RDP 8602: ShortTerm Interest Rates, Weekly Money Announcements and Rational Forecasts 4. The Forecasting Equations
May 1986
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The Money Market Services survey was first published to coincide with the money announcement that was made on 29 September 1977. The estimation period starts two weeks after this to allow for the calculation of lagged values. It ends on 10 February 1984 which is the date of the last money announcement that was made on a weekendingWednesday basis, and the last one made before the Fed's switch to contemporaneous reserve accounting. There are 331 weekly observations in this estimation period. To allow tests for possible breaks in the estimated equations following changes in Fed operating procedures and the other factors that were discussed in Section 2, this period is divided into a number of subperiods. These are given in Table 1, where the dates are the dates of the relevant money announcements and n is the number of observations.
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 
A further complication is introduced by the available data on the “final” money supply. Each March, the Fed publishes the “Money Stock Revisions” document. This provides the only available source of revisions to the seasonally adjusted, weekly money stock data. unfortunately, the March 1983 issue is the most recent one that is available on a weekendingWednesday basis. This provides data up to and including the statement week ending 29 December 1982 which corresponds to the announcement made on 7 January 1983. Hence, Period 5 is truncated by 57 observations when estimating the parameters of the “final” forecasting equations.^{[29]}
The information set available to agents before the announcement, I_{t} , contains thirty variables, including the constant term. It is clear from Table 1 that projections on this information set cannot be calculated for Period 2, and that those for Period 4 would have few degrees of freedom. Fourteen of these variables are the two sets of lagged values of the announced revisions to each of the seven previous changes (i.e., the lagged R_{i}'s). If these could be dropped from the information set there would be a significant increase in the degrees of freedom available for Period 4.
Table 2 presents the relevant Ftests for the null hypothesis that the coefficients of these fourteen variables are jointly zero. They are calculated for the forecasting equations for the final change given information just after the announcement (FA), and for the final change (F), the announced change (A) and the revisions (the R_{i}'s) based on information just before the announcement.^{[30]} Of the thirty Fstatistics in Table 2, only one is significant at the 5% level of significance. This suggests that the fourteen lagged revisions may be dropped from the information set I_{t}.
Period  df1  df2  FA^{2,3}  F^{3}  A  R_{1}  R_{2}  R_{3}  R_{4}  R_{5}  R_{6}  R_{7}  

1  14  74  .19  .12  .84  .50  .79  .37  .96  .18  .21  .27  
23  14  88  .61  .54  .62  .43  .41  .39  .63  .52  .38  .53  
45  14  79  .52  .67  1.65°  1.41  1.43  .92  .76  1.89^{*}  .74  .75  
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 Fdistribution with df1 and df2 degrees of freedom. An * (°)
denotes rejection of the null hypothesis at the 5% (10%) level of significance,
respectively. 
The Fstatistics for testing for the presence of breaks in the forecasting equations (once the lagged revisions have been dropped) are presented in Table 3.^{[31]} The FA equation (the forecast of the final change after the announcement has been made), and a number of the revision equations, exhibit evidence of a break between Periods 3 and 4. This was when the monetary aggregate was switched from MIB to MI. Given that this change was essentially a renaming of the aggregate and not a redefinition, the explanation seems to lie in the effect this had on validating the wider definition of MI. The F equation (for the final change expected prior to the announcement) appears to have no breaks during the estimation period. The forecasting equation for the announcement, the A equation, shows evidence of a break at the end of Period 1. This coincides with the Fed's move to a nonborrowed reserves policy. No equation has a break between Periods 4 and 5 at the 5% level of significance. Equations for two of the revisions are the only ones to exhibit a break between Periods 2 and 3, when the release date was changed to the following day and the monetary aggregate was expanded to MIB. However, in both cases the break is less serious than the one between Periods 1 and 2.
H_{0}  df1  df2  FA^{2}  F  A  R_{1}  R_{2}  R_{3}  R_{4}  R_{5}  R_{6}  R_{7}  

1=12  17  88  .49^{3}  .40  1.27  3.17^{*}  1.76^{*}  .18  .18  .07  .17  6.28^{*}  
23=3  17  85  .97^{3}  .50  .34  2.22^{*}  .65  .10  .13  .13  .11  3.40^{*}  
1=23  16  190  .73  1.39  2.35^{*}  .87  .36  .28  .71  .31  .28  .12  
3=4  16  107  2.32^{*}  .99  .80  .43  1.93^{*}  3.96^{*}  3.48^{*}  1.26  1.04  2.21^{*}  
23=4  16  124  2.13^{*}  1.00  .78  .47  2.13^{*}  4.63^{*}  4.10^{*}  1.35  1.19  1.74^{*}  
4=5  16  77  1.30^{4}  .58^{5}  1.45  .67  1.38  1.51  1.58°  .89  1.44  1.75°  
23=45  16  195  2.12^{*}^{6}  1.01^{6}  1.04  .74  2.29^{*}  4.45^{*}  4.01^{*}  2.09^{*}  1.73^{*}  2.14^{*}  
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 Fdistribution with df1 and df2
degrees of freedom. An * (°) denotes rejection of the null hypothesis at the 5%
(10%) level of significance, respectively. 
Based on these results, the estimates obtained from Period 1, Period 23 (the union of the Periods 2 and 3) and Period 45 (likewise), will be used as the forecasting equations. Their properties are summarised in Tables 4 and 5.
Period  R_{1}  R_{2}  R_{3}  R_{4}  R_{5}  R_{6}  R_{7}  

