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A friction model of the prime
Journal of Banking and Finance 13 (1989) 127-135. North-Holland
A FRICTION MODEL OF THE PRIME* Shawn M. FORBES and Lucille S. MAYNE Case Western Reserve University, Cleveland, OH 44106, USA Received December 1986, final version received February 1988 This paper presents and estimates the parameters of a model whose basis is the rigidity of the prime rate, a friction model of the prime. Unlike traditional methods employed to analyze the prime, the estimator used here is asymptotically consistem and unbiased.
1. Introduction
Although the rates on most large loans to businesses are now tied directly to market rates, the greatest number of loans in the commercial and industrial category are under $1,000,000 and typically priced at some spread over prime [Brady (1985)]. That rate also serves as the basis for revolving loan commitments, consumer and construction loans, and syndicated Eurocurrency loans. Thus the prime is one of the most important indicators of lending rates. The distinguishing feature of the prime is its rigidity: the tendency to remain unchanged despite movements in exogenous rates. Explanations for this behavior are varied. For example, Goldberg (1982) has hypothesized, . . . without a dominant price leader, banks may be unwilling to initiate prime rate changes for modest changes in current money market conditions out of fear of being rebuffed. This unwillingness can be attributed to banks perceiving that their rivals will match any price decrease, but not match any increase, i.e., banks may perceive they have a kinked demand curve (p. 284). Alternatively, Hac!j,.'michalakis (1981) has suggested that the prime is rigid because prime-loan customers are a heterogenous group and there is no external standard for rating them per se, slowing the dissemination of information in that market. Further, since there is uncertainty over the *We wish to express our sincere thanks to two anonymous reviewers for their constructive comments and insights on earlier versions of this paper.
0378--4266/89/$3.50 ~) I989, Elsevier Science Publishers B.V. (Nol'th-HoUand)
S.M. Forbes and LS. Mayne, Friction model of the prime
128
course of future in',erest rates and many loans have floating rates tied to the prime, th~ potential costs of eliminating excess supply or demand through adjustment~ in the prime are greater than for auction markets. The combined effect of these factors is to make the prime slower to respond to changes in external conditions than market rates. Fischer (1982, p. 105) argues that ' . . . the prime exists not as a precise figure, which will be charged on a given loan, but as a benchmark or, as some have termed it, list price'. As a list price the prime may exhibit considerable rigidity without ~ecessarily affecting bank pricing flexibility or profitability once rate concessions, balance requirements, and fees and credits for other banks services are considered. ~ Although widely recognized, the rigidity of the prime has yet to be explicitly accounted for in the specification e f a statistical model of the setting of that rate. This study presents a statistical model appropriate to tke observed behavior of the prime; a model in which the dependent variable is, over some range, not related to the independent variables. Unlike traditional techniques that have been used to analyze the prime, the estimator presented here is consistent and unbiased. 2. Model The general specification commonly employed in analyzing the prime is of the form P, = ~ + r~fl:Rj + l~,,
(l)
whece the Ri are exogenous interest rates, 0~ is a spread, and #t is a random disturbance term. For example, Goldberg (1982, 1984) proposes Pt--°t + floCD, + fllCDt- I q-fl2CDt- 2 q'~t,
(2)
where CD, is the secondary market rate on large, negotiable certificates of deposit which serves as a proxy for the marginal cost of funds~ and 0c iS the desired spread over cost. There is, however, a serious statistical problem'with these models. The tendency of the prime to remain unchanged while exogenous rates fluctuate is inconsistent with the continuous density specification of (1). Therefore, the usual least-squares estimator is clearly inappropriate being biased and ;,nconsistent.z In looking for an altema0,ve model, it should be recognized that the prime's characteristic fig/dity means that no t For a discussion of list pri~ rigidity under oligopolistic competition characterized by barometric firms, see StigJer (1968, pp. 226-228). 2For a brief di,;cussion of the estimation problems of limited dependen~ variables see Judge et al. (1982, pp. 526-528).
