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Board of Governors of the Federal Reserve System

International Finance Discussion Papers

Number 948, September 2008 --- Screen Reader
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NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. References in publications to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors. Recent IFDPs are available on the Web at http://www.federalreserve.gov/pubs/ifdp/. This paper can be downloaded without charge from the Social Science Research Network electronic library at http://www.ssrn.com/.

Abstract:

Exchange rates have raised the ire of economists for more than 20 years. The problem is that few, if any, exchange rate models are known to systematically beat a naive random walk in out of sample forecasts. Engel and West (2005) show that these failures can be explained by the standard-present value model (PVM) because it predicts random walk exchange rate dynamics if the discount factor approaches one and fundamentals have a unit root. This paper generalizes the Engel and West (EW) hypothesis to the larger class of open economy dynamic stochastic general equilibrium (DSGE) models. The EW hypothesis is shown to hold for a canonical open economy DSGE model. We show that all the predictions of the standard-PVM carry over to the DSGE-PVM. The DSGE-PVM also yields an unobserved components (UC) models that we estimate using Bayesian methods and a quarterly Canadian-U.S. sample. Bayesian model evaluation reveals that the data support a UC model that calibrates the discount factor to one implying the Canadian dollar-U.S. dollar exchange rate is a random walk dominated by permanent cross-country monetary and productivity shocks.

Keywords: Exchange rates, present-value model and fundamentals, random walk, DSGE model, unobserved components model, Bayesian model comparison

JEL classification: E31, E37, F41

The search for satisfactory exchange rate models continues to be
elusive. This paper studies a workhorse theory of currency market
equilibrium determination, the present-value model (PVM) of
exchange rates, in the spirit of Engel and West (2005). Starting with the PVM and using
uncontroversial assumptions about fundamentals and the discount
factor, Engel and West (EW) hypothesize that the PVM generates an
approximate random walk in exchange rates if the PVM discount
factor approaches one and fundamentals are *I*(1). An
important implication of the EW hypothesis is that fundamentals
have no power to forecast future exchange rates, even with the PVM
dictating equilibrium in the currency market. EW support their
hypothesis with a key theorem and empirical and simulation
evidence.

This paper complements Engel and West (2005) by generalizing their main hypothesis
in two ways. First, the EW hypothesis is generalized using a
canonical two-country monetary dynamic stochastic general
equilibrium (DSGE) model. Its linearized uncovered interest parity
(UIP) and money demand equations yield the DSGE-PVM that coincides
with the standard PVM of the exchange rate. Second, we show the
standard- and DSGE-PVMs make equivalent predictions for exchange
rates. The predictions are summarized in five propositions: (1) the
exchange rate and fundamental cointegrate [Campbell and Shiller
(1987)], (2) the PVM yields an error
correction model (ECM) for currency returns in which the lagged
cointegrating relation is the only regressor, (3) the PVM predicts
a limiting economy (*i.e.*, the PVM discount factor approaches
one from below) in which the exchange rate is a martingale, (4)
given fundamental growth depends only on the lagged cointegrating
relation, the exchange rate and fundamental have a common
trend-common cycle decomposition [Vahid and Engle (1993)], and (5) the EW hypothesis is also
satisfied when the exchange rate and fundamental share a common
feature and the PVM discount factor approaches one. A corollary to
(5) is that the exchange rate is unpredictable when the PVM
discount factor goes to one.

We report evidence from vector autoregression (VARs) about the
propositions using quarterly floating rate Canadian-, Japanese-,
and U.K.-U.S. samples. The VAR evidence rejects cointegration and
reveals substantial serial correlation for the exchange rate and
the fundamental. There is also evidence that a common feature
exists between the Canadian dollar-, Yen-, and Pound-U.S. dollar
exchange rates and the relevant fundamentals. Nonetheless, the VAR
approach is unable to address the EW hypothesis question of whether
the PVM discount factor approaches one.^{1}

The DSGE-PVM possesses a deep structure tied to the primitives of the underlying open economy unlike the standard-PVM. Rather than rely on the entire set of DSGE optimality and equilibrium condition, we give empirical content to the DSGE-PVM by placing restrictions on its fundamentals (cross-country money and consumption). We restrict these fundamentals with permanent-transitory decompositions. This decomposition allows us to cast the DSGE-PVM as a tri-variate unobserved components (UC) model in the exchange rate and observed fundamentals. The UC model also incorporates DSGE-PVM cross-equation restrictions conditional on whether the discount factor is calibrated or estimated. Three UC models calibrate the discount factor to one, which disconnects the exchange rate from the transitory component(s) of fundamentals. Transitory fundamentals restrict the exchange rate in three other UC models in which the DSGE-PVM discount factor is estimated.

We estimate six UC models on a Canadian-U.S. sample running from
1976Q1 to 2004Q4. The UC models
yield state space systems for the DSGE-PVM, which allows us to
recruit the Kalman filter to evaluate likelihoods. We compute
likelihoods of the UC models using the Metropolis-Hastings (MH)
simulator described by Rabanal and Rubio-Ramírez (2005) to draw Markov chain Monte Carlo
(MCMC) replications from posteriors. We conduct model comparisons
using marginal posterior likelihoods of the six UC models to find
which is favored by the data. We find that the data favors the UC
model that calibrates the discount factor to one and in which
cyclical fluctuations are driven only with the transitory shock to
cross-country consumption. Favored next is the UC model with the
same transitory shock and in which the estimated posterior mean of
the DSGE-PVM discount factor is 0.9962. The posterior of this UC
model reveals that permanent shocks to fundamentals dominate
exchange rate fluctuations. Thus, the data prefer UC models that
are consistent with the EW hypothesis. Moreover, we find that the
data fail to support UC models that tie the exchange rate to the
transitory monetary shock. Rogoff (2007) also notes that exchange rates appear
disconnected from 'mean reverting monetary fundamentals'. These
results stand in contrast to those of open economy DSGE models
which assign key roles to nominal rigidities, UIP shock
persistence, and monetary disturbances.^{2}

The next section constructs the standard- and DSGE-PVMs of the exchange rate. Section 3 presents five propositions that generalize the EW hypothesis. Our Bayesian econometric strategy is discussed in section 4. Section 5 reports estimates of six UC models. We conclude in section 6.

This section fleshes out the standard PVM, in which the equilibrium exchange rate is determined by melding a liquidity-money demand function, UIP condition, purchasing power parity (PPP), and flexible prices. This is a workhorse exchange rate model used by, among others, Dornbusch (1976), Bilson (1978), Frankel (1979), Meese (1986), Mark (1995), and Engel and West (2005). This section also develops a PVM of the exchange rate derived from a canonical optimizing two-country monetary DSGE model. We show that the EW hypothesis generalizes to this wider class of models.

The standard-PVM of the exchange rate starts with the liquidity-money demand function

(1) |

where , , and denote the home country's natural logarithm of money stock, price level, output, and the level of the nominal interest rate. The parameter measures the income elasticity of money demand. Since the nominal interest rate is in its level, is the interest rate semi-elasticity of money demand. Define cross-country differentials , , , and , where denotes the foreign country. Assuming PPP holds, , where is the log of the (nominal) exchange rate in which the U.S dollar is the home country's currency.

Under UIP, the law of motion of the exchange rate is approximately

(2) |

Substitute for in the law of motion of
the exchange rate
(2)
with the money demand function
(1) and
impose PPP to produce the Euler equation
, where the standard-PVM discount factor is
and
is the standard-PVM
fundamental, which nets cross-country money with its income demand.
Iterate on the Euler equation through date *T* and
recognize that the transversality condition
to obtain the standard PVM relation

(3) |

The standard PVM
(3)
sets the log exchange rate equal to the annuity value of the
fundamental
at the standard-PVM
discount factor .^{3}

The optimizing monetary DSGE model consists of the preferences
of domestic and foreign economies and their resource constraints.
For the home (*h*) and foreign (*f*)
countries, the former objects take the form

(4) |

where and represent the th country's consumption and the th country's holdings of its money stock. The resource constraint of the home country is

(5) |

where
,
, ,
, , and
denote the *i*th country's
nominal holding of its own bonds at the end of date *t*,
the *i*th country's nominal holding of the
th country's bonds at the end of date
*t*, the return on the *i*th
country's bond, the return on the th
country's bond, the output level of the *i*th country,
and the level of the exchange rate. The two-country DSGE model is
closed with
. This
condition forces the world stock of nominal debt to be in zero net
supply, period-by-period, along the equilibrium path.

In section 2, analysis of the standard-PVM relies on *I*(1) fundamentals. Likewise, we assume that the processes for
labor-augmenting total factor productivity (TFP), , and satisfy

**ASSUMPTION 1**:
and
.

**ASSUMPTION 2**: Cross-country TFP and money
stock differentials are *I*(1) and do not
cointegrate.

Assumptions 1 and 2 impose stochastic trends on the two-country DSGE model.

The home country maximizes its expected discounted lifetime utility over uncertain streams of consumption and real balances,

subject to (5).
The first-order necessary conditions of economy *i* yield optimality conditions that describe UIP and money demand. The
utility-based UIP condition of the home country is

(6) |

where
is the marginal utility
of consumption of the home country at date *t*. Given
the utility specification
(4),
the exact money demand function of country *i* is

(7) |

The consumption elasticity of money demand is unity, while the interest elasticity of money demand is a nonlinear function of the steady state bond return.

The UIP condition (6) and money demand equation (7) can be stochastically detrended and then linearized to produce an equilibrium DSGE-law of motion for the exchange rate. Begin by combining the utility function (4) and the UIP condition (6) to obtain

where
is the utility level of
country *i* at date *t*. Prior to
stochastically detrending the previous expression, define
,
,
,
,
,
,
, and
. Note that
is the transitory
component of consumption of the th economy,
is the TFP
(money) growth rate of country , and the
cross-country TFP (money stock) differential
are *I*(1).
Applying the definitions, the stochastically detrended UIP
condition becomes

A log linear approximation of the stochastically detrended UIP condition yields

(8) |

where, for example, and denotes the steady state world real rate.

