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Board of Governors of the Federal Reserve System
International Finance Discussion Papers
Number 948, September 2008 --- Screen Reader Version*

Exchange Rates and Fundamentals: A Generalization^*

James M. Nason and John H. Rogers^†

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publications to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors.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

1  Introduction

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.

2  Two Present-Value Models of Exchange Rates

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.

2.A  The Standard Present-Value Model of Exchange Rates

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

$$m_{h,t} \; - \; p_{h,t} \; \; = \; \; \psi\hspace{0.01in} y_{h,t} \; - \; \phi\hspace{0.01in} r_{h,t}, \quad0 \; < \; \psi, \, \phi,$$ (1)

where $m_{h,t}$ $\,p_{h,t}$, $\,y_{h,t}$, and $r_{h,t}$ denote the home country's natural logarithm of money stock, price level, output, and the level of the nominal interest rate. The parameter $\psi$ measures the income elasticity of money demand. Since the nominal interest rate is in its level, $\phi$ is the interest rate semi-elasticity of money demand. Define cross-country differentials $m_{t}$ $=$ $m_{h,t}$ $-$ $m_{f,t}$, $\,p_{t}$ $=$ $p_{h,t}$ $-$ $p_{f,t}$, $\,y_{t}$ $=$ $y_{h,t}$ $-$ $y_{f,t}$, and $\,r_{t}$ $=$ $r_{h,t}$ $\,-\,$ $r_{f,t}$, where $f$ denotes the foreign country. Assuming PPP holds, $e_{t}$ $=$ $p_{t}$, where $e_{t}$ 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

$$\mathbf{E}_{t} e_{t+1} \, - \, e_{t} \; \; = \; \; r_{t}.$$ (2)

Substitute for $r_{t}$ in the law of motion of the exchange rate (2) with the money demand function (1) and impose PPP to produce the Euler equation $e_{t} - \omega\mathbf{E}_{t} e_{t+1} = (1 - \omega) \bigl ( m_{t} - \psi y_{t} \bigr )$ , where the standard-PVM discount factor is $\omega\equiv\frac{\displaystyle \phi}{ \displaystyle 1+\phi}$ and $m_{t} - \psi y_{t}$ 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 $\lim_{T \, \rightarrow\infty }\omega^{T+1} \mathbf{E}_{t} e_{t+T} = 0$ to obtain the standard PVM relation

$$e_{t} \; \; = \; \; (1 \, - \, \omega) \sum_{j=0}^{\infty} \omega^{j} \mathbf{E}_{t} \left\{ m_{t+j} \, - \, \psi y_{t+j} \right\} .$$ (3)

The standard PVM (3) sets the log exchange rate equal to the annuity value of the fundamental $m_{t} - \psi y_{t}$ at the standard-PVM discount factor $\omega$.³

2.B  The DSGE Model

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

$$\mathcal{U}\left( C_{i,t}, \, \, \frac{M_{i,t}}{P_{i,t}} \right) \; \; = \; \; \frac{\left[ C_{i,t}^{ \displaystyle \nu}\left( \frac {\displaystyle M_{i,t} }{\displaystyle P_{i,t}} \right) ^{\displaystyle (1 - \nu)}\right] ^{ \displaystyle (1 - \varphi)}}{1 - \varphi}, \; \; \; \; 0 \, < \, \nu\, < 1, \; \; \; 0 \, < \, \varphi,$$ (4)

where $C_{i,t}$ and $M_{i,t}$ represent the $i$th country's consumption and the $i$th country's holdings of its money stock. The resource constraint of the home country is

$$\qquad B^{h}_{h,t} \, + \, s_{t} B^{f}_{h,t} \, + \, P_{h,t} C_{h,t} \, + \, M_{h,t} \; = \; (1 + r_{h,t-1}) B^{h}_{h,t-1} \, + \, s_{t} (1 + r_{f,t-1}) B^{f}_{h,t-1} \, + \, M_{h,t-1} \, + \, P_{h,t} Y_{h,t},$$ (5)

where $B^{i}_{i,t}$, $B^{\ell}_{i,t}$, $r_{i,t-1}$, $r_{\ell ,t-1}$, $Y_{i,t}$, and $s_{t}$ denote the ith country's nominal holding of its own bonds at the end of date t, the ith country's nominal holding of the $\ell$th country's bonds at the end of date t, the return on the ith country's bond, the return on the $\ell$th country's bond, the output level of the ith country, and the level of the exchange rate. The two-country DSGE model is closed with $B^{h}_{h,t}$ $\, + \,$ $B^{f}_{h,t}$ $\, + \,$ $B^{h}_{f,t}$ $\, + \,$ $B^{f}_{f,t}$ $= 0$. 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), $A_{i,t}$, and $M_{i,t}$ satisfy

ASSUMPTION 1: $\, ln[A_{i,t}]$ and $ln[M_{i,t}]$ $\sim I(1), \; i \; = \; h, \, f$.

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.

