Alternative Hypotheses About the Excess Return on Dollar Assets, 1980–84

Eduardo Borensztein

doi:10.5089/9781451956740.024.A002

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Alternative Hypotheses About the Excess Return on Dollar Assets, 1980–84

Author: Mr. Eduardo Borensztein

Publication Date:: 01 Jan 1987

eISBN:: 9781451956740

ISBN:: 9781451956740

Language:: English

Keywords:: excess return; U.S. dollar; dollar assets; return differential; pound sterling; Real exchange rates; Currencies; Exchange rates; Asset prices; Asset bubbles; probability distribution

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During 1980-84, the observed returns on financial assets denominated in U.S. dollars were persistently much higher than those on similar assets denominated in the currencies of other major industrial countries. This persistence of excess returns, in the order of 12-18 percentage points annually, is puzzling from the viewpoint of the theory of efficient financial markets because it suggests that some portfolio (one that would hold dollar-denominated securities and go short on securities denominated in European currencies) could earn extraordinary profits. This paper investigates the empirical plausibility of, and finds some support for, some nonstandard hypotheses about the probability distribution of the exchange rate that can provide an explanation of the observations that is consistent with the existence of efficient financial markets.

Abstract

The increase in the return differential in favor of dollar assets can be appreciated from the data presented in Table 1. The top panel displays average values of the dollar excess return over three periods since the advent of generalized floating. The middle panel shows the frequency of observations of positive values of the excess return over the same periods. The excess return was computed as the difference between the rate of interest on 90-day Eurodollar deposits and the ex post return, in dollars, on similar deposits of other currency denomination. Monthly observations were used. It is evident that, before the most recent period, the excess returns on dollar-denominated assets tended to take positive and negative values with similar frequency and were not—on average—very different from zero. But from October 1980 through December 1984, the excess returns were persistently positive and achieved average values of 10-18 percentage points on an annual basis.¹

Table 1.

Performance of Return on U.S. Dollar Assets, 1974–84

(In percent)

^{^a}Excess of the interest rate on U.S. dollar-denominated deposits over the sum of the interest rate on other currency-denominated deposits and the rate of depreciation of the dollar with respect to that currency. Monthly observations; dates refer to beginning of period.

Table 1.

Performance of Return on U.S. Dollar Assets, 1974–84

(In percent)

	January 1974-	October 1977-	October 1980-
Currency	September 1980	September 1980	December 1984
Average ex post excess return on 90-day Eurodollar deposits^a
Over deutsche mark	-3.31	0.73	16.44
Over French franc	-5.03	-2.89	15.19
Over pound sterling	0.64	-10.47	18.18
Over Japanese yen	3.16	0.03	10.12
Over Swiss franc	-7.56	1.56	17.29
Frequency of positive U.S. dollar excess returns
Over deutsche mark	43.4	50.0	76.5
Over French franc	38.2	38.9	74.5
Over pound sterling	35.5	22.2	84.3
Over Japanese yen	53.9	50.0	72.5
Over Swiss franc	46.1	61.1	86.3
Average nominal interest rate differential on 90-day Eurodollar deposits
Over deutsche mark	1.72	4.76	4.64
Over French franc	-4.39	-0.64	-3.26
Over pound sterling	-5.78	-1.83	0.97
Over Japanese yen	7.56	5.32	5.59
Over Swiss franc	2.69	7.89	6.97

Table 1.

Performance of Return on U.S. Dollar Assets, 1974–84

(In percent)

	January 1974-	October 1977-	October 1980-
Currency	September 1980	September 1980	December 1984
Average ex post excess return on 90-day Eurodollar deposits^a
Over deutsche mark	-3.31	0.73	16.44
Over French franc	-5.03	-2.89	15.19
Over pound sterling	0.64	-10.47	18.18
Over Japanese yen	3.16	0.03	10.12
Over Swiss franc	-7.56	1.56	17.29
Frequency of positive U.S. dollar excess returns
Over deutsche mark	43.4	50.0	76.5
Over French franc	38.2	38.9	74.5
Over pound sterling	35.5	22.2	84.3
Over Japanese yen	53.9	50.0	72.5
Over Swiss franc	46.1	61.1	86.3
Average nominal interest rate differential on 90-day Eurodollar deposits
Over deutsche mark	1.72	4.76	4.64
Over French franc	-4.39	-0.64	-3.26
Over pound sterling	-5.78	-1.83	0.97
Over Japanese yen	7.56	5.32	5.59
Over Swiss franc	2.69	7.89	6.97

In contrast, the differentials in nominal interest rates do not exhibit a similar pattern. As can be seen from the bottom panel of Table 1, the interest rate differential in favor of dollar assets showed a very modest tendency to increase relative to the pound sterling and the Japanese yen, but not relative to the rest of the currencies. Even in those cases in which there has been an increase in the interest rate differential, the magnitude of the increase is much smaller than the change observed in ex post returns.² Therefore, the persistent appreciation of the U.S. dollar exchange rate was primarily responsible for the change in the pattern of the ex post return differential. During the last of the three periods considered, the dollar rose, in effective terms, at an average rate of some 12 percent annually. The U.S. dollar exchange rate is also the major driving force of the short-run variability of the excess return. A (sample-moments) variance decomposition of the ex post excess returns shows that over 90 percent of the variance of the excess return is explained by the variance of the exchange rate.

To summarize the stylized facts, the data in Table 1 suggest two observations. First, there has been a substantial increase in the mean value of the ex post difference in returns in favor of U.S. dollar-denominated assets, a process that started in the last months of 1980. Second, the increase in the excess return has been the result of the combination of relatively constant nominal interest rate differentials with a persistent appreciation of the dollar.

To find a theoretical justification for these observations, there are two main lines of reasoning that can be followed. The first is to assume that the ex post excess returns serve as unbiased estimates of the expected returns, an assumption that leads to the conclusion that there has been a considerable increase in the risk premium associated with dollar assets.³ The second is to consider that there has been a sustained divergence between the expected and the realized values of the returns, a fact that would likely arise when the probability distribution of the exchange rate is asymmetric.

The plausibility of the first explanation—an increase in the risk premium as the main determinant—is questionable. Although the increase in the supply of dollar-denominated assets—which came mostly as a consequence of the large U.S. federal deficits over the last period—could provide some rationalization for a higher risk premium on those assets, an increase of 10-18 percentage points in the risk premium seems to be excessively high. For example, the excess return of the stock market over treasury bills in the United States was, on average, about 6 percent for the period 1889-1978, as computed by Grossman and Shiller (1981; data presented by Mehra and Prescott (1985)). In addition, the increase in the risk premium that results from a change in the relative asset supplies can theoretically be calculated on the basis of the capital asset price model. Frankel (1985), using sample moments to estimate the variance-covariance matrix and assuming a value of 2 for the coefficient of risk aversion, calculated that an increase of 1 percentage point in the share of U.S. government bonds in the portfolio of world asset holders implies an increase of only 2 annual basis points in the yield differential favoring dollar assets. Then, given an estimated increase in the share of U.S. assets of between 10 and 15 percentage points since 1980, this factor could account only for less than 1 annual percentage point of excess return, less than a tenth of the observed increase in return differentials. Although these computations probably underestimate the required changes in risk premiums, they still fall considerably short of explaining increments of 10-18 percent.⁴

Therefore, this paper investigates the second explanation—that is, the hypothesis that there has been a systematic difference between the observed return differentials and their expected values. This situation would arise when the probability distribution of the exchange rate is asymmetric and includes a small probability of a very large change in the value of the exchange rate. This change would be associated with an event such as a shift in policy regimes or a sudden change in the expectations of market participants. If such an event has a low probability of occurrence, it is likely that the change will not be observed during long sample periods; if the implied change in the exchange rate is large enough, however, the potential change will still have a considerable effect on the observed excess return.

There is also some casual evidence supporting the idea that the exchange rate was following a process of that type, particularly near the end of the sample period. For example, there seemed to be a widespread belief among economists that a large drop in the value of the dollar was eventually inevitable, either because of the action of market forces or because of political decisions. This is evident, for example, from the Symposium on the Exchange Rate held at the Brookings Institution (see the proceedings in the volume containing Frankel (1985)). Furthermore, in a communique issued after a meeting in September 1985, the Group of Five major industrial countries expressed the view that the U.S. dollar was above the value that corresponded to economic fundamentals and started to coordinate some policy actions intended to bring its value down. Anticipations of actions of this type, even if the actions fail to take place for some time, affect the return on financial assets and could well have generated the high ex post excess return on dollar assets that was observed during the period under study.

There are two families of models that can explain a situation such as the one described: a “peso-problem” model and a speculative bubble model. A peso problem arises when a potential change in the current regime or policy stance has a significant influence over the expected value of the exchange rate, but when this change does not take place over some relevant length of time.⁵ A speculative bubble occurs when the price of an asset follows a rational expectations path but diverges from the fundamental value of the asset; that is, from the intrinsic value of the asset according to economic considerations. Although both of these formulations would tend to influence the foreign exchange market in a similar way, their underlying driving forces are different. In the peso-problem case it is the anticipation of policy actions that drives the market, whereas in the case of a speculative bubble it is purely the mood or the whims of economic agents that motivate the market⁶.

Several authors have considered these types of alternative hypotheses as plausible explanations of the behavior of the dollar exchange rate, although without attempting to test them empirically. Dornbusch (1982), for example, considers the applicability of these families of models and explains what he calls the “dollar problem.” He sees the tightening of monetary policy in the United States since 1979 as a random event, in the sense that its continuation was regarded by the public as less than certain. In addition, there are several empirical studies—Cumby and Obstfeld (1984), Hodrick and Srivastava (1984), and Fama (1984), for example—that mention the peso problem as a possible reason for the failure of tests of asset price equations, but that do not pursue the idea further.

