, 0),
ψq =
∂q
. ∂ψ
ψx =
(¯
x, 0, 0),
∂x′
. ∂ψ
ψw =
(¯
x, 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 x
¯ 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 ≈ x
¯ + qX1,t .
In particular, the approximating process on the right-hand side has x
¯ + 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 also applies to several of the example models analyzed in this
chapter. In Section 8.8 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.
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 ) +
(23)
+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 (¯
x, 0, 0) and formed using
the construction of the derivative matrices described in Appendix A.2. As noted by Schmitt-Groh´e
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
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.
24
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 (q) ≈ x
¯ + 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 (16) given in Section 8.1. When the shock vector Wt is a multivariate standard normal,
we can utilize results from Section 8.1 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.
8.4
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)
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
Yt ≈ Y0,t + qY1,t +
q2
Y2,t
2
(24)
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 zeroth-order
contribution to Y is
.
Y0,t+1 − Y0,t = κ(¯
x, 0, 0) = κ
¯;
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 (¯
x, 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 )
25
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 (17) that we
introduced in Section 8.1.
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 (t¯
κ)
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. 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.
8.5
Approximating shock elasticities
We consider two alternative approaches to approximating shock elasticities of the form:
ε(x, t) = η(x) ·
E [Mt W1 | X0 = x]
.
E [Mt | X0 = x]
(25)
Recall that we produced this formula by localizing the risk exposure and computing a (logarithmic)
derivative.
8.5.1
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
ε(X0 , t) ≈ ε0 (t) + qε1 (t) +
26
q2
ε2 (X1,0 , X2,0 , t)
2
where only the second-order component is state-dependent. First, observe that the zeroth-order
approximation is
ε0 (t) = 0
because the zeroth-order contribution in the numerator of (25) is
E [exp(t¯
κ)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) = η(¯
x) · E [Y1,t W1 | F0 ] = η(¯
x) · 
t−1
X
j=1
′
κx (ψx )j−1 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:
i
o
n h
ε2 (X1,0 , X2,0 , t) = η(¯
x) · E (Y1,t )2 W1 + Y2,t W1 | F0 − 2E [Y1,t W1 | F0 ] E [Y1,t | F0 ] +
∂η
+2
(¯
x) X1,0 · E [Y1,t W1 | F0 ] .
∂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 (25), 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) = η(x) ·
E [M1 W1 | X0 = x]
= qη(x) · α(x).
E [M1 | X0 = x]
27
We may use our previous formulas or perform a direct calculation to show that
ε1 (1) =
ε2 (X1,0 , X2,0 , 1) =
η(¯
x) · α(¯
x)
′
′
′ ∂α
′ ∂η
x) + 2(X1,0 )
x)
2(X1,0 )
(¯
x) α(¯
(¯
x) η(¯
∂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 ]
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.
8.5.2
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 8.1. 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
∂α
∂η
x) + q ′ (¯
x)X1,0 · α(¯
x)X1,0 .
ε(x, 1) ≈ q η(¯
x) + q ′ (¯
∂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
28
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.
8.6
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 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. In Boroviˇcka and Hansen (2013), we also provide
an expansion method with an alternative scaling of the stochastic discount factor, motivated by
preferences that feature aversion to model misspecification as in Hansen and Sargent (2001, 2008).
8.7
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 8.1.
8.8
Recursive 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.
29
8.8.1
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 +
i
h
exp(−δ)
log E (Vt+1 )1−γ |Ft .
1−γ
(26)
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 Vt − log Ct =
log E
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:
St+1
St
Ct
Ct+1
(Vt+1 )1−γ
i
E (Vt+1 )1−γ | Ft
1−γ
1−γ Ct+1
(U
)
t+1
Ct
Ct
,
= exp(−δ)
1−γ
Ct+1
E (Ut+1 )1−γ CCt+1
|
F
t
t
= exp(−δ)
h
(27)
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 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
30
robustness” interpretation of the utility recursion (26). 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
1−γ
(Ut+1 )1−γ CCt+1
t
i= 1−γ
1−γ Ct+1
| Ft
| Ft
E (Ut+1 )
Ct
1−γ
Set+1
(Vt+1 )
≡ h
Set
E (Vt+1 )1−γ
in the stochastic discount factor ratio (27) 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.
8.8.2
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:
log Ut ≈ log U0,t + q log U1,t +
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 (24). 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.