We use the linear approximate law of motion of the exchange rate (8), and a stochastically detrended version of the money demand equation (7) to produce the DSGE-PVM. When linearized, the unit consumption elasticity-money demand equation (7) produces . Impose PPP on the stochastically detrended version of the money demand equation and combine it with the law of motion (8) of the transitory component of the exchange rate to find

Solving this stochastic difference equation forward gives a present value relation for the transitory component of the exchange rate

(9) |

where the relevant tranversality conditions are invoked and the DSGE-PVM discount factor . Note that the DSGE-PVM and permanent income hypothesis discount factors are equivalent.

The DSGE-PVM relation (9) is the equilibrium law of motion of the cyclical component of the exchange rate. Transitory movements in the exchange rate are equated with the future discounted expected path of cross-country money and TFP growth and the (negative of the) annuity-value of the transitory component of cross-country consumption. The DSGE model identifies the exchange rate's unobserved time-varying risk premium with the expected path of cross-country TFP growth and transitory consumption, which suggest additional sources of exchange rate fluctuations.

The DSGE model produces a present value relation that resembles the standard-PVM (3). The DSGE-PVM follows from unwinding the stochastic detrending of the present value (9)

(10) |

Thus, the standard-PVM (3) and DSGE-PVM (10) are identical up to differences in their discount factors and real fundamentals. The standard-PVM discount factor is tied to the interest rate semi-elasticity of money demand, , while the DSGE-PVM sets to the inverse of the gross steady state real world interest rate, . For the standard-PVM (DSGE-PVM), the real fundamental is cross-country output (consumption ). Table 1 summarizes the notable elements of the standard- and DSGE-PVMs.

This section presents five propositions that generalize the EW
hypothesis. This allows a broader empirical analysis of the EW
hypothesis, and does so using standard time series tools. The
propositions apply to the standard-PVM *and* the DSGE-PVM
because their present value relations coincide. Thus, we generalize
the EW hypothesis to the large class of two-country monetary DSGE
models.

We collapse the differences in the discount factor and real fundamental of the standard-PVM (3) and DSGE-PVM (10) to stress their mutual predictions in this section. These differences are put aside by defining a PVM discount factor equal to either or , while the fundamental is equivalent to either or . With these assumptions, the focus is on the PVM

(11) |

which subsumes the standard- and DSGE-PVMs. The PVM (11) provides several predictions given

**ASSUMPTION 3**:
.

**ASSUMPTION 4**:
has a Wold
representation,
, where
.^{4}

Engel and West (2005) employ Assumption 3, but they do not require restrictions as strong as Assumption 4. However, Assumption 4 is standard for linear rational expectation models; see Hansen, Roberds, and Sargent (1991). Assumption 4 is also an implication of a linear approximate solution of the open economy DSGE model, while Assumption 3 is consistent with Assumptions 1 and 2.

The first prediction is that and share a common trend. This follows from subtracting the latter from both sides of the equality of the present-value relation (11) and combining terms to produce the exchange rate-fundamental cointegrating relation

(12) |

Equation (12) reflects the forces - expected discounted value of fundamental growth - that push the exchange rate toward long-run PPP. The explanation is

**PROPOSITION 1**: *If satisfies Assumptions 3 and 4,
forms
a cointegrating relation with cointegrating vector
, where
*.

The proposition is a variation of results found in Campbell and
Shiller (1987). We interpret the
cointegration relation
as the 'adjusted' exchange
rate because movements in fundamentals are eliminated from it.
According to the cointegration present value relation
(12), the
'adjusted' exchange rate is stationary and forward-looking in
fundamental growth. Moreover, the cointegration relation
is an infinite-order moving
average, MA equal to
, where
and
under Assumptions 3 and 4 (*i.e.*, is
and its growth rate has a Wold
representation). Thus, the 'adjusted' exchange rate is a "*cycle
generator*" - as defined by Engle and Issler (1995) - because shocks to serially
correlated fundamental growth create persistent PPP deviations.

The standard- and DSGE-PVM require Assumptions 3 and 4 to satisfy Proposition 1. Rather than these assumptions, we can construct a cointegration relation from the DSGE model using Assumptions 1 and 2 because is implied by the balanced growth restriction, , where and . In this case, PPP deviations arise from the DSGE-PVM because of restrictions the present-value relation (9) places on the transitory component of the exchange rate, .

The second PVM prediction is that currency returns depend only on the lagged 'adjusted' exchange rate and fundamental forecast innovation. We show this by first rewriting the PVM of (11) as . Differencing this equation produces, . Next, add and subtract inside the brackets, and substitute with the cointegration-present-value relation (12) to obtain

(13) |

In equilibrium, currency return are generated by the lagged cointegration relation, , and the expected annuity value of the forecast innovations of the fundamental. The lagged cointegration relation is the error correction mechanism of (13) that reflects the only force that restores currency returns to equilibrium and PPP in response to the shock innovation . These ideas are summarized by

**PROPOSITION 2**: *Under Proposition 1, the
PVM predicts that the equilibrium currency return is an error
correction mechanism in which the lagged 'adjusted' exchange rate
(or cointegration relation) is the only factor that drives the
exchange rate to PPP in response to fundamental shock
innovations.*

Equation
(13) is an ECM
that regresses currency returns only on the lagged `adjusted'
exchange rate. The regression is
with factor loading
and currency return forecast error
.^{5}

Proposition 2 relies on
* *< 1 to define short- to
medium-run currency return dynamics. This raises the question of
the impact of relaxing this bound.

**PROPOSITION 3**: *The exchange rate
approaches a martingale (in the strict sense) as
,
according to the present-value relation
*(13)* assuming
Proposition 1*.

Proposition 3 relies on
to produce the
martingale
and random walk behavior in the
exchange rate.^{6} This behavior suggests an equilibrium
path for in which its best forecast is
, given relevant information, because
the source of serial correlation,
disappears as
.^{7}

Engel and West (2005) show that
the PVM of the exchange rate yields an approximate random walk as
approaches one. This section
affirms the EW hypothesis, but unlike Proposition 3 does not rely
on Proposition 2. Rather than follow the EW proof exactly, we
invoke Assumptions 3 and 4, the present-value relation
(3), the
Weiner-Kolmogorov prediction formula, and the *conjecture*
`a` to find that currency returns are unpredictable.

The EW hypothesis is . Its hypothesis test begins by noting , which is obtained from the present-value relation (3). Use this equation to construct , given Assumptions 3 and 4 and the Weiner-Kolmogorov prediction formula. The PVM of (11) also sets currency returns equal to the annuity value of fundamental growth, . The last two equations yield

(14) |

By letting
, the random
walk hypothesis of EW is verified independent of the ECM of
Proposition 2 (and cointegration prediction of Proposition
1).^{8}

The ECM (13) and Proposition 2 maps into the EW currency return generating equation (14). First, apply the change of index to the present value of (14) to obtain the present-value cointegration relation (12) lagged once. For the ECM (13), its present value equals subsequent to evoking Assumptions 3 and 4 and the Weiner-Kolmogorov prediction formula. Thus, when the PVM discount factor is arbitrarily close to one, the EW hypothesis predicts which is consistent with currency returns following an ECM with no own lags or lags of fundamental growth. Since the standard- and DSGE-PVMs produce the ECM, the EW hypothesis is generalized to the larger class of two-country monetary DSGE models.

Proposition 2 predicts an ECM for currency returns that is
consistent with the EW currency return generating equation
(14).
These results rely, at most, on assumptions 3 and 4 under which
fundamentals are *I*(1) and have a Wold representation
in growth rates. However, empirical work on exchange rates often
employ multivariate time series models (*i.e.*, VARs) instead
of the deeper notion of a Wold representation.

This section studies the impact on the bivariate exchange rate-fundamental process, of endowing an ECM on fundamental growth. In this case, forms a VECM(0)

(15) |

where is the factor loading on for and is its forecast innovation. Pre-multiplying the VECM(0) by creates the common feature

(16) |

The vector satisfies the Engle and Kozicki (1993) notion of a common feature because it creates a linear combination of and that is unpredictable conditional on their history. Given this common feature restriction and the cointegration relation of Proposition 1, Vahid and Engle (1993) provide a method to construct a Stock and Watson (1988) multivariate Beveridge and Nelson (1981) common trend-common cycle decomposition. We summarize these results with

**PROPOSITION 4**: *Assume fundamental
growth is the ECM process
, where the forecast
innovation
is Gaussian. When Proposition
2 holds, has a common feature,
,
in the sense of Engle and Kozicki *(1993)*, where
. The cointegrating and common feature vectors and
restrict the trend-cycle
decomposition of , as described by Vahid and
Engle *(1993)*.*

The common feature of Proposition 4 endows
with a
common trend and a common cycle Beveridge-Nelson-Stock-Watson
(BNSW) decomposition. Vahid and Engle (1993) provide an example in which the
cointegration and common feature vectors restrict the trend of
to
, which gives trend and cycle components
and
, respectively.^{9} The BNSW decomposition imposes a
common cycle on and in
the short-, medium-, and long-run, which restricts the exchange
rate to be unpredictable at all forecast horizons. This prediction
is at odds with the empirical evidence of Mark (1995).

The common feature relation (16) also provides another approach to verify the EW hypothesis, .

**PROPOSITION 5**: *Let the exchange rate
and fundamental have the VECM(0)
*(15)*.
Then, the EW hypothesis requires currency returns and fundamental
growth to share a common feature defined by
and that
0 or
.*

Proposition 5 differs from other approaches to the EW
hypothesis. First, the common feature relation
(16)
imposes cross-equation restrictions on
because its cycle generator,
the lagged cointegrating relation
, is annihilated by
. Having
eliminated
, the EW hypothesis
decouples the exchange rate from fundamental growth and its
forecast innovation
(
). Finally,
observe that when
0 (or
,
. This leaves only the forecast
innovation
to generate movements in
. Thus, the EW hypothesis is
affirmed by Proposition 5.^{10}

A corollary of Proposition 5 is that changes in fundamentals do not Granger cause currency returns as . Only if , do movements in fundamentals have predictive power for currency returns according to the PVM. However, currency returns Granger cause growth in the fundamental as long as it is predicted by its own lagged forecast innovations. The equilibrium currency return generating equation (13) and Proposition 2 shows that this holds even if .

