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Dynamic implicit cost and Discount Window borrowing
Journal
of Monetary
Economics
32 (1993)
1055120.
North-Holland
Dynamic implicit cost and discount window borrowing An empirical
investigation
Donald H. Dutkowsky* S~racusrUnivemity, S)‘racuse. NY Received
May 1991. final version
13244, USA received
May 1993
This paper investigates dynamic implicit cost in aggregate Discount Window borrowing. We utilize Goodfriend’s formulation, with nonprice rationing based upon the bank’s current request and its past borrowing record. Estimation by Generalized-Method-of-Moments reveals a positive, systematic implicit cost structure which dampens over time. Significance extends back to nine biweekly reserve maintenance intervals. We also specify implicit cost as a geometrically declining distributed lag of past borrowing and derive a semi-reduced form for borrowed reserve demand. Empirical results support this specification as well.
Key words: Discount
window
borrowing;
Central
bank policy; Operating
target
1. Introduction
Aggregate Discount Window borrowing continues to play a central role in the Federal Reserve’s conduct of monetary policy. As stated in publications such as Sternlight et al. (1991), the Fed has generally maintained the borrowed reserve operating procedure begun in October 1982. In recent years, though, widespread concern has been expressed regarding a breakdown in the ability to predict aggregate borrowed reserves. Particular attention has focused on the relationship between aggregate Discount Window
Correspondence fo: Donald H. Dutkowsky, Department ship and Public Affairs, Syracuse University, Syracuse,
of Economics, Maxwell School of CitizenNY 13244-1090. USA.
*I wish to thank Yucong Zhao for research assistance as well as an anonymous referee and the editors for numerous substantial comments on previous drafts. This paper also benefited from discussions with Josh Feinman, Bill Foote, Gary Gillum, Duke Kao, Masao Ogaki. Steve Peristiani, and John Wenninger. However, the views are solely those of the author and do not necessarily reflect official positions of the Federal Reserve Bank of New York or the Federal Reserve System.
0304-3932/93/$06.00
(c) 1993-Elsevier
Science Publishers
B.V. All rights reserved
borrowing and the spread between the Federal Funds rate and the Discount rate, Sternlight et al. (1989, p. 98) write: ‘Examination of the relationship back to 1984 shows that a series of shifts has . . . resulted in reduced discount window borrowing for given spreads of the funds rate over the discount rate.’ Garfinkel (1989, p. 1) states: ‘The recent instability . . is also demonstrated by noting that the correlation between the spread and adjustment and seasonal borrowings over the past year was 0.07, down considerably from the previous year’s correlation of 0.53.’ Garfinkel (I 990, p. 19) writes: ‘Toward the end of 1988 and continuing into 1989, however, this [operating] procedure was compIicated by the demand for borrowed reserves and the federal funds rate. Specifically, the willingness of depository institutions to borrow reserves, for given federal funds and discount rates, was declining unexpectedly.’ Sternlight et al. (1991) state that this difficulty continued throughout 1990. At the same time, Christian0 and Eichenbaum (1992) place Discount Window borrowing at the heart of their resolution of an empirical puzzle involving the liquidity effect of monetary policy. Using impulse function analysis, the authors find that positive innovations in nonborrowed reserves cause sharp declines in short-term interest rates, even though the same interest rates increase in response to positive shocks in the monetary base or Ml. They attribute the strikingly different dynamic movements to the behavior of Discount Window borrowing, showing that innovations in the nonborrowed base generate an interest rate response similar to those of nonborrowed reserves. Christian0 and Eichenbaum contend that an integrated explanation of the liquidity effect must include a description of the linkage between borrowed reserves and interest rates within a detailed model of Discount Window borrowing. In this regard, Marvin Goodfriend’s (1983) study offers an intuitively appeaiing means of understanding bank borrowing from the Discount Window. Based upon well-known Federal Reserve attitudes, Goodfriend argues that implicit cost due to nonprice rationing at the Discount Window depends upon the past record of borrowing as well as the magnitude of the current request. Consequently borrowing choice becomes a dynamic optimization problem, involving past and expected future borrowing along with the interest rate spread. The Goodfriend study has opened up several veins of research which employ versions of the model to examine monetary policy, interest rates, and banking. This paper empirically investigates the role of dynamic implicit cost in aggregate Discount Window borrowing. Despite the Goodfriend model’s substantial insights, no studies have reported empirical estimates. Our mode! generalizes Goodfriend (1983) to consider bank reluctance stemming from its borrowing record over a number of past periods. We estimate the model by Generalized-Method-of-Moments (GMM), with biweekly reserve maintenance data spanning 1985-89. The findings produce evidence supporting a positive, systematic implicit cost structure due to past borrowing which dampens over time. Significance extends
back to nine reserve maintenance intervals. The structure of implicit cost implies a borrowing function in which changes in aggregate Discount borrowing are positively affected by changes in the interest rate spread. Borrowing changes are also inversely related to changes in banks’ past borrowing as well as changes in expected future borrowing. The above results suggest that implicit cost can alternatively be specified as an infinite, geometrically declining distributed lag of past borrowing. Within this formulation, a semi-reduced form for borrowed reserve demand can be analytically derived. Estimation by GMM provides evidence supporting this specification as well. The paper proceeds as follows. Section 2 presents the extended Goodfriend model. Issues involved with GMM estimation of aggregate Discount Window borrowing are discussed in section 3. Section 4 reports estimation findings associated with the extended model. The mode1 with geometrically declining implicit cost is presented in section 5. Section 6 concludes the paper.
2. The extended Goodfriend model We begin by specifying the Discount Window borrowing decision. Along the lines of Goodfriend (1983) the representative bank uses the Discount Window to maximize the expected total discounted profit of borrowing. Banks form expectations rationally. The ‘revenue’ of borrowing from the Federal Reserve consists of interest saved by not having to pay the more expensive Federal Funds rate. The cost of Discount Window borrowing has two components. Besides interest based upon the discount rate, banks face implicit cost due to Federal Reserve nonprice rationing at the Discount Window. Described comprehensively in Goodfriend (1983) implicit cost reflects bank reluctance to borrow from the Federal Reserve due to increased surveillance and administrative counseling. Regulations and writings regarding adjustment borrowing indicate that the Fed discourages continuous borrowing as well as large requests. The optimization problem calls for choosing a sequence of Discount Window borrowing to maximize
V, = E, f h’([fiRB, 1=0
- idisc,RB,
- C,),
(1)
where V0 denotes expected total discounted profits at date 0, RB, refers to the level of date t Discount Window borrowing, rg is the date t Federal Funds rate, idisc, denotes the Discount rate at date t, C, refers to date t implicit cost, h is the
D.H. Dutkowsky, Discount window borrowing
108
rate of time discount with 0 < b < 1, and E, = E( IQ,), the conditional expectations operator based upon information available at date t. Implicit cost stems from both the current borrowing request and past borrowing extending back n reserve maintenance periods. With this assumption, date t implicit cost can be modeled as C, = aoRB, + c,RB:
+ 2coRB,
2
diRB,_i,
i=l
co,di > 0,
i = 1,2, . . , PI.
(2)
The parameters a0 and co measure marginal implicit cost due to the current request. The a, parameter may have either positive or negative sign. But positive marginal implicit cost implies that its magnitude is dominated by the co and d parameters. The di parameter describes reluctance due to borrowing i periods in the past. Greater emphasis attached to more recent borrowing implies a declining lag structure among the d parameters, i.e., dl > d2 > . > d,. This specification of implicit cost generalizes Goodfriend’s model, in keeping with his basic theme of ‘progressive pressure’ in nonprice rationing at the Discount Window. Assuming that the representative bank borrows from the Federal Reserve in each period, substituting (2) into (1) and performing the optimization yields the following Euler equation for date t borrowing: K, = a0 - 2co
RB, + i:
diRB,-i
+ ~ b’diE,RB,+i
(3)
i=l
i=l
where K, = iflt - idisc, is the interest rate spread.’ The bank equates the marginal implicit cost of date t borrowing with the marginal monetary saving. Marginal implicit cost is based upon the current magnitude, reluctance due to past borrowing, and the discounted effect of current borrowing on future requests. Note that (3) can be rearranged in terms of date t borrowing as follows: RB, = - ao/(2co) + K,/(~c,)
-
f: diRB,-i i=l
Within this structural equation, current borrowing date t spread. The function also reveals a distributed
-
i
b’diE,RB,+i.
(4)
i=l
depends positively on the lag on past borrowing and
‘Despite the fact that relatively few banks utilize the Discount Window during any particular period, we follow Goodfriend (1983) and subsequent theoretical work on borrowing that emphasizes intertemporal behavior by assuming interior solutions.
