数理解析研究所講究録
第 1788 巻 2012 年 107-126

107

Macroeconomics and Finance: How Dynamic
Macroeconomic Models Treat Finance
Ippei Fujiwara
Crawford School of Economics and Government#132 Lennox Crossing
Australian National University
Canberra, ACT 0200*

AUSTRALIA
January 20121

Introduction

1.1

Focus of Macroeconomics$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$

The Great Depression
ily equilibrate.

$\Rightarrow$

Keynesian view that markets may not read-

The Great Ination
importance of aggregate supply shocks andspurred real business cycle research.
$\Rightarrow$

The Great $Di\sin flation\Rightarrow$ New Keynesianism recognizing the potencyof monetary policy.

The Great Moderation
dynamic stochastic general equilibrium(DSGE) model as a macroeconomic orthodoxy.
$\Rightarrow$

The Great Panic and Recession of 2008 and $2009\Rightarrow??$ ?

Australian National University, CAMA, EABCN, and Globalization and Monetary
Policy Insititute. Email: ippe . fujiwara@anu.edu.au.
$*$

$i$

108
1.2
$\bullet$$\bullet$$\bullet$

1.3$\bullet$$\bullet$$\bullet$$\bullet$Aim of This Presentation
To show a subjective summary of macroeconomic attempts to explain
risk premium, particularly equity premium and term premium in dynamic macroeconomic framework.

A key feature of recent events has been the close feedback between thereal economy and financial conditions.
Joint analysis of macroeconomy and risk premium, in addition to thefinancial frictions and macroeconomy, can be a central focus of macroeconomics.

Structure of Presentation
Introduction to the Dynamic Macroeconomics

Equity Premium Puzzle
Macro-Finance: Joint Explanation of Term Structure and Macroeconomy

Summary

2

DSGE Model

2.1

History$\bullet$Neoclassical growth model, namely Ramsey (1928) –Cass (1965)
Koopmans (1965) model (stochastic extension, Brock and Mirman,
optimal saving1972), that is Solow (1956) –Swan (1956) model
choice, was thought to be very theoretical.

-$+$$\bullet$

Kydland and Prescott (1982) use this neoclassical growth model for
positive analysis.

$\bullet$$\bullet$Neoclassical model calibrated to explain the data is then called RealBusiness Cycle (RBC) models.

Macroeconomics has been increasingly paying more attention to fitting
the data.

109$\bullet$

RBC models has developed to explain the data better by incorporatingsuch mechanisms as indivisible labor (Hansen, 1985, Rogerson, 1988),
tax distortions (Braun, 1994, McGrattan, 1994), capacity utilization(Greenwood, Hurcowitz and Hoffman, 1988), labor hoarding (Burnside,
Eichenbaum and Rebelo, 1996), home production (Benhabib, Rogerson
and Wright, 1991), or Labor search and matching (Merz, 1995, Andolfatto, 1996).

$\bullet$

New Keynesian Model, further incorporating nominal rigidities, is now
the workhorse model (especially, Christiano, Eichenbaum and Evans,
2005, and Smets and Wouters, 2003) in many policy institutions.

Simplest Example

2.2$\bullet$

Simple RBC Model
The social planner maximizes$E_{0}\sum_{t=0}^{\infty}\beta^{t}[\log(C_{t})-\frac{\chi}{2}h_{t}^{2}]$,subject to$Y_{t}=[\exp(z_{t})h_{t}]^{1-\alpha}K_{t}^{\alpha}$$C_{t}+I_{t}=Y_{t}$,,$K_{t+1}=(1-\delta)K_{t}+I_{t}$,and$z_{t}=\rho_{z}z_{t-1}+u_{t}$$\bullet$$\bullet$$\bullet$,

Uniqueness of the equilibrium path in this economy can be proved byapplying contraction mapping theorem to functional space.

Note that by specifying functional forms and shocks, the model can fit
the data.
First order necessary conditions are$\frac{1}{C_{t}}=\beta E_{t}\frac{1}{C_{t+1}}\{(1-\delta)+\alpha[\exp(z_{t+1})h_{t+1}]^{1-\alpha}K_{t+1}^{a-1}\}$$(1-\alpha)[\exp(z_{t})h_{t}]^{-\alpha}\exp(z_{t})K_{t}^{a}=\chi h_{t}C_{t}$,and$C_{t}+K_{t+1}-(1-\delta)K_{t}=[\exp(z_{t})h_{t}]^{1-\alpha}K_{t}^{a}$

.

