RDP 8602: Short-Term Interest Rates, Weekly Money Announcements and Rational Forecasts Appendix

Robert G. Trevor

May 1986

ANALYSIS IN GROWTH RATES

Table 1: Estimation Periods
Period n Start End Reasons for Break from Previous Period
1 104 10/13/77 10/04/79  
2 17 10/11/79 01/31/80 Fed switched to nonborrowed reserves
3 101 02/08/80 01/08/82 Switch to MIB, release day now Fridays
4 38 01/15/82 10/01/82 Definition changed to “new” MI
5 71 10/08/82 02/10/84 Fed switched to borrowed reserves
Table 2: Lagged Revisions in Forecasting Equations1
Period df1 df2 FA2,3 F3 A R1 R2 R3 R4 R5 R6 R7
1 14 74 .19 .12 .85 .50 .78 .37 1.01 .19 .22 .25
23 14 88 .62 .56 .63 .46 .46 .46 .67 .52 .43 .57
45 14 79 1.22 1.06 1.64° 1.48 1.48 .98 .82 1.85* .82 .82

Notes: 1. These are the F statistics for testing the null hypothesis that the coefficients of the revisions, lagged once and twice, are zero in each of the forecasting equations for each period indicated. Under the null, these statistics have an F-distribution with df1 and df2 degrees of freedom. An * (°) denotes rejection of the null hypothesis at the 5% (10%) level of significance, respectively.
2. These statistics have df1 and (df2–8) degrees of freedom.
3. These statistics are not calculated for the truncated part of period 45. For this period, the second degree of freedom is reduced by 57.
Table 3: Breaks in Forecasting Equations1
H0 df1 df2 FA2 F A R1 R2 R3 R4 R5 R6 R7
1=12 17 88 .403 .33 1.17 2.79* 1.51 .15 .19 .07 .15 5.13*
23=3 17 85 1.043 .51 .39 2.48* .70 .10 .15 .13 .11 3.30*
1=23 16 190 .65 1.22 2.49* .94 .41 .31 .82 .32 .28 .14
3=4 16 107 2.03* .88 .71 .37 1.71 3.75* 3.32* 1.19 .91 1.93*
23=4 16 124 1.79* .85 .66 .43 1.86* 4.37* 3.92* 1.27 1.04 1.49
4=5 16 77 1.264 .555 1.45 .70 1.32 1.48 1.53 .93 1.40 1.69°
23=45 16 195 1.85*6 .916 1.05 .70 2.09* 4.35 3.95* 1.98* 1.53° 1.88*

Notes: 1. These are the F statistics for testing the null hypothesis that the coefficients of each forecasting equation are the same in the two periods indicated. Under the null, these statistics have an F-distribution with df1 and df2 degrees of freedom. An * (°) denotes rejection of the null hypothesis at the 5% (10%) level of significance, respectively.
2. These statistics have (df1+8) and (df2–16) degrees of freedom.
3. The first degree of freedom is 17.
4. This statistic is not calculated for the truncated part of period. It has degrees of freedom 14 and 14.
5. This statistic is not calculated for the truncated part of period 5. It has degrees of freedom 14 and 22.
6. These statistics are not calculated for the truncated part of period 5. The second degree of freedom is reduced by 57.
Table 4: Forecasting the Revisions1
Period R1 R2 R3 R4 R5 R6 R7
1 R2
dw		 .10
2.02		 .17
1.76°		 .07
2.07		 .14
2.01		 .05
2.01		 .04
2.04		 .04
2.04
23 R2
dw		 .19°
2.09		 .06
1.94		 .08
2.12		 .11
2.00		 .17
2.10		 .15
2.05		 .03
2.41°
45 R2
dw		 .19
2.01		 .34*
2.14		 .44*
2.29°		 .45*
2.14		 .36*
1.94		 .46*
2.20°		 .47*
2.24°

Notes: 1. An * (°) next to an R2 denotes rejection of the null hypothesis that the explanatory power of the equation 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 inconclusive region.
Table 5: Forecasting the Money Supply1
        F: Final before announcement2,4   A: Announced2
Period   df2   R2 dw H01 H02 H03   R2 dw H01 H02 H03
  df1       16 15 14       16 15 14
1   88   .11 2.66* 7.76* 8.03* .42   .58* 1.86 2.09* 1.71° 1.71°
23   102   .31* 2.05 9.52* 9.90* 3.33*   .46* 1.97 2.11* 2.20* 2.02*
45   93   .41 2.29° 3.85* 3.70* 1.02   .58* 2.06 1.82° 1.63° 1.73°
        FA: Final after announcement3,4  
Period   df2   R2 dw H04 H05 H06 H07  
  df1       24 23 15 14  
1   80   .22 2.81* 14.01* 14.39* .25 .23  
23   94   .50* 2.13 21.25* 22.17* 2.93* 1.70°  
45   28   .79* 2.11 5.64* 5.88* 1.41 1.12  