D.H. Dutkowsky. Discount window borrowing
109
a distributed lead on expected future borrowing. Implicit cost implies a negative relationship with current borrowing for both the leads and lags, with the effects gradually dampening out. Notably, progressive pressure as applied by the Fed implies that lag structures for past and future borrowing are identical, except for the rate of time discount. When n = 1, the model reduces to the formulation in Goodfriend (1983). In this case, (3) can be solved for date t borrowed reserve demand as
W = glRB,-
1 -
Cg,/@cod,)l
f
i=O
(l/gz)‘C- aO + EtKt+i19
(5)
where g1 = [-
1 + (1 - 4hd:)“‘]/(2hdI)
and
g2 = l/bg,.
(6)
If 1 > (1 + b)dr , then the roots to the Euler equation have the desired stability property, g2 < - 1 < g, < 0 [see, e.g., Sargent (1987)]. While borrowing is positively affected by an increase in the current spread, a rise in the expected future spread causes current borrowing to fall. Since the bank would anticipate greater borrowing in date t + 1 due to the favorable spread, it borrows less now to reduce expected future implicit cost.
3. Econometric issues Given the dynamic behavior implied by Goodfriend’s model, GMM emerges as a leading procedure to estimate borrowed reserve demand.2 The technique is designed to consistently estimate parameters of Euler equations arising from intertemporal optimization under rational expectations. Assuming there are no shocks to the implicit cost function, the procedure gets by without specifying additional processes for expectations determination or solving for a semireduced form.3 We illustrate how to implement GMM using the model with single-period implicit cost. To sidestep possible difficulties with nonstationarity, we begin by first differencing the Euler equation. This transformation expresses the Euler equation in terms of stationary variables while preserving the behavioral
‘Past econometric research on aggregate Discount Window borrowing emanates from Goldfeld and Kane (1966). Most work features a borrowing function positively related to the current-period interest rate spread, derived from a static optimization framework. Extensions of the Goldfeld and Kane model focus on nonlinearity and heteroskedasticity between aggregate borrowing and the spread within the borrowing function [see, e.g., Dutkowsky (1984). Dutkowsky and Foote (1985), Dutkowsky and Foote (1988). and Peristiani (1991)]. Frost and Sargent (1970) examine seasonal effects using spectral analysis. ‘See Garber
and King
(1983).
110
D. H. Durkowsky,
Discount
window
borrowing
relationship. A Dickey-Fuller test indicates that the adjustment borrowing series is stationary, but it cannot reject the null hypothesis of a unit root in the spread. The test performed on changes in the spread rejects the unit root. Setting n equal to 1 in (4), differencing the equation, and rearranging, we have - DK,/(2c0)
+ DRB, + d, DRB,_,
+ bd, DRB,, , = De,+ ,,
(7)
where
De,+1 = hd, C(W+
I
-
E,RB,+,) - (RB, - E,-,RB,)I,
(8)
and D denotes the differencing operator, Dx, = x, - xf_, . GMM can be applied to consistently estimate the parameter vector 0 = (b, co, d, )‘, since rational expectations imply that E,_ , .x,_, De,+ 1 = 0 for and any stationary variable .x,- , in Qf_, . Given a sample of T observations a q-dimensional vector of instruments from !L?_, denoted by Z,_ 1, with q greater than or equal to the dimensionality of the parameter vector, the procedure computes the vector of sample first moments of Z,_ I De,+, . Substituting for De,+, using (8) the q x 1 vector is given by r+(O)
i Z,.. , [t=I
= (l/T)
DK,/(2c,)
+ DRB,
+ d, DRB,- , + bd, DRB,, ,I. The GMM
estimator
JT(@o, I’) =
of 0, denoted
by Oo, minimizes
(9) the function
C~~,(~o)l’~-‘CT~,(~o)l,
(10)
where I/ is a q x q positive definite matrix further described in Gallant (1987). When the dimensionalities of the instrument and parameter vectors are equal and the partial derivative matrix of mr(O) with respect to 0 has full rank, the function is exactly identified. In this instance, GMM imposes orthogonality and generates estimates by solving mr(Oo) = 0. For the case in which the number of instruments exceeds the number of parameters, the minimized JT value can be utilized to test for the overidentifying restrictions. As Gallant (1987) explains, one rejects the hypothesis that the model is correctly specified when the statistic exceeds the upper LYx 100% critical point of the chi-squared distribution with degrees of freedom equal to the difference in dimensionality between the instrument and parameter vectors.
4. The estimated model The sample consists of biweekly data covering July 3, 1985 through from December 19, 1984 to November 1989. We also use observations
July 12, 15, 1989
to form the series for lagged and future borrowing. The two-week interval corresponds to the Federal Reserve’s reserve maintenance period instituted on February 15, 1984. The sample falls entirely within the borrowed reserve targeting procedure and begins after the substantial adjustment borrowing, later coverted to extended credit, during mid-1984 related to the Continental Bank of Illinois. The interest rate spread K is formed by subtracting the discount rate from the Federal Funds rate. The variable is greater than zero throughout the sample period except for two observations prior to July 1985, suggesting that nonlinearity in the borrowings-spread relationship does not present a concern [see, e.g., Dutkowsky and Foote (1985) Peristiani (1991)]. We investigated nonlinearity by estimating the models with changes in the spread squared and cubed; in all cases, the additional terms were insignificant. Aggregate adjustment borrowing from the Discount Window serves as the measure of RB. Despite the fact that the Fed targets adjustment plus seasonal borrowing, our formulation focuses on borrowing due to short-term reserve adjustment. Spread and adjustment borrowings data are reported in the appendix. A review of the adjustment borrowing series reveals several instances of large borrowing within an individual period, even though borrowing in adjacent dates follows usual patterns. A special borrowing situation for the reserve maintenance period ending December 4,1985, has been documented by the Fed. The large borrowing resulted from the massive securities failure of the Bank of New York, along with the problem on settlement night with the Clearing House Interbank Payments System. Following Federal Reserve estimates described in Sternlight et al. (1986) we subtract 1.6 billion from adjustment borrowing for this observation. Since we encounter no other accounts of unusual borrowing, we leave the remaining observations unchanged.4 Table I reports GMM estimates of the implicit cost parameters in eq. (4). The computer algorithm utilizes PROC MATRIX in Version 5.18 of SAS and follows the GMM example in Gallant (1987). The rate of time discount is restricted to equal 0.999. We provide empirical results for a lag length of nine reserve maintenance intervals.” The instrument vector contains 27 elements: a constant term, lagged borrowing changes ranging from two to thirteen periods, and current and lagged changes in the spread up to and including a lag of thirteen. We do not include
4Three other spikes are the July 13, 1988, January 13, 1988, and December 31, 1986 intervals. The latter date is the only distinctive observation for the interest rate spread, a 160-basis-point increase followed by a nearly 100.basis-point drop. Pearce (1990) provides an explanation for these outliers, contending that banks may borrow extra amounts at the end of the year in order to ‘window dress’ their balance sheets. Since this behavior does not fit easily into the intertemporal framework and the role of dummy variables within the Euler equation is unclear, we do not consider it here. ‘Estimated
implicit
cost parameters
for lag lengths
beyond
nine periods
were insignificant.
112
Table 1 Implicit
~.
~-~
L‘0
--
~
cost estimates: Extended ..____ .__ ~ Extended Goodfriend model
Goodfriend -... -.. _
model.” ~~~ _-.___ Test for progressive
4.929
pressure
4.539 (2.754)
(I ,826) d,
0.855 (0.043)
0.826 (0.096)
0.883 (0.114)
dz
0.742 (0.062)
0.702 (0.100)
0.796 (0.208)
d3
0.698 (0.059)
0.682 (0.097)
0.900 (0.238)
d,
0.571 (0.058)
0.545 (0.109)
0.874 (0. t 94)
ds
0.446 (0.063)
0.403 (0.109)
0.630 (0.203)
&
0.370 (0.063)
0.381 (0.092)
0.539 (0.238)
d,
0.267 (0.054)
0.355 (O.lcto)
0.568 (0.247)
ds
0.207 (0.059)
0.286 (0.121)
0.550 (0.245)
do
O.llI (0.037)
0.121 (0.092)
0.271 (0.168)
Jr statistic
I 1.020 (0.856)
6654 (0.574)
“The table reports GMM estimates or the implicit cost parameters defined in (2). estimated within the Euler equation (4) in first differenced form. The test for progressive pressure presents estimates of the model with unrestricted lead and lag coefficients. The rate of time discount (h) is restricted to equal 0.999. The sample consists of biweekly data from 7/3/X5 to 7/12/89. Asymptotic standard errors appear in parentheses under the estimated parameters. P-values are in parentheses under the JT statistics. which test for overidentifying restrictions.
the change in borrowing lagged one period, as the variable may not be in the date t - 1 information set. Following Goodfriend (1992), since the Fed reports aggregate borrowing data with a lag, it is unlikely that individual banks have access to contemporaneous information for this variable. The accessability of interest rate data suggests that this problem does not occur with the spread.’
