15.2 Estimating the coefficients of an exchange rate basket
In this section we compare the adaptive estimator with standard procedures which
have been designed to cope with time varying regressor coefficients.
A simple solution to this problem consists in applying a window estimator, i.e.
an estimator which only uses the most recent
observations:
 |
(15.7) |
where the value of
is specified by the practitioner.
Another, more refined technique, consists in describing the coefficients
as an unobserved stochastic process:
,
see Elliot et al. (1995).
Apart from the cases when there is some knowledge about the data generating process
of
, the most common specification is as a multivariate random walk:
 |
(15.8) |
In this context, equations (15.8) and
(15.1) can be regarded as a state space model,
where equation (15.8) is the state equation (the signal)
and equation (15.1) is the measurement equation and it
plays the role of a noisy observation of
. A Kalman
filter algorithm can be used for the estimation, see Cooley and Prescott (1973).
The Kalman filter algorithm requires the initialization of two variables:
and
and its recursions read as follows, see Chui and Chen (1998):
The question of the initialization of the Kalman filter will be discussed in the
next section together with the Thai Baht basket example.
In the notation above, the index
denotes the estimate performed
using all the observation before time
(forecasting estimate), while
refers to the estimate performed using all the observations up to
time
(filtering estimate).
The four estimators described above: the adaptive, the recursive, the window and the
Kalman filter Estimator are now applied to the data set of the Thai Baht basket. For
deeper analysis of these data see Christoffersen and Giorgianni (2000) and Mercurio and Torricelli (2001).
15.2.1 The Thai Baht basket
An exchange rate basket is a form of pegged exchange rate regime
and it takes place whenever the domestic currency can be expressed as
a linear combination of foreign currencies. A currency basket can be therefore
expressed in the form of equation (15.1), where:
is set constantly
equal to one and is
taken as numeraire,
represents the home currency exchange rate with respect
to the numeraire, and
is the amount of currency
per unit of currency
, i.e. the
cross currency exchange rate. The above relationship usually holds only on the average,
because the central bank cannot control the exchange rate exactly, therefore the
error term
is added.
Because modern capital mobility enables the investors to exploit the interest
rate differentials which may arise between the domestic and the foreign currencies,
a pegged exchange rate regime can become an incentive to speculation and eventually
lead to destabilization of the exchange rate, in spite of the fact that its purpose
is to reduce exchange rate fluctuations, see Eichengreen et al. (1999).
Indeed, it appears that one of the causes which have led to the Asian crisis of 1997
can be searched in short term capital investments.
From 1985 until its suspension on July 2, 1997 (following a speculative
attack) the Bath was pegged
to a basket of currencies consisting of Thailand's main trading
partners. In order to gain greater
discretion in setting monetary policy, the Bank of Thailand neither
disclosed the currencies in the basket nor the weights. Unofficially, it was known
that the currencies composing the basket were: US Dollar,
Japanese Yen and German Mark.
The fact that the public was not aware of the values of the basket weights, also
enabled the monetary authorities to secretly adjust their values in order to react to
changes in economic fundamentals and/or speculative pressures.
Therefore one could express the USD/THB exchange rate in the following way:
This exchange rate policy had provided Thailand with a good stability of the
exchange rate as it can be seen in Figure 15.3.
During the same period, though, the interest rates had maintained constantly higher
than the ones of the countries composing the basket, as it is shown in Figure 15.4.
Figure:
Interest rates time series: German (thick dotted line), Japanese
(thin dotted line), American (thick straight line), Thai (thin straight
line).
XFGbasket.xpl
|
This facts suggest the implementation of a speculative strategy, which consists in borrowing
from the countries with a lower interest rate and lending to the ones with an
higher interest rate.
A formal description of the problem can be made relying on a mean-variance hedging approach, see
Musiela and Rutkowski (1997). The optimal investment strategy
is obtained by the minimization of the quadratic cost function below:
The solution is:
for  |
|
It can be seen that, when the interest rates in Thailand
are sufficiently high with respect to the foreign interest rates
the following inequality holds
 |
(15.9) |
This means that an investment in Thailand is cheaper than an investment
with the same expected revenue in the countries composing the basket.
In the empirical analysis we find out that the relationship (15.9)
is fulfilled during the whole period under investigation for any of the four
methods that we use to estimate the basket weights.
