I. Introduction

The substantial nominal and real appreciation of the dollar during the first half of the 1980s and its subsequent depreciation after 1985 has been difficult to explain. ^1/ According to one interpretation, the dollar’s strength in the early 1980s reflected a substantial shift in the U.S. policy mix toward relatively tight monetary policy and large budget deficits that pushed up nominal and real interest rates causing large inflows of capital. Subsequently, as the monetary stance was loosened and the fiscal deficit narrowed, these pressures reversed themselves leading to declines in the dollar. According to another interpretation, changes in the incentives to invest in U.S. real assets in the early 1980s together with political uncertainties and financial deregulation abroad led to a massive but temporary increase in the demand for the dollar during the first half of the 1980s. With the unwinding of these factors in the latter part of the decade, the dollar then reversed its previous gains.

Notwithstanding a large amount of empirical work it has been very difficult to distinguish between these alternative interpretations of the dollar’s movements during the 1980s. Most structural exchange rate models have explained only a small proportion of the dollar’s movements, and have typically exhibited periods of instability. ^2/ Moreover, the building blocks for many of these models including interest rate parity and purchasing power parity, have been shown to have been violated in the 1970s and 1980s.

The empirical regularity that has emerged is that the nominal and real exchange rates of the dollar in terms of other major currencies are closely approximated by random walks, and that forecasts from a random walk outperform those from most structural exchange rate models. ^3/ Some investigators see the random walk property of exchange rates as a natural outcome of efficient financial markets in which prices fully reflect all available information. ^4/ According to others, the random walk carries strong implications for identifying the kinds of shocks that have driven exchange rates, with the presumption that most of these shocks have been expected to be permanent, and the models that are appropriate for analyzing their behavior. ^5/

This paper addresses a number of questions about the time series processes followed by exchange rates with a view to shedding light on the factors behind the permanent or longer-run movements in the dollar. Rather than estimate a particular structural model, the paper focuses on the stochastic processes for exchange rates implied by wide classes of structural models, and whether the data are consistent with these processes. In addition, using cointegration techniques, the paper examines whether the long-run variance of exchange rates can be accounted for by the economic variables identified in structural exchange rate models.

The paper is organized as follows. Section II discusses the stochastic processes for exchange rates implied by structural exchange rate models and identifies two sources of systematic movement in exchange rates and hence departures from a random walk—those related to predictable movements in forcing variables (extrinsic dynamics) and those related to the internal dynamics of models (intrinsic dynamics). Section III examines the actual time series processes followed by nominal and real exchange rates of the dollar to see if there is evidence of mean reversion and departures from a random walk. Section IV identifies the time series properties of the forcing variables appearing in structural exchange rate models and tests for long-run cointegrating relationships between these variables and dollar exchange rates. Section V provides some concluding remarks.

II. Structural Exchange Rate Models

This section reviews several classes of exchange rate models with a view to identifying the time series processes they predict for exchange rates. The section begins by considering the solutions for exchange rates implied by flexible and fixed price monetary models, and then discusses the dynamic process contained in portfolio models. ^6/ Two sources of systematic exchange rate movements are identified: those caused by predictability in the forcing variables included in exchange rate models (extrinsic dynamics) and those due to dynamic adjustment to shocks (intrinsic dynamics).

Monetary models. The monetary models are based on parity conditions linking interest rates on assets denominated in different currencies (equations (1) and (2)); a parity condition linking domestic and foreign prices, with the long-run real exchange rate constant under purchasing power parity (equation (3)); and equilibrium in financial markets, as represented by equality between the stock demands and supplies of domestic and foreign money (equations (4) and (5)). ^7/ The monetary models assume residents of each country hold only their own money.

Monetary models have been applied in different ways over the recent floating rate period. ^8/ In ‘flexible-price’ applications, the models determine nominal exchange rates on the basis of given values for real variables such as relative outputs and the real exchange rate. ^9/ In “sticky-price” applications, equations (1)-(5) describe the long-run behavior of nominal exchange rates, with departures from these relationships in the short run. ^10/

Under the assumption of flexible prices, the solution to equations (1)-(5) has a familiar form in which the nominal exchange rate depends on the current and expected future values of its determinants: here, relative money supplies, relative outputs, and the real exchange rate ^11/

The significance of events in the future in equation (6) is determined by the size of the discount factor α/(1 + α), which depends on the (semi-interest) elasticity of money demand. If the elasticity of money demand is very high, events expected far into the future have almost the same weight as those in the present.

The forward-looking nature of equation (6) does not mean that exchange rates should be entirely unpredictable and follow driftless random walks (or martingales), conditions that are sometimes regarded as an implication of market efficiency. ^12/ The only efficiency conditions in the monetary model relate to the forward exchange market and imply that forward exchange rates should be unbiased predictors of expected future spot exchange rates. Given the assumption that residents in each country only hold their own money, there is no efficiency condition for the spot exchange market and exchange rates can be characterized by systematic movements in the monetary model.

The amount of systematic exchange rate movement in the flexible-price model is determined by the size of the (semi) interest elasticity of money demand and the systematic movement in the forcing variables. Table 1 shows the solutions for the nominal exchange in the flexible-price model under alternative assumptions about the stochastic processes for relative money supplies, including those with inter-temporally correlated components. ^13/ Interpretation of the solutions is facilitated by decomposing the change in the nominal exchange rate between any two periods into a component that is unpredictable as of period t-1 (unsystematic component) and a component that is predictable (systematic component) ^14/

Table 1.

Flexible Price Monetary Model

Note:

For each money supply process the systematic (or predictable) change in the exchange rate over the next period is given by

(E_ts_t+1 - s_t)

Table 1.

