# A New Test for Market Efficiency
and Uncovered Interest Parity

Richard T. Baillie
Michigan State University, USA
King’s College, University of London, UK

Francis X. Diebold
University of Pennsylvania, USA

George Kapetanios
King’s College, University of London, UK

Kun Ho Kim
Concordia University, Canada

First Draft: September 26, 2021

This Draft:

Abstract: We suggest a new single-equation test for Uncovered Interest Parity
($UIP$) based on a dynamic regression approach. The method provides consistent
and asymptotically efficient parameter estimates, and is not dependent on
assumptions of strict exogeneity. This new approach is asymptotically more
efficient than the common approach of using $OLS$ with $HAC$ robust standard
errors in the static forward premium regression. The coefficient estimates
when spot return changes are regressed on the forward premium are all positive
and remarkably stable across currencies. These estimates are considerably
larger than those of previous studies, which frequently find negative
coefficients. The method also has the advantage of showing dynamic effects of
risk premia, or other events that may lead to rejection of $UIP$ or the
efficient markets hypothesis.

JEL Classification: C22, C31.

Keywords: Dynamic regressions, forward premium anomaly, rational expectations,
efficient markets, robust standard errors.

Acknowledgments: For helpful comments we thank the co-editor and two referees.
The usual disclaimer applies.

##  1 Introduction

Considerable past research in international finance tests the major parity
conditions and/or models the role of risk premia and informational
inefficiency in currency markets. This paper introduces a new and simple
single-equation approach for testing uncovered interest parity ($UIP$), which
also allows for the inclusion of other variables that could represent time-
varying risk premia. Our approach is based on a single-equation dynamic
regression model and is widely applicable to situations where the maturity
time of a forward contract exceeds the sampling period of the data. It (1)
avoids the need to use inefficient $HAC$ robust inference and automatically
delivers consistent and asymptotically efficient estimates of the dynamic
regression parameters, (2) provides evidence on both short- and long-run
adjustments to the $UIP$ condition, and (3) facilitates incorporating
additional restrictions on the error process implied by the efficient markets
hypothesis ($EMH$) and rational expectations in foreign exchange spot and
forward markets.

$UIP$ asserts that the interest rate differential between two countries, or
equivalently the forward premium, is an efficient predictor of spot exchange
rate returns. This is requires the existence of rational expectations and a
constant risk premium. A widespread and important situation occurs when the
sampling frequency of the data exceeds the maturity time of the forward
contract. In this case the forward rate is a multi-step prediction of the
future spot rate, so the errors in $(UIP)$ regressions of spot returns on
forward premia are generally serially correlated.

Hansen and Hodrick (1980) noted that the consistency of the $GLS$ estimator
commonly invoked to correct for serial correlation requires strictly exogenous
regressors, which is unlikely to hold in $\mathit{UIP}$ regressions, and they
therefore recommended using $OLS$ with a $HAC$ robust estimated covariance
matrix. The $OLS$ estimator would then be a consistent, albeit inefficient,
estimator of the regression parameters, and this has led to a plethora of
$HAC$ regression methods (e.g., Newey and West, 1987).

An alternative approach has been to estimate a vector autoregression, ($VAR$)
and then to test cross-equation restrictions that correspond to the $EMH$; see
Hakkio (1981), Baillie, Lippens and McMahon (1983) and Levy and Nobay (1986).
Some comparisons between the different methodologies are given in Hodrick
(1987) and Baillie (1989). The potential advantage with the $VAR$ approach is
that it generally provides increased asymptotic efficiency compared with the
single equation approach. The disadvantage is that it requires the
specification and estimation of a full multi-equation $VAR.$

In this paper we propose a different approach, based on a single dynamic
regression, which we call $DynReg$. It requires only $OLS$ estimation and does
not require strict exogeneity, yet we show both theoretically and in
simulations that it is consistent and asymptotically efficient, and that
associated hypothesis tests have good finite-sample size and power.

We apply the $DynReg$ method to $32$ years of weekly data, regressing spot
returns on the lagged forward premium. We find clear rejections of the $UIP$
hypothesis (a $UIP$ regression parameter, $\beta$, of unity), consistent with
the presence of time-varying risk premia, yet our coefficient estimates are
more reasonable than those of many earlier studies. In particular, they are
remarkably stable across currencies and all positive, whereas previous studies
often found large negative $\beta$’s. We also provide rolling $DynReg$
estimates, which indicate quite stable and relatively similar estimated
$\beta$ coefficients across time and currencies. Similar analysis is also
provided for the forward rate forecast error regressed on past errors. These
results are more volatile over time and include periods when the $UIP$
condition cannot be rejected.

The plan of the rest of the paper is as follows. Section 2 describes the
formulations of the $UIP$ hypothesis, reviews previous econometric tests,
describes the $DynReg$ procedure, and shows to implement it in the context of
$UIP$ tests. Section 3 presents simulation evidence documenting the fine
performance of $DynReg$ estimates of $UIP$ regressions compared to $OLS/HAC$.
Section 4 describes the results of a $DynReg$ $UIP$ analysis of several
floating exchange rates. Section 5 provides a brief conclusion.

##  2 UIP and EMH

Here we develop both economic and econometric aspects of uncovered interest
parity and the efficient markets hypothesis.