The typical 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 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 ≡ u
¯=
exp (−δ)
(log C0,t+1 − log C0,t )
1 − exp (−δ)
31
(28)
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 ] +
i
h
+(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 8.4. 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.7
Under the recursive utility preferences, 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 ]
7
In related work we derive an alternative approximation for stochastic discount factors motivated by a concern
for robustness and calibrations of that concern. This change in perspective alters the expansion in a substantive way
by allowing for the robustness concern to present in lower order terms.
32
9
Continuous-time framework
This is under construction and will exploit results from Boroviˇcka et al. (2011) and Boroviˇcka et al.
(2014b).
33
10
Empirical evidence
Still under construction.
34
11
An example: Models with financial constraints
In an arbitrage-free, complete market environment, there exists a unique stochastic discount factor
that, given a probability measure, prices all traded securities. In economies with financial market
imperfections and constraints, this is no longer true.
In this section, we investigate a class of models that introduces frictions into the operations of
the financial sector. These models became immensely popular for the study of financial crises and
their impact on the macroeconomy, and serve as a toolbox for policy analysis.
There are two key features that are of interest to us. First, financial markets in these economies
are segmented, and different investors can own specific subsets of assets. These implies the existence
of alternative stochastic discount factors for individual investors that have to agree only on prices of
assets traded between given investors. Second, assets are valuable not only for their cash flows but
also because their ownership can relax or tighten financial constraints faced by individual investors.
When these constraints are binding, investors’ optimality conditions include ‘shadow prices’ of these
constraints that distort equilibrium asset prices.
11.1
Stochastic discount factors
As a laboratory, we consider the financial frictions model by Gertler and Kiyotaki (2010). In this
model, households and firms are separated by a financial sector populated by banks that accept
deposits from households and finance firm capital. The financing is subject to a constraint that
restricts the leverage of the bank. Banks accumulate own net worth that alleviates the constraints
but are facing random ‘termination’ shocks that assure that the financial sector on aggregate never
completely outsaves these constraints.
Here, we focus on the key relationships of the model and implications for asset pricing. A
complete description of the model is in Appendix C and we also refer to Gertler and Kiyotaki
(2010).
11.1.1
Households
Households in the model have preferences over consumption and leasure with internal habit formation:


∞ X
χ
.
Ut (C, L) = E 
log (Ct+j − hCt+j−1 ) −
(Lt+j )1+ε | Ft  .
1+ε
j=0
This implies the stochastic discount factor of the households of the form
h
St+1
Uc,t+1
=β
h
Uc,t
St
where Uc is the marginal utility of consumption
Uc,t
1
1
− hβE
| Ft .
=
Ct − hCt−1
Ct+1 − hCt
35
d ,
The only assets the households trade are riskless deposits that pay gross return (deposit rate) Rt+1
and therefore the stochastic discount factor S of the household must satisfy the equilibrium Euler
equation
St+1 d
1=E
R
| Ft .
St t+1
11.1.2
Financial sector
The banks in the financial sector trade three assets. They hold firm capital kt with unit price Qt
financed through own net worth nt , deposits dt and riskless borrowing in the interbank market bt :
Qt kt = nt + dt + bt .
(29)
Given the capital structure (kt , bt , dt ) at the end of period t, the net worth at the beginning of
period t + 1 satisfies
k
b
d
nt+1 = Rt+1
Qt kt − Rt+1
bt − Rt+1
dt
(30)
b
d
k
where Rt+1
and Rt+1
are riskless returns. The return on capital Rt+1
satisfies
k
Rt+1
= ψt+1
Zt+1 + (1 − δ) Qt+1
Qt
where Zt+1 is the dividend payment, Qt+1 the period t + 1 price of capital, δ the depreciation rate
and ψt+1 an aggregate shock to the quality of capital that follows a Markov process.
Denote V (kt , bt , dt ) the value of the bank. Capital market imperfections imply the financial
constraint
V (kt , bt , dt ) ≥ θ k Qt kt − θ b bt .
(31)
Gertler and Kiyotaki (2010) motivate this constraint through a moral hazard problem in which
the banker has the option to divert a fraction θ k of firm capital net of a fraction θ b of interbank
borrowing but the literature provided other similar microfoundations for this type of constraints
as well. The moral hazard problem is thus severe when θ k is high and θ b is low, i.e., when the
banker can divert a substantial share of capital without having to repay much of the debt. In these
situations, the financial constraint is more likely to be tight and borrowing is limited in equilibrium.