‘Although the change in the current spread is not an element of s2,_ r, we experience difficulty in finding suitable instruments that capture movement in the first difference of the spread. Removing DK, from the instrument set results in a noticeable loss of efficiency on the co estimate, although the remaining parameters are not qualitatively affected. GMM estimates with and without the DK, variable do not show noticeable change in the Jr statistic.
Estimates of the implicit cost parameters appear in the first column of table 1. The second two subcolumns report estimates of the model with unrestricted lead and lag coefficients. The findings for the restricted estimates reveal strong evidence of dynamic implicit cost. All the estimated parameters for reluctance due to past borrowing are greater than zero and significant. The rate of decline in marginal implicit cost is roughly constant over the lag structure, equaling approximately 0.800 per period. The estimated parameter for marginal implicit cost due to current-period borrowing (co) is also significantly positive. Estimation without restricting the discount factor results in nearly identical implicit cost estimates. Although the estimated rate of time discount exceeds one, the estimate is insignificant from unity. The last row reports the JT statistic, the minimized value of the objective function described in (10). To test overidentifying restrictions for the models in table 1, the statistic is compared to chi-squared with 27 - m degrees of freedom, where m denotes the dimensionality of the parameter vector. The p-values reported in parentheses indicate that the test cannot reject the null hypothesis for either model at any reasonable level of significance. This result holds for all the estimated models reported here. The empirical results support the dynamic optimization framework centering around progressive pressure. As mentioned above, the last two subcolumns of table 1 report estimates in which effects of past and expected future borrowing are not constrained to be equal. Estimated implicit cost parameters for past borrowing (left subcolumn) follow closely the restricted estimates of the nine-lag model. Implicit cost estimates for expected future borrowing (right subcolumn) exceed their counterparts for past borrowing and have higher standard errors, but all have positive sign and nearly all are significantly different from zero. Moreover, simple t tests cannot reject at any reasonable level of significance the individual null hypotheses that the d parameter for borrowing i periods in the past equals the effect of expected borrowing i periods in the future, for i= I,2 )...) 9. Global tests for identical lag structures corroborate this result. The D test discussed in Newey and West (1987) equals the difference between the Jr statistics for the restricted and unrestricted models. Under the null hypothesis the statistic follows a chi-squared distribution with degrees of freedom equal to the difference in dimensionality between the parameter vectors. Performing the test with the models in table 1 yields a statistic of 4.366, indicating the progressive pressure cannot be rejected at any reasonable significance level (for example, the IO percent critical point of the chisquared distribution with 9 degrees of freedom equals 14.68). Gallant (1987) argues that the test should be conducted by estimating both equations with the V matrix of the unrestricted model and subtracting the resulting minimized values. This test generates a statistic of 3.084, also insignificant at the IO percent level.
In examining robustness, we find that the implicit cost estimates show noticeable sensitivity when we estimate the Euler equations using levels of the spread and borrowing. But the results involving changes in RB and K stand up well to estimation with more conventional procedures. Similar findings are obtained when we estimate the borrowing function by Two-Stage-LeastSquares using the same instrument set as the GMM estimations and the h = 0.999 restriction. OLS estimation produces more striking results. Even without restricting the rate of time discount, significantly negative and nearly identical lag structures emerge for changes in past and future borrowing.
5. Borrowing under geometrically
declining implicit cost
The estimated model of progressive pressure, with its nine-period leads and lags, does not easily solve analytically for a demand function for Discount Window borrowing. But characteristics of the implicit cost structure indicated by the previous empirical results ~ the long lag length along with positive and declining reluctance - suggest an extension that produces a neat closed form solution for borrowing. Specify implicit cost as a geometrically decreasing function of past borrowing given by C, = a,RB, + coRBf + 2c,,RB, i
dbRB,_i,
(11)
i=l
where d, refers to the constant rate of decay parameter resulting Euler equation can then be expressed as
with 0 < d,,
I. The
E,K, - a,, - 2c0(l - d,B)-‘E,RB, - 2cobdoB-‘(1
- bd,B-l)mlE,RB,=
0.
with B denoting the backstep operator, B’&x,+~ = E,x,+~_~. Operating through by (1 - d,B)( 1 - hd,, B - ’ ) and performing braic rearrangement, we obtain
RB, = ho + h,E,K,+,
+ h,K, + hjK,m1,
(12)
some alge-
(13)
where ho = - a,,(1 - d,)(l h, =
- hd0)/[2c0(l
- bdi)],
- MolC2co(l -
bd;)l,
(14)
(15)
(1 +
h2 = h, = h2 > 0
hd;)l[2c,u - hG)l, - &/[2co(l
and
-
bdi?)l,
(16) (17)
h,, h3 < 0.
A highly manageable semi-reduced form results. The solution reveals that Discount Window borrowing responds positively to increases in the date t spread. Negative effects of the expected date t + 1 spread as well as the date r - 1 spread reflect dynamic implicit cost. This formulation implies that only the past and expected future spreads in adjacent periods enter into current borrowing. An estimated rate of time discount can be computed as h = h,/h,. Estimates of this formulation of the borrowing function appear in table 2.’ The time discount parameter is again restricted to equal 0.999. The model with borrowing and the spread expressed in levels (column 1) generates estimates which conform to a priori signs and are significantly different from zero. The results hold up reasonably for the estimated equation in first differences. Reported in the second column, the findings also indicate a significantly positive current spread effect and significantly negative coefficients for the past and expected future spread. Despite the similarity of the estimated equations in table 2, some further examination gives an edge to the model expressed in levels. From (14))( 17) and the coefficients in table 2, the model in levels generates plausible estimates of the structural parameters of a, = -3.470, c0 = 2.097, and do = 0.597. Real-valued estimates could not obtained from the model in first differences.* In addition, we computed the estimated interest rate spread forecast error for both models using (13) and checked for serial correlation using time series methods. Based upon several diagnostic criteria within PROC ARIMA of SAS, we find no distinctive evidence to counter the hypothesis of white noise for the model estimated in levels. The evidence indicates that the estimated forecast error from the model in first differences exhibits serial correlation, primarily reflecting a partial autocorrelation of -0.48 between the error and its one-period lag.9
‘When estimating the model with levels, we add the change in borrowing lagged one period to the instrument set. In examining (13) for GMM estimation it is easily shown that for the function with borrowings and the spread expressed in levels, orthogonality holds for any variable in 0,. ‘This results because the solution for do comes from a quadratic equation. and complex roots are obtained when we substitute the estimates of the model in first differences. We report the smaller root for the model in levels. ‘The specific criteria consist of a scan of the partial autocorrelations, an autoregression of lags from l-24 periods. and the Lyung-Box statistic calculated for lags of 6, 12, 18 and 24 periods. Although serial correlation violates the conceptual property of the forecast error as predicted by rational expectations, GMM still produces consistent parameter estimates [see, e.g., Gallant (1987)].
D.H. Dutkowsky. Discount window borrowing
116
Table 2 Estimated
borrowing
function:
Geometrically
declining
implicit cost.”
Units of RB and K
ho
Levels
First differences
0.209 (0.03 1)
0.004 (0.010)
h,
- 0.221 (0.042)
- 0.244 (0.055)
h2
0.502 (0.045)
0.343 (0.052)
h3
- 0.221 (0.042)
- 0.245 (0.055)
19.639 (0.768)
19.837 (0.706)
Jr statistic
“The table reports GMM estimates restricted to equal 0.999. The sample standard errors appear in parentheses under the JT statistics, which test for
of the borrowing function (13). The rate of time discount (b) is consists of biweekly data from 7/3/85 to 7/12/89. Asymptotic under the estimated parameters. P-values are in parentheses overidentifying restrictions.
6. Conclusion
This study provides empirical evidence that banks’ rational response to dynamic implicit cost is a key determinant of borrowing from the Federal Reserve Discount Window. Our results reveal a significantly positive, systematically declining, and plausible structure of implicit cost due to past borrowing which extends back to nine biweekly reserve maintenance intervals. The findings indicate that changes in aggregate adjustment borrowing vary positively with respect to interest rate spread changes, and are negatively affected by changes in past as well as expected future borrowing. Further tests fail to reject identical lag structures in past and expected future borrowing, providing additional evidence for the dynamic optimization framework based upon progressive pressure. A version of the model restricting implicit cost to a geometrically declining distributed lag also fits the data well. This formulation produces a closed form solution, with borrowing positively affected by the current spread and negatively related to the adjacent past and expected future spreads. Estimation by GMM generates empirical support for the equation expressed in either levels or first differences of borrowing and the spread. Overall, this study offers considerable empirical validation for the dynamic behavior of Discount Window borrowing put forth by Goodfriend (1983).
