Efthymios G. Pavlidis, Ivan Paya and David A. Peel 1053sup
r∈[r 0 ,1]ADFr⇒ sup
r∈[r 0 ,1]∫r0WdW
∫r0W^2, (22.77)whereWdenotes Brownian motion andr∈[r 0 ,1]. If the null hypothesis is rejected
then confidence intervals for the parameterφcan be constructed on the basis of
the work of Phillips and Magdalinos (2007) regarding the asymptotic distribution
theory for mildly explosive processes.
Application of these methods to exchange rates and their fundamental deter-
minants as well as forward premia would seem to be of interest. It might also be
interesting to examine the power of their tests against certain nonlinear processes,
given that the forward premium should embody any rational bubble and that it
has been modeled by such processes as discussed in section 22.3.
Another important property of Markovian bubbles was proved initially by Lux
and Sornette (2002). The standard empirical finding is that the distribution of asset
returns belongs to the class of so-called fat-tailed distributions with hyperbolic
decline of probability mass in the tails. They derived the implications of rational
bubbles of the Blanchard–Watson type for the unconditional distribution of prices,
price changes and returns. They proved that the Blanchard–Watson (1982) bubble
exhibited a tail index of less than unity (see, e.g., Koedijket al., 1990; Loretan
and Phillips, 1994; Huismanet al., 2001; Wagner and Marsh, 2005).^62 Yoon (2005)
proved the same result for the Evans (1991) bubble and this property was transferred
to asset returns. In fact, the empirical results for exchange rates typically gener-
ate tail estimates of around 2–6, suggesting the absence of bubbles. However, the
results of Phillipset al.(2006) are relevant here. For instance, the standard ADF test
suggested the absence of bubbles when applied to the full sample of Nasdaq price
data that they considered – February 1973 to June 2005. However, their new test
procedure detects the presence of a bubble in June 1995 continuing until July 2001.
This suggests that one could, in principle, employ the Phillips procedure to indi-
cate the potential presence of bubbles and use the indicated bubble samples to
obtain tail estimates. It would also seem of interest to examine the properties of
the cointegrating residuals using the Phillipset al.(2006) tests as well as their tail
properties, perhaps estimating the cointegrating vector by the Phillipset al.(1996)
FM-LAD estimator, the properties of which seem ideal in this context.
22.6 Exchange rates, economic fundamentals and forecasting
In their landmark paper, Meese and Rogoff (1983a) employed rolling regressions in
order to generate forecasts for the level of the spot rate based on a comprehensive
range of exchange rate determination models: the flexible-price monetary model
(Frenkel–Bilson), the sticky-price monetary model (Dornbusch–Frankel), and the
sticky-price asset model (Hooper–Morton). At the time there was widespread opti-
mism about the potential of monetary models to explain the fluctuations of floating
exchange rates.