During their whole lifetime, bankers are separated from households but every period, with
probability 1 − σ, they face a random idiosyncratic termination shock. If the bank is terminated,
the banker returns to the household with a dividend equal to the current net worth. Consequently,
the Bellman equation has to satisfy the recursion
V (kt , bt , dt ) =
max
(kt+1 ,bt+1 ,dt+1 )
E
St+1
[(1 − σ) nt+1 + σV (kt+1 , bt+1 , dt+1 )] | Ft
St
where the choice of the capital structure in period t + 1 has to satisfy the constraint (31).
Assigning the Lagrange multipliers σ (St+1 /St ) µt+1 to the balance sheet constraint (29) and
36
σ (St+1 /St ) λt+1 to the financial constraint (31) for period t + 1, optimal choice of capital structure
implies Euler equations for the pricing of capital, deposits and interbank borrowing. The Euler
equation for the capital choice is
Vk,t
St+1
k
=E
(1 − σ + σµt+1 ) Rt+1
| Ft
Qt
St
(32)
The left-hand side is the the consumption-equivalent value of one unit of investment into physical
k . With probability 1 − σ, the banker is
capital. Each unit of investment earns a return Rt+1
terminated and this return is channeled to the household as dividends. With probability σ, the
investment remains in the bank, with marginal value µt+1 . This marginal value is given by
µt+1
Vk,t+1
+ λt+1
=
Qt+1
Vk,t+1
−θ .
Qt+1
(33)
Since the banker is allowed to change the capital structure freely, subject to the balance sheet
constraint (29), the value of µt+1 has to be equalized across all traded assets.
The first term on the right-hand side of (33) is the continuation value of the investment, and
the second term is the additional benefit arising from the ability of the asset to relax the financing
constraint. Equation (32) can then be rewritten as
Qt = E
1 + λt
St+1
(1 − σ + σµt+1 )
ψt+1 (Zt+1 + (1 − δ) Qt+1 ) | Ft
St
µt + λt θ k
{z
}
|
{z
}|
payoff in period t + 1
‘stochastic discount factor’
.
(34)
Notice that each unit of capital owned at the end of period t, with price Qt , is a claim on a
dividend cash flow given by
Gt+j = Zt+j (1 − δ)
j−1
j
Y
ψt+m ,
j ≥ 1,
m=1
and thus we the cash flow corresponds to a multiplicative functional with
log Gt+1 − log Gt = log (1 − δ) + log ψt+1 + log Zt+1 − log Zt .
Equation (34) is an Euler equation that prices this cash flow. Define
St+1 θ k . St+1
1 + λt
=
(1 − σ + σµt+1 )
k
St (θ )
St
µt + λt θ k
(35)
This is the ‘stochastic discount factor’ that is used to discount cash flows from the capital stock
k. Notice that is not a stochastic discount factor in the true sense because it is asset-specific; it
incorporates the non-monetary benefit of capital in the form of its ability to relax the financial
constraint. This ability is the stronger, the smaller the value of θ k . When the financial constraint
37
CRRA preferences
consumption shock-exposure
investment shock-exposure
0.025
0.075
0.02
0.05
0.025
0.015
0
0.01
−0.025
0.005
0
−0.05
0
10
20
30
40
50
60
70
80
90
100
−0.075
0
10
20
0.04
0.04
0.02
0.02
0
0
−0.02
0
10
20
30 40 50 60 70
maturity (quarters)
40
50
60
70
80
90
100
80
90
100
profit shock-exposure
dividend shock-exposure
0.06
−0.02
30
80
90
100
−0.04
0
10
20
30 40 50 60 70
maturity (quarters)
Figure 1: Shock-exposure elasticities for the Gertler and Kiyotaki (2010) model. The blue solid
line corresponds to the capital quality shock ψ, the red dashed line to the neutral technology shock
a. Shock elasticities are annualized.
is not binding, λt = 0, and the value of θ k is irrelevant. On the other hand, when λt > 0, identical
cash flows with heterogeneous parameters θ k will be priced differently. This does not constitute an
arbitrage opportunity because of the binding financial constraint. Finally, in a frictionless world,
we would have λt ≡ 0 and µt ≡ 0, and thus the stochastic discount factor that the bank is using to
price assets would be identical to the household’s stochastic discount factor.
