Money Announcements, The Demand for Bank Reserves, and the Behavior of the Federal Funds Rate within the Statement Week

The effect of money stock announcements on the Federal funds rate has been attributed informally to the information conveyed by the announcements about aggregate reserve demand. This "Aggregate Information Hypothesis" explains the effect without reference to Federal Reserve intervention in the funds market. In this paper I provide a formal model of the Aggregate Information Hypothesis under lagged reserve accounting.The model relies on imperfect information in the funds market, and on imperfect bank arbitrage of reserve demand between days of the week. Some stylized facts are presented about funds rate behavior in the period 1980-1983.


Introduction
This paper documents two puzzling features of Federal funds rate behavior under lagged reserve accounting in the early 1980's, and presents a stylized model of the funds market which explains these features.
The first noteworthy fact which is analyzed is that day-to-day changes in the rate within the statement week may be significantly positive or negative on average across weeks, and are serially correlated. This fact is puzzling, as pointed out by Shiller, Campbell and Schoenholtz [1983], because banks must hold reserves to meet an average requirement over the statement week. The funds rate is the daily price of holding reserves, and one would expect banks to adjust their reserve holdings across days to eliminate any predictable changes in the funds rate through the week. That is, one would expect the funds rate to follow a martingale within the statement week.1 Secondly, there has been a significant effect of Friday Ml money stock announcements on the Federal funds rate. When the money stock, which is announced with a two week lag, is announced to have been surprisingly large, the funds rate has tended to rise.
The announcement effect on longer term interest rates may be explained as the result of expected policy reactions (when Ml is high, the Fed is expected to tighten and raise real rates), or an expected change in policy (a positive Ml surprise indicates a higher long-run money growth rate and higher expected inflation).2 It is harder to explain the announcement ef-fect on the funds rate in this manner.
The inflation story seems implausible for an overnight rate, while the tightening story requires that after a high money announcement the Federal Reserve reduces nonborrowed reserves at the end of a statement week relative to the beginning of the week. It is hard to see why the Federal Reserve should behave in this way; after all, it has some foreknowledge of the announcement, so that it could tighten early in the week. This would reduce the interest rate response to the announcement, something which the Federal Reserve has publicly said would be desirable. Shiller, Campbell and Schoenholtz [1983] proposed an alternative explanation for the money announcement effect on the Federal funds rate.
They pointed out that under lagged reserve accounting, the announcement of Ml from two weeks ago is an announcement of current aggregate demand for bank reserves.3 Before the announcement, banks know their own reserve requirements but not the reserve demand of other banks. If banks collectively underestimate aggregate reserve demand, but know the position of the supply curve for reserves, then the money announcement will raise the expected end-of-week funds rate. By the arbitrage argument given above, the funds rate will rise immediately. Nichols and Small [1985] have criticized this explanation, which they call the "Aggregate Information Hypothesist' (AIH), on the following ground.
If the supply curve for reserves is known, as postulated by the AIH, then the Federal funds rate in the first part of the week reveals the demand for reserves in that part of the week. Nichols and Small argue that this is In fact Ml contains public currency holdings and thus is not equivalent to reservable deposits. For simplicity this distinction will be ignored in what follows. tantamount to revealing aggregate reserve demand for the week as a whole, so that the money stock announcement conveys no new information under the AIH.
-In this paper I argue to the contrary, that a money announcement may convey information even if there is no uncertainty about the position of the supply curve for reserves. This is possible because banks may be uncertain about the timing of other banks' reserve demands within the statement week; thus the funds rate in the first part of the week reveals "early" reserve demand but does not necessarily reveal "late" reserve demand.
The formal model which illustrates this point also generates predictable changes in the funds rate through the week. Such deviations from martingale behavior are a necessary (but not sufficient) condition for the Aggregate Information Hypothesis as modelled here.L The model of this paper should not be interpreted as a complete description of all the factors which affect the Federal funds market. Rather, it is a highly stylized representation of one aspect of Federal funds rate determination. Relevant empirical results on Federal funds rate behavior in 1980-83 are briefly presented in Section 2 of the paper. Section 3 develops the formal model for lagged reserve accounting, and Section 4 concludes. ' Walsh [1983] presented a formal model of intraweek bank reserve demand with some similarities to the one worked out here. The most important difference is that in Walsh's model banks are uncertain about the supply of reserves within the week, so that the model is not a pure illustration of the Aggregate Information Hypothesis. Also Walsh focuses on aggregate uncertainty of banks, rather than the individual bank's inference from private and public information which is studied in this paper.