The plan of the paper is the following. Section I investigates the plausibility of the peso-problem explanation for the high ex post return on dollar assets. The strategy is to take standard models of asset pricing and to test whether the restrictions imposed by such models can be rejected in a way that conforms to the predictions of a peso-problem model. The evidence shows that variables that are indicators of expectations of potential policy actions are correlated with the observed values of the excess returns, which is consistent with the peso-problem explanation (although not exclusively so). Section II discusses a speculative bubble model that is consistent with the behavior displayed by the variables during this period and presents some statistical tests of that specification. Again, the evidence is in general favorable, although some of the parameters are not significant; however, as with the peso-problem model, the tests performed do not rule out other interpretations of the data. Section III draws some conclusions.

I. A “Peso-Problem” Model

This model describes a situation in which the returns on financial assets are heavily influenced by a potential change in the policy stance that would bring about a substantial depreciation of the exchange rate. Although the probability of such a policy shift occurring in any given month is low, the depreciation of the exchange rate that this policy shift would generate is considerable. The observed high excess return on dollar assets is then a consequence of the lack of occurrence of this policy change: if the change were to occur, the dollar would depreciate, and the excess return would be highly negative; but as long as the change does not occur, the excess return is persistently positive. From the point of view of the effects on the excess return, the exact nature of the policy change involved is unimportant. The only requirements are that it implies a substantial depreciation of the dollar and that the probability of its occurrence is low, as perceived by the market participants.

The econometric implications of this type of situation for the excess returns will first be obtained for a simple case, which is when domestic and foreign assets are perfect substitutes. The same approach will then be followed to perform the empirical tests on the basis of more sophisticated formulations of asset pricing models. The econometric implications of the model are obtained under the assumption that, during the sample, the policy change did not take place. This is somewhat different from the original approach followed by Krasker (1980), who focused in the speed of convergence of the estimators when the probability distribution of the residuals is asymmetric. That approach seems to be more appropriate for a situation in which the economy repeatedly alternates between the two different regimes, which is not the situation suggested by the data in this case. In addition, the econometric difficulties caused by this situation are not merely a small-sample problem: by the very nature of the problem, the exchange rate will not in general have a stationary probability distribution. For example, if the changes in policy regime are expected to be more than ephemeral, after an actual change in regime takes place the probability of occurrence of the two regimes must change, which means that the probability distribution function of the exchange rate process must also change.⁷

The simplest version of this family of models can be developed under the assumptions that there are two countries and two assets (one denominated in each currency), that individuals are neutral to risk, and that the prices of goods are nonstochastic. In such a world, under the usual assumptions about portfolio selection, all assets are perfect substitutes, and the financial market equilibrium condition is that the expected nominal return on all assets be the same. Denoting by i the (continuously compounded) interest rate on the home currency assets, by i* the same variable in reference to the foreign country, and by x the logarithm of the exchange rate (units of domestic currency per one unit of foreign currency), one may define the excess return y as

\begin{matrix} y_{t + 1} = i_{t} - i_{t}^{*} - x_{t + 1} + x_{t} . & (1) \end{matrix}

Under rational expectations, the equalization of expected returns implies that the excess return must be unpredictable on the basis of information known at the time that the asset prices are determined, or

\begin{matrix} E (y_{t + 1} | I_{t}) = 0, & (2) \end{matrix}

where E[()|I_t] refers to the mathematical expectation conditional on the values of all relevant variables dated at time t or earlier.

These constraints on the conditional expectations of returns can be used directly to perform econometric tests of the validity of the constraints. In particular, a testable proposition is that the excess returns should exhibit no correlation with any variable that is part of the information known to the agents at the time of determining the price of the asset. The failure of this property to hold implies that the market is not using all the available information to predict asset returns, and therefore the model specification can be rejected; however, the test does not discriminate which particular assumption fails to hold.

Now suppose that, at any time t, the system can find itself in either of two states, which are represented by the variable ft. In the context of the described situation, Ω indicates the policy regime or stance in relation to the exchange rate. If Ω = 1, the stochastic model that determines the exchange rate is $x_{t} = \bar{e} + u_{t}$ , with E(u_tI I_t-1) = 0, $var (u_{t}) = σ_{u}^{2}$ . If instead Ω = 2, the exchange rate follows a model $x_{t} = \bar{s} + u_{t}$ . For simplicity it is assumed that the random term u_t is the same in both states (so that the states differ only in the expected values of the exchange rate, e and s). This assumption is not crucial for the qualitative results.

The probability of state 1 is P(Ω = 1) = II. The same rational expectations equilibrium condition should still apply; that is, E (y_t+1I I_t) = 0. To satisfy this condition, the excess return must follow

\begin{matrix} y_{t + 1} = [Π \bar{e} + (1 - Π) \bar{s}] - x_{t + 1} . & (3) \end{matrix}

Suppose that during the sample period the system has been permanently in state 1. Then an econometrician testing the properties of the excess return will face the fact that E (y_t+1|I_t, Ω = 1) is equal to (1 - Π)(s—e) instead of zero. This fact will lead the econometrician to reject the model (or the market efficiency assumption), even though that conclusion is obviously unwarranted.

To test the applicability of the peso-problem formulation, the best alternative would be to develop a full general equilibrium model that would specify how the values of Π;, e, and s are determined as a function of the policy regime and other variables. This paper, however, pursues a more modest goal and will conduct a search for evidence that the behavior of the excess return has been roughly consistent with a model of the peso-problem variety. As shown above, the excess return, instead of being unpredictable, would be a function of the differential value of the exchange rate under the two regimes and of the corresponding probabilities. Neither Π nor s is observable, however, and in general neither one will be constant over time. Furthermore, a glance at equation (3) reveals that it is not possible to identify both Π and s. But if there exist some variables Z that are correlated with Π and s., these other variables could be used as instruments or indicators of the unobservable variables. More specifically, the assumption is that

\begin{matrix} {[(1 - Π) (\bar{s} - \bar{e})]}_{t + 1} = Z_{t} β + \in_{t + 1} . & (4) \end{matrix}

The notation on the left-hand side above is meant to represent the probability of a change in regime at time t + 1 times the difference between the expected values of the exchange rate under states 1 and 2 at t +1 (these values are no longer assumed to be constant). From equation (3), equation (4) implies that the following regression could be estimated:

\begin{matrix} y_{t + 1} = Z_{t} β + \in_{t + 1} . & (5) \end{matrix}

The instruments Z must be correlated with the probability—as evaluated ex ante by market participants—of a change in regime. Therefore, variables such as the appreciation of the U.S. dollar real exchange rate, the U.S. trade deficit, or indicators of the fiscal and monetary stances, could play that role. If the vector of coefficients β is nonzero, the evidence would be consistent with the peso-problem formulation. Such a test would represent a rejection of the rational expectations conditions (equation (2)) because Z_t, is obviously part of I_t although the reasons for this rejection could be multiple. According to the peso-problem explanation, such rejection is not due to a failure of the theoretical structure of the model; instead, it is a consequence of a particular random distribution of the exchange rate.

The tests conducted here are based on the two most popular asset pricing models: the consumption-based asset price model (which has been used in the context of international finance, for example, by Hansen and Hodrick (1983)), and the traditional (or static) capital asset pricing model, or CAPM (which has been applied and tested in the case of international finance, for example, by Frankel (1982)). The strategy is the following. The null hypothesis will be that the corresponding equilibrium asset price model holds in “pure” form; that is, with the usual assumption that the exchange rate has a stationary probability distribution. The alternative hypothesis will be that a peso-problem situation exists and that some instruments Z_t, which satisfy the requirements explained above, are available. As above, the test that the coefficients of those variables are nonzero will serve as indirect evidence in favor of the validity of the alternative formulation of the exchange rate process.

The Consumption-Based Model

This model is based on the optimal time path of consumption and portfolio selection that an individual chooses in a dynamic stochastic environment. From the viewpoint of asset prices, the main implication is that the equilibrium returns on each of the different assets should be perfectly correlated with the marginal utility of consumption enjoyed by any consumer-investor in this economy. (See Merton (1973) or Breeden (1979); a derivation is presented in Appendix I.)

A particularly convenient version of this model for empirical purposes is obtained under two assumptions: first, that the utility function belongs in the class of constant relative risk aversion; second, that the logarithm of the marginal utility of consumption, the logarithms of the returns on the different assets, and the logarithms of the other relevant variables follow a joint Gaussian vector autoregression process. Under this specification, one implication of the theory is that the expected value of the difference in the logarithms of any two nominal returns (which has been defined above as the excess return) should equal a constant that is determined by the variance of the returns and the covariances between returns and marginal utility (see Appendix I).⁸ By a standard rational expectations argument, it follows that the sample observations can be represented as

\begin{matrix} y_{t}^{j} = k^{j} + η_{t}, & (6) \end{matrix}

where y^j represents the ex post excess return of dollar assets over the foreign asset j. Furthermore, η has the property that E (η_t|I_t-1) = 0, where I_t-1 indicates a set that contains past values of the relevant variables, and var $var (η_{t}) = σ_{η}^{2}$ .

For the alternative hypothesis, the assumption is that there are two possible states of the economy, related to different policy regimes. During some span of time the economy found itself continuously under one regime, but the probability of a change of regime was not insignificant. There exist some instruments Z_t that are correlated with the probability of a regime change and with the magnitude of the depreciation that would follow. Then, as shown in Appendix I,

E (η_{t + 1} | I_{t}, Ω_{t + 1} = 1) = Z_{t} β .