Flexible Price Monetary Model

General Solution
$\begin{array}{c}{\mathrm{s}}_{\mathrm{t}}=\frac{1}{\left(1+\mathrm{\alpha }\right)}\underset{\mathrm{i}=0}{\overset{\mathrm{\infty }}{\mathrm{\Sigma }}}{\mathrm{E}}_{\mathrm{t}}\left[\frac{\mathrm{\alpha }}{\left(1+\mathrm{\alpha }\right)}\right]\stackrel{\mathrm{i}}{\cdot }\left[\left({\mathrm{m}}_{\mathrm{t}+\mathrm{i}}-{\mathrm{m}}_{\mathrm{t}+\mathrm{i}}^{*}\right)-\mathit{\beta }\left({\mathrm{y}}_{\mathrm{t}+\mathrm{i}}-{\mathrm{y}}_{\mathrm{t}+\mathrm{i}}^{*}\right)+{\mathrm{q}}_{\mathrm{t}+\mathrm{i}}\right]\end{array}$
In what follows it is assumed that
	${\mathrm{y}}_{\mathrm{t}+\mathrm{i}}-{\mathrm{y}}_{\mathrm{t}+\mathrm{i}}^{*}={\mathrm{q}}_{\mathrm{t}+\mathrm{i}}=0$ for all i
Case I:	Money supplies follow random walks around random trends. (Money Supplies are I(2) processes.)
	mt = mt-1 + gt + dt	${\mathrm{m}}_{\mathrm{t}}^{*}={\mathrm{m}}_{\mathrm{t}-1}^{*}+{\mathrm{g}}_{\mathrm{t}}^{*}+{\mathrm{d}}_{\mathrm{t}}^{*}$
	gt = gt-1 + kt	${\mathrm{g}}_{\mathrm{t}}^{*}={\mathrm{g}}_{\mathrm{t}-1}^{*}+{\mathrm{k}}_{\mathrm{t}}^{*}$
	where ${\mathrm{d}}_{\mathrm{t}},{\mathrm{d}}_{\mathrm{t}}^{*},{\mathrm{k}}_{\mathrm{t}},{\mathrm{k}}_{\mathrm{t}}^{*}$ are white noise errors.
$\begin{array}{l}{\mathrm{s}}_{\mathrm{t}}=\left({\mathrm{m}}_{\mathrm{t}-1}-{\mathrm{m}}_{\mathrm{t}-1}^{*}\right)+\left({\mathrm{d}}_{\mathrm{t}}-{\mathrm{d}}_{\mathrm{t}}^{*}\right)+\left(1+\mathrm{\alpha }\right)\left({\mathrm{g}}_{\mathrm{t}-1}-{\mathrm{g}}_{\mathrm{t}-1}^{*}\right)+\left(1+\mathrm{\alpha }\right)\left({\mathrm{k}}_{\mathrm{t}}-{\mathrm{k}}_{\mathrm{t}}^{*}\right)\\ {\mathrm{E}}_{\mathrm{t}}{\mathrm{s}}_{\mathrm{t}+\mathrm{i}}=\left({\mathrm{m}}_{\mathrm{t}-1}-{\mathrm{m}}_{\mathrm{t}-1}^{*}\right)+\left({\mathrm{d}}_{\mathrm{t}}-{\mathrm{d}}_{\mathrm{t}}^{*}\right)+\left(\mathrm{i}+1+\mathrm{\alpha }\right)\left({\mathrm{g}}_{\mathrm{t}-1}-{\mathrm{g}}_{\mathrm{t}-1}^{*}\right)+\left(\mathrm{i}+1+\mathrm{\alpha }\right)\left({\mathrm{k}}_{\mathrm{t}}-{\mathrm{k}}_{\mathrm{t}}^{*}\right)\\ {\mathrm{E}}_{\mathrm{t}}{\mathrm{s}}_{\mathrm{t}+1}-{\mathrm{s}}_{\mathrm{t}}=\left({\mathrm{g}}_{\mathrm{t}-1}-{\mathrm{g}}_{\mathrm{t}-1}^{*}\right)+\left({\mathrm{k}}_{\mathrm{t}}-{\mathrm{k}}_{\mathrm{t}}^{*}\right)\end{array}$
Case II:	Money supplies are subject to temporary and permanent shocks. (Money supplies are I(1) processes.)
	${\mathrm{m}}_{\mathrm{t}}={\overline{\mathrm{m}}}_{\mathrm{t}}+{\mathrm{v}}_{\mathrm{t}}$	${\mathrm{m}}_{\mathrm{t}}^{*}={\overline{\mathrm{m}}}_{\mathrm{t}}^{*}+{\mathrm{v}}_{\mathrm{t}}^{*}$
	${\overline{\mathrm{m}}}_{\mathrm{t}}={\overline{\mathrm{m}}}_{\mathrm{t}-1}+{\mathit{\mu }}_{\mathrm{t}}$	${\overline{\mathrm{m}}}_{\mathrm{t}}^{*}={\overline{\mathrm{m}}}_{\mathrm{t}-1}^{*}+{\mathit{\mu }}_{\mathrm{t}}^{*}$
	where, ${\mathrm{v}}_{\mathrm{t}},{\mathrm{v}}_{\mathrm{t}}^{*},{\mathit{\mu }}_{\mathrm{t}},{\mathit{\mu }}_{\mathrm{t}}^{*}$ are white noise errors.
$\begin{array}{l}{\mathrm{s}}_{\mathrm{t}}=\left({\overline{\mathrm{m}}}_{\mathrm{t}-1}-{\overline{\mathrm{m}}}_{\mathrm{t}-1}^{*}\right)+\left({\mathit{\mu }}_{\mathrm{t}}-{\mathit{\mu }}_{\mathrm{t}}^{*}\right)+{\left(1+\mathrm{\alpha }\right)}^{-1}\left({\mathrm{v}}_{\mathrm{t}}-{\mathrm{v}}_{\mathrm{t}}^{*}\right)\\ {\mathrm{E}}_{\mathrm{t}}{\mathrm{s}}_{\mathrm{t}+\mathrm{i}}=\left({\overline{\mathrm{m}}}_{\mathrm{t}-1}-{\overline{\mathrm{m}}}_{\mathrm{t}-1}^{*}\right)+\left({\mathit{\mu }}_{\mathrm{t}}-{\mathit{\mu }}_{\mathrm{t}}^{*}\right)\\ {\mathrm{E}}_{\mathrm{t}}{\mathrm{s}}_{\mathrm{t}+1}-{\mathrm{s}}_{\mathrm{t}}=-{\left(1+\mathrm{\alpha }\right)}^{-1}\left({\mathrm{v}}_{\mathrm{t}}-{\mathrm{v}}_{\mathrm{t}}^{*}\right)\end{array}$
Case III:	Money supplies are subject to intertemporally correlated shocks. (Money supplies are I(0) processes.)
	${\mathrm{m}}_{\mathrm{t}}={\overline{\mathrm{m}}}_{\mathrm{t}}+{\mathrm{v}}_{\mathrm{t}}$	${\mathrm{m}}_{\mathrm{t}}^{*}={\overline{\mathrm{m}}}^{*}+{\mathrm{v}}_{\mathrm{t}}^{*}$
	vt = ρvt-1+ ∑t	${\mathrm{v}}_{\mathrm{t}}^{*}={\mathrm{\rho v}}_{\mathrm{t}-1}^{*}+{\mathrm{\Sigma }}_{\mathrm{t}}^{*}$
	ρ ∈ (0,1)	ρ ∈ (0,1)
	where ${\mathrm{v}}_{\mathrm{t}},{\mathrm{v}}_{\mathrm{t}}^{*},{\mathrm{\Sigma }}_{\mathrm{t}},{\mathrm{\Sigma }}_{\mathrm{t}}^{*}$ are white noise errors.
$\begin{array}{l}{\mathrm{s}}_{\mathrm{t}}=\left(\overline{\mathrm{m}}-{\overline{\mathrm{m}}}^{*}\right)+\mathrm{\lambda }\left({\mathrm{v}}_{\mathrm{t}}-{\mathrm{v}}_{\mathrm{t}}^{*}\right)\\ {\mathrm{E}}_{\mathrm{t}}{\mathrm{s}}_{\mathrm{t}+\mathrm{i}}=\left(\overline{\mathrm{m}}-{\overline{\mathrm{m}}}^{*}\right)+\mathrm{\lambda \rho }\stackrel{\mathrm{i}}{\cdot }\left({\mathrm{v}}_{\mathrm{t}}-{\mathrm{v}}_{\mathrm{t}}^{*}\right)\end{array}$
	where λ = [1 + α(1 - ρ)]-1
${\mathrm{E}}_{\mathrm{t}}{\mathrm{s}}_{\mathrm{t}+1}-{\mathrm{s}}_{\mathrm{t}}=\mathrm{\lambda }\left(\mathit{\rho }-1\right)\left({\mathrm{v}}_{\mathrm{t}}-{\mathrm{v}}_{\mathrm{t}}^{*}\right)$

