Are exchange rate fluctuations justified by changes in their “fundamental” determinants? In an efficient market with rational investors, exchange rates are forward-looking prices that reflect anticipated changes of the relative demand and supply of two moneys. Hence, their volatility should reflect investors’ expectations of changes in the determinants of money stocks, such as incomes and interest rates. Given a model of exchange rate determination, market efficiency places restrictions on the relative volatility of exchange rates and of their determinants, which can be tested to yield insight on the validity of the underlying model. We apply a family of such tests in this paper, to assess whether some popular exchange rate models are capable of matching observed patterns of exchange rate volatilities over the post-Bretton Woods period.
Our interest in exchange rate volatility is motivated by both analytical and policy concerns. Exhibit A in policy discussions of exchange rate volatility is often represented by a chart such as Figure 1, pointing to the dramatic increase in exchange rate volatility for major currencies since the breakdown of the Bretton Woods system. This evidence is often accompanied by an expression of concern that “private markets may not always anchor their behavior to economic fundamentals, thus making their responses susceptible to contagion and bandwagon effects that may be disruptive and detrimental to economic performance” (Mussa and others (1994), p. 18). Excessive exchange rate volatility is also often advocated as grounds for throwing sand in the wheels of currency markets (see, for instance, Eichengreen, Tobin, and Wyplosz (1995)).
Before policies to inhibit market response can be advocated, however, one ought to verify that the volatility of exchange rates does not simply reflect that of their underlying determinants. Also, as these tests are typically joint tests of market efficiency and of a specific exchange rate model, the results must be conditioned on the particular model that they are based on.
Given the prominence in research on exchange rates of the monetary model and of its variants, early studies of exchange rate volatility, including Huang (1981), Vander Kraats and Booth (1983), and Wadhwani (1987), followed Shiller’s (1981) work on stock price volatility to construct “variance bounds” tests of the monetary model of the exchange rate. Invariably, these studies found the volatility of exchange rates since the breakdown of the Bretton Woods system to exceed the model’s predictions, leading their authors and several commentators (see, for instance, Levich (1985)), to assert either the inefficiency of foreign exchange markets or the invalidity of the underlying model, or both.
Later studies, however, questioned those results. Diba (1987), for instance, showed that the tests of Huang (1981) and Vander Kraats and Booth (1983) were vitiated by an erroneous transformation of annual semi-elasticities of money demand to interest rates into their high-frequency counterparts. When correctly calibrated, those tests failed to reject the monetary model’s predictions, although that failure was likely to reflect the weak restrictions placed by those tests on the model. Early volatility tests of asset price models were also shown to suffer from serious statistical biases, leading them to reject the underlying model too often in finite samples (see, for instance, Kleidon (1986) and Marsh and Merton (1986)). To address some of these concerns, Gros (1989) used improved volatility inequalities to present new evidence of excessive exchange rate volatility. His analysis, however, was not based on a formal statistical test, was subject to the same calibration problems pointed out by Diba (1987), and involved measuring exchange rate volatility in a way that was likely to overstate his finding of excessive volatility. In summary, after 15 years of research, evidence on the ability of the most popular exchange rate models to match the observed variability of exchange rates rests largely on mis-specified and weak statistical tests. Not surprisingly, several authors have expressed a perception of futility when reviewing the inconclusive state of research on exchange rate volatility (see, for instance, Frankel and Meese (1987), pp. 134–6).
Figure 1.Germany, Japan, and the United States: Volatility of Nominal Exchange Rates
Dotted line indicates approximate date of breakdown of the Bretton Woods System. Source: IMF, International Financial Statistics,
In this paper we use recent volatility-based tests of asset price models to provide new, clearer evidence on the ability of some popular exchange rate models to match observed patterns of exchange rate volatilities over the post-Bretton Woods period. We apply the methodology developed by Mankiw, Romer, and Shapiro (1991) in the context of stock price models, and use data for the eight major currencies from 1973 to 1994 to test the textbook flex-price version of the monetary model as well as more general models with sticky prices, sluggish adjustment of money stocks, and time-varying risk premiums.
Our tests show that these models are broadly rejected by the data, but that this rejection can—in general—be attributed not to excessive exchange rate volatility, but rather to the inconsistency of these models with the assumed efficiency of currency markets. Hence, our tests confirm the wisdom from standard regression-based tests of these models (see Hodrick (1987) for a survey). However, since our tests possess better statistical properties than previous regression-based tests (in particular, they are unbiased in small samples), they provide even stronger evidence against the joint hypotheses of the monetary model (or of its extensions) and of market efficiency. Furthermore, because they are explicitly constructed as volatility tests, our tests help clarify the role played by exchange rate volatility in the models’ rejection.
