Appendix I: Specification and Estimation of the ARCH-Type Models
Appendix II: Calculation of Asset Returns
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Mr. Temel, a doctoral candidate in applied economics at the University of Minnesota, was a summer intern in 1991 in the Treasurer’s Department. We would like to thank Lawrence K. Duke, Szeina Lurie, Inci Otker, Anthony Richards, Orlando Roncesvalles, George Tavlas, George Tsetsekos, and Michael Wattleworth for helpful comments and suggestions.
Although the estimated coefficient a1 for the average composite index (ACI) is lower than that for the SDR, the sum of the estimated coefficients ai, i=1, …, 5, for the ACI process is greater than the estimated coefficient a1 for the SDR.
This stylized fact is often referred in the relevant literature as “volatility clustering.”
That is, an AR(1) model means that yt depends on yt-1 but not on earlier (yt-2, yt-3, …) observations. An ARCH(1, 1) model means that yt depends on yt-1 but not on earlier (yt-2, yt-3 …) observations, and that ht depends on
In calculating returns, we have omitted capital gains or losses resulting from changes in the prices of bills on the assumption that such short-term securities generally have relatively small price variations over the course of a week.
Standard normal distributions have a kurtosis equal to three. Distributions with a kurtosis higher than three are leptokurtic in that they have more observations in the tails, away from the mean, when compared with a normal distribution.
The Bera-Jarque test statistic is approximately distributed as a central x2 (2) under the null hypothesis of normality in the underlying distribution of returns (see Hendry (1989), pp. 32-33).
The augmented Dickey-Fuller test is a stationarity test, where the coefficient of the lagged dependent variable is tested to see whether it is equal or greater than unity (unit root test). A coefficient that is less than one indicates a stationary time series.
The implied annualized average weekly returns are calculated as
Accordingly, the implied annualized average weekly returns for U.S. dollar investments are 4.84 percent, the Japanese yen investments 11.34 percent, the pound sterling investments 9.30 percent, the French franc investments 11.15 percent, SDR investments 6.56 percent, ECU investments 7.33 percent, the Average Composite index investments 1.23 percent, and gold 0.96 percent, during the period February 1982 to December 1991.
It represents the average share of currencies in total official holdings of foreign exchange reserves for the period 1981-87.