Note:

For each money supply process the systematic (or predictable) change in the exchange rate over the next period is given by

(E_ts_t+1 - s_t)

View Table

Table 1.

Flexible Price Monetary Model

General Solution
$\begin{array}{c}{\mathrm{s}}_{\mathrm{t}}=\frac{1}{\left(1+\mathrm{\alpha }\right)}\underset{\mathrm{i}=0}{\overset{\mathrm{\infty }}{\mathrm{\Sigma }}}{\mathrm{E}}_{\mathrm{t}}\left[\frac{\mathrm{\alpha }}{\left(1+\mathrm{\alpha }\right)}\right]\stackrel{\mathrm{i}}{\cdot }\left[\left({\mathrm{m}}_{\mathrm{t}+\mathrm{i}}-{\mathrm{m}}_{\mathrm{t}+\mathrm{i}}^{*}\right)-\mathit{\beta }\left({\mathrm{y}}_{\mathrm{t}+\mathrm{i}}-{\mathrm{y}}_{\mathrm{t}+\mathrm{i}}^{*}\right)+{\mathrm{q}}_{\mathrm{t}+\mathrm{i}}\right]\end{array}$
In what follows it is assumed that
	${\mathrm{y}}_{\mathrm{t}+\mathrm{i}}-{\mathrm{y}}_{\mathrm{t}+\mathrm{i}}^{*}={\mathrm{q}}_{\mathrm{t}+\mathrm{i}}=0$ for all i
Case I:	Money supplies follow random walks around random trends. (Money Supplies are I(2) processes.)
	mt = mt-1 + gt + dt	${\mathrm{m}}_{\mathrm{t}}^{*}={\mathrm{m}}_{\mathrm{t}-1}^{*}+{\mathrm{g}}_{\mathrm{t}}^{*}+{\mathrm{d}}_{\mathrm{t}}^{*}$
	gt = gt-1 + kt	${\mathrm{g}}_{\mathrm{t}}^{*}={\mathrm{g}}_{\mathrm{t}-1}^{*}+{\mathrm{k}}_{\mathrm{t}}^{*}$
	where ${\mathrm{d}}_{\mathrm{t}},{\mathrm{d}}_{\mathrm{t}}^{*},{\mathrm{k}}_{\mathrm{t}},{\mathrm{k}}_{\mathrm{t}}^{*}$ are white noise errors.
$\begin{array}{l}{\mathrm{s}}_{\mathrm{t}}=\left({\mathrm{m}}_{\mathrm{t}-1}-{\mathrm{m}}_{\mathrm{t}-1}^{*}\right)+\left({\mathrm{d}}_{\mathrm{t}}-{\mathrm{d}}_{\mathrm{t}}^{*}\right)+\left(1+\mathrm{\alpha }\right)\left({\mathrm{g}}_{\mathrm{t}-1}-{\mathrm{g}}_{\mathrm{t}-1}^{*}\right)+\left(1+\mathrm{\alpha }\right)\left({\mathrm{k}}_{\mathrm{t}}-{\mathrm{k}}_{\mathrm{t}}^{*}\right)\\ {\mathrm{E}}_{\mathrm{t}}{\mathrm{s}}_{\mathrm{t}+\mathrm{i}}=\left({\mathrm{m}}_{\mathrm{t}-1}-{\mathrm{m}}_{\mathrm{t}-1}^{*}\right)+\left({\mathrm{d}}_{\mathrm{t}}-{\mathrm{d}}_{\mathrm{t}}^{*}\right)+\left(\mathrm{i}+1+\mathrm{\alpha }\right)\left({\mathrm{g}}_{\mathrm{t}-1}-{\mathrm{g}}_{\mathrm{t}-1}^{*}\right)+\left(\mathrm{i}+1+\mathrm{\alpha }\right)\left({\mathrm{k}}_{\mathrm{t}}-{\mathrm{k}}_{\mathrm{t}}^{*}\right)\\ {\mathrm{E}}_{\mathrm{t}}{\mathrm{s}}_{\mathrm{t}+1}-{\mathrm{s}}_{\mathrm{t}}=\left({\mathrm{g}}_{\mathrm{t}-1}-{\mathrm{g}}_{\mathrm{t}-1}^{*}\right)+\left({\mathrm{k}}_{\mathrm{t}}-{\mathrm{k}}_{\mathrm{t}}^{*}\right)\end{array}$
Case II:	Money supplies are subject to temporary and permanent shocks. (Money supplies are I(1) processes.)
	${\mathrm{m}}_{\mathrm{t}}={\overline{\mathrm{m}}}_{\mathrm{t}}+{\mathrm{v}}_{\mathrm{t}}$	${\mathrm{m}}_{\mathrm{t}}^{*}={\overline{\mathrm{m}}}_{\mathrm{t}}^{*}+{\mathrm{v}}_{\mathrm{t}}^{*}$
	${\overline{\mathrm{m}}}_{\mathrm{t}}={\overline{\mathrm{m}}}_{\mathrm{t}-1}+{\mathit{\mu }}_{\mathrm{t}}$	${\overline{\mathrm{m}}}_{\mathrm{t}}^{*}={\overline{\mathrm{m}}}_{\mathrm{t}-1}^{*}+{\mathit{\mu }}_{\mathrm{t}}^{*}$
	where, ${\mathrm{v}}_{\mathrm{t}},{\mathrm{v}}_{\mathrm{t}}^{*},{\mathit{\mu }}_{\mathrm{t}},{\mathit{\mu }}_{\mathrm{t}}^{*}$ are white noise errors.
$\begin{array}{l}{\mathrm{s}}_{\mathrm{t}}=\left({\overline{\mathrm{m}}}_{\mathrm{t}-1}-{\overline{\mathrm{m}}}_{\mathrm{t}-1}^{*}\right)+\left({\mathit{\mu }}_{\mathrm{t}}-{\mathit{\mu }}_{\mathrm{t}}^{*}\right)+{\left(1+\mathrm{\alpha }\right)}^{-1}\left({\mathrm{v}}_{\mathrm{t}}-{\mathrm{v}}_{\mathrm{t}}^{*}\right)\\ {\mathrm{E}}_{\mathrm{t}}{\mathrm{s}}_{\mathrm{t}+\mathrm{i}}=\left({\overline{\mathrm{m}}}_{\mathrm{t}-1}-{\overline{\mathrm{m}}}_{\mathrm{t}-1}^{*}\right)+\left({\mathit{\mu }}_{\mathrm{t}}-{\mathit{\mu }}_{\mathrm{t}}^{*}\right)\\ {\mathrm{E}}_{\mathrm{t}}{\mathrm{s}}_{\mathrm{t}+1}-{\mathrm{s}}_{\mathrm{t}}=-{\left(1+\mathrm{\alpha }\right)}^{-1}\left({\mathrm{v}}_{\mathrm{t}}-{\mathrm{v}}_{\mathrm{t}}^{*}\right)\end{array}$