2.C  Optimizing UIP and Money Demand

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

$$\mathbf{E}_{t} \left\{ \sum^{\infty}_{j=0} (1 + \rho)^{-j} \mathcal{U}\left( C_{h,t+j}, \, \, \frac{M_{h,t+j}}{P_{h,t+j}} \right) \right\} , \quad0 \, \, < \, \, \rho,$$

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

$$\mathbf{E}_{t} \left\{ \frac{\mathcal{U}_{C,h,t+1}}{P_{h,t+1}} \right\} (1 \, + \, r_{h,t}) \; \; = \; \; \mathbf{E}_{t} \left\{ \frac{\mathcal{U} _{C,h,t+1}}{P_{f,t+1}} \right\} \frac{(1 \, + \, r_{f,t})}{s_{t} },$$ (6)

where $\mathcal{U}_{C,h,t}$ 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

$$\frac{M_{i,t}}{P_{i,t}} \; \; = \; \; C_{i,t} \left( \frac{1 \, - \, \nu} {\nu} \right) \frac{1 \, + \, r_{i,t}}{r_{i,t}}, \; \quad\; i \; = \; h, \, f.$$ (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

$$\mathbf{E}_{t} \left\{ \frac{\mathcal{U}_{h,t+1}}{P_{h,t+1} C_{h,t+1}} \right\} (1 \, + \, r_{h,t}) \; \; = \; \; \mathbf{E}_{t} \left\{ \frac{ \mathcal{U}_{h,t+1}} {P_{f,t+1}C_{h,t+1}} \right\} \frac{(1 \, + \, r_{f,t}) }{s_{t}},$$

where $\mathcal{U}_{i,t}$ is the utility level of country i at date t. Prior to stochastically detrending the previous expression, define $\widehat{\mathcal{U}}_{i,t} = \mathcal{U}_{i,t}/A_{i,t}$, $\, \widehat {P}_{i,t} = P_{i,t} A_{i,t}/M_{i,t}$, $\, \widehat{C}_{i,t} = C_{i,t}/A_{i,t}$, $\gamma_{A,i,t} = A_{i,t}/A_{i,t-1}$, $\gamma_{M,i,t} = M_{i,t}/M_{i,t-1}$, $\, \widehat{s}_{t} = s_{t} A_{t} /M_{t}$, $\, A_{t} = A_{h,t}/A_{f,t}$, and $M_{t} = M_{h,t}/M_{f,t}$. Note that $\widehat{C} _{i,t}$ is the transitory component of consumption of the $i$th economy, $\gamma_{A,i,t} (\gamma _{M,i,t})$ is the TFP (money) growth rate of country $i$, and the cross-country TFP (money stock) differential $A_{t} \, (M_{t})$ are I(1). Applying the definitions, the stochastically detrended UIP condition becomes

$$\quad\mathbf{E}_{t} \left\{ \frac{\widehat{\mathcal{U}}_{h,t+1} \gamma^{1 - \varphi}_{A,h,t+1}}{\gamma_{M,h,t+1}\widehat{P}_{h,t+1} \widehat{C}_{h,t+1} } \right\} (1 \, + \, r_{h,t}) \; \; = \; \; \mathbf{E}_{t} \left\{ \frac{ \widehat{\mathcal{U}}_{h,t+1} \gamma_{A,f,t+1}} {\gamma^{\varphi}_{A,h,t+1} \gamma_{M,f,t+1} \widehat{P}_{f,t+1} \widehat{C}_{h,t+1}} \right\} \frac{(1 \, + \, r_{f,t})}{\widehat{s}_{t}}.$$

A log linear approximation of the stochastically detrended UIP condition yields

$$\mathbf{E}_{t} \widetilde{e}_{t+1} \; - \; \widetilde{e}_{t} \; \; = \; \; \frac{r^{\ast}}{1 + r^{\ast}} \widetilde{r}_{t} \; + \; \mathbf{E}_{t} \left\{ \widetilde{\gamma}_{A,t+1} \, - \, \widetilde{\gamma}_{M,t+1} \right\} ,$$ (8)

where, for example, $\widetilde{e}_{t} = \ln[\widehat{s}_{t}] - \ln[s^{\ast}]$ and $r^{\ast} (= r_{h}^{\ast} = r_{f}^{\ast})$ denotes the steady state world real rate.

2.D  A DSGE-PVM of the Exchange 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 $\, -\widetilde{p}_{t}$ $=$ $\widetilde{c}_{t} - \frac{\displaystyle 1 \; \;} {\displaystyle 1 + r^{\ast}} \widetilde{r}_{t}$ . 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

$$\left[ 1 \; - \; \frac{1 \; \;}{1 + r^{\ast}} \mathbf{E}_{t} \mathbf{L}^{-1} \right] \widetilde{e}_{t} \; \; = \; \; \frac{1 \; \;}{1 + r^{\ast}} \mathbf{\ E}_{t} \left\{ \widetilde{\gamma}_{M,t+1} \, - \, \widetilde {\gamma}_{A,t+1} \right\} \; \; - \; \; \frac{r^{\ast}}{1 + r^{\ast}} \widetilde{c}_{t}.$$

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

$$\widetilde{e}_{t} \; \; = \; \; \sum_{j=1}^{\infty} \kappa^{j}\mathbf{E}_{t} \left\{ \widetilde{\gamma}_{M,t+j} \, - \, \widetilde{\gamma}_{A,t+j} \right\} \; \; - \; \; (1 - \kappa) \sum_{j=0}^{\infty} \kappa^{j} \mathbf{E}_{t} \widetilde{c}_{t+j},$$ (9)

where the relevant tranversality conditions are invoked and the DSGE-PVM discount factor $\kappa$ $\equiv$ $\frac{\displaystyle 1 \; \;}{\displaystyle 1+r^{\ast}}$. 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)

$$e_{t} \; \; = \; \; (1 - \kappa) \sum_{j=0}^{\infty} \kappa^{j} \mathbf{E}_{t} \left\{ m_{t+j} \, - \, c_{t+j} \right\} .$$ (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 $\omega$ is tied to the interest rate semi-elasticity of money demand, $\phi$, while the DSGE-PVM sets $\kappa$ to the inverse of the gross steady state real world interest rate, $1 + r^{\ast}$. For the standard-PVM (DSGE-PVM), the real fundamental is cross-country output $y_{t}$ (consumption $c_{t}$). Table 1 summarizes the notable elements of the standard- and DSGE-PVMs.