1  R^{2} dw 
.10 2.03 
.17 1.77° 
.08 2.07 
.14 2.00 
.05 2.01 
.04 2.04 
.04 2.04 

23  R^{2} dw 
.20° 2.07 
.06 1.93 
.08 2.12 
.11 1.99 
.18 2.09 
.16 2.04 
.03 2.41° 

45  R^{2} dw 
.18 2.01 
.33^{*} 2.14 
.43^{*} 2.29° 
.44^{*} 2.14 
.34^{*} 1.91 
.46^{*} 2.20° 
.46^{*} 2.24° 

Notes: 1. An * (°) next to an R^{2} 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 DurbinWatson (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. 
F: Final before announcement^{2,4}  A: Announced^{2}  

Period  df2  R^{2}  dw  H_{0}1  H_{0}2  H_{0}3  R^{2}  dw  H_{0}1  H_{0}2  H_{0}3  
df1  16  15  14  16  15  14  
1  88  .12  2.63^{*}  7.99^{*}  8.29^{*}  .46  .58^{*}  1.86  2.15^{*}  1.76°  1.76°  
23  102  .31^{*}  2.05  9.88^{*}  10.26^{*}  3.31^{*}  .46^{*}  1.97  2.07^{*}  2.16^{*}  1.96^{*}  
45  93  .41  2.28°  3.87^{*}  3.71^{*}  1.04  .58^{*}  2.06  1.68°  1.52  1.59°  
FA: Final after announcement^{3,4}  
Period  df2  R^{2}  dw  H_{0}4  H_{0}5  H_{0}6  H_{0}7  
df1  24  23  15  14  
1  80  .23  2.79^{*}  14.82^{*}  15.91^{*}  .29  .26  
23  94  .51*  2.14  22.55^{*}  23.51^{*}  3.03^{*}  1.74°  
45  28  .79^{*}  2.12  5.39^{*}  5.62^{*}  1.38  1.10  
Notes: 1. An * (°) next to an R^{2} 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 DurbinWatson (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 Fdistribution with df1 and df2 degrees of freedom. An * (°)
denotes rejection of the null hypothesis at the 5% (10%) level of significance,
respectively. 
Table 4 presents the R^{2} and Durbin Watson statistics for the forecasting equations for the seven revision variables. None of the equations display any serial correlation in the residuals. However, only at the end of the sample period do the equations have any explanatory power. This suggests that the initial revisions are largely unforecastable.
The results for the announced and final change equations are presented in Table 5. The announced change is reasonably well forecastable from the information set. However, the hypothesis that the survey is a rational expectation of the announcement (hypothesis H_{0}1) is rejected at the 10% level of significance for each period and at the 5% level for two of the three subperiods.^{[32]} When the hypothesis is relaxed slightly to allow for a nonzero intercept (hypothesis H_{0}2 that the survey is a biased but efficient predictor), it is rejected at the 5% level for Period 23. The same result is obtained for the hypothesis that the other variables are orthogonal (hypothesis H_{0}3), which allows for a nonunit coefficient on the survey variable as well as a nonzero intercept term. These results are in marked contrast to the usual assumption that the survey incorporates all the available information about the announcement. They show that Model A is preferred to Model S (the standard model) as a means of measuring the announcement effect.
The results for the final change variable are even more pronounced. Before the announcement, the final change is almost unpredictable.^{[33]} The equation has forecasting power at the 5% level of significance in only one of the periods. Adding the announced change to the information set improves the predictability of the final change. Nevertheless, the announcement is not a rational expectation (hypothesis H_{0}4), nor an efficient predictor (hypothesis H_{0}5). However, including the survey in the equation does not in general improve the forecast it provides (hypothesis H_{0}6). Nor does the inclusion of the other variables in the information set I_{t} (hypothesis H_{0}7).
Since the survey is not a rational expectation of either the announced or final changes, and the announced change is not a rational expectation of the final change, the F models are the appropriate vehicles for measuring the announcement effect. The results for these models are presented in the next section.
Footnotes
Out of sample forecasts of the final change predicted by these equations are used to estimate the interest rate equations over the full length of Period 5. [29]
The equations cannot be estimated separately for Periods 2 or 4. Other Ftests on the equations suggested that the breaks occurred between Periods 1 and 2, and between Periods 3 and 4. This issue is addressed in more detail, below, for the equations after the lagged revisions have been dropped. [30]
The left most column of this table gives the hypothesis being tested. For example, “H_{0}: 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 of the forecasting equation in each distinct period. [31]
This is the joint test that the intercept is zero, the coefficient on the survey is unity, and that the other variables have zero coefficients. In each period, the coefficient on the survey variable is not significantly different from unity. However, this is a necessary, not a sufficient, condition for rationality. [32]
The presence of serial correlation in some of the final equations does not allow the predictability to be improved by adding lagged values of the dependent variable to the equation. This is because these were not in agents' information sets at that time. [33]