2.
Empirical Evidence, 1980-83 In this section I briefly present some evidence that the Federal funds rate has not followed a martingale within the week, and that it was affected by money stock surprises in the lagged reserve accounting period. Rather than rely on one or the other interpretation of the funds rate data, I show that the martingale hypothests for the funds rate can be rejected under either interpretation. The first four rows of Table 1 present summary statistics for daily changes in the funds rate within the statement week. Under either interpretation of the daily average data, if the funds rate were a martingale mean daily changes would all be insignificantly different from zero. However in the full sample and each subsample two out of four mean daily changes are significantly different from zero at the 5% level.
There is a predominance of negative signs, indicating that the funds rate tends to fall through the statement week.
The last four rows of Table 1 summarize some regressions which describe the dynamics of funds rate behavior. In rows 5 to 7 I regress the change in the funds rate on the previous day's change. Standard errors in parentheses are the heteroskedasticity-consistent standard errors proposed by MacKinnon and White [1984]. These have desirable finite-sample as well as asymptotic properties. They are uniformly larger than the unadjusted standard errors, and thus represent a conservative evaluation of the significance of the coefficients.
In row 5 the previous dayts funds rate change is shown to have a highly significant positive effect on the funds rate change from Tuesday to Wednesday, in the full sample and both subsamples. The lagged change is almost as significant in row 6, with the exception of the second subsample.
innovation variance. The variances of daily average funds rate changes alter systematically through the week, as described below. This complication is ignored in what follows.
In row 7, however, the change from Friday to Monday is shown to have an insignificant correlation with the change from Thursday to Friday. The regression of row 5 rejects the martingale hypothesis at the 5b/0 level for either interpretation of the daily average data; the regression of row 6 rejects it only if the daily averages are equivalent to point-in-time data.6 These results are consistent with those reported by Cornell [1983] for a sample period roughly equivalent to the first subsample here.7 Row 8 shows that there is a significant contemporaneous reaction of the funds rate to money announcement surprises, in the full sample and both subsamples.8 The daily averaging problem does not affect the money announcement regressions, since the announcement occurs between trading days.
6 Results which are not reported, for lack of space, show that there is significant second-order serial correlation of funds rate changes. This is inconsistent with the martingale hypothesis, even if the daily average data suffer from the Working problem.
Cornell stressed that there is negative serial correlation between the change on the last day of one statement week, and the change on the first day of the next week. This is not inconsistent with the martingale hypothesis.