3  Generalizing the Engel-West Hypothesis

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 $\mathcal{B}$ equal to either $\omega$ or $\kappa$, while the fundamental $z_{t}$ is equivalent to either $m_{t}$ $-$ $\psi y_{t}$ or $m_{t}$ $-$ $c_{t}$. With these assumptions, the focus is on the PVM

$$e_{t} \; \; = \; \; (1 - \mathcal{B}) \sum_{j=0}^{\infty} \mathcal{B}^{j} \mathbf{E}_{t} z_{t+j},$$ (11)

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

ASSUMPTION 3: $\, z_{t} \sim I(1)$.

ASSUMPTION 4: $\, (1 - \mathbf{L}) z_{t}$ has a Wold representation, $(1 - \mathbf{L}) z_{t} \; = \; \Delta z^{\ast} \, + \, \zeta(\mathbf{L}) \, \upsilon_{t}$ , where $\mathbf{L}\,z_{t} = z_{t-1}$.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.

3.A  Cointegration Restrictions

The first prediction is that $e_{t}$ and $z_{t}$ 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

$$e_{t} \; - \; z_{t} \; \; = \; \; \sum^{\infty}_{j=1} \mathcal{B}^{j} \mathbf{E}_{t} \Delta z_{t+j}, \quad\Delta\; \equiv\; 1 - \mathbf{L} .$$ (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 $z_{t}$ satisfies Assumptions 3 and 4, $\mathcal{X}_{t} = \beta^{\prime} q_{t}$ forms a cointegrating relation with cointegrating vector $\beta^{\prime} = [1 \; \; -1]$, where $q_{t} \equiv[e_{t} \; \; \; z_{t}]^{\prime}$.

The proposition is a variation of results found in Campbell and Shiller (1987). We interpret the cointegration relation $\mathcal{X}_{t}$ 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 $\mathcal{X}_{t}$ is an infinite-order moving average, MA$(\infty)$ equal to $\mathcal{B} \left( \mathbf{L} \right) \zeta _{\mathcal{B}} \left( \mathbf{L} \right)$ $\hspace{-1mm} \upsilon_{t}$, where $\mathcal{B} \left( \mathbf{L} \right)$ $=$ $\sum^{\infty}_{j=0} \mathcal{B}^{j} \mathbf{L}^{j}$ and $\zeta_{\mathcal{B}} \left( \mathbf{L} \right)$ $=$ $\sum^{\infty}_{j=0} (\mathcal{B} \zeta)^{j} \mathbf{L}^{j-1}$ under Assumptions 3 and 4 (i.e., $z_{t}$ is $I(1)$ 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 $\mathcal{X}_{t}$ is implied by the balanced growth restriction, $e_{t} \equiv\ln[s_{t}] = \widetilde{e}_{t} + m_{t} - a_{t}$, where $m_{t} = \ln[M_{t}]$ and $a_{t} = \ln[A_{t}]$. 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, $\widetilde{e}_{t}$.

3.B  Equilibrium Currency Return Dynamics

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 $e_{t}$ $-$ $(1 - \mathcal{B}) z_{t}$ $= (1 - \mathcal{B}) \sum^{\infty}_{j=1} \mathcal{B}^{j} \mathbf{E}_{t} z_{t+j}$ . Differencing this equation produces, $\Delta e_{t} - (1 - \mathcal{B}) \Delta z_{t} = (1 - \mathcal{B}) \displaystyle \sum^{\infty}_{j=1} \mathcal{B}^{j} \left[ \mathbf{E}_{t} z_{t+j} - \mathbf{E}_{t-1} z_{t+j-1} \right]$ . Next, add and subtract $\mathbf{E}_{t-1} z_{t+j}$ inside the brackets, and substitute with the cointegration-present-value relation (12) to obtain

$$\Delta e_{t} \; - \; \frac{1 - \mathcal{B}}{\mathcal{B}} \mathcal{X}_{t-1} \; \; = \; \; (1 - \mathcal{B}) \sum^{\infty}_{j=0} \mathcal{B}^{j} \bigl[ \mathbf{E}_{t} \, - \, \mathbf{E}_{t-1} \bigr] z_{t+j} .$$ (13)

In equilibrium, currency return are generated by the lagged cointegration relation, $\mathcal{X}_{t-1}$, 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 $u_{\Delta e, t}$. 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 $\Delta e_{t}$ $=$ $\, \vartheta\mathcal{X}_{t-1}$ $+$ $u_{\Delta e, t}$ with factor loading $\vartheta$ $=$ $\frac{\displaystyle 1 - \mathcal{B} }{ \displaystyle \mathcal{B}}$ and currency return forecast error $u_{\Delta e,t}$ $=$ $(1 - \mathcal{B}) \sum^{\infty}_{j=0} \mathcal{B}^{j} \bigl[ \mathbf{E}_{t} \, - \, \mathbf{E}_{t-1} \bigr] z_{t+j}$ .⁵

3.C  A Limiting Model of Exchange Rate Determination

Proposition 2 relies on $\mathcal{B}$ < 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 $\mathcal{B} \, \longrightarrow\, 1$, according to the present-value relation (13) assuming Proposition 1.