###  2.1 Conceptual Formulations

The natural logarithm of the spot exchange rate at time $t$ is denoted by
$s_{t},$which is denominated in terms of the amount of foreign currency per
one numeraire US dollar. While $f_{t}$ is the natural logarithm of the
corresponding forward exchange rate at time $t$ with maturity time, or
forecast horizon, of $k\geq 1$. On denoting the domestic nominal interest rate
as $i_{t}$ and the corresponding foreign interest rate as $i_{t}^{\ast}$, then
the theory of Uncovered Interest Parity ($UIP$) implies that

$E_{t}(s_{t+k}-s_{t})=\left(i_{t}-i_{t}^{\ast}\right),$ (1)

where $E_{t}$ represents the conditional expectation based on a sigma field of
information available at time $t$. Hence $UIP$ requires the twin assumptions
of rational expectations and a constant or zero risk premium. Given the no
arbitrage condition, Covered Interest Parity $(CIP)$ condition implies that
$\left(i_{t}-i_{t}^{\ast}\right)=\left(f_{t}-s_{t}\right)$ and will hold as an
identity, and as an empirical matter CIP does indeed hold almost exactly
(Frenkel and Levich, 1975; Taylor, 1987). Hence the $UIP$ condition in
equation (1) is also frequently expressed as

$E_{t}(s_{t+k}-s_{t})=\left(f_{t}-s_{t}\right).$ (2)

The condition can be tested from the regression

$\left(s_{t+k}-s_{t}\right)=\alpha+\beta\left(f_{t}-s_{t}\right)+u_{t+k},$ (3)

so that the $k$ period rate of appreciation of the spot rate is predictable
from the forward premium. A test of $UIP$ or the $EMH$, is that
$H_{0}:\alpha=0$ and $\beta=1$ and the error process is subject to the
restriction

$Cov\left(u_{t+k}u_{t+k-j}\right)=0\text{\ \ for \ }j>k.$ (4)

Bilson (1981) and Fama (1984) analyzed the $k=1$ case with the sampling
frequency matching the maturity time of the forward contract, so that a
natural test of $UIP$ and $EMH$ was to estimate the regression

$\Delta s_{t+1}=\alpha+\beta\left(f_{t}-s_{t}\right)+u_{t+1},$ (5)

where $UIP$ implies that that $H_{0}:\alpha=0$ and $\beta=1$ and $u_{t+1}$ is
a serially uncorrelated white noise process. It has been noted by Fama (1984)
and many subsequent studies that the estimated slope coefficient is frequently
$\beta<0.$ This implies a violation of $UIP$ with the country with the higher
rate of interest having an appreciating currency rather than a depreciating
currency; which is known as the Forward Premium Anomaly.

Another way of testing $UIP$ is to express the condition as the forward rate
forecast error being unpredictable and to estimate the model

$\left(s_{t+k}-f_{t}\right)=\alpha+\beta\left(s_{t}-f_{t-k}\right)+u_{t+k},$
(6)

and to test $H_{0}:\alpha=0$ and $\beta=0$ and was tested by Hansen and
Hodrick (1980).

Early work by Frenkel (1977, 1979) tested the hypothesis $f_{t}=E_{t}s_{t+1}$
by estimating the regression

$s_{t+1}=\alpha+\beta f_{t}+u_{t+1},$ (7)

and testing that $\alpha=0,$ $\beta=1$ and $u_{t+1}$ serially uncorrelated.
These early studies which used monthly data with $30$ day forward rates so
that the maturity time of the forward contract exactly matched the sampling
interval of the data, generally found that the $EMH$ could not be rejected.
However, equation (3) is complicated by the fact that the variables in
question are non stationary. In particular, see Baillie and Bollerslev (1989),
Husted and Rush (1990) and Corbae and Ouliaris (1988) who all found strong
evidence that nominal spot and forward rates are well represented as $I(1)$
processes, which also appear to be cointegrated with the forward premium
$\left(s_{t}-f_{t}\right)$ being stationary. Hence either equations (1) or (2)
provide the natural economic theory to be tested.

It was also realized that more powerful tests of $UIP$ and the $EMH$ could be
obtained by using higher-frequency data where the maturity time of the forward
contract exceeds the sampling interval of the data; so that $k>1$. This
initially led to weekly data being used by Hansen and Hodrick (1980), Hakkio
(1981), Baillie, Lippens and McMahon (1983), bi weekly data in Hansen and
Hodrick (1983); and daily data in Baillie and Osterberg (1997). The
availability of higher frequency data then led to the development of a variety
of other testing procedures. Both the specifications of the tests for $UIP$
and $EMH$ in equations (3) and (6) provides the interesting complication,
given in equation (4) that a valid linear model for $u_{t+k}$ would be an
$MA(k-1)$ process, with the possibility of additional forms of non-linearity.
The question now arises as how equations (3) and (6) should be estimated.

###  2.2 Econometric Tests

Both equations (3) and (6) can be expressed as linear regressions,

$y_{t+k}=\alpha+\beta x_{t}+u_{t+k}.$ (8)

The estimation of equation (3) proceeds by setting $\
y_{t+k}=\left(s_{t+k}-s_{t}\right)$ and $x_{t}=\left(f_{t}-s_{t}\right)$, and
the estimation of equation (6) has $y{{}_{t+k}=}\left({s}_{t+k}-f_{t}\right)$
and $x_{t}=y{{}_{t}=(s}_{t}-f_{t-k})$. In both cases the error process is
defined in equation (4), with the precise $MA$ representation to be given
later. Both models have overlapping data with $k>1$, and both have error
processes where weak exogeneity is not in doubt, because
$E\left\\{x_{t}u_{t+k}\right\\}=0$.