The propositions suggest testable restrictions on exchange rates and fundamentals. Table 3 describes details of the tests and summarizes results. Fisrt, if the lag length of the levels VAR of the exchange rate and fundamental exceeds one, the VECM (15) is rejected. Second, cointegration tests are sufficient to examine Proposition 1. Finally, common feature tests are used, following Vahid and Engel (1993) and Engel and Issler (1995), that yield information about Proposition 4.

We estimate VARs of foreign currency-U.S. dollar exchange rates
and fundamentals using Canadian, Japanese, U.K., and U.S. data on a
1976Q1 - 2004Q4 sample.^{11} VAR
lag lengths are chosen using likelihood ratio (LR) statistics,
given a VAR(8), , VAR(1).^{12} As
described in Table 3, the Canadian-, Japanese-, and U.K.-U.S.
samples yield a VAR(8), VAR(5), and VAR(4), respectively.^{13}
Thus, the Canadian, Japanese, U.K., and U.S. data reject the VECM
(15)
because
has more serial correlation
than explained by the lagged cointegration relation
.

Table 3 also presents Johansen (1991, 1994) trace and max test statistics that fail to confirm the cointegration prediction of Proposition 1 for the Canadian-, Japanese-, and U.K.-U.S. samples. This finding is consistent with Engel and West (2005), who argue there is little evidence that exchange rates and fundamentals cointegrate.

Finally, the common feature test is described in Table 3. This
uses squared canonical correlations of currency returns and
fundamental growth. The common feature null is that the smallest
correlation equals zero. We use a
statistic of Vahid and Engle (1993)
and a *F*-statistic developed by Rao (1973) to test this null. The tests reject
the null for the largest canonical correlation, but not for the
smaller one in the three samples. This is evidence that currency
returns and fundamental share a common feature in the Canadian-,
Japanese-, and U.K.-U.S. samples. Given a common feature, the
exchange rate approximates a random walk when
. The next
section explores the empirical content of this hypothesis in the
Canadian-U.S. data.

Propositions 1-5 broaden our understanding of the EW hypothesis, which is also generalized to hold for the DSGE-PVM. Although the previous section discusses VAR methods that yield evidence about the joint behavior of the exchange rate and standard-PVM fundamentals, this approach is not informative about estimates of the PVM discount factor.

This section presents methods to estimate a PVM discount factor
and test the EW hypothesis. Instead of relying on VARs, we employ
unobserved components (UC) models to estimate the DSGE-PVM and test
the EW hypothesis using Bayesian methods. A brief example motivates
our approach. Consider the PVM
(11)
where the fundamental has the
permanent-transitory decomposition
,
,
,
0,
,
, and
0 for all and
.^{14} Combining the PVM
(11)
and the permanent-transitory decomposition of gives an equilibrium permanent-transitory decomposition
of the exchange rate,
, where
is a 1 row vector with a first element of
one and zeros elsewhere and
is the companion
matrix of the AR of
. The exchange rate trend
is identified with the random walk of under
its permanent-transitory decomposition. Transitory exchange rate
fluctuations are driven by the fundamental cyclical component,
, which is the common
dynamic factor of the exchange rate and observed fundamental. The
permanent-transitory decomposition of the exchange rate is useful
for the EW hypothesis because it becomes possible to estimate
, along with the coefficients of
the permanent-transitory decomposition of .
Note also that as
approaches one, the permanent
component comes to dominate exchange rate
fluctuations as predicted by the EW hypothesis.

We use Bayesian methods to estimate multivariate UC models of
the DSGE-PVM. The models represent different combinations of
restrictions imposed by the DSGE-PVM on the exchange rate,
cross-country money, and cross-country consumption. For example,
is estimated for three UC models,
which ties the exchange rate to the transitory component(s) of
fundamentals. The exchange rate is disconnected from transitory
shocks in remaining three UC models because is
calibrated to one. We cast the UC models in state space form to
evaluate numerically the likelihoods. We use a sample of the
Canadian dollar-U.S. dollar
(*CDN*$/*US*$) exchange rate and the
Canadian-U.S. money and consumption differentials from
1976Q1-2004Q4. The random walk
MH simulator is used to generate MCMC draws from the UC model
posterior distributions conditional on this sample. We compute
model moments, such as parameter means, unconditional variance
ratios, permanent-transitory decompositions, and forecast error
variance decompositions (FEVDs), from the posterior distributions.
Model comparisons are based on marginal likelihoods, which we
construct by integrating the likelihood function of each model
across its parameter space where the weighting function is the
model prior.

The state space systems of the six UC models begin with the balanced growth restriction the DSGE model imposes on the exchange rate. This restriction is equivalent to the permanent-transitory decomposition . The DSGE-PVM (9) places cross-equation restrictions on the stationary component of the exchange rate, .

Cross-equation restrictions are conditioned on the permanent and transitory components of cross-country money and cross-country consumption. The permanent components of money and consumption are , , and , , respectively. Note that and are the deterministic trend growth rates of cross-country money and TFP. We assume is a MA , , where and . For , we employ a AR , , where . Put these elements together to form the balanced growth version of the DSGE-PVM

(17) |

which satisfies the DSGE balanced growth path restrictions. The balanced growth DSGE-PVM (17) implies the cointegrating relation of Proposition 1. Thus, the exchange rate responds only to trends in cross-country money, , and TFP, , in the long-run. Serial correlation in the exchange rate is produced by the transitory components of cross-country money and consumption, and . Also, if a common cycle generates these transitory components, the exchange also shares the restriction. Thus, the permanent and transitory components of cross-country money and consumption drive exchange rate fluctuations, which give rise to cross-equation restrictions in the UC models.

The UC models are classified according to whether there are two cycles or a common cycle and whether is calibrated to one or estimated. Three UC models follow from solving the DSGE-PVM (17) given MA and AR or a common cycle is imposed using either the MA or AR . The three UC models are estimated when is calibrated to one. We also use the UC models to estimate . The six UC models have in common the cross-country money trend, , and TFP trend, .

A rich set of cross-equation restrictions arises in the 2-trend, 2-cycle UC model with . In part, its state space system consists of the observation equations

(18) |

where , factor loadings on and its lags are

(19) |

factor loadings on , , are elements of the row vector

(20) |

and is the companion matrix of the AR of . The system of first-order state equations is

(21) |

with covariance matrix , where .

We also study UC models that impose one common transitory factor on and . When the common component is , the response of to is denoted . This implies and gives rise to the 2-trend, money cycle UC model. Identifying the common transitory component with defines which restricts the 2-trend, consumption cycle UC model. The appendix describes the state space systems of the 2-trend, money cycle and 2-trend, consumption cycle UC models.

The three remaining UC models set . The restriction on the state space of the 2-trend, 2-cycle UC model is that the exchange rate is decoupled from transitory cross-country money and consumption shocks. Similar restrictions arise in the observer equation of the 2-trend, money cycle and 2-trend, consumption cycle UC models. Thus, we are able to compare DSGE-PVMs in which is estimated to those in which is calibrated to one. This provides an empirical appraisal of the EW hypothesis.

We label the 2-trend, 2-cycle UC model with
. Likewise,
and
denote the
2-trend, money cycle and 2-trend, consumption cycle,
UC models. The state
space system of
is
(18)
and
(21), while the
appendix presents these systems for
and
. These state
space systems represent the dynamics of
restricted by the DSGE-PVM and permanent-transitory specifications
of and . We calibrate
in
,
, and
. The state
space systems are mapped into the Kalman filter to evaluate
likelihood functions as proposed by Harvey (1989) and Hamilton (1994).^{15} Denote the likelihood
, where 2,
,
is either calibrated to one or
estimated, and
is the parameter vector
of
.

The largest parameter vector is
. It contains
elements,
. We add the parameters
,
,
, and
to
to better fit
to the data. For example,
the Canadian-U.S. TFP differential exhibits more variation than
if the correlation coefficient of
innovations to and
,
, is negative.^{16} The remaining three parameters allow
for an unrestricted exchange rate intercept, , a linear exchange rate trend, , and a factor loading on the Canadian-U.S. TFP
differential, , rather than set the
(1, 2) element in the matrix of the
observation system
(18)
to negative one.^{17} We estimate
to ask if the data supports the cointegration-balanced growth path
restriction imposed on the DSGE-PVM
(17).

The parameter vectors of the other five UC models are smaller. The model drops two plus parameters from , while adding the factor loading on for , . The factor loading enters the parameter vector of , while and are dropped from . The parameter vectors of the UC models , , and are identical to , , and except that .

The sample runs from 1976Q1 to
2004Q4, *T* = 116. We have
observations on the Canadian dollar-U.S. dollar exchange rate
(average of period). The Canadian monetary aggregate is M1 in
current Canadian dollars, while for the U.S. it is the Board of
Governors monetary base (adjusted for changes in reserve
requirements) in current U.S. dollars. Consumption is the sum of
non-durable and services expenditures in constant local currency
units.^{18} The aggregate quantity data is
converted to per capita units. The data is logged and multiplied by
100, but is neither demeaned nor detrended.

The likelihood functions of the UC models do not have analytic
solutions. We approximate the likelihoods
and
with posterior distributions of
and
, generated by the
MCMC replications of the random walk MH simulator. Our estimates of
and
and marginal
likelihoods build on the Bayesian estimation tools of
Fernández-Villaverde and Rubio-Ramírez (2004), Rabanal and Rubio-Ramírez
(2005), Geweke (1999, 2005), An and
Schorfheide (2007), and Gelman,
Carlin, Stern, and Rubin (2004).
The MH simulator creates 1.5 million MCMC draws from the posterior.
The initial 750,000 draws are treated as a burn-in sample and
therefore discarded. We base our estimates on the remaining 750,000
draws from the posteriors of the
,
,
,
,
, and
models.^{19}

The second column of table 4 (5) list the priors of , 2, , . Under a normal prior, the first element is the degenerate mean and second its standard deviation. The inverse-gamma priors are parameterized by its degrees of freedom, the first element, and its mean, the second element. The left and right end points of a uniform prior is denoted by its first and second elements.

