## Applications of ARCH to foreign exchange rate data

The characterization of exchange rate movements, including second-order dynamics, have important implications for many issues in international finance. In addition to international asset pricing theories along the lines discussed in the previous two sections for domestic assets, international portfolio management obviously depends on expected exchange rate movements through time. Several policy-oriented questions relating to the impact of the exchange rate on different macroeconomic variables also require an understanding of the exchange rate dynamics.

### 5.1. ARCH effects and model specification

As for other speculative prices, traditional time series models have not been able to capture the stylized facts of short-run exchange rate movements, such as their contiguous periods of volatility and stability together with their leptokurtic unconditional distributions; see, e.g., Mussa (1979) and Friedman and Vandersteel (1982). As discussed above, the ARCH class of models is ideally suited to modeling such behavior. Whereas stock returns have been found to exhibit some degree of asymmetry in their conditional variances, the two-sided nature of the foreign exchange market makes such asymmetries less likely. In the absence of any structural model for the conditional variances, the linear GARCH(p,q) model in (7) therefore is a natural candidate for modeling exchange rate dynamics.

For example, using daily data on five different nominal U.S. dollar rates, Hsieh (1988a) argues that the conditional distributions of the daily nominal returns are changing through time, as evidenced by highly significant autocorrelations for the squared returns, but that an ARCH(12) model with linearly declining lag structure captures most of the nonlinear stochastic dependencies present; see also Milh0j (1987a), Diebold (1988), and Diebold and Nerlove (1989).27 These findings are corroborated in the later papers by Hsieh (1989a, b) using GARCH(1,1) type formulations.28 Interestingly, judged on the basis of the BDS test for nonlinear dependencies discussed in section 3.1 and the Ljung-Box test for the standardized squared residuals, the simple GARCH(1,1) model does better in describing the data than the ARCH(12) model estimated in Hsieh (1988a). Similar conclusions are reached in the studies by Taylor (1986), McCurdy and Morgan (1988), Kugler and Lenz (1990), and Papell and Sayers (1990).

Of course, as for other speculative prices, it is possible that the significant ARCH effects could be due to misspecified first-order dynamics resulting in dependence in the higher-order conditional moments. However, if such nonlinear dependence is present in the conditional mean it should be exploitable for forecasting purposes. Interestingly, in a detailed nonparamet-ric analysis using locally-weighted regression techniques for ten weekly U.S. dollar exchange rates, Diebold and Nason (1990) find that forecasts based on these nonparametric estimates lead to no improvement in forecast accuracy when compared to the forecasts from a simple martingale model, consistent with the idea that any significant dependencies in short-run exchange rate movements work through the conditional variance and higher even-ordered conditional moments only. Similar conclusions are reached in the studies by

27Tsay (1987), on using a generalization of the time-varying parameter formulation of the standard linear ARCH(g) model as discussed in section 2.1, finds that when allowing for cross-parameter correlations the estimates from this model with weekly data on the British pound/U.S. dollar exchange rate are very close to the results obtained with a conventional linear ARCH(12) model.

Only for the British pound, as analyzed further in Gallant, Hsieh, and Tauchén (1989) using seminonparametric methods, is there any substantial evidence against the GARCH(1,1) model including deterministic vacation effects in the conditional variance as a simple parsimonious representation of the daily nominal rates.

Meese and Rose (1991) and Kim (1989), though Taylor (1990b) surprisingly argues that the conditional mean can be predicted well enough to obtain net trading profits.

While ARCH effects are highly significant with daily and weekly data, both Diebold (1988) and Baillie and Bollerslev (1989) have noted that ARCH effects tend to weaken with less frequently sampled data. For example, in Baillie and Bollerslev (1989) the average Ljung-Box portmanteau test for the first ten autocorrelations for the squared logarithmic first difference of the exchange rates averaged across the six currencies decreases gradually from a highly significant 130.6 for daily data to an insignificant 10.6 for data sampled monthly. This is in accordance with the asymptotic results in Diebold (1986b, 1988), and as shown in Drost and Nijman (1991) the actual parameter estimates obtained by Baillie and Bollerslev (1989) for the GARCH(1,1) models with less frequently sampled data may also be explained by aggregation effects. For most domestic assets the empirical evidence pertaining to temporal aggregation is less clear, possible due to compounding higher-order nonlinear dependencies. A detailed empirical study of these issues across different asset categories seems worthwhile.

