RDP 8906: A Random Walk Around the $A: Expectations, Risk, Interest Rates and Consequences for External Imbalance Appendix A

Here, technical details concerning the estimation of equation (3) are discussed. The results in Table 1 with k = four weeks use the one month forward rate to generate the variable fd_t,k, while the results with k = thirteen weeks use the three month forward rate. The fact that the forward rates are defined for a slightly different time length than the change in the spot rate makes minimal difference for our purposes. For example, for the first regression in Table 1, it amounts to ignoring the difference between s_{t + 28} and s_{t + 30} (with t measured in days). Viewed from time t, s_{t + 30} − s_{t + 28} is very closely modelled as a random variable with zero mean, and so may be included in the error term in equation (3). This timing issue is, however, critical for alternative tests of the efficiency of the foreign exchange market (see Tease, 1988).

Recursive least squares regression of equation (3) with k = four weeks using data beginning on 6 Jan 84 shows that the point estimate of the coefficient β is strongly positive (as large as 17) and unstable up to the beginning of 1985, after which it becomes negative and fairly stable^[42] to the end of the sample (21 Apr 89). Therefore in Table 1, we report regressions starting in Jan 84 and in Feb 85. The latter date is chosen to correspond as closely as possible to Tease (1988).

Using data on the exchange rates of five countries against the US and assuming H₀: α = 0, β = 1, Cumby and Obstfeld (1984) strongly reject the assumption of conditional homoscedasticity for the errors in equation (3) for four of the exchange rates. Applying their test to our problems leads to test statistics of 2.02 and 2.93 for k = four weeks and 1.73 and 0.897 for k = thirteen weeks. The test statistic is asymptotically distributed χ²(2) which has a critical value of 5.99 at the 5% level. Thus, with this test, we cannot reject the null hypothesis of conditional homoscedasticity in all cases.

Footnotes

Despite this fair degree of stability, estimates of equation (3) for sub-periods can produce rather different results – see equation (3) in Table 2. [42]