Notes: 1. An * (°) next to an R2 denotes rejection of the null hypothesis that the explanatory power of the equation 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 inconclusive region. The remaining entries are F statistics for testing the relevant null hypothesis. Under the null, these statistics have an F-distribution with df1 and df2 degrees of freedom. An * (°) denotes rejection of the null hypothesis at the 5% (10%) level of significance, respectively.
2. H01 is the hypothesis that the survey provides a rational expectation of the final or announced change, H02 is the hypothesis that the survey provides an efficient forecast, and H03 is the hypothesis that all other information is orthogonal.
3. H04 is the hypothesis that the announcement provides a rational expectation of the final change, H05 is the hypothesis that the announcement provides an efficient forecast, H06 is the hypothesis that all other information is orthogonal, and H07 is the hypothesis that all information other than the announcement and the survey is orthogonal.
4. These statistics are not calculated for the truncated part of period 45. For this period, the second degree of freedom is reduced by 57.
Table 6: Breaks in Interest Rate Equations1
    Federal Funds Rate T-Bill Rate
H0 m, k F2 FR3 A2 AR3 S2 F2 FR3 A2 AR2 S2
df1     17 17 17 17 17 17 17 17 17 17
1=12 101, 14 39.63* 34.00* 40.76* 34.48* 40.15* 7.43* 5.13* 7.37* 4.86* 7.66*
23=3 98, 14 .58 .94 .66 .95 .62 .65 .77 .53 .65 .72
df1     3 17 3 17 3 3 17 3 17 3
1=23 216, 28 2.81* 1.54 3.36* 1.84* 1.93 5.12* 2.00* 5.21* 1.58° 8.79*
3=4 133, 28 1.30 .88 .05 .75 .32 6.48* 2.45* 1.20 1.54° 1.62
23=4 150, 28 1.62 .92 .11 .77 .34 6.40* 2.25* 1.22 1.46 1.27
4=5 103, 28 4.29* 1.19 2.03 1.25 2.97* 4.15* 2.45* .97 1.76* 1.27

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 .50 1.05 .45 1.05 .75 .35 .37 .36
23 101 1.33 1.14 1.09 1.10 2.07° 1.04 2.88* .57
4 21 .69 .36 .52 .37 1.21 .79 1.20 .69
5 54 .73 .20 .80 2.50*2 .33 .32 .666 1.112

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 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
 		   −.002 [.018] .058 [.104] .034 [.100] .00
 		 1.93
 		   .021 [.019] .313* [.109] .010 [.109] .08*
 		 1.71
 
F 23
 		   115
 		   .033 [.078] 1.761* [.455] .319 [.374] .12*
 		 2.50*
 		   −.005 [.040] 1.176* [.227] .663* [.191] .26*
 		 2.17
 
F 4
 		   35
 		   .089 [.128] .857* [.333] −.525 [.467] .19*
 		 2.56°
 		   .068 [.097] .585* [.243] −. 478 [.377] .20*
 		 1.81
 
F 5
 		   68
 		   .094° [.055]2 .062 [.082] −.112 [.157]2 .02
 		 1.61
 		   .004 [.030]2 .086° [.448] .009 [.086]2 .05
 		 2.01
 
A 1
 		   101
 		   .004 [.009] .001 [.024] −.005 [.020] .00
 		 1.90
 		   .023* [.010] .058* [.025] −.009 [.022] .05°
 		 1.70
 
A 23
 		   115
 		   .067 [.067] .489* [.134] .021 [.152] .10*
 		 2.44*
 		   .065° [.036] .367* [.069] .058 [.081] .20*
 		 1.91
 
A 4
 		   35
 		   .022 [.100] .508* [.202] −.071 [.193] .16°
 		 2.67*
 		   .020 [.074] .276° [.151] −.150 [.144] .12°
 		 1.76
 
A 5
 		   68
 		   .055* [.027] .093 [.065] −.013 [.060] .03
 		 1.69
 		   .005 [.018] .109* [.034] −.044 [.039] .15*
 		 2.09
 
S 1
 		   101
 		   .010 [.011] .013 [.021] − .025 [.026] .01
 		 1.89
 		   .039* [.011] .062* [.022] −.044 [.027] .09*
 		 1.69°
 
S 23
 		   115
 		   .055 [.066] .354* [.121] −.005 [.253] .07*
 		 2.44*
 		   .079* [.031] .394* [.057] −.356* [.119] .31*
 		 2.08
 
S 4
 		   35
 		   −.014 [.080] .479* [.160] −.237 [.175] .24*
 		 2.79*
 		   −.006 [.060] .248* [.121] −.264° [.132] .20*
 		 1.84
 
S 5
 		   68
 		   .048° [.025] .083 [.062] −.028 [.071] .03
 		 1.73
 		   −.003 [.013] .114* [.031] −.090* [.036] .20*
 		 2.06
 

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 9: Money Supply Measures1
    Final Announced Survey
Period obs mean variance mean variance mean variance
1 104 1.547 7.306 .935 35.208 2.068 12.419
23 118 1.148 9.827 1.064 40.600 .654 6.608
4 38 1.240 14.886 1.256 42.912 .130 20.254
5 71 2.2372 11.4922 1.983 30.172 1.297 11.382

Notes: 1. These values refer to the percentage change in the money supply multiplied by a factor of ten.
2. These statistics are calculated for the truncated part of period 5. The number of observations is reduced by 57.