That October 1980 is tentatively considered the starting date for this situation suggests running the following regression for each return differential favoring dollar-denominated assets:

\begin{matrix} y_{t + 1} = k + D_{t} Z_{t} β + μ_{t + 1}, & (7) \end{matrix}

where D_t= 0 before October 1980 and D_t = 1 since October 1980.

As before, because the Z variables are in the information set they should not help predict the excess return under the null hypothesis. The test that the β coefficients are nonzero, although only formally rejecting the model, also provides evidence in favor of a peso-problem specification.⁹

Empirical Results

The main difficulty in the empirical estimation of equation (7) is the selection of appropriate instruments for the unobservable variables involved in the equation. Potential candidates are variables that somehow measure the degree of “overvaluation” of the U.S. dollar: the higher is the “overvaluation” of the U.S. dollar, the more likely it is that the authorities will shift the policy stance (and, trivially, the larger would be the devaluation that would take place if such a policy change were enacted). The variables that provided the best empirical results were the real exchange rate of the dollar and the U.S. trade deficit. The other variables that were considered as candidates were the interest rate differential and the U.S. budget deficit.

Selection of the nominal interest rates also involves some complication. It would be preferable to use the nominal return on assets that are completely default-free, so that the exchange rate could be considered to be the only source of uncertainty. A standard procedure (for example, in Mishkin (1984)) is to use Eurocurrency deposit rates, under the assumption that the differences in expected rates of return are not affected by the default risk, since that risk is identical for deposits of different currency denominations. This reasoning is not, however, entirely correct. The difference in expected returns is not affected by the default risk only when that risk—in addition to being the same for all assets—is uncorrelated with variations in the exchange rate.¹⁰ That is not likely to be the case if changes in exchange rates have some impact on the financial position of banks operating in the Eurocurrency market. This situation could arise either directly, when the net asset position of banks is not perfectly balanced in every currency, or indirectly through the effect that sharp changes in exchange rates may have over the solvency of the borrowers in this market. It is assumed, however, that the default risk in the Eurocurrency market is of sufficiently small magnitude to be ignored safely. Furthermore, given the predominance of exchange rate changes in the determination of the actual excess returns, it is unlikely that small differences in the measures of interest rates would bring about a substantial difference in the empirical results.

The results of estimating equation (7) by ordinary least squares are presented in Table 2. The sample period runs from June 1974 through February 1985. Five excess returns were examined independently: the return of U.S. dollar deposits versus deutsche mark, French franc, pound sterling, yen, and Swiss franc deposits. The interest rate variable is the 30-day Eurocurrency deposit rate. Lagged values of the real exchange rate of the dollar (based on normalized labor costs) and the ratio of the U.S. trade deficit to gross national product (GNP) were used as indicators of the “overvaluation” of the dollar. Although these two variables had significant coefficients when included alone, there was a considerable loss of significance when both were included simultaneously, probably as a consequence of their high collinearity.

Table 2

Test of the Consumption-Beta Model, One-Month Eurocurrency Returns

Note: The dependent variable is the excess return over each currency, expressed in annual percentage terms. Ordinary least-squares standard errors appear in parentheses. The sample period is June 1974 through February 1985. DW is the Durbin-Watson test statistic; SEE is the standard error of estimate;

^{^**}indicates significance at the 1 percent level;

^{^*}indicates significance at the 5 percent level.

^{^a}Test of the significance of all coefficients. Critical value (5 percent) is 3.0.

^{^b}Test of the null hypothesis of homoskedasticity, distributed as x² with two degrees of freedom.

^{^c}International Monetary Fund, International Financial Statistics (IFS), line 65umc 110, interpolated using nominal exchange rates as benchmarks to obtain monthly data.

Table 2

Test of the Consumption-Beta Model, One-Month Eurocurrency Returns

Currency	Constant	Z	DW	F^a	SEE	Test^b
Z₁= Real effective exchange rate
of U.S. dollar, based on labor costs^c
Deutsche mark	-0.12	1.09^*	2.17	6.04	3.20	0.24
	(0.37)	(0.44)
French franc	-0.28	1.10^*	2.16	6.36	3.14	1.21
	(0.36)	(0.44)
Pound sterling	-0.26	1.13^*	1.84	6.61	3.18	6.28
	(0.37)	(0.44)
Japanese yen	0.18	0.42	1.82	0.86	3.26	1.12
	(0.38)	(0.45)
Swiss franc	-0.26	1.27^*	1.95	5.94	3.76	0.36
	(0.43)	(0.52)
Z₂ = Ratio of U.S. trade deficit
to gross national product (GNP) in corresponding quarter
Deutsche mark	0.08	6.58^*	2.14	5.41	3.16	1.51
	(0.34)	(2.83)
French franc	-0.04	6.15^*	2.11	4.94	3.09	1.79
	(0.33)	(2.77)
Pound sterling	-0.14	8.87 ^**	1.96	11.67	2.90	5.59
	(0.31)	(2.60)
Japanese yen	0.26	2.47	1.81	0.71	3.27	1.92
	(0.35)	(2.93)
Swiss franc	-0.02	0.73^*	1.91	4.85	3.72	1.78
	(0.40)	(3.34)

^{^**}indicates significance at the 1 percent level;

^{^*}indicates significance at the 5 percent level.

^{^a}Test of the significance of all coefficients. Critical value (5 percent) is 3.0.

^{^b}Test of the null hypothesis of homoskedasticity, distributed as x² with two degrees of freedom.

^{^c}International Monetary Fund, International Financial Statistics (IFS), line 65umc 110, interpolated using nominal exchange rates as benchmarks to obtain monthly data.

Table 2

Test of the Consumption-Beta Model, One-Month Eurocurrency Returns

Currency	Constant	Z	DW	F^a	SEE	Test^b
Z₁= Real effective exchange rate
of U.S. dollar, based on labor costs^c
Deutsche mark	-0.12	1.09^*	2.17	6.04	3.20	0.24
	(0.37)	(0.44)
French franc	-0.28	1.10^*	2.16	6.36	3.14	1.21
	(0.36)	(0.44)
Pound sterling	-0.26	1.13^*	1.84	6.61	3.18	6.28
	(0.37)	(0.44)
Japanese yen	0.18	0.42	1.82	0.86	3.26	1.12
	(0.38)	(0.45)
Swiss franc	-0.26	1.27^*	1.95	5.94	3.76	0.36
	(0.43)	(0.52)
Z₂ = Ratio of U.S. trade deficit
to gross national product (GNP) in corresponding quarter
Deutsche mark	0.08	6.58^*	2.14	5.41	3.16	1.51
	(0.34)	(2.83)
French franc	-0.04	6.15^*	2.11	4.94	3.09	1.79
	(0.33)	(2.77)
Pound sterling	-0.14	8.87 ^**	1.96	11.67	2.90	5.59
	(0.31)	(2.60)
Japanese yen	0.26	2.47	1.81	0.71	3.27	1.92
	(0.35)	(2.93)
Swiss franc	-0.02	0.73^*	1.91	4.85	3.72	1.78
	(0.40)	(3.34)

^{^**}indicates significance at the 1 percent level;

^{^*}indicates significance at the 5 percent level.

^{^a}Test of the significance of all coefficients. Critical value (5 percent) is 3.0.

^{^b}Test of the null hypothesis of homoskedasticity, distributed as x² with two degrees of freedom.

^{^c}International Monetary Fund, International Financial Statistics (IFS), line 65umc 110, interpolated using nominal exchange rates as benchmarks to obtain monthly data.

The results are supportive of the peso-problem hypothesis, except for the case of the excess returns over yen-denominated assets. In all but that case, the coefficients of the indicator variables are significant at least at the 5 percent level. Because the right-hand-side variables are all part of the information set, their significance constitutes a clear rejection of the model in its pure form; that is, when the exchange rate is generated by a covariance-stationary process. In addition, all the coefficients have the expected signs, and their values are not unreasonable. When the real exchange rate is used as the indicator variable, the results imply that a 10 percent real appreciation of the U.S. dollar increases the return differential in favor of dollar assets by an annual rate of approximately 1.7-2.0 percent ¹¹ When the trade account is used, the coefficients imply that an increase in the commercial deficit of $40 billion causes the excess return on dollar assets to increase by 6-8 percentage points relative to most other currencies (but only less than 1 percentage point relative to the Swiss franc).¹²

Almost every empirical study of returns on assets of different currency denomination has found evidence of heteroskedasticity—even when different specifications, sample periods, and data were used (compare, for example, Hsieh (1984), Cumby and Obstfeld (1984), and Hodrick and Srivastava (1984)). Apart from the estimation and testing biases that arise, heteroskedasticity is also a signal of misspecification of the model being estimated. In particular, it constitutes a violation of the covariance-stationarity assumption. Therefore, a test of the homo-skedasticity assumption was performed. This test is basically the extension to time-series regressions of White’s (1980) procedure, which has been applied in most of the previous studies. Except for one of the regressions of the excess return over pound sterling, where the test statistic is marginally significant at 5 percent, heteroskedasticity is amply rejected. This finding is clear support for the alternative specification and may provide an explanation of why other studies have systematically encountered heteroskedasticity.

Given that many previous studies have focused on the return on 90-day Eurocurrency deposits, and considering that that might be a more relevant decision period than the 30-day term used above, the same regressions were run using data on (overlapping) three-month ex post returns. Under ordinary least squares, the results are very similar—indeed, the significance of the coefficients improves. These results are reported in Table 3. The ordinary least-squares estimates of the standard errors, however, are inconsistent in this case. The problem that arises here is that, since the interval between observations is shorter than the term of the deposits, some autocorrelation will be present in the residuals because they represent, at least in part, prediction errors. Consider the prediction error that is discovered at time t. It will affect the return differential on deposits made at t - 3 and due at t, but also the return differential on deposits made at t - 2 and t - 1 that mature at t + 1 and t + 2. The residuals will therefore tend to follow a second-order moving average process. Standard corrections, such as generalized least squares, are not useful in this case and would actually render the estimates of the regression coefficients inconsistent. The inconsistency arises because the regressors are not exogenous but merely predetermined, and because generalized least-squares involves the use of transformed right-hand-side variables that are correlated with the transformed residuals.