However, as noted by Hansen and Hodrick (1980), consistency of time series
versions of $GLS$ techniques require the strict econometric exogeneity of the
$x$ process in equation (8), in the sense that $E(u_{t+k}\mid x_{t}$ ,
$x_{t-1},x_{t+1},......)=0,$, so that $x$ is uncorrelated with all past and
future values of $u$. $GLS$ estimation of $\beta$ implicitly filters the data,
which distorts orthogonality conditions and renders $GLS$ inconsistent in the
absence of strict exogeneity.

Because of the possible lack of strict exogeneity in equation (8), producing
inconsistency of $GLS$, Hansen and Hodrick (1980) recommend the use of $OLS$
rather than $GLS$. $OLS$ is consistent but inefficient when disturbances are
serially correlated, and the usual OLS standard errors are inconsistent. One
can, however, work out the correct standard error. In particular, the
consistent but asymptotically inefficient $OLS$ estimator is

$\widehat{\beta_{OLS}}=\left(\sum_{t=1}^{T}x_{t}x_{t}^{/}\right)^{-1}\left(\sum_{t=1}^{T}x_{t}y_{t+k}\right),$

with limiting distribution

$T^{1/2}(\ \widehat{\beta_{OLS}}-\beta)\rightarrow N(0,M),$ (9)

where $\beta$ is the true value of $\beta$ and $M=Q^{-1}\Omega Q^{-1}$, with

$Q=p\lim\left(T^{-1}\sum_{t=1}^{k}x_{t}x_{t}^{/}\right)=p\lim\left(T^{-1}X^{/}X\right).$
(10)

The practical use of the above result depends on the estimated covariance
matrix of the error process, so that

$\widehat{M}=Q^{-1}\widehat{\Omega}Q^{-1}$ (11)

Hansen and Hodrick (1980) recommended estimating $\Omega$ by a $k$-dimensional
band diagonal matrix, which would allow for an $MA(k-1)$ error process.
Subsequently there has been a vast literature focusing on the estimation of
$\Omega$, which then leads to the use of robust (“HAC”) standard errors with
OLS-estimated regression parameters. The method of Newey and West (1987) has
become particularly influential.

It is worth noting that the above complications and considerations do not
arise in the $VAR$ approach where the hypothesis that one variable is a
$k$-step-ahead prediction of another variable can be handed by a set of non-
linear restrictions on the $VAR$ parameters. However, we will not pursue this
issue here since this has been previously discussed by Baillie (1989) and our
aim in this paper is to focus on an alternative to the above robustness
approach in single equation estimation.

Before explaining an alternative single equation procedure that delivers
asymptotically efficient parameter estimates and tests (unlike OLS/HAC), we
first note some additional restrictions to the theory of $EMH.$ Due to the
$k$-1 period overlap in sequential $k$-step-ahead forecasts, $u_{t+k}$ can be
expected to be an $MA(k-1)$ process,

$u_{t+k}=\varepsilon_{t+k}-\theta_{1}\
\varepsilon_{t+k-1}-....-\theta_{k-1}\varepsilon_{t+1}=\theta(L)\varepsilon_{t+k},$
(12)

where $\varepsilon_{t+k}$ are white noise and $\
\theta(L)=\left(1-\theta_{1}L-....-\theta_{k-1}L^{k-1}\right)$. Following the
standard approach of previous literature in using weekly data, the forward
rate is generally measured on the Tuesday of each week and the spot rate on
the Thursday. This method of defining the data produces an average of $22$
days in the forward contract, which implies a maturity time of four weeks and
two days, or $(22/5)$, or $4.40$ weeks. On assuming $k=4$ in equation (1) then
$y{{}_{t+k}}$ in equation (7) would have an autocorrelation pattern of
$\rho_{1}=17/22=0.77$, and $\rho_{2}=12/22=0.55$, $\rho_{3}=7/22=0.32$,
$\rho_{4}=2/22=0.09$ and $\rho_{k}=0,$ for $k\geq 5.$ These population
autocorrelations imply a unique, invertible, $MA(4)$ process:

$\theta(L)=\left(1+0.8366L+0.7728L^{2}+0.6863L^{3}+0.2577L^{4}\right),$ (13)

with roots of $\left(0.1909\pm 1.1724i\right)$ and $\left(-1.5266\pm
0.6575i\right)$, and autoregressive representation $\pi(L)=\theta(L)^{-1}$.

###  2.3 The DynReg Approach

An attractive alternative to the Hansen-Hodrick $OLS/HAC$ approach – in part
because it delivers efficient as opposed to merely consistent parameter
estimates – is a single-equation Dynamic Regression (“$DynReg$”)
approach.111Here we give a basic sketch; for a more complete treatment see
Baillie, Diebold, Kapetanios and Kim (2022). Consider the $UIP$ equation (3)
from the perspective of a vector process
$\mathbf{z}_{t}=\left\\{y_{t},x_{t}\right\\}$. $z$ is assumed to be covariance
stationary with a Wold Decomposition of