Case III:	Money supplies are subject to intertemporally correlated shocks. (Money supplies are I(0) processes.)
	${\mathrm{m}}_{\mathrm{t}}={\overline{\mathrm{m}}}_{\mathrm{t}}+{\mathrm{v}}_{\mathrm{t}}$	${\mathrm{m}}_{\mathrm{t}}^{*}={\overline{\mathrm{m}}}^{*}+{\mathrm{v}}_{\mathrm{t}}^{*}$
	vt = ρvt-1+ ∑t	${\mathrm{v}}_{\mathrm{t}}^{*}={\mathrm{\rho v}}_{\mathrm{t}-1}^{*}+{\mathrm{\Sigma }}_{\mathrm{t}}^{*}$
	ρ ∈ (0,1)	ρ ∈ (0,1)
	where ${\mathrm{v}}_{\mathrm{t}},{\mathrm{v}}_{\mathrm{t}}^{*},{\mathrm{\Sigma }}_{\mathrm{t}},{\mathrm{\Sigma }}_{\mathrm{t}}^{*}$ are white noise errors.
$\begin{array}{l}{\mathrm{s}}_{\mathrm{t}}=\left(\overline{\mathrm{m}}-{\overline{\mathrm{m}}}^{*}\right)+\mathrm{\lambda }\left({\mathrm{v}}_{\mathrm{t}}-{\mathrm{v}}_{\mathrm{t}}^{*}\right)\\ {\mathrm{E}}_{\mathrm{t}}{\mathrm{s}}_{\mathrm{t}+\mathrm{i}}=\left(\overline{\mathrm{m}}-{\overline{\mathrm{m}}}^{*}\right)+\mathrm{\lambda \rho }\stackrel{\mathrm{i}}{\cdot }\left({\mathrm{v}}_{\mathrm{t}}-{\mathrm{v}}_{\mathrm{t}}^{*}\right)\end{array}$
	where λ = [1 + α(1 - ρ)]-1
${\mathrm{E}}_{\mathrm{t}}{\mathrm{s}}_{\mathrm{t}+1}-{\mathrm{s}}_{\mathrm{t}}=\mathrm{\lambda }\left(\mathit{\rho }-1\right)\left({\mathrm{v}}_{\mathrm{t}}-{\mathrm{v}}_{\mathrm{t}}^{*}\right)$

Note:

For each money supply process the systematic (or predictable) change in the exchange rate over the next period is given by

(E_ts_t+1 - s_t)

Given that there are no intrinsic dynamics in the flexible-price monetary models, systematic exchange rate movements only occur when there are predictable elements in the forcing variables. The most obvious source of systematic movements arises from differences in monetary growth rates across countries: countries with faster monetary growth rates will tend to have continuously depreciating currencies (Table 1). ^15/ But systematic movement can also be the result of temporary or inter-temporally correlated shocks to money supplies. ^16/ For example, a temporary increase in the domestic money supply will lead to a temporary depreciation of the domestic currency, followed by expectations of appreciation. The only case in which the nominal exchange rate follows a driftless random walk in the flexible-price model is when the forcing variables are described by driftless random walks. ^17/

The solutions in Table 1 also imply that any nonstationarity in nominal exchange rates derives from nonstationarity in the forcing variables. In each solution, the order of integration of the exchange rate matches that of the money supply process. ^18/ If, for example, relative money supplies are integrated of order two, exchange rates have the same order of integration. The real exchange rate is exogenous in this version of the flexible-price monetary model. If purchasing power parity holds in the long run, the real exchange rate will be a stationary variable. However, if there are permanent real shocks, such as changes in tastes or productivity, it will be nonstationary.

An important extension of the monetary model allows for short-run price rigidities. In an early paper Dornbusch(1976) modified the flexible-price monetary model to allow for price stickiness and attempted to rationalize the high degree of volatility in nominal exchange rates over the floating rate period. The major features of Dornbusch’s sticky-price model are illustrated in Table 2 using a simple stochastic model that includes a Phillip’s curve and an aggregate demand equation. ^19/ Solutions to the model are shown for the case in which the domestic money supply follows a first-order auto-regressive process and all other forcing variables are constant. ^20/ In contrast to the flexible-price model, the Dornbusch model includes intrinsic dynamics associated with slowly adjusting commodity prices.