S~M. Forbes and L.S. Mayne, Friction model of the prime '
I
40La2 '~,
II
2o1-
y
I
~
I
'~,_
RANGE OF NO CHANGE
,,~
0
129
i
I
50
=
I
I
I ! I =
100 (Pt-1- CDt )
~',~ "~'~,. I
150
=
'~
200
Fig. 1. Hypothetical relationship between changes in the prime and (.P,_l - CO,).
continuous density can explain its conditional distribution. Rather, the prime must be represented as a limited dependent variable whose value is at times unrelated to exogenous rates. In the model presented below, changes in the prime are used as the dependent variable rather than the level of that rate. This serves to focus attention on the incidence and magnitude of that which is noteworthy" i~ the prime, its relatively infrequent changes. It is assumed that the desirability and amount of any change are determined by a comparison of the spread between relevant exogenous rates and the previous period's prime? While this spread is within tolerable bounds the prime remains unchanged. If, however, conditions change sufficiently over time to cause the spread to lie outside this tolerable range, the prime is adjusted to restore the spread to withia the tolerable bounds. For example, iacreases in exogenous rates if large enough or frequent enough could cause the spread to fa~l below the lower threshold, triggering an increase in the prime. Conversely, decreases in exogenous rates would increase the spread and at some point trigger a reduction. Fig. 1 is a depiction of this hypothetical relationship, assuming that changes in the prime are li~ear functions of the spread between the lagged prime and exogenous rates. The main feature of this relationship is a range of spread values separating increases from decreases for which the prime remains unchanged. Rosett (1959; developed a model of relationships in which the dependent variable behaves as the prime does that he termed a 3We are grateful to an anonymous refere~ for pointing out that relevant market rates are simply those interest rates which reflect chang~~s in the supply and demand cc~nditions for shortterm bank loans.
S.M. Forbes and L.$. Mayne, Friction model of the prime
130
friction model. Letting APt represent the change in the prim,~ from period t - 1 to t, a friction model of the prime can be stated as
AP,= Ytt+e,,
if (Y~,+8,>0)
(3)
APfffiO
if (Yt,+stO)
(4)
APt = Yzt + st if ( Y2, + ~, < 0),
(5)
where st is a random error with zero mean and variance ¢2. That is, the positive segment of (¥1:=~-~(Pt_l-Rt)) represents expected increases in the prime, the negative segment of (Yzt= a s - / l ( P t - t-Rt)) expected decreases, and the difference between the intercepts oct and oc2 is a measure of the hypothesized discontinuity. Rosett presented a likelihood function for his friction model assuming a normally distributed error. However, his presentation is complicated by having the limiting value of the dependent variable potentially different for each observation and by"~reating the intercepts separately from other parameters. In the present study the use of changes in the prime makes only a single mass point necessary, zero, representing an unchanging prime. This simplification and the integration of the intercept terms into the parameter vector reduce the complexity of the likelihood function. Therefore, the likelihood function used in this paper differs from his and is presented in Appendix A. Since the likelihood function is the product of discrete probabilities and density functions, the first-order derivatives of the loglikelihood function are non-linear. Consequently, the likelihood equation must be solved in an iterative fashion. The well-known method of Newton is used here:
3. Data In this study the secondary-macket rate on 90.day CDs is used as the relevant exogenous rate. It is chosen over other rates, such .as those on 30day CDs, Fed funds, and T-bills, because of its success in a preliminary testing of the friction model and its predominance in other empirical work on the prime. Since it has ~ e n suggested that the prime is set reiativ¢ to a weighted average over time of such rates (e.g., Goldberg), the gjt of (3), (4), and (5) are expressed as
Yit=~-~Pt-t +AoCDt+AICD,_t+A~CD~_2,
j= l,2.
(6)
4The first and second derivatives necessary to the method of Newton are available from the authors upon request.
S.M. Forbes and L.S. Mayne, Friction model of the prime
131
20 ~ " PRIME
18 Z
z
<1:
- - - 9 0 - D A Y CD
:I ~t/-q
!6
/?,
ie
n- 14 LU
~ t
o. 12 IZ
,
,,, 10 nuJ I:L
V
8 6
i
.
.p
,
1 0 1 7 9 - 10182
4
I I I I I I I I I I l l l l l l l l
FED FUNDS RATE
NON-BORROWED RESERVES
BORROWED RESERVES
M O N E T A R Y REGIMES SOURCE: FEDERAL RESERVE BULLETIN, VARIOUS ISSUES
Fig. 2. Average monthly prime and 90-day CD rate: January 1977-August 1987.