$\mathbf{z}_{t}=\sum_{k=0}^{\infty}\Psi_{k}\mathbf{w}_{t-k}\text{\ }$ (14)

and a corresponding $VAR$ representation of

$\mathbf{z}_{t}=\sum_{k=1}^{\infty}\Pi_{k}\mathbf{z}_{t-k}+\mathbf{w}_{t},$
(15)

where $\mathbf{\Psi}_{k}$ and $\mathbf{\Pi}_{k}$ are absolutely summable
sequences of non-stochastic $2x2$ matrices with
$\mathbf{\Psi}_{0}=\mathbf{I.}$ It is further assumed that
$E\left(\mathbf{w}_{t}\mid Z_{t-1}^{w}\right)=0$ a.s. and
$E\left(\mathbf{w}_{t}\mathbf{w}_{t}^{/}\mid
Z_{t-1}^{w}\right)=\mathbf{\Omega}_{w}$ a.s. with
$\left|\mathbf{\Omega}_{w}\right|>0$ and
$\left|\left|\mathbf{\Omega}_{w}\right|\right|<\infty$ and$\
sup_{t}(\left|\left|\mathbf{w}_{t}\right|\right|^{4})<\infty$ with
$Z_{t-1}^{w}$ being the $\sigma$ sigma field generated by
$\left\\{\mathbf{w}_{s};s\leq t\right\\}.$

A single equation of the $VAR$ in equation (15) can be conveniently expressed
as

$y_{t}=\sum_{j=1}^{p}\phi_{j}y_{t-j}+\sum_{i=1}^{k}\sum_{j=0}^{q}\beta_{i,j}x_{i,t-j}+\varepsilon_{t},$
(16)

or

$\phi(L)y_{t}=\sum_{i=1}^{k}\beta_{i}(L)x_{i,t}+\varepsilon_{t}.$ (17)

This single equation is the dynamic regression ($DynReg$) of interest. Its
parameters are $\phi(L)=\left(1-\phi_{1}L-...-\phi_{p}L^{p}\right)$ and $\
\beta_{i}(L)=\left(\beta_{i,0}-\beta_{i,1}L-...-\beta_{i,q}L^{q}\right)$, and
there are $\left(k+p+kp\right)$ parameters in total222In the simulations and
empirical applications of subsequent sections we set $p=q$ and select $p$
using the the Schwarz (1978) ($BIC$) criterion,
$ln(\widehat{\sigma}_{u,h}^{2})+\left\\{k(1+p)+p\right\\}T^{-1}\ln(T)$, The
full set of parameters are denoted by
$\theta^{/}=\left(\phi_{1},..\phi_{p},\beta_{1,0}..\beta_{1,q},\beta_{2,0},...\beta_{k,q}\
\right)$.

The $OLS$ estimates of the $DynReg$ parameters are denoted by
$\widehat{\theta}$, and following the standard assumptions in Grenander (1981)
and Hannan and Deistler (1988), then as $T\rightarrow\infty$, we have

$\widehat{\theta}=\theta+O_{p}(T^{-1/2}),$

and moreover that

$T^{1/2}(\widehat{\theta}-\theta)\rightarrow N(0,Q^{-1}),$

where $\theta$ is the true value of the parameters and $Q$ is analogous to the
definition in equation (10) and is

$Q=p\lim T^{-1}\sum_{t=1}^{k}z_{t}z_{t}^{/},$ (18)

where $\
z_{t}^{/}=\left(y_{t-1},..y_{t-p},x_{1,t},...x_{k,t},x_{1,t-1},...x_{k,t-p}\right)$.

Under the null hypothesis of $EMH$ with Rational Expectations and constant
risk premium, we wish to estimate the model in equation (8), subject to the
restriction of having the $MA(k)$ error process defined in equation (12);
namely $u_{t}=\theta(L)\varepsilon_{t}$. The estimation of the general model
in equation (8) can be specialized to either the Fama regression in equation
(3), or the forward rate forecast error model in equation (6). On
premultipying through equation (8) by the filter $\theta(L)^{-1}$, we obtain

$\left\\{\theta(L)^{-1}y_{t+k}\right\\}=\alpha^{\ast}+\beta\left\\{\theta(L)^{-1}x_{t}\right\\}+\varepsilon_{t+k}$
(19)

where the new intercept is $\ \alpha^{\ast}=$ $\alpha\theta(1)^{-1}$. The
filtered explanatory variable is uncorrelated with current and future
innovations, $\varepsilon_{t+k}$, so that strict exogeneity is satisfied. Then
estimation of equation (19) by $OLS$ will produce consistent and
asymptotically efficient estimates of the regression parameters. In practice
it is convenient to use the approximation $\ \theta(L)^{-1}\approx\pi(L)$
where $\pi(L)=\left(1-\pi_{1}L-...-\pi_{p}L^{p}\right)$ and is a polynomial in
the lag operator of order $p$ and has all its roots lying outside the unit
circle.333Again, the choice of $p$ can based on $BIC$. The $DynReg$ will then
be

$\pi(L)y_{t+k}=\alpha^{\ast}+\beta\pi(L)x_{t}+\varepsilon_{t+k}\ $ (20)

which is a restricted version of the general dynamic regression in equation
(16) and can also be estimated by restricted $OLS$ and now contains $(k+1)p$
parameters.

For the case of weekly data, with $k=4$, and from equation (13), then

$\theta(L)=\left(1+0.8366L+0.7728L^{2}+0.6863L^{3}+0.2577L^{4}\right)$

and on inverting $\theta(L)$ we find $\pi(L)$ such that $\pi_{1}=-0.84$,
$\pi_{2}=-0.07$, $\pi_{3}=0.02$, $\pi_{4}=0.38,$ $\pi_{5}=0.08$, etc. and the
weights quickly decay to zero after eleven lags. The above restricted $DynReg$
model or $RDynReg$ model, can be contrasted with the unrestricted $DynReg$ in
equation (16). Both the restricted and unrestricted dynamic regressions are
reported in the following simulation results and also the tests of the $EMH$
based on the estimated models. We also report Likelihood Ratio ($LR$) tests to
compare the $OLS$ model with the $DynReg$ and also to compare the $DynReg$
model to the $RDynReg$ model, which is based on the full set of $EMH$
restrictions.