Table 3

Test of the Consumption-Beta Model, Three-Month Eurocurrency Returns

Note: See “Note” to Table 2. The sample period is June 1974 through December 1984.

^{^a}Test of significance of all coefficients, based on the consistent variance-covariance matrix, distributed as x² with two degrees of freedom. Critical value (5 percent) is 5.99.

^{^b}Standard errors based on a consistent estimate of MA2 variance-covariance matrix.

Table 3

Test of the Consumption-Beta Model, Three-Month Eurocurrency Returns

Currency	Constant	Z	DW	SEE	Test^a
	Z₁= Real effective exchange rate
	of U.S. dollar, based on labor costs
Deutsche mark	-0.10	1.14	0.75	1.72	0.10
	(0.20)	(0.24)
	(3.68)^b	(4.18)^b
French franc	-0.25	1.14	0.69	1.68	0.08
	(0.19)	(0.24)
	(3.46)^b	(4.32)^b
Pound sterling	0.29	1.37	0.74	1.74	0.14
(0.20)	(0.25)
	(3.84)^b	(4.17)^b
Japanese yen	0.12	0.50	0.54	1.94	0.03
	(0.22)	(0.27)
	(4.80)^b	(5.22)^b
Swiss franc	-0.26	1.32	0.62	2.22	0.08
	(0.25)	(0.31)
	(5.32)^b	(5.61)^b
	Z₂ = Ratio of U.S. trade deficit
	to GNP in corresponding quarter
Deutsche mark	0.10	5.73	0.74	1.78	0.12
	(0.19)	(1.60)
	(3.64)^b	(21.6)^b
French franc	0.02	4.77	0.65	1.76	0.07
	(0.19)	(1.59)
	(3.64)^b	(21.8)^b
Pound sterling	-0.08	7.41	0.76	1.8	0.24
	(0.19)	(1.62)
	(3.75)^b	(20.1)^b
Japanese yen	0.26	1.8	0.53	1.96	0.02
	(0.21)	(1.76)
	(4.43)^b	(28.0)^b
Swiss franc	-0.03	6.92	0.62	2.27	0.12
	(0.24)	(2.04)
	(5.04)^b	(27.2)^b

Note: See “Note” to Table 2. The sample period is June 1974 through December 1984.

^{^a}Test of significance of all coefficients, based on the consistent variance-covariance matrix, distributed as x² with two degrees of freedom. Critical value (5 percent) is 5.99.

^{^b}Standard errors based on a consistent estimate of MA2 variance-covariance matrix.

Table 3

Test of the Consumption-Beta Model, Three-Month Eurocurrency Returns

Currency	Constant	Z	DW	SEE	Test^a
	Z₁= Real effective exchange rate
	of U.S. dollar, based on labor costs
Deutsche mark	-0.10	1.14	0.75	1.72	0.10
	(0.20)	(0.24)
	(3.68)^b	(4.18)^b
French franc	-0.25	1.14	0.69	1.68	0.08
	(0.19)	(0.24)
	(3.46)^b	(4.32)^b
Pound sterling	0.29	1.37	0.74	1.74	0.14
(0.20)	(0.25)
	(3.84)^b	(4.17)^b
Japanese yen	0.12	0.50	0.54	1.94	0.03
	(0.22)	(0.27)
	(4.80)^b	(5.22)^b
Swiss franc	-0.26	1.32	0.62	2.22	0.08
	(0.25)	(0.31)
	(5.32)^b	(5.61)^b
	Z₂ = Ratio of U.S. trade deficit
	to GNP in corresponding quarter
Deutsche mark	0.10	5.73	0.74	1.78	0.12
	(0.19)	(1.60)
	(3.64)^b	(21.6)^b
French franc	0.02	4.77	0.65	1.76	0.07
	(0.19)	(1.59)
	(3.64)^b	(21.8)^b
Pound sterling	-0.08	7.41	0.76	1.8	0.24
	(0.19)	(1.62)
	(3.75)^b	(20.1)^b
Japanese yen	0.26	1.8	0.53	1.96	0.02
	(0.21)	(1.76)
	(4.43)^b	(28.0)^b
Swiss franc	-0.03	6.92	0.62	2.27	0.12
	(0.24)	(2.04)
	(5.04)^b	(27.2)^b

Note: See “Note” to Table 2. The sample period is June 1974 through December 1984.

^{^a}Test of significance of all coefficients, based on the consistent variance-covariance matrix, distributed as x² with two degrees of freedom. Critical value (5 percent) is 5.99.

^{^b}Standard errors based on a consistent estimate of MA2 variance-covariance matrix.

A solution to this problem is presented by Cumby, Huizinga, and Obstfeld (1983). It consists of a two-step procedure: the first is to estimate the coefficients by some consistent method (ordinary least squares in this case, since all the right-hand-side variables are predetermined); the second is to use the fitted residuals to obtain an estimate of their variance-covariance matrix, imposing the serial correlation structure that is known to exist. This estimate of the variance-covariance matrix is used to carry out testing of the hypothesis. The results of applying this procedure, also presented in Table 3, are not so favorable to the alternative hypothesis. The significance of the coefficients disappears in all cases.¹³ This finding is puzzling because the behavior of the one-month and the three-month excess returns look remarkably similar at first glance. Furthermore, if the estimation is carried out with nonover-lapping data (a procedure that is in principle inferior because it disregards some information), the results are basically similar to those obtained by applying ordinary least squares, as can be seen from Table 4. Therefore, it is the procedure followed to correct the variance-covariance matrix, rather than the data themselves, that is responsible for the lack of significance. A possible explanation may come from analyzing the error term in equation (7). Because the indicator variables do not provide an optimal forecast of the unobservable probability of a change in regime, the residuals in this regression will not be only the pure prediction error. It is a situation similar to the case of omitted variables. Then, if these “omitted variables” have a strong serial correlation, the variance-covariance matrix will, as a result, appear somehow “inflated.”

Table 4

Three-Month Eurocurrency Returns, Nonoverlapping Data

Note: See “Note” to Table 2 The sample period is June 1974 through December 1984.

^{^a}Test of the significance of all coefficients. Critical value (5 percent) is 3.0.

^{^b}Test of the null hypothesis of homoskedasticity, distributed as X²with two degrees of freedom.

^{^c}IFS line 65umc 110, interpolated using nominal exchange rates as bench-marks to obtain monthly data.

Table 4

Three-Month Eurocurrency Returns, Nonoverlapping Data

Currency	Constant	Z	DW	F^a	SEE	Test^b
	Z₁ = Real effective exchange rate
	of U.S. dollar, based on labor costs^c
Deutsche mark	-0.05	1.08^*	2.04	5.84	1.79	0.24
	(0.37)	(0.45)
French franc	-0.16	1.02^*	1.96	5.28	1.78	1.21
	(0.37)	(0.45)
Pound sterling	-0.27	1.18^*	1.64	7.48	3.18	6.28
(0.36)	(0.43)
Japanese yen	0.03	0.55	1.66	1.34	1.91	1.12
	(0.40)	(0.48)
Swiss franc	-0.09	1.17^*	2.00	4.31	2.25	0.36
	(0.47)	(0.56)
	Z₂ = Ratio of U.S. trade deficit
	to GNP in corresponding quarter
Deutsche mark	0.12	7.32^*	1.97	4.58	1.82	1.51
	(0.34)	(3.42)
French franc	2.62	6.38	1.83	3.48	1.82	1.79
	(0.35)	(3.42)
Pound sterling	0.05	5.69	1.52	2.75	1.82	5.59
(0.35)	(3.43)
Japanese yen	0.20	2.37	1.65	0.42	1.94	1.92
	(0.37)	(3.65)
Swiss franc	0.08	7.98	2.00	3.47	2.27	1.78
	(0.43)	(4.28)

Note: See “Note” to Table 2 The sample period is June 1974 through December 1984.

^{^a}Test of the significance of all coefficients. Critical value (5 percent) is 3.0.

^{^b}Test of the null hypothesis of homoskedasticity, distributed as X²with two degrees of freedom.

^{^c}IFS line 65umc 110, interpolated using nominal exchange rates as bench-marks to obtain monthly data.

Table 4

Three-Month Eurocurrency Returns, Nonoverlapping Data

Currency	Constant	Z	DW	F^a	SEE	Test^b
	Z₁ = Real effective exchange rate
	of U.S. dollar, based on labor costs^c
Deutsche mark	-0.05	1.08^*	2.04	5.84	1.79	0.24
	(0.37)	(0.45)
French franc	-0.16	1.02^*	1.96	5.28	1.78	1.21
	(0.37)	(0.45)
Pound sterling	-0.27	1.18^*	1.64	7.48	3.18	6.28
(0.36)	(0.43)
Japanese yen	0.03	0.55	1.66	1.34	1.91	1.12
	(0.40)	(0.48)
Swiss franc	-0.09	1.17^*	2.00	4.31	2.25	0.36
	(0.47)	(0.56)
	Z₂ = Ratio of U.S. trade deficit
	to GNP in corresponding quarter
Deutsche mark	0.12	7.32^*	1.97	4.58	1.82	1.51
	(0.34)	(3.42)
French franc	2.62	6.38	1.83	3.48	1.82	1.79
	(0.35)	(3.42)
Pound sterling	0.05	5.69	1.52	2.75	1.82	5.59
(0.35)	(3.43)
Japanese yen	0.20	2.37	1.65	0.42	1.94	1.92
	(0.37)	(3.65)
Swiss franc	0.08	7.98	2.00	3.47	2.27	1.78
	(0.43)	(4.28)

Note: See “Note” to Table 2 The sample period is June 1974 through December 1984.

