1. The monetary model of the exchange rate

The standard specification of the monetary model involves two equations describing domestic and foreign money demands, a purchasing power parity equation, and an uncovered interest parity equation. Assuming, as is standard, that the domestic and foreign money demand parameters are the same, the model can be written as:

where m_t, p_t, and y_t, denote (log) domestic money supply, prices, and real income, respectively, and i_t denotes the interest rate at time t on deposits maturing at time t+1. Foreign variables are denoted with an asterisk. The (log) exchange rate, defined as the domestic price of a unit of foreign currency, is denoted by s_t, while E_t[.] denotes the rational expectation operator conditional on information available at time t.

Equations (1)-(4) can be collapsed into

where

Equations (5) and (6) express the exchange rate as the sum of current fundamentals, f_t, plus a linear function of its own value at time t+1. Solving (5) forward up to time t+h, yields

The standard assumption in the literature is to assume the absence of exchange rate bubbles, i.e., that ${\lim }_{i\to \infty }{\left(\frac{1}{1+\mathrm{\alpha }}\right)}^i{\mathrm{E}}_t\left[f_{t+i}\right]\text{=}0,$ and solve equation (7) forward solely in terms of fundamentals:

The tests considered in this paper, however, are robust to the presence of bubbles, and can be performed directly on (7): if s_t+h, incorporates a bubble term (i.e., a capital gain that reflects solely the anticipation of a future currency transaction), so does s_t, and equation (7) remains valid. In most of our tests, the investment’s holding period h is set at three, as forward markets tend to be most liquid for three-month maturities, and the power of our tests falls when h increases much above three. Tests for different values of h are discussed below.

2. Volatility tests of the monetary model

Let us now define the perfect-foresight (or fundamental) exchange rate, $s_t^{*}$ as the value that the exchange rate would take if investors could predict with certainty future fundamentals, as well as the exchange rate at t+h. The fundamental exchange rate $s_t^{*}$ is obtained from equation (7) by dropping the expectation operator:

so that, by definition, $s_t={\mathrm{E}}_t\left[s_t^{*}\right]$ under the assumptions of the monetary model.

We must now choose a benchmark exchange rate, denoted by $s_t^{\mathrm{o}}$ about which to measure the volatility of both market and fundamental exchange rates. The need for a benchmark rate reflects analytical and statistical considerations. First, there is simply not a unique way to define “exchange rate volatility” except with respect to a reference benchmark (be it its sample mean or other variables). Second, the notorious nonstationarity of exchange rates requires measuring exchange rate movements with respect to a specific (stochastic) trend.

There are two main requirements for the choice of $s_t^{\mathrm{o}}$: that it be known to investors at time t, and that the differences s_t-$s_t^{\mathrm{o}}$ and $s_t^{*}-s_t^{\mathrm{o}}$ be stationary. There are many alternative ways to choose $s_t^{\mathrm{o}}$, however, so as to satisfy these requirements. For instance, $s_t^{\mathrm{o}}$ could be defined as the model’s prediction at time t-1 of the exchange rate at time t, which can be obtained by lagging equation (5) once and solving for E_t-1[s_t]:

When $s_t^{\mathrm{o}}$ is chosen in this fashion, the series $s_t^{*}-s_t^{\mathrm{o}}$ and s_t-$s_t^{\mathrm{o}}$ describe fundamental and actual exchange rate surprises, based on the predictions of the monetary model.

Alternatively, exchange rate volatility could be measured by setting $\begin{array}{c}{s}_t^{\mathrm{o}}=s_{t-L}^{\mathrm{o}}\end{array}$ where L is a suitable lag. For instance, it is customary to focus on conditional volatilities, defined, with L=1, as the volatility of the first difference of exchange rates, s_t-s_t-1. Our task, in this case, would be to assess the consistency of the model’s predictions of market and fundamental exchange rate changes from the last known realization of the exchange rate.

