Given the nature of technical analysis, i.e. mainly predicting the direction of future exchange rate changes, we derive sign predictions on the basis of the above rational learning process⁴. For expositional purposes, assume for the moment that information about fundamentals is immediately available. If µ₀ < µ₁ implies an exchange rate appreciation, a rational forecasting device signals a future increase at time t > τ if the probability of the regime shift exceeds 0.5, or P_{1, t} > P_{0, t}. With eq. (9) we can reformulate this condition to

(13.1)

or

(13.2)

The forecasting device (13.2) simply states that the exchange rate is expected to rise whenever the average of the t – τ + 1 observations of z is closer to µ₁ than to µ₀. Of course, restricting expectations to the direction of future exchange rate movements is an inferior forecasting strategy when data on fundamentals is available to market participants. Thus, we now assume that observations of z are available only with a lag of δ periods.

At time τ, the exchange rate may respond to an announcement that a switch has occurred. In the following δ periods, the market’s forward-looking expectations remain fixed until time τ + δ due to the lack of new external information. Thus, the exchange rate is driven by contemporaneous realizations of z alone:⁵

(14)

Since the exchange rate is a linear transformation of the unobservable fundamental, technical analysis infers information about z through past movements of the exchange rate. To derive a rational sign prediction at time τ + κ, the Bayesian approach of (13) is now applied to past exchange rates:

(15)

From eq. (15) we find that the exchange rate is expected to rise if the average of the last κ + 1 observations exceeds the average of the pre- and post-shock equilibrium value. The filter rule (15) can be examined further assuming that the market is fully agnostic about the current state of the fundamental so that agents perceive one regime to be as likely as another (P_{1, t} = P_{0, t}). Taking into account that with a rising number of exchange rate observations ( κ), the impact of the fundamental’s innovations cancels out, we find:

(16)

and

(17)

The difference between (16) and (17) thus converges to

(18)

i.e. the extent to which the mean of the exchange rate switches due to contribution of contemporaneous realizations of z. It is important to note that if the regime shift has not occurred, i.e. the mean of z is still µ₀, the difference between the two averages is zero, implying that the filter rule (15) does not signal a switch.

Although this strategy seems to be suitable at first glance, the practical implementation of the moving average rule is not straightforward. First, to make use of a filter rule like (15) it is necessary to calculate

Since the pre- and post-shift means of the underlying process are generally not known by market participants with certainty, agents need an approximation. A solution to this problem is based on the fact that the exchange rate at time t < τ follows eq. (4), implying that = µ₀, and at time t = t the exchange rate follows eq. (14) so that = µ_1. If the probability assigned to the regime shift is close to one then the average of the last ? = 2 d observations is an unbiased approximation for (17). Since, however, the market is uncertain about the regime shift, P1, τ is significantly lower than unity and the span of the average ( ρ) has to be smaller: (19)

We can, therefore, reformulate the filter rule (15) to

(20)

The filter rule (20) represents the so-called moving average trading rule (Murphy, 1999) or oscillator model (Schulmeister, 1987), which is a standard forecasting device of financial market practitioners.

A second problem may arise from the fact that information about the date of the possible regime shift is generally not available. In this case, the span of both the long and the short moving average cannot be exactly determined. Arbitrary values for ρ and δ produce biased results in the sense of Eq. (19), but as long as ρ > δ, a signal can only arise if the regime shift has indeed occurred. Thus, it may be concluded that the performance of trading rules will not significantly vary across different spans of the moving averages, which is empirically examined in the following section.

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4 It is important to mention that we do not propose a trading model of foreign exchange, but instead focus on the information content of technical analysis within the outlined learning model.

5 In eq. (14) the case where the market bets on fads can be considered by introducing µ₀ instead of µ₁.

Prof. Stefan Reitz

Next: Empirical Evidence

Summary: Index