^{^a}Test of the significance of all coefficients. Critical value (5 percent) is 3.0.

^{^b}Test of the null hypothesis of homoskedasticity, distributed as X²with two degrees of freedom.

^{^c}IFS line 65umc 110, interpolated using nominal exchange rates as bench-marks to obtain monthly data.

To summarize, it is clear that—with the exception of the Japanese yen—the pure form of the intertemporal models of asset prices can be rejected. Furthermore, the finding that the higher is the degree of “over-valuation” of the dollar, the higher is the excess return on dollar assets, is an indication that a peso-problem type of situation of exchange rate expectations may be at least part of the explanation of the observed excess returns.

The Capital Asset Pricing Model

The CAPM is certainly the most popular model of asset pricing in the finance literature. It is a static or one-period model that is based on the asset demands of savers who behave as mean-variance optimizers. Mean-variance optimization can be considered a local approximation of the more general assumption of expected utility maximization. Although the assumptions on which the CAPM is based seem to be more restrictive than those of the intertemporal model considered above, the simplicity and intuitive appeal of the CAPM’s predictions are compensations. In addition, if the random process generating the returns is time separable in a well-defined sense, or if the utility of the agents is logarithmic, the intertemporal problem collapses to the static one (Merton (1982)). Furthermore, the empirical implementation of the CAPM requires fewer additional assumptions, which is perhaps a reason why Mankiw and Shapiro (1984) found that it does a better job in explaining a cross section of returns than does the consumption-beta model. For international finance, an additional attractive feature of this model is that it provides a microeconomic foundation for the asset market approach to the exchange rate.

A useful formulation for empirical applications is the one proposed by Kouri (1977) and Dornbusch (1983), which focuses on asset demands. These turn out to be proportional to the expected rates of return, with the factor of proportionality involving the variances and covariances of all the returns. The derivation is shown in Appendix I. This implies that the excess returns will satisfy

\begin{matrix} E (y_{t + 1}) = α + w_{t} δ, & (8) \end{matrix}

where w_t is a vector containing the shares of the different assets in the aggregate portfolio. The coefficients 8 are functions of the variance-covariance matrix of the returns, but they will be assumed to be constant, as is necessary to obtain an estimable form.

The approach will be entirely parallel to the one followed with the intertemporal model. It is assumed that the probability distribution of the exchange rate depends, as above, on which of the two possible policy regimes is governing the economy. For any period during which the system is in state 1, equation (3) implies that the excess returns will satisfy

\begin{matrix} y_{t + 1} - E (y_{t + 1} | I_{t}) = {[(1 - Π) (\bar{s} - \bar{e})]}_{t + 1} + u_{t + 1}, & (9) \end{matrix}

where, in this case, E (y_t+1|I_t) is given by equation (8). If, as before, there exist instruments Z_t for the first term in the right-hand side of equation (9), the following regression can be run:

\begin{matrix} y_{t + 1} = α + w_{t + 1} δ + D_{t} Z_{t} β + u_{t + 1}, & (10) \end{matrix}

where D_t is the dummy variable that equals unity from October 1980 onward.¹⁴ Once again, a rejection of the hypothesis that β = 0 will constitute a rejection of the static CAPM and a suggestion of a peso-problem type of situation.

Empirical Results

The most serious data problem in this case is to select and measure the securities to be included in the aggregate portfolio. The standard procedure, followed here, is to consider only the outstanding stock of government bonds, in the spirit of most empirical work relating to the asset market approach to the exchange rate.¹⁵ The specific procedure followed to construct those stocks is detailed in Appendix II. The ex post real interest rates were computed by using an inflation rate measured as a weighted average of consumer price indices, with the weights determined by the size of the respective GNPs. Only the 30-day returns were used.

The results, displayed inTable 5, indicate that when the real exchange rate is used as the indicator variable, the null hypothesis—the pure form of this model—is rejected in all cases but that of Japan. For Japan the entire equation does not seem to fit well, since the value of the F test of significance of all coefficients is about unity, whereas the critical value (5 percent) is 2.1. The U.S. trade deficit did not provide successful results in this model, and it is not reported. The point estimates of the share coefficients seem to be roughly in line with previous studies that have used different samples and definitions of asset quantities. The estimated coefficients imply that a 10 percent increase in the real exchange rate causes an increase of about 4-8 percentage points in the yield differential (see footnote 11 above for the derivation of these values).

Table 5

Test of the CAPM Model

(y = α+ Wδ + Zβ+U)^a

Note: See “Note” to Table 2. The sample period is June 1974 through September 1984.

^{^a}The dependent variable is the difference in monthly ex post real interest rates (on each currency minus that on the U.S. dollar). A weighted average of consumer price indices was used as deflator. The explanatory variables W are a constant and the portfolio share of each currency-denominated bond in the following order: deutsche mark, French franc, pound sterling, Japanese yen, and Swiss franc; Z is the dollar real exchange rate lagged one period.

^{^b}Test of the significance of all coefficients. Critical value (5 percent) is 2.1.

Table 5

Test of the CAPM Model

(y = α+ Wδ + Zβ+U)^a

Currency	α	δ₁	δ₂	δ₃	δ₄	δ₅	β	DW	F^b
Deutsche mark	-1.8	-209.5**	-30.3	55.8	419.4**	288.3	-5.77**	2.10	3.69
	(6.2)	(65.9)	(55.4)	(32.9)	(117.6)	(342.2)	(1.7)
French franc	-1.6	-197.8**	-41.2	64.1*	415.2**	135.9	-6.07**	2.14	4.22
	(6.0)	(63.7)	(53.5)	(31.8)	(113.7)	(330.8)	(1.6)
Pound sterling	-6.3	-143.4*	51.4	28.7	278.8*	495.1	-4.27**	2.15	3.55
	(5.8)	(62.1)	(52.2)	(31.0)	(110.8)	(322.5)	(1.6)
Japanese yen	7.8	-60.1	-68.9	-14.1	191.9	-129.0	-2.96	1.80	0.99
	(6.8)	(72.1)	(60.6)	(36.0)	(128.7)	(374.6)	(1.8)
Swiss franc	-3.3	-263.0**	-68.6	105.5**	576.6**	216.9	-8.37**	1.91	5.59
	(6.9)	(74.2)	(62.4)	(37.1)	(132.4)	(385.5)	(1.9)

Note: See “Note” to Table 2. The sample period is June 1974 through September 1984.

^{^b}Test of the significance of all coefficients. Critical value (5 percent) is 2.1.

Table 5

Test of the CAPM Model

(y = α+ Wδ + Zβ+U)^a

Currency	α	δ₁	δ₂	δ₃	δ₄	δ₅	β	DW	F^b
Deutsche mark	-1.8	-209.5**	-30.3	55.8	419.4**	288.3	-5.77**	2.10	3.69
	(6.2)	(65.9)	(55.4)	(32.9)	(117.6)	(342.2)	(1.7)
French franc	-1.6	-197.8**	-41.2	64.1*	415.2**	135.9	-6.07**	2.14	4.22
	(6.0)	(63.7)	(53.5)	(31.8)	(113.7)	(330.8)	(1.6)
Pound sterling	-6.3	-143.4*	51.4	28.7	278.8*	495.1	-4.27**	2.15	3.55
	(5.8)	(62.1)	(52.2)	(31.0)	(110.8)	(322.5)	(1.6)
Japanese yen	7.8	-60.1	-68.9	-14.1	191.9	-129.0	-2.96	1.80	0.99
	(6.8)	(72.1)	(60.6)	(36.0)	(128.7)	(374.6)	(1.8)
Swiss franc	-3.3	-263.0**	-68.6	105.5**	576.6**	216.9	-8.37**	1.91	5.59
	(6.9)	(74.2)	(62.4)	(37.1)	(132.4)	(385.5)	(1.9)

Note: See “Note” to Table 2. The sample period is June 1974 through September 1984.

^{^b}Test of the significance of all coefficients. Critical value (5 percent) is 2.1.

The importance of this model in this context is that it contains an explicit formulation of the risk-premium explanation. That is, the increase in the outstanding stock of dollar securities associated with the U.S. federal deficit is included as an explanatory variable in this specification. Nevertheless, there is still evidence that variables measuring the degree of “overvaluation” of the dollar are able to explain, to some extent, the behavior of the excess return on dollar assets.

II. A Speculative Bubble Model

The price of an asset is in a speculative bubble equilibrium when it follows a rational expectations path that diverges from the fundamental value of the asset. This means that the expectations of the market participants are “correct” (that is, they are equal to the mathematical expectation of the asset price), but these expectations are not equal to the fundamental value of the asset (that is, to its intrinsic economic value). In the case of the exchange rate, the fundamental value seems a bit more difficult to define than in the case of a real asset, since the fundamental value of the currencies involved is itself a less precise concept. Because currencies lack consumption value, their market value necessarily depends on the expectation that they will be accepted as means of payments by other agents in the future. It is still possible, however, to define the fundamental value of the exchange rate—for example, as the value that corresponds to the equilibrium path that converges to a steady state.

Are “rational” bubbles a possible equilibrium in foreign exchange markets? From the literature (see Blanchard and Watson (1982) and Tirole (1985), for example), the answer seems to be in the affirmative, if a reasonable description of the foreign exchange market could be one in which agents with finite lives trade a perpetual asset, the supply of which is determined by outside forces. In addition, given that assessments of the fundamental value of the exchange rate seem to be quite “noisy,” misperceptions or self-fulfilling prophecies might be likely in these markets. The question of the general equilibrium implications of a bubble on the exchange rate is a more difficult one, however, since all the repercussions on the rest of the economy should be considered.