The data are from various issues of the Federal Reserve Bulletin and cover slightly more than a decade, January 1977 through August 1987.5 The prime is the percent per annum reported as an average of the daily rates in effect over a given month. Per-annum rates on 90-day CDs are monthly averages of daily rates in the secondary market. The study period encompasses three distinct intervals in terms of Federal Reserve policy regimes: January 1977 through September 1979 when the operating target for implementing monetary policy was the fed funds rate; October 1979 through October 1982 when the target was non-borrowed reserves; and November 1982 through the remainder of the study period with borrowed reserves serving as the target. These changes in the implementation of monetary policy are discussed in detail in Axilrod (1985), Gilbert (1985), Mayer (1987), and Wallieh (1984). What is important for this study is that each regime has had implications for the level and volatility of interest rates as illustrated in fig. 2 and the following summary data:
Dates 01/77--09/79 10/79-10/82 1i/82-O8/87
Range of fluctuation (basis points) Prime CD 665 724 938 1,016 550 587
Number and type of changes in average monthly prime + 0 19 ll 2 14 4 19 10 28 19
5January 1977 is chosen as the beginning dale because it was then that the present form of the prime rate and CD series were introduced into the Federal Reserve Bulletin.
S.M. Forbes and L.5. Mayne, Friction model of the prime
132
Table 1 Estimates from the friction model. Monetary regimes Parameter
Fed funds rate .Jan. 77-Sep. 79
Non-borrowed reserves Oct. 79-Oct. 82
Borrowed reserves Nov. 82-Aug. 87
~t
41.928 (20.14) b
- 11.66 (51.24)
45.39a (19.65)
0~2
73.83 ° (27.11)
13.63 (50.63)
93.98" (22.66)
//
-0.488" (0.124)
-0.581" (0.134)
-0.611" (0.098)
go
0.531" (0.067)
0.558" (0.043)
0.609" (0.091)
,~t
-0.006 (0.098)
0.224" (0.087)
0.027 (0.116)
~.2
-0.074 (0.092)
-0.103 (0.089)
0.011 (0.089)
¢2
251.4(P
1,528.39~
295.36'
( 88.43)
(368.77)
(91.28)
.31.91" (11.05)
25.29 (14.17)
48.59a (10.78)
• 2 - 0ct
"Denotes significantly different from 0 .~t the 5% level, t'Standard errors are in parentheses.
In both the first and third intervals the average monthly prime exhibits rigidity, with rates generally rising in the first and falling in the latter. During the middle period the prime rose to historically high levels and was unusue!ly volatile, being unchanged in only four of thirty-seven months. 4. Results
The estimated coefficients for each of the three intervals are reported in table 1. In all cases ~2>a~, /~<0, and ,~o>0, consistent with the model's specification. The estimates obtained from the middle interval, however, differ markedly from those of the other two. 6 In those the differences between a~ and ~2 is significantly di~erent from zero, and each 0cj is also significantly different from zero. This is consistent with a range of no-change bounded on one end by a smaller positive threshold for increases and on the other by a larger one for decreases. In the middle interval, however, there is no significant difference between the ~j, nor are the 0c~significantly different from 6The hypothesig of a single set of parameters being appropriate to both the first and third periods cannot ~: rejected at the 5% level by means of a likelihood ratio test,
S.M. Forbes and L.S. Mayne, Friction model of the prime A
133
500
b-. W Z 0---
&
300
& ACTUAL O FORECAST
o
~ aA o
100
tum ~
~-UJ _z -
6
6 oO &
0 oo
A
I
. a
o
A
aA
oA
W
o>. z ~ -300 oZ
0 -500
D
10179I " 1 / 7 7 - 9179 -10182 JJJ.UJJJJJJ2~ FED FUNDS NON-SORROWED RATE RESERVES
11182 - 8/87 BORROWED RESERVES
MONETARY REGIMES
Fig. 3. Actual and forecast changes: January !977-August 1987.