We choose degenerate priors for the lag lengths of the MA
of
and AR
of
that set
.
Normal priors for the MA (
and
) and AR (
and
) coefficients allow for disparate
transitory behavior in
and
. The prior means of
,
,
, and
guarantee that the relevant
eigenvalues are strictly less than one. The eigenvalues of the
MA(2) (AR(2)) of
(
) are 0.60 ± 0.20*i* (0.95 and -0.10). The standard
deviation of the normal priors of the MA and AR coefficients
provide for a wide set of realizations for
,
,
, and
. However, when a draw generates
an eigenvalue greater than one (in absolute value) for either the
MA or AR coefficients, the draw is discarded. Nonetheless, the MA
and AR priors admit transitory cycles in cross-country money and
consumption that allow for power at the business cycle frequencies,
if the data wants.

We opt for priors of and that rely on the Canadian-U.S. money stock and consumption differentials samples. Since and represent deterministic trend growth, we ground the priors on normal distributions. The prior standard deviations of and match sample moments.

Priors on the standard deviations of the shock innovations reflect standard practice for estimating DSGE models with Bayesian methods. For example, Adolfson, Laséen, Lindé, and Villani (2007) employ inverse-gamma priors for the standard deviations of the shock innovations of their sticky price open economy DSGE model. However, there is a lack of good information about , , , and . This explains why we impose a prior with two degrees of freedom, which forces these standard deviations to be positive. On the other hand, we attach a normally distributed prior to the correlation of innovations to and , . Its mean is negative to capture our prior that the TFP differential, , is smoother than cross-country consumption, . Since we have no information about the extent of the smoothness, the mean is -0.5 with a standard deviation of 0.2 that places draws close to negative one or zero in the 95 percent coverage interval of the prior. Draws greater than one or less than negative one are ignored. The correlation of innovations to and is fixed at zero, reflecting our assumption that the sources and causes of permanent and transitory monetary shocks are orthogonal.

The exchange rate intercept and linear time trend priors are set according to a linear regression of the exchange rate on these objects. This motivates our choice of normally distributed priors for and and of their degenerate means and standard deviations.

The remaining factor loadings have priors that reflect a dearth of information on our part. The uniform priors of , , and are wide and include zero. If, for example, is small it indicates the inadequacy of the balanced growth restriction and the impact of permanent fluctuations in Canadian-U.S. TFP differentials on the exchange rate. The same holds for the response of to transitory movements in the Canadian-U.S. money stock (consumption) differential.

The
models have only one
'economic' parameter, the DSGE-PVM discount factor
, in common. We adopt the Engel and West (2005) prior for . They
argue that it is necessary for
to generate an
approximate random walk exchange rate from the standard-PVM. Hence,
our prior on is constructed to provide
information about the EW hypothesis from the posteriors of the
models. We impose an
inverse-gamma prior on the DSGE-PVM discount factor , which follows Del Negro and Schorfheide (2006). The degenerate prior means of
and
exp([*a** = 0.158]/400) imply an annual
average real world interest rate of about five percent. Although a
five percent real world interest rate is large for the floating
rate period, the standard deviation of 0.038 guarantees draws for
that cover a wide interval. However,
MCMC draws from the random walk MH simulator of the
models obey the EW prior
because we ignore draws for which
.

This section presents the results of implementing our empirical
strategy. Tables 4 and 5 provide the posterior means and standard
deviations of
and
vectors, *i* 2,
,
, for the six UC models. We
include marginal likelihoods of the six UC models, using methods
described by Fernández-Villaverde and Rubio-Ramírez
(2004), Rabanal and
Rubio-Ramírez (2005), and
Geweke (1999), to conduct inference
across these models. We present densities of the prior and
posteriors of for
,
, and
in figure 1.
Tables 6, 7, 8, and 9 report factor loadings on the exchange rate
of the transitory components of money and consumption
differentials, unconditional variance ratios of the present
discounted value of the shock innovations to the exchange rate,
FEVDs of the trend-cycle decomposition of the exchange rate with
respect to these shocks, and summary statistics of trend-cycle
decompositions, respectively. Figures 2, 3, and 4 plot the
trend-cycle decomposition of the
*CDN*$/*US*$ exchange rate.

Tables 4 and 5 list the posterior means and standard deviations
of the parameters of the six UC models. Estimates of
,
,
appear in
table 4. These three models exhibit persistence in the transitory
components of the money and consumption differentials,
and
. For example, the
model
yields AR (MA) estimates that imply the half life of a shock to
is 17 (7) years.^{20} However, only
is persistent in the
model. The half life of
a shock to
is less than two quarters,
while for
it is between nine and ten
years. Note also that the priors and posterior means of the MA
coefficients,
and
, only differ for the
model.
Although the posterior means of the AR coefficients have moved away
from the prior means, a one standard deviation of the posterior of
covers zero for the
and
models.

The posterior means of and show that Canada experiences slower (faster) trend money (TFP) growth than the U.S. over the sample. Trend U.S. money growth is on average about 0.05 percent higher annually according to the posteriors of the , , and models. Across these models, indicates Canadian deterministic trend TFP growth dominates its U.S. counterpart by about 0.06 percent at an annual rate.

The , , and models show differences across estimates of the posterior means of the shock innovation standard deviations. Only the estimated impulse structure of the model is dominated by movements in the permanent innovations of the money shock, . The converse is that this model yields the smallest posterior means of the standard deviation of the TFP differential shock and shock innovations, and . The model yields the largest estimates of and , but these posterior means are about the same magnitude. Note also that the correlation of the innovations to the TFP shock and shock is estimated to be and by the and models, respectively. Thus, these models are consistent with the TFP trend being more volatile than observed Canadian-U.S. consumption.

Estimates of the exchange rate intercept and linear time trend
indicate that the
model provides
the largest value for the *US*$ in steady state
and the largest deterministic growth rate for the
*CDN*$/*US*$ exchange rate. The posterior
means of and
imply that the steady state
*CDN*$/*US*$ exchange rate is 1.23 with a
deterministic annual growth rate of about 0.8 percent. For the
model,
the analogous values are 1.10 (1.03) and 0.3 (0.2) percent per
annum. Thus, the
model places
more emphasis on deterministic elements to fit the data compared to
the other two UC models that calibrate to
one.

The remaining coefficients are the factor loadings , , and . Posterior mean estimates of and -9.02 reveal that there are statistically and economically large deviations from the balanced growth path by the and models. The model is closer to satisfying the balanced growth hypothesis that . This UC model has a posterior mean of -0.72 for whose two standard deviation interval contains the balanced growth restriction. The response of the Canadian-U.S. money stock differential to is also close to negative one for the model because the posterior mean of with a standard deviation of 0.21. The model reveals that a one percent rise in results in a 4.4 percent rise in the Canadian-U.S. consumption differential.

The key economic parameter of the DSGE-PVM is its discount factor . Table 5 lists the posterior means and standard deviations of the and , and vectors that include estimates of . Aside from the inclusion of the prior, posterior mean, and standard deviation at the top of table 5, the posterior means and standard deviations of the remaining coefficients resemble those reported in table 4. The only notable exceptions are that the posterior means of , , , and are smaller for the model compared to its cousin with the calibration . The result is that is the largest innovation shock standard deviation of the model. Also, this UC model and the data produce an estimate of whose one standard deviation coverage interval contains negative one. Thus, the model is closer to the balanced growth hypothesis and relies to a greater extent on permanent shocks to the Canadian-U.S. money stock differential.

The posterior means of range from 0.966 for the model, to 0.974 for the model, to the largest estimate of 0.9962 for the model. These estimates are consistent with annual world real interest rates of 15.1, 11.4, and 1.7 percent given the posteriors of the , and models, respectively. Although the , models have posteriors that suggest unreasonably large world real interest rates, these UC models yield 95 percent coverage intervals whose upper end is 0.999. The model produces a posterior of with a 95 percent coverage interval whose lower end equals 0.987. This value of is greater than the posterior means of for the and models. Thus, the model generates a posterior distribution of that is to the right of those produced by the and models.

Figure 1 reinforces the view that the model posteriors yield estimates of that are to the right of those of the and models. Posterior densities of appear in figure 1 for these UC models, along with the density of the inverse-gamma prior restricted to the EW prior of . The solid (black) line is the prior density. It is close to the posterior density of derived from the model, which is the dashed (blue) line. The model generates a posterior density of , the dot-dash (green) plot, that moves off the prior by placing less weight on s less than 0.97 and more weight above it. The dot-dot (red) plot is the density of from the model posterior. This density is deflated by ten percent to ease comparison to the other densities. A striking feature of figure 1 is that the model posterior pushes off of its prior because its mass lays between 0.98 and 0.999.

Table 6 contains the posterior means of the exchange rate factor
loadings with respect to
and
, the
s and
s.^{21} A
striking aspect of the estimates of
,
, and
is that the
response of the
*CDN*$/*US*$ exchange rate to innovations in
is economically small for
either the
or
models. The
large posterior standard errors on these factor loading also
indicate the imprecision the Canadian-U.S. data give to these
estimates. The data yield a more precise estimate of
for the
model. The posterior mean of
this factor loading shows that the exchange rate falls by 0.6
percent given a one percent increase in
. These estimates drop to
-0.33 for the
model. Also, the
associated 95 percent coverage interval contains zero. In summary,
the
,
, and
models have
posteriors in which there is either a negligible exchange rate
response to
shocks or an economically
large negative reaction by the
*CDN*$/*US*$ exchange rate to
fluctuations. However, the
latter exchange rate response is sometimes estimated
imprecisely.