Therefore it is possible to construct a mean self-financing strategy
which produces a positive expected payoff:
- at time
- borrow the portfolio
from the countries composing the basket,
- lend
to Thailand,
- invest the difference
in the numeraire currency at the risk-free rate
,
- at time
- withdraw the amount
from Thailand,
- pay back the loan of
,
- keep the difference.
The expression for the profit and for its expected value are:
15.2.2 Estimation results
For the implementation of the investment strategy described above one needs the
estimate of the, possibly time-varying, basket weights. The precision of the estimation has
a direct impact on the economic result of the investment. Therefore,
we compare four different estimators of the basket weights: the adaptive, the recursive,
the window and the Kalman filter estimator using economic criteria
for a one month and for a three month investment horizon.
In particular we compute the
average expected profit and the average realized profit.
Figure 15.5:
Estimated exchange rate basket weights: adaptive
(straight line), recursive (thine
dotted line), window (thick dotted line).
|
The adaptive estimation procedure requires three parameters:
,
and
.
The choice of
does not influence the results
very much and it can be reasonably set to
. This value represents the minimal amount of data
which are used for the estimation, and in the case of a structural break, the minimal delay before
having the chance of detecting the change point.
The selection of
and
is more critical. These two values determine the sensitivity
of the algorithm. Small values would imply a fast reaction to changes in the regressor
coefficients, but but they would also lead to the selection of intervals of homogeneity which
are possibly too small.
Large values would imply a slower reaction and consequently the selection of intervals which can be
too large.
To overcome this problem we suggest the following approach.
The main idea is that small changes in the values of
and
should not
affect the estimation results. Therefore
we restrict our attention on a set
of
possible pairs
.
In the present context we chose all the even number between
and
:
Then we compare the
pairs with the following criterion at each time
:
Finally, we estimate the value of
with the selected pair
.
The appeal of the above selection criterion consists of the fact that it leads to the
choice of the pair
which has provided the least quadratic hedging
costs over the past trading periods.
Notice that in general we have different results depending
on the length of the forecasting horizon: here one and three month. Figure 15.5 shows
the results for the three month horizon. It is interesting to see that the adaptive estimate
tends to coincide with the recursive estimate during
the first half of the sample, more or less, while during the second half of the sample it tends to follow the
rolling estimate.
We remark that the problem of selecting free parameters is not specific to the adaptive estimator.
The window estimator requires the choice of the length of the window:
, while
the Kalman filter needs the specification of the data generating process of
and the determination of
and
. In this application
is set equal to 250,
and
are estimated recursively from the data using the OLS, while
and
are initialized using the first 350 observations
which are then discarded. We remark that this choice is consistent with the one of
Christoffersen and Giorgianni (2000).
Table 15.2:
Summary statistics of the profits.
ONE MONTH HORIZON |
Recursive |
Window |
KF |
Adaptive |
Average Expected Profits |
.772 |
.565 |
.505 |
.553 |
Average Realized Profit |
.403 |
.401 |
.389 |
.420 |
Standard errors |
(.305) |
(.305) |
(.330) |
(.333) |
THREE MONTH HORIZON |
Recursive |
Window |
KF |
Adaptive |
Average Expected Profits |
1.627 |
1.467 |
1.375 |
1.455 |
Average Realized Profit |
1.166 |
1.141 |
1.147 |
1.182 |
Standard errors |
(.464) |
(.513) |
(.475) |
(.438) |
|
Table 15.2 shows the result of the simulated investment.
The investments are normalized such that at each trading day we take
a short position of 100 USD in the optimal portfolio of the hard
currencies.
The result refers to the period April 9 1993 to February 12 1997
for the one month horizon investment and June 7 1993 to February 12
1997 for
the three month horizon investment.
Notice first
that the average realized profits are
positive and, as far as the three month investment
horizon is concerned, they are significantly larger than zero
among all methods. This provides a clear evidence for the fact that arbitrage profits were
possible with in the framework of the Thai Bath basket for the period
under study.
The comparison of the estimator also show the importance of properly accounting for
the time variability of the parameters. The recursive estimator shows modest result
as far as the realized profits are concerned and the largest bias between expected
the realized profit. On one side, the bias is reduced by the window estimator and by
the Kalman filter, but on the other side these two methods provide a worse performance
as far as the realized profit are concerned. Finally, the adaptive estimator appears
to be the best one, its bias is much smaller than the one of the recursive estimator
and it delivers the largest realized profits for both investment horizons.