1072 The Econometrics of Exchange Rates
takes the form:
Bt= b
b+θ
EtBt+ 1 −(^1 −θ)b
b+θ
Et− 1 Bt+(^1 −θ)b
b+θ
Bt− 1.
Forθ=1 we obtain the bubble solution as in (22.68) above. IfEt+ 1 Bt=δBtthen the
analysis can proceed as above. The magnitude ofδ>1 could be important as to whether
the bubble is asymptotically stationary (see Yoon, 2005).
- In the latter case, the form of the bubble depends on the time series process followed by
the fundamentals. For the geometric process:
Bt=csβf,tlnsf,t−lnsf,t− 1 =μ+ωt,
whereωt∼N(0,σh^2 )andμis a drift term,βis the solution ofβμ+0.5β^2 σh^2 +lnλ=0 and
cis an arbitrary constant. Minford and Peel (2002) provide a textbook treatment, while
Bidarkota and Dupoyet (2007) extend the analysis.
- Diba and Grossman (1988) tested for bubbles by applying unit root tests to the asset price,
real stock prices and dividends. If the fundamental is an integrated process, sayI( 1 ), then
from (22.67) the bubble will imply rejection of cointegration. The important insight of
Evans was to show via simulation that standard unit root and cointegration tests have
little power to detect his periodically collapsing bubble. Yoon (2005) demonstrates that
Evans bubbles tail indices are less than one, a property we comment on below. Gurkaynak
(2005) has a useful survey of many of the empirical tests for bubbles.
- Phillipset al.(2006) demonstrate that Evans bubbles with aπas low as 0.25 may be
detectable.
- Note that, for comparison, the tail index for the studentt-distribution is its degrees of
freedom.
- The latter justification is in line with recent empirical evidence which suggests that the
relationship between the exchange rate and the fundamental value is characterized by
significant nonlinearities (Taylor and Peel, 2000; Tayloret al., 2001).
- However, spurious regression problems may rise even at short horizons. Fersonet al.(2003)
decompose asset returns into expected returns and an unpredictable noise component. In
this setting, although the dependent variable is not a persistent process, the possibility of
persistent expected returns and explanatory variables may lead to spurious results.
- Hence, Mark’s conjecture that the mixed evidence on long-horizon predictability is due
to the small sample size is not supported by the data.
- Abhyankaret al.(2005) adopt a different approach which utilizes the realized end-of-
period wealth so as to evaluate forecasts based on monetary fundamentals. Their findings
suggest that there is evidence of economic value to exchange rate predictability, especially
at long horizons.
- Four hundred tests are implemented since four models, four bilateral exchange rates, five
forecast horizons and five test statistics are considered.
- Aq-value is defined as the minimum possible false discovery rate for which the null
hypothesis is rejected (Storey, 2003). The false discovery rate is the ratio of the num-
ber of tests for which we reject the null,F, over the total number of tests,S, given that
the null is true,E(F/S). A detailed description of the computation ofq-values and their
properties is provided in McCracken and Sapp (2005) and the references therein.
- The particular ESTAR specification remains the same under the null and the alternative,
and is determined in the preliminary analysis on the basis of the goodness-of-fit, the sig-
nificance of the coefficients and residual diagnostics (Eitrheim and Teräsvirta, 1996). The
bootstrap procedure is similar to the VECM bootstrap of Kilian (1999) and was described
in section 22.6.1.1.
- Taylor and Kilian (2003) conduct Monte Carlo experiments to investigate the power and
size properties of the tests. Their findings indicate that in-sample tests have substantially
higher power than out-of-sample tests.