D. H. Dulkowsky, Discouni window borrowing
117
Appendix Biweekly
aggregate
discount
borrowing
(RB) and interest
rate spread
(K) data.a
Month
Date
Year
RB
K
DEC JAN JAN JAN FEB FEB MAR MAR APR APR MAY MAY JUN JUN JUL JUL JUL AUG AUG SEP SEP OCT OCT NOV NOV DEC DEC JAN JAN JAN FEB FEB MAR MAR APR APR MAY MAY JUN JUN JUL JUL JUL AUG AUG SEP SEP OCT OCT NOV NOV
19 2 16 30 13 27 13 27 IO 24 8 22 5 19 3 17 31 14 28 II 25 9 23 6 20 4 18
1984 1985 1985 1985 1985 1985 1985 1985 1985 1985 1985 1985 1985 I985 1985 1985 1985 1985 1985 1985 1985 1985 1985 1985 1985 1985 1985 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986
0.342 0.572 0.202 0.320 0.321 0.49 1 0.572 0.353 0.360 0.262 0.389 0.905 0.434 0.369 0.394 0.649 0.227 0.258 0.253 0.528 0.303 0.573 0.301 0.326 0.553 2.342 0.264 0.815 0.116 0.333 0.139 0.529 0.165 0.166 0.222 0.127 0.255 0.164 0.2C5 0.091 0.236 0.213 0.282 0.252 0.244 0.371 0.278 0.239 0.195 0.330 0.306
_ 0.155 0.207 0.255 0.320 0.515 0.485 0.575 0.565 0.565 0.075 0.270 0.102 0.175 _ 0.130 0.260 0.420 0.260 0.315 0.445 0.340 0.405 0.480 0.585 0.595 0.540 0.600 0.540 1.290 0.570 0.350 0.410 0.330 0.419 0.360 0.220 0.052 0.375 0.345 0.400 0.380 0.440 0.404 0.370 0.330 0.380 0.325 0.345 0.415 0.370 0.440 0.555
I 15 29 12 26 12 26 9 23 7 21 4 18 2 16 30 13 27 10 24 8 22 5 19
1986
I18
Month DEC DEC DEC JAN JAN FEB FEB MAR MAR APR APR MAY MAY JUN JUN JUL JUL JUL AUG AIJG SEP SEP OCT OCT NOV NOV DEC DEC DEC JAN JAN FEB FEB MAR MAR APR APR MAY MAY JUN JUN JUN JUL JUL AUG AUG SEP SEP OCT OCT NOV NOV NOV DEC
D.H. Dutknwsky,
Date
17 31 I4 28 11 25 11 25 8 22 6 20 17 15 29 12 26 9 23 21 4 18 2 I6 30 I3 27 10 24 9 23 6 20 4 18
15 29 I3 27 10 24 21 5 19 2 16 30 14
Discount window borrowing
Year
RB
1986 1986 1986 1987 1987 1987 1987 1987 1987 1987 1987 1987 1987 1987 1987 1987 1987 1987 1987 1987 1987 1987 1987 1987 1987 1987 1987 1987 1987 1988 1988 1988 1988 1988 1988 1988 1988 1988 1988 1988 1988 1988 1988 1988 1988 1988 1988 1988 1988 1988 1988 1988 1988 1988
0.179 0.170 0.868 0.262 0.427 0.104 0.300 0.108 0.170 0.295 0.579 0.953 0.365 0.57 1 0.152 0.269 0.164 0.226 0.156 0.306 0.233 0.245 0.497 0.342 0.119 0.102 0.104 0.079 0.253 1.394 0.122 0.089 0.117 0.i 72 0.116 0.201 0.217 0.246 O.i36 0.313 0.192 0.183 0.964 0.215 0.184 0.143 0.179 0.488 0.301 0.186 0.137 0.215 0.521 0.355
K 0.625 0.635 2.255 1.315 0.570 0.680 0.580 0.590 0.610 0.670 0.835 1.400 I .260 1.225 1.225 1.200 1.080 1.080 1.165 1.250 1.181 i ,235 I.495 1.480 0.730 0.125 0.835 0.710 0.780 0.915 0.775 0.575 0.645 0.555 0.560 0.720 0.870 0.835 1.030 1.275 1.400 1.585 I .700 I.815 1.724 1.605 1.650 i ,650 1.810 1.770 1.830 1.785 1.885 2.050
D.H.
Dutkowsky.
Discount
Month
Date
Year
DEC JAN JAN FEB FEB MAR MAR APR APR MAY MAY MAY JUN JUN JUL JUL AUG AUG SEP SEP OCT OCT NOV NOV
28 II 25 8 22 8 22 5 19 3 17 31 14 28 12 26 9 23 6 20 4 18 I 15
1988 1989 1989 1989 1989 1989 1989 1989 1989 1989 1989 1989 I989 1989 1989 1989 1989 1989 1989 I989 1989 1989 1989 1989
“Federal Reserve: Discount Window borrowings spread data are reported in percent per year.
window borrowing
RB
0.243 0.747 0.438 0.400 0.267 0.435 0.287 0.335 0.423 0.316 0.197 0.128 0.08 1 0.212 0.088 0.082 0.124 0.220 0.03 1 0.155 0.420 0.292 0.042 0.105 are measured
in billions of dollars;
119
K
2.365 2.650 2.595 2.630 2.830 2.85 1 2.845 2.740 2.885 2.870 2.790 2.790 2.405 2.530 2.445 2.190 1.895 2.025 1.930 2.005 2.100 1.850 1.760 1.675 interest rate
References Christiano, L.J. and MS. Eichenbaum, 1992, Liquidity effects, monetary policy, and the business cycle, Discussion paper no. 70 (Federal Reserve Bank of Minneapolis, Minneapolis, MN). Dutkowsky, D.H., 1984, The demand for borrowed reserves: A switching regression model, Journal of Finance 39, 4077424. Dutkowsky, D.H. and W.G. Foote, 1985, Switching, aggregation, and the demand for borrowed reserves, Review of Economics and Statistics 67. 331-335. Dutkowsky, D.H. and W.G. Foote, 1988, Forecasting discount window borrowing, International Journal of Forecasting 4, 5933603. Frost, P.A. and T.J. Sargent, 1970, Money-market rates, the discount rate, and borrowing from the Federal Reserve, Journal of Money, Credit and Banking 2, 56-82. Gallant, A.R., 1987, Nonlinear statistical models (Wiley, New York, NY). Garber, P.M. and R.G. King, 1983, Deep structural excavation? A critique of Euler equation methods, Working paper no. 83314 (University of Rochester, Rochester, NY). Garfinkel, M.R., 1989, Monetary trends (Federal Reserve Bank of St. Louis, St. Louis, MO). Garfinkel, M.R.. 1990, The FOMC in 1989: Walking a tightrope, Federal Reserve Bank of St. Louis Review 72, 17~ 35. Goldfeld, SM. and E.J. Kane. 1966. The determinants of member bank borrowing: An econometric study, Journal of Finance 21.499-5 14. Goodfriend, M.S., 1983, Discount window borrowing, monetary policy, and the post-October 6, 1979 Federal Reserve operating procedure, Journal of Monetary Economics 12, 3433356. Goodfriend, M.S., 1992, Information-aggregation bias, American Economic Review 82, 5088519.
Newey, W.K. and K.D. West, 1987, Hypothesis testing with efficient method of moments estimation, International Economic Review 28, 777-787. Pearce, D.K., 1990, Discount window borrowing and Federal Reserve operating procedures, Working paper (North Carolina State University, Raleigh, NC). Peristiani, S., 1991, The model structure of discount window borrowing, Journal of Money. Credit and Banking 23, 14.~34. Sargent, T.D.. 1987, Macroeconomic theory (Academic Press, New York, NY). Sternlight, P.D., C. Edwards, R.S. Hilton, and A. Meulendyke. 1991, Monetary policy and open market operations during 1990, Federal Reserve Bank ofNew York Quarterly Review 16.52-78. Sternlight, P.D., J. Gluck, C. Edwards, and A. Meulendyke, 1989, Monetary policy and open market operations during 1988, Federal Reserve Bank of New York Quarterly Review 14. 83- 102. Sternlight, P.D., A. Meulendyke, S. Krieger, R. VanWicklen. D. Schindewolf, and L. Raftery. 1986, Monetary policy and open market operations during 1985, Federal Reserve Bank of New York Quarterly Review I I, 34-53.
of Monetary
Economics
32 (1993)
1055120.