##  3 Simulation Results

The simulation work was based on observed weekly spot exchange rates, and an
artificially generated error process from equation (13). Hence the
artificially generated forward rate is

$f_{t}={s_{t+4}}-u_{t+4}$

and is generated to satisfy the null hypothesis of rational expectations and a
time invariant risk premium. The innovations $\varepsilon_{t}$ are generated
from an assumed $NID(0,\sigma^{2})$ process, where from equation (13) it can
be seen that $\ 2.8345\sigma^{2}=Var(u_{t})$, where the $Var(u_{t})$ is
calculated for each currency from an initial forward premium regression. The
artificial forward rates are then used to construct $y_{t+k}$ and $x_{t}$ in
equation (8). The weekly spot exchange rates were from January $1989$ through
April $2021$, for the six major currencies of Australia, Canada, Japan, New
Zealand. Switzerland and $UK$ against the numeraire $US$ dollar. The spot
rates were recorded on the Thursday of each week and realized $T=1,941$
observations and were obtained from Bloomberg. Monte Carlo results for the
unrestricted and restricted $DynReg$ are presented in Table 1. The first six
rows are from estimation by $OLS$ of the traditional static regression in
equation (6), with the first row reporting conventional $OLS$ robust standard
errors; while the next five rows use different $HAC$ covariance matrices. In
order, the methods are: Hansen and Hodrick (1980), Newey West (1987), Andrews
(1991), Kiefer-Vogelsang (2001) and finally the Equally Weighted Cosine
($EWC$) method of Lazarus et al (2018).

The seventh and eighth rows of Table 1, in contrast, provide results from
using the $DynReg$ and $RDynReg$ approaches. The $DynReg$ method has an
unrestricted parameterization as in equation (16), while the $RDynReg$ method
imposes the restrictions associated with $UIP$ In all the estimated models the
lag order, $p$, is selected by $BIC$ for each simulation replication.

The $DynReg$ and $RDynReg$ estimators of $\beta$ clearly have substantially
reduced biases and $MSE$s. This result holds for all six simulation designs,
corresponding to the six different spot rates. Hence the inclusion of lagged
information in estimation makes a large difference compared with static HAC
estimation of equation (6).

Table 1 also presents estimates of the empirical test size, which is the
probability of rejecting the null hypothesis when it is true. $OLS$ clearly
has poor size properties, and all other test statistics offer massive
improvement, with $DynReg$ and $RDynReg$ faring slightly better than the HAC
alternatives.

Finally, the simulation results of Figures 1–3 we show size-corrected power
curves for sample sizes $T=250$, $T=500$ and $T=1,000$, respectively, for
$\beta\in[-0.3,0.3]$. $DynReg$ clearly dominates, for all six currencies. The
high $DynReg$ test power is a natural consequence of its higher estimation
efficiency.

In summary, Table 1 and Figures 1–3 clearly indicate that the $DynReg$ and
$RDynReg$ methods improve on all competitors in all dimensions.

##  4 Empirical Results for Six Currencies

The above methodology was also implemented on the same weekly spot exchange
rate data between January $1989$ through April $2021$ and were complemented
with the actual $30$ day forward rate data which was recorded on the Tuesday
of each week. This provides $T=1,941$ observations for each bi-variate system
for each of the six currencies. In practice, due to the occurrence of
holidays, religious festivals, and weekends, all of which produce market
closures, the length of time between a forward rate and its corresponding spot
rate in the data set is between $19$ and $25$ days.

There are several sets of results; each of which includes both $OLS/HAC$ and
$DynReg$ estimation. Table 2 presents results for the model in equation (6),
where the forward rate forecast error is regressed on its lagged value. The
$OLS/HAC$ results have positive but small values for the estimated $\beta$
with significant rejections of the $\beta=0$ null for all countries apart from
Switzerland. The $DynReg$ results uniformly do not reject the null.

The more interesting results appear in Table 3. They are based on the classic
Fama forward premium regression (3), which has more economic and financial
intuition. The $OLS/HAC$ $\beta$ estimates are between 0.07 for Canada and
-0.11 for Japan. Three of the currencies have an estimated $\beta<0$, which is
the case originally emphasized by Fama (1984). None of these six estimated
$\beta$ coefficients are significantly different from zero at conventional
levels. However, the results of $DynReg$ estimation indicate long-run $\beta$
in the range of 0.24 to 0.31 while the restricted $RDynReg$ are in the range
of 0.30 to 0.40 for all six currencies. Hence the $DynReg$ results all
indicate significant risk premia but less than those of early studies with
monthly data where the $\beta<0.$

The appropriate Likelihood Ratio ($LR$) test statistic for the hypothesis of
$UIP$ is denoted by $\lambda_{LR}$ and shows overwhelming rejections of the
$UIP$ and $EMH$ for all six currencies. Hence this indicates the importance of
information in the lagged forward rate errors, which is likely due to time
variation in the risk premium.

Some further insights into testing the $UIP$ condition are obtained by
estimating the above models with five years of observations in each rolling
sample. The results are reported graphically in Figures 4 through 7. Figures 4
and 5 show the estimates of $\beta$ from the forward rate forecast error
regressions. The $OLS$ estimates in the left hand panel are considerably more
jagged and rough than those of the long run beta estimated by $DynReg$ in the
right hand set of panels. The estimates of long run $\beta$ do not
significantly depart from zero in any case. The 95% confidence bands for
$DynReg$ almost entirely contain the null in the Hansen-Hodrick model (i.e.
$\beta=0$) for Australian Dollar, Canadian Dollar, Japanese Yen and New
Zealand Dollar (i.e. Figs 4 and 5). For Swiss Franc and UK Pound, the band
mostly covers the null value. This is clearly not the case with OLS.