### 5.2. Nonnormal conditional densities

While the simple symmetric linear GARCH(1,1) model may provide a good description of the second-order dynamics for most exchange rate series over the post-1973 free float, the assumption of conditional normality does not capture all the excess kurtosis observed in daily or weekly data; see McCurdy and Morgan (1987), Milh0j (1987a), Hsieh (1989a), and Baillie and Bollerslev (1989). As discussed in section 2.3 the resulting QML estimates obtained under the assumption of conditional normality are generally consistent and asymptotically normally distributed but the asymptotic covariance matrix of the parameter estimates will have to be appropriately modified. However, in many applications, including options pricing, a complete characterization of the distribution for the spot rates and not just the conditional variance are of interest.

Following the discussion in the previous sections, several alternative conditional error distributions have consequently been employed in the literature. Baillie and Bollerslev (1989) find that the Student-i distribution compares favorably to the power exponential and captures the excess kurtosis for most of the rates. The Student-? distribution is also estimated by Hsieh (1989a), together with the generalized error distribution, a normal-Poisson, and a normal-lognormal mixture distribution. It is also worth noting the results in Jorion (1988), where the jump-diffusion ARCH(l) model discussed in section 3.2 above is estimated for weekly data on the Deutschemark/U.S. dollar rate for the 1974-1985 period. Based on a standard likelihood ratio test, both the jump process parameters and the ARCH parameters are jointly significant, consistent with the presence of excess kurtosis in the standardized residuals from conventional ARCH models.

In a related context, Lastrapes (1989) finds, not surprisingly, that including dummy variables in the conditional variance to allow for changes in the policy of the FED reduces the degree of leptokurtosis in the standardized residuals.29 Similarly, McCurdy and Morgan (1988) find that departures from conditional normality tend to be associated with a few specific policy events. Further work trying to endogenously determine the timing of major exchange rate movements and changes in regimes would be interesting and could help explain part of the remaining leptokurtosis; see also Engel and Hamilton

5.3. Nonlinear and nonparametric ARCH

As discussed in the previous section, several authors have noted deviations from normality in the standardized residuals from estimated linear GARCH(p,g) models, and successfully proceeded to characterize these deviations by some parametric leptokurtic density. Alternatively, following the discussion in section 2.4, a nonparametric procedure could be employed. This is the approach taken by Gallant, Hsieh, and Tauchen (1989), where the seminonparametric technique of Gallant and Tauchen (1989) is used in estimating a model for the 'recalcitrant' British pound/U.S. dollar rate analyzed in Hsieh (1989a). The leading term in the expansion for the conditional density resembles the conventional linear ARCH model, and contrary to other speculative prices, the response of the conditional variance to negative and positive surprises is virtually symmetric. However, the estimated conditional density has interesting hump-shaped tails. This same shape is also evident in the results reported in Engle and Gonzalez-Rivera

(1991), where nonparametric density estimation is used in characterizing the distribution of the standardized residuals from a GARCH(1,1) model for the same rate and sample period. It is likely that this particular pattern is influenced by a few observations, and therefore peculiar to the given period. In fact, Bollerslev (1987) and Baillie and Bollerslev (1989) on analyzing data for the British pound for 1980-1985, i.e., excluding data from the 1970's, find little evidence against the simple GARCH(1,1) model with i-distributed errors.

