RDP 8603: Risk Premia, Market Efficiency and the Exchange Rate: Some Evidence Since the Float 3. Econometric Methodology

To maximise the sample size, overlapping data are used. That is, data on spot rates and 15-day, 30-day and 90-day forward rates are sampled weekly. The econometric consequences of adopting this sampling procedure can be demonstrated by the following model. Consider the forecasting equation

where Χ_t is a vector of elements of Φ_t and β is a vector of parameters.

To test for rationality, (9) can be estimated as

where u_t+n = Z_t+n − E(Z_t+n|Φ_t), is the forecast error which becomes observable at period t+n.

Using data sampled more finely than the forecast interval n will result in serially correlated errors. In particular, it will be found that

and

This result obtains because the future values Z_t+1, Z_t+2, … Z_t+n−1 are unobservable at period t, the time the forecast is made. Consequently, the corresponding forecast errors u_t+n−k = Z _t+n−k − E(Z_t+n−k|Φ_t−k) for k=1, 2…n−1 are unobservable and are thus not elements of Φ_t. Since u_t+n−k for k ≤ n−1 are not elements of Φ_t it is possible that they are correlated with u_t+n. Because of this serial correlation, OLS estimation of (9'), will yield consistent coefficient estimates, but inconsistent estimates of the standard errors.

To overcome this problem, Hansen and Hodrick (1980) estimate β by OLS and follow Hansen (1979) to estimate a consistent asymptotic covariance matrix.^[6]

Hansen (1979) shows that

Hansen and Hodrick (1980) show that a consistent estimate of θ is obtained from

where is a T×T symmetric matrix with non-zero elements being the serial covariances of the OLS residuals. The results reported in the following section will follow this procedure.

Footnote

GLS is inappropriate in (9′) since variables typically contained in Χ_t are not strictly exogenous. GLS estimates of β thus do not satisfy the orthogonality condition resulting in inconsistent estimates. Hansen and Hodrick (1980 p 833) note that their procedure is not fully efficient but is computationally more tractable than alternative procedures. [6]