The results for the Fama regression in (5), shown by Figs 6 and 7, are
particularly interesting and indicate considerable stability over the rolling
sample from the $DynReg$ estimates. These estimates are all positive and are
typically around 0.4 instead of the $\beta=1$ implied by $UIP$. Given that the
confidence bands for the $DynReg$ estimates in Figs 6 and 7 stay above zero
for all six currencies, the estimates are statistically different to zero for
virtually all sets of rolling regressions, which is not the case with $OLS$.
Switzerland has slightly increased $\beta$ during the financial crisis and is
otherwise quite stable. New Zealand has a slightly lower $\beta$ value than
the other currencies.

Hence there is considerable evidence that the $UIP$ condition needs to be
appended with risk premium terms, or possibly some measure of informational
inefficiency. Models with appropriate variables could potentially be included
in the modeling framework introduced in this paper.

##  5 Conclusions

This paper has suggested a new single-equation test for $UIP$ and $EMH$ based
on $OLS$ estimation of a dynamic regression. The approach provides consistent
and asymptotically efficient parameter estimates, and is not dependent on
assumptions of strict exogeneity. This new approach has the advantage of being
asymptotically more efficient than the common approach of using $HAC$ robust
standard errors in the static forward premium regression. The method also has
advantages of showing dynamic effects of risk premia, or other events that may
lead to rejection of $UIP$ and $EMH$. The empirical results when spot returns
are regressed on the lagged forward premium are all positive and remarkably
stable across currencies.

## References

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Table 1: Performance of Tests of $\beta=0$ in Dynamic Regressions versus $OLS$
with robust standard errors in the Hansen-Hodrick (1980) model:

$s_{t+k}-f_{t}=\alpha+\beta(s_{t}-f_{t-k})+u_{t+k}.$

Under $EMH$ that $H_{0}:\,\alpha=0$ and $\beta=0$ and
$Cov(u_{t+k},\,u_{t+k-j})=0$ for $j>k$,

|  | Australian Dollar |  | Canadian Dollar |
---|---|---|---|---|---
|  | Bias | MSE | Level |  | Bias | MSE | Level |
OLS |  | -0.0054 | 0.0019 | 0.2630 |  | -0.0034 | 0.0018 | 0.2620 |
OLS-HH |  | – | – | 0.0660 |  | – | – | 0.0560 |
OLS-NW |  | – | – | 0.0810 |  | – | – | 0.0760 |
OLS-Andrews |  | – | – | 0.0760 |  | – | – | 0.0720 |
OLS-KV |  | – | – | 0.0520 |  | – | – | 0.0540 |
OLS-EWC |  | – | – | 0.0590 |  | – | – | 0.0660 |
$DynReg$ |  | -0.0004 | 0.0012 | 0.0590 |  | 0.0010 | 0.0011 | 0.0570 |
$RDynReg$ |  | -0.0018 | 0.0006 | 0.0550 |  | -0.0004 | 0.0006 | 0.0460 |
|  | Japanese Yen |  | New Zealand Dollar |
|  | Bias | MSE | Level |  | Bias | MSE | Level |
OLS |  | -0.0054 | 0.0017 | 0.2470 |  | -0.0021 | 0.0021 | 0.2880 |
OLS-HH |  | – | – | 0.0650 |  | – | – | 0.0580 |
OLS-NW |  | – | – | 0.0790 |  | – | – | 0.0780 |
OLS-Andrews |  | – | – | 0.0760 |  | – | – | 0.0740 |
OLS-KV |  | – | – | 0.0540 |  | – | – | 0.0620 |
OLS-EWC |  | – | – | 0.0680 |  | – | – | 0.0660 |
$DynReg$ |  | -0.0017 | 0.0011 | 0.0450 |  | -0.0002 | 0.0012 | 0.0430 |
$RDynReg$ |  | -0.0010 | 0.0006 | 0.0590 |  | 0.0004 | 0.0007 | 0.0550 |

Table 1: (Continued) Performance of Tests of $\beta=0$ in Dynamic Regressions
versus $OLS$ with robust standard errors in the Hansen-Hodrick (1980) model:

$s_{t+k}-f_{t}=\alpha+\beta(s_{t}-f_{t-k})+u_{t+k}.$

Under $EMH$ that $H_{0}:\,\alpha=0$ and $\beta=0$ and
$Cov(u_{t+k},\,u_{t+k-j})=0$ for $j>k$,

