1050 The Econometrics of Exchange Rates
They set out a battery of diagnostic tests for the model, some of which we discussed
in section 22.2. The empirical results reported from employing daily data for the
Swedish and Norwegian krone suggest the STARTZ model provides a parsimonious
representation of the behavior of the Swedish krone between 1985 and 1991, and
for the Norwegian krone between 1989 and 1990. The estimates accord with the
theoretical models. We suggest that it would be useful to apply the STARTZ model to
the daily datasets examined by Taylor and Iannizzotto. It would also be interesting
to find out the properties of standard unit root or fractional tests from a simulated
STARTZ model, where one expects that such tests will exhibit low power.
22.5 Speculative bubbles
22.5.1 Theory
Consider the discrete time stochastic differential equation that occurs in asset
market exchange rate models (see Engel and West, 2005):
st=( 1 −b)a′ 1 xt+ba′ 2 xt+bEtst+ 1 , b∈(0, 1). (22.61)The above equation states that the exchange rate depends upon the current level
of economic fundamentalsxtplus the discounted expected spot rate next period,
wherebis the discount factor. In the absence of rational bubbles, the forward
solution to the above equation is:
st=( 1 −b)Et⎛
⎝
∑∞j= 0bja 1 xt+j⎞
⎠+bEt⎛
⎝
∑∞j= 0bja 2 xt+j⎞
⎠. (22.62)The logarithm of the exchange rate can be written as the discounted sum of current
and expected future fundamentals, such as interest rates, prices, money supplies
and income. This is a general form of several exchange rate determination models
based on macroeconomic fundamentals that can provide several insights concern-
ing the empirical findings of studies on exchange rate forecasting discussed in
section 22.6. By assuming Cagan-style money demand functions for the home and
foreign countries with common parameters, we obtain:
st=mt−m∗t+yt−γ(wt−w∗t)+λ(it−i∗t), (22.63)wheremtis the log of the money supply,wtis the log of real income,itis the
short-term interest rate,γandλdenote the income elasticity and interest rate
semi-elasticity of money demand, andytis the real exchange rate. An asterisk
denotes foreign quantities. The deviations from UIP,rpt=it−i∗t−Et
(
st+ 1)
, can
be thought of as unobserved fundamentals. Substituting into (22.63) yields:
st=mt−m∗t+yt−γ(wt−w∗t)+λrpt+λEt(
st+ 1). (22.64)