5.4. Sources of intermarket and intramarket volatility

Maintaining market efficiency, the pronounced ARCH effects present with high-frequency data could be due to the amount of information or the quality

2lfAlso, the degree of persistence in the conditional variance is diminished.

of the information reaching the market in clusters, or from the time it takes market participants to fully process the information; see, e.g., Diebold and Nerlove (1989) and Gallant, Hsieh, and Tauchen (1989). In order to show that information processing is the source of the volatility clustering, Engle, Ito, and Lin (1990a) define four separate market locations: Europe, New York, Pacific, and Tokyo. If the information arrivals in one market are uncorrelated with the information arrivals in any other market, a test of whether increased volatility in one market causes an increase in volatility in another market is in effect a test of information processing as the source of volatility clustering. The results in Engle, Ito, and Lin (1990a) with intraday observations on the Japanese yen/U.S. dollar rate show that, except for the Tokyo market, each market's volatility is significantly affected by changes in volatility in the other markets, so that volatility is transmitted through time and different market locations as a 'meteor shower', lending support to the information processing hypothesis. Information processing as the main determinant behind the volatility spillovers is also consistent with the evidence reported in Engle, Ito, and Lin (1990b), who rule out the influence of stochastic policy coordination on the basis of equally important volatility spillovers in the early 1980's, a period known for little international policy coordination. Lin (1989) on applying a multivariate factor ARCH model reaches a similar conclusion.

Using hourly data on four major U.S. currencies during the first half of 1986, Baillie and Bollerslev (1991) also examine the causal relationship between returns and volatility. Significant evidence for the 'meteor shower' hypothesis is again evident. Interestingly, however, Baillie and Bollerslev (1991) also report some evidence for market-specific volatility, after taking account of deterministic patterns across the trading day. Furthermore, the volatility during the day is found to exhibit a very distinct and remarkably similar pattern for all four rates, with increases occurring around the opening and closing of each of the three major world markets, i.e., London, New York, and Tokyo. Consistent with the findings in Whistler (1988), the U.S. market is overall the most volatile, followed by the European market.

The implementation and tests of more structural models consistent with the empirical findings discussed above would clearly be of interest and could help in gaining some further understanding about the underlying market micro structure theories at work. The empirical analysis of higher-frequency data, such as the continuously recorded bid and ask quotations described in Goodhart (1990), also hold the promise of important insights along these lines.

### 5.5. Volatility persistence

In accordance with the findings for stock returns and interest rates, the persistence of volatility shocks in the foreign exchange market is also very high. For instance, Engle and Bollerslev (1986), on estimating a GARCH(1,1) model for weekly data on the U.S. dollar vis-a-vis the Swiss franc, finds

+ /3X= 0.996, providing a motivation for the Integrated GARCH, or IGARCH, class of models discussed above. Very similar results are reported in Bollerslev (1987), McCurdy and Morgan (1987, 1988), Hsieh (1988a), Kim (1989), Baillie and Bollerslev (1989), Hsieh (1989a), and Taylor (1990a).30

Even though many different currencies may exhibit IGARCH-type behavior, it is certainly possible that this persistence is common across different rates.31 The presence of such co-persistence among the variances has many important practical implications (e.g., in optimal portfolio allocation decisions involving a trade-off between future expected returns and the associated risk). The empirical relevance of this idea has been illustrated within the context of a bivariate GARCH(1,1) model by Bollerslev and Engle (1990), where it is found that most of the persistence in the conditional covariance matrix for the Deutschemark and the British pound/U.S. dollar rates derives from some common set of underlying forcing variables, and that the corresponding bilateral Deutschemark/British pound rate has much less persistent volatility shocks. In addition to further theoretical work along these lines, extensions of the limited empirical evidence to other currencies and asset categories would be desirable.

### 5.6. ARCH-M models and the risk premium

A growing body of literature has found that the forward rate is not an unbiased predictor of the corresponding future spot rate; see, e.g., Hakkio (1981), Hsieh (1984), Baillie (1989), and McCurdy and Morgan (1991a). Assuming that expectations are rational, a risk premium can reconcile this observation with market efficiency, and several theoretical models have been formulated which generate risk premia in foreign exchange markets.32 Examples include Hodrick and Srivastava (1984), Domowitz and Hakkio (1985), Diebold and Pauly (1988a), and Kendall (1989). According to most of these theories, the risk premium depends on some function of the conditional probability distribution of the future spot rate. Given the evidence in the previous sections pertaining to the time-varying nature of the conditional

■"'Somewhat puzzling, for the hourly GARCH(24,1) models with hourly dummy variables in the conditional variances reported by Baillie and Bollerslev (1991), the estimates indicate much less persistence, with aj +«24 + j3, between 0.374 and 0.771 only.