Tables 7 and 8 present unconditional variance ratios and FEVDs
computed using the posteriors of the
,
, and
models. We
calculate the variances of the present discounted values (PDVs) of
the money, TFP, and consumption shock innovations using the
DSGE-PVM version of the equilibrium currency return generating
equation
(14)
and UC model restrictions when is estimated.
The variance ratios are these values divided by the sample variance
of the
*CDN*$/*US*$ exchange rate ( 2.04). According to the unconditional variance ratios, only
permanent shocks to the Canadian-US money differential,
, and the TFP
differential,
, explain variation in the
*CDN*$/*US*$ exchange rate. The variances
of the PDVs of shock innovations to
and
are small and lack
precision. Note that except for the
model, the
variance of the PDV of
is larger than that of
.

We report FEVDs in table 8 with implications similar to the
unconditional variance ratios.^{22} The top panel of figure 8 shows
that the posterior of the
model yields a FEVD in which
the TFP shock
makes a large and
increasing contribution to exchange rate fluctuations at longer
horizons. The money shock
remains economically
important for exchange rate movements out to a three to five year
horizon, but shocks to
and
are unimportant at any
horizon. Much the same is true for the FEVDs found using the
model posterior.
However, the relative shares of the
and
shocks are unchanged at a
two-thirds/one-third split from the one quarter to ten year
horizons.

The posterior of the model imbues a sluggish dynamic to the exchange rate FEVDs found in the bottom panel of table 8. The and shocks are responsible for about 60 and 40 percent, respectively, of fluctuations in the exchange rate at short horizons. At a 10 year horizon, the contribution of ( ) only falls (rises) to 55 (45) percent. Thus, only the posterior of the model predicts that permanent shocks to money dominate exchange rate movements at longer horizons.

Trend-cycle decompositions of the
*CDN*$/*US*$ exchange rate and
Canadian-U.S. money and consumption differentials are plotted in
figures 2 and 3 with summary statistics given in table 9. We run
,
, and
model
posteriors through the Kalman smoother to create trend-cycle
decompositions and summary statistics. Figures 2 and 3 and table 9
contain moments that are averages over 750,000 draws from UC model
posteriors. Trend exchange rate growth is labeled
in table 9.

The top window of figure 2 contains plots of the exchange rate
and smoothed trends taken from the posteriors of the
and
models.^{23} The
solid (black) line is , the log of the actual
exchange rate. The smoothed trends of the
and
models are the
dashed (blue) and dotted (red) plots, respectively. Note that these
UC models generate smoothed exchange rate trends that are more
volatile than the actual exchange rate. The top row of table 9
indicate that the posteriors of the
and
models generate
standard deviations of
equal to 2.66 and 2.44,
respectively. The standard deviation of
equals 2.04.

The smoothed exchange rate cycles appear in the bottom window of figure 2. The dotted (blue) line is the smoothed exchange rate cycle, , based on the posterior of the model, while the dotted (red) line is associated with the model. Although the former exhibits more variability than the latter (the standard deviations are 3.68 and 2.44), these s are persistent with AR1 correlation statistics of 0.97 and 0.98. Note that only the posterior of the model yields a (close to) non-zero correlation for and , according to table 9.

Figure 3 depicts smoothed permanent-transitory decompositions of the Canadian-U.S. money and consumption differentials. The actual differentials and smoothed trends appear in the top row of windows, while smoothed cycles are found in the bottom row of windows. The money (consumption) differentials are the right (left) side windows. The posterior of the model produces a money trend, that almost perfectly mimics the actual Canadian-U.S. money differentials, as shown in the top left window of figure 3. The result is that smoothed is much less volatile, with a standard deviation of 0.68 compared to a standard deviation of 1.62 for . The bottom left window of figure 3 shows a saw-toothed pattern in , conditional on the posterior of the model. This explains the AR1 correlation statistic of -0.68 for (middle of the second column of table 9).

Table 9 reveals that the posterior of model produces a smoothed money trend, , that is about as volatile as in the model. The relevant standard deviations are 1.62 and 1.71 (second and fourth columns of table 9). However, the smoothed and have a positive correlation of 0.62 only in the posterior of the model (bottom half of the fourth column of table 9).

The posteriors of the and models yield qualitatively similar plots for the smoothed TFP trend differential, . These plots appear in the top right window of figure 3 as dashed (blue) and dotted (red) lines for the and models, respectively, where observed cross country consumption is the solid (black) line. The smoothed TFP growth differential, , is 50 percent more volatile for the model than it is for the model. The posteriors of these two UC models also produce correlations of -0.71 and -0.85 between and . Thus, a rising U.S. TFP is associated with an appreciation of the U.S. dollar. The late 1970s is one such period because Canada experienced a greater relative productivity slowdown. By the 1980s, Canadian TFP is growing more rapidly than in the U.S., which continues into the early 1990s. Subsequently, U.S. TFP recovers relative to Canadian TFP. At the end of the sample, the Canadian-U.S. TFP differential is expanding once more.

The bottom right window of figure 3 presents the smoothed of the and models. The former cycle is the dashed (blue) line and the latter is the dotted (red) plot. These cycles are persistent, with AR1 correlation statistics of 0.97 and 0.98, but the model generates a third less volatility in smoothed than found for the model.

The smoothed has peaks and troughs that coincide with several U.S.-Canadian business cycle dates. For example, troughs in the posterior mean of appear in 1981 and 1990, which also represent recessions dates in the U.S. and Canada. Since the end of the 1990-1991 recession, the rise in points to a persistent, but transitory, rise in U.S. consumption relative to Canada. Nonetheless, has been falling rapidly since a peak in late 2001, which corresponds to the end of the last U.S. recession.

The bottom row of table 9 shows that
and
are perfectly negatively
correlated in the posteriors of the
and
models. The
negative correlation of the transitory component of the exchange
rate with
helps to interpret
exchange rate fluctuations. Peaks in the transitory component of
the exchange rate occur either at or shortly after the end of
recessions. For example, the smoothed exchange rate cycles have a
tendency to peak and trough around dates usually associated with
U.S. and Canadian business cycle dates (*i.e.*, the late
1970s, early 1990s, and 2001). A specific case is the peak in
during the 1990-1991
recession in the U.S., which is a moment at which the Canadian
dollar approached par against the U.S. dollar. An exception is the
end of the 2001 recession when the Canadian dollar reached a low of
nearly 0.62 to the U.S. dollar.

The bottom row of table 4 reports the log marginal likelihoods,
, of the
,
, and
models. These
marginal likelihoods show that our Canadian-U.S. sample gives most
support to the
model. The
difference between this model and the
model is about 29, so that
the Bayes factor prefers the UC model with only transitory
consumption. For the data to give more credence to the latter
model, its prior probability must be raised by the prior
probability of the
model
multiplied by 4.7 × 10^{12}[=exp(29.18)]. Since
the magnitude of this factor is large, it seems unreasonable to
include the transitory money shock in the UC model when is calibrated to one.

The last row of table 5 contains the log marginal likelihoods of the , , and models. The ranking of these models matches that of the UC models with the calibration. The model dominates the and models. A key reason is that the posteriors of these models yield economically implausible estimates of the DGSE-PVM discount factor .

This raises the question of whether it is difficult to choose
between the
and
models. Our
sample favors the
and
models compared
to the other four. The
model has the
largest marginal likelihood, which suggests that the data support
it over the
model. This
choice relies on the belief that scaling up the prior probability
of the
model by 167.3 = exp(5.12)
is too large to be
justified. If, on the other hand, this factor is regarded as
inconclusive in rejecting the
model, it could
be argued that our Canadian-U.S. sample cannot pick between the
and
models.^{24}
Nonetheless, these UC models support for the EW hypothesis.

Engel and West (2005) argue that the exchange rate will approximate a random walk when the discount factor is close to one and fundamentals have a unit root. Propositions 3 and 5 also predict that will collapse to random walk as .

We extract evidence about the EW hypothesis from the and model posteriors. The focus is on these UC models rather than the model because it attributes all exchange rate movements to permanent shocks. We conduct this comparison with s at the 16th and 84th percentiles, along with the largest s, from the and vectors. For the ( ) model, the 16th percentile, 84th percentile, and largest s are 0.9425, 0.9883, and 0.9990 (0.9943, 0.9987, and 0.9990), respectively. Fixing at these values, we simulate the and models extracting 2000 draws from the posteriors, discard the first 1000, run the Kalman smoother on the remaining 1000, and average the ensemble to generate exchange rate cycles that respect the rational expectations hypothesis.

Figure 4 plots the smoothed exchange rate cycles. The top (bottom) window contains the created from the posterior of the ( ) model. The dot-dash (blue), dotted (green), and dotted (red) lines are conditional on the 16th percentile, 84th percentile, and largest s, respectively. Across the top and bottom windows, the volatility of is compressed as approaches 0.999. This is reflected in the standard deviations of that equal 4.44, 2.84, and 0.57 moving from the smallest to largest for the model. The equivalent standard deviations are 3.74, 1.64, and 1.30 for the model. Although the model generates exchange rate cycles that are smoother than at its posterior mean only for the largest s, this UC model is able to produce smoother exchange rate cycles at the 84th percentile and largest s. Thus, pushing increases the smoothness of the exchange rate cycle. This is evidence that lends credence to the EW hypothesis.

Economists have little to say about the impact of policy on currency markets without an equilibrium theory of exchange rate determination that is empirically relevant. According to Engel and West (2005), the near random walk behavior of exchange rates explains the failure of equilibrium models to fit the data or to find any model that systematically beats it at out-of-sample forecasting. They conjecture that the standard-present value model (PVM) of exchange rates yields the random walk prediction when fundamentals are persistent and the discount factor is close to one.

This paper generalizes the Engel and West (EW) hypothesis by constructing a PVM from a two-country monetary dynamic stochastic general equilibrium (DSGE) model. The standard- and DSGE-PVMs yield identical predictions for the exchange rate. These predictions are summarized by five propositions. Thus, we generalize the EW hypothesis to the larger class of open economy DSGE models.