zero. The results from that period also differ with respect to the significance of the coefficients on CD rates. In the first and third intervals, only Ao is significantly different from zero, indicating that it is the spread between the contemporaneous CD rate alone and the prime which triggers adjustments. In the middel interval both ;to and At are significantly different from zero. These results suggest that the prime exhibited a continuous, lagged response to the unusually volatile interest-rate environment of that period. Goldberg [(1982) and (1984)] has suggested that a distributed lag characterizes the relationship betwen the prime and exogenous rates; however, no allowance was made for the rigidity of the prime in formulating the statistical models of his studies. The lack of significance of At and A, in the intervals when the prime exhibits its characteristic stickiness indicates that including an explicit measure of the rigidity of the prime makes lagged exogenous rates redundant. The estimated change in the prime under the friction model is given by E(aP,)-- Y, , I'F( +
,, ( : )I + ( , f ( -
(,2),
(7)
where f(*) and F(*) are the normal density and cumulative density functions respectively. Fig. 3 is a graph of both the estimated and actual changes in the prime for each interval. The graph provides further evidence of the ability of the friction model to capture ~he rigidity of this rate.
134
S.M. Forbes and L.S. Mayne, Friction model of the prime
5. Conclusion The friction model of the prime presented in this study appears to be appropriate in periods of both rising and falling rates when the prime exhibits its characteristic rigidity. Upper and lower thresholds are identified which, when exceeded, are associated with changes in the prime. Moreover, the estimated parameters of the model provide evidence that contemporaneous rather than lagged exogenous rates are important in forecasting changes in the prime.
Appendix A: Likelihood and log-i~kelihood functions Let y,=Ap,, (/)Xlt= Ylt, ~x2t= Y2I. The likelihood function for a sample of changes in the prime which has p positive, r negative, and q zero observations is written: P
I
L = 1-]
g e-(l/2a2}O,.-4,x~.)2
n"l q
× H
[F(~)X2m,~y2)-F(f~Xtm,cY2)"]
m=l
-ff-.,ff
× |l
- -
e --( l / 2a2)(yk -- ~x2k)2
k= Jt ~
where ~xj,
1
In the following log-likelihood function, F(dpxj,, ~r2) will be written as F~t: q
log(L)= ~ log(F2,,,-Fl,n) .=1 P
n=I
(/~+r-----)log27ta2 2 i"
k=l
References Axilrod, A.H., 1985, U.S. ~q~,,netary policy in recent years: An overview, Federal Reserve B-Iletin 71, no. 1, 14-24. Brady, T.F., 1985, Loan pricing and busines:~ lending at commercial banks, Federal Reserve Bulletin 71, no. 1, 1-13.
S.M. Forbes and L.S. Mayne, Friction model of the prime
135
Fischer, G.C., 1982, The prime: Myth and reality (School of Business Administration, Temple University, Philadelphia, PA). Gilbert, R.A., 1985, Operating procedures for conducting monetary policy, Federal Reserve Bank of St. Louis Review, Feb., 13-21. Goldberg, M.A., 1982, The pricing of the prime rate, Journal of Banking and Finance 6, 277-296. Goldberg, M.A., 1984, The sensitivity of the prime rate to money market conditions, The Journal of Financial Research VII, no. 4, 269-280. Hadjimichalakis, K.G., 1981, Symmetric versus asymmetric interest rate adjustment mechanisms: The prime rate, below-prime lending and the commercial paper rate, Economics Letters 7, 257-264. Judge, G.C., W.E. Griffiths, R.C. Hill and T.C. Lee, 1982, Introduction to the theory and practice of econometrics (Wiley, New York). Mayer, T., 1987, Disclosing monetary policy, Monograph Series in Finan~ and Economics, no. 1 (Salomon Brothers Center for the Study of Financial Institutions, Graduate School of Business Administration, New York University, New York). Rosett, R.N., 1959, A statistical model of friction in economics, Econometrica 26, 263-267. Stigler, G.J., 1968, The organization of industry (Richard D. Irwin, Homewood, IL). Wallich, H.C., 1984, Recent techniques of monetary policy, Federal Reserve Bank of Kansas City Economic Review, May, 21-30.
A FRICTION MODEL OF THE PRIME* Shawn M. FORBES and Lucille S. MAYNE Case Western Reserve University, Cleveland, OH 44106, USA Received December 1986, final version received February 1988 This paper presents and estimates the parameters of a model whose basis is the rigidity of the prime rate, a friction model of the prime. Unlike traditional methods employed to analyze the prime, the estimator used here is asymptotically consistem and unbiased.