Proposition 3 relies on $\mathcal{B} \longrightarrow1$ to produce the martingale $\mathbf{E}_{t} e_{t+1}$ $=$ $e_{t}$ and random walk behavior in the exchange rate.⁶ This behavior suggests an equilibrium path for $e_{t+1}$ in which its best forecast is $e_{t}$, given relevant information, because the source of serial correlation, $\mathcal{X}_{t}$ disappears as $\mathcal{B} \longrightarrow1$.⁷

3.D  PVM Exchange Rate Dynamics Redux

Engel and West (2005) show that the PVM of the exchange rate yields an approximate random walk as $\mathcal{B}$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 $e_{t} \, =$ a$z_{t}$ to find that currency returns are unpredictable.

The EW hypothesis is $plim_{\displaystyle \mathcal{B} \longrightarrow\displaystyle 1} [\Delta e_{t} - \mbox{\texttt{a}} \zeta(\mathbf{1}) \upsilon_{t}] = 0$ . Its hypothesis test begins by noting $e_{t}$ $\, = \,$ $z_{t-1}$ $\, + \,$ $$\sum_{j=0}^{\infty}\mathcal{B}^{j}\mathbf{E}_{t} \Delta z_{t+j}$$ , which is obtained from the present-value relation (3). Use this equation to construct $\Delta e_{t} - \mathbf{E}_{t-1} \Delta e_{t}$ $=$ $\zeta\left( \mathcal{B} \right) \upsilon_{t}$, 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, $\Delta e_{t}$ $=$ $(1 - \mathcal{B}) \displaystyle \sum_{j=0}^{\infty} \mathcal{B} ^{j}$ $\mathbf{E}_{t} \Delta z_{t+j}$. The last two equations yield

$$\Delta e_{t} \; \; = \; \; \zeta\left( \mathcal{B} \right) \upsilon_{t} \; + \; (1 - \mathcal{B}) \sum_{j=0}^{\infty} \mathcal{B}^{j}\mathbf{E}_{t-1} \Delta z_{t+j}.$$ (14)

By letting $\mathcal{B} \longrightarrow1$, 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 $j = i - 1$ to the present value of (14) to obtain the present-value cointegration relation (12) lagged once. For the ECM (13), its present value $(1 - \mathcal{B}) \sum^{\infty}_{j=0} \mathcal{B}^{j} \bigl[ \mathbf{E}_{t} - \mathbf{E}_{t-1} \bigr] z_{t+j}$ equals $\zeta\left( \mathcal{B} \right)$ $\hspace{-0.75mm} \upsilon_{t}$ subsequent to evoking Assumptions 3 and 4 and the Weiner-Kolmogorov prediction formula. Thus, when the PVM discount factor $\mathcal{B}$ is arbitrarily close to one, the EW hypothesis predicts $\Delta e_{t}$ $=$ $\zeta\left( \mathbf{1} \right)$ $\hspace{-0.75mm} \upsilon_{t}$ 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.

3.E  A Common Trend-Common Cycle Model of Exchange Rates and Fundamentals

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, $q_{t} = \bigl[e_{t} \; \; z_{t} \bigr]^{\prime}$ of endowing an ECM on fundamental growth. In this case, $\Delta q_{t}$ forms a VECM(0)

$$\Delta q_{t} \; \; = \; \; \left[ \begin{array}[c]{c} \vartheta\\ \eta \end{array} \right] \mathcal{X}_{t-1} \; + \; \left[ \begin{array}[c]{c} u_{\Delta e,t}\\ u_{\Delta z,t} \end{array} \right] ,$$ (15)

where $\eta$ is the factor loading on $\mathcal{X}_{t-1}$ for $\Delta z_{t}$ and $u_{\Delta z,t}$ is its forecast innovation. Pre-multiplying the VECM(0) by $\overline{\beta}^{\, \prime} = \Bigl [1 \; \; -\hspace{-0.65mm}\frac{\displaystyle \vartheta}{\displaystyle \eta} \Bigr]$ creates the common feature

$$\overline{\beta}^{\, \prime} \Delta q_{t} \; \; = \; \; \overline{\beta}^{\, \prime} \left[ u_{\Delta e, t} \; \; \; \; u_{\Delta z,t} \right] ^{\prime }.$$ (16)

The vector $\overline{\beta}^{\, \prime}$ satisfies the Engle and Kozicki (1993) notion of a common feature because it creates a linear combination of $\Delta e_{t}$ and $\Delta z_{t}$ 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 $\, \Delta z_{t}$ $=$ $\eta\mathcal{X}_{t-1}$ $+$ $u_{\Delta z,t}$, where the forecast innovation $u_{\Delta z,t}$ is Gaussian. When Proposition 2 holds, $\, q_{t}$ has a common feature, $\overline{\beta }^{\, \prime} \Delta q_{t}$, in the sense of Engle and Kozicki (1993), where $\overline{\beta}^{\, \prime} = \Bigl [1 \; \; -\hspace{-0.65mm}\frac{\displaystyle \vartheta}{\displaystyle \eta} \Bigr]$ . The cointegrating and common feature vectors $\beta$ and $\overline{ \beta}$ restrict the trend-cycle decomposition of $q_{t}$, as described by Vahid and Engle (1993).