Specifically, our tests point to the importance of choosing an economically meaningful definition of exchange rate volatility as a basis for the tests, so as to avoid disguising a model’s rejection as evidence of excessive volatility. The tests show that when exchange rate volatility is defined—traditionally—as the average of conditional or unanticipated exchange rate changes, then there is no evidence that exchange rates may have been excessively volatile with respect to the predictions of even the most restrictive version of the monetary model. Evidence of excessive exchange rate volatility, however, may emerge on the basis of alternative definitions of volatility.
Our findings, we hope, will contribute to future research on exchange rate volatility being more clearly defined in its scope and perhaps encourage greater caution in making claims of “excessive” exchange rate volatility. Our imposition on the data of a monetary model straitjacket should strengthen our conclusions: if exchange rates do not appear to be excessively volatile even with respect to a framework predicting their close movement with money and income alone, the likelihood that evidence of excess volatility would be uncovered based on more flexible models appears even more remote.
I. Volatility Tests of the Monetary Model
It is easiest to introduce our volatility tests based on a standard specification of the monetary model with flexible prices and instantaneous adjustment of money stocks. We outline the general testing methodology in this section and apply it to more general models in the next section.
The Monetary Model of the Exchange Rate
The standard specification of the monetary model includes 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 mt, pt, and yt denote (log) domestic money supply, prices, and real income, respectively, and it 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 st while Et[.] denotes the rational expectation operator conditional on information available at time t, The exogenous terms yt and mt may incorporate a stochastic component; alternatively, a stochastic (“velocity”) term can be added to introduce uncertainty in the right-hand side of equation (1). For ease of exposition, we shall adhere to the convention that mt already incorporates an exogenous stochastic shock.
It is standard in the literature to assume the absence of exchange rate bubbles, that is, that
The tests considered in this paper, however, are robust to the presence of bubbles, and can be performed directly on equation (7): if st+h incorporates a bubble term (i.e., a capital gain that reflects solely the anticipation of a future currency transaction), so does st and equation (7) remains valid. In most of our tests, the investment’s holding period, h, is set at 3, as forward markets tend to be most liquid for 3-month maturities, and the power of our tests falls when h increases much above 3. Tests for different values of h are discussed below.
Volatility Tests of the Monetary Model
Let us now define the perfect-foresight (or “fundamental”) exchange rate,
so that, by definition,
We must now choose a “benchmark” exchange rate, denoted by
There are two main requirements for the choice of
Alternatively, exchange rate volatility could be measured by setting
Yet another possibility would be to choose 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 (and, in this case, under the no-bubble condition that
With rational expectations, the forecast error
Therefore, squaring both sides of the identity
taking expectations, and using equation (11), yields
Therefore, the monetary model implies the testable restriction qt = 0, and hence the restriction E[qt] = 0: the sample mean of the qts, , should be close to zero if the model correctly describes the dynamics of exchange rates. As noted above, this test is robust to the presence of exchange rate bubbles: if the market exchange rate st incorporates a bubble term, so do
Thus, we can construct a test of the monetary model as follows. First, we can compute the fundamental exchange rate,
whose variance is estimated by
Equation (13) also implies that
where the expectations are now taken unconditionally. Should either one of these inequalities be violated, then Et[qt] < 0 (although the opposite is not necessarily true), thus suggesting the use of inequalities (16A) and (16B) as diagnostics for the model. Inequality (16A), in particular, states that the market exchange rate, st, should forecast the behavior of the fundamental rate,
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 – st and st –
regression coefficient is zero. A nonzero
In comparison with (17), the volatility tests implemented in this paper reject the model when the mean of the qts is zero. Now, since can be written as
a link between the regression-based and the volatility-based methods is apparent. Both methods involve testing whether
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 Monte Carlo simulations showing that the finite-sample distribution of a sample-mean 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
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 . The usefulness of the rejection of the model, in particular, depends on being a useful benchmark for the measurement of exchange rate volatility. Establishing the best benchmark is a task 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 depends on agents’ attitudes toward risk, the nature of adjustment costs, and other such factors. 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. For similar reasons, few scientists seem to find useful a description of stars’ and planets’ positions in space with respect to the moon, nor do they find useful the (formally correct) statement that the sun is more volatile than is the earth with respect to 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—we follow an eclectic approach in our statistical analysis. We present results for a variety of choices of , 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 in which the benchmark rate is set at the value that the exchange rate would take if investors believed that fundamentals behave as random walks. The tests can be applied in a straightforward way to other definitions of .
Though our tests’ dependence on the benchmark 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 rates’ volatility—just as there is no unique way to measure planets’ 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.
