In a relatively simplistic version of competitive exchange markets with no transaction costs, assuming that economic agents are risk neutral and that the information is used rationally, the foreign exchange market efficient hypothesis implies that the forward exchange rate is an unbiased and efficient predictor of future spot exchange rates. Therefore, the forward rate forecast error is random and unrelated to the information set available to market participants at the time expectations are formed.

However, empirical studies have concluded that the forward rate is often a biased predictor of future spot rates. A number of models have attributed this bias to the risk premium; that is, currency traders will demand a high return for holding the currency. Hence, market participants may not be risk neutral but risk averse. They will exploit a profit opportunity resulting from a difference between their expected future spot rate and the current forward rate only if the expected return is sufficiently large.

In what follows we sketch the basic ideas behind the unbiased forward exchange rate hypothesis. Let s t be the log spot exchange rate, f k t the log k-periods forward exchange rate (domestic price of the foreign currency). Agents’ risk neutrality implies that there is not risk premium, that is

Recall that is the expected excess return from forward exchange speculation, where E t ∙ is the conditional expectation of the spot rate at time t + k formed on the basis of the information available at time t and f k , t p = f k , t - s t is the k-period forward premium. The assumption that agents use rationally all the available information implies no gains for speculators on the expected returns, hence

Thus, the traditional approach to test the foreign exchange market efficiency is based on equations (1) and (2),

s t + k = α + β f k , t + ε k , t (3)

where α and β are the parameters, and ε k , t is the error term⁴.

Previous to the surge of co-integration modeling framework, concerns that non-stationary spot and forward rate would lead to the wrong inference in OLS regressions let some researchers to induce stationarity by differencing them; see for instance Bilson (1981), Longworth (1981), Hansen and Hodrick (1983), Fama (1984) and Huang (1984). The stationary version of the unbiased hypothesis states that the k-period forward exchange premium f k , t p provides an unbiased forecast of the future spot exchange rate, that is, . This hypothesis is tested empirically using the following regression equation,

s t + k - s t = α + β f k , t p + ε k , t (4)

Given the assumption of risk neutral agents and perfectly competitive markets, the foreign exchange market efficient hypothesis implies the following joint hypothesis for both equations (3) and (4):

H 0 : α = 0   a n d   β = 1

According to Fama’s market efficiency’s forms, if H 0 is true then the unbiased forward rate hypothesis holds in its strongest form. The assumption of rational expectations implies that α = 0 . Additionally, if the forward premium, f k , t p , is an unbiased predictor of future changes in the spot rate, that is, if the equilibrium condition between forward and spot holds, the estimates of β should not be significantly different from one. But instead of finding β = 1 , the empirical coefficients obtained are not only systematically below unity, but even predominately negative.

We now recall to the model we shall use to test the efficient market hypothesis. In an efficient market, historic forward rate forecast errors s t + k - f k , t should not be useful for explaining current forward rate forecast errors. Hansen and Hodrick (1980) studied this issue by pointing out that the hypothesis implies that the forecast error is uncorrelated with information available at time t. That is, for the joint hypothesis H 0 : α = 0   a n d   β = 1 , equation (2) implies that the forecast error s t + k - f t , k is orthogonal to the information set that is available at the time the expectation is made. This proposition is often tested empirically using the following regression equation,

s t + k - f k , t = β 0 + ∑ i β i s t - f k , t - i + ε t + k (5)

where the forecast error is made a function of past forecast errors. No significant values of β 0 and the β i combined with a white noise error process ε t + k would suggest that the efficient market hypothesis holds in the short run. In summary, the exchange market will be efficient in its weak-form when the spot and forward rates are co-integrated. If the factor of co-integration equals 1, then the hypothesis holds in the semi-strong definition of market efficiency. Finally, the strong-form follows, in addition to the conditions for the previous forms, that the forecast error must be a white noise process.

In this section, we briefly describe the econometric models used in the empirical analysis. First, we consider Hansen and Hodrick’s (1980) model and label it “Model A”

s t + k - f k , t = β 0 + β 1 s t - f k , t - 1 + β 2 s t - 1 - f k , t - 2 + ε t + k ;     ε t I t - 1 ~ N 0 , σ 2 (6)

where I t is the information set available to agents at time t. In this equation, we regress the forecast errors on a constant and lagged forecast errors. Note that Model A assumes that the conditional variance is a constant. However, as it is well known the conditional variance of many economic and financial time series are time-varying.

The null hypothesis for testing the efficient market hypothesis in Model A is therefore,

H 0 :   β 0 =   β 1 =   β 2 = 0

As noted previously, Clarida et al (2003), Spagnola et al (2005) and Chen (2010), favor the use of a nonlinear Markov switching model because it is better fit to test the market efficiency hypothesis when the market has faced several disturbing shocks which may induce time varying coefficients.

Markov-Switching has become one of the most popular nonlinear time series modeling technique. Roughly speaking, it involves multiple structures that characterize the time series behavior during different regimes. By allowing the model to switch between these structures, this representation is able to capture relatively complex dynamic patterns. An important feature of this kind of model is that the switching mechanism is controlled by an unobservable state variable that follows a first-order Markov chain structure. The Markovian property regulates the process in such a way that the current value of the state variable depends on its immediate past value. As such, a given structure may prevail for a random period of time, and it is replaced by another structure when switching takes place.