The common feature of Proposition 4 endows $q_{t} = [e_{t} \; \; z_{t}]^{\prime}$ 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 $q_{t}$ to $\mathbf{I}_{2}$ $-$ $\overline{\beta} (\beta^{\prime} \, \overline{\beta})^{-1} \beta^{\prime}$ , which gives trend and cycle components $\frac{\displaystyle -\mathcal{B} \eta }{\displaystyle 1 - \mathcal{B}(1 + \eta)} \overline{\beta}^{\, \prime} q_{t}$ and $\frac{\displaystyle 1 - \mathcal{B}}{\displaystyle 1 - \mathcal{B} (1 + \eta)} \beta^{\prime} q_{t}$ , respectively.⁹ The BNSW decomposition imposes a common cycle on $e_{t}$ and $z_{t}$ 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, $plim_{\displaystyle \mathcal{B} \longrightarrow\displaystyle 1} [\Delta e_{t} - \mbox{\texttt{a}} \zeta(\mathbf{1}) \upsilon_{t}] = 0$ .

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 $\overline{\beta}^{\, \prime} = [1 \; \; -\hspace{-0.65mm} \frac{\displaystyle \vartheta}{\displaystyle \eta}]$ and that $\vartheta$ $\longrightarrow$ 0 or $\mathcal{B} \longrightarrow1$.

Proposition 5 differs from other approaches to the EW hypothesis. First, the common feature relation (16) imposes cross-equation restrictions on $\Delta q_{t}$ because its cycle generator, the lagged cointegrating relation $\mathcal{X}_{t-1}$, is annihilated by $\overline {\beta}^{\, \prime}$. Having eliminated $\mathcal{X}_{t-1}$, the EW hypothesis decouples the exchange rate from fundamental growth and its forecast innovation $u_{\Delta z, t}$ ($=$ $\zeta(\mathbf{1})\upsilon_{t}$). Finally, observe that when $\vartheta$ $\longrightarrow$ 0 (or $\mathcal{B} \longrightarrow1)$, $\overline{\beta}^{\, \prime}$ $\longrightarrow$ $[1 \; \; 0]$. This leaves only the forecast innovation $u_{\Delta e, t}$ to generate movements in $\Delta e_{t}$. 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 $\mathcal{B} \longrightarrow1$. Only if $\mathcal{B} \in(0, \, 1)$, 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 $\mathcal{B} \longrightarrow1$.

3.F  Reduced Form Evidence

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), $\, \ldots$, 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 $\Delta q_{t}$ has more serial correlation than explained by the lagged cointegration relation $\mathcal{X} _{t-1}$.

Table 3 also presents Johansen (1991, 1994) trace and $\lambda-$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 $\chi^{2}$ 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 $\mathcal{B} \longrightarrow1$. The next section explores the empirical content of this hypothesis in the Canadian-U.S. data.

4  Econometric Models and Methods

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 $z_{t}$ has the permanent-transitory decomposition $z_{t}$ $=$ $\tau_{t}$ $+$ $\widetilde {z}_{t}$, $\tau_{t+1}$ $=$ $\tau_{t}$ $+$ $\varepsilon_{\tau,t+1}$, $(1 - \sum_{i=1}^{p_{z}} \mathcal{A}_{\widetilde{z},i} \mathbf{L}^{i})$ $\widetilde{z}_{t}$ $=$ $\varepsilon_{\widetilde{z},t}$, $\mathbf{E}_{t} \varepsilon_{\tau,t+1}$ $=$ $\mathbf{E}_{t} \varepsilon_{\widetilde{z},t+1}$ $=$ 0, $\mathbf{E}_{t} \varepsilon_{\tau,t+1}^{2}$ $=$ $\sigma_{\tau}^{2}$, $\mathbf{E}_{t} \varepsilon_{\widetilde{z},t+1}^{2}$ $=$ $\sigma _{\widetilde{z}}^{2}$, and $\mathbf{E}_{t} \varepsilon_{\tau,t+i} \, \varepsilon_{\widetilde{z},t+j}$ $=$ 0 for all $i$ and $j$ .¹⁴ Combining the PVM (11) and the permanent-transitory decomposition of $z_{t}$ gives an equilibrium permanent-transitory decomposition of the exchange rate, $e_{t} = \tau_{t} + (1 - \mathcal{B}) \iota_{\widetilde{z}} \left[ \mathbf{I} \, - \, \mathcal{B} \mathcal{A} _{\widetilde{z}} \right] ^{-1} \widetilde{z}_{t}$ , where $\iota _{\widetilde{z}}$ is a 1 $\times$ $p_{z}$ row vector with a first element of one and zeros elsewhere and $\mathcal{A}_{\widetilde{z}}$ is the companion matrix of the AR of $\widetilde{z}_{t}$. The exchange rate trend is identified with the random walk of $z_{t}$ under its permanent-transitory decomposition. Transitory exchange rate fluctuations are driven by the fundamental cyclical component, $\widetilde{z}_{t}$, 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 $\mathcal{B}$, along with the coefficients of the permanent-transitory decomposition of $z_{t}$. Note also that as $\mathcal{B}$ approaches one, the permanent component $\tau_{t}$ 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, $\kappa$ 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 $\kappa$ 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.

4.A  State Space Systems of the UC Models

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 $e_{t} = m_{t} - a_{t} + \widetilde{e}_{t}$. The DSGE-PVM (9) places cross-equation restrictions on the stationary component of the exchange rate, $\widetilde{e}_{t}$.