|  | Swiss Franc |  | UK Pound |
---|---|---|---|---|---
|  | Bias | MSE | Level |  | Bias | MSE | Level |
OLS |  | -0.0047 | 0.0017 | 0.2420 |  | -0.0029 | 0.0018 | 0.2350 |
OLS-HH |  | – | – | 0.0640 |  | – | – | 0.0650 |
OLS-NW |  | – | – | 0.0720 |  | – | – | 0.0740 |
OLS-Andrews |  | – | – | 0.0710 |  | – | – | 0.0740 |
OLS-KV |  | – | – | 0.0440 |  | – | – | 0.0530 |
OLS-EWC |  | – | – | 0.0610 |  | – | – | 0.0660 |
$DynReg$ |  | -0.0001 | 0.0011 | 0.0470 |  | -0.0008 | 0.0010 | 0.0390 |
$RDynReg$ |  | -0.0015 | 0.0006 | 0.0480 |  | -0.0003 | 0.0006 | 0.0460 |
Key: The first six test statistics are based on $OLS$ estimation of $\beta$
with standard errors based on estimated parameters estimated covariance matrix
computed by (i) regular $OLS$, (ii) $HH$, method of Hansen and Hodrick, (iii)
$NW$, method of Newey and West, (iv) $Andrews$, method of Andrews, (v) $KV$,
method Kiefer and Vogelsang, (vi) $EWC$, Equally Weighted Cosine. The $DynReg$
statistics are based on estimation of the unrestricted dynamic regression in
equation (16) and $RDynReg$ is the restricted dynamic regression in equation
(19) that constrains the error to be an $MA$(k-1) process as defined in
equation (12).

|  |
---|---|---
|  |
|  |
Figure 1: Size-corrected power for OLS (green), OLS-NW (blue), OLS-KV (yellow)
and $RDynReg$ (red): A 5% t-test is conducted for the sample size T=250. The
underlying model is the Hansen-Hodrick model (1980). The null value is
$\beta=0$ and the alternatives are $\beta=\pm 0.1,\pm 0.2,\pm 0.3$.

|  |
---|---|---
|  |
|  |
Figure 2: Size-corrected power for OLS (green), OLS-NW (blue), OLS-KV (yellow)
and $RDynReg$ (red): A 5% t-test is conducted for the sample size T=500. The
underlying model is the Hansen-Hodrick model (1980). The null value is
$\beta=0$ and the alternatives are $\beta=\pm 0.1,\pm 0.2,\pm 0.3$.