1. Introduction
Although the rates on most large loans to businesses are now tied directly to market rates, the greatest number of loans in the commercial and industrial category are under $1,000,000 and typically priced at some spread over prime [Brady (1985)]. That rate also serves as the basis for revolving loan commitments, consumer and construction loans, and syndicated Eurocurrency loans. Thus the prime is one of the most important indicators of lending rates. The distinguishing feature of the prime is its rigidity: the tendency to remain unchanged despite movements in exogenous rates. Explanations for this behavior are varied. For example, Goldberg (1982) has hypothesized, . . . without a dominant price leader, banks may be unwilling to initiate prime rate changes for modest changes in current money market conditions out of fear of being rebuffed. This unwillingness can be attributed to banks perceiving that their rivals will match any price decrease, but not match any increase, i.e., banks may perceive they have a kinked demand curve (p. 284). Alternatively, Hac!j,.'michalakis (1981) has suggested that the prime is rigid because prime-loan customers are a heterogenous group and there is no external standard for rating them per se, slowing the dissemination of information in that market. Further, since there is uncertainty over the *We wish to express our sincere thanks to two anonymous reviewers for their constructive comments and insights on earlier versions of this paper.
0378--4266/89/$3.50 ~) I989, Elsevier Science Publishers B.V. (Nol'th-HoUand)
S.M. Forbes and LS. Mayne, Friction model of the prime
128
course of future in',erest rates and many loans have floating rates tied to the prime, th~ potential costs of eliminating excess supply or demand through adjustment~ in the prime are greater than for auction markets. The combined effect of these factors is to make the prime slower to respond to changes in external conditions than market rates. Fischer (1982, p. 105) argues that ' . . . the prime exists not as a precise figure, which will be charged on a given loan, but as a benchmark or, as some have termed it, list price'. As a list price the prime may exhibit considerable rigidity without ~ecessarily affecting bank pricing flexibility or profitability once rate concessions, balance requirements, and fees and credits for other banks services are considered. ~ Although widely recognized, the rigidity of the prime has yet to be explicitly accounted for in the specification e f a statistical model of the setting of that rate. This study presents a statistical model appropriate to tke observed behavior of the prime; a model in which the dependent variable is, over some range, not related to the independent variables. Unlike traditional techniques that have been used to analyze the prime, the estimator presented here is consistent and unbiased. 2. Model The general specification commonly employed in analyzing the prime is of the form P, = ~ + r~fl:Rj + l~,,
(l)
whece the Ri are exogenous interest rates, 0~ is a spread, and #t is a random disturbance term. For example, Goldberg (1982, 1984) proposes Pt--°t + floCD, + fllCDt- I q-fl2CDt- 2 q'~t,
(2)
where CD, is the secondary market rate on large, negotiable certificates of deposit which serves as a proxy for the marginal cost of funds~ and 0c iS the desired spread over cost. There is, however, a serious statistical problem'with these models. The tendency of the prime to remain unchanged while exogenous rates fluctuate is inconsistent with the continuous density specification of (1). Therefore, the usual least-squares estimator is clearly inappropriate being biased and ;,nconsistent.z In looking for an altema0,ve model, it should be recognized that the prime's characteristic fig/dity means that no t For a discussion of list pri~ rigidity under oligopolistic competition characterized by barometric firms, see StigJer (1968, pp. 226-228). 2For a brief di,;cussion of the estimation problems of limited dependen~ variables see Judge et al. (1982, pp. 526-528).
S~M. Forbes and L.S. Mayne, Friction model of the prime '
I
40La2 '~,
II
2o1-
y
I
~
I
'~,_
RANGE OF NO CHANGE
,,~
0
129
i
I
50
=
I
I
I ! I =
100 (Pt-1- CDt )
~',~ "~'~,. I
150
=
'~
200
Fig. 1. Hypothetical relationship between changes in the prime and (.P,_l - CO,).