,

110
Usually, the model is log-linearly approximated as$\bullet$$E_{t}\hat{C}_{t+1}$$\hat{C}_{t}$$=$,$+(1-\alpha)[1-\beta(1-\delta)](E_{t}z_{t+1}+E_{\eta}\hat{h}_{t+1}-\hat{K}_{t+1})$,$\alpha\hat{K}_{t}-\hat{C}_{t}-(1+\alpha)\hat{h}_{t}+(1-\alpha)z_{t}=0$

and$\frac{1-\beta(1-\delta)-\alpha\beta\delta}{\alpha\beta}\hat{C}_{t}+\hat{K}_{t+1}-(1-\delta)\hat{K}_{t}$$\frac{1-\beta(1-\delta)}{\alpha\beta}[(1-\alpha)z_{t}+(1-\alpha)\hat{h}_{t}+\alpha\hat{K}_{t}]$

$=$

,

where
$\log(\frac{X_{t}}{X})\approx\frac{X_{t}-X}{X}\equiv\hat{x}_{t}$$\bullet$.

This system can be expressed in the state space form:$A$,$(\begin{array}{l}E_{\eta}\hat{C}_{t+l}\hat{K}_{t+1}\end{array})=B(\begin{array}{l}\hat{C}_{t}\hat{K}_{t}\end{array})+CE_{t}z_{t+1}+Dz_{t}$or$(\begin{array}{l}E_{t}\hat{C}_{t+1}\hat{K}_{t+l}\end{array})=A^{-1}B(\begin{array}{l}\hat{C}_{t}\hat{K}_{t}\end{array})+A^{-1}CE_{t}$$\bullet$銑 $+$ 1 $+A^{-1}Dz_{t}$With eigendecomposition:$A^{-1}B=(\begin{array}{ll}v_{1l} v_{12}v_{2l} v_{22}\end{array})(\begin{array}{ll}\lambda_{l} 00 \lambda_{2}\end{array})(\begin{array}{ll}v_{ll} v_{12}v_{21} v_{22}\end{array})$

we now have$(\begin{array}{ll}v_{l1} v_{l2}v_{21} v_{22}\end{array})(\begin{array}{l}E_{\eta}\hat{C}_{t+l}\hat{K}_{t+l}\end{array})$$=(\begin{array}{ll}\lambda_{l} 00 \lambda_{2}\end{array})(\begin{array}{ll}v_{11} v_{l2}v_{2l} v_{22}\end{array})(\begin{array}{l}\hat{C}_{t}\hat{K}_{t}\end{array})$,$+(\begin{array}{l}c_{l}c_{2}\end{array})E_{t}z_{t+1}+(\begin{array}{l}d_{l}d_{2}\end{array})z_{t}$

,

.

111
where
$(\begin{array}{l}c_{1}c_{2}\end{array})=(\begin{array}{ll}v_{1l} v_{12}v_{21} v_{22}\end{array})A^{-1}C$,$(\begin{array}{l}d_{1}d_{2}\end{array})=(\begin{array}{ll}v_{11} v_{12}v_{2l} v_{22}\end{array})A^{-1}D$.

and$\bullet$

The first row is$v_{11}E_{t}\hat{C}_{t+1}+v_{12}\hat{K}_{t+1}=\lambda_{1}(v_{11}\hat{C}_{t}+v_{12}\hat{K}_{t})+c_{1}E_{t}z_{t+1}$$\bullet$.

If$|\lambda_{1}|>1$(1),$(v_{11}\hat{C}_{t}+v_{12}\hat{K}_{t})$$=- \frac{c_{1}}{\lambda_{1}}E_{t}z_{t+1}-\frac{c_{1}}{\lambda_{1}^{2}}E_{t}z_{t+2}-\ldots-\lim_{Tarrow\infty}\frac{c_{1}}{\lambda_{1}^{T}}E_{t}z_{t+T}$$=- \frac{\lambda_{1}c_{1}}{\lambda_{1}-\rho_{z}}z_{t}$,and therefore.