11.2
Asset pricing implications
Figures 1 and 2 show the shock-exposure and shock-price elasticities for equilibrium cash flows in
the model.
The first panel in Figure 1 shows the shock-exposure elasticity of aggregate consumption. The
model produces the typical hump-shaped responses, and the shock-exposure elasticity for consumption is an input into the household’s stochastic discount factor.
Figure 2 plots the shock-price elasticities for the cash flows priced under the two stochastic
discount factors. The household’s stochastic discount factor produces small shock-price elasticities,
which is consistent with the low consumption volatility and the low level of risk aversion in the
38
bank’s stochastic discount factor for capital
household’s stochastic discount factor
0.1
0.1
0.08
0.08
0.06
0.06
0.04
0.04
0.02
0.02
0
0
10
20
30
40
50
60
70
80
90
0
100
0
10
20
30
40
50
60
70
80
90
100
Figure 2: Shock-price elasticities for the Gertler and Kiyotaki (2010) model. The blue solid line
corresponds to the capital quality shock ψ, the red dashed line to the neutral technology shock a.
The left panel depicts the shock-price elasticities constructed using the stochastic discount factor
S h of the household, while the right panel shows the shock-price elasticities for the stochastic
discount factor S(θ k ) used by the bank to price the claim to physical capital. Shock elasticities are
annualized.
model. Consistent with the absence of a permanent shock, the shock-price elasticities decay to zero
as maturity increases. The household’s stochastic discount factor has no permanent component in
the martingale factorization constructed in Section 6.
The bank’s stochastic discount factor tells a very different story. Shock-price elasticities are
much higher and do not decay to zero. The reason is the binding financial constraint whose
Lagrange multiplier is embedded in the bank’s stochastic discount factor. First, this constraint
magnifies the volatility of the stochastic discount factor, and second, it introduces a permanent
component to valuation through a nonseparability in the stochastic discount factor.
39
Appendix
A
Exponential-quadratic framework
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 (16)
X1,t+1
= Θ10 + Θ11 X1,t + Λ10 Wt+1
(36)
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 (17):
Yt+1 − Yt
= Γ0 + Γ1 X1,t + Γ2 X2,t + Γ3 (X1,t ⊗ X1,t ) +
(37)
+Ψ0 Wt+1 + Ψ1 (X1,t ⊗ W1,t+1 ) + Ψ2 (Wt+1 ⊗ Wt+1 ) ,
we can write the conditional expectation of M as
¯0
log E [Mt | F0 ] = Γ
t
¯1
+ Γ
t
¯2
X1,0 + Γ
t
¯3
X2,0 + Γ
t
(X0 ⊗ X0 )
(38)
¯ i are constant coefficients to be determined.
where Γ
t
The dynamics given by (36)–(37) embeds the perturbation approximation constructed in Section 8.2 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 (¯
x, 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 .
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.
40
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 ) .
(39)
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
n
(BW ) ⊗ (AX) = [B ⊗ A•j ]j=1 (X ⊗ W )
where
n
[B ⊗ A•j ]j=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 individual
components with k-th row
′
∂ 2 f (k)
(x,
w)
[fij (x, w)]k• = vec
.
∂j∂i′
It follows from formula (39) that, for example,
′
x
′
∂ 2 f (k)
∂ 2 f (k)
(x, w) w = vec
(x, w) (x ⊗ w) = [fxw (x, w)]k• (x ⊗ w) .
∂x∂w′
∂w∂x′
41
A.3
Conditional expectations
Notice that a complete-the squares argument implies that, for a 1 × k vector A, a 1 × k 2 vector B, and a
scalar function f (w),
E [exp (B (Wt+1 ⊗ Wt+1 ) + AWt+1 ) f (Wt+1 ) | Ft ] =
(40)
1 ′
W
(matk,k (2B)) Wt+1 + AWt+1 f (Wt+1 ) | Ft
= E exp
2 t+1
1
−1/2
−1 ′
˜ [f (Wt+1 ) | Ft ]
= |Ik − sym [matk,k (2B)]|
exp
A (Ik − sym [matk,k (2B)]) A E
2
where ˜· is a measure under which
−1
−1
.