Table 2.

Sticky-Price Monetary Model

Table 2.

Sticky-Price Monetary Model

	mt - pt = βȳt - αRt
	${\mathrm{R}}_{\mathrm{t}}={\mathrm{R}}_{\mathrm{t}}^{*}+{\mathrm{E}}_{\mathrm{t}}{\mathrm{s}}_{\mathrm{t}+1}-{\mathrm{s}}_{\mathrm{t}}$
	Pt+1 = Pt + π(yt - ȳt)
	yt = θ(st - pt)	${\mathrm{p}}_{\mathrm{t}}^{*}=0$
General solution
	$\begin{array}{l}\left({\mathrm{s}}_{\mathrm{t}}-{\mathrm{p}}_{\mathrm{t}}\right)=\frac{\left({\mathrm{\lambda }}_1-1\right)}{\mathit{\pi \theta }}{\mathrm{p}}_{\mathrm{t}}+\frac{\left({\mathrm{\lambda }}_1-{\mathrm{\lambda }}_2\right)}{\mathit{\pi \theta }}\underset{\mathrm{i}=0}{\overset{\mathrm{\infty }}{\mathrm{\Sigma }}}{\mathrm{\lambda }}_2^{-\left(\mathrm{i}+1\right)}\mathrm{\Omega }{\mathrm{E}}_{\mathrm{t}}\left({\mathrm{z}}_{\mathrm{t}-1+\mathrm{i}}\right)\\ {\mathrm{p}}_{\mathrm{t}}={\mathrm{\lambda }}_1{\mathrm{p}}_{\mathrm{t}-1}-\mathrm{\pi }{\overline{\mathrm{y}}}_{\mathrm{t}-1}+\left({\mathrm{\lambda }}_1-{\mathrm{\lambda }}_2\right)\underset{\mathrm{i}=0}{\overset{\mathrm{\infty }}{\mathrm{\Sigma }}}{\mathrm{\lambda }}_2^{-\left(\mathrm{i}+1\right)}\mathrm{\Omega }{\mathrm{E}}_{\mathrm{t}}\left({\mathrm{z}}_{\mathrm{t}-1+\mathrm{i}}\right)\end{array}$
Here, λ1 and λ2 are the eigenvalues given by
	$\begin{array}{c}{\mathrm{\lambda }}_1=1-\frac{\mathit{\pi \theta }}{2}-\frac{1}{2}{\left[{\left(\mathit{\pi \theta }\right)}^2+4{\mathit{\theta \pi \alpha }}^{-1}\right]}^{\mathrm{½}}\\ {\mathrm{\lambda }}_2=1-\frac{\mathit{\pi \theta }}{2}+\frac{1}{2}{\left[{\left(\mathit{\pi \theta }\right)}^2+4{\mathit{\theta \pi \alpha }}^{-1}\right]}^{\mathrm{½}}\end{array}$
so that λ2 > λ1. It is assumed that λ1 lies inside the unit circle and that λ1 lies outside it. zt is the vector given by
	${\mathrm{z}}_{\mathrm{t}}=\left[{\mathrm{m}}_{\mathrm{t}},{\mathrm{R}}_{\mathrm{t}}^{*},{\overline{\mathrm{y}}}_{\mathrm{t}}\right]$
and Ω is a coefficient vector given by
$\mathrm{\Omega }=\left[\frac{-{\mathit{\pi \theta \alpha }}^{-1}}{{\mathrm{\lambda }}_2-{\mathrm{\lambda }}_1},\frac{\mathit{\pi \theta }}{{\mathrm{\lambda }}_2-{\mathrm{\lambda }}_1},\frac{\mathit{\pi }\left[\mathit{\theta }\left({\mathit{\alpha }}^{-1}\mathit{\beta }+\mathit{\pi }\right)-\left(1-{\mathrm{\lambda }}_1\right)\right]}{{\mathrm{\lambda }}_2-{\mathrm{\lambda }}_1}\right]$
The particular solution when the money supply follows an AR1 process is given below:
	mt = ρmt-1 + μt	where μ is white noise.
	${\mathrm{R}}_{\mathrm{t}}^{*}=0$
	ȳt = 0
When ρ ∈ (0,1), mt follow a stationary stochastic process. When ρ = 1, mt follows a random walk.
	$\begin{array}{l}\left({\mathrm{s}}_{\mathrm{t}}-{\mathrm{p}}_{\mathrm{t}}\right)=\frac{\left({\mathrm{\lambda }}_1-1\right)}{\mathit{\pi \theta }}{\mathrm{p}}_{\mathrm{t}}+\frac{{\mathit{\alpha }}^{-1}}{{\mathrm{\lambda }}_2-\mathit{\rho }}{\mathrm{m}}_{\mathrm{t}}\\ {\mathrm{p}}_{\mathrm{t}}={\mathrm{\lambda }}_1{\mathrm{p}}_{\mathrm{t}-1}+\frac{{\mathit{\pi \theta \alpha }}^{-1}}{{\mathrm{\lambda }}_2-\mathit{\rho }}{\mathrm{m}}_{\mathrm{t}-1}\\ {\mathrm{s}}_{\mathrm{t}}=\frac{\left({\mathrm{\lambda }}_1-1\right)}{\mathit{\pi \theta }}{\mathrm{p}}_{\mathrm{t}}+{\mathrm{\lambda }}_1{\mathrm{p}}_{\mathrm{t}-1}+{\mathit{\alpha }}^{-1}\frac{\left(\mathit{\rho }+\mathit{\pi \theta }\right)}{{\mathrm{\lambda }}_2-\mathit{\rho }}{\mathrm{m}}_{\mathrm{t}-1}+\frac{{\mathit{\alpha }}^{-1}}{{\mathrm{\lambda }}_2-\mathit{\rho }}{\mathit{\mu }}_{\mathrm{t}}\end{array}$

View Table

Table 2.