Our empirical results support the view that the Canadian-U.S. data prefer a random walk exchange rate and a DSGE-PVM with a discount factor calibrated to one. At the same time we obtain evidence on the nature of the shocks driving exchange rates. Bayesian estimates of the DSGE-PVM indicate that the Canadian dollar-U.S. dollar exchange rate is dominated by permanent shocks, whether the discount factor is estimated or calibrated to one, which supports the EW hypothesis. Our evidence is also consistent with the recent VAR literature suggesting that monetary policy shocks have only a minor impact on exchange rate fluctuations. Monetary policy shocks are also found to be unimportant for exchange rate movements by Lubik and Schorfheide (2006) within the context of an estimated open economy DSGE model. Whether this result holds across a wider set of open economy DSGE models is a worthy goal of future research.

Adolfson, M., S.
Laséen, J. Lindé, and M. Villani. 2007. Bayesian
Estimation of an Open Economy DSGE Model with Incomplete
Pass-Through. *Journal of International Economics* 72,
481-511.

An, S., F. Schorfheide. 2007.
Bayesian analysis of DSGE models. *Econometric Reviews* 26,
113-172.

Beveridge, S., C.R. Nelson.
1981. A New Approach to Decomposition of Economic Time Series into
Perman-
ent and Transitory Components with Particular Attention to
Measurement of the Business Cycle. *Journal of Monetary
Economics* 7, 151-174.

Bilson, J.F.O. 1978. Rational Expectations and the Exchange Rate. in Frenkel, J.A., H.G. Johnson, THE ECONOMICS OF EXCHANGE RATES: SELECTED STUDIES, Addison-Wesley, Reading, MA.

Campbell, J.Y., R. Shiller.
1987. Cointegration and Tests of Present Value Models. *Journal
of Political Economy* 93, 1062-1088.

Del Negro, M., F. Schorfheide. 2006. Forming Priors for DSGE Models (and How it Affects the Assessment of Nominal Rigidites). Working Paper 2006-16, Federal Reserve Bank of Atlanta.

Dornbusch, R. 1976.
Expectations and Exchange Rate Dynamics. *Journal of Political
Economy* 84, 1161-1176.

Eichenbaum, M., C. Evans,
1995. Some Empirical Evidence on the Effects of Monetary Policy
Shocks on Exchange Rates. *Quarterly Journal of Economics*
110, 975-1010.

Engel, C., K.D. West, 2005.
Exchange Rates and Fundamentals. *Journal of Political
Economy* 113, 485-517.

Engle, R.F., J.V. Issler,
1995. Estimating Common Sectoral Cycles. *Journal of Monetary
Economics* 35, 83-113.

Engle, R.F., S. Kozicki, 1993.
Testing for Common Features. *Journal of Business and Economics
Statistics* 11, 369-395.

Faust, J., J. Rogers, 2003.
Monetary Policy's Role in Exchange Rate Behavior. *Journal of Monetary Economics* 50, 1403-1424.

Fernández-Villaverde, J., J.F.
Rubio-Ramírez. 2004. Comparing Dynamic Equilibrium Models to
Data: A Bayesian Approach. *Journal of Econometrics* 123,
153-187.

Frankel, J.A., 1979. On the
Mark: A Theory of Floating Exchange Rates Basedon Real Interest
Rate Differentials. *American Economics Review* 69,
610-622.

Gelman, A., J.B. Carlin, H.S. Stern, D.B. Rubin, 2004, BAYESIAN DATA ANALYSIS, SECOND EDITION, Chapman and Hall, New York, NY.

Geweke, J., 1999. Using
Simulation Methods for Bayesian Econometric Models: Inference,
Development and Communication. *Econometric Reviews* 18,
1-126.

Geweke, J., 2005. CONTEMPORARY BAYESIAN ECONOMETRICS AND STATISTICS, J. Wiley and Sons, Inc., Hoboken, NJ.

Hamilton, J.D., 1994, TIME SERIES ANALYSIS, Princeton University Press, Princeton, NJ.

Hansen, L.P., W. Roberds, T.J. Sargent. 1991. Time Series Implications of Present Value Budget Balance and of Martingale Models and Consumption and Taxes. in Hansen, L.P., T.J. Sargent (eds.), RATIONAL EXPECTATIONS ECONOMETRICS, Westview Press, Boulder, CO.

Harvey, A.C., FORECASTING, STRUCTURAL TIME SERIES MODELS AND THE KALMAN FILTER, Cambridge University Press, Cambridge, England.

Harvey, A.C., T.M. Trimbur,
H.K. van Dijk. 2007. Trends and Cycles in Economic Time Series: A
Bayesian Approach. *Journal of Econometrics* 140, 618-649.

Hecq, A., F.C. Palm, J-P.
Urbain. 2000. Permanent-Transitory Decomposition in VAR Models with
Cointegration and Common Cycles. *Oxford Bulletin of Economics
and Statistics* 62, 511-532.

Jeffreys, H., 1998. THEORY OF PROBABILITY, Oxford University Press, Oxford, UK.

Johansen, S. 1991. Estimation
and Hypothesis Testing of Cointegration Vectors in Gaussian Vector
Auto-
regressive Models. *Econometrica* 59, 1551-1580.

Johansen, S. 1994. The Role of
the Constant and Linear Terms in Cointegration Analysis and
Nonstationary Variables. *Econometric Reviews* 13,
205-229.

Kim, S., N. Roubini. 2000.
Exchange Rate Anomalies in the Industrial Countries: A Solution
with a Structural VAR Approach. *Journal of Monetary
Economics* 45, 561-586.

Lubik, T., F. Schorfheide. 2006. A Bayesian Look at New Open Economy Macroeconomics. in Gertler, M., K. Rogoff (eds.), NBER MACROECONOMICS ANNUAL 2005. MIT Press, Cambrigde, MA.

MacKinnon, J.G., A.A. Haug,
L. Michelis. 1999. Numerical Distributions of Likelihood Ratio
Tests of Cointegration. *Journal of Applied Econometrics* 14,
563-577.

Maheswaran, S., C.A. Sims. 1993. Empirical Implications of Arbitrage-Free Asset Markets. in Phillips, P.C.B. (ed.), MODELS, METHODS AND APPLICATIONS OF ECONOMETRICS, Blackwell Publishing, Oxford, UK.

Mark, N. 1995. Exchange Rates
and Fundamentals: Evidence on Long-Horizon Predictability.
*American Economics Review* 85, 201-218.

Mark, N., D. Sul, 2003.
Cointegration Vector Estimation by Panel DOLS and Long-Run Money
Demand. *Oxford Bulletin of Economics and Statistics* 65,
665-680.

Meese, R.A., 1986. Testing
for Bubbles in Exchange Markets: A Case of Sparkling Rates.
*Journal of Political Economy* 94, 345-373.

Morley, J.C., C.R. Nelson,
E. Zivot. 2003. Why Are Beveridge-Nelson and Unobserved Component
Decompositions of GDP So Different? *Review of Economics and
Statistics* 85, 235-234.

Osterwald-Lenum, M. 1992.
Quantiles of the Asymptotic Distribution of the Maximum Likelihood
Cointegration Rank Test Statistics, *Oxford Bulletin of Economics
and Statistics* 54, 461-472.

Rabanal, P., J.F.
Rubio-Ramírez. 2005. Comparing New Keynesian Models of the
Business Cycle: A Bayesian Approach. *Journal of Monetary
Economics* 52, 1151-1166.

Rao, C.S. 1973. LINEAR STATISTICAL INFERENCE, Wiley, Ltd., New York, NY.

Rogers, J.H., 2000. Monetary
Shocks and Real Exchange Rates. *Journal of International
Economics* 49, 269-288.

Rogoff, K. 2007. Comments on: Engel, Mark, and West's Exchange Rate Models Are Not as Bad as You Think. in Acemoglu, D., K. Rogoff, M. Woodford (eds.), NBER MACROECONOMICS ANNUAL 2007, VOL. 22, MIT Press, Cambridge, MA.

Rossi, B. 2007. Comments on: Engel, Mark, and West's Exchange Rate Models Are Not as Bad as You Think. in Acemoglu, D., K. Rogoff, M. Woodford (eds.), NBER MACROECONOMICS ANNUAL 2007, VOL. 22, MIT Press, Cambridge, MA.

Scholl, A., Uhlig, H., 2005. New Evidence on the Puzzles. Results from Agnostic Identification on Monetary Policy and Exchange Rates. SFB 649 Discussion Paper No. 2005-037.

Sims, C.A. 1980.
Macroeconomics and Reality. *Econometrica* 48, 1-48.

Sims, C.A., Stock, J.H.,
M.W. Watson. 1990. Inference in Linear Time Series Models with Some
Unit Roots. *Econometrica* 58, 113-144.

Stock, J.H., M.W. Watson.
1988. Testing for Common Trends. *Journal of the American
Statistical Association* 83, 1097-1107.

Vahid, F., R.F. Engle. 1993.
Common Trends and Common Cycles. *Journal of Applied
Econometrics* 8, 341-360.