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 $\mu_{t+1} = \mu^{\ast} + \mu_{t} + \varepsilon_{\mu,t+1}$, $\, \varepsilon_{\mu,t+1} \sim\mathcal{N}(0, \sigma^{2}_{\varepsilon_{\mu}})$ , and $a_{t+1} = a^{\ast} + a_{t} + \varepsilon_{a,t+1}$, $\varepsilon_{a,t+1} \sim\mathcal{N}(0, \sigma^{2}_{\varepsilon_{a}})$ , respectively. Note that $\mu^{\ast}$ and $a^{\ast}$ are the deterministic trend growth rates of cross-country money and TFP. We assume $\widetilde{m}_{t}$ is a MA $(k_{\widetilde{m}})$, $\widetilde{m}_{t} = \sum^{k_{\widetilde{m}}}_{j=0} \alpha_{j} \varepsilon _{\widetilde{m},t-j}$ , where $\alpha_{0} \equiv1$ and $\varepsilon _{\widetilde{m},t} \sim\mathcal{N}(0, \sigma^{2}_{\varepsilon_{\widetilde{m}} })$ . For $\widetilde{c}_{t}$, we employ a AR $(k_{\widetilde{c}})$, $\widetilde{c}_{t} = \sum^{k_{\widetilde{c}}}_{j=1} \theta_{j} \widetilde {c}_{t-j} + \varepsilon_{\widetilde{c},t}$ , where $\varepsilon_{\widetilde {c},t} \sim\mathcal{N}(0, \sigma^{2}_{\varepsilon_{\widetilde{c}}})$ . Put these elements together to form the balanced growth version of the DSGE-PVM

$$e_{t} \; = \; \mu_{t} \, - \, a_{t} \, + \, (1 - \kappa) \sum_{j=0}^{\infty} \kappa^{j} \mathbf{E}_{t} \left\{ \widetilde{m}_{t+j} \, - \, \widetilde {c}_{t+j} \right\} ,$$ (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, $\mu_{t}$, and TFP, $a_{t}$, in the long-run. Serial correlation in the exchange rate is produced by the transitory components of cross-country money and consumption, $\widetilde{m}_{t}$ and $\widetilde {c}_{t}$. 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 $\kappa$ is calibrated to one or estimated. Three UC models follow from solving the DSGE-PVM (17) given $\widetilde{m}_{t} \sim$ MA $(k_{\widetilde{m}})$ and $\widetilde{c}_{t} \sim$ AR $(k_{\widetilde{c}})$ or a common cycle is imposed using either the MA $(k_{\widetilde{m}})$ or AR $(k_{\widetilde{c}})$. The three UC models are estimated when $\kappa$ is calibrated to one. We also use the UC models to estimate $\kappa$. The six UC models have in common the cross-country money trend, $\mu_{t}$, and TFP trend, $a_{t}$.

A rich set of cross-equation restrictions arises in the 2-trend, 2-cycle UC model with $\kappa\in(0, \, 1)$. In part, its state space system consists of the observation equations

$$\qquad\left[ \begin{array}[c]{c} e_{t}\\ m_{t}\\ c_{t} \end{array} \right] \; \; = \; \; \left[ \begin{array}[c]{cccccccccc} 1 & -1 & \delta_{\widetilde{m},0} & \delta_{\widetilde{m},1} & \ldots & \delta_{\widetilde{m},k_{\widetilde{m}}} & \delta_{\widetilde{c},0} & \delta_{\widetilde{c},1} & \ldots & \delta_{\widetilde{c},k_{\widetilde{c}} -1}\\ 1 & 0 & 1 & \alpha_{1} & \ldots & \alpha_{k_{\widetilde{m}}} & 0 & 0 & \ldots & 0\\ 0 & 1 & 0 & 0 & \ldots & 0 & 1 & 0 & \ldots & 0\\ & & & & & & & & & \end{array} \right] \mathcal{S}_{\widetilde{m},c,t},$$ (18)

where $\mathcal{S}_{\widetilde{m},c,t} = \left[ \mu_{t} \; a_{t} \; \varepsilon_{\widetilde{m},t} \; \varepsilon_{\widetilde{m},t-1} \, \ldots\, \varepsilon_{\widetilde{m},t-k_{\widetilde{m}}} \; \; \widetilde {c}_{t} \; \, \widetilde{c}_{t-1} \, \ldots\, \widetilde{c}_{t-k_{\widetilde {c}}+1} \right] ^{\prime}$ , factor loadings on $\varepsilon_{\widetilde{m} ,t}$ and its lags are

$$\quad\delta_{\widetilde{m},i} \; \; = \; \; (1 \, - \, \kappa) \displaystyle \sum^{k_{\widetilde{m}}}_{j=i} \kappa^{j-i} \alpha_{j}, \quad i \; = \; 0, \, \ldots, \, k_{\widetilde{m}},$$ (19)

factor loadings on $\widetilde{c}_{t}$, $\ldots$, $\widetilde {c}_{t-k_{\widetilde{c}}}$ are elements of the row vector

$$\delta_{\widetilde{c}} \; \; = \; \; -s_{\widetilde{c}} (1 \, - \, \kappa) \biggl [\mathbf{I}_{k_{\widetilde{c}}} \, \, - \, \, \kappa\Theta\biggr]^{-1}, \quad s_{\widetilde{c}} \; = \; [1 \; \; \, \mathbf{0}_{1 \times k_{\widetilde{c}}-1}],$$ (20)

and $\Theta$ is the companion matrix of the AR $(k_{\widetilde{c} })$ of $\widetilde{c}_{t}$. The system of first-order state equations is