31 For example, in a study similar to Lamoureux and Lastrapes (1990a), Connolly (1990) finds that using comtemporancous volume in the conditional variance equation for the yen/dollar spot rate tends to decrease the measured persistence, suggesting a common forcing variable for volume and volatility.

32An alternative explanation consistent with market efficiency would be the restriction imposed by limit moves. However, in a detailed empirical analysis, Kodres (1990) finds little support for this hypothesis.

distribution of spot exchange rates, this may therefore result in a time-varying risk premium. Several different specifications and proxies for this risk premium have been used empirically, many of which depend directly on the conditional variance of the spot rate; see Hodrick (1987) for an excellent survey of this literature.

The first attempts by Domowitz and Hakkio (1985) and Diebold and Pauly (1988a) at modeling such a time-varying risk premium in the forward foreign exchange market within a univariate ARCH-M framework were largely unsuccessful. Several explanations for this are possible. For example, the problem of determining who is compensated for risk in an exchange economy might argue against the constancy of the ARCH-M parameter, leading to insignificant results.33 An alternative explanation is that both studies use monthly data, which as noted in section 5.1 generally shows only minimal ARCH effects, thereby leading to insignificant findings.34

Indeed, Kendall and McDonald (1989) on using weekly data for the Australian/U.S. dollar and a GARCH(1,1)-M model obtain a significant estimate for the ARCH-M parameter. Conversely, the results in McCurdy and Morgan (1988) with daily and weekly futures data, and in Kendall (1989) with weekly spot data do not support a significant simple mean-variance tradeoff. However, since the conditional variance merely serves as a proxy for the risk premium, a more structural based multivariate approach is likely to be superior from a theoretical perspective.

Before we turn to a discussion of the implementation of such multivariate models, it is also worth noting the analysis in Hodrick (1989), where an ARCH-M-type model is used to examine how the exchange rate is affected by the uncertainty in the inflation rate, monetary policy, and income growth. A two-step procedure is employed in which the exogenous conditional variances are estimated from a set of linear ARCH(l) models, and subsequently used as regressors to explain the monthly movements in the U.S. exchange rate for Japan, West Germany, and the United Kingdom. As the conditional variance estimates again show little temporal variation on a monthly basis, the results for the formal monetary cash-in-advance model are somewhat disappointing, but holds the promise of important future insight.

31A1so, Frankel (1986) argues that the risk premium must be small because it is determined by the conditional variance of the difference between the change in the spot rate and the forward discount, which is bounded by the unconditional variance. However, as Pagan (1988) points out, this argument is not true if the conditional variance is changing through time. Thus as noted in Frankel (1988), only the average risk premium must be small.

34In contrast, Pagan and Ullah (1988) find strong support for the presence of a time-varying risk premium in the Canadian/U.S. dollar market with monthly data over the earlier time period from 1970-1978. The risk premium here is proxied by a simple linear function of a nonparamet-ric estimate for the conditional variance obtained from a normal kernel; cf. section 2.4. However, this might be driven by the influence of the Quebec crisis.

Of course, from a more technical point of view, the indirect two-step procedure is subject to the same criticism as discussed in section 2.8 above.

### 5.7. Multivariate ARCH models and asset pricing

Several authors have speculated that the weak results that have been found in the foreign exchange market using univariate ARCH-M models to estimate time-varying risk premia might be due to the conditional variances being poor proxies for risk; see, e.g., Domowitz and Hakkio (1985), McCurdy and Morgan (1987, 1988), Diebold and Pauly (1988a), Lee (1988), Thomas and Wickens (1989), and Baillie and Bollerslev (1990). In particular, the premium might be better approximated by a function of the time-varying cross-currency conditional covariances and not just the own conditional variance.