Wt+1 ∼ N (Ik − sym [matk,k (2B)]) A′ , (Ik − sym [matk,k (2B)])
We start by utilizing formula (40) to compute
Y¯ (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
−1
− log |Ik − sym [matk,k (2Ψ2 )]| + µ′ (Ik − sym [matk,k (2Ψ2 )]) µ
2
2
=
with µ defined as
µ = Ψ′0 + [matk,n (Ψ1 )] X1,t .
Reorganizing terms, we obtain
¯0 + Γ
¯ 1 X1,t + Γ
¯ 2 X2,t + Γ
¯ 3 (X1,t ⊗ X1,t )
Y¯ (Xt ) = Γ
(41)
where
¯0
Γ
=
¯1
Γ
¯2
Γ
=
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 )]
=
Γ2
¯3
Γ
=
Γ0 −
(42)
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 (42) define a mapping
P¯ = E¯ (P) ,
¯ j = 0. We now substitute the law of motion for X1 and X2 to produce Y¯ (Xt ) = Y˜ (Xt−1 , Wt ). It
with all Ψ
42
is just a matter of algebraic operations to determine that
Y˜ (Xt−1 , Wt ) =
=
log E [exp (Yt+1 − Yt ) | Ft ] =
˜0 + Γ
˜ 1 X1,t−1 + Γ
˜ 2 X2,t−1 + Γ
˜ 3 (X1,t−1 ⊗ X1,t−1 )
Γ
˜ 0 Wt + Ψ
˜ 1 (X1,t−1 ⊗ Wt ) + Ψ
˜ 2 (Wt ⊗ Wt )
+Ψ
where
˜0
Γ
˜1
Γ
=
˜2
Γ
˜3
Γ
=
˜0
Ψ
=
˜1
Ψ
=
˜2
Ψ
=
=
=
¯0 + Γ
¯ 1 Θ10 + Γ
¯ 2 Θ20 + Γ
¯ 3 (Θ10 ⊗ Θ10 )
Γ
¯ 1 Θ11 + Γ
¯ 2 Θ21 + Γ
¯ 3 (Θ10 ⊗ Θ11 + Θ11 ⊗ Θ10 )
Γ
¯ 2 Θ22
Γ
¯ 2 Θ23 + Γ
¯ 3 (Θ11 ⊗ Θ11 )
Γ
¯ 1 Λ10 + Γ
¯ 2 Λ20 + Γ
¯ 3 (Θ10 ⊗ Λ10 + Λ10 ⊗ Θ10 )
Γ
h
in ¯ 2 Λ21 + Γ
¯ 3 Θ11 ⊗ Λ10 + Λ10 ⊗ (Θ11 )
Γ
•j
j=1
¯ 2 Λ22 + Γ
¯ 3 (Λ10 ⊗ Λ10 ) .
Γ
This set of equations defines the mapping
A.4
(43)
P˜ = E˜ P¯ .
Iterative formulas
We can write the conditional expectation in (38) recursively as
Mt
| F1 | F0 .
log E [Mt | F0 ] = log E exp (Y1 − Y0 ) E
M1
˜ we can therefore express the coefficients P¯ in (38) using the recursion
Given the mappings E¯ and E,
P¯t = E¯ P + E˜ P¯t−1
where the addition is by coefficients and all coefficients in P¯0 are zero matrices.
A.5
Coefficients Φ∗i
In the above calculations, we constructed a recursion for the coefficients in the computation of the conditional
expectation of the multiplicative functional M . A single iteration of this recursion can be easily adapted
to compute the coefficients Φ∗i , i = 0, . . . 3, in the conditional expectation in equation (19) for an arbitrary
function log f (x).
1. Associate log f (xt+1 ) = Y (xt+1 ) from equation (41), i.e., set Γi , i = 0, . . . , 3, in equation (41) equal
to the desired Φi from equation (18). These are the coefficients in set P.
e i , i = 0, . . . , 3 and Ψ
e i , i = 0, 1, 2 using (43). This yields the
2. Apply the mapping Ee P , i.e., compute Γ
e
function log fe(xt , wt+1 ) ≡ log f (xt+1 ), with coefficient set P
e i and Ψ
e i the corresponding coefficients Γi and Ψi of Yt+1 − Yt from equation
3. Add to these coefficients Γ
(37), i.e., form coefficient set P + Ee P .
43
4. Apply the mapping E P + Ee P , i.e., compute (42) where on the right hand side the coefficients
Γi and Ψi (coefficient set P) are replaced with coefficients computed in the previous step, i.e., set
P + Ee P .