Yet another possibility would be to choose $s_t^{\mathrm{o}}$ as some “naive” exchange rate forecast, for instance as the value that the exchange rate would take if investors expected fundamentals to evolve as a random walk. Under this assumption, solving Equation (7) forward with E_t[f_t+i] = f_t, as implied by the random walk hypothesis, yields s_t = f_t, i.e., the exchange rate itself should follow a random walk under the assumptions of the monetary model. Note that this “naive” forecast need not be a rational one (although there is, indeed, a considerable amount of evidence that exchange rates may be well approximated by random walks): as long as s_t = f_t is known to investors at time t, and s_t-$s_t^{\mathrm{o}}$ and $s_t^{*}-s_t^{\mathrm{o}}$ are stationary, then $s_t^{\circ }\equiv f_t$ is an acceptable benchmark for our volatility tests. Indeed, Mankiw, Romer, and Shapiro (1985, 1991) have suggested a very similar benchmark when testing for excess stock price volatility, by assuming stock dividends to follow a random walk. Gros (1989) followed their lead, measuring exchange rates with respect to a random walk benchmark. These issues, and their implications for the volatility tests performed in this paper, are further discussed in the following.

Now, with rational expectations, the forecast error $s_t^{*}-{\mathrm{E}}_t\left[s_t^{*}\right]=s_t^{*}-s_t$ should be uncorrelated with variables known at time t, including s_t and $s_t^{\mathrm{o}}$. This implies that we should have

Therefore, squaring both sides of the identity

taking expectations, and using equation (11), yields

or

Therefore, the monetary model implies the testable restriction q_t = 0, and hence the restriction E[q_t]=0: the sample mean of the q_t’s, q, should be close to zero if the model describes correctly the dynamics of exchange rates. As noted above, this test is robust to the presence of exchange rate bubbles, since if the market exchange s_t incorporates a bubble term, so do $s_t^{*}$ and $s_t^{\mathrm{o}}$, and equation (14) remains valid.

Thus, we can construct a test of the monetary model as follows. First, we can compute the fundamental exchange rate, $s_t^{\mathrm{*}}$, and the benchmark exchange rate, $s_t^{\mathrm{o}}$, from (9) and (10), after calibrating the money demand parameters α and β. Then we can use an asymptotic distribution of the sample mean q and reject the model when q is significantly different from zero. A Generalized Method of Moments distribution of q (see Bollerslev and Hodrick, 1992) is a normal distribution

with variance estimated by $\hat{V}=C\left(0\right)+2{\mathrm{\Sigma }}_{j=1}^{h-1}C\left(j\right)$, where $C\left(j\right)=\frac{h-j}{h}{\mathrm{\Sigma }}_{t=j+1}^{\tau }\frac{q_{t+h}{q}_{t+h-j}}{T}$, which is robust to serial correlation and heteroskedasticity in the error terms.

Equation (13) also implies

where the expectations are now taken unconditionally. Should either one of these inequalities be violated, then E_t[q_t]<0 (although the opposite is not necessarily true), thus suggesting to use (16A-B) as diagnostics for the model. Inequality (16A), in particular, states that the market exchange rate, s_t, should forecast the behavior of the fundamental rate, $s_t^{\mathrm{*}}$, better than the benchmark rate, $s_t^{\mathrm{o}}$, in terms of the usual mean-square error criterion. Inequality (16B) states that the market exchange rate should be less volatile around the benchmark rate than the fundamental rate. This latter inequality provides an ‘excess volatility’ test of the model, and its intuition is simple: the fundamental exchange rate should deviate from the benchmark rate by as much as the market rate does, plus a forecast error $s_t^{\mathrm{*}}$-s_t. If markets are efficient and the monetary model correctly describes exchange rate dynamics, this forecast error should not be systematically related to information available to investors. The volatility of the fundamental exchange rate around the benchmark rate, therefore, should exceed that of the market rate.

3. Relation to alternative tests of the model

Two points should be noted about the test procedure just outlined. First, there is a close relationship between the volatility-based tests presented here and previous regression-based tests (see Hodrick, 1987, for a survey). Recall that the starting point of our analysis is the condition that $s_t^{\mathrm{*}}$-s_t and s_t-$s_t^{\mathrm{o}}$ should be uncorrelated if exchange rates reflect information available to market participants and the monetary model correctly describes the dynamics of exchange rates. Using regression analysis, the natural test of this hypothesis would be to regress $s_t^{\mathrm{*}}$-s_t on s_t-$s_t^{\mathrm{o}}$, and test if the regression coefficient

is zero. A non-zero ${\hat{\pi }}_1$, implies that the prediction error $s_t^{\mathrm{*}}$-s_t can be forecast at time t. This, among other things, implies that s_t+1-E_t[s_t+1] is forecastable, since $s_t^{*}-s_t=\frac{\mathrm{\alpha }}{1+\mathrm{\alpha }}(s_{t+1}-{\mathrm{E}}_t\left[s_{t+1}\right])$, i.e., that there is a systematic bias between exchange rate predictions and realizations.