Indirect support for this hypothesis is provided by Lee (1988), who finds that the conditional covariance between the Deutschemark and the Japanese yen/U.S. dollar spot rates, as modeled by a bivariate ARCH(12) model, helps explain the weekly movements in the yen/U.S. dollar rate. The results in Baillie and Bollerslev (1990) with weekly data and a four-dimensional GARCH(1,1) model for the one-month-forward rate forecast error for four European currencies also indicate highly significant contemporaneous correlations. However, the time-varying conditional covariances do not yield any improvement in forecast accuracy beyond the MA(4) correlation structure in the overlapping forward rate forecast errors implied from a simple martingale model for the spot rates.

More formal tests for mean-variance efficiency and alternative pricing formulations have also found their implementation in the foreign exchange market. Attanasio and Edey (1988), Mark (1988), Engel and Rodrigues (1989), and Giovannini and Jorion (1989) all estimate and test specifications of the international CAPM in Frankel (1982), while explicitly allowing for a time-varying conditional covariance matrix. Modeling the temporal dependence in the second-order moments generally leads to significantly better performance of the model and a more precise estimate of the coefficient of relative risk aversion. Nonetheless, both Engel and Rodrigues (1989) using five monthly U.S. exchange rates and Giovannini and Jorion (1989) with weekly data on three U.S. currencies and a stock market index, formally reject the restrictions implied by the CAPM. An alternative structural based approach is taken by Kaminsky and Peruga (1990), who estimate a version of the intertemporal consumption-based CAPM in which the risk premium is a function of the time-varying conditional covariances between the future spot rate and consumption. Using monthly data together with a multivariate ARCH(l) formulation little support for the model is forthcoming. Of course, the model may still provide a good description over shorter time intervals than one month, but the availability of data complicates such an analysis; see McCurdy and Morgan (1991a). In fact, using weekly foreign exchange rates, McCurdy and Morgan (1991b) find evidence of a significant time-varying risk premium in deviations from uncovered interest rate parity, where the risk premium is given by the conditional covariance with a benchmark portfolio set equal to the return on a worldwide equity index.

While the studies discussed above have highlighted the importance of accounting for short-lived temporal variation in both conditional variances and covariances, a completely satisfactory model for the time-varying risk premium in the forward foreign exchange market has yet to be formulated.

5.8. Multivariate ARCH models, policy analysis, and dynamic hedging

Multivariate ARCH models have also been useful in addressing various policy issues related to the foreign exchange market. For instance, Diebold and Pauly (1988b) and Bollerslev (1990) study the effect on short-run exchange rate volatility following the creation of the European Monetary System (EMS). Both studies find an increase in the conditional variances and covariances among the different European rates after the 1979 inception of the EMS. At the same time, Bollerslev (1990), on estimating a multivariate GARCH(1,1) model with constant conditional correlations, argues that the coherence also increased over the EMS period, possibly as a result of the increased policy coordination among the member countries.

In a series of recent papers, the effect of central bank interventions on foreign exchange dynamics have been analyzed in the context of a GARCH formulation by Connolly and Taylor (1990), Humpage and Osterbert (1990), and Mundaca (1990). A common finding across these studies concerns the positive correlation between current intervention and exchange rate volatility. However, further analysis regarding the simultaneous determination of exchange rates and intervention policies seems warranted.

Other macroeconomic motivated applications include Kroner and Lastrapes (1991), who use a multivariate GARCHG, 1)-M model to show that exchange rate uncertainty significantly affects the level and the price of trade in the economy. In a related context, Kroner and Claessens (1991) present a dynamic multiple hedging model based on the intertemporal CAPM in which the optimal hedging portfolio is a function of the time-varying variances and covariances. Using a multivariate GARCH(1,1) model, the optimal debt portfolios for Indonesia are estimated.

Given the substantial increase in international portfolio diversification by many investors and institutions in recent years coupled with the complex second-order dynamics of short-run exchange rate movements, it would be very interesting to extend the analysis in Kroner and Claessens (1991) and Cecchetti, Cumby, and Figlewski (1988), discussed in section 4.3 above, to optimal dynamic hedging strategies for the currency risk involved with direct short-term investment in foreign assets. The results in Kroner and Sultan (1991) pertaining to the yen and Baillie and Myers (1991) for different commodities are encouraging.

## Forex Trading Manual

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