North-Holland
Dynamic implicit cost and discount window borrowing An empirical
investigation
Donald H. Dutkowsky* S~racusrUnivemity, S)‘racuse. NY Received
May 1991. final version
13244, USA received
May 1993
This paper investigates dynamic implicit cost in aggregate Discount Window borrowing. We utilize Goodfriend’s formulation, with nonprice rationing based upon the bank’s current request and its past borrowing record. Estimation by Generalized-Method-of-Moments reveals a positive, systematic implicit cost structure which dampens over time. Significance extends back to nine biweekly reserve maintenance intervals. We also specify implicit cost as a geometrically declining distributed lag of past borrowing and derive a semi-reduced form for borrowed reserve demand. Empirical results support this specification as well.
Key words: Discount
window
borrowing;
Central
bank policy; Operating
target
1. Introduction
Aggregate Discount Window borrowing continues to play a central role in the Federal Reserve’s conduct of monetary policy. As stated in publications such as Sternlight et al. (1991), the Fed has generally maintained the borrowed reserve operating procedure begun in October 1982. In recent years, though, widespread concern has been expressed regarding a breakdown in the ability to predict aggregate borrowed reserves. Particular attention has focused on the relationship between aggregate Discount Window
Correspondence fo: Donald H. Dutkowsky, Department ship and Public Affairs, Syracuse University, Syracuse,
of Economics, Maxwell School of CitizenNY 13244-1090. USA.
*I wish to thank Yucong Zhao for research assistance as well as an anonymous referee and the editors for numerous substantial comments on previous drafts. This paper also benefited from discussions with Josh Feinman, Bill Foote, Gary Gillum, Duke Kao, Masao Ogaki. Steve Peristiani, and John Wenninger. However, the views are solely those of the author and do not necessarily reflect official positions of the Federal Reserve Bank of New York or the Federal Reserve System.
0304-3932/93/$06.00
(c) 1993-Elsevier
Science Publishers
B.V. All rights reserved
borrowing and the spread between the Federal Funds rate and the Discount rate, Sternlight et al. (1989, p. 98) write: ‘Examination of the relationship back to 1984 shows that a series of shifts has . . . resulted in reduced discount window borrowing for given spreads of the funds rate over the discount rate.’ Garfinkel (1989, p. 1) states: ‘The recent instability . . is also demonstrated by noting that the correlation between the spread and adjustment and seasonal borrowings over the past year was 0.07, down considerably from the previous year’s correlation of 0.53.’ Garfinkel (I 990, p. 19) writes: ‘Toward the end of 1988 and continuing into 1989, however, this [operating] procedure was compIicated by the demand for borrowed reserves and the federal funds rate. Specifically, the willingness of depository institutions to borrow reserves, for given federal funds and discount rates, was declining unexpectedly.’ Sternlight et al. (1991) state that this difficulty continued throughout 1990. At the same time, Christian0 and Eichenbaum (1992) place Discount Window borrowing at the heart of their resolution of an empirical puzzle involving the liquidity effect of monetary policy. Using impulse function analysis, the authors find that positive innovations in nonborrowed reserves cause sharp declines in short-term interest rates, even though the same interest rates increase in response to positive shocks in the monetary base or Ml. They attribute the strikingly different dynamic movements to the behavior of Discount Window borrowing, showing that innovations in the nonborrowed base generate an interest rate response similar to those of nonborrowed reserves. Christian0 and Eichenbaum contend that an integrated explanation of the liquidity effect must include a description of the linkage between borrowed reserves and interest rates within a detailed model of Discount Window borrowing. In this regard, Marvin Goodfriend’s (1983) study offers an intuitively appeaiing means of understanding bank borrowing from the Discount Window. Based upon well-known Federal Reserve attitudes, Goodfriend argues that implicit cost due to nonprice rationing at the Discount Window depends upon the past record of borrowing as well as the magnitude of the current request. Consequently borrowing choice becomes a dynamic optimization problem, involving past and expected future borrowing along with the interest rate spread. The Goodfriend study has opened up several veins of research which employ versions of the model to examine monetary policy, interest rates, and banking. This paper empirically investigates the role of dynamic implicit cost in aggregate Discount Window borrowing. Despite the Goodfriend model’s substantial insights, no studies have reported empirical estimates. Our mode! generalizes Goodfriend (1983) to consider bank reluctance stemming from its borrowing record over a number of past periods. We estimate the model by Generalized-Method-of-Moments (GMM), with biweekly reserve maintenance data spanning 1985-89. The findings produce evidence supporting a positive, systematic implicit cost structure due to past borrowing which dampens over time. Significance extends
back to nine reserve maintenance intervals. The structure of implicit cost implies a borrowing function in which changes in aggregate Discount borrowing are positively affected by changes in the interest rate spread. Borrowing changes are also inversely related to changes in banks’ past borrowing as well as changes in expected future borrowing. The above results suggest that implicit cost can alternatively be specified as an infinite, geometrically declining distributed lag of past borrowing. Within this formulation, a semi-reduced form for borrowed reserve demand can be analytically derived. Estimation by GMM provides evidence supporting this specification as well. The paper proceeds as follows. Section 2 presents the extended Goodfriend model. Issues involved with GMM estimation of aggregate Discount Window borrowing are discussed in section 3. Section 4 reports estimation findings associated with the extended model. The mode1 with geometrically declining implicit cost is presented in section 5. Section 6 concludes the paper.
2. The extended Goodfriend model We begin by specifying the Discount Window borrowing decision. Along the lines of Goodfriend (1983) the representative bank uses the Discount Window to maximize the expected total discounted profit of borrowing. Banks form expectations rationally. The ‘revenue’ of borrowing from the Federal Reserve consists of interest saved by not having to pay the more expensive Federal Funds rate. The cost of Discount Window borrowing has two components. Besides interest based upon the discount rate, banks face implicit cost due to Federal Reserve nonprice rationing at the Discount Window. Described comprehensively in Goodfriend (1983) implicit cost reflects bank reluctance to borrow from the Federal Reserve due to increased surveillance and administrative counseling. Regulations and writings regarding adjustment borrowing indicate that the Fed discourages continuous borrowing as well as large requests. The optimization problem calls for choosing a sequence of Discount Window borrowing to maximize
V, = E, f h’([fiRB, 1=0
- idisc,RB,
- C,),
(1)
where V0 denotes expected total discounted profits at date 0, RB, refers to the level of date t Discount Window borrowing, rg is the date t Federal Funds rate, idisc, denotes the Discount rate at date t, C, refers to date t implicit cost, h is the
D.H. Dutkowsky, Discount window borrowing
108
rate of time discount with 0 < b < 1, and E, = E( IQ,), the conditional expectations operator based upon information available at date t. Implicit cost stems from both the current borrowing request and past borrowing extending back n reserve maintenance periods. With this assumption, date t implicit cost can be modeled as C, = aoRB, + c,RB:
+ 2coRB,
2
diRB,_i,
i=l
co,di > 0,
i = 1,2, . . , PI.
(2)
The parameters a0 and co measure marginal implicit cost due to the current request. The a, parameter may have either positive or negative sign. But positive marginal implicit cost implies that its magnitude is dominated by the co and d parameters. The di parameter describes reluctance due to borrowing i periods in the past. Greater emphasis attached to more recent borrowing implies a declining lag structure among the d parameters, i.e., dl > d2 > . > d,. This specification of implicit cost generalizes Goodfriend’s model, in keeping with his basic theme of ‘progressive pressure’ in nonprice rationing at the Discount Window. Assuming that the representative bank borrows from the Federal Reserve in each period, substituting (2) into (1) and performing the optimization yields the following Euler equation for date t borrowing: K, = a0 - 2co
RB, + i:
diRB,-i
+ ~ b’diE,RB,+i
(3)
i=l
i=l
where K, = iflt - idisc, is the interest rate spread.’ The bank equates the marginal implicit cost of date t borrowing with the marginal monetary saving. Marginal implicit cost is based upon the current magnitude, reluctance due to past borrowing, and the discounted effect of current borrowing on future requests. Note that (3) can be rearranged in terms of date t borrowing as follows: RB, = - ao/(2co) + K,/(~c,)
-
f: diRB,-i i=l
Within this structural equation, current borrowing date t spread. The function also reveals a distributed
-
i
b’diE,RB,+i.
(4)
i=l
depends positively on the lag on past borrowing and
‘Despite the fact that relatively few banks utilize the Discount Window during any particular period, we follow Goodfriend (1983) and subsequent theoretical work on borrowing that emphasizes intertemporal behavior by assuming interior solutions.