II. More General Specifications of the Model
The test procedure described in the previous section can be applied to a variety of exchange rate models. We discuss here some extensions of the monetary model that allow for slow adjustment of prices and money holdings, and for violations of the interest parity condition. The point here is not to insist on the empirical accuracy of these specifications, although we draw from the empirical literature in calibrating our models. Rather, the aim is to illustrate the flexibility of the testing procedure, and to provide preliminary evidence of the robustness of our results. Our qualitative findings turn out to be rather robust, suggesting that greater effort to incorporate empirically accurate models, while welcome, is unlikely to overturn our main results.
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
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 is expected to converge to PPP in a single period.
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.
Sluggish Money Adjustment
We can also relax the assumption that real money balances adjust instantaneously to their long-run equilibrium value. Partial adjustment equations of the form
have been widely used in the literature on money demand functions. The terms
Thus, the coefficients β and α in equations (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.
Except for this redefinition, the test procedure remains the same as described above.
Time-Varying Risk Premiums
Finally, we can allow for a time-varying residual, xt in the interest parity equation (4):
For our purposes, an ex post estimate of the residual xt can be obtained by de-trending
One possible interpretation of xt 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 model3 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.
III. Empirical Results
Data and Calibration of the Tests
The full sample covers data for Canada, France, Germany, Italy, Japan, Switzerland, the United Kingdom, and the United States between January 1973 and 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 IMF’s International Financial Statistics, except for a few series that were incomplete in the International Financial Statistics, which were obtained from the IMF’s Current Economic Indicators database.
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 of 1 to 4 (see, for instance, 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 α (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, independent 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 of 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 = 0.06 and λ = 0.01). 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 = 0.1 and θ = 0.5 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 the theoretical concepts of “output,” “money,” and “prices,” suggesting allowance for a degree of freedom in scaling raw data. After 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 (that is, so that
Table 1 reports results of tests of the textbook flex-price instantaneous-money-adjustment version of the monetary model, for different values of the elasticities β and α. Several important features are apparent from these results.
First, the model is strongly rejected in almost all cases, with test statistics often exceeding their 99 percent critical values of
|α =0.1||α = 6||α = 0.1||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|
|Canadian dollar||4.88||4.95||2.41||2.75||0.94||1.17||1.76||1.83||— 4.92AB||–5.01AB||–3.00B||–3.65B|
λ = 1
|λ = 0.5||λ =0.1||Sticky|
λ = 1
|λ = 0.5||λ = 0.1||Sticky|
λ = 1
|λ = 0.5||λ =0.1||Sticky|
|Variable risk premiums||1973||1984||Variable risk premiums||1973||1984||Variable risk premiums||1973||1984|
|ρ = 14,400||ρ = 1,600||1983||1994||ρ = 14,400||ρ = 1,600||1983||1994||ρ = 14,400||ρ = 1,600||1983||1994|
|Canadian dollar||0.61||–0.75||3.03||3.85||–1.26||– 1.94||0.11||1.58||–3.20B||–2.65B||–3.68B||–4.09B|
|Japanese yen||2.17||0.79||2.09||2.89||0.71||–0.40||0.44||0.76||–2.35B||– 1.44||–2.68B||–3.39B|
There is also no evidence of excessive exchange rate volatility when the lagged exchange rate is used as a benchmark
Tables 2 and 3 confirm the previous discussion for alternative specifications of the model. Table 2 presents results for the baseline set of parameters (α = β = λ = θ = 1), for the model with partial adjustment of money stocks (λ = 0.5 and λ = 0.1), and for the sticky price model. Predictably, the models with sluggish money and price adjustment can be rejected with somewhat less confidence than the baseline model.5 More careful modeling of the money and price processes is likely to weaken the evidence against the model even further. For the purpose of the present study, however, it suffices to note that these extensions tend to make the model’s rejection marginally less significant, while not affecting the volatility inequalities: there is no evidence of excess volatility when exchange rates are measured with respect to their lagged values or with respect to the trends anticipated by the model; there is evidence of excess volatility, however, when exchange rates are measured with respect to a random walk benchmark.
Table 3 presents test results for the model with time-varying risk premiums and for the baseline model over two half-samples. Allowing for violations of the interest parity condition sharply weakens the overall evidence against the model, particularly when the Hodrick-Prescott parameter p is set to such a low value (ρ = 1,600) that much of the interest parity residuals are incorporated into fundamentals. While these tests are clearly skewed in favor of the model (little structure is imposed on the premiums xt, except that they should be smooth over time), our previous discussion of the volatility inequalities (16A–B) requires little change. Indeed, these results suggest that more accurate modeling of the interest parity condition is likely to weaken the evidence of excess volatility even based on a random walk benchmark.