Hence, “Model B” is a two-state Markov switching model that differentiates between the state in which the unbiased forward exchange rate hypothesis is fulfilled and the state in which it is not. To complete the description of the Markov-Switching model we point out that the unobservable realization of the regime w t ∈ 1,2 , is governed by a discrete-time, discrete-state Markov stochastic process, which is defined by transition probabilities as follows:

p i j = Pr ⁡ w t + 1 = j w t = i ,       ∑ j = 1 2 p i j = 1       f o r   a l l       i , j   ∈ 1,2 (7)

where p i j denotes the probability that state i will be followed by state j, and these are collected into a transition probability matrix P given by

The model is useful to make probabilistic inferences about the unobserved state w t based on estimates of the transition probabilities p i j . Two types of inference can be made: (i) about the smoothed probability, Pr ⁡ w t = j I N , which is the probability of being in state j based on the entire observed information set, and (ii) about the filtered probability, denoted as Pr ⁡ w t = j I t , which is the best guess about w t inferred from information in the sample data up to time t < N. Therefore, Model B is defined as follows:

s t + k - f k , t = β 0 w t + β 1 w t s t - f k , t - 1 + β 2 w t s t - 1 - f k , t - 2 + ε t + k ;     ε t I t - 1 ~ N 0 , σ w t 2 (9)

where β i w t = β i 1   i f   w t = 1 ,   a n d   β i w t = β i 2   i f   w t = 2 ,   f o r   i = 0,1 , 2 while σ w t 2 = σ 1 2   i f   w t = 1 and σ w t 2 = σ 2 2   i f   w t = 2 .

For the model specification test, where H 0 s s is the single state (SS) null hypothesis, we follow Engel and Hamilton (1990) to prove the following hypothesis:

H 0 s s : β 01 = β 02 ,     β 11 = β 12 ,     β 21 = β 22 ,     σ 1 2 ≠ σ 2 2

If we cannot reject H 0 s s , then it is implied that the true data generating process comes from a single state, as opposed to two states. The test statistics for H 0 s s hypothesis is the Wald statistic, and it has the χ ( 3 ) 2 .

For testing the hypothesis of market efficiency in the two-state Markov switching model, the hypothesis for state 1 and that for state 2 are:

H 0 w 1 : β 01 =   β 11 =   β 21 = 0

H 0 w 2 : β 02 =   β 12 =   β 22 = 0

The Wald statistics for testing H 0 w 1   o r   H 0 w 2 is as follows

where R is a 3x3 restriction matrix; β ^ w = β 0 w , β 1 w ,   β 2 w T and Θ is the asymptotic variance.

The data set used in our empirical analysis consists of weekly data on the natural logarithm for the Mexican/US foreign exchange spot rate and forward rate, namely those for the 30 and 90 days forward rates. Following the covered interest rate parity, forward rates for the 30 and 90 days were valued the Thursday of each week. For the domestic risk free instruments we use the 28 and 90 days secondary market interest rates of Mexican certificates of deposit (CETES); while for the foreign risk free instruments, the four weeks Treasury Bills and the three month Treasury Bills were used. The spot rate and CETES are from the Mexican Central Bank statistics (http://www.banxico.org.mx) and Treasury Bills are from the US-Federal Reserve Bank of ST. Louis statistics (https://fred.stlouisfed.org). Sample periods were determined on the availability of data. Therefore, data for the 90-day forward rate run from March 28, 2002 to January 5, 2017; while data for the 30-days forward rate run from January 31, 2002 to January 5, 2017.

In table 1 we present the estimated regressions of the forecast error on a constant and two lagged forecast errors. We also present hypothesis tests for model selection; model generating process originated by a single state versus two states and market efficiency in the two-state Markov switching model. All numerical computations were carried out with WinRATS version 9.1 (www.estima.com)

As we can observe from table 1, the Likelihood Ratio (LR) test for model selection indicates that the two-state Markov switching model is preferable to the one state linear regression model, for both, the 30 and 90 day forward rate. As Krolzing (1997) argues, there is no general test to compare two models with different number of regimes. The issue is that the asymptotic theory cannot be used here because there are unidentified nuisance parameters as well as violation of the non-singularity conditions. However, most researchers still use the LR to obtain useful supporting evidences. Throughout this paper, the LR tests are considered in this way

Moreover, the test results for H 0 s s : β 01 = β 02 ,   β 11 = β 12 ,   β 21 = β 22 ,   σ 1 2 ≠ σ 2 2 (the true data generating process comes from a single state) indicates that, again, the two-state Markov switching model is preferable to the single-state model.

Next, we test the efficient market hypothesis ( H 0 , H 0 w 1 and H 0 w 2 ). For the linear Model A, we reject the efficiency of the markets for both, the 30 and 90 days forward rate at the 5% level. We now turn to the results from Model B, the two-state Markov switching model. For the 30-days forward rate, the parameters estimates in state 1, the constant and the two lagged forecast errors are significantly equal to zero, but in state 2 the constant and the two lagged forecast error are significantly different from zero. The χ 2 statistics for the 30-days forward rate in Model B are 6.824 with the p-value equal to 0.1864 in state 1, while 63.938 with the p-value equal to 0.000 in state 2. Therefore, we cannot reject the null hypothesis of efficiency in state 1, but we reject the hypothesis in state 2 at the 1% level. The results suggest that state1 is the efficient state and state 2 is the inefficient state.

Finally, as shown in Table 1, for the 90-day forward rate, the χ 2 statistics indicates that the efficient market hypothesis is rejected at the 1% level, for state 1 and 2, which indicates that both of the states are inefficient, implying that the lagged forecast error have explanatory power to predict the current forecast error for the 90-day forward rate.

Table 1

Numbers in parenthesis are standard errors for parameter estimates, while p-values for testing hypothesis. * denotes significance at the 5% level