D.H. Dutkowsky. Discount window borrowing
109
a distributed lead on expected future borrowing. Implicit cost implies a negative relationship with current borrowing for both the leads and lags, with the effects gradually dampening out. Notably, progressive pressure as applied by the Fed implies that lag structures for past and future borrowing are identical, except for the rate of time discount. When n = 1, the model reduces to the formulation in Goodfriend (1983). In this case, (3) can be solved for date t borrowed reserve demand as
W = glRB,-
1 -
Cg,/@cod,)l
f
i=O
(l/gz)‘C- aO + EtKt+i19
(5)
where g1 = [-
1 + (1 - 4hd:)“‘]/(2hdI)
and
g2 = l/bg,.
(6)
If 1 > (1 + b)dr , then the roots to the Euler equation have the desired stability property, g2 < - 1 < g, < 0 [see, e.g., Sargent (1987)]. While borrowing is positively affected by an increase in the current spread, a rise in the expected future spread causes current borrowing to fall. Since the bank would anticipate greater borrowing in date t + 1 due to the favorable spread, it borrows less now to reduce expected future implicit cost.
3. Econometric issues Given the dynamic behavior implied by Goodfriend’s model, GMM emerges as a leading procedure to estimate borrowed reserve demand.2 The technique is designed to consistently estimate parameters of Euler equations arising from intertemporal optimization under rational expectations. Assuming there are no shocks to the implicit cost function, the procedure gets by without specifying additional processes for expectations determination or solving for a semireduced form.3 We illustrate how to implement GMM using the model with single-period implicit cost. To sidestep possible difficulties with nonstationarity, we begin by first differencing the Euler equation. This transformation expresses the Euler equation in terms of stationary variables while preserving the behavioral
‘Past econometric research on aggregate Discount Window borrowing emanates from Goldfeld and Kane (1966). Most work features a borrowing function positively related to the current-period interest rate spread, derived from a static optimization framework. Extensions of the Goldfeld and Kane model focus on nonlinearity and heteroskedasticity between aggregate borrowing and the spread within the borrowing function [see, e.g., Dutkowsky (1984). Dutkowsky and Foote (1985), Dutkowsky and Foote (1988). and Peristiani (1991)]. Frost and Sargent (1970) examine seasonal effects using spectral analysis. ‘See Garber
and King
(1983).
110
D. H. Durkowsky,
Discount
window
borrowing
relationship. A Dickey-Fuller test indicates that the adjustment borrowing series is stationary, but it cannot reject the null hypothesis of a unit root in the spread. The test performed on changes in the spread rejects the unit root. Setting n equal to 1 in (4), differencing the equation, and rearranging, we have - DK,/(2c0)
+ DRB, + d, DRB,_,
+ bd, DRB,, , = De,+ ,,
(7)
where
De,+1 = hd, C(W+
I
-
E,RB,+,) - (RB, - E,-,RB,)I,
(8)
and D denotes the differencing operator, Dx, = x, - xf_, . GMM can be applied to consistently estimate the parameter vector 0 = (b, co, d, )‘, since rational expectations imply that E,_ , .x,_, De,+ 1 = 0 for and any stationary variable .x,- , in Qf_, . Given a sample of T observations a q-dimensional vector of instruments from !L?_, denoted by Z,_ 1, with q greater than or equal to the dimensionality of the parameter vector, the procedure computes the vector of sample first moments of Z,_ I De,+, . Substituting for De,+, using (8) the q x 1 vector is given by r+(O)
i Z,.. , [t=I
= (l/T)
DK,/(2c,)
+ DRB,
+ d, DRB,- , + bd, DRB,, ,I. The GMM
estimator
JT(@o, I’) =
of 0, denoted
by Oo, minimizes
(9) the function
C~~,(~o)l’~-‘CT~,(~o)l,
(10)
where I/ is a q x q positive definite matrix further described in Gallant (1987). When the dimensionalities of the instrument and parameter vectors are equal and the partial derivative matrix of mr(O) with respect to 0 has full rank, the function is exactly identified. In this instance, GMM imposes orthogonality and generates estimates by solving mr(Oo) = 0. For the case in which the number of instruments exceeds the number of parameters, the minimized JT value can be utilized to test for the overidentifying restrictions. As Gallant (1987) explains, one rejects the hypothesis that the model is correctly specified when the statistic exceeds the upper LYx 100% critical point of the chi-squared distribution with degrees of freedom equal to the difference in dimensionality between the instrument and parameter vectors.
4. The estimated model The sample consists of biweekly data covering July 3, 1985 through from December 19, 1984 to November 1989. We also use observations
July 12, 15, 1989
to form the series for lagged and future borrowing. The two-week interval corresponds to the Federal Reserve’s reserve maintenance period instituted on February 15, 1984. The sample falls entirely within the borrowed reserve targeting procedure and begins after the substantial adjustment borrowing, later coverted to extended credit, during mid-1984 related to the Continental Bank of Illinois. The interest rate spread K is formed by subtracting the discount rate from the Federal Funds rate. The variable is greater than zero throughout the sample period except for two observations prior to July 1985, suggesting that nonlinearity in the borrowings-spread relationship does not present a concern [see, e.g., Dutkowsky and Foote (1985) Peristiani (1991)]. We investigated nonlinearity by estimating the models with changes in the spread squared and cubed; in all cases, the additional terms were insignificant. Aggregate adjustment borrowing from the Discount Window serves as the measure of RB. Despite the fact that the Fed targets adjustment plus seasonal borrowing, our formulation focuses on borrowing due to short-term reserve adjustment. Spread and adjustment borrowings data are reported in the appendix. A review of the adjustment borrowing series reveals several instances of large borrowing within an individual period, even though borrowing in adjacent dates follows usual patterns. A special borrowing situation for the reserve maintenance period ending December 4,1985, has been documented by the Fed. The large borrowing resulted from the massive securities failure of the Bank of New York, along with the problem on settlement night with the Clearing House Interbank Payments System. Following Federal Reserve estimates described in Sternlight et al. (1986) we subtract 1.6 billion from adjustment borrowing for this observation. Since we encounter no other accounts of unusual borrowing, we leave the remaining observations unchanged.4 Table I reports GMM estimates of the implicit cost parameters in eq. (4). The computer algorithm utilizes PROC MATRIX in Version 5.18 of SAS and follows the GMM example in Gallant (1987). The rate of time discount is restricted to equal 0.999. We provide empirical results for a lag length of nine reserve maintenance intervals.” The instrument vector contains 27 elements: a constant term, lagged borrowing changes ranging from two to thirteen periods, and current and lagged changes in the spread up to and including a lag of thirteen. We do not include
4Three other spikes are the July 13, 1988, January 13, 1988, and December 31, 1986 intervals. The latter date is the only distinctive observation for the interest rate spread, a 160-basis-point increase followed by a nearly 100.basis-point drop. Pearce (1990) provides an explanation for these outliers, contending that banks may borrow extra amounts at the end of the year in order to ‘window dress’ their balance sheets. Since this behavior does not fit easily into the intertemporal framework and the role of dummy variables within the Euler equation is unclear, we do not consider it here. ‘Estimated
implicit
cost parameters
for lag lengths
beyond
nine periods
were insignificant.
112
Table 1 Implicit
~.
~-~
L‘0
--
~
cost estimates: Extended ..____ .__ ~ Extended Goodfriend model
Goodfriend -... -.. _
model.” ~~~ _-.___ Test for progressive
4.929
pressure
4.539 (2.754)
(I ,826) d,
0.855 (0.043)
0.826 (0.096)
0.883 (0.114)
dz
0.742 (0.062)
0.702 (0.100)
0.796 (0.208)
d3
0.698 (0.059)
0.682 (0.097)
0.900 (0.238)
d,
0.571 (0.058)
0.545 (0.109)
0.874 (0. t 94)
ds
0.446 (0.063)
0.403 (0.109)
0.630 (0.203)
&
0.370 (0.063)
0.381 (0.092)
0.539 (0.238)
d,
0.267 (0.054)
0.355 (O.lcto)
0.568 (0.247)
ds
0.207 (0.059)
0.286 (0.121)
0.550 (0.245)
do
O.llI (0.037)
0.121 (0.092)
0.271 (0.168)
Jr statistic
I 1.020 (0.856)
6654 (0.574)
“The table reports GMM estimates or the implicit cost parameters defined in (2). estimated within the Euler equation (4) in first differenced form. The test for progressive pressure presents estimates of the model with unrestricted lead and lag coefficients. The rate of time discount (h) is restricted to equal 0.999. The sample consists of biweekly data from 7/3/X5 to 7/12/89. Asymptotic standard errors appear in parentheses under the estimated parameters. P-values are in parentheses under the JT statistics. which test for overidentifying restrictions.
the change in borrowing lagged one period, as the variable may not be in the date t - 1 information set. Following Goodfriend (1992), since the Fed reports aggregate borrowing data with a lag, it is unlikely that individual banks have access to contemporaneous information for this variable. The accessability of interest rate data suggests that this problem does not occur with the spread.’