(2)$\hat{C}_{t}=-\frac{v_{12}}{v_{11}}\hat{K}_{t}-\frac{\lambda_{1}c_{1}}{v_{11}(\lambda_{1}-\rho_{z})}z_{t}$$\bullet$The second row is.

$v_{21}E_{t}\hat{C}_{t+1}+v_{22}\hat{K}_{t+1}=\lambda_{2}(v_{21}\hat{C}_{t}+v_{22}\hat{K}_{t})+d_{2}z_{t}$$\bullet$If$|\lambda_{2}|<1$$\hat{K}_{t+1}=\omega_{2}\hat{K}_{t}+\frac{v_{11}d_{2}-v_{21}\lambda_{2}c_{1}}{v_{11}v_{22}-v_{21}v_{12}}z_{t}$$\bullet$$\bullet$

(3)

,

.

(4)

Equations (2) and (4) are solutions. Conditions (1) and (3) are usuallysatisfied thanks to $\beta<1$ .

Parameters can be estimated by applying Kalman filter for the state
space model consisting of equations (2) and (4).

112

2.3
$\bullet$$\bullet$$\bullet$$\bullet$

Implications to Finance-Villaverde and RubioThere exist several exceptions such as Fern
positive
analysis do not conRam\’irez (2005), but standard models forsider higher order terms.
$\mathfrak{W}deZ$

Therefore, certainty equivalence holds and no risk premium is considered in the model.

Unconditional expected returns for any asset must be equated.
Several new challenges are made recently with increasing importanceof understanding risk premium.

Equity Premium Puzzle

3$\bullet$$\bullet$$\bullet$$\bullet$

Mehra and Prescott (1985) argue a particular empirical problem of the
RBC model.

According to Kocherlakota (1996), “Over the last one hundred years,
the average real return to stocks in the United States has been about
six percent per year higher that on Reasury bills. At the same time,
the average real return on treasury bills has been about one percent
per year.”
Returns on assets can be different depending on the equity premium,
namely the degree to which each asset retum covaries with the consumption.
Yet, the size of the observed equity premium is only justified when risk

aversion is set incredibly high.

Risk Free Rate Puzzle$\bullet$$\bullet$

Weil (1989) shows that this fact also implies the second puzzle.

According to Kocherlakota (1996), “although Treasury bills offer only
a low rate of return, individuals defer consumption at a sufficiently
fast rate to generate average per capita consumption growth of around
two percent per year... although individual like consumption to bevery smooth, and although the risk free rate is very low, they still saveenough that per capita consumption grows rapidly.”

113
3.1
$\bullet$

CRRA Preference
Under CRRA preference, which has been intensively used in macroeconomics,$u(C_{t})= \frac{C_{t}^{1-\alpha}}{1-\alpha}$where$\bullet$$\alpha$,is the coefficient of relative risk aversion.

Then, we maximize the life-time welfare:$E_{0}\sum_{t=0}^{\infty}\beta^{t}u(C_{t})$,subject to the budget constraint:$C_{t}+s_{t}+B_{t}\leq R_{\tau}^{s}S_{t-1}+$$\bullet$瑠 $B_{t-1}+Y_{t}$ .

Then, we can derive the optimality (absence of arbitrage) conditions
for bond and stock holdings are expressed$E_{t}[(\frac{C_{t+1}}{C_{t}})^{-\alpha}(R_{\iota+1}^{s}-R_{t+1}^{b})]=0$

,

(5)

and
$E_{t}\beta[(\frac{C_{t+1}}{C_{t}})^{-\alpha}R_{t+1}^{b}]=1$,

where the pricing kernel or stochastic discount factor is defined as$M_{t,t+1}= E_{t}\beta(\frac{C_{t+1}}{C_{t}})^{-\alpha}$$\bullet$.

Test results on equations (5) and (6) are shown as below.

(6)

114

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me $Rarrow^{\dot}$ ti$!m 1^{1}$エ–$r\cdot n----\cdot\propto\infty$$oe_{*}-1)A\cdot\approx ARdarrow*$$-$ $-$
伽』$-$ 轟の–

– —$\cdot\cdot$.

