**
Around and around: The expectations hypothesis
**

We show how to construct models of the term structure of interest
rates in which the expectations hypothesis
holds. McCulloch (1993) presents such a model, thereby
contradicting an assertion by Cox, Ingersoll, and Ross (1981),
but his example is
Gaussian and falls outside the class of finite-dimensional Markovian
models. We generalize McCulloch's model in three ways: (*i*) We
provide an arbitrage-free characterization of the unbiased
expectations hypothesis in terms of forward rates; (*ii*) we
extend this characterization to a whole class of expectations
hypotheses; and (*iii*) we show how to construct
finite-dimensional Markovian and non-Gaussian examples.

**
Apparent arbitrage
**

The doubling strategy does not produce an arbitrage in
the space of signed measures equipped with the weak-* topology.
(A signed measure represents a payout.)
The absence of arbitrage opportunities is guaranteed by the
existence of a valuation operator (a strictly
positive continuous linear functional).
Nevertheless,
the sequence of signed measures generated by the doubling strategy
produces what appears to be an arbitrage
from the perspective of convergence almost everywhere
of the corresponding sequence of
Radon-Nikodym derivatives taken with respect to a fixed positive numeraire
measure.

**
Forces that shape the yield curve: Parts I and II
**

The yield curve is shaped by (i) expectations of
the future path of short-term interest rates and (ii)
uncertainty about the path. Uncertainty affects the yield curve
through two channels: (i) Investors attitudes
toward risk as reflected in risk premia, and
(ii) the nonlinear relation between yields
and bond prices (known as convexity). The way in
which these forces simultaneously work to shape the yield curve can be understood
in terms of the conditions that guarantee the
absence of arbitrage opportunities.

**
Modeling the state-price deflator and the term structure of interest rates
**

Generalizing the results of Kazemi (1992),
we decompose the state-price deflator into the product of
a positive martingale (the permanent component) and the inverse of the
price of the very-long discount (VLD) bond (a trend-stationary
process). We further decompose the permanent component into
a *term-structure martingale* correlated with bond returns and a
*neutrino factor* uncorrelated with bond returns. We analyze
the implications of the constancy of one or both of these permanent
components. When the term-structure martingale is constant,
the risk premia for all bonds are determined by the covariance
with the return on the VLD bond. When the neutrino factor is constant,
risk premia for all assets are determined by the price of term structure
risk. When both permanent components are constant,
the state-price deflator is trend-stationary
and all assets are priced by the VLD bond. Since exchange rates can
be modeled as the ratio of two state-price deflators, expected
exchange-rate depreciation depends on the ratio of the
two neutrino factors. When neutrino factors are constant, an
important source of variation is missing. We apply our analysis to
Constantinides (1992) and Rogers (1997), and we provide a number of
exponential-affine examples.

**
Modeling negative autocorrelation in continuous time
**

A continuous-time first-order autoregressive process cannot display
negative autoregression at any sampling horizon, but a first-order
system of two processes where only one
is observed can produce a discretely sampled AR(1) with
negative first-order autocorrelation.

**
Consumption and asset prices with homothetic recursive preferences
**

When preferences are homothetic, utility can be expressed in terms of
current consumption and a variable that captures all information
about future opportunities. We use this observation to express
the differential equation that characterizes utility as a restriction on the
information variable in terms of the dynamics of consumption. We derive the
supporting price system and returns process and thereby characterize optimal
consumption and portfolio decisions. We provide a fast and accurate numerical
solution method and illustrate its use with a number of Markovian models.
In addition, we provide insight by changing the numeraire from
units of consumption to units of the consumption process. In terms of the new
units, the wealth-consumption ratio (which is closely related to the information
variable) is the value of a coupon bond and the existence of an infinite-horizon
solution depends on the positivity of the asymptotic forward rate.

**
Consumption and asset prices with recursive preferences:
Continuous-time approximations to discrete-time models
**

This paper presents tractable and efficient numerical solutions to general
equilibrium models of asset prices and consumption where the representative
agent has recursive preferences. It provides a discrete-time presentation of the
approach of Fisher and Gilles (1999), treating continuous-time representations as approximations
to discrete-time "truth." First, exact discrete-time solutions are derived,
illustrating the following ideas: (*i*) The price-dividend ratio (such as
the wealth-consumption ratio) is a perpetuity (the canonical
infinitely-lived asset), the value of which is the sum of dividend-denominated bond
prices, and (*ii*) the positivity of the dividend-denominated asymptotic
forward rate is necessary and sufficient for the convergence of
value function iteration for an important class of models.
Next, continuous-time approximations are introduced.
By assuming the size of the time step is small, first-order
approximations in the stepsize provide the same analytical flexibility to discrete-time
modeling as Ito's lemma provides in continuous time. Moreover, it is shown
that differential equations provide an efficient platform for value function iteration.
Last, continuous-time normalizations are adopted, providing an efficient
solution method for recursive preferences.