In comparison with (17), the volatility tests implemented in this paper reject the model when the mean of the q_t’s is zero. Now, since q be written as

a link between the regression-based and the volatility-based methods is apparent. Both methods involve testing whether $\frac{1}{T}\underset{t}{\mathrm{\Sigma }}(s_t-s_t^{\mathrm{o}})(s_t^{*}-s_t)$, suitably standardized, is significantly different^Tft^-om zero, and both tests provide—when statistically significant—evidence of excess returns’ forecastability, i.e., that either markets are inefficient, or the assumed exchange rate model is invalid, or both. There are two main differences between the two methods, however.

The first difference is that the test used in this paper is directly linked to measures of exchange rate volatility and to corresponding volatility inequalities. These inequalities allow testing whether “excessive” volatility (suitably defined) is indeed a cause of failure of the model.

The second difference is more technical: the volatility-based tests used here have been shown to exhibit better statistical properties than the corresponding regression-based tests. Specifically, Mankiw, and Shapiro (1986) and Mankiw, Romer, and Shapiro (1991) present Montecarlo simulations showing that the finite-sample distribution of a sample-mean q is well approximated by its asymptotic distribution, namely, that the volatility-based test tends to reject the underlying model on average the right number of times. In contrast, the finite-sample distribution of the regression coefficient ${\hat{\pi }}_1$, is poorly approximated by its asymptotic distribution. In particular, if the dependent and the independent variables are correlated, and the independent variable is itself serially correlated—which is certainly the case in our tests, where the same “fundamentals” drive both the dependent and independent variables—then regression-based tests are systematically biased, tending to reject the underlying model too often in finite samples.

Next, the ability of the test to reject the model, as well as the interpretation of the test results, depend crucially on the particular choice of $s_t^{\mathrm{o}}$. While it impinges on the model to pass tests with respect to all suitable $s_t^{\mathrm{o}}$s, the usefulness of the test depends on $s_t^{\mathrm{o}}$ being a useful benchmark for the measurement of exchange rate volatility. Establishing the best benchmark is a task that goes well beyond the scope of this paper, particularly since different benchmarks are bound to be useful in different contexts. In welfare analysis, for instance, the appropriate definition of “exchange rate volatility” will depend on agents’ attitudes toward risk, the nature of adjustment costs, etc. Nevertheless, it is clear that the usefulness of a particular benchmark is likely to depend on whether exchange rates tend to gravitate, in some loose sense, around it. Few scientists seem to find useful a description of stars’ and planets’ position in space with respect to the moon, nor to find useful the (formally correct) statement that the sun is more volatile than the earth about the moon.

As the issue of choosing a suitable benchmark cannot be resolved in a statistical context—only on the basis of a well specified economic model—in our statistical analysis we follow an eclectic approach. We present results for a variety of choices of $s_t^{\mathrm{o}}$, drawing from common usage in research and policy analysis, in order to highlight the implications of alternative definitions of “exchange rate volatility.” We focus, in particular, on the conditional volatility of exchange rates and on the volatility of their unanticipated movements. We present tests that use the model’s one-period-ahead prediction of the exchange rate as a benchmark (see equation (10)), to test whether unanticipated changes in exchange rates are consistent with movements in their determinants. We also present tests that use the lagged exchange rate as a benchmark, to test whether total changes in exchange rates are justified by movements in their determinants. Finally, we present tests where the benchmark rate is set at the value that the exchange rate would take if investors believed fundamentals to behave as random walks. The tests can be applied in a straightforward way to other definitions of $s_t^{\mathrm{o}}$.

Though our tests’ dependence on the benchmark $s_t^{\mathrm{o}}$ may seem unfortunate, this dependence simply reflects the intrinsic ambiguity of the concept of “exchange rate volatility”: there is simply no unique way to measure exchange rate volatility—just as there is no unique way to measure a planet’s movements—except with respect to a specific benchmark. This is the main point we wish to emphasize in this paper, that this ambiguity should be explicitly recognized.

1. Sticky prices

Following Mussa (1985) and Gros (1989), we adopt a sticky-price version of Purchasing Power Parity (PPP), much along the lines of the standard Dornbusch model. In this model, the real exchange rate $s_t+p_t^{*}-p_t$, moves toward its PPP value at the rate θ, as in

In equation (19), when θ=0 the real exchange rate follows a random walk with no tendency to revert to PPP. When θ=1 the exchange rate converges to its PPP value in a single period.

