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 $ k$ observations:

$\displaystyle \widehat\theta_t = \left(\sum_{s = t - k}^t X_s X_s^{\top}\right)^{-1}\sum_{s= t-k}^t X_sY_s,$ (15.7)

where the value of $ k$ is specified by the practitioner. Another, more refined technique, consists in describing the coefficients $ \theta $ as an unobserved stochastic process: $ (\theta_t)_{t\in \mathbb{N}}$, see Elliot et al. (1995). Apart from the cases when there is some knowledge about the data generating process of $ \theta_t$, the most common specification is as a multivariate random walk:

$\displaystyle \theta_t = \theta_{t-1} + \zeta_t \qquad \zeta_t\sim \textrm{N}(0, \Sigma).$ (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 $ \theta_t$. 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: $ \widehat\theta_{0\vert} $ and $ P_{0\vert} = \mathop{\hbox{Cov}}(\widehat\theta_{0\vert})$ and its recursions read as follows, see Chui and Chen (1998):

$\displaystyle \left\{ \begin{array}{lcl} P_{0\vert} & = & \mathop{\hbox{Cov}}(\...
... t-1} + G_t(Y_t - X_t^{\top} \widehat\theta_{t\vert t-1} ). \end{array} \right.$    

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 $ t\vert t-1$ denotes the estimate performed using all the observation before time $ t$ (forecasting estimate), while $ t\vert t$ refers to the estimate performed using all the observations up to time $ t$ (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: $ X_{1,t}$ is set constantly equal to one and is taken as numeraire, $ Y_t$ represents the home currency exchange rate with respect to the numeraire, and $ X_{j,t}$ is the amount of currency $ 1$ per unit of currency $ j$, 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 $ \varepsilon _t$ 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:

$\displaystyle Y_{USD/THB,t}= \theta_{USD,t} + \theta_{DEM,t} X_{USD/DEM,t} + \theta_{JPY,t} X_{USD/JPY,t} +
\sigma\varepsilon_t.
$

This exchange rate policy had provided Thailand with a good stability of the exchange rate as it can be seen in Figure 15.3.

Figure: Exchange rate time series. 30337 XFGbasket.xpl
\includegraphics[width=1.1\defpicwidth]{dm.ps} \includegraphics[width=1.1\defpicwidth]{yen.ps} \includegraphics[width=1.1\defpicwidth]{tb.ps}

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). 30342 XFGbasket.xpl
\includegraphics[width=1.1\defpicwidth]{1mr.ps} \includegraphics[width=1.1\defpicwidth]{3mr.ps}

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 $ \xi_1^*,\ldots,\xi_p^*$ is obtained by the minimization of the quadratic cost function below:

$\displaystyle <tex2html_comment_mark>1320 E\left\{\left(Y_{t+h} - \sum_{j=1}^{p}\xi_{j} X_{j,t+h} \right)^2\bigg\vert\mathcal{F}_t \right\}.$    

The solution is:

$\displaystyle \xi_j^*= E(\theta_{j,t+h}\vert\mathcal{F}_t)$    for $\displaystyle j =1,\ldots,p.$    

It can be seen that, when the interest rates in Thailand $ (r_0)$ are sufficiently high with respect to the foreign interest rates $ (r_j,\; j=1,\dots,p)$ the following inequality holds

$\displaystyle (1+r_0)^{-1} Y_t < \sum_{j=1}^{p} (1+r_j)^{-1}E(\theta_{j,t+h}\vert\mathcal{F}_t) X_{j,t}.$ (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: The expression for the profit and for its expected value are:
$\displaystyle \Pi_{t+h}$ $\displaystyle =$ $\displaystyle Y_{t+h} - \sum_{j=1}^{p} E(\theta_{j,t+h}\vert\mathcal{F}_t)
X_{j,t+h}$  
    $\displaystyle + (1+r_1)
\left(\sum_{j=1}^{p} (1+r_j)^{-1} E(\theta_{j,t+h}\vert\mathcal{F}_t)X_{j,t}-
(1+r_0)^{-1} Y_t\right)$  
$\displaystyle E(\Pi_{t+h}\vert\mathcal{F}_t)$ $\displaystyle =$ $\displaystyle (1+r_1)
\left(\sum_{j=1}^{p} (1+r_j)^{-1} E(\theta_{j,t+h}\vert\mathcal{F}_t)X_{j,t} - (1+r_0)^{-1}
Y_t\right).$  


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).
\includegraphics[width=1.1\defpicwidth]{wus.ps} \includegraphics[width=1.1\defpicwidth]{wdm.ps} \includegraphics[width=1.1\defpicwidth]{wyen.ps}

The adaptive estimation procedure requires three parameters: $ m$, $ \lambda$ and $ \mu$. The choice of $ m_0$ does not influence the results very much and it can be reasonably set to $ 30$. 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 $ \lambda$ and $ \mu$ 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 $ \lambda$ and $ \mu$ should not affect the estimation results. Therefore we restrict our attention on a set $ \mathcal{S}$ of possible pairs $ (\lambda,\,\mu)$. In the present context we chose all the even number between $ 2$ and $ 8$:

$\displaystyle \mathcal{S} = \{(\lambda,\,\mu)\vert\;\lambda,\,\mu \in\,\{2,\,4,\,6,\,8\} \}$    

Then we compare the $ 16$ pairs with the following criterion at each time $ t$:

$\displaystyle (\lambda^*, \mu^*) = \underset{(\lambda,\mu)\in\mathcal{S}}{\arg\...
...
\left(Y_{s} -
\sum_{j=1}^{d}\widehat\theta_{j,s\vert s-h} X_{j,s} \right)^2.
$

Finally, we estimate the value of $ \widehat\theta_{t+h\vert t}$ with the selected pair $ (\lambda^*, \mu^*)$. The appeal of the above selection criterion consists of the fact that it leads to the choice of the pair $ (\lambda,\,\mu)$ 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: $ k$, while the Kalman filter needs the specification of the data generating process of $ \theta_t$ and the determination of $ \Sigma$ and $ \sigma$. In this application $ k$ is set equal to 250, $ \Sigma$ and $ \sigma$ are estimated recursively from the data using the OLS, while $ \widehat\theta_{0\vert} $ and $ P_{0\vert}$ 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.