Sticky-Price Monetary Model

	mt - pt = βȳt - αRt
	${\mathrm{R}}_{\mathrm{t}}={\mathrm{R}}_{\mathrm{t}}^{*}+{\mathrm{E}}_{\mathrm{t}}{\mathrm{s}}_{\mathrm{t}+1}-{\mathrm{s}}_{\mathrm{t}}$
	Pt+1 = Pt + π(yt - ȳt)
	yt = θ(st - pt)	${\mathrm{p}}_{\mathrm{t}}^{*}=0$
General solution
	$\begin{array}{l}\left({\mathrm{s}}_{\mathrm{t}}-{\mathrm{p}}_{\mathrm{t}}\right)=\frac{\left({\mathrm{\lambda }}_1-1\right)}{\mathit{\pi \theta }}{\mathrm{p}}_{\mathrm{t}}+\frac{\left({\mathrm{\lambda }}_1-{\mathrm{\lambda }}_2\right)}{\mathit{\pi \theta }}\underset{\mathrm{i}=0}{\overset{\mathrm{\infty }}{\mathrm{\Sigma }}}{\mathrm{\lambda }}_2^{-\left(\mathrm{i}+1\right)}\mathrm{\Omega }{\mathrm{E}}_{\mathrm{t}}\left({\mathrm{z}}_{\mathrm{t}-1+\mathrm{i}}\right)\\ {\mathrm{p}}_{\mathrm{t}}={\mathrm{\lambda }}_1{\mathrm{p}}_{\mathrm{t}-1}-\mathrm{\pi }{\overline{\mathrm{y}}}_{\mathrm{t}-1}+\left({\mathrm{\lambda }}_1-{\mathrm{\lambda }}_2\right)\underset{\mathrm{i}=0}{\overset{\mathrm{\infty }}{\mathrm{\Sigma }}}{\mathrm{\lambda }}_2^{-\left(\mathrm{i}+1\right)}\mathrm{\Omega }{\mathrm{E}}_{\mathrm{t}}\left({\mathrm{z}}_{\mathrm{t}-1+\mathrm{i}}\right)\end{array}$
Here, λ1 and λ2 are the eigenvalues given by
	$\begin{array}{c}{\mathrm{\lambda }}_1=1-\frac{\mathit{\pi \theta }}{2}-\frac{1}{2}{\left[{\left(\mathit{\pi \theta }\right)}^2+4{\mathit{\theta \pi \alpha }}^{-1}\right]}^{\mathrm{½}}\\ {\mathrm{\lambda }}_2=1-\frac{\mathit{\pi \theta }}{2}+\frac{1}{2}{\left[{\left(\mathit{\pi \theta }\right)}^2+4{\mathit{\theta \pi \alpha }}^{-1}\right]}^{\mathrm{½}}\end{array}$
so that λ2 > λ1. It is assumed that λ1 lies inside the unit circle and that λ1 lies outside it. zt is the vector given by
	${\mathrm{z}}_{\mathrm{t}}=\left[{\mathrm{m}}_{\mathrm{t}},{\mathrm{R}}_{\mathrm{t}}^{*},{\overline{\mathrm{y}}}_{\mathrm{t}}\right]$
and Ω is a coefficient vector given by
$\mathrm{\Omega }=\left[\frac{-{\mathit{\pi \theta \alpha }}^{-1}}{{\mathrm{\lambda }}_2-{\mathrm{\lambda }}_1},\frac{\mathit{\pi \theta }}{{\mathrm{\lambda }}_2-{\mathrm{\lambda }}_1},\frac{\mathit{\pi }\left[\mathit{\theta }\left({\mathit{\alpha }}^{-1}\mathit{\beta }+\mathit{\pi }\right)-\left(1-{\mathrm{\lambda }}_1\right)\right]}{{\mathrm{\lambda }}_2-{\mathrm{\lambda }}_1}\right]$
The particular solution when the money supply follows an AR1 process is given below:
	mt = ρmt-1 + μt	where μ is white noise.
	${\mathrm{R}}_{\mathrm{t}}^{*}=0$
	ȳt = 0
When ρ ∈ (0,1), mt follow a stationary stochastic process. When ρ = 1, mt follows a random walk.
	$\begin{array}{l}\left({\mathrm{s}}_{\mathrm{t}}-{\mathrm{p}}_{\mathrm{t}}\right)=\frac{\left({\mathrm{\lambda }}_1-1\right)}{\mathit{\pi \theta }}{\mathrm{p}}_{\mathrm{t}}+\frac{{\mathit{\alpha }}^{-1}}{{\mathrm{\lambda }}_2-\mathit{\rho }}{\mathrm{m}}_{\mathrm{t}}\\ {\mathrm{p}}_{\mathrm{t}}={\mathrm{\lambda }}_1{\mathrm{p}}_{\mathrm{t}-1}+\frac{{\mathit{\pi \theta \alpha }}^{-1}}{{\mathrm{\lambda }}_2-\mathit{\rho }}{\mathrm{m}}_{\mathrm{t}-1}\\ {\mathrm{s}}_{\mathrm{t}}=\frac{\left({\mathrm{\lambda }}_1-1\right)}{\mathit{\pi \theta }}{\mathrm{p}}_{\mathrm{t}}+{\mathrm{\lambda }}_1{\mathrm{p}}_{\mathrm{t}-1}+{\mathit{\alpha }}^{-1}\frac{\left(\mathit{\rho }+\mathit{\pi \theta }\right)}{{\mathrm{\lambda }}_2-\mathit{\rho }}{\mathrm{m}}_{\mathrm{t}-1}+\frac{{\mathit{\alpha }}^{-1}}{{\mathrm{\lambda }}_2-\mathit{\rho }}{\mathit{\mu }}_{\mathrm{t}}\end{array}$

The solutions to the sticky-price model presented in Table 2 have two important implications for short-run exchange rate behavior. The first is that nominal and real exchange rates may overshoot in response to innovations in the money supply. A sufficient condition for nominal exchange rate overshooting in the model presented in Table 2 is that the money supply follows a random walk so that all changes in the money supply are expected to be permanent. ^21/ The second implication is that there can be predictable nominal and real exchange rate movements associated with the slow adjustment of commodity prices. These can arise with the money supply itself following a random walk (with ρ = 1 in Table 2), implying that deviations of the exchange rate from a random walk occur even if forcing variables are entirely unpredictable, and reflect the intrinsic dynamics of the model.

If purchasing power parity is assumed to hold in the long run, the real exchange rate is a stationary variable in the sticky-price model. It will, however, be closely correlated with the nominal exchange rate in the short run and display predictable movements. If there are permanent real shocks, changing for example the level of potential output, the real exchange rate will be nonstationary in this model.