$arrow e\sim r--\sim$$H4*\mapsto-arrow\mapstoarrow b\infty$$-arrowarrow\ddot{arrow}\epsilon_{\ddot{r}--}^{nm}$$\underline{--}\underline{r_{\infty m}-}\cdot\cdot--\sim--\underline{t}$$rhr\alpharightarrow m-\sim\cdot-\cdotarrow$$-arrow\infty\sim$ 幽– $Rr\infty$$\bullet$$\bullet$$\underline{r1}-\underline{\infty}--_{\dot{rightarrow}d}r_{\infty}$

Correlation between consumption growth and equity returns observed
from the data is too small for the sizable equity premium.
There have been several attempts to solve equity premium puzzle.

1. Preference Modification

2. Low Requency Event
3. Incomplete Market

3.2$\bullet$$\bullet$Preference Modification
Habit Formation
Epstein-Zin Preference

Habit$\bullet$$\bullet$

What happens if we become more risk averse when the level of consumption decreases in a recession?
A model of time varying risk aversion, where the risk aversion depends
on the level of consumption, may resolve the equity premium and therisk free rate puzzles.

115$\bullet$Abel (1990) and Gali (1994) propose a preference with habit formationas$U_{t}= \frac{c_{t}^{1-\alpha}C_{t}^{\gamma}C_{t-1}^{\lambda}}{1-\alpha}$

,

where
is the aggregate consumption and taken as given when optimizing. We assume that agents tend to be jealous and. Therefore, this preference reflects the idea by Duesenberry (1949), “keeping
up with Joneses.”$C_{t}$$\gamma,$$\bullet$$\lambda<0$

Under this preference, optimality conditions for bond as well as stock
holdings in equations (5) and (6) are now rewritten as,$E_{t}[(\frac{C_{t+1}}{C_{t}}I^{\gamma-\sigma}(\frac{C_{t}}{C_{t-1}})^{\lambda}(R_{t+1}^{s}-R_{t+1}^{b})]=0$

and$E_{t}\beta[(\frac{C_{t+1}}{C_{t}})^{\gamma-\sigma}(\frac{C_{t}}{C_{t-1}})^{\lambda}R_{t+1}^{b}]=1$$\bullet$$\bullet$.

With reasonable parameters, the model solves the equity premium puzzle but not the risk free rate puzzle.
When , which is supposed to be negative, is large in absolute value,
marginal utility of own consumption is highly sensitive to fluctuations
in per capita consumption and therefore strongly negatively correlated
to stock returns.
$\gamma$

3.3$\bullet$$\bullet$Epstein-Zin PreferenceWith the standard CRRA preference, the risk aversion is the reciprocalof the IES.
Epstein and Zin (1989, 1991) propose a preference which disentangles
the risk aversion and the IES :$\psi$$\gamma$$U_{t}=\{C_{t}^{1-\frac{1}{\psi}}+\beta[E_{t}(U_{t+1}^{1-\gamma})]^{\frac{1-\varpi 1}{1-\gamma}}\}^{\frac{1}{1_{\phi}^{1}-}}$$\bullet$.

When the risk aversion equals the reciprocal of the IES
lapses to the CRRA utility function.
$\gamma$$\frac{1}{\psi}$, this col-

116$\bullet$$\bullet$

High risk aversion toghether with high IES is possible. This is a goodnews for risk free rate puzzle.

Optimality conditions for bond and stock holdings in equations (5) and(6) are now$E_{t}[U_{t1}^{\frac{1}{\psi+}-\gamma}(\frac{C_{t+1}}{C_{t}})^{-\frac{1}{\psi}}(R_{t+1}^{s}-R_{t+1}^{b})]=0$,

and$E_{t}\beta[(E_{t}U_{t+1}^{1-\gamma})^{\frac{\gamma-\not\simeq}{1-\gamma}}U_{t+1}^{\frac{1}{\psi}-\gamma}(\frac{C_{t+1}}{C_{t}})^{-\frac{1}{\psi}}R_{t+1}^{b}]=1$$\bullet$$\bullet$.

The existence of the unobservable expected utility may solve both puzzles.

Kocherlakota (1996), however, claims that if consumption growth isnot predictable, equations (5) and (6) hold again under unconditional
expectation.