Table 1: Summary of Standard PVM and DSGE-PVM: Panel A: Standard-PVM

ECM(0): | (13) . |
---|---|

EW Equation: | (14) . |

Parameters: |
Discount Factor, Money Demand Interest Rate Semi-Elasticity, Money Demand Income Elasticity. |

Fundamentals: |
, , Cross Country Money, Cross-Country Output. |

Table 1: Summary of Standard PVM and DSGE-PVM: Panel B: DSGE-PVM

ECM(0): |
(13)
. |
---|---|

EW Equation: |
(14)
. |

Parameters: |
Discount Factor, Steady State Real World Interest Rate. |

Fundamentals: |
, Cross Country Money, Cross-Country Output. |

Table 2: Summary of Propositions for Standard- and DSGE-PVMs

Proposition
1: |
PVM Predicts Exchange Rate and Fundamentals Cointegrate; Campbell and Shiller (1987). |
---|---|

Proposition
2: |
Currency Returns Are an ECM(0). |

Proposition
3: |
Exchange Rate Approximates a Martingale as . |

Proposition
4: |
VECM(0) Imply Common Trend and Common Cycle for Exchange Rate and Fundamental. |

Proposition
5: |
EW's (2005) Hypothesis Needs Currency Returns and Fundamental Growth Share a Co-Feature and . |

Table 3: Tests of Propositions 1, 3, and 5: Sample 1976Q1 - 2004Q4: Panel A: Proposition 3: VECM(0)

Statistic | Canada & U.S. | Japan & U.S. | U.K. & U.S. |
---|---|---|---|

Levels VAR Lag Length | 8 | 5 | 4 |

LR statistic p-value | (0.02) | (0.01) | (0.09) |

Table 3: Tests of Propositions 1, 3, and 5: Sample 1976Q1 - 2004Q4: Panel B: Proposition 1: Common Trend Cointegration Tests

Model | Canada & U.S. Case 2* | Japan & U.S. Case 1 | U.K. & U.S. Case 1 |
---|---|---|---|

λ-Max statistic: Two Unit Root | 4.86 | 0.20 | 2.27 |

λ-Max statistic: One Unit Root | 17.28 | 4.64 | 12.32 |

Trace statistic: Two Unit Root | 4.86 | 0.20 | 2.27 |

Trace statistic: One Unit Root | 12.42 | 4.43 | 10.04 |

Table 3: Tests of Propositions 1, 3, and 5: Sample 1976Q1 - 2004Q4: Panel C: Proposition 5: Common Cycle

Statistic | Canada & U.S. | Japan & U.S. | U.K. & U.S. |
---|---|---|---|

Sq. Canonical Correlations | 0.30, 0.09 | 0.44, 0.08 | 0.19, 0.07 |

χ^{2}-statistic p-value | (0.01), (0.69) | (0.00), (0.21) | (0.00), (0.12) |

F-statistic p-value | (0.00), (0.61) | (0.00), (0.19) | (0.00), (0.11) |

The level of fundamentals equals cross-country money netted with
cross-country output calibrated to a unitary income elasticity of
money demand. The money stocks (outputs) are measured in current
(constant) local currency units and per capita terms. A constant
and linear time trend are included in the level VARs. The LR
statistics employ the Sims (1980)
correction and have standard asymptotic distribution according to
results in Sims, Stock, and Watson (1990). The case 2* and
case 1 model definitions are based on Osterwald-Lenum (1992). MacKinnon, Haug, and Michelis
(1999) provide five percent
critical values of 8.19 (8.19) and 18.11 (15.02) for the case
2* model λ-max
(trace) tests and 3.84 (3.84) and 15.49 (14.26) for the case 1
model. The common feature tests compute the canonical correlations
of Δ*e _{t}* and Δ

Table 4: UC Model Posterior Means, *κ* = 1

Parameter | Priors | |||
---|---|---|---|---|

Normal | -1.1906 | -0.8691 | - | |

: [parameters] (std. devs) | [-1.2, 0.10] | (0.0613) | (0.0470) | - |

Normal | 0.4133 | 0.9501 | - | |

: [parameters] (std. devs) | [0.40, 0.17] | (0.1092) | (0.0528) | - |

Normal | 0.9407 | - | 0.9830 | |

: [parameters] (std. devs) | [0.85, 0.10] | (0.0491) | - | (0.0296) |

Normal | 0.0403 | - | 0.0069 | |

: [parameters] (std. devs) | [0.10, 0.15] | (0.0488) | - | (0.0291) |

Normal | -0.1260 | -0.1258 | -0.1258 | |

: [parameters] (std. devs) | [-0.126, 0.015] | (0.0150) | (0.0150) | (0.0120) |

Normal | 0.1615 | 0.1645 | 0.1571 | |

: [parameters] (std. devs) | [0.158, 0.025] | (0.0229) | (0.0213) | (0.0199) |

Inv-Gamma | 1.8838 | 2.4629 | 1.6784 | |

: [parameters] (std. devs) | [2.0, 1.5] | (0.1436) | (0.1507) | (0.1264) |

Inv-Gamma | 1.0471 | 0.3971 | 1.9461 | |

: [parameters] (std. devs) | [2.0, 0.4] | (0.2206) | (0.0345) | (0.3831) |

Inv-Gamma | 0.6002 | 0.4728 | - | |

: [parameters] (std. devs) | [2.0, 0.6] | (0.0899) | (0.1168) | - |

Inv-Gamma | 1.2874 | - | 2.0135 | |

: [parameters] (std. devs) | [2.0, 0.7] | (0.2354) | - | (0.3823) |

Normal | -0.8758 | - | -0.9475 | |

: [parameters] (std. devs) | [-0.5, 0.2] | (0.0462) | - | (0.0234) |

Normal | 80.1528 | 138.8984 | 62.9442 | |

: [parameters] (std. devs) | [100.0, 15.0] | (6.8283) | (5.7281) | (2.9991) |

Normal | 0.7038 | 1.9366 | 0.3831 | |

: [parameters] (std. devs) | [1.0, 0.5] | (0.1710) | (0.1106) | (0.1306) |

Uniform | -2.6822 | -9.0223 | -0.7208 | |

: [parameters] (std. devs) | [-10.0, 0.0] | (0.6316) | (0.5377) | (0.1886) |

Uniform | - | 4.3973 | - | |

: [parameters] (std. devs) | [-2.0, 7.5] | - | (1.1057) | - |

Uniform | - | - | -0.8985 | |

: [parameters] (std. devs) | [-7.5, 2.0] | - | - | (0.2099) |

- | -53.95 | -226.76 | -24.76 |

Table 5: UC Model Posterior Means, *κ* ∈ [0.9, 0.999]

Parameter | Priors | |||
---|---|---|---|---|

Inv-Gamma | 0.9658 | 0.9738 | 0.9962 | |

: [parameters] (std. devs) | [0.988, 0.038] | (0.0219) | (0.0196) | (0.0046) |

Normal | -1.1892 | -0.8828 | - | |

: [parameters] (std. devs) | [-1.20, 0.10] | (0.0673) | (0.0369) | - |

Normal | 0.4131 | 0.8465 | - | |

: [parameters] (std. devs) | [0.40, 0.17] | (0.1179) | (0.0223) | - |

Normal | 0.9396 | - | 0.9799 | |

: [parameters] (std. devs) | [0.85, 0.10] | (0.0431) | - | (0.0315) |

Normal | 0.0421 | - | 0.0018 | |

: [parameters] (std. devs) | [0.10, 0.15] | (0.0422) | - | (0.0314) |

Normal | -0.1256 | -0.1260 | -0.1260 | |

: [parameters] (std. devs) | [-0.126, 0.015] | (0.0148) | (0.0149) | (0.0147) |

Normal | 0.1621 | 0.1630 | 0.1578 | |

: [parameters] (std. devs) | [0.158, 0.025] | (0.0227) | (0.0205) | (0.0243) |

Inv-Gamma | 1.8914 | 2.4241 | 1.7188 | |

: [parameters] (std. devs) | [2.0, 1.5] | (0.1484) | (0.1663) | (0.1112) |

Inv-Gamma | 1.0842 | 0.4043 | 1.5738 | |

: [parameters] (std. devs) | [2.0, 0.4] | (0.2477) | (0.0378) | (0.2727) |

Inv-Gamma | 0.6068 | 0.5752 | - | |

: [parameters] (std. devs) | [2.0, 0.6] | (0.0865) | (0.0975) | - |

Inv-Gamma | 1.3828 | - | 1.6742 | |

: [parameters] (std. devs) | [2.0, 0.7] | (0.2346) | - | (0.2772) |

Normal | -0.8990 | - | -0.9256 | |

: [parameters] (std. devs) | [-0.5, 0.2] | (0.0409) | - | (0.0291) |

Normal | 85.2271 | 135.7241 | 67.8714 | |

: [parameters] (std. devs) | [100.0, 15.0] | (6.7620) | (6.3363) | (3.7506) |

Normal | 0.7955 | 1.9076 | 0.4526 | |

: [parameters] (std. devs) | [1.0, 0.5] | (0.1774) | (0.1211) | (0.1112) |

Uniform | -3.2825 | -8.8023 | -1.2605 | |

: [parameters] (std. devs) | [-10.0, 0.0] | (0.6426) | (0.5900) | (0.3087) |

Uniform | - | 4.4186 | - | |

: [parameters] (std. devs) | [-2.0, 7.5] | - | (0.7952) | - |

Uniform | - | - | -1.0380 | |

: [parameters] (std. devs) | [-7.5, 2.0] | - | - | (0.2179) |

- | -53.94 | -253.03 | -29.88 |

Table 6: UC Model Posterior Means, *κ* ∈ [0.9, 0.999], Factor Loadings on Money and Consumption Cycles

Parameter | |||
---|---|---|---|

0.0086 | -0.0841 | - | |

std. dev. | (0.0096) | (0.0676) | - |

-0.0269 | 0.0064 | - | |

std. dev. | (0.0188) | (0.0082) | - |

0.0143 | -0.0762 | - | |

std. dev. | (0.0108) | (0.0624) | - |

-0.0040 | -0.1542 | - | |

std. dev. | (0.0202) | (0.1237) | - |

-0.6044 | - | -0.3252 | |

std. dev. | (0.2042) | - | (0.2052) |

-0.0238 | - | -0.0004 | |

std. dev. | (0.0261) | - | (0.0111) |

-0.6283 | - | -0.3256 | |

std. dev. | (0.2117) | - | (0.2053) |

^{†} The factor loadings
and
equal zero for the
model.