continuous density can explain its conditional distribution. Rather, the prime must be represented as a limited dependent variable whose value is at times unrelated to exogenous rates. In the model presented below, changes in the prime are used as the dependent variable rather than the level of that rate. This serves to focus attention on the incidence and magnitude of that which is noteworthy" i~ the prime, its relatively infrequent changes. It is assumed that the desirability and amount of any change are determined by a comparison of the spread between relevant exogenous rates and the previous period's prime? While this spread is within tolerable bounds the prime remains unchanged. If, however, conditions change sufficiently over time to cause the spread to lie outside this tolerable range, the prime is adjusted to restore the spread to withia the tolerable bounds. For example, iacreases in exogenous rates if large enough or frequent enough could cause the spread to fa~l below the lower threshold, triggering an increase in the prime. Conversely, decreases in exogenous rates would increase the spread and at some point trigger a reduction. Fig. 1 is a depiction of this hypothetical relationship, assuming that changes in the prime are li~ear functions of the spread between the lagged prime and exogenous rates. The main feature of this relationship is a range of spread values separating increases from decreases for which the prime remains unchanged. Rosett (1959; developed a model of relationships in which the dependent variable behaves as the prime does that he termed a 3We are grateful to an anonymous refere~ for pointing out that relevant market rates are simply those interest rates which reflect chang~~s in the supply and demand cc~nditions for shortterm bank loans.
S.M. Forbes and L.$. Mayne, Friction model of the prime
130
friction model. Letting APt represent the change in the prim,~ from period t - 1 to t, a friction model of the prime can be stated as
AP,= Ytt+e,,
if (Y~,+8,>0)
(3)
APfffiO
if (Yt,+st
(4)
APt = Yzt + st if ( Y2, + ~, < 0),
(5)
where st is a random error with zero mean and variance ¢2. That is, the positive segment of (¥1:=~-~(Pt_l-Rt)) represents expected increases in the prime, the negative segment of (Yzt= a s - / l ( P t - t-Rt)) expected decreases, and the difference between the intercepts oct and oc2 is a measure of the hypothesized discontinuity. Rosett presented a likelihood function for his friction model assuming a normally distributed error. However, his presentation is complicated by having the limiting value of the dependent variable potentially different for each observation and by"~reating the intercepts separately from other parameters. In the present study the use of changes in the prime makes only a single mass point necessary, zero, representing an unchanging prime. This simplification and the integration of the intercept terms into the parameter vector reduce the complexity of the likelihood function. Therefore, the likelihood function used in this paper differs from his and is presented in Appendix A. Since the likelihood function is the product of discrete probabilities and density functions, the first-order derivatives of the loglikelihood function are non-linear. Consequently, the likelihood equation must be solved in an iterative fashion. The well-known method of Newton is used here:
3. Data In this study the secondary-macket rate on 90.day CDs is used as the relevant exogenous rate. It is chosen over other rates, such .as those on 30day CDs, Fed funds, and T-bills, because of its success in a preliminary testing of the friction model and its predominance in other empirical work on the prime. Since it has ~ e n suggested that the prime is set reiativ¢ to a weighted average over time of such rates (e.g., Goldberg), the gjt of (3), (4), and (5) are expressed as
Yit=~-~Pt-t +AoCDt+AICD,_t+A~CD~_2,
j= l,2.
(6)
4The first and second derivatives necessary to the method of Newton are available from the authors upon request.
S.M. Forbes and L.S. Mayne, Friction model of the prime
131
20 ~ " PRIME
18 Z
z
<1:
- - - 9 0 - D A Y CD
:I ~t/-q
!6
/?,
ie
n- 14 LU
~ t
o. 12 IZ
,
,,, 10 nuJ I:L
V
8 6
i
.
.p
,
1 0 1 7 9 - 10182
4
I I I I I I I I I I l l l l l l l l
FED FUNDS RATE
NON-BORROWED RESERVES
BORROWED RESERVES
M O N E T A R Y REGIMES SOURCE: FEDERAL RESERVE BULLETIN, VARIOUS ISSUES
Fig. 2. Average monthly prime and 90-day CD rate: January 1977-August 1987.