|  |
---|---|---
|  |
|  |
Figure 3: Size-corrected power for OLS (green), OLS-NW (blue), OLS-KV (yellow) and $RDynReg$ (red): A 5% t-test is conducted for the sample size T=1000. The underlying model is the Hansen-Hodrick (1980) model. The null value is $\beta=0$ and the alternatives are $\beta=\pm 0.1,\pm 0.2,\pm 0.3$. Table 2: Estimation of the Hansen-Hodrick (1980) model $s_{t+k}-f_{t}=\alpha+\beta(s_{t}-f_{t-k})+u_{t+k}$ for actual weekly spot and forward exchange rate data from January 1989 through April 2021.  |  | Australian Dollar |  | Canadian Dollar |
---|---|---|---|---|---
|  | $\hat{\beta}$ | $s.e.(\hat{\beta})$ | $\lambda_{LR}$ |  | $\hat{\beta}$ | $s.e.(\hat{\beta})$ | $\lambda_{LR}$ |
OLS |  | 0.1393 | 0.0241 | – |  | 0.1210 | 0.0243 | – |
OLS-HH |  | – | 0.0664 | – |  | – | 0.0536 | – |
OLS-NW |  | – | 0.0606 | – |  | – | 0.0488 | – |
OLS-Andrews |  | – | 0.0620 | – |  | – | 0.0479 | – |
OLS-KV |  | – | 0.0158 | – |  | – | 0.0069 | – |
OLS-EWC |  | – | 0.0566 | – |  | – | 0.0448 | – |
$DynReg$ |  | -0.0776 | 0.0320 | 2408.6 |  | -0.0167 | 0.0320 | 2226.6 |
$RDynReg$ |  | 0.0024 | 0.0245 | – |  | -0.0003 | 0.0246 | – |
|  | Japanese Yen |  | New Zealand Dollar |
|  | $\hat{\beta}$ | $s.e.(\hat{\beta})$ | $\lambda_{LR}$ |  | $\hat{\beta}$ | $s.e.(\hat{\beta})$ | $\lambda_{LR}$ |
OLS |  | 0.1503 | 0.0242 | – |  | 0.0955 | 0.0255 | – |
OLS-HH |  | – | 0.0533 | – |  | – | 0.0598 | – |
OLS-NW |  | – | 0.0483 | – |  | – | 0.0566 | – |
OLS-Andrews |  | – | 0.0486 | – |  | – | 0.0582 | – |
OLS-KV |  | – | 0.0098 | – |  | – | 0.0089 | – |
OLS-EWC |  | – | 0.0497 | – |  | – | 0.0571 | – |
$DynReg$ |  | 0.0120 | 0.0324 | 2479.5 |  | -0.0555 | 0.0338 | 2386.3 |
$RDynReg$ |  | 0.0192 | 0.0245 | – |  | -0.0236 | 0.0258 | – |
|  | Swiss Franc |  | UK Pound |
|  | $\hat{\beta}$ | $s.e.(\hat{\beta})$ | $\lambda_{LR}$ |  | $\hat{\beta}$ | $s.e.(\hat{\beta})$ | $\lambda_{LR}$ |
OLS |  | 0.0373 | 0.0244 | – |  | 0.0799 | 0.0244 | – |
OLS-HH |  | – | 0.0474 | – |  | – | 0.0598 | – |
OLS-NW |  | – | 0.0446 | – |  | – | 0.0540 | – |
OLS-Andrews |  | – | 0.0455 | – |  | – | 0.0542 | – |
OLS-KV |  | – | 0.0333 | – |  | – | 0.0158 | – |
OLS-EWC |  | – | 0.0438 | – |  | – | 0.0576 | – |
$DynReg$ |  | -0.0415 | 0.0322 | 2315.3 |  | -0.0527 | 0.0321 | 2391.8 |
$RDynReg$ |  | 0.0280 | 0.0245 | – |  | -0.0584 | 0.0245 | – |
Key: See key to Table 1. The Likelihood Ratio test statistic $\lambda_{LR}$ is
from a test of the static $OLS$ regression model against the DynReg model.
Table 3: Estimation of the Fama (1984) model $s_{t+k}-s_{t}=\alpha+\beta(f_{t}-s_{t})+u_{t+k}$ for actual weekly spot and forward exchange rate data from January 1989 through April 2021.  |  | Australian Dollar |  | Canadian Dollar |
---|---|---|---|---|---
|  | $\hat{\beta}$ | $s.e.(\hat{\beta})$ | $\lambda_{LR}$ |  | $\hat{\beta}$ | $s.e.(\hat{\beta})$ | $\lambda_{LR}$ |
OLS |  | 0.0115 | 0.0815 | – |  | 0.0727 | 0.0762 | – |
OLS-HH |  | – | 0.1077 | – |  | – | 0.1124 | – |
OLS-NW |  | – | 0.1055 | – |  | – | 0.1141 | – |
OLS-Andrews |  | – | 0.1043 | – |  | – | 0.1129 | – |
OLS-KV |  | – | 0.0300 | – |  | – | 0.0270 | – |
OLS-EWC |  | – | 0.1068 | – |  | – | 0.1057 | – |
$DynReg$ |  | 0.2440 | 0.0503 | 2852.4 |  | 0.3038 | 0.0495 | 2669.7 |
$RDynReg$ |  | 0.3696 | 0.0340 | – |  | 0.3972 | 0.0325 | – |
|  | Japanese Yen |  | New Zealand Dollar |
|  | $\hat{\beta}$ | $s.e.(\hat{\beta})$ | $\lambda_{LR}$ |  | $\hat{\beta}$ | $s.e.(\hat{\beta})$ | $\lambda_{LR}$ |
OLS |  | -0.1148 | 0.0815 | – |  | -0.0320 | 0.0842 | – |
OLS-HH |  | – | 0.0971 | – |  | – | 0.0847 | – |
OLS-NW |  | – | 0.0927 | – |  | – | 0.0898 | – |
OLS-Andrews |  | – | 0.0907 | – |  | – | 0.0902 | – |
OLS-KV |  | – | 0.0171 | – |  | – | 0.0639 | – |
OLS-EWC |  | – | 0.0842 | – |  | – | 0.0877 | – |
$DynReg$ |  | 0.2750 | 0.0503 | 2847.1 |  | 0.2392 | 0.0514 | 2845.9 |
$RDynReg$ |  | 0.3462 | 0.0334 | – |  | 0.3398 | 0.0351 | – |
|  | Swiss Franc |  | UK Pound |
|  | $\hat{\beta}$ | $s.e.(\hat{\beta})$ | $\lambda_{LR}$ |  | $\hat{\beta}$ | $s.e.(\hat{\beta})$ | $\lambda_{LR}$ |
OLS |  | 0.0277 | 0.0770 | – |  | -0.0737 | 0.0861 | – |
OLS-HH |  | – | 0.1545 | – |  | – | 0.1066 | – |
OLS-NW |  | – | 0.1491 | – |  | – | 0.1075 | – |
OLS-Andrews |  | – | 0.1489 | – |  | – | 0.1058 | – |
OLS-KV |  | – | 0.0983 | – |  | – | 0.0725 | – |
OLS-EWC |  | – | 0.1486 | – |  | – | 0.1142 | – |
$DynReg$ |  | 0.2427 | 0.0477 | 2796.6 |  | 0.3054 | 0.0503 | 2797.2 |
$RDynReg$ |  | 0.3012 | 0.0326 | – |  | 0.3566 | 0.0365 | – |
Key: See key to Table 1. The Likelihood Ratio test statistic $\lambda_{LR}$ is
from a test of the static $OLS$ regression model against the $DynReg$ model.
|  |
---|---|---
Australian Dollar | Australian Dollar |
|  |
Canadian Dollar | Canadian Dollar |
|  |
Japanese Yen | Japanese Yen |
Figure 4: The blue curve is the 5-year rolling OLS (left)/$RDynReg$ (right) estimate of $\beta$ in the Hansen-Hodrick (1980) model; The null is $\beta=0$ and the dashed ones are the 95% confidence bands. |  |
---|---|---
New Zealand Dollar | New Zealand Dollar |
|  |
Swiss Franc | Swiss Franc |
|  |
UK Pound | UK Pound |
Figure 5: The blue curve is the 5-year rolling OLS (left)/$RDynReg$ (right) estimate of $\beta$ in the Hansen-Hodrick (1980) model; The null is $\beta=0$ and the dashed ones are the 95% confidence bands. |  |
---|---|---
Australian Dollar | Australian Dollar |
|  |
Canadian Dollar | Canadian Dollar |
|  |
Japanese Yen | (f) Japanese Yen |
Figure 6: The blue curve is the 5-year rolling OLS (left)/$RDynReg$ (right) estimate of $\beta$ in the Fama (1984) model; The null is $\beta=1$ and the dashed ones are the 95% confidence bands. |  |
---|---|---
New Zealand Dollar | New Zealand Dollar |
|  |
Swiss Franc | Swiss Franc |
|  |
UK Pound | UK Pound |
Figure 7: The blue curve is the 5-year rolling OLS (left)/$RDynReg$ (right)
estimate of $\beta$ in the Fama (1984) model; The null is $\beta=1$ and the
dashed ones are the 95% confidence bands.