Replacing Equation (3) with Equation (19) in the model, some tedious but straightforward algebra yields the ‘fundamental’ sticky-price exchange rate, solved forward up to time t+h:

where $\gamma =\frac{(1-\theta )\mathrm{\alpha }}{1+\mathrm{\alpha }}$, and ${\tilde{s}}_t^{*}$ denotes the fundamental exchange rate for the flex-price model (see equation (9)). The expectation at t-1 of the market exchange rate at time t is

For θ=1, the sticky-price model degenerates into the flex-price model; for smaller values of θ, the sticky-price model predicts greater volatility of exchange rates in response to changes in their determinants. Except for this redefinition of variables, the test procedure remains the same as described in the previous section.

2. Slow money adjustment

We can also relax the assumption that real money balances adjust instantaneously to their equilibrium value. Partial adjustment equations of the form

have been widely used in the literature on money demand functions. The terms (1-λ) (m_t-1-p_t) and $(1-\lambda )(m_{t-1}^{*}-p_t^{*})$ include lagged money terms and contemporaneous price terms, reflecting an adjustment process of money balances specified in nominal terms. Alternatively, the adjustment term could have been specified as (1-λ) (m_t-1-p_t) (and similarly for the foreign equation), if the assumed slow response were of real money balances. We use the former specification because it allows the familiar forward solution of the exchange rate in terms of fundamentals, and because of its superior empirical performance (see Fair, 1987). ^2/

Thus, the coefficients β and α in (22) and (23) describe the long-run income elasticity and long-run interest semi-elasticity of money demand, respectively; λ measures the speed of adjustment of real money balances to their steady state, and takes values between zero and one: when λ=1, real money balances adjust immediately to their steady state; when λ=0, money demand does not respond to interest rates and income, and the model has no steady-state solution.

Replacing (1) and (2) with (22) and (23) yields

Except for this redefinition, the test procedure remains the same as described above.

3. Time-varying risk premia

Finally, we can allow for a time-varying residual, x_t, in the interest parity equation (4):

For our purposes, an ex post estimate of the residual x_t can be obtained by detrending $i_t-i_t^{*}-(s_{t+1}-s_t)$, i.e., $(1+\mathrm{\alpha })s_t-\mathrm{\alpha }{s}_{t+1}-(m_t-m_t^{*})+\mathrm{\beta }(y_t-y_t^{*})$ with a Hodrick-Prescott filter. The test procedure then remains the same as that described in Section 3, with “fundamentals” redefined as

One possible interpretation of x_t is as a risk-premium and, for simplicity, we shall use this label in our discussion. Nevertheless, our simple treatment must be viewed only as an ad hoc way to relax the interest parity equation, not as a structural model of a risk premium. Our aim, once again, is not to provide a satisfactory exchange rate model, but only to verify the robustness of our results to relaxation of the basic assumptions of the monetary model. ^3/ Hence, in our tests we have used a variety of values for the Hodrick-Prescott smoothing coefficient, ρ, summarized by the choices ρ=14,400 (which, following the literature, equals 100 times the square of the sample frequency), and ρ=1,600, a lower value that allows the model to better fit the data.

1. Data and calibration of the tests

The full sample covers data for the United States, Japan, Germany, France, the United Kingdom, Italy, Canada, and Switzerland during the period January 1973-September 1994. We use the same proxies for the monetary model’s variables used in previous studies: monthly data for narrow and broad money supply, GDP, industrial production, consumer price indices, and exchange rates against the U.S. dollar (end-of-period data). All data are from the IFS of the IMF, except for a few incomplete IFS series that were obtained from the Current Economic Indicator data-base of the IMF.

We calibrated our tests using parameters estimated in previous empirical studies, choosing wide ranges to encompass both estimated and plausible values for each parameter.

Most estimates of the annualized semi-elasticity of money demand to short term interest rates, for money demand functions with instantaneous adjustment, fall in the range 1 to 4 (see, for instance, Bilson, 1978, and Laidler, 1993). Estimated exchange rate models tend to yield smaller values of α (see, for instance, Flood, Rose, and Mathieson, 1991), while money demand equations with partial adjustment tend to yield somewhat higher values of a (see, for instance, Fair, 1987, and Goldfeld and Sichel, 1990). We took the values α=0.1 and α=6 (to be multiplied by 12 in monthly tests) as spanning the plausible range of α, using α=1 as a baseline.

