RDP 8602: Short-Term Interest Rates, Weekly Money Announcements and Rational Forecasts 5. The Interest Rate Equations

Robert G. Trevor

May 1986

Given the forecasting equations estimated for Periods 1, 23 and 45, one needs first to determine the appropriate estimation periods for the interest rate equations, and then to determine whether the models should be extended to include the revision (the “R” models). The F-tests for the relevant hypotheses are presented in Tables 6 and 7.

Table 6: Breaks in Interest Rate Equations1
      Federal Funds Rate T-Bill Rate
H0 m, k F2 FR3 A2 AR3 S2 F2 FR3 A2 AR3 S2
df1     17 17 17 17 17 17 17 17 17 17
1=12 101, 14 39.18* 33.87* 40.73* 34.42* 40.06* 7.38* 4.92* 7.35* 4.67* 7.64*
23=3 98, 14 .58 .93 .65 .94 .62 .64 .81 .53 .67 .71
df1     3 17 3 17 3 3 17 3 17 3
1=23 216, 28 2.16° 1.28 2.62° 1.54° 1.53 3.88* 1.70* 4.21* 1.29 7.06*
3=4 133, 28 1.43 .86 .05 .68 .29 7.24* 2.65* 1.24 1.55° 1.95
23=4 150, 28 1.79 .91 .11 .70 .30 7.22* 2.40* 1.26 1.45 1.62
4=5 103, 28 4.34* 1.24 2.16° 1.17 3.31* 4.15* 2.70* .95 1.87* 1.55

Notes: 1. The entries are F statistics for testing the null hypothesis that the coefficients of each equation are the same in the two periods indicated. An * (°) denotes rejection of the null hypothesis at the 5% (10%) level of significance, respectively.
2. Under the null, these statistics have an F-distribution with df1 and m degrees of freedom.
3. Under the null, these statistics have an F-distribution with df1 and (m-k) degrees of freedom.
Table 7: Role of Revisions1
    Federal Funds Rate T-Bill Rate
    Model AR Model FR Model AR Model FR
Period df2 H08 H09 H08 H09 H08 H09 H08 H09
1 87 .49 1.06 .43 1.06 .73 .35 .36 .36
23 101 1.30 1.07 1.05 1.03 1.93° 1.06 2.70* .62
4 21 .61 .39 .45 .41 1.21 .81 1.22 .66
5 54 .78 .21 .87 2.54*2 .33 .28 .63 .732

Notes: 1. The entries are F statistics for testing the relevant null hypothesis. Under the null, these statistics have an F-distribution with 7 and df2 degrees of freedom for H08, and 7 and (df2+8) degrees of freedom for H09. H08 is the hypothesis that the surprise parts of the revisions are jointly orthogonal, H09 is the hypothesis that the expected parts of the revisions are jointly orthogonal. An * (°) denotes rejection of the null hypothesis at the 5% (10%) level of significance, respectively. (These tests are independent.)
2. These statistics are biased towards the rejection of the null hypothesis, and have 7 and df2 degrees of freedom.

Table 6 suggests that breaks occur between Periods 1 and 2, Periods 3 and 4, and Periods 4 and 5.[34] Surprisingly, no breaks are detected between Periods 2 and 3 (when MI was replaced by MI B) for any of the models. Thus the results in Table 7 are presented for Periods 1, 23, 4 and 5. The only definite rejection of the null hypothesis that the revisions play no role in the interest rate equation, occurs for the T-Bill rate in Period 23, when Model FR is preferred to Model F. In all cases, Model F is the appropriate one, although the results in Models A and S will also be presented for purposes of comparison.

Looking again at Table 6, the breaks in the preferred models can be evaluated. For the T-Bill rate, Model F exhibits a break between Period 1 and Period 23, as do Models A and S. Hence, there was a significant change in the announcement effect following the Fed's move to a nonborrowed reserves policy.[35] Given that this was a move away from a policy of targeting the Federal Funds rate, it is a little surprising that the break between Period 1 and Period 23 in the equation for this interest rate is only significant at the 10% level.

The T-Bill rate equation also exhibits breaks between Periods 23 and 4 and between Periods 4 and 5. The latter is explained by the Fed's switch from a nonborrowed reserves policy to one focusing on borrowed reserves. The break at the end of Period 3 is a little puzzling. As in the case of the forecasting equations discussed above, the most likely explanation is that agents interpreted the “renaming” of the MI aggregate as confirmation of a policy shift to a broader money measure. It is interesting to note that neither of the other models (A or S) detect either of these breaks in the announcement effect on the T-Bill rate.