Finally, overall evidence against the model remains strong when the tests are performed over split samples, and the qualitative aspects of the excess volatility tests are also unchanged.
A variety of alternative specifications of the tests, for which results are available upon request, gave very similar results. We relaxed the assumption of long-run PPP by allowing for a time-varying trend in the real exchange rate (estimated, as in the case of a time-varying risk premium, by passing
IV. Concluding Remarks
The volatile behavior of exchange rates since the breakdown of the Bretton Woods system has led scholars and policymakers to question the consistency of exchange rate fluctuations with movements in their underlying determinants, with complaints of unjustified (or irrational) market turbulence rising loud during episodes of exchange rate instability. Several previous studies have presented evidence that exchange rates since 1973 have been excessively volatile with respect to the predictions of a variety of popular exchange rate models. However, for reasons ranging from calibration errors, small-sample biases, and lack of a proper testing procedure, these studies failed to present convincing evidence of the inability of these models to match the observed pattern of exchange rate volatilities over the post-Bretton Woods period.
We have tried in this paper to provide more solid evidence on the ability of a popular family of exchange rate models to match observed patterns of exchange rate changes, by applying recent tests of asset price volatility. Two main points have emerged from the analysis.
First, consistent with previous regression-based tests of the monetary model, our volatility-based tests strongly reject the monetary model in its textbook flex-price format and in more general versions that allow for sluggish price and money adjustments: excess returns in currency markets appear to be too forecastable for these markets to be efficient and exchange rates to be governed by the rules of the monetary model. As the volatility-based tests that we employ have been shown to possess better statistical properties, than previous regression-based tests, our evidence against the monetary model should be viewed as stronger than that previously available in the literature. Nevertheless, rejection of the monetary model is hardly a novelty, and this negative outcome would be, on its own account, of secondary interest.
Second, and more interesting, our tests highlight the ambiguity of the concept of exchange rate volatility, and the implications of this ambiguity for claims of “excess” exchange rate volatility. Measuring exchange rate deviations with respect to different benchmarks leads to different conclusions on whether major currencies’ exchange rates have been excessively volatile over the post-Bretton Woods period. While we have made no attempt to resolve this ambiguity, we have showed that based on certain (odd, in our view) definitions of exchange rate volatility, claims that major exchange rates may have been too volatile over the post-Bretton Woods period may be formally justified. However, based on definitions of exchange rate volatility common in the exchange rate literature (which focus on the volatility of exchange rate surprises, and on the conditional volatility of exchange rates), major exchange rates over the post-Bretton Woods period do not appear to violate the predictions of even the most restrictive version of the monetary model: their volatility is consistent with the anticipated volatility of their determinants.
Our results should lead to greater skepticism of future claims of “excessive” exchange rate volatility, and our focus on a model as restrictive as the monetary model should reinforce this skepticism: models that do not tie down exchange rate movements to those of such a narrow set of macroeconomic variables (for instance, portfolio balance models with imperfect substitutability, asset-price models with variable rates of discount, etc.) should provide even less evidence of excess volatility. (Formally, for a perfectly-fitting model,
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Research for this paper was conducted when Leonardo Bartolini was an Economist in, and Gordon Bodnar was visiting, the Research Department of the IMF. Leonardo Bartolini is now in the Research Department of the Federal Reserve Bank of New York. Gordon Bodnar is at the Wharton School of the University of Pennsylvania. Both authors hold a Ph.D. from Princeton University. They thank Eswar Prasad for providing them a Gauss code of the Hodrick-Prescott filter, Peter Isard, Peter Clark, Carol Osier, and participants in seminars at the IMF and at the Federal Reserve Bank of New York for comments, and Susanna Mursula for technical assistance.
Serial correlation is often a problem when using overlapping observations. In our case, in particular, observations overlap over the holding period h,
We limited our exploration of money demand equations to specifications with instantaneous or partial adjustment of money stocks. Recent literature has considered buffer-stock and error-correction models that imply even greater departure from the standard specification of the monetary model (see Boughton (1992) for a survey). These specifications are difficult to integrate in equilibrium exchange rate models because of their data-dependent parameterization and their tendency to involve lags of interest rates at different maturities and to predict long-run price non-homogeneity.
In particular, since we use all the model’s equations to obtain an ex post estimate of xt, the model would be identically satisfied (i.e.,
A more complete set of tests is available from the authors, together with the data and a copy of the GAUSS program used for the tests.
Given the existing econometric evidence on money demand equations, it is also not surprising that the data favor an extremely low value of λ. See the discussion of the calibration of λ above.
Exceptions included moving to quarterly data, which caused most tests to become insignificant, likely as a result of a drastic fall in the degrees of freedom; and increasing the holding period beyond six months and the lag