‘Although the change in the current spread is not an element of s2,_ r, we experience difficulty in finding suitable instruments that capture movement in the first difference of the spread. Removing DK, from the instrument set results in a noticeable loss of efficiency on the co estimate, although the remaining parameters are not qualitatively affected. GMM estimates with and without the DK, variable do not show noticeable change in the Jr statistic.
Estimates of the implicit cost parameters appear in the first column of table 1. The second two subcolumns report estimates of the model with unrestricted lead and lag coefficients. The findings for the restricted estimates reveal strong evidence of dynamic implicit cost. All the estimated parameters for reluctance due to past borrowing are greater than zero and significant. The rate of decline in marginal implicit cost is roughly constant over the lag structure, equaling approximately 0.800 per period. The estimated parameter for marginal implicit cost due to current-period borrowing (co) is also significantly positive. Estimation without restricting the discount factor results in nearly identical implicit cost estimates. Although the estimated rate of time discount exceeds one, the estimate is insignificant from unity. The last row reports the JT statistic, the minimized value of the objective function described in (10). To test overidentifying restrictions for the models in table 1, the statistic is compared to chi-squared with 27 - m degrees of freedom, where m denotes the dimensionality of the parameter vector. The p-values reported in parentheses indicate that the test cannot reject the null hypothesis for either model at any reasonable level of significance. This result holds for all the estimated models reported here. The empirical results support the dynamic optimization framework centering around progressive pressure. As mentioned above, the last two subcolumns of table 1 report estimates in which effects of past and expected future borrowing are not constrained to be equal. Estimated implicit cost parameters for past borrowing (left subcolumn) follow closely the restricted estimates of the nine-lag model. Implicit cost estimates for expected future borrowing (right subcolumn) exceed their counterparts for past borrowing and have higher standard errors, but all have positive sign and nearly all are significantly different from zero. Moreover, simple t tests cannot reject at any reasonable level of significance the individual null hypotheses that the d parameter for borrowing i periods in the past equals the effect of expected borrowing i periods in the future, for i= I,2 )...) 9. Global tests for identical lag structures corroborate this result. The D test discussed in Newey and West (1987) equals the difference between the Jr statistics for the restricted and unrestricted models. Under the null hypothesis the statistic follows a chi-squared distribution with degrees of freedom equal to the difference in dimensionality between the parameter vectors. Performing the test with the models in table 1 yields a statistic of 4.366, indicating the progressive pressure cannot be rejected at any reasonable significance level (for example, the IO percent critical point of the chisquared distribution with 9 degrees of freedom equals 14.68). Gallant (1987) argues that the test should be conducted by estimating both equations with the V matrix of the unrestricted model and subtracting the resulting minimized values. This test generates a statistic of 3.084, also insignificant at the IO percent level.
In examining robustness, we find that the implicit cost estimates show noticeable sensitivity when we estimate the Euler equations using levels of the spread and borrowing. But the results involving changes in RB and K stand up well to estimation with more conventional procedures. Similar findings are obtained when we estimate the borrowing function by Two-Stage-LeastSquares using the same instrument set as the GMM estimations and the h = 0.999 restriction. OLS estimation produces more striking results. Even without restricting the rate of time discount, significantly negative and nearly identical lag structures emerge for changes in past and future borrowing.
5. Borrowing under geometrically
declining implicit cost
The estimated model of progressive pressure, with its nine-period leads and lags, does not easily solve analytically for a demand function for Discount Window borrowing. But characteristics of the implicit cost structure indicated by the previous empirical results ~ the long lag length along with positive and declining reluctance - suggest an extension that produces a neat closed form solution for borrowing. Specify implicit cost as a geometrically decreasing function of past borrowing given by C, = a,RB, + coRBf + 2c,,RB, i
dbRB,_i,
(11)
i=l
where d, refers to the constant rate of decay parameter resulting Euler equation can then be expressed as
with 0 < d,,
I. The
E,K, - a,, - 2c0(l - d,B)-‘E,RB, - 2cobdoB-‘(1
- bd,B-l)mlE,RB,=
0.
with B denoting the backstep operator, B’&x,+~ = E,x,+~_~. Operating through by (1 - d,B)( 1 - hd,, B - ’ ) and performing braic rearrangement, we obtain
RB, = ho + h,E,K,+,
+ h,K, + hjK,m1,
(12)
some alge-
(13)
where ho = - a,,(1 - d,)(l h, =
- hd0)/[2c0(l
- bdi)],
- MolC2co(l -
bd;)l,
(14)
(15)
(1 +
h2 = h, = h2 > 0
hd;)l[2c,u - hG)l, - &/[2co(l
and
-
bdi?)l,
(16) (17)
h,, h3 < 0.
A highly manageable semi-reduced form results. The solution reveals that Discount Window borrowing responds positively to increases in the date t spread. Negative effects of the expected date t + 1 spread as well as the date r - 1 spread reflect dynamic implicit cost. This formulation implies that only the past and expected future spreads in adjacent periods enter into current borrowing. An estimated rate of time discount can be computed as h = h,/h,. Estimates of this formulation of the borrowing function appear in table 2.’ The time discount parameter is again restricted to equal 0.999. The model with borrowing and the spread expressed in levels (column 1) generates estimates which conform to a priori signs and are significantly different from zero. The results hold up reasonably for the estimated equation in first differences. Reported in the second column, the findings also indicate a significantly positive current spread effect and significantly negative coefficients for the past and expected future spread. Despite the similarity of the estimated equations in table 2, some further examination gives an edge to the model expressed in levels. From (14))( 17) and the coefficients in table 2, the model in levels generates plausible estimates of the structural parameters of a, = -3.470, c0 = 2.097, and do = 0.597. Real-valued estimates could not obtained from the model in first differences.* In addition, we computed the estimated interest rate spread forecast error for both models using (13) and checked for serial correlation using time series methods. Based upon several diagnostic criteria within PROC ARIMA of SAS, we find no distinctive evidence to counter the hypothesis of white noise for the model estimated in levels. The evidence indicates that the estimated forecast error from the model in first differences exhibits serial correlation, primarily reflecting a partial autocorrelation of -0.48 between the error and its one-period lag.9
‘When estimating the model with levels, we add the change in borrowing lagged one period to the instrument set. In examining (13) for GMM estimation it is easily shown that for the function with borrowings and the spread expressed in levels, orthogonality holds for any variable in 0,. ‘This results because the solution for do comes from a quadratic equation. and complex roots are obtained when we substitute the estimates of the model in first differences. We report the smaller root for the model in levels. ‘The specific criteria consist of a scan of the partial autocorrelations, an autoregression of lags from l-24 periods. and the Lyung-Box statistic calculated for lags of 6, 12, 18 and 24 periods. Although serial correlation violates the conceptual property of the forecast error as predicted by rational expectations, GMM still produces consistent parameter estimates [see, e.g., Gallant (1987)].
D.H. Dutkowsky. Discount window borrowing
116
Table 2 Estimated
borrowing
function:
Geometrically
declining
implicit cost.”
Units of RB and K
ho
Levels
First differences
0.209 (0.03 1)
0.004 (0.010)
h,
- 0.221 (0.042)
- 0.244 (0.055)
h2
0.502 (0.045)
0.343 (0.052)
h3
- 0.221 (0.042)
- 0.245 (0.055)
19.639 (0.768)
19.837 (0.706)
Jr statistic
“The table reports GMM estimates restricted to equal 0.999. The sample standard errors appear in parentheses under the JT statistics, which test for
of the borrowing function (13). The rate of time discount (b) is consists of biweekly data from 7/3/85 to 7/12/89. Asymptotic under the estimated parameters. P-values are in parentheses overidentifying restrictions.
6. Conclusion
This study provides empirical evidence that banks’ rational response to dynamic implicit cost is a key determinant of borrowing from the Federal Reserve Discount Window. Our results reveal a significantly positive, systematically declining, and plausible structure of implicit cost due to past borrowing which extends back to nine biweekly reserve maintenance intervals. The findings indicate that changes in aggregate adjustment borrowing vary positively with respect to interest rate spread changes, and are negatively affected by changes in past as well as expected future borrowing. Further tests fail to reject identical lag structures in past and expected future borrowing, providing additional evidence for the dynamic optimization framework based upon progressive pressure. A version of the model restricting implicit cost to a geometrically declining distributed lag also fits the data well. This formulation produces a closed form solution, with borrowing positively affected by the current spread and negatively related to the adjacent past and expected future spreads. Estimation by GMM generates empirical support for the equation expressed in either levels or first differences of borrowing and the spread. Overall, this study offers considerable empirical validation for the dynamic behavior of Discount Window borrowing put forth by Goodfriend (1983).