Most estimates of the income elasticity of money demand, β, range from about 0.5 to about 1.5, independently of the assumed speed of money adjustment (see, for instance, Fair, 1987, and Goldfeld and Sichel, 1990). We begin by considering the values β=0.2 and β=2, and then fix β at unity for the rest of the analysis.

The speed of adjustment of money stocks to their steady state, λ, is more difficult to calibrate. Annualized estimates of λ from pre-1974 (quarterly) data typically range around 0.3 - 0.5, suggesting a half-life of money shocks between 6 and 18 months—already a surprisingly slow response. In fact, estimates from post-1974 data typically yield near-zero estimates λ, implying no response of money demand to changes in its determinants, and a breakdown of the steady-state solution of the model. It is now well understood that the estimated unstable behavior of money balances from post-1974 data reflects more the difficulty of capturing with simple dynamic specifications the financial innovations and institutional changes that occurred since 1974, than an implausibly low speed of portfolio adjustment (see Goldfeld and Sichel, 1990, and Boughton, 1992, for a discussion). We report results for the baseline case of instantaneous adjustment, λ=1, and for the cases of λ=0.5 and λ=0.1 (corresponding to monthly values of about λ=.06 and λ=.01). We also present results of tests conducted over split samples, as a preliminary check of robustness of our inference to structural breaks.

Estimated half-lives of price shocks in PPP equations range anywhere from one to six years, depending on currency and sample (see, for instance, Hakkio, 1992, and Lothian and Taylor, 1993). Hence, (annualized) values of θ should range between 0.1 and 0.5. We report tests with θ set at 0.3; results for tests calibrated with θ=0.5 and θ=0.1 were very similar.

Finally, “fundamental” data for output, money, and prices must be normalized. First, data are reported only in index number form. Second, econometricians can only guess which macro-aggregates best capture “output”, “money”, and “prices” in theoretical money demand functions, suggesting allowance for a degree of freedom in scaling raw data. Upon allowing for a multiplicative factor in raw data (i.e., for an additive constant in log-transformed data), the model suggests a logical way to normalize the series: by scaling raw data so that the model holds on average over the sample (i.e., so that ${\overline{s}}_t^{*}={\overline{s}}_t$).

2. Results

Tables 1-3 report a sample of the volatility tests that we implemented. ^4/

Table 1.

Flexible Price Model

Notes: The reported values are the standard normal statistics for the null hp. that q_t=0, where q_t is defined by equation (14) in the text. Newey-West standard errors, with a lag truncation of 6-(n/255)^1/4, where n is the number of observations, were used. A superscript A indicates that inequality (16A) is violated, while a superscript B indicates that inequality (16B) is violated. Monthly data for broad money and industrial production are used in these tests. The sample goes from April 1973 to September 1994. The number of observations is 255.

Table 1.

Flexible Price Model

	$s_t^{\mathrm{o}}={\mathrm{E}}_{t-1}\left[s_t\right]$				$s_t^{\mathrm{o}}=s_{t-1}$				$s_t^{\mathrm{o}}=f_t$
	α=.1		α=6		α=.1		α=6		α=0.1		α=6
	β=0.2	β=2	β=0.2	β=2	β=0.2	β=2	β=0.2	β=2	β=0.2	β=2	β=0.2	β=2
Pound sterling	4.56	4.70	2.52	2.58	-0.39	-0.25	2.29	2.24	-4.60 AB	-4.74 AB	-2.07 B	-2.52 B
French franc	3.62	4.22	2.83	2.84	-0.25	0.33	2.76	2.79	-3.69 AB	-4.36 AB	-2.19 B	-2.28 B
Deutsche mark	4.09	4.44	2.76	2.96	-0.30	-0.24	2.58	2.63	-4.14 AB	-4.47 AB	-2.23 B	-2.43 B
Italian lira	3.98	4.33	2.53	2.42	-0.37	-0.86 A	2.37	2.32	-4.01 AB	-4.65 AB	-1.38 B	-1.12 B
Swiss franc	5.41	4.74	2.78	2.93	-0.18	0.09	2.59	2.61	-5.46 AB	-4.79 AB	-1.87 B	-2.40 B
Canadian dollar	4.88	4.95	2.41	2.75	0.94	1.17	1.76	1.83	-4.92 AB	-5.01 AB	-3.00 B	-3.65 B
Japanese yen	5.09	4.14	2.63	2.45	-1.25 A	-1.43 A	2.05	2.07	-5.07 AB	-4.15 AB	-2.14 B	-1.64 B

Notes: The reported values are the standard normal statistics for the null hp. that q_t=0, where q_t is defined by equation (14) in the text. Newey-West standard errors, with a lag truncation of 6-(n/255)^1/4, where n is the number of observations, were used. A superscript A indicates that inequality (16A) is violated, while a superscript B indicates that inequality (16B) is violated. Monthly data for broad money and industrial production are used in these tests. The sample goes from April 1973 to September 1994. The number of observations is 255.