3.4$\bullet$$\bullet$

Low Frequency Event
Long-Run Risk

Disaster ShockLong-Run Risk: Bansal and Yaron (2004)$\bullet$Consider again Epstein-Zin Preference$U_{t}=\{C_{t}^{\frac{1-\gamma}{\theta}}+\beta[E_{t}(U_{t+1}^{1-\gamma})]^{\frac{1}{\theta}}\}^{\frac{\theta}{1-\gamma}}$

,

where
$\theta=\frac{1-\gamma}{1-\frac{1}{\psi}}$$\bullet$$\bullet$.

When $\theta=1$ , the agent is indifferent to the timing of the resolution of
the uncertainty of the consumption path.
When $\theta<1$ , namely risk aversion is larger than the reciprocal of the, the agent prefers early resolution.
IES$\gamma$$1/\psi$$\bullet$When $\theta>1$ , namely risk aversion is smaller than the reciprocal of, the agent prefers late resolution.
the IES$\gamma$$1/\psi$

117$\bullet$

The logarithm of the stochastic discount factor for the$\log(C_{t})$ , is$m_{t,t+1}=\log(\beta)+\triangle c_{t+1}$$\bullet$$\log$

utility: $U_{t}=$.

On the other hand, that of Epstein-Zin preference is expressed as(7)$m_{t,t+1}$$=$$\log(\beta)+\triangle c_{t}$$-( \gamma-1)\sum_{i=0}^{\infty}(E_{t+1}-E_{t})\triangle c_{t+j+1}$$- \frac{1}{2}(\gamma-1)’ var_{t}[\sum_{i=0}^{\infty}\beta^{j}(E_{t+1}-E_{t})\Delta c_{t+j+1}]$$\bullet$$\bullet$$\bullet$.

Bansal and Yaron (2004) suppose that the agent prefers early resolution
in their model and specify the long-run risks, that is the shocks to thelong-run component of consumption, and time varying risk.
This long-run model can resolve both the equity premium and risk
free rate puzzles thanks to the effects from expected growth rate of
consumption and time varying risks on the stochastic discount factor
in equation (7).
Question is whether we can statistically distinguish stochastic specffication with long-run risk from one without it.

Disaster Shock: Rietz (1988)$\bullet$$\bullet$$\bullet$$\bullet$

Mehra and Prescott (1985) assume that consumption growth rates are
symmetric about their mean and they fall above their mean as often as
they fall below.

Rietz (1988) supposes a situation when consumption fall drastically andequity returns are far below average by incorporating a low-probability,
depression-like state as an additional state in the Markov chain.
Introduction of disaster risk has now become more interested than before due to the recent financial crisis.
Robert Barro and his coauthors now estimate the disaster risk from
historical data.

118
3.5
$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$

3.6$\bullet$

Incomplete Market
When individual cannot write contracts against any contingency, individual consumption growth can be more covary with stock returns thanper capita consumption growth. This may solve the equity premium
puzzle.

Under such situation, precautionary saving motive stemming from selfinsurance may result in lower natural rate of interest. This may solve
the risk free rate puzzle.
Previous studies, however, report that incomplete market qualitatively
alleviate these puzzles but not solve them quantitatively.

Huggett (1993) and Heaton and Lucas (1995) show that under realistic
calibration, the difference between the incomplete markets interest rate
and the complete markets interest rate is small. Most individuals are
sufficiently far from the debt ceiling.

Constantinides and Duffie (1995) show analytical solution under incomplete market by assuming that the idiosyncratic shocks are permanent.
This could reduce equilibrium real interest rates and increase equity
premium.
Heaton and Lucas (1995), however, find again with the model calibrated
to match the data that difference in interest rates between complete andincomplete markets is small.

Marcet and Singleton (1991), Telmer (1993), Lucas (1994), den Haan
(1994), and Heaton and Lucas (1995) show that allocations under incomplete markets are almost the same as those under complete markets.
Krueger and Lustig (2010) provide various examples of economies inwhich uninsurable income shocks do not matter for the equity premium.

Rich Households
According to Ait-Sahalia, Parker and Yogo (2004), “The risk aversion
implied by the consumption of luxury goods is more than an order ofmagnitude less than that implied by national accounts data. For thevery rich, the equity premium is much less of a puzzle.”