Table 7: US Model Posterior Means, *κ* ∈ [0.9, 0.999], Variance(PDV-*ε*) / Variance(Δ*e*)

Parameter | |||
---|---|---|---|

0.92 | 1.48 | 0.71 | |

std. dev. | (0.15) | (0.22) | (0.09) |

3.04 | 0.36 | 0.96 | |

std. dev. | (0.73) | (0.06) | (0.53) |

0.00 | 0.00 | - | |

std. dev. | (0.00) | (0.00) | - |

0.22 | - | 0.12 | |

std. dev. | (0.16) | - | (0.22) |

Table 8: UC-Models, *κ* ∈ [0.9, 0.999], Exchange Rate FEVDs^{†}: Panel A: Model

Forecast Horizon | |||
---|---|---|---|

1 | 0.23 | 0.77 | 0.00 |

4 | 0.23 | 0.77 | 0.00 |

12 | 0.21 | 0.79 | 0.00 |

20 | 0.19 | 0.80 | 0.01 |

40 | 0.15 | 0.82 | 0.03 |

Table 8: UC-Models, *κ* ∈ [0.9, 0.999], Exchange Rate FEVDs^{†}: Panel B: Model

Forecast Horizon | |||
---|---|---|---|

1 | 0.32 | 0.68 | - |

4 | 0.32 | 0.68 | - |

12 | 0.32 | 0.68 | - |

20 | 0.32 | 0.68 | - |

40 | 0.32 | 0.68 | - |

Table 8: UC-Models, *κ* ∈ [0.9, 0.999], Exchange Rate FEVDs^{†}: Panel C: Model

Forecast Horizon | |||
---|---|---|---|

1 | 0.59 | 0.40 | 0.01 |

4 | 0.58 | 0.41 | 0.01 |

12 | 0.57 | 0.42 | 0.01 |

20 | 0.57 | 0.43 | 0.01 |

40 | 0.55 | 0.45 | 0.01 |

^{†} The summary statistics are the
means of the ensemble of FEVDs with respect to permanent and
transitory Canadian-U.S. money differential and Canadian-U.S.
consumption differential shocks generated from the
,
, and
model
posterior distributions.

Table 9: UC-Models, *κ* ∈ [0.9, 0.999], Summary of the Trend-Cycle Decomposition^{†}

Parameter | |||
---|---|---|---|

2.66 | 104.28 | 2.44 | |

3.68 | 106.84 | 2.55 | |

0.97 | 0.55 | 0.98 | |

-0.06 | -0.37 | -0.02 | |

1.62 | 4.54 | 1.71 | |

0.68 | 3.03 | - | |

-0.68 | 0.13 | - | |

0.26 | -0.24 | - | |

0.99 | 12.18 | 1.56 | |

5.73 | - | 7.73 | |

0.97 | - | 0.98 | |

-0.16 | - | -0.14 | |

0.01 | -0.40 | 0.62 | |

-0.85 | -0.93 | -0.71 | |

0.52 | 0.55 | 0.10 | |

0.04 | -0.73 | - | |

-1.00 | - | -1.00 |

^{†} The summary statistics are the
means of the ensemble of
*CDN*$/*US*$ exchange rate,
Canadian-U.S. money differential, and Canadian-U.S. consumption
differential trends and cycles generated from the
,
, and
model
posterior distributions.

Figure 1: Prior and Posterior PDFs of DSGE-PVM Discount Factor

Figure 2: CDN$/US$ Exchange Rate Trend and Cycle, 1976Q1 - 2004Q4

Figure 3: CDN-US Money, Consumption Trends and Cycles, 1976Q1 - 2004Q4

Figure 4: CDN$/US$ Ex Rate Cycles at Different DSGE-PVM Discount Factors

* We wish to thank Toni Braun, Fabio Canova, Menzie Chinn, Frank Diebold, John Geweke, Fumio Hayashi, Sharon Kozicki, Adrian Pagan, Juan F. Rubio-Ramírez, Tom Sargent, Pedro Silos, Ellis Tallman, Noah Williams, Farshid Vahid, Kenji Wada, Ken West, Tao Zha, the Federal Reserve Bank of Atlanta macro lunch study group, and seminar participants at the 2006 NBER Summer Institute Working Group on Forecasting and Empirical Methods in Macroeconomics and Finance, the 2007 Norges Bank Workshop on, "Prediction and Monetary Policy in the Presence of Model Uncertainty", the 2007 Bank of Canada-European Central Bank Exchange Rate Modeling Workshop, the 2007 Reserve Bank of Australia Research Workshop, "Monetary Policy in Open Economies", Ohio State, Houston, Tokyo, and Washington University for useful comments. The views in this paper represent those of the authors and are not those of either the Federal Reserve Bank of Atlanta, the Board of Governors of the Federal Reserve System, or any of its staff. Errors in this paper are the responsibility of the authors. Return to text

^{†} Authors' contact details: James M. Nason, Research Department, Federal Reserve Bank of Atlanta, 1000 Peachtree St., N.E., Atlanta, GA 30309, email: jim.nason@atl.frb.org. John H. Rogers, International Finance Division, Board of Governors of the Federal Reserve System, Washington, D.C. 20551, email: john.h.rogers@frb.gov. Return to text

1. Actual data most often rejects the standard-PVM. Typical are tests Meese (1986) reported that employed the first ten years of the floating rate regime. He finds that exchange rates are infected with persistent deviations from fundamentals, which reject the standard-PVM and its cross-equation restrictions. However, Meese is unable to uncover the source of the rejections. Instead of a condemnation of the standard-PVM, we view results such as Meese's as a challenge to update and deepen its analysis. Return to text

2. The open economy VAR literature offers mixed evidence on the importance of various shocks for the exchange rate. Early papers including Eichenbaum and Evans (1995), Rogers (2000) and Kim and Roubini (2000) found some significance for identified monetary shocks. Recent contributions, however, suggest that monetary policy shocks have only a minor impact on exchange rate fluctuations, consistent with Rogoff's view, for example, see Faust and Rogers, (2003) and Scholl and Uhlig (2005). Return to text

3. The present-value relation (3) yields the weak prediction that Granger-causes . Engel and West (2005) and Rossi (2007) report that this prediction is often not rejected in G -7 data. Return to text

4. The restrictions on the moving average are is linearly deterministic, , is an infinite order lag polynominal with roots outside the unit circle, the s are square summable, and is mean zero, homoskedastic, linearly independent given history and is serially uncorrelated with itself and the past of . Return to text

5. The error is also justified if the econometrician's information set is strictly within that of currency traders. Return to text

6. Maheswaran and Sims (1993) show that the martingale restriction has little empirical content for tests of asset pricing models when data is sampled at discrete moments in time. Return to text

7. Hansen, Roberds, and Sargent (1991) study linear rational expectations models that anticipate Proposition 3. Return to text

8. This analysis matches equations A.3 - A.11 and the surrounding discussion of Engel and West (2005). Return to text

9. Vahid and Engle show a *n*-dimension VAR(1) with *d* cointegrating
relations has *n* - *d* common feature relations. Return to
text

10. Proposition 5 can also be cast as an implication of the BNSW representation of . In this case, removes the vector MA in and from the BNSW representation of . Only a linear combination of pure forecast innovations, and , are left to drive . Let 0 to obtain the random walk exchange rate with innovation . Return to text

11. Fundamentals equal cross-country money minus cross-country output, which implies an income elasticity of money demand, , calibrated to one. This calibration is consistent with estimates reported by Mark and Sul (2003). The money stocks (outputs) are measured in current (constant) local currency units and per capita terms. Return to text

12. The VARs include a constant and linear time trend. The LR statistics employ the Sims (1980) correction and have standard asymptotic distribution according to results in Sims, Stock, and Watson (1990). Return to text

13. The Canadian-U.S. and Japanese-U.S.
VARs are selected when the *p*-value of the LR
test is five percent or less. Since the U.K.-U.S. VAR offers
ambiguous results, we settle on a VAR(4). Return to text

14. We thank Farshid Vahid for suggesting this example. Return to text

15. A related example is Harvey, Trimbur, and van Dijk (2007) who use Bayesian methods to estimate permanent-transitory decompositions of aggregate time series, but without rational expectations cross-equation restrictions. Return to text

16. Morley, Nelson, and Zivot (2003) show that this restriction applied to an univariate UC model resolves its differences with the Beverage and Nelson (1981) decomposition. Return to text

17. The factor loading on the permanent component of remains (normalized to) one. Return to text

18. The appendix discusses the data and explains, for example, that Canadian consumption includes semi-durable expenditures. Return to text

19. The posterior distributions are
based on acceptance rates of between 25 and 36 percent. Besides the
750,000 MCMC draws used to compute the moments reported below, four
more sequences of 750,000 MCMCs are generated from disparate
starting values to assess across chain and within chain
convergence. We compute the
statistic of Gelman, Carlin,
Stern, and Rubin (2004) to
evaluate across chain across and the separated partial means test
of Geweke (2005) convergence, which
is distributed asymptotically . Across the
77 parameters of the six UC models, the two largest
s are 1.20 and 1.04, while
Gelman, et al suggest a
of about 1.10. On five
subsamples, the Geweke separated partial means test has no
*p*-value smaller than 0.21 across the six UC
models and five MCMC simulation sequences. Return to text

20. The half life equals log[0.5]/log[*q*], where *q* is the largest modulus of the companion matrix of the AR or MA
coefficients. Return to text

21. For the and models, the relevant factor loadings are multiplied by or . Return to text

22. The FEVDs are computed using the VECM implied by equation (13). The VECM is placed in state space form as outlined by Heqc, Palm, and Urbain (2000) and iterated to create the FEVDs of table 8. Return to text

23. We do not present the trend-cycle decompositions based on the posterior of the model because its log marginal likelihood is far below those of the other UC models. Table 9 includes standard deviations of and from the posterior of the model that are larger by a factor of 30 or compared to these statistics from the and models. This signals the lack of acceptance of the model by our Canadian-U.S. sample. Return to text

24. Jeffreys (1998, p. 432) contends that Bayes factors differing by 3.16 is evidence about the two models just between 'not worth more than a bare mention' and substantially in favor of the model with the larger marginal likelihood. Return to text

This version is optimized for use by screen readers. Descriptions for all mathematical expressions are provided in LaTex format. A printable pdf version is available. Return to text