The Fed's switch to borrowed reserves targeting (at the beginning of Period 5) also has a significant impact on the announcement effect for the Federal funds rate. This break is also detected by Model S, but not by Model A, at the 5% level of significance.

More information on the nature of these breaks in the interest rate equations is given by the parameter estimates presented in Table 8.[36] The change in Fed policy at the beginning of period 2 lead to breaks in Model F for both interest rates. Table 8 shows that it produced a sizeable increase in the announcement effect in both cases. The break that occurs in Model F at the end of Period 3 seems to be related to the role of the revisions and the expected change in the money supply. More will be said about this when the parameter estimates for the preferred model (Model FR) are presented.

Table 8: Parameter Estimates1
Model&
Period 		      Federal Funds Rate   T-Bill Rate
  n-k   α β γ R2 dw   α β γ R2 dw
F 1
 		   101
 		   −.005
[.018]		 .018
[.029]		 .014
[.028]		 .01
 		 1.94
 		   .016
[.019]		 .090*
[.031]		 .012
[.030]		 .08*
 		 1.73
 
F 23
 		   115
 		   .028
[.079]		 .421*
[.112]		 .087
[.093]		 .12*
 		 2.49*
 		   −.008
[.040]		 .293*
[.056]		 .168*
[.047]		 .27*
 		 2.18
 
F 4
 		   35
 		   .089
[.127]		 .189*
[.074]		 −.114
[.109]		 .19*
 		 2.56°
 		   .068
[.096]		 .129*
[.054]		 −.103
[.082]		 .20*
 		 1.80
 
F 5
 		   68
 		   .095
[.054]2		 .013
[.017]		 −.025
[.033]2		 .02 1.61°   .007
[.030]2		 .018°
[.009]		 −.001
[.018]2		 .05
 		 2.02
 
A 1
 		   101
 		   .004
[.009]		 .001
[.007]		 − .001
[.006]		 .00
 		 1.90
 		   .023*
[.010]		 .017*
[.007]		 −.002
[.006]		 .05°
 		 1.70°
 
A 23
 		   115
 		   .066
[.067]		 .115*
[.033]		 .008
[.037]		 .10*
 		 2.43*
 		   .066°
[.036]		 .091*
[.017]		 .013
[.020]		 .20*
 		 1.91
 
A 4
 		   35
 		   .021
[.100]		 .111*
[.045]		 −.013
[.043]		 .15°
 		 2.67*
 		   .019
[.074]		 .059°
[.034]		 −.031
[.031]		 .11
 		 1.76
 
A 5
 		   68
 		   .056*
[.027]		 .019
[.013]		 −.004
[.012]		 .03
 		 1.69
 		   .006
[.018]		 .022*
[.007]		 −.101
[.008]		 .16*
 		 2.08
 
S 1
 		   101
 		   .010
[.011]		 .004
[.006]		 −.007
[.007]		 .01
 		 1.89
 		   .038*
[.011]		 .018*
[.006]		 −.012
[.008]		 .09*
 		 1.68°
 
S 23
 		   115
 		   .056
[.066]		 .085*
[.030]		 .001
[.062]		 .07*
 		 2.43*
 		   .079*
[.031]		 .097*
[.014]		 −.087*
[.029]		 .31*
 		 2.09
 
S 4
 		   35
 		   −.014
[.080]		 .106*
[.036]		 −.052
[.039]		 .24*
 		 2.79*
 		   −.006
[.060]		 .055*
[.027]		 −.059°
[.029]		 .20*
 		 1.85
 
S 5
 		   68
 		   .049°
[.025]		 .017
[.012]		 −.007
[.014]		 .03
 		 1.73
 		   −.002
[.013]		 .023*
[.006]		 −.019*
[.007]		 .20*
 		 2.05
 

Notes: 1. Estimated standard errors are given in square brackets. An * (°) next to a parameter estimate or R2 denotes rejection of the null hypothesis that the corresponding population value is zero at the 5% (10%) level of significance, respectively. An * next to a Durbin Watson (dw) statistic denotes rejection of the null hypothesis of no autocorrelation at the 5% level of significance; a ° indicates that the dw statistic falls within the inconculsive region.
2. These estimated standard errors are biased towards zero.