119$\bullet$$\bullet$4

Parker and Vissing-Jorgensen (2009) document that the consumptionof rich households is over five times more volatile than aggregate consumption, which may help to explain average premiums in financial
markets.
Chien, Cole, and Lustig (2009) build a model in which a large fractionof households do not rebalance their portfolios in response to aggregate
shocks. As a consequence, households who do rebalance need to sellmore stocks in good times and buy more stocks in bad times. Thismechanism generates time variation in risk premiums.

Macro-Finance$\bullet$$\bullet$Macro-finance research has examined the relationship between the term
structure of interest rates and the macro-economy in an interdisciplinary fashion.
Belows are only selective survey of this field:

1. Structural VAR Models
2. Affine Term Structure Models
3. DSGE Models
4. Nelson-Siegel Models

4.1$\bullet$

Structural VAR Model
Evans and Marshall (1988) estimate a structural VAR model as$(\begin{array}{ll}a 0c 1\end{array})(\begin{array}{l}Z_{t}R_{j,t}\end{array})$$=$$(\begin{array}{ll}A(L) 0C(L) D(L)\end{array})(\begin{array}{l}Z_{t}R_{j,t}\end{array})$$+\sigma(\begin{array}{l}\epsilon_{t}^{Z}d_{t}\end{array})$$\bullet$is a vector of endogenous variables including policy interest rates.
is a bond yield for maturity . Variables inare determined independently frombased on the idenfiticaitons imposed by previous
studies.
$Z_{t}$$R_{j,t}$$j$$Z_{t}$$R_{j,t}$$\bullet$They assume that yields should be linearly estimated by macroeconomic variables.

120
4.2
$\bullet$$\bullet$$\bullet$$\bullet$Affine Term Structure Model
Since Vasicek (1977), various extensions have been made to affine term
structure model to better explain the data.
Multi unobserved factor model proposed by Duffie and Kan (1996) is
widely used with high explanatory power of the yield curve.

Ang and Piazzessi (2003) incorporate macroeconomic variables in addition to unobserved factors and show that fit is improved.
Ichiue and Ueno (2006) extend Black (1995) and propose a term structure model with explicit considerations to the zero bound constraint.

Duffie and Kan (1996)$\bullet$

Assume that spot risk free rate is expressed as the affine function ofstate variables:$r_{t}=\delta_{0}+\delta_{1}’X_{t}$$\bullet$

Assume that state variables under bothGaussian processes as

$P$

.

(8)

and

$Q$

$X_{t+1}=K^{Q}X_{t}+\Sigma_{X}\epsilon_{t+1}^{Q}$

measures follows such,

and$X_{t+1}=a^{P}+K^{P}X_{t}+\Sigma_{X}\epsilon_{t+1}^{P}$$\bullet$.

(9)

By modeling the market price of risk also as the affine function:$\Lambda_{t}=\lambda_{0}+\Lambda_{1}X_{t}$,we have$a^{P}=\Sigma_{X}\lambda_{0},$$K^{P}=K^{Q}+\Sigma_{X}\Lambda_{1}$,since the market price of risk should satisfy$\epsilon_{t}^{P}=\epsilon_{t}^{Q}-\Lambda_{t}dt$$\bullet$.

The Bond prices are also expressed as the affine function:$p_{t}^{(m)}=\exp(A_{m}+B_{m}’X_{t})$and therefore, the yields are$y_{t}^{(m)}=- \frac{A_{m}}{m}-\frac{B_{m}’X_{t}}{m}$

.

,

121
where
$B_{0}=0$,$B_{m}=-\delta_{1}’+K’B_{m-1}$$A_{0}=0$

,

,

and
$A_{m}=A_{m-1}- \delta_{0}+\frac{1}{2}B_{m}’\Sigma_{X}\Sigma_{X}’B_{m}$$\bullet$.

Note that we can compute term premium because we have yields underboth $P$ and measures.
$Q$

Kalman Filter$\bullet$Observation equation is defined as$y_{t}=A+B’X_{t}+\eta_{t}$$\bullet$.

For simplicity, when we aim at fitting 10 and 20 year zero coupon yields
with two unobserved factors, although five factor model is popular,$(\begin{array}{l}y_{t}^{(10)}y_{t}^{(20)}\end{array})$$=$$-( \frac A_{240}\frac{1}{24018^{0}}A_{120})$$-( \frac B_{240}\frac{1}{2401201}B_{120}’)(x_{t}^{2}x_{t}^{1})+\eta_{t}$$\bullet$.

