In the BS and LS model, it is assumed that agents form their expectations rationally, that capital is perfectly mobile and that if any risk premium exists, it is negligible. Given these assumptions, the interest rate differential equals the expected change in the exchange rate
2.1) i - i* = Et (ds) / dt
that is Uncovered Interest Parity holds.
However, as it is assumed that the target zone is not fully credible, the expected change of the exchange rate, lit (ds) / dt, is the sum of two terms: first, the expected change of the exchange rate within the band, lit (dx) / dt, where x = s - c and c is the central parity, and second the expected rate o f realignment, Et (dc) / dt. So
2.2) Et <ds) / dt s lit (dx) / dl + Et (dc) / dt
and. given the assumption that UIP holds
2.3) i - i* = lit (dx) / dt + lit (dc) / dt
BS observe that equation 2.3) implies that in order to get an estimate of the expected realignment, it is sufficient to get an estimate of the change of the exchange rate within the band, and then subtract it from the interest rate differential. Svensson (1991), however, observes that such estimation is made difficult by the fact that usually the exchange rate within the band, x, jumps at the time of a realignment. This difficulty is overcome by conditioning the expected exchange rate within the band upon no realignment and hence rewriting 2.3) as follows:
where gt, the expected rate of devaluation, is the product of the probability of a realignment per unit of time, vt , and the expected devaluation size conditional upon realignment, Et (dc I r) + Et (dx I r) - Et (dx I nr)
such that
2.4) gt = vt [ Et (dc I r) + Et (dx I r) - Et (dx I nr)]
(for details see Svensson (1993)).
Even before testing the hypothesis that the expected devaluation follows a random walk, it is possible to argue against it. The argument which follows, as well as our empirical analysis, is conducted in a discrete time framework, while the theoretical analysis is developed in a continuous time framework. The approximation is acceptable, however, if one uses high frequency data, as we do in this paper.
Given that the Brownian motion process has its counterpart in discrete time in the random walk (with or without drift), it is our purpose to show that, on the basis of the assumptions made in the BS and ES models, the expected rate of devaluation, defined in equation 2.4), cannot follow such a process.
The argument can be divided into three steps:
i) Unless the exchange rate follows an integrated process of order greater than one, the first difference is a stationary process. The empirical literature usually shows that the exchange rate, both spot and forward, follows an integrated process of the first order (see Meese and Singleton 1982, Meese and Rogoff 1983, Baillie and Bollerslev 1989, Baillie and McMahon 1989, lie Vries 1994) and hence, by definition, its first difference is a stationary process.
ii) If the realised changes of the exchange rale (both total and inside the band) are stationary, also their expected counterparts are, if we assume, as BS and ES do, that the market is efficient. Indeed, according to the hypothesis of rational expectations
(where lt is the information set available at time t), i.e. the realised value of the exchange rate at time t+j, st+j, differs from its expected value for an error, ul+j, which follows a white noise process if the horizon of the expectations j matches the sampling frequency of the data; however, if the frequency of the data is finer than the expectation horizon, then ut+j is autocorrelated and follows a moving average process, which is, by definition stationary. Given this, the order of integration of the realised series has to match the order of integration of the expected series. If this is true, then, as the exchange rate follows at most an 1(1) process, also the expected future exchange rate does and the first difference is a stationary process.
It is then possible to state that both Et(lJs) and Et(Dx) follow an 1(0) process:
2.6) Et(Ds)~ 1(0), Et(Dxlnr)~l(0)
where Ds = S(+ i - st and Dx = xl+ | - xl
iii) f rom the discrete time counterpart o f equations 2.2), 2.3) and 2.3') it also follows that:
2.7) Et(Dxlnr) + gt ~ 1(0)
This is possible only if either Et(l>xlnr) and gt are both l( I) and coinlegrated or if they are both 1(0). Hut we have shown above that Et(l)xlnr)~ 1(0), hence gt cannot be an integrated process of the first order.
We show empirically that gt is not an I( 1) process in the remainder of the chapter, by exploiting the implication that, conditional on the validity of the assumptions staled above (i.e. that agents are rational and that the risk premium is negligible), the order of integration of the expected rate of devaluation has to match the order of integration of the interest rate differential (remembering that the expected change of the exchange
rale within the band, conditional upon non-realignment, cannot be an integrated process) in order to show that gt is actually a stationary process.
Other studies have recently tried to test directly for the order of integration of the expected rate of devaluation (see Lindberg, Soderlind and Svensson (1993) and Rose and Svensson (1991)). The procedure followed in order to estimate the devaluation risk has been proposed by Bertola and Svensson (1993); it has been defined drift- adjustment method', as the interest rale differential is adjusted by the drift of the exchange rate inside the band in order to get an estimate of the expected devaluation. The procedure involves the estimation of the expected change of the exchange rate inside the band, Lt(dx)/dt, which is then subtracted from the interest rate differential in order to get the expected rale of devaluation. Although in principle the relation between the expected change of the exchange rate inside the band and the current exchange rate is non-linear, BS suggest that a linear approximation can be acceptable for reasonable parameter values. It then follows that, according to the authors, the expected change of the exchange rate inside the band can be estimated by applying standard econometrics techniques (for applications of this method, see Lindberg, Soderlind, Svensson 1993, Svensson 1993, Rose and Svensson 1991).
In order to test for the order of integration of the devaluation expectation, Lindberg, Soderlind and Svensson (1993) and Rose and Svensson (1991) apply unit root tests to the series obtained by implementing the ’drift adjustment method'.
This procedure has the weakness that there is no agreement on how to measure expected future exchange rates inside the band, while the results of the tests do in fact depend heavily on how the expected rate of devaluation is estimated. In contrast, in this chapter we lest for the order of integration of the expected rale of devaluation by using the interest rale differential, which can be observed directly. It must be noted that while the theoretical analysis is developed in a continuous time framework, the empirical analysis is conducted in a discrete lime framework. The approximation is however acceptable if one uses high f requency data, as we do here.