D. H. Dulkowsky, Discouni window borrowing
117
Appendix Biweekly
aggregate
discount
borrowing
(RB) and interest
rate spread
(K) data.a
Month
Date
Year
RB
K
DEC JAN JAN JAN FEB FEB MAR MAR APR APR MAY MAY JUN JUN JUL JUL JUL AUG AUG SEP SEP OCT OCT NOV NOV DEC DEC JAN JAN JAN FEB FEB MAR MAR APR APR MAY MAY JUN JUN JUL JUL JUL AUG AUG SEP SEP OCT OCT NOV NOV
19 2 16 30 13 27 13 27 IO 24 8 22 5 19 3 17 31 14 28 II 25 9 23 6 20 4 18
1984 1985 1985 1985 1985 1985 1985 1985 1985 1985 1985 1985 1985 I985 1985 1985 1985 1985 1985 1985 1985 1985 1985 1985 1985 1985 1985 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986
0.342 0.572 0.202 0.320 0.321 0.49 1 0.572 0.353 0.360 0.262 0.389 0.905 0.434 0.369 0.394 0.649 0.227 0.258 0.253 0.528 0.303 0.573 0.301 0.326 0.553 2.342 0.264 0.815 0.116 0.333 0.139 0.529 0.165 0.166 0.222 0.127 0.255 0.164 0.2C5 0.091 0.236 0.213 0.282 0.252 0.244 0.371 0.278 0.239 0.195 0.330 0.306
_ 0.155 0.207 0.255 0.320 0.515 0.485 0.575 0.565 0.565 0.075 0.270 0.102 0.175 _ 0.130 0.260 0.420 0.260 0.315 0.445 0.340 0.405 0.480 0.585 0.595 0.540 0.600 0.540 1.290 0.570 0.350 0.410 0.330 0.419 0.360 0.220 0.052 0.375 0.345 0.400 0.380 0.440 0.404 0.370 0.330 0.380 0.325 0.345 0.415 0.370 0.440 0.555
I 15 29 12 26 12 26 9 23 7 21 4 18 2 16 30 13 27 10 24 8 22 5 19
1986
I18
Month DEC DEC DEC JAN JAN FEB FEB MAR MAR APR APR MAY MAY JUN JUN JUL JUL JUL AUG AIJG SEP SEP OCT OCT NOV NOV DEC DEC DEC JAN JAN FEB FEB MAR MAR APR APR MAY MAY JUN JUN JUN JUL JUL AUG AUG SEP SEP OCT OCT NOV NOV NOV DEC
D.H. Dutknwsky,
Date
17 31 I4 28 11 25 11 25 8 22 6 20 17 15 29 12 26 9 23 21 4 18 2 I6 30 I3 27 10 24 9 23 6 20 4 18
15 29 I3 27 10 24 21 5 19 2 16 30 14
Discount window borrowing
Year
RB
1986 1986 1986 1987 1987 1987 1987 1987 1987 1987 1987 1987 1987 1987 1987 1987 1987 1987 1987 1987 1987 1987 1987 1987 1987 1987 1987 1987 1987 1988 1988 1988 1988 1988 1988 1988 1988 1988 1988 1988 1988 1988 1988 1988 1988 1988 1988 1988 1988 1988 1988 1988 1988 1988
0.179 0.170 0.868 0.262 0.427 0.104 0.300 0.108 0.170 0.295 0.579 0.953 0.365 0.57 1 0.152 0.269 0.164 0.226 0.156 0.306 0.233 0.245 0.497 0.342 0.119 0.102 0.104 0.079 0.253 1.394 0.122 0.089 0.117 0.i 72 0.116 0.201 0.217 0.246 O.i36 0.313 0.192 0.183 0.964 0.215 0.184 0.143 0.179 0.488 0.301 0.186 0.137 0.215 0.521 0.355
K 0.625 0.635 2.255 1.315 0.570 0.680 0.580 0.590 0.610 0.670 0.835 1.400 I .260 1.225 1.225 1.200 1.080 1.080 1.165 1.250 1.181 i ,235 I.495 1.480 0.730 0.125 0.835 0.710 0.780 0.915 0.775 0.575 0.645 0.555 0.560 0.720 0.870 0.835 1.030 1.275 1.400 1.585 I .700 I.815 1.724 1.605 1.650 i ,650 1.810 1.770 1.830 1.785 1.885 2.050
D.H.
Dutkowsky.
Discount
Month
Date
Year
DEC JAN JAN FEB FEB MAR MAR APR APR MAY MAY MAY JUN JUN JUL JUL AUG AUG SEP SEP OCT OCT NOV NOV
28 II 25 8 22 8 22 5 19 3 17 31 14 28 12 26 9 23 6 20 4 18 I 15
1988 1989 1989 1989 1989 1989 1989 1989 1989 1989 1989 1989 I989 1989 1989 1989 1989 1989 1989 I989 1989 1989 1989 1989
“Federal Reserve: Discount Window borrowings spread data are reported in percent per year.
window borrowing
RB
0.243 0.747 0.438 0.400 0.267 0.435 0.287 0.335 0.423 0.316 0.197 0.128 0.08 1 0.212 0.088 0.082 0.124 0.220 0.03 1 0.155 0.420 0.292 0.042 0.105 are measured
in billions of dollars;
119
K
2.365 2.650 2.595 2.630 2.830 2.85 1 2.845 2.740 2.885 2.870 2.790 2.790 2.405 2.530 2.445 2.190 1.895 2.025 1.930 2.005 2.100 1.850 1.760 1.675 interest rate
References Christiano, L.J. and MS. Eichenbaum, 1992, Liquidity effects, monetary policy, and the business cycle, Discussion paper no. 70 (Federal Reserve Bank of Minneapolis, Minneapolis, MN). Dutkowsky, D.H., 1984, The demand for borrowed reserves: A switching regression model, Journal of Finance 39, 4077424. Dutkowsky, D.H. and W.G. Foote, 1985, Switching, aggregation, and the demand for borrowed reserves, Review of Economics and Statistics 67. 331-335. Dutkowsky, D.H. and W.G. Foote, 1988, Forecasting discount window borrowing, International Journal of Forecasting 4, 5933603. Frost, P.A. and T.J. Sargent, 1970, Money-market rates, the discount rate, and borrowing from the Federal Reserve, Journal of Money, Credit and Banking 2, 56-82. Gallant, A.R., 1987, Nonlinear statistical models (Wiley, New York, NY). Garber, P.M. and R.G. King, 1983, Deep structural excavation? A critique of Euler equation methods, Working paper no. 83314 (University of Rochester, Rochester, NY). Garfinkel, M.R., 1989, Monetary trends (Federal Reserve Bank of St. Louis, St. Louis, MO). Garfinkel, M.R.. 1990, The FOMC in 1989: Walking a tightrope, Federal Reserve Bank of St. Louis Review 72, 17~ 35. Goldfeld, SM. and E.J. Kane. 1966. The determinants of member bank borrowing: An econometric study, Journal of Finance 21.499-5 14. Goodfriend, M.S., 1983, Discount window borrowing, monetary policy, and the post-October 6, 1979 Federal Reserve operating procedure, Journal of Monetary Economics 12, 3433356. Goodfriend, M.S., 1992, Information-aggregation bias, American Economic Review 82, 5088519.
Newey, W.K. and K.D. West, 1987, Hypothesis testing with efficient method of moments estimation, International Economic Review 28, 777-787. Pearce, D.K., 1990, Discount window borrowing and Federal Reserve operating procedures, Working paper (North Carolina State University, Raleigh, NC). Peristiani, S., 1991, The model structure of discount window borrowing, Journal of Money. Credit and Banking 23, 14.~34. Sargent, T.D.. 1987, Macroeconomic theory (Academic Press, New York, NY). Sternlight, P.D., C. Edwards, R.S. Hilton, and A. Meulendyke. 1991, Monetary policy and open market operations during 1990, Federal Reserve Bank ofNew York Quarterly Review 16.52-78. Sternlight, P.D., J. Gluck, C. Edwards, and A. Meulendyke, 1989, Monetary policy and open market operations during 1988, Federal Reserve Bank of New York Quarterly Review 14. 83- 102. Sternlight, P.D., A. Meulendyke, S. Krieger, R. VanWicklen. D. Schindewolf, and L. Raftery. 1986, Monetary policy and open market operations during 1985, Federal Reserve Bank of New York Quarterly Review I I, 34-53.