Given the potential importance of real shocks for exchange rates, the flexible-price monetary model has been adapted to allow a more explicit role for real factors, giving rise to inter-temporal equilibrium models in which nominal and real exchange rates are simultaneously determined. ^22/ These applications continue to assume price flexibility but allow for shocks such as productivity improvements or increases in real government spending to explicitly influence nominal and real exchange rates. In addition, the models include intrinsic dynamics associated with factors such as slow adjustment of investment spending and inter-temporal labor substitution.

Portfolio models, The key assumptions of the portfolio models are that (interest-earning) assets denominated in different currencies are imperfect substitutes and residents of each country have a preference at the margin for assets denominated in their own currencies. ^23/ An important source of systematic movement in nominal and real exchange rates in these models derives from intrinsic dynamics associated with current account imbalances. When residents of a country have a preference at the margin for their own commodities and assets, countries with current account surpluses may have continually appreciating real and nominal exchange rates due to the need to offset incipient excess demands for domestic assets and goods. ^24/ More generally, these models suggest that nominal exchange rates will depend on the accumulation of outside domestic assets (money and bonds) and (net) foreign bonds. Given that outside domestic assets are created through the fiscal balance while (net) foreign assets are acquired by running current account surpluses, the models imply that the evolution of the exchange rate may be associated with fiscal and current account balances. The stochastic processes nominal and real exchange rates follow then depend on the stochastic processes describing asset supplies—money, public interest-bearing debt, and foreign assets, as well as the intrinsic dynamics of the models. ^25/

Neither the monetary nor portfolio models reviewed in this section implies that nominal or real exchange rates should be characterized by random walks. Predictable movements in exchange rates can arise either from extrinsic dynamics in the forcing variables or from intrinsic dynamics that spread the adjustment to shocks over time. If purchasing power parity is assumed to hold in the long run, real exchange rates will display predictable fluctuations around a fixed mean. But if allowance is made for permanent real shocks, such as changes in tastes or technology, real exchange rates will be nonstationary. The stochastic process for nominal exchange rates also displays predictable movements in these models which can either be around a fixed or a varying mean, depending on the processes followed by forcing variables. Differences in monetary policy across countries are one reason why nominal exchange rates might be nonstationary. Divergences in fiscal policy as well as current account imbalances are another potential source of nonstationarity. Both nominal and real exchange rates are expected therefore to deviate from random walks on the basis of standard exchange rate models. ^26/

The only models that predict a (driftless) random walk for nominal or real exchange rates, regardless of the stochastic process describing forcing variables, are the currency substitution and financial arbitrage models. Neither of these models, however, appears to have found its way into the mainstream of structural exchange rate models. In currency substitution models, the requirement that the expected pecuniary returns from holding domestic and foreign monies be equal when monies are perfect substitutes implies that the change in the nominal exchange rate should be unpredictable. ^27/ In financial arbitrage models, the equalization of expected real interest rates on domestic and foreign bonds is interpreted as requiring that the expected change in the real exchange rate be unpredictable. ^28/ It is unclear, however, why real interest rates measured in terms of different consumption baskets should be equalized, suggesting that the financial arbitrage models may not provide a firm basis for expecting real exchange rate changes to be entirely unpredictable.

III. Univariate Time Series Tests

This section presents evidence on the time series processes followed by nominal and real exchange rates of the dollar in terms of other major currencies. ^29/ After briefly examining measures of the variability of dollar exchange rates over the recent floating rate period, the section presents standard tests for the stationarity of exchange rates and departures from a random walk. This is then followed by the application of new and more powerful tests intended to detect evidence of mean reversion in exchange rates over long periods. ^30/

An examination of the time series processes followed by exchange rates is useful for a number of reasons. First, in the absence of speculative bubbles, any nonstationarity in exchange rates must ultimately be determined by nonstationarity in the forcing variables appearing in structural exchange rate models. Explaining the long-run behavior of exchange rates then requires identifying other economic variables with similar orders of nonstationarity. Second, particular processes such as the random walk are frequently regarded as carrying strong implications for the shocks that have driven exchange rates, and the economic models that are appropriate for analyzing their behavior. If one is to view the exchange rate as an equilibrium exchange rate, then the random walk property implies that all shocks driving them must be permanent. ^31/ Further, for the real exchange rate such a finding argues for the prevalence of real rather than nominal shocks, since the latter would be expected to have only transitory effects. As noted earlier, moreover, the random walk finding argues against the applicability of models—such as the sticky-price monetary model of Dornbusch(1976)—that ascribe a major role to short-run disequilibrium dynamics, since such dynamics induce systematic movements in exchange rates. Third, specific hypotheses such as purchasing power parity can be tested by looking for evidence of mean reversion in real exchange rates. ^32/

The data series for the study comprise the four bilateral spot exchange rates between the U.S. dollar and the deutsche mark, Japanese yen, pound sterling, and Canadian dollar. In order to ensure comparability with other studies, real exchange rates are measured using consumer prices. The data are sampled quarterly over the period 1974:1 through 1989:4 and are taken from International Financial Statistics.

Before applying the statistical tests, it is useful to consider some summary statistics for the data series. Confirming the view that exchange rates behave like asset prices, Tables 3a and 3b show that the variance of quarterly changes in nominal exchange rates has been considerably larger than that in relative consumer prices. ^33/ For some bilateral rates, it is almost 75 times the variance of changes in commodity prices. As a result, the variance of changes in the real exchange rate is dominated by that of changes in the nominal exchange rate. In fact, the variance of changes in real and nominal exchange rates are essentially equal, and there is almost perfect contemporaneous correlation between them. ^34/ The table also shows that while there is some evidence of drift in exchange rates, in no case is it statistically significant.

Table 3a.

Summary Statistics for the Log of U.S. Dollar Exchange Rates and Relative Price Levels Quarterly, 1974:1 to 1989:4

Table 3a.

Summary Statistics for the Log of U.S. Dollar Exchange Rates and Relative Price Levels Quarterly, 1974:1 to 1989:4

		Levels	First Differences
		Standard deviation	Mean	t-statistic for mean	Standard deviation
Spot rate, sx
X =	UK	0.186	-0.00634	-0.91	0.0556
	GR	0.173	0.00629	0.81	0.0618
	CA	0.107	-0.00277	-0.99	0.0222
	JA	0.268	0.01039	1.35	0.0609
Relative Price level, PX - PUS
X =	UK	0.137	0.00867	4.68	0.0147
	GR	0.143	-0.00740	-7.79	0.0075
	CA	0.045	0.00213	2.84	0.0060
	JA	0.128	-0.00480	-3.74	0.0102
Real exchange rate, qX
X =	UK	0.143	0.00232	0.33	0.0552
	GR	0.139	-0.00111	-0.14	0.0618
	CA	0.080	-0.00064	-0.23	0.0219
	JA	0.192	0.00559	0.71	0.0628

View Table

Table 3a.