View Table

Table 1.

Flexible Price Model

	$s_t^{\mathrm{o}}={\mathrm{E}}_{t-1}\left[s_t\right]$				$s_t^{\mathrm{o}}=s_{t-1}$				$s_t^{\mathrm{o}}=f_t$
	α=.1		α=6		α=.1		α=6		α=0.1		α=6
	β=0.2	β=2	β=0.2	β=2	β=0.2	β=2	β=0.2	β=2	β=0.2	β=2	β=0.2	β=2
Pound sterling	4.56	4.70	2.52	2.58	-0.39	-0.25	2.29	2.24	-4.60 AB	-4.74 AB	-2.07 B	-2.52 B
French franc	3.62	4.22	2.83	2.84	-0.25	0.33	2.76	2.79	-3.69 AB	-4.36 AB	-2.19 B	-2.28 B
Deutsche mark	4.09	4.44	2.76	2.96	-0.30	-0.24	2.58	2.63	-4.14 AB	-4.47 AB	-2.23 B	-2.43 B
Italian lira	3.98	4.33	2.53	2.42	-0.37	-0.86 A	2.37	2.32	-4.01 AB	-4.65 AB	-1.38 B	-1.12 B
Swiss franc	5.41	4.74	2.78	2.93	-0.18	0.09	2.59	2.61	-5.46 AB	-4.79 AB	-1.87 B	-2.40 B
Canadian dollar	4.88	4.95	2.41	2.75	0.94	1.17	1.76	1.83	-4.92 AB	-5.01 AB	-3.00 B	-3.65 B
Japanese yen	5.09	4.14	2.63	2.45	-1.25 A	-1.43 A	2.05	2.07	-5.07 AB	-4.15 AB	-2.14 B	-1.64 B

Notes: The reported values are the standard normal statistics for the null hp. that q_t=0, where q_t is defined by equation (14) in the text. Newey-West standard errors, with a lag truncation of 6-(n/255)^1/4, where n is the number of observations, were used. A superscript A indicates that inequality (16A) is violated, while a superscript B indicates that inequality (16B) is violated. Monthly data for broad money and industrial production are used in these tests. The sample goes from April 1973 to September 1994. The number of observations is 255.

Table 2.

Baseline, Sluggish Money Adjustment, and Sticky Prices

Notes: The reported values are the standard normal statistics for the null hp. that q_t=0, where q_t is defined by equation (14) in the text. Newey-West standard errors, with a lag truncation of 6-(n/255)^1/4, where n is the number of observations, were used. A superscript A indicates that inequality (16A) is violated, while a superscript B indicates that inequality (16B) is violated. Monthly data for broad money and industrial production are used in these tests. The baseline model uses α=1, β=1, λ=1, θ=1; the sticky price model uses θ=.3. The sample goes from April 1973 to September 1994. The number of observations is 255.

Table 2.

Baseline, Sluggish Money Adjustment, and Sticky Prices

	$s_t^{*}={\mathrm{E}}_{t-1}\left[s_t\right]$				$s_t^{*}=s_{t-1}$				$s_t^{*}=f_t$
	Baseline λ=1	λ=0.5	λ=0.1	Sticky Prices	Baseline λ=1	λ=0.5	λ=0.1	Sticky prices	Baseline λ=1	λ=0.5	λ=0.1	Sticky Prices
Pound sterling	3.80	4.00	3.48	3.60	0.88	0.71	-0.14	0.84	-4.35 B	-4.51 B	-4.11 B	-4.23 B
French franc	2.90	2.96	2.78	2.51	1.81	2.14	2.52	2.04	-3.33 B	-3.65 B	-4.60 B	-2.97 B
Deutsche mark	3.74	3.76	3.43	3.59	1.28	1.11	0.16	1.36	-4.11 B	-4.07 B	-2.44 B	-3.68 B
Italian lira	2.91	3.17	3.17	2.37	1.33	1.72	2.46	1.73	-3.60 B	-3.93 B	-4.74 B	-3.07 B
Swiss franc	4.13	4.14	3.02	3.13	1.63	2.23	2.41	1.49	-4.82 B	-4.13 B	-2.21 B	-4.32 B
Canadian dollar	4.37	3.86	2.46	4.32	1.44	1.17	0.33	1.21	-4.81 B	-4.55 B	-4.43 B	-4.90 B
Japanese yen	4.54	4.64	3.02	2.98	0.18	0.44	1.04	0.47	-4.57 B	-4.75 B	-4.49 B	-3.77 B