Table 6 showed that the second change in Fed policy, at the beginning of Period 5, produced a significant break in both interest rate equations for Model F. Examination of the parameter estimates shows that the effect of the announcement on these rates changed back to an effect of similar magnitude to that present under the original Fed policy during Period 1. This is in direct contrast to the results obtained from the standard model (Model S) for the T-Bill rate. Model S suggests that the expected change predicted by the survey had a separate impact on this interest rate in Period 5. comparison with Model A indicates that this is probably due to the result that the survey is not a rational predictor of the announced changes.[37]

The results for the preferred model, Model F, suggest that the null hypothesis of market efficiency (i.e., that α and γ are both zero) cannot be rejected, except in the case of the T-Bill equation for Period 23 when Model FR is the preferred model. The relevant estimates for Model FR are,

The estimates of β and γ are similar to those obtained from Model F and reported in Table 8. However, the unexpected components of the just announced revisions to the announced change in the money supply published three, four and six weeks earlier, also have an effect on the T-Bill rate in this period. None of the expected components of the revisions are significantly different from zero. However, the estimate of γ (the coefficient of EΔM) is significant. The hypothesis (H010) that all the expected values have zero coefficients, is rejected at the 5% level of significance by the appropriate F-test. This suggests that either the T-Bill market was inefficient during this period, that there is other information that needs to be taken into account when estimating the forecasting equations, or that a Type 1 error has occurred.

A comparison of the parameter estimates for Models A and S with those for Model F is interesting. For the Federal Funds rate, there is little difference in the parameter estimates of Models A and S. Model F, however, shows that the impact of the Fed's change to a nonborrowed reserves policy was much larger than would otherwise be detected. (Notice that the Period 1 announcement effect is zero in each model. Now compare the models for Period 23.)[38] For the T-Bill equations, Model S rejects the hypothesis of market efficiency for three of the four estimation periods.[39]

The most noticeable difference between Model F and the others is the estimates of β for both interest rates for Period 23. Table 9 shows that this is not simply due to the relative magnitudes of the final, announced and survey changes. The final and survey data are of a similar order of magnitude and the announced data are a lot noisier. However, there is nothing special about the relative variance sizes for Period 23. This suggests that the Fed's change from targeting the Federal Funds rate to a nonborrowed reserve policy, had a much larger effect on financial markets than has previously been appreciated from examination of the announcement effect.

Table 9: Money Supply Measures
    Final Announced Survey
Period obs mean variance mean variance mean variance
1 104 .551 .872 .324 4.374 .727 1.522
23 118 .473 1.594 .426 6.821 .263 1.118
4 38 .558 3.020 .555 8.573 .053 4.092
5 71 1.0501 2.5611 .966 7.401 .646 2.761

Notes: 1. These statistics are calculated for the truncated part of period 5. The number of observations is reduced by 57.

Furthermore, the Model S results suggest that if the previous studies had tested for parameter change rather than comparing point estimates, they would only have detected a change in the T-Bill equation in response to the first change in Fed policy. Model F, on the other hand, suggests that breaks occurred in response to the shift to the “new” MI aggregate, as well as to the second change in Fed operating targets.

Footnotes

The left most column of this table gives the hypothesis being tested. For example, “H0: 1=23” is the hypothesis that the parameters are the same in Period 1 as they are in Period 23. The overlap between the periods on each side of the equality in the first two rows of the table, (e.g., 1=12) indicates a Chow test due to insufficient degrees of freedom for separate estimation. [34]

Loeys (1985) is the only other study to provide tests for parameter change. He also detected a significant break in the announcement effect on the T-Bill rate following this policy change. However, his tests are based on a variant of Model S where the constant term (α) is constrained to be the same in each period, and the expectation term (γ) is constrained to be zero. These assumptions produce biases in addition to those involved in using Model S. While they have not performed hypothesis tests, other authors have obtained results that have a similar pattern to the ones described here. [35]

It has already been shown that Model FR is preferred in Period 23. These estimates will be presented separately. Those for Model F for this period and for Models A and S for all periods are presented for comparative purposes. The presence of serial correlation in the residuals of some of the equations for the Federal Funds rate presents a problem. However, under the hypothesis of market efficiency, they can not simply be re-estimated with lagged values of the interest rate as explanatory variables. [36]

Although, somewhat paradoxically, Table 5 suggests that it was "more" of a rational predictor in this period than in the earlier periods. [37]

This is consistent with the familiar econometric result that measurement errors tend to bias parameter estimates towards zero. [38]

Similar results for market efficiency were obtained (using Model S) by Cornell (1985), Falk and Orazem (1985), Gavin and Karamouzis (1984), Grossman (1981), Roley (1983) and Urich and Wachtel (1981). [39]