5. The resulting coefficients Γi , i = 0, . . . , 3, are the desired coefficients Φ∗i .
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 8.1.1, utilize the formulas derived in Appendix A to deduce the
one-period change of measure
log L1,t = log M1 + log E
Mt
Mt
| X1 − log E M1 E
| X1 | X0 = x .
M1
M1
In particular, following the set of formulas (43), define
µ0,t
µ1,t
µ2,t
′
Ψ1 + Φ∗1,t−1 Λ1,0 + Φ∗2,t−1 Λ20 + Φ∗3,t−1 (Θ10 ⊗ Λ10 + Λ10 ⊗ Θ10 )
h
in = matk,n Ψ1 + Φ∗2,t−1 Λ21 + Φ∗3,t−1 Θ11 ⊗ Λ10 + Λ10 ⊗ (Θ11 )•j
j=1
¯
¯
= sym matk,k Ψ2 + Γ2 Λ22 + Γ3 (Λ10 ⊗ Λ10 ) .
=
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 (40) then implies that
e [W1 | F0 ] =
E [L1,t W1 | F0 ] = E
−1
= (Ik − 2µ2,t )
(µ0,t + µ1t X1,0 ) .
The variance of W1 under the ˜· measure satisfies
B.2
¯ 2 Λ22 + Γ
¯ 3 (Λ10 ⊗ Λ10 ) −1 .
e t = Ik − 2sym matk,k Ψ2 + Γ
Σ
Approximation of the shock elasticity function
In Section 8.5, we constructed the approximation of the shock elasticity function ε (x, t). The first-order
approximation is constructed by differentiating the elasticity function under the perturbed dynamics
E [Mt (q) W1 | X0 = x] d
η(X0 (q)) ·
= η (¯
x) · E [Y1,t W1 | X0 = x] .
ε1 (X1,0 , t) =
dq
E [Mt (q) | X0 = x] q=0
44
The first-derivative process Y1,t can be expressed in terms of its increments, and we obtain a state-independent
function
′

t−1
X
j−1
ε1 (t) = η (¯
x) · E 
κx (ψx )
ψw + κw 
j=1
where κx , ψx , κw , ψw are derivative matrices evaluated at the steady state (¯
x, 0).
Continuing with the second derivative, we have
ε2 (X1,0 , X2,0 , t) =
=
E [Mt (q) W1 | X0 = x] d2
η(X
(q))
·
=
0
dq2
E [Mt (q) | X0 = x] q=0
i
o
n h
η (¯
x) · E (Y1,t )2 W1 + Y2,t W1 | F0 − 2E [Y1,t W1 | F0 ] E [Y1,t | F0 ] +
∂η
+2
(¯
x) X1,0 · E [Y1,t W1 | F0 ] .
∂x′
However, notice that
h
E (Y1,t )2 W1 | F0
i
=
′


t−1
t−1
X
X
κx (ψx )j−1 ψw + κw 
2
κx (ψx )j X1,0  
j=1
j=0
E [Y1,t W1 | F0 ] =
E [Y1,t | F0 ] =


t−1
X
j−1
κx (ψx )
j=1
t−1
X
j
′
ψw + κw 
κx (ψx ) X1,0
j=0
and thus
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
∂η
ε2 (X1,0 , X2,0 , t) = η (¯
x) · E [Y2,t W1 | F0 ] + 2
(¯
x) X1,0 · E [Y1,t W1 | F0 ] .
∂x′
(44)
The expression for the first term on the right-hand side is

E [Y2,t W1 | F0 ] = E 
+2
t−1
X
j=0

(Y2,j+1 − Y2,j ) W1 | F0  = 2matk,n (κxw ) X1,0 +
t−1 h
X
j=1
+2
j−1
′
ψw
(ψx′ )
ii
h
matn,n (κxx ) (ψx )j + matk,n κx (ψx )j−1 ψxw X1,0
j−1 h
t−1 X
h
i
i
X
k−1
j−k−1
k
′
ψw
(ψx′ )
matn,n κx (ψx )
ψxx (ψx ) X1,0 .
j=1 k=1
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 .
45
C
Gertler and Kiyotaki (2010) model with financial constraints
Full description of Gertler and Kiyotaki (2010).
46
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