Notes: The reported values are the standard normal statistics for the null hp. that q_t=0, where q_t is defined by equation (14) in the text. Newey-West standard errors, with a lag truncation of 6-(n/255)^1/4, where n is the number of observations, were used. A superscript A indicates that inequality (16A) is violated, while a superscript B indicates that inequality (16B) is violated. Monthly data for broad money and industrial production are used in these tests. The baseline model uses α=1, β=1, λ=1, θ=1; the sticky price model uses θ=.3. The sample goes from April 1973 to September 1994. The number of observations is 255.

View Table

Table 2.

Baseline, Sluggish Money Adjustment, and Sticky Prices

	$s_t^{*}={\mathrm{E}}_{t-1}\left[s_t\right]$				$s_t^{*}=s_{t-1}$				$s_t^{*}=f_t$
	Baseline λ=1	λ=0.5	λ=0.1	Sticky Prices	Baseline λ=1	λ=0.5	λ=0.1	Sticky prices	Baseline λ=1	λ=0.5	λ=0.1	Sticky Prices
Pound sterling	3.80	4.00	3.48	3.60	0.88	0.71	-0.14	0.84	-4.35 B	-4.51 B	-4.11 B	-4.23 B
French franc	2.90	2.96	2.78	2.51	1.81	2.14	2.52	2.04	-3.33 B	-3.65 B	-4.60 B	-2.97 B
Deutsche mark	3.74	3.76	3.43	3.59	1.28	1.11	0.16	1.36	-4.11 B	-4.07 B	-2.44 B	-3.68 B
Italian lira	2.91	3.17	3.17	2.37	1.33	1.72	2.46	1.73	-3.60 B	-3.93 B	-4.74 B	-3.07 B
Swiss franc	4.13	4.14	3.02	3.13	1.63	2.23	2.41	1.49	-4.82 B	-4.13 B	-2.21 B	-4.32 B
Canadian dollar	4.37	3.86	2.46	4.32	1.44	1.17	0.33	1.21	-4.81 B	-4.55 B	-4.43 B	-4.90 B
Japanese yen	4.54	4.64	3.02	2.98	0.18	0.44	1.04	0.47	-4.57 B	-4.75 B	-4.49 B	-3.77 B

Notes: The reported values are the standard normal statistics for the null hp. that q_t=0, where q_t is defined by equation (14) in the text. Newey-West standard errors, with a lag truncation of 6-(n/255)^1/4, where n is the number of observations, were used. A superscript A indicates that inequality (16A) is violated, while a superscript B indicates that inequality (16B) is violated. Monthly data for broad money and industrial production are used in these tests. The baseline model uses α=1, β=1, λ=1, θ=1; the sticky price model uses θ=.3. The sample goes from April 1973 to September 1994. The number of observations is 255.

Table 3.

Time-Varying Risk Premia and Various Subsamples

Notes: The reported values are the standard normal statistics for the null hp. that q_t=0, where q_t is defined by equation (14) in the text. Newey-West standard errors, with a lag truncation of 6-(n/255)^1/4, where n is the number of observations, were used. A superscript A indicates that inequality (16A) is violated, while a superscript B indicates that inequality (16B) is violated. Monthly data for broad money and industrial production are used in these tests. The coefficient ρ is the smoothing coefficient for the Hodrick-Prescott filter. The sample goes from April 1973 to September 1994 unless mentioned. The number of observations in the four samples is 255, 255, 128, and 126, respectively.

Table 3.