**
A simple model of the failure of the term structure of interest rates
**

This paper presents a simple exponential-affine model of the term
structure that replicates the failure of the expectations hypothesis
quite well. The model has two Gaussian state variables. Under the
equivalent martingale measure, both state variables (the short rate
and its stochastic mean) determine the shape of the yield curve, while
under the physical measure the short rate is Markovian. Thus
instantaneous forward rates have a classical errors-in-variables
structure: They are moved by two variables (that are independent under
the physical measure), but the short-term interest rate is not moved
by one of them. This model shows that stochastic volatility is not
required to model the failure of the expectations hypothesis. The key
is stochastic risk premia.

**
The absence of arbitrage and general equilibrium in continuous time:
An overview
**

We review the absence-of-arbitrage restrictions in a Brownian motion
setting, relying on the existence of a *state-price
deflator*. We show how to find useful representations by either
changing the numeraire or by changing the *deflator-asset* and
its associated equivalent martingale measure. We examine the relation
between utility and the state-price deflator, and the
absence-of-arbitrage condition for a perpetuity is applied to finding
(*i*) the optimal relationship between consumption and wealth and
(*ii*) the nominal interest rate.

**
Risk-neutral pricing and risk neutrality:
A tale of two equivalent martingale measures
**

Two properties of asset prices have often been associated with
risk-neutrality: (a) the expected return on all assets equals
the risk-free rate (the expected-return property) and
(b) the value of an asset equals the present value of the
expectation (under the physical measure) of its payoff (the
certainty-equivalent property). In this paper, we show the
following: (i) the two properties are equivalent when interest
rates are deterministic but mutually exclusive when interest rates are
stochastic, (ii) an economy of risk-neutral investors who
consume only at a single point of time in the future will support the
certainty-equivalent property, (iii) the distribution that can
be uncovered using options prices is associated with the
certainty-equivalent property, (iv) the distribution associated
with the expected-return property cannot be uncovered using option
prices when interest rates are stochastic, although they can be
uncovered from futures prices on options that expire on the delivery
date. We provide an empirical investigation of the relationship
between these measures in the context of multi-factor models of the
term structure.

**
Estimating exponential-affine models of the term structure
**

We show how to estimate any model of the term structure of interest rates
in the affine-exponential class, which includes the Vasieck (1977),
Cox, Ingersoll, and Ross (1985), and Longstaff and Schwartz (1992) models,
among many others. For
most models in this class, analytical expressions for both bond prices and
the conditional distribution for the state variables are not available.
However, there are (*a*) fast and accurate numerical solutions to
the bond-price partial differential equation and (*b*) closed-form
expressions for the first two conditional moments for the state variables.
We show how to construct a quasi-maximum likelihood estimator using
(*a*) and (*b*) based on the maximum likelihood estimator of
Chen and Scott (1993). We discuss extensions to other estimation
techniques.

**
The term structure of repo spreads
**

We characterize the term structure of repo spreads in terms of forward
prices and forward rates. The analysis builds on Duffie (1996),
who showed that expected future overnight repo spreads are capitalized
into a security's price. We show how to calculate correctly the
forward price of a bond on special; we derive an absence-of-arbitrage
restriction on the drift of the process for forward repo term spreads
in the spirit of Heath, Jarrow, and Morton (1992); and we characterize an
extension of the unbiased expectations hypothesis in terms of that
restriction. Finally, we describe the average behavior of the term
structure of repo spreads for U.S. Treasury securities over the
auction cycle, and we examine empirically the unbiased expectations
hypothesis of the term structure of repo spreads.

**
Term premia in exponential-affine models of the term structure
**

We derive expressions for various term premia in the context of the
class of affine-exponential models of the term structure. In
addition, we show how regression tests of the expectations hypothesis
can be understood in terms of these models. In particular, we derive
expressions for the regression coefficients in terms of the parameters
of the models. Moreover, these expressions can used to define a GMM
estimator of the parameters in a given non-Gaussian affine-exponential
model.

**
The term structure of tax-exempt spreads: The effect of convexity
**

When taxable yields are tied to tax-exempt yields at the short end of
the yield curve, the volatility of the before-tax short-term interest
rate will be greater than the volatility of the after-tax short-term
interest rate, and the relative shapes of the two term structures will
be affected by this relationship. In particular, the yield on a
default-free tax-exempt zero-coupon bond will be greater than the
"after-tax yield" on a default-free taxable zero-coupon
bond. Estimates of the size and variability of this effect are
provided.

**
Fitting the term structure of interest rates with smoothing splines
**

We describe a technique for fitting the term structure of interest
rates using smoothing splines, which incorporate a "roughness"
penalty. An increase in the penalty reduces the effective number of
parameters. We use generalized cross validation to choose adaptively
the penalty and hence the effective number of parameters. We show how
our technique can be used to spline an arbitrary transformation of the
discount function, using a B-spline bases. Our Monte Carlo simulations
and estimation results suggest that fitting a smoothing spline to the
forward rate curve using generalized cross validation produces the
best results.

**
Happy-hour economics
**

Seasonal increases in demand may cause price to fall in markets with
travel or search cost. These costs limit the effective
substitutability among products and brands. When individual demand
increases, the benefits from additional travel or search increase.
Increased willingness to travel or search can translate into more
elastic demand curves facing monopolistic competitors, in which
case--as long as marginal cost does not increase too quickly--price
falls.

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