Results of the SLR

This section will test whether BAX contracts represent unbiased predictors of future three-month BA rates. A version of equation (5) will be used, modified slightly to use the observed yields on the BAX contracts as the independent variable and the three-month BA rate at the contract’s settlement as the dependent variable. A dummy variable will again be included to adjust for the impact of 11 September 2001. The equation will also contain a term to represent the time to settlement for the contract. This will allow the estimated risk premiums to change as the various contracts move closer to maturity. The specific equation is as follows:

, (6)

where 3mBA_m is the three-month BA rate at the time of the front BAX contract’s settlement, 3mBA_tis the current three-month BA rate at time t, BAX_tis the BAX contract yield at time t, is the risk premium, and d is the time to the contract’s settlement (in days).²³

The equation is estimated using daily data for both the pre- and post-FAD periods to determine whether the predicted change in three-month BA yields (BAX_t-3mBA_t) is an unbiased predictor of the actual change (3mBA_m-3mBA_t). The null hypothesis is again . Table 8 shows the regression results.

22. The volume data are from the Montreal Exchange.

23. Days to settlement are divided by 100 in the estimation. This allows all the independent variables to be of similar magnitude.

3mB A_m–3mB A_t = –α+β₁(BAX_t–3mB A_t) β+ ₂( ) βd + ₃(Θ_sep11) ε+

α

β = 1

The results of the regressions show that, for the entire sample period, the EH cannot be rejected at the 90 per cent confidence level for the first and third contracts, and at the 95 per cent confidence level for the second contract. The value of is significant only for the front contract, indicating that, for it, the risk premium declines as the contract moves closer to its settlement date. The estimate of for the front contract is zero, which shows that the entire risk premium is captured in the estimate of . For the second and third contracts, the value of the coefficients is not significantly different from zero, while the estimates of are all positive and significantly different from zero, which indicates a risk premium for these contracts that does not vary

significantly over the term of the contract. This makes intuitive sense, because pricing of the front

Table 8: BAX Contract EH Regression Results

Independent

instrument d R² SEE White

test^a DW^b Chow a. The White test statistic indicates the presence of heteroscedasticity in the residuals. The p-value (in

parenthe-ses) represents the probability that the null hypothesis of no heteroscedasticity cannot be rejected.

b. The Durbin-Watson statistics are all well below 2, which indicates a serial correlation of residuals. The residu-als are residu-also heteroscedastic. The Newey-West HAC standard-error terms are shown in parentheses and are robust to both the serial correlation and heteroscedasticity of the residuals.

c. The Chow breakpoint test is used to determine whether there is a structural break in the estimated equations. In all cases, the p values are 0.00, which indicates a significant structural break.

d. The p values are shown in parentheses. Values of that are not significantly different from one at the 10 per cent confidence level are indicated by an asterisk.

α β Θ_sep11 β = 1

contract is heavily influenced by the fact that its yield must converge to the three-month BA rate over a relatively short period of time (between 1 and 91 days). The closer to settlement the contract is, the more certainty there is regarding its final value. For the second and third contracts, their longer time to settlement (92 to 181 days for the second and 182 to 273 days for the third) make them less influenced by this convergence. The predictive power of these contracts is relatively strong, with adjusted R² values ranging from 47 per cent to 73 per cent.

The results for the subperiods are much less satisfactory. The estimates of are very volatile, with the EH being rejected in both subperiods for the front contract, in the pre-FAD period for the second contract, and in the post-FAD period for the third contract. This behaviour is likely due to a small-sample-size problem and large expectational errors (given the longer term of these assets).

The path of interest rates within the two subperiods helps to demonstrate this problem. As Figure 10 shows, the pre-FAD period (to the left of the vertical line) is marked by generally rising interest rates, with BAX yields climbing from approximately 3 per cent to 6 per cent. In the post-FAD period, on the other hand, rates fall rapidly from 6 per cent to 2 per cent. It is possible that market expectations consistently underestimate the degree of tightening in the pre-FAD period, and consistently underestimate the degree of easing in the post-FAD period.²⁴ This is indicative of a small-sample-size problem, and over an entire interest rate cycle the directional forecast errors net out.

Figure 10: BAX Yields

24. Market commentary over these periods is consistent with the proposition that the pace of changes in the overnight rate is in excess of that expected by market participants.

β

It is also possible that the shift to the FAD regime has less of an impact on the BAX market than it does on the shorter-maturity BA and treasury bill markets. The pricing of the one- and three-month instruments examined earlier is very sensitive to the specific timing of changes in the overnight rate. For a one-month asset, the exact day of a change in the overnight rate makes a significant impact on the pricing of the instrument. The BAX contracts, however, measure

expectations over a longer horizon (from six to twelve months). This makes them less sensitive to the actual timing of the moves than to the general overall trend of monetary policy.

The full-period results of the regressions that are shown in Table 8 do support the hypothesis that, over an entire interest rate cycle, the front three BAX futures contracts are rational estimators of future three-month BA rates. The EH cannot be rejected at the 90 per cent confidence level for the first and third contracts, nor at the 95 per cent level for the second contract. These results are less robust than those obtained using one- and three-month assets, and it is possible that the maturity spectrum has moved far enough along that the time-varying component of the risk premium has become significant. Nevertheless, the results are sufficiently strong that they warrant the inclusion of the BAX contracts into the expectations model. The additional information content, combined with the fact that these contracts are widely used by market participants as a means of hedging and speculating on future changes in the overnight rate, outweighs the increased uncertainty generated by moving further along the time-to-maturity spectrum.

Given that the BAX contracts will be included in the expectations model, the regression equations are re-estimated to provide estimates of the values of the various risk premiums. Since the EH cannot be rejected, the equations are re-estimated with the value of set to 1.0. As well, for the second and third contracts, the coefficient is set to zero. Table 9 shows the resulting term premium estimates.

Table 9: BAX Term Premium Estimates

Contract Term premium estimate^a (relative to three-month BA)

a. Standard-error estimates appear in parentheses.

Total term premium^b

b. The total term premium consists of the term premium estimated for each BAX con-tract relative to the three-month BA rate, plus the estimated three-month BA term pre-mium (11 basis points).

Front 0.11 * (days to settle) (0.07)

The initial term premium values represent the premiums for the various BAX contracts relative to the value of a three-month BA at the time of the contract’s settlement, whereas the total term premium values are relative to the overnight target. Figure 11 shows an estimated range of values (plus/minus one standard error) for the term premium on the various instruments.

Figure 11: Term Premium Ranges

As Figure 11 shows, the estimates for the values of the term premiums range from very small and stable (one-month BAs, which have an estimate of 4 basis points and a standard error of 0.5 basis points) to fairly large and uncertain (the third BAX contract, which has an estimate of 39 basis points and a standard error of almost 6 basis points). Clearly, as the time horizon is extended, measures of expectations become increasingly uncertain. Estimates obtained using one- and three-month BAs appear to be quite precise. Once BAX contracts are introduced, these estimates become increasingly uncertain.