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against the rationality of the survey expectations. In addition, the slope estimates typically exceed 1 indicating that survey respondents evidently place too much weight on the forward rate when predicting the future spot. That is, an increase in the forward premium predicts that the survey forecast will exceed the future spot rate.

2.Estimates of β2 are generally insigniÞcantly di erent from 1. This suggests that survey respondents do not believe that there is a risk premium in the forward foreign exchange rate. Respondents use the forward rate as a predictor of the future spot. They are putting too much weight on the forward rate and are forming their expectations irrationally in light of the empirically observed forward rate bias.

We should point out that some economists are skeptical about the accuracy of survey data and therefore about the robustness of results obtained from the analyses of these data. They question whether there are su cient incentives for survey respondents to truthfully report their predictions and believe that you should study what market participants do, not what they say.

6.5The ‘Peso Problem’

On the surface, systematic forecast errors suggests that market participants are repeatedly making the same mistake. It would seem that people cannot be rational if they do not learn from their past mistakes. The ‘peso problem’ is a rational expectations explanation for persistent and serially correlated forecast errors as typiÞed in the survey data. Until this point, we have assumed that economic agents know with complete certainty, the model that describes the economic environment. That is, they know the processes including the parameter values governing the exogenous state variables, the forms of the utility functions and production functions and so forth. In short, they know and understand everything that we write down about the economic environment.

In ‘peso problem’ analyses, agents may have imperfect knowledge about some aspects of the underlying economic environment. Like applied econometricians, rational agents have observed an insu cient number of data points from which to exactly determine the true structure of the economic environment. Systematic forecast errors can arise as a small sample problem.

A Simple ‘Peso-Problem’ Example.

The ‘peso problem’ was originally studied by Krasker [87] who observed a persistent interest di erential in favor of Mexico even though the nominal exchange rate was Þxed by the central bank. By covered interest arbitrage, there would also be a persistent forward premium, since if i is the US interest rate and i is the Mexican interest rate, it −it = ft −st < 0. If the Þx is maintained at t + 1, we have a realization of ft < st+1, and repeated occurrence suggests systematic forward rate forecast errors.

Suppose that the central bank Þxes the exchange rate at s0 but the peg is not completely credible. Each period that the Þx is in e ect, there is a probability p that the central bank will abandon the peg and devalue the currency to s1 > s0 and a probability 1 −p that the s0 peg will be maintained. The process governing the exchange rate is

p > 0, we will observe the sequence of systematic, serially correlated, but rational forecast errors

If the forward exchange rate is the market’s expected future spot rate,

we have a rational explanation for the forward premium bias. Although (119) the forecast errors are serially correlated, they are not useful in predict-

ing the future depreciation.

on Et(st+1) which is a function of the currently available information set, It. Since ft is the only exogenous variable and the model is linear, it is reasonable to conjecture that the solution has form

Now you need to determine the coe cients π0 and π1 that make (6.35) the solution. From (6.34), the one-period ahead forecast of the fundamentals is, Etft+1 = δ0 + ft. If (6.35) is the solution, you can advance time by one period and take the conditional expectation as of date t to get

Substitute (6.35) and (6.36) into (6.33) to obtain

In order for (6.37) to be a solution, the coe cients on the constant and on ft on both sides must be equal. Upon equating coe cients, you see that the equation holds only if π0 = λδ0 and π1 = 1. The no bubbles solution for the exchange rate when the fundamentals follow a random walk with drift δ0 is therefore

A possible regime shift. Now suppose that market participants are told at date t0 that the drift of the process governing the fundamentals may have increased to δ1 > δ0. Agents attach a probability p0t = Prob(δ = δ0|It) that there has been no regime change and a probability p1t = Prob(δ = δ1|It) that there has been a regime change where It is the information set available to agents at date t. Agents use new information as it becomes available to update their beliefs about the true drift. At time t, they form expectations of the future values of the fundamental according to

= λ(δ1 − δ0)[p1t+1 − p1t] + δ1 + vt+1 − [δ0 + (δ1 − δ0)p1t].

The regime probabilities p1t and the updated probabilities p1t+1 − p1t are serially correlated during the learning period. The rational forecast error therefore contains systematic components and is serially correlated, but the forecast errors are not useful for predicting the future depreciation. To determine explicitly the sequence of the agent’s belief probabilities, we use,

Bayes’ Rule: for events Ai, i = 1, . . . , N that partition the sample space S, and any event B with Prob(B) > 0

P(Ai)P(B|Ai) P(Ai|B) = PN P(Aj)P(B|Aj).

j=1

To apply Bayes rule to the problem at hand, let news of the possible

regime shift be released at t = 0. Agents begin with the unconditional (121) probability, p0 = P(δ = δ0), and p1 = P(δ = δ1). In the period after the announcement t = 1, apply Bayes’ Rule by setting B = (∆f1), A1 = δ1,

A2 = δ0 to get the updated probabilities

192CHAPTER 6. FOREIGN EXCHANGE MARKET EFFICIENCY

The updated probabilities p0t = P(δ0|∆ft, . . . , ∆f1) are called the posterior probabilities. An equivalent way to obtain the posterior probabilities is

How long is the learning period? To start things o , you need to specify an initial prior probability, p0 = P(δ = δ0).11 Let δ0 = 0, δ1 = 1, and let v have a discrete probability distribution with the probabilities,

We generate the distribution of posterior probabilities, learning times, and forecast error autocorrelations by simulating the economy 2000 times. Figure 6.3 shows the median of the posterior probability distribution when the initial prior is 0.95. The distribution of learning times and autocorrelations is not sensitive to the initial prior. The learning time distribution is quite skewed with the 5, 50, and 95 percentiles of the distribution of learning times being 1, 14, and 66 periods respectively. Judging from the median of the distribution, Bayesian updaters quickly learn about the true economy. Since the forecast errors are serially correlated only during the learning period, we calculate the autocorrelation of the forecast errors only during the learning period. The median autocorrelations at lags 1 through 4 of the forecast

11Lewis’s approach is to assume that learning is complete by some date T > t0 in the future at which time p0,T = 0. Having pinned down the endpoint, she can work backwards to Þnd the implied value of p0 that is consistent with learning having been completed by T .