This altogether with the state equation (9) are jointly estimated via
maximum likelihood.

Ang and Piazzessi (2003)$\bullet$$\bullet$

Short-rate is one of the most important driver of yield curves but reflects macroeconomic variables such as output and inflation via monetary policy rule.
Equation (8) is now rewritten as$r_{t}=\delta_{0}+\delta_{1}’X_{t}^{o}+\delta_{2}’X_{t}^{u}$$\bullet$$\bullet$.

The short rate dynamics of the term structure model can be interpreted
as a version of the Taylor rule with the errors as unobserved factors.
In other words, the pricing kernel is driven by shocks to both observed
macro factors and (uncorrelated) unobserved factors.
“Macro factors primarily explain movements at the short end and middle of the yield curve while unobservable factors still account for mostof the movement at the longend of the yield curve.”

122
4.3
$\bullet$$\bullet$$\bullet$

DSGE Models for Bond Pricing
DSGE Models for Bond Pricing is still a very much on-going project.
H\"ordahl, TYistani and Vestin (2006) combines the DSGE model withthe affine term structure model.
-Villaverde,
Rudebusch and Swanson (2008, 2009) and Binsbergen,
explaininterrelationshipbe
Koijen, and Rubio-Ramfrez (2010) aim to
tween macroeconomy and term premium in theoretically consistent$Fern\mathfrak{W}dez$manner.
$\bullet$

De Graeve, Emiris and Wouters (2009) claim that if the DSGE model
is rich enough to explain the data, first order approximated model can
explain the yield curve without relying on the term premium.

H\"ordahl, $Rtani$ and Vestin (2006)$\bullet$$\bullet$H\"ordahl, Tkistani and Vestin (2006) combine the affine term structure
model with the new Keynesian model.
They suppose that the macroeconomy is depicted by a standard newKeynesian model with lags:$\pi_{t}=\frac{\mu_{\pi}}{12}\sum_{i=1}^{12}E_{t}\pi_{t+i}+(1-\mu_{\pi})\sum_{i=1}^{3}\delta_{\pi i}\pi_{t-i}+\delta_{x}x_{t}+\epsilon_{t}^{\pi}$,12$X_{t}$$=$$\frac{\mu_{x}}{12}\sum E_{t}x_{t+i}$

$i=1$

3
$+(1- \mu_{x})\sum_{i=1}\zeta_{xi^{X_{t-i}}}+\zeta_{r}(r_{t}-E_{t}\pi_{t+11})+\epsilon_{t}^{x}$

,

and
.

$r_{t}=(1-\rho)[\beta E_{t}\pi_{t+11}-\pi_{t}^{*}+\gamma x_{t}]+\rho r_{t-1}+\eta_{t}$$\bullet$

It is also assumed unobserved inflation target:$\pi_{t}^{*}=\phi_{\pi}\pi_{t-1}^{*}+u_{\pi,t}$$\bullet$.

This system can be solved as in equations (2) and (4). Therefore, theshort-rate can be expressed by the state variables obtained from the
new Keynesian model:$r_{t}=\delta_{0}+\delta_{1}’Z_{t}$

.

(10)

123
$\bullet$4.4$\bullet$$\bullet$$\bullet$Hordahl, Tristani and Vestin (2006) jointly estimate the new Keynesian model and the affine term structure model based on equation (10)together.

Rudebusch and Swanson (2008)
Models explained above and below separate the determination of macroeconomic variables and the term premium. Macroeconomic variables
are determined irrespective of term premium.
Only consumption based model approach can theoretically explain the
joint determination of premium and macroeconomic variables.

Rudebusch and Swanson (2008) simulate a new Keynesian model withinternal habit formation with higher order approximation based on perturbation method.

$\bullet$$\bullet$$\bullet$They conclude that “standard DSGE models, even with nominal rigidities, labor market frictions, and consumption habits, appear to fallshort of being able to price nominal bonds.”
Yet, Rudebusch and Swanson (2009) report that if Epstein-Zin preference is employed, “the DSGE model is able to fit the asset pricing
facts without compromising its ability to fit the macroeconomic data.

Binsbergen, Fern\’andez-Villaverde, Koijen, and Rubio-Ram\’irez (2010)
extends this analysis with endogenous capital etc.