$$\mathcal{S}_{\widetilde{m},c,t+1} \; \; = \; \; \left[ \begin{array}[c]{c} \mu^{\ast}\\ a^{\ast}\\ 0\\ \vdots\\ 0\\ \vdots \end{array} \right] \; \; + \; \; \left[ \begin{array}[c]{ccccccc} 1 & 0 & \ldots & 0 & 0 & \ldots & 0\\ 0 & 1 & \ldots & 0 & 0 & \ldots & 0\\ 0 & 0 & \ldots & 0 & 0 & \ldots & 0\\ \vdots & \vdots & \mathbf{I}_{k_{\widetilde{m}}} & \vdots & \vdots & & \vdots\\ 0 & 0 & \ldots & 0 & \theta_{1} & \ldots & \theta_{k_{\widetilde{c}}}\\ \vdots & \vdots & & \vdots & & \mathbf{I}_{k_{\widetilde{c}}-1} & \mathbf{0}_{(k_{\widetilde{c}}-1) \times1}\\ & & & & & & \end{array} \right] \, \mathcal{S}_{\widetilde{m},c,t} \; \; + \; \; \left[ \begin{array}[c]{c} \varepsilon_{\mu,t+1}\\ \varepsilon_{a,t+1}\\ \varepsilon_{\widetilde{m},t+1}\\ \mathbf{0}_{k_{\widetilde{m}} \times1}\\ \varepsilon_{\widetilde{c},t+1}\\ \mathbf{0}_{(k_{\widetilde{c}}-1) \times1}\\ \end{array} \right] ,$$ (21)

with covariance matrix $\Omega_{m,\widetilde{c}} = \varepsilon _{\widetilde{m},c,t} \varepsilon_{\widetilde{m},c,t}^{\prime}$ , where $\varepsilon_{\widetilde{m},c,t} = [\varepsilon_{\mu,t+1} \, \, \varepsilon _{a,t+1} \, \, \varepsilon_{\widetilde{m},t+1} \, \, \mathbf{0}_{k_{\widetilde {m}} \times1} \, \, \varepsilon_{\widetilde{c},t+1} \, \, \mathbf{0} _{(k_{\widetilde{c}}-1) \times1}]^{\prime}$ .

We also study UC models that impose one common transitory factor on $m_{t}$ and $c_{t}$. When the common component is $\widetilde{m}_{t}$, the response of $c_{t}$ to $\widetilde{m}_{t}$ is denoted $\pi_{m,\widetilde{c}}$. This implies $\widetilde{c}_{t}$ $=$ $\pi _{m,\widetilde{c}} \widetilde{m}_{t}$ and gives rise to the 2-trend, money cycle UC model. Identifying the common transitory component with $\widetilde{c}_{t}$ defines $\widetilde{m}_{t}$ $=$ $\pi_{m,\widetilde{c}} \widetilde{c}_{t}$ 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 $\kappa= 1$. 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 $\kappa$ is estimated to those in which $\kappa$ is calibrated to one. This provides an empirical appraisal of the EW hypothesis.

4.B  The UC Model and Its Likelihood Function

We label the 2-trend, 2-cycle UC model with $\kappa\in(0, \, 1)$ $UC_{2,2,\kappa}$. Likewise, $UC_{2,\widetilde{m},\kappa }$ and $UC_{2,\widetilde{c},\kappa}$ denote the 2-trend, money cycle and 2-trend, consumption cycle, $\kappa\in(0, \, 1)$ UC models. The state space system of $UC_{2,2,\kappa}$ is (18) and (21), while the appendix presents these systems for $\, UC_{2,\widetilde{m},\kappa}$ and $UC_{2,\widetilde{c},\kappa}$. These state space systems represent the dynamics of $\mathcal{Y}_{t} = \bigl [e_{t} \; \; m_{t} \; \; c_{t} \bigr ]^{\prime}$ restricted by the DSGE-PVM and permanent-transitory specifications of $m_{t}$ and $c_{t}$. We calibrate $\kappa= 1$ in $UC_{2,2,\kappa=1}$, $\, UC_{2,\widetilde{m},\kappa=1}$, and $UC_{2,\widetilde{c},\kappa=1}$. 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 $\mathcal{L} \bigl(\mathcal{Y}_{t} \vert\, \, \Gamma_{2,i,\kappa}, \, \, UC_{2,i,\kappa} \bigr)$ , where $i$ $=$ 2, $\, \widetilde{m}, \, \widetilde{c}$, $\kappa$ is either calibrated to one or estimated, and $\Gamma_{2,i,\kappa }$ is the parameter vector of $UC_{2,i,\kappa}$.