To account for the observed behavior of the excess return on dollar assets in recent years, the exchange value of the U.S. dollar could be thought to have been on a bubble path (along which its value was appreciating) for most recent years; however, there was always a small probability that the dollar would crash and suffer a large depreciation that would restore the fundamental equilibrium. This would produce persistently positive excess returns on dollar assets even under equalization of ex ante returns. The duration of the period with high positive excess returns may be too long, however, to be explained by a single episode (over four years); this possibility suggests that the model might apply only to a subset of the sample considered.

Despite the different underlying forces that drive the process of appreciation of the exchange rate, there are some formal similarities between the speculative bubble model and the peso-problem models. Both models could explain persistent differences in the ex post returns on assets of different currency denomination and a sustained appreciation of the exchange rate. In addition, the statistical results reported in the preceding section do not necessarily rule out the speculative bubble hypothesis. Consider, for example, a situation in which the exchange rate is being driven entirely by speculative forces; all the same, the more over-valued the dollar becomes, the larger is the depreciation necessary to bring it back to the fundamental level. Under a speculative bubble equilibrium, excess returns would then also tend to show some correlation with the degree of overvaluation of the dollar.

The model will be developed under the assumption that the assets are perfect substitutes. This is certainly not the best assumption, but it will suffice to explore the explanatory power of this approach for the behavior of excess returns. There are two possible states of the world: under state 1 (Ω= 1), the exchange rate is on the bubble path; under state 2 (Ω= 2), the exchange rate is on its fundamental path. With slightly changed notation, e_t will denote the bubble exchange rate, and s_t will denote the fundamental exchange rate. The probability of a bubble state occurring is constant over time and is denoted as Π. From equations (1) and (2) it follows that, if the economy is on a bubble equilibrium at time t,

\begin{matrix} F_{t} = Π E (e_{t + 1}) + (1 - Π) E (s_{t + 1}) - e_{t}, & (11) \end{matrix}

where $F_{t} = i_{t} - i_{t}^{*}$ . Equation (11) can be solved forward to obtain the equilibrium value of the bubble exchange rate:

\begin{matrix} e_{t} = Σ_{k = 0}^{\infty} Π^{k} E [(1 - Π) s_{t + k + 1} - F_{t + k} | I_{t}] . & (12) \end{matrix}

In equation (12), e_t is the only value that satisfies the dynamic rational expectations relation (11) and that does not start off on an explosive or implosive path. It is not, however, the only rational expectations solution (see, for example, Shiller (1978)), and it does not conform very well to stylized facts that were described earlier. If F is expected to remain roughly constant (as it has been), and so is the fundamental exchange rate s (as is assumed to be approximately true), then the bubble exchange rate e will also be roughly constant, which openly contradicts the behavior of the dollar in the past five years. A definite, marked trend in e requires that either F or s display a similar trend. Therefore, other solutions to equation (11) will be considered, which are of the form:

\begin{matrix} e_{t} = Σ_{k = 0}^{\infty} Π^{k} E [(1 - Π) s_{t + k + 1} - F_{t + k} | I_{t}] + A Π^{- t}, & (13) \end{matrix}

where A is an arbitrary constant that should be negative in the case of the dollar in recent years. It can be checked that equation (13) is a solution by using it to substitute for e_t and e_t+1 in equation (11). This type of solution implies that the bubble exchange rate is following an explosive path; that is, as t→∞,e_t→0. In the long run, however, the bubble will burst with probability 1. This is because the probability of a crash at or before time T is given by $Σ_{k = 0}^{T} Π^{k} (1 - Π)$ , which approaches unity as T approaches infinity. Despite movement of the price of foreign exchange toward a value of zero, the chances of its actually getting close to that value are insignificant. Indeed, the expected duration of this bubble is only 1/(1 - Π).¹⁶ This model can therefore serve as at least a local approximation of the exchange rate process, although it would cease to be a sensible one if the bubble were to last for an unexpectedly long time.

Empirical Results

To proceed to the empirical estimation it is necessary to solve the infinite sums on the right-hand side of equation (13). This is achieved by specifying what kind of processes the fundamental exchange rate and the interest rate differential follow. For the fundamental exchange rate, it is assumed that it remained constant during the sample period. For the interest rate differential, it is assumed that it follows a univariate ARIMA process. Following Box-Jenkins identification procedures, an AR(2) specification was chosen for the interest rate differential (an ARMA(1,1) specification also seemed to fit the data well). Under these assumptions, applying techniques developed in Hansen and Sargent (1980) allows the following system to be obtained:

\begin{matrix} e_{t} = \bar{s} - θ_{1} F_{t} - θ_{2} F_{t - 1} + A Π^{- t} + \in_{1 t} & (14) \end{matrix}

\begin{matrix} F_{t} = Ψ_{1} F_{t - 1} + Ψ_{2} F_{t - 2} + \in_{2 t}, & (15) \end{matrix}

with the following cross-equation restrictions:

\begin{matrix} θ_{1} = 1 / (1 - Ψ_{1} Π - Ψ_{2} Π^{2}), & (16) \end{matrix}

\begin{matrix} θ_{2} = Ψ_{2} Π θ_{1} . & (17) \end{matrix}

There are two interesting tests that can be performed on this system. One is the significance of the term AΠ^-t, which represents a test of the existence of an explosive bubble. The other is the set of cross-equation restrictions (16) and (17), which represents a test of rationality in the formation of expectations on future interest rate differentials. A feature that may be problematic in a solution such as equation (13) is that equation (13) implies that the exchange rate will appreciate at an increasing rate, even with a constant nominal interest differential. This fact can be seen by noting that, in the case of a constant F, E (e_t–e_t-1II_t-1) =A (1–Π)Π^-t. This acceleration in the process derives, however, from the general structure of the problem and not from any particular assumption made here. As the dollar becomes more overvalued, the loss in the case of a bursting bubble and crash becomes larger; therefore, the expected rate of appreciation under the bubble state must be increasing too. This is the principal reason why this type of bubble is not well suited to explain a long-lasting episode.

The results, which are presented in Table 6, are in general favorable for the model. Only the case of the dollar versus deutsche mark was considered. Because the model is nonlinear in both the variables and the parameters (even in its unrestricted form), a minimum-distance method of estimation was used (the MINDIS routine of the RAL system). Given the plausibility of bubbles of shorter duration, three nested sample periods were considered: (1) from January 1983 through February 1985, (2) from November 1981 through February 1985, and (3) from November 1980 through February 1985. The coefficients of the model are jointly highly significant; the key parameters have the correct sign; and in general the parameter values are in line with prior beliefs. Moreover, the results do not differ considerably for the different sample periods.

Table 6

Model of a Speculative Bubble on the Exchange Rate: Estimation of Equations (14) and (15)

Note: The case tested was the U.S. dollar-deutsche mark exchange rate. Numbers in parentheses are standard errors of the coefficients.

^{^a}Likelihood-ratio test of the joint significance of A and II. Distributed as x². Critical values are (5 percent) 5.99 and (1 percent) 9.21.

^{^b/}Likelihood-ratio test of the parameter cross-equation restrictions (equations (16) and (17) in the text). Distributed as X²-Critical values are the same as for the first test.

^{^c/}Restricted to obtain convergence.

Table 6

Model of a Speculative Bubble on the Exchange Rate: Estimation of Equations (14) and (15)

Parameter	ψ₁	ψ₂	s	θ₁	θ₂	A	Π	Test^a	Test^b
	Sample period: January 1983 through February 1985
Unrestricted	1.07	-0.27	-0.81	21.9	-18.4	-0.019	0.95	65.2
	(0.18)	(0.18)	(0.08)	(9.96)	(9.97)	(0.027)	(0.02)
Restricted	1.06	-0.25	-0.78			-0.031	0.95		6.8
	(0.17)	(0.17)	(0.08)			(0.034)	(0.02)
	Sample period: November 1981 through February 1985
Unrestricted	0.89	-0.20	-0.79	12.1	-10.5	-0.028	0.95	99.3
	(0.15)	(0.16)	(0.04)	(6.08)	(6.11)	(0.017)	(0.01)
Restricted	0.88	-0.20	-0.78			-0.031	0.95		0.4
	(0.15)	(0.16)	(0.04)			(0.018)	(0.01)
	Sample period: December 1980 through February 1985
Unrestricted	0.83	-0.33	-0.69	2.5	1.55	-0.092	0.97	108.6
	(0.12)	(0.12)	(0.06)	^c	(3.77)	(0.050)	(0.01)
Restricted	0.83	-0.33	-0.69			-0.09	0.97		0.6
	(0.12)	(0.12)	(0.06)			(0.05)	(0.01)

Note: The case tested was the U.S. dollar-deutsche mark exchange rate. Numbers in parentheses are standard errors of the coefficients.

^{^a}Likelihood-ratio test of the joint significance of A and II. Distributed as x². Critical values are (5 percent) 5.99 and (1 percent) 9.21.

^{^b/}Likelihood-ratio test of the parameter cross-equation restrictions (equations (16) and (17) in the text). Distributed as X²-Critical values are the same as for the first test.

^{^c/}Restricted to obtain convergence.