Time-Varying Risk Premia and Various Subsamples

	$s_t^{*}={\mathrm{E}}_{t-1}\left[s_t\right]$				$s_t^{*}=s_{t-1}$				$s_t^{*}=f_t$
	Variable Risk Premia		1973.04 to 1983.12	1984.01 to 1994.09	Variable Risk Premia		1973.04 to 1983.12	1984.01 to 1994.09	Variable Risk Premia		1973.04 to 1983.12	1984.01 to 1994.09
	ρ=14,400	ρ=1,600			ρ=14,400	ρ=1,600			ρ=14,400	ρ=1,600
Pound sterling	2.18	1.53	2.22	2.83	1.07	0.36	1.17	0.88	-3.53 B	-2.63 B	-2.62 B	-3.38 B
French franc	2.58	1.53	2.35	2.18	1.29	0.41	0.98	1.33	-3.13 B	-1.66	-3.36 B	-2.59 B
Deutsche mark	2.51	1.20	2.73	3.05	0.91	-0.13	0.78	1.31	-3.37 B	-2.31	-3.53 B	-3.41 B
Italian lira	2.18	1.11	2.28	2.46	1.07	0.36	0.29	1.32	-2.87 B	-1.48	-3.20 B	-2.98 B
Swiss franc	2.51	1.20	3.12	2.45	1.09	0.13	1.07	1.32	-2.88 B	-1.96	-3.55 B	-2.93 B
Canadian dollar	0.61	-0.75	3.03	3.85	-1.26	-1.94	0.11	1.58	-3.20 B	-2.65 B	-3.68 B	-4.09 B
Japanese yen	2.17	0.79	2.09	2.89	0.71	-0.40	0.44	0.76	-2.35 B	-1.44	-2.68 B	-3.39 B

Notes: The reported values are the standard normal statistics for the null hp. that q_t=0, where q_t is defined by equation (14) in the text. Newey-West standard errors, with a lag truncation of 6-(n/255)^1/4, where n is the number of observations, were used. A superscript A indicates that inequality (16A) is violated, while a superscript B indicates that inequality (16B) is violated. Monthly data for broad money and industrial production are used in these tests. The coefficient ρ is the smoothing coefficient for the Hodrick-Prescott filter. The sample goes from April 1973 to September 1994 unless mentioned. The number of observations in the four samples is 255, 255, 128, and 126, respectively.

View Table

Table 3.

Time-Varying Risk Premia and Various Subsamples

	$s_t^{*}={\mathrm{E}}_{t-1}\left[s_t\right]$				$s_t^{*}=s_{t-1}$				$s_t^{*}=f_t$
	Variable Risk Premia		1973.04 to 1983.12	1984.01 to 1994.09	Variable Risk Premia		1973.04 to 1983.12	1984.01 to 1994.09	Variable Risk Premia		1973.04 to 1983.12	1984.01 to 1994.09
	ρ=14,400	ρ=1,600			ρ=14,400	ρ=1,600			ρ=14,400	ρ=1,600
Pound sterling	2.18	1.53	2.22	2.83	1.07	0.36	1.17	0.88	-3.53 B	-2.63 B	-2.62 B	-3.38 B
French franc	2.58	1.53	2.35	2.18	1.29	0.41	0.98	1.33	-3.13 B	-1.66	-3.36 B	-2.59 B
Deutsche mark	2.51	1.20	2.73	3.05	0.91	-0.13	0.78	1.31	-3.37 B	-2.31	-3.53 B	-3.41 B
Italian lira	2.18	1.11	2.28	2.46	1.07	0.36	0.29	1.32	-2.87 B	-1.48	-3.20 B	-2.98 B
Swiss franc	2.51	1.20	3.12	2.45	1.09	0.13	1.07	1.32	-2.88 B	-1.96	-3.55 B	-2.93 B
Canadian dollar	0.61	-0.75	3.03	3.85	-1.26	-1.94	0.11	1.58	-3.20 B	-2.65 B	-3.68 B	-4.09 B
Japanese yen	2.17	0.79	2.09	2.89	0.71	-0.40	0.44	0.76	-2.35 B	-1.44	-2.68 B	-3.39 B

Notes: The reported values are the standard normal statistics for the null hp. that q_t=0, where q_t is defined by equation (14) in the text. Newey-West standard errors, with a lag truncation of 6-(n/255)^1/4, where n is the number of observations, were used. A superscript A indicates that inequality (16A) is violated, while a superscript B indicates that inequality (16B) is violated. Monthly data for broad money and industrial production are used in these tests. The coefficient ρ is the smoothing coefficient for the Hodrick-Prescott filter. The sample goes from April 1973 to September 1994 unless mentioned. The number of observations in the four samples is 255, 255, 128, and 126, respectively.