A Rational pctations Model of Reserve Demand
Under Lagged Reserve Accounting In this section I develop a model of bank reserve demand which helps to explain the observed features of funds rate behavior under lagged reserve accounting.9 In particular, the model generates predictable differences between the funds rate early and late in the statement week, and it allows the existence of a money announcement effect on the funds rate out assuming that the Fed reacts to announcements within the week.
The formal model treats the Federal funds rate as the cost to a bank of holding its reserves on a particular day of the week. The justification for this treatment is that the funds rate is the cost to an individual bank of increasing reserves at one margin, or the benefit to it of reducing reserves at that margin. There are of course other margins where banks can increase or reduce reserves; for example, they can borrow at the discount window or sell securities. However at the optimum costs are equated at all margins and the model uses the funds rate as an indicator of these costs.
Furthermore the funds rate is assumed to be the same for all banks; it would be easy to generalize the model to allow for a constant difference in funds rates across banks.
Under lagged reserve accounting, each bank j knows its overall reserve requirement for the statement week, R., at the beginning of the week. The bank must choose its reserve holdings for the first day of the week knowing R. and the first-day federal funds rate, r1, but without knowing the funds rate that will prevail later in the week.
Details of calculations are in an Appendix available from the author.
For simplicity I divide the statement week into only two "days".'°T hen the bank must choose first-and second-day reserve holdings, R1. and R ., subject to R . + R . R.. If there were no costs associated with adlj 2j j justing reserves between the first and second days, the bank would simply hold reserves on the day with the lowest (expected) federal funds rate.
This behavior on the part of all banks would eliminate any predictable change in the rate from the first to the second day.
However there are predictable changes in the funds rate, and one can imagine a number of reasons why banks find it costly to adjust the timing of reserve holdings within the week. For example, there may be an inventory motive for reserve holdings, to finance stochastic inflows to and outflows from a bank's reserve account. Rather than model this explicitly, I simply introduce a first-day reserve "target", T1., together with a quadratic cost of deviation from the target. Bank j chooses R1. to minimize the cost function 2 (1) Mm C = r1R1.
The first-order condition for this problem is (2) R1.
It is apparent from (2) that >O is necessary if expected differences between r2 and r1 are not to generate unbounded shifts in reserve demand.
10 A money announcement is then modelled as a change from one equilibrium to another in the first day, rather than as a real-time event occurring between two days of the week. Walsh [1983] used a three-day model of the statement week. This enabled him to model money announcements in real time, at the cost of greater complexity in intertemporal aspects of his model.
The first-order condition (2) contains the bankts expectation of the second-day federal funds rate, E. [r2]. In order to model this expectation as rational, we must state explicitly the information problem of the bank.
On the demand side, the bank knows R. and T1.. Suppose that (3) R.a+u+s. The first-day reserve target T1. is some constant fraction b of the bank j total reserve requirement, plus an aggregate shock v and an idiosyncratic shock i.. v and T. are normally distributed with zero means and variances V and V respectively. u, v, . and 11. are all independent. I assume (l/J) X ii. = 0. Then the average first-day reserve target across banks is T1 hR + v.
On the supply side, the Federal Reserve supplies reserves to the banking system as a whole according to some rule which I assume is constant across the two days of the statement week, and known to individual banks at the beginning of the week.11 For simplicity, assume this rule is linear in Note that the supply function for reserves includes borrowed reserves; thus the implicit penalty for discount window borrowing is incorporated the Federal funds rate: (5) R. = c + dr. i1,2 1 -1 where R. = (l/J)R... This supply schedule is consistent with the analysis of Hetzel [1982], so long as the funds rate exceeds the discount rate.  (7) and (8), Next postulate that bank fs expectation is linear in its information: The perfect information case is solved by substituting (8) and (9) into (6). One obtains an equation which must hold for arbitrary u and v. These results are fairly intuitive. A shock to overall reserve demand, u, raises both r1 and r2: it raises them equally if b = 1/2, for in this case banks wish to distribute their additional reserve demand equally across days of the week, or if a = 0. In these cases u does not affect the difference between r1 and r2. A shock to the first-day reserve target, v, raises r1 and lowers r2 so that the average of the two rates is unaffected.
v has no effect on the funds rates if a = 0, that is if banks are infinitely willing to arbitrage across days of the week. When b = 1/2 or a 0, the constant terms simplify to [a-2c]/2d for each equation. When b > 1/2 for nonzero a, r1 tends to be greater than r2 on average across weeks, as observed in the 1980-83 period.
The imperfect information case is more complex. The first step in solving it is to find the values of the coefficients which are consistent with given values of the u coefficients. Aggregating equation (10) across banks, and substituting it and equation (8) into equation (6), one finds 1b2+1r1/(l+ad-Tr3) and 2=(a+'ii2)/(l+ad-iu3). The value of is the same as 3* above.
The next step is to obtain the ii coefficients as functions of the coefficients. Since r , R., T . and r are multivariate normal, one can The importance of equations (12) is that they show that l falls as the equilibrium becomes noisy, while 2 increases. Intuitively, when the equilibrium is noisy banks confuse u and v shocks. A u shock to overall reserve demand is thought by each bank to be partly a v shock to the firstday reserve target, which will lower the funds rate on the second day of the week. Banks respond by shifting reserve demand to the second day, lowering the first-day funds rate; 'l is lower than under full information.
Similarly a v shock, raising the first-day reserve target for all banks, is thought by each bank to be partly a u shock to overall reserve demand throughout the statement week. Accordingly banks do not fully anticipate the decline in reserve demand and the funds rate that will in fact occur on the second day of the week, and they shift less of their reserve demand towards the second day. This keeps the funds rate high on the first day, so is higher than under full information.
Confusion of this sort can explain why there is a money announcement effect on the federal funds rate, without any need to appeal to intra-week shifts in reserve supply. We have noted that under lagged reserve accounting, a money announcement is an announcement of current aggregate reserve demand R = a + u. Accordingly it shifts the funds market from the noisy equilibrium to the fully informed equilibrium, without changing any real conditions in the market.
One can derive some propositions about the money announcement effect even without the imperfect information model solution (12). First, from the aggregated first-order condition (6) Next, we note from equation (7)  For such an infinitesimally noisy equilibrium, the effect of the announcement, A, on the first-period funds rate is Since l* and 2* are both positive, a positive money surprise occurs when there is a high value of u and a low value of v; the low v "disguises" the high u by offsetting its effect on the first-day funds rate.

Conclusion
In this paper I have argued that random shifts in banks' desired timing of reserve holdings can explain some otherwise puzzling features of Federal funds rate behavior within the statement week. I have developed a model for the lagged reserve accounting regime, in which such shifts generate predictable changes in the funds rate from one part of the week to another. They also create confusion so that the funds rate does not perfectiy aggregate private information, and an announcement of the money stock moves the market to a more informed equilibrium. In this model, a money announcement may alter the Federal funds rate even when there is no uncertainty about Federal Reserve supply behavior within the statement week. Note: = significant at the 5% level, ** = significant at the 1% level.