The largest parameter vector is $\Gamma _{2,2,\kappa}$. It contains $11 + k_{\widetilde{m}} \, + \, k_{\widetilde{c}}$ elements, $\, \Gamma_{2,2,\kappa} = \bigl [\kappa$ $\, \, \alpha_{1}$ $\, \, \ldots\, \, \alpha_{k_{\widetilde{m}}}$ $\, \, \theta_{1} \, \, \ldots\, \, \theta_{k_{\widetilde{c}}}$ $\, \, \mu^{\ast}$ $\, \, a^{\ast}$ $\, \, \sigma_{\mu} \; \, \, \sigma_{a}$ $\, \, \sigma_{\widetilde{m}} \; \, \, \sigma_{\widetilde{c}} \, \, \; \varrho_{a,\widetilde{c}} \; \, \, \pi_{e,0} \; \, \, \pi_{e,t} \; \, \, \pi_{e,a} \bigr]^{\prime}$ . We add the parameters $\varrho_{a,\widetilde{c}}$, $\, \pi_{e,0}$, $\, \pi_{e,t}$, and $\pi_{e,a}$ to $\Gamma_{2,2,\kappa}$ to better fit $UC_{2,2,\kappa}$ to the data. For example, the Canadian-U.S. TFP differential exhibits more variation than $c_{t}$ if the correlation coefficient of innovations to $a_{t}$ and $\widetilde{c}_{t}$, $\mathbf{E} \{ \varepsilon_{a,t} \, \varepsilon _{\widetilde{c},t} \} = \varrho_{a,\widetilde{c}}$ , is negative.¹⁶ The remaining three parameters allow for an unrestricted exchange rate intercept, $\pi_{e,0}$, a linear exchange rate trend, $\pi_{e,t}$, and a factor loading on the Canadian-U.S. TFP differential, $\pi_{e,a}$, rather than set the (1, 2) element in the matrix of the observation system (18) to negative one.¹⁷ We estimate $\pi_{e,a}$ 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 $UC_{2,\widetilde{m},\kappa}$ model drops two plus $k_{\widetilde{c}}$ parameters from $\Gamma_{2,\widetilde{m},\kappa}$ $=$ $\bigl [\kappa$ $\, \, \alpha_{1}$ $\, \, \ldots\, \, \alpha_{k_{\widetilde {m}}}$ $\; \, \, \mu^{\ast}$ $\, \, a^{\ast}$ $\, \, \sigma_{\mu} \; \, \, \sigma_{a}$ $\, \, \sigma_{\widetilde{m}} \; \, \, \pi_{e,0} \; \, \, \pi_{e,t} \; \, \, \pi_{e,a} \; \, \, \pi_{c,\widetilde{m}} \bigr]^{\prime}$ , while adding the factor loading on $\widetilde{m}_{t}$ for $c_{t}$, $\pi_{c,\widetilde{m}}$. The factor loading $\pi_{m,\widetilde{c}}$ enters the parameter vector of $UC_{2,\widetilde{m},\kappa}$, while $\alpha_{1}$ $\, \, \ldots\, \, \alpha_{k_{\widetilde{m}}}$ and $\sigma_{\widetilde{m}}$ are dropped from $\, \Gamma_{2,\widetilde{c},\kappa} = \bigl [\kappa$ $\, \, \theta_{1} \, \, \ldots\, \, \theta_{k_{\widetilde{c}}}$ $\, \, \mu^{\ast}$ $\, \, a^{\ast}$ $\, \, \sigma_{\mu} \; \, \, \sigma_{a}$ $\; \, \, \sigma_{\widetilde{c}} \, \, \; \varrho_{a,\widetilde{c}} \; \, \, \pi_{e,0} \; \, \, \pi_{e,t} \; \, \, \pi_{e,a} \; \, \, \pi_{m,\widetilde{c}} \bigr]^{\prime}$ . The parameter vectors of the UC models $UC_{2,2,\kappa=1}$, $\, UC_{2,\widetilde{m},\kappa=1}$, and $UC_{2,\widetilde{c},\kappa=1}$ are identical to $\, \Gamma_{2,2,\kappa}$, $\, \Gamma_{2,\widetilde{m},\kappa}$, and $\Gamma_{2,\widetilde{c},\kappa}$ except that $\kappa= 1$.

4.C  The Data

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.

4.D  Estimation Methods

The likelihood functions of the UC models do not have analytic solutions. We approximate the likelihoods $\mathcal{L} (\mathcal{Y}_{t} \vert\, \, \Gamma_{2,i,\kappa=1}, \, \, UC_{2,i,\kappa=1})$ and $\mathcal{L}(\mathcal{Y}_{t} \vert\, \, \Gamma_{2,i,\kappa}, \, \, UC_{2,i,\kappa})$ with posterior distributions of $\, \Gamma_{2,i,\kappa=1}$ and $\, \Gamma_{2,i,\kappa}$, generated by the MCMC replications of the random walk MH simulator. Our estimates of $\, \Gamma_{2,i,\kappa=1}$ and $\, \Gamma_{2,i,\kappa}$ 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 $\, UC_{2,2,\kappa=1}$, $\, UC_{2,\widetilde{m} ,\kappa=1}$, $\, UC_{2,\widetilde{c},\kappa=1}$, $\, UC_{2,2,\kappa}$, $\, UC_{2,\widetilde{m},\kappa}$, and $UC_{2,\widetilde{c},\kappa}$ models.¹⁹

4.E  Priors

The second column of table 4 (5) list the priors of $\, \Gamma_{2,i,\kappa=1}$ $\, (\Gamma_{2,i,\kappa})$, $i$ $=$ 2, $\, \widetilde{m}$, $\, \widetilde{c}$. 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 $(k_{\widetilde{m}})$ of $\widetilde{m}_{t}$ and AR $(k_{\widetilde{c}})$ of $\widetilde{c}_{t}$ that set $k_{\widetilde{m}} = k_{\widetilde{c}} = 2$. Normal priors for the MA ( $\alpha_{1}$ and $\alpha_{2}$) and AR ( $\theta_{1}$ and $\theta_{2}$) coefficients allow for disparate transitory behavior in $\widetilde{m}_{t}$ and $\widetilde{c}_{t}$. The prior means of $\alpha_{1}$, $\alpha_{2}$, $\, \theta_{1}$, and $\theta_{2}$ guarantee that the relevant eigenvalues are strictly less than one. The eigenvalues of the MA(2) (AR(2)) of $\widetilde{m}_{t}$ ( $\widetilde{c}_{t}$) are 0.60 ± 0.20i (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 $\alpha_{1}$, $\alpha_{2}$, $\, \theta_{1}$, and $\theta_{2}$. 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 $\mu^{\ast}$ and $a^{\ast}$ that rely on the Canadian-U.S. money stock and consumption differentials samples. Since $\mu^{\ast}$ and $a^{\ast}$ represent deterministic trend growth, we ground the priors on normal distributions. The prior standard deviations of $\mu^{\ast}$ and $a^{\ast}$ 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 $\sigma_{\mu}$,