Table 6

Model of a Speculative Bubble on the Exchange Rate: Estimation of Equations (14) and (15)

Parameter	ψ₁	ψ₂	s	θ₁	θ₂	A	Π	Test^a	Test^b
	Sample period: January 1983 through February 1985
Unrestricted	1.07	-0.27	-0.81	21.9	-18.4	-0.019	0.95	65.2
	(0.18)	(0.18)	(0.08)	(9.96)	(9.97)	(0.027)	(0.02)
Restricted	1.06	-0.25	-0.78			-0.031	0.95		6.8
	(0.17)	(0.17)	(0.08)			(0.034)	(0.02)
	Sample period: November 1981 through February 1985
Unrestricted	0.89	-0.20	-0.79	12.1	-10.5	-0.028	0.95	99.3
	(0.15)	(0.16)	(0.04)	(6.08)	(6.11)	(0.017)	(0.01)
Restricted	0.88	-0.20	-0.78			-0.031	0.95		0.4
	(0.15)	(0.16)	(0.04)			(0.018)	(0.01)
	Sample period: December 1980 through February 1985
Unrestricted	0.83	-0.33	-0.69	2.5	1.55	-0.092	0.97	108.6
	(0.12)	(0.12)	(0.06)	^c	(3.77)	(0.050)	(0.01)
Restricted	0.83	-0.33	-0.69			-0.09	0.97		0.6
	(0.12)	(0.12)	(0.06)			(0.05)	(0.01)

Note: The case tested was the U.S. dollar-deutsche mark exchange rate. Numbers in parentheses are standard errors of the coefficients.

^{^a}Likelihood-ratio test of the joint significance of A and II. Distributed as x². Critical values are (5 percent) 5.99 and (1 percent) 9.21.

^{^b/}Likelihood-ratio test of the parameter cross-equation restrictions (equations (16) and (17) in the text). Distributed as X²-Critical values are the same as for the first test.

^{^c/}Restricted to obtain convergence.

The first hypothesis—the significance of the explosive bubble term—stands up well in the data. The values of likelihood-ratio tests of the exclusion of the term AΠ^-t are reported in the penultimate column of Table 6 The tests easily reject such exclusion. The t-statistics for A itself are only marginally significant, however, although those for II are highly significant. The values of Π range between 0.95 and 0.97, which implies a probability of a crash of between 3 and 6 percent for each month. As the sample period considered becomes longer, the estimates of II become higher because, for a bubble of longer durability, a lower probability of a crash is required. The estimates also imply that if the bubble had burst at any point in the sample, the dollar-deutsche mark exchange rate would have collapsed to a value that ranges between DM 2.0 and DM 2.3 according to the sample period considered. That value is the fundamental exchange rate implied by the estimates.

The second hypothesis—the cross-equation parameter restrictions—is also accepted. A likelihood-ratio test statistic of the functional relationships between parameters (16) and (17) is reported in the last column of Table 6. The restrictions cannot be rejected at the 5 percent level of significance (although in the shortest sample the restrictions would be rejected at the 1 percent level). The implication of this hypothesis is that expectations of interest rate differentials are formed rationally, considering the autoregressive process followed by interest rate differentials. This result should be taken with caution because some of the coefficients involved are not statistically significant; restrictions among their values are therefore less meaningful.

In sum, these results are almost entirely consistent with the speculative bubble model and suggest that this approach might be valuable; a possible extension would be to apply this kind of model to data of higher frequency. A final qualification should be repeated. The success of the estimation of this model does not necessarily preclude alternative explanations of the exchange rate process. For example, the explosive bubble term could be a proxy for any omitted variable, such as a marked increase in the risk premium.

III. Conclusions

In reality, a complete explanation of the high excess return of dollar assets during the period 1980-84 might involve a combination of different reasons. Some increase in the risk premium may have occurred, and some of the observations might have been an unanticipated sequence of new information that favored a dollar appreciation. In addition, the two alternative hypotheses considered in this paper—the “peso-problem” and the speculative bubble models—were shown to be consistent with the data.

The observed correlation between excess returns and some proposed measures of the degree of “overvaluation” of the dollar provides the basis for a formal rejection of the theoretical asset pricing models in their “pure” form and suggests that the dollar was suffering some sort of peso problem. In addition, this type of hypothesis may help to explain the frequent rejections of pure asset price models and the evidence of heteroskedasticity that is reported in most empirical applications. The correlation mentioned does not, however, seem to hold in the case of the excess return on dollar-denominated assets over yen-denominated assets.

The speculative bubble model that was postulated showed some empirical success. The relevant statistical tests were favorable, and the coefficients had correct signs and plausible values. This hypothesis presents a major shortcoming, however, in that it predicts an increasing rate of appreciation. This acceleration makes the model somewhat inappropriate for a prolonged period of appreciation. Nevertheless, it is an appealing model that might be well suited for runs of shorter duration.

APPENDIX I: Asset Pricing Models

This appendix provides the derivation of the two models presented in Section I.

The Consumption-Based Model

The intertemporal capital asset pricing model is based on the optimal decisions of an agent who chooses a path of consumption and asset demands from a menu of securities that offer uncertain returns. These choices can be represented as the solution to the following problem:

J (W_{t}) = \max u (c_{t}) + β E_{t} [J (W_{t + 1})],

subject to

\begin{matrix} W_{t + 1} = (W_{t} - c_{t} + Ψ_{t}) [R_{t + 1}^{1} + Σ_{j = 2}^{n} w^{j} (R_{t + 1}^{j} - R_{t + 1}^{1})], & (18) \end{matrix}

where W represents (nonhuman) wealth, R^j(j = 1,…, n) is the random variable that equals unity plus the rate of return on asset j,Ψ is nonstochastic labor income, and w^j (j = 2,…, n) is the share of asset j in the agent’s portfolio. Note that the condition that the sum of w^j equals unity has been imposed, but some of the w’smay be negative (short-selling of the asset). (The notation applied here is slightly different from the one in the text.)

The solution to this problem gives the following first-order necessary condition:

\begin{matrix} u_{c}^{'} (c_{t}) = β E_{t} [u_{c}^{'} (c_{t + 1}) R_{t}^{j}] & (19) \end{matrix}

for j=1, …, n. Letting $S_{t + 1} = β u_{c}^{'} (c_{t + 1}) / u_{c}^{'} (c_{t}),$ , equation (19) can be written as

\begin{matrix} E_{t} [S_{t + 1} R_{t}^{j}] = 1 . & (20) \end{matrix}

To have an estimable form, the model must be complemented by some statistical assumptions. The following ones closely resemble those in Hansen and Singleton (1983). Let $L e t R_{t} = {(R_{t}^{1}, R_{t}^{2}, …, R_{t}^{n})}^{'}$ . Let h_t = In (S^t, R_t, Z_t)’, where Z_t are other economic variables that have some significant relationship with consumption and asset returns (in the applications in the text, these variables were the real exchange rate and the trade deficit). It is then assumed that

\begin{matrix} h_{t} = A_{0} + A (L) h_{t - 1} + U_{t}, & (21) \end{matrix}

where A(L) is a matrix polynomial in the lag operator and U is multivariate normal (0,Σ). Consider the set $I_{t}^{h} = (h_{t}, h_{t - 1}, …) .$ . Conditional on $I_{t}^{h}$ , the distribution of lnS_{t + 1} + lnR_{t +1}is normal $(μ_{t}^{j}, σ^{j 2});$ μ ^Jt is a function of the past values of h; and σ^j2 is constant. By the properties of the log-normal distribution, it follows that

\begin{matrix} E (S_{t + 1} R_{t + 1}^{j} | I_{t}^{H}) = \exp (μ_{t}^{j} + \frac{1}{2} σ^{j 2}) . & (22) \end{matrix}

Given that , is a subset of I _t by the law of iterated expectations, $E (S_{t + 1} R_{t + 1}^{j} | I_{t}^{h}) = 1 .$ Then

\begin{matrix} μ_{t - 1} = - \frac{1}{2} σ^{j 2}, & (23) \end{matrix}

which means that

\begin{matrix} E (\ln S_{t + 1} + \ln R_{t + 1}^{j} | I_{t}^{h}) = - \frac{1}{2} σ^{j 2} . & (24) \end{matrix}

Therefore, for two returns i and j,

\begin{matrix} E (\ln R_{t + 1}^{j} - \ln R_{t + 1}^{i} | I_{t}^{h}) = \frac{1}{2} (σ^{i 2} - σ^{j 2}) . & (25) \end{matrix}

Equation (25) implies that

\begin{matrix} \ln R_{t + 1}^{j} - \ln R_{t + 1}^{i} = y_{t + 1}^{i} = k^{i j} + η_{t + 1}, & (26) \end{matrix}

with $E (η_{t + 1} | I_{t}^{h}) = 0 .$ The first equality follows from the fact that the difference in the logarithms of the real returns is equal to the difference in the logarithms of the nominal returns. It can be seen that these distributional assumptions produce a constant risk premium.

Turning to the alternative hypothesis, under state 1 (see the text), the exchange rate has an expected value of E (x|Ω = 1) = s; under state 2 the expected value is E(x|Ω = 2) = e. Recall that $y_{t + 1}^{i} = i_{t} - i_{t}^{*} - (x_{t + 1} - x_{t});$ ; it can then be seen that

\begin{matrix} E (y_{t + 1}^{i} | I_{t}^{h}, Ω_{t + 1} = 1) = k^{i} + {(1 - Π)}_{t + 1} ({\bar{e}}_{t + 1} - {\bar{s}}_{t + 1}) = Z_{t} β, & (27) \end{matrix}

which is the basis for the tests carried out in the text.