Summary Statistics for the Log of U.S. Dollar Exchange Rates and Relative Price Levels Quarterly, 1974:1 to 1989:4

		Levels	First Differences
		Standard deviation	Mean	t-statistic for mean	Standard deviation
Spot rate, sx
X =	UK	0.186	-0.00634	-0.91	0.0556
	GR	0.173	0.00629	0.81	0.0618
	CA	0.107	-0.00277	-0.99	0.0222
	JA	0.268	0.01039	1.35	0.0609
Relative Price level, PX - PUS
X =	UK	0.137	0.00867	4.68	0.0147
	GR	0.143	-0.00740	-7.79	0.0075
	CA	0.045	0.00213	2.84	0.0060
	JA	0.128	-0.00480	-3.74	0.0102
Real exchange rate, qX
X =	UK	0.143	0.00232	0.33	0.0552
	GR	0.139	-0.00111	-0.14	0.0618
	CA	0.080	-0.00064	-0.23	0.0219
	JA	0.192	0.00559	0.71	0.0628

Table 3b.

Correlation of Changes in U.S. Dollar Real and Nominal Exchange Rates

Quarterly, 1974:1 to 1989:4

^1/The reported correlations are between contemporaneous values of the first difference of real exchange rate and 0 to 8 lags of the first difference of nominal exchange rates.

Table 3b.

Correlation of Changes in U.S. Dollar Real and Nominal Exchange Rates

Quarterly, 1974:1 to 1989:4

Lag length 1/	0	1	2	3	4	5	6
UK	0.97	0.21	-0.11	0.16	0.16	-0.19	0.04
GR	0.99	0.17	-0.10	0.11	0.18	0.06	-0.13
CA	0.96	0.09	-0.06	0.17	0.00	0.01	0.23
JA	0.99	0.18	0.06	0.03	0.00	-0.19	-0.15

^1/The reported correlations are between contemporaneous values of the first difference of real exchange rate and 0 to 8 lags of the first difference of nominal exchange rates.

View Table

Table 3b.

Correlation of Changes in U.S. Dollar Real and Nominal Exchange Rates

Quarterly, 1974:1 to 1989:4

Lag length 1/	0	1	2	3	4	5	6
UK	0.97	0.21	-0.11	0.16	0.16	-0.19	0.04
GR	0.99	0.17	-0.10	0.11	0.18	0.06	-0.13
CA	0.96	0.09	-0.06	0.17	0.00	0.01	0.23
JA	0.99	0.18	0.06	0.03	0.00	-0.19	-0.15

^1/The reported correlations are between contemporaneous values of the first difference of real exchange rate and 0 to 8 lags of the first difference of nominal exchange rates.

The first set of tests we apply are for unit-root nonstationarity in the time series processes for nominal and real exchange rates and relative prices (Table 4). ^35/ As was noted earlier, none of the structural exchange rate models carries strong predictions as to whether we would expect to find stationarity in nominal exchange rates, with the issue an empirical one. Differences in monetary growth rates across countries as well as current account imbalances, however, suggest that it is quite likely nominal exchange rates will be nonstationary. Whether real exchange rates would be expected to be stationary is also an empirical question. Only in the case of models built on purchasing power parity is there a presumption that real exchange rates will be stationary.

Table 4.

Test Statistics for the Null Hypothesis of a Unit Root ^1/

Quarterly, 1974:1 to 1989:4

^1/All regressions include a constant and three seasonals.

^2/Regression includes two lags.

^3/For residuals from the augmented Dickey-Fuller regression.

^4/See Hall and Henry (1988).

Table 4.

Test Statistics for the Null Hypothesis of a Unit Root ^1/

Quarterly, 1974:1 to 1989:4

		Dickey- Fuller Statistic	Augmented Dickey-Fuller Statistic 2/	Durbin- Watson 3/
Spot rate, sx
X =	UK	-1.58	-1.75	1.94
	GR	-1.04	-1.43	1.95
	CA	-1.81	-1.76	1.97
	JA	-0.52	-1.13	2.02
Relative price levels, PX - PUS
X =	UK	-5.50***	-4.60***	2.09
	GR	-2.24	-0.96	2.07
	CA	-0.78	-1.33	2.04
	JA	1.51	-0.36	2.18
Real exchange rates, qX
X =	UK	-1.48	-1.60	1.91
	GR	-1.25	-1.37	1.94
	CA	-1.36	-1.39	2.00
	JA	-1.18	-1.79	2.03
Critical values 4/
1%	***	-4.07	-3.77
5%	**	-3.37	-3.17
10%	*	-3.03	-2.84

^1/All regressions include a constant and three seasonals.

^2/Regression includes two lags.

^3/For residuals from the augmented Dickey-Fuller regression.

^4/See Hall and Henry (1988).

View Table

Table 4.

Test Statistics for the Null Hypothesis of a Unit Root ^1/

Quarterly, 1974:1 to 1989:4

		Dickey- Fuller Statistic	Augmented Dickey-Fuller Statistic 2/	Durbin- Watson 3/
Spot rate, sx
X =	UK	-1.58	-1.75	1.94
	GR	-1.04	-1.43	1.95
	CA	-1.81	-1.76	1.97
	JA	-0.52	-1.13	2.02
Relative price levels, PX - PUS
X =	UK	-5.50***	-4.60***	2.09
	GR	-2.24	-0.96	2.07
	CA	-0.78	-1.33	2.04
	JA	1.51	-0.36	2.18
Real exchange rates, qX
X =	UK	-1.48	-1.60	1.91
	GR	-1.25	-1.37	1.94
	CA	-1.36	-1.39	2.00
	JA	-1.18	-1.79	2.03
Critical values 4/
1%	***	-4.07	-3.77
5%	**	-3.37	-3.17
10%	*	-3.03	-2.84

^1/All regressions include a constant and three seasonals.

^2/Regression includes two lags.

^3/For residuals from the augmented Dickey-Fuller regression.

^4/See Hall and Henry (1988).