De Graeve, Emiris and Wouters (2009)$\bullet$$\bullet$

De Graeve, Emiris and Wouters (2009) estimate the standard DSGE
model based on Christiano, Eichenbaum and Evans (2005) or Smets andWouters (2003), which incorporates nominal price and wage rigidities,
consumption habit, investment growth adjustment cost, endogenous
capacity utilization and structurally decompose the US yield curves.
They conclude that contrary to the previous studies, the US yield curveis consistent with the expectations hypothesis under the rational expectation of the standard DSGE models.

124

4.5
$\bullet$$\bullet$Nelson-Siegel Model
Yield curves based on Nelson and Siegel (1987) model show remarkablefit.
Recent paper by Christensen, Diebold and Rudebusch (2009) construct
the Arbitrage.Free Nelson-Siegel model. With this model, we can understand the underlying factor in the context of “level,” “slope,” and“curvature.”
Litterman and Sheinkman (1991)$\bullet$$\bullet$

4.6$\bullet$Litterman and Sheinkman (1991) show with the principle component
analysis that most of the variations in the Treasury yields are explainedby only three factors.
They are interpreted as “level,” “slope,” and “curvature.”

Nelson and Siegel (1987)
Nelson and Siegel (1987) propose an yield curve model as$y(m)=L+S( \frac{1-e^{-m\lambda}}{m\lambda})+C(\frac{1-e^{-m\lambda}}{m\lambda}-e^{-m\lambda})$$\bullet$.

The factor loading for the first term is constant, that for the second
term starts at 1 and decays monotonically to and that for the third
term starts at , increases and decreases to . Therefore, these terms
are considered to represent “level,” “slope,” and “curvature.”
$0$

$0$

$\bullet$$0$

We can estimate four parameters, $L,$ $S,$

$C$

and

$\lambda$

for each period.

Diebold and Li (2006)$\bullet$Diebold and Li (2006) propose a dynamic Nelson-Siegel model:$y_{t}(m)=L_{t}+S_{t}( \frac{1-e^{-m\lambda}}{m\lambda})+C_{t}(\frac{1-e^{-m\lambda}}{m\lambda}-e^{-m\lambda})$where factors,
processes.

$L_{t},$$S_{t}$

and$C_{t}$,, are assumed to follow autoregressive

125
4.7
$\bullet$$\bullet$$\bullet$$\bullet$

Christensen, Diebold and Rudebusch (2009)
Above models based on Nelson and Siegel (1987) are not from theory
just from mere fitting.
Christensen, Diebold and Rudebusch (2009) combine Nelson-Siegel
model with theory, namely no arbitrage.
We can understand how such factors as “level,” “slope,” and “curvature” affect yields in a theoretically consistent manner.
Christensen, Diebold and Rudebusch (2009) assume that yields are
explained by three factors,and .
$L_{t},$$\bullet$$C_{t}$$S_{t}$

Instantaneous risk-free rate is given by$r_{t}=L_{t}+S_{t}$while state equations are defined under$(\begin{array}{l}dL_{t}dS_{t}dC_{t}\end{array})$

$=$

,

$Q$

measure as$(\begin{array}{lll}0 0 00 -\lambda \lambda 0 0 -\lambda\end{array})(\begin{array}{l}L_{t}S_{t}C_{t}\end{array})dt$$-(\begin{array}{lll}\sigma_{L} 0 00 \sigma_{S} 00 0 \sigma_{C}\end{array})(\begin{array}{l}dW_{t}^{Q,L}dW_{t}^{Q,S}dW_{t}^{Q,C}\end{array})$$\bullet$.

Under these settings, yields can be expressed as Nelson-Siegel model:$y_{t}(m)$$=$$L_{t}+S_{t}( \frac{1-e^{-m\lambda}}{m\lambda})$$+C_{t}( \frac{1-e^{-m\lambda}}{m\lambda}-e^{-m\lambda})$$+ \frac{A(m)}{m}$,where $A(m)$ is a yield adjustment factor, which is time-invariant but
depends only on the maturity.

Summary5$\bullet$Puzzles in the financial markets have not been solved completely.

126$\bullet$A model with a preference disentangling risk aversion and intertemporal substitution together with the low frequency events seem to bepromising in solving puzzles in financial markets.