The Static Capital Asset Pricing Model

The static CAPM determines the equilibrium returns that would prevail if the market were made up of savers who try to maximize the expected utility they derive from the value of their wealth. A useful approximation is to consider that the agents maximize a function of the mean and the variance of wealth. (The approximation is exact if utility is quadratic or the returns have a normal distribution.) The basic specification (identical to that in Frankel (1982), for example) is

\max F [E_{t} (W_{t + 1}), V_{t} (W_{t + 1})]

s u b j e c t t o W_{t + 1} = W_{t} (1 + R_{t + 1}^{1} + w_{t}^{1} y_{t + 1}),

where W represents the value of (nonhuman) wealth, E_t and V_t are the conditional first and second moments, R¹ is unity plus the real rate of return on asset 1, y = [R²—R¹,…, Rⁿ—R^*]^’ and w is the vector of shares of assets 2 through n in the agent’s portfolio. As before, the shares are constrained to add to unity, but they can be negative.

A first-order condition for this optimization is

\begin{matrix} F_{1}^{'} W_{t} E_{t} (y_{t + 1}) + F_{2}^{'} W_{t}^{2} [2 Σ w_{t} + 2 cov (y_{t + 1}, R_{t + 1}^{1})] = 0, & (28) \end{matrix}

where $F_{1}^{'} a n d F_{2}^{'}$ indicate the derivatives of the utility function with respect to its first and second argument, and Σ is the variance-covariance matrix of the excess returns y. With the coefficient of risk aversion ρ approximated by $- 2 W F_{2}^{'} / F_{1}^{'},$ equation (11) implies:

\begin{matrix} E_{t} (y_{t + 1}) = K + ρ Σ w_{t}, & (29) \end{matrix}

which is the null hypothesis of this model. Now consider that there are two possible states of the world. Under state 1, the excess returns follow the random process $y_{t + 1} = {\bar{y}}_{t}^{1} + u_{t + 1}$ and in state 2 they satisfy $y_{t + 1} = {\bar{y}}_{t}^{2} + u_{t + 1}$ Then $E_{t} (y_{t + 1}) = Π {\bar{y}}_{t}^{1} + (1 - Π) {\bar{y}}_{t}^{2}$ where I is the probability of state 1. It follows that, in state 1, the ex post excess returns will be given by

\begin{matrix} y_{t + 1} = E_{t} (y_{t + 1}) + (1 - Π) ({\bar{y}}_{t}^{2} - {\bar{y}}_{t}^{1}) + u_{t + 1} . & (30) \end{matrix}

Substituting the expected value of the excess returns from equation (12) yields the alternative hypothesis of this model that is tested in the text.

APPENDIX II: Computation of the Aggregate Portfolio Composition

The outstanding stocks of bonds issued by the different governments were computed by applying the following methodology: for the United States, International Monetary Fund, International Financial Statistics (IFS), lines 88 minus 88aa minus 88ad; for the Federal Republic of Germany, IFS lines 88 minus 12a; for France, IFS lines 88b minus 12a; for the United Kingdom, the flow of budget deficit was cumulated over a starting value of £57.8 billion at end-1972 (the deficit was estimated as IFS lines 84a minus 84aa minus 84ab plus 85ac plus 85ad; because only quarterly data are available, interpolation was used to obtain monthly figures); for Japan, Bank of Japan, Monthly Statistics (Tokyo), Table 79 (column corresponding to “National Governments Debts” held by “Other Holders or Lenders”); for Switzerland, the flow of budget deficit was cumulated over a starting value of Sw F 9.6 billion at end-1972 (the deficit measure was IFS line 84a).

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Mr. Borensztein, an economist in the Research Department, is agraduate of the Universidad de Buenos Aires and the Massachusetts Institute of Technology. Although the author retains sole responsibility for any remaining errors, he would like to acknowledge helpful comments received from Rudiger Dornbusch, Stanley Fischer, Julio Rotemberg, and his colleagues in the Fund.

The time intervals are more or less arbitrary. In the last quarter of 1980 there was a sudden increase in the excess return, and after that it stayed high on average through March 1985. The data used in this study end in December 1984. The other division (at October 1977) was chosen so as to divide in half the number of observations since January 1974.

This behavior of the nominal interest rate differential may be sensitive to the dates and the measure of interest rates chosen. In any case, movements in interest rate differentials have been one order of magnitude smaller than exchange rate changes.

A different, but related, explanation of recent exchange rate developments is the “safe haven” hypothesis, which explains the real appreciation of the dollar as the consequence of a change in perceptions about country risks (Dooley and Isard (1985)). In this context, however, the shift in asset demands would imply a lower rather than higher expected return on dollar-denominated assets because this explanation implies a lower relative risk on dollar assets.

Several comments on Frankel’s paper (1985) pointed out the implausibly low value of the risk premium that is predicted by these estimates. The critical problem seems to be the values of the variance-covariance matrix, for which the sample moments might be a poor estimate if the processes are not stationary.

The origin of the term refers to the Mexican situation of the 1970s, when the peso was permanently at a forward discount despite a fixed exchange rate system that had been in place for years. See Krasker (1980).

A third possibility that has been suggested is that the large appreciation of the dollar was caused by an increase in the long-term interest differential. If the exchange rate of the dollar is expected to return eventually to its starting value, a 5 percentage point interest differential on ten-year securities requires a logarithmic initial appreciation of the exchange rate of 50 percent. This interpretation, however, does not help to explain the large interest rate differential on short-term assets.

In more technical terms, the observations would be neither independent nor identically distributed.

Note that this model implies that the risk premium is constant.

That the β coefficients are different from zero could also be consistent with other alternative hypotheses. For example, it could be argued that the variables Z are acting as predictors of a time-varying risk premium. As argued in the introduction to the paper, however, this explanation does not appear to be sufficient to justify the magnitude of the return differentials.

To see this, consider the following example. Let α be the random variable that indicates the proportion of a deposit that is recovered (α will equal unity in normal times and will be less than unity in the event of bank failure). Then α(1+ i) is the final value of a dollar-denominated asset, and α(1 + i*)x is the yield of, say, a deposit denominated in deutsche mark, where x indicates the rate of depreciation. The expected return differential will be (1 + i) E(α) -(1 + i*)E (αx). Then, if for example cov(α,x)>0, the difference in expected returns will tilt in favor of deutsche mark deposits.

These values result from multiplying the coefficients reported in Table 3 by the mean of the right-hand-side variable (1.3) and by 12 (to annualize the return).

In this case the coefficients directly give the value of the change because the explanatory variable is the trade deficit at monthly rate as a percentage of GNP at annual rate.

Another estimation was done using the variance-covariance matrix suggested by Hansen and Hodrick (1980), which is consistent but not heteroskedasticity-resistant. The results did not differ significantly.

Note that, although the regression based on the consumption-beta model—equation (7)—seems to be nested into equation (10), it can be seen that this is not the case when all the restrictions implied by the models are considered. For example, as shown in Appendix I, the coefficients α and δ in equation (10) are a function of covariances among asset returns, and of the variance-covariance matrix of the residuals u, whereas the constant term k in equation (7) is a function of the covariances of asset returns with consumption. (These restrictions were not tested in this paper.)

The practice is based on the assumption that government bonds are outside assets. On the basis of financial theory, however, the correlations with all the traded securities are relevant; for example, stocks and private bonds should also be included in the model.

This model was first developed (in a different context) by Blanchard (1979).

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Lessons from Empirical Models of Exchange Rates: Volume 34 No. 1

Author: International Monetary Fund. Research Dept.

Volume/Issue:: Volume 1987: Issue 001

Publisher:: International Monetary Fund

ISBN:: 9781451956740

ISSN:: 1020-7635

Pages:: 194

DOI:: https://doi.org/10.5089/9781451956740.024

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Table 6

Model of a Speculative Bubble on the Exchange Rate: Estimation of Equations (14) and (15)

Parameter	ψ₁	ψ₂	s	θ₁	θ₂	A	Π	Test^a	Test^b
	Sample period: January 1983 through February 1985
Unrestricted	1.07	-0.27	-0.81	21.9	-18.4	-0.019	0.95	65.2
	(0.18)	(0.18)	(0.08)	(9.96)	(9.97)	(0.027)	(0.02)
Restricted	1.06	-0.25	-0.78			-0.031	0.95		6.8
	(0.17)	(0.17)	(0.08)			(0.034)	(0.02)
	Sample period: November 1981 through February 1985
Unrestricted	0.89	-0.20	-0.79	12.1	-10.5	-0.028	0.95	99.3
	(0.15)	(0.16)	(0.04)	(6.08)	(6.11)	(0.017)	(0.01)
Restricted	0.88	-0.20	-0.78			-0.031	0.95		0.4
	(0.15)	(0.16)	(0.04)			(0.018)	(0.01)
	Sample period: December 1980 through February 1985
Unrestricted	0.83	-0.33	-0.69	2.5	1.55	-0.092	0.97	108.6
	(0.12)	(0.12)	(0.06)	^c	(3.77)	(0.050)	(0.01)
Restricted	0.83	-0.33	-0.69			-0.09	0.97		0.6
	(0.12)	(0.12)	(0.06)			(0.05)	(0.01)

Note: The case tested was the U.S. dollar-deutsche mark exchange rate. Numbers in parentheses are standard errors of the coefficients.

^{^a}Likelihood-ratio test of the joint significance of A and II. Distributed as x². Critical values are (5 percent) 5.99 and (1 percent) 9.21.

^{^b/}Likelihood-ratio test of the parameter cross-equation restrictions (equations (16) and (17) in the text). Distributed as X²-Critical values are the same as for the first test.

^{^c/}Restricted to obtain convergence.

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Abstract

I. A “Peso-Problem” Model

The Consumption-Based Model

Empirical Results

The Capital Asset Pricing Model

Empirical Results

II. A Speculative Bubble Model

Empirical Results

III. Conclusions

APPENDIX I: Asset Pricing Models

The Consumption-Based Model

The Static Capital Asset Pricing Model

APPENDIX II: Computation of the Aggregate Portfolio Composition

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