The data are from various issues of the Federal Reserve Bulletin and cover slightly more than a decade, January 1977 through August 1987.5 The prime is the percent per annum reported as an average of the daily rates in effect over a given month. Per-annum rates on 90-day CDs are monthly averages of daily rates in the secondary market. The study period encompasses three distinct intervals in terms of Federal Reserve policy regimes: January 1977 through September 1979 when the operating target for implementing monetary policy was the fed funds rate; October 1979 through October 1982 when the target was non-borrowed reserves; and November 1982 through the remainder of the study period with borrowed reserves serving as the target. These changes in the implementation of monetary policy are discussed in detail in Axilrod (1985), Gilbert (1985), Mayer (1987), and Wallieh (1984). What is important for this study is that each regime has had implications for the level and volatility of interest rates as illustrated in fig. 2 and the following summary data:
Dates 01/77--09/79 10/79-10/82 1i/82-O8/87
Range of fluctuation (basis points) Prime CD 665 724 938 1,016 550 587
Number and type of changes in average monthly prime + 0 19 ll 2 14 4 19 10 28 19
5January 1977 is chosen as the beginning dale because it was then that the present form of the prime rate and CD series were introduced into the Federal Reserve Bulletin.
S.M. Forbes and L.5. Mayne, Friction model of the prime
132
Table 1 Estimates from the friction model. Monetary regimes Parameter
Fed funds rate .Jan. 77-Sep. 79
Non-borrowed reserves Oct. 79-Oct. 82
Borrowed reserves Nov. 82-Aug. 87
~t
41.928 (20.14) b
- 11.66 (51.24)
45.39a (19.65)
0~2
73.83 ° (27.11)
13.63 (50.63)
93.98" (22.66)
//
-0.488" (0.124)
-0.581" (0.134)
-0.611" (0.098)
go
0.531" (0.067)
0.558" (0.043)
0.609" (0.091)
,~t
-0.006 (0.098)
0.224" (0.087)
0.027 (0.116)
~.2
-0.074 (0.092)
-0.103 (0.089)
0.011 (0.089)
¢2
251.4(P
1,528.39~
295.36'
( 88.43)
(368.77)
(91.28)
.31.91" (11.05)
25.29 (14.17)
48.59a (10.78)
• 2 - 0ct
"Denotes significantly different from 0 .~t the 5% level, t'Standard errors are in parentheses.
In both the first and third intervals the average monthly prime exhibits rigidity, with rates generally rising in the first and falling in the latter. During the middle period the prime rose to historically high levels and was unusue!ly volatile, being unchanged in only four of thirty-seven months. 4. Results
The estimated coefficients for each of the three intervals are reported in table 1. In all cases ~2>a~, /~<0, and ,~o>0, consistent with the model's specification. The estimates obtained from the middle interval, however, differ markedly from those of the other two. 6 In those the differences between a~ and ~2 is significantly di~erent from zero, and each 0cj is also significantly different from zero. This is consistent with a range of no-change bounded on one end by a smaller positive threshold for increases and on the other by a larger one for decreases. In the middle interval, however, there is no significant difference between the ~j, nor are the 0c~significantly different from 6The hypothesig of a single set of parameters being appropriate to both the first and third periods cannot ~: rejected at the 5% level by means of a likelihood ratio test,
S.M. Forbes and L.S. Mayne, Friction model of the prime A
133
500
b-. W Z 0---
&
300
& ACTUAL O FORECAST
o
~ aA o
100
tum ~
~-UJ _z -
6
6 oO &
0 oo
A
I
. a
o
A
aA
oA
W
o>. z ~ -300 oZ
0 -500
D
10179I " 1 / 7 7 - 9179 -10182 JJJ.UJJJJJJ2~ FED FUNDS NON-SORROWED RATE RESERVES
11182 - 8/87 BORROWED RESERVES
MONETARY REGIMES
Fig. 3. Actual and forecast changes: January !977-August 1987.
zero. The results from that period also differ with respect to the significance of the coefficients on CD rates. In the first and third intervals, only Ao is significantly different from zero, indicating that it is the spread between the contemporaneous CD rate alone and the prime which triggers adjustments. In the middel interval both ;to and At are significantly different from zero. These results suggest that the prime exhibited a continuous, lagged response to the unusually volatile interest-rate environment of that period. Goldberg [(1982) and (1984)] has suggested that a distributed lag characterizes the relationship betwen the prime and exogenous rates; however, no allowance was made for the rigidity of the prime in formulating the statistical models of his studies. The lack of significance of At and A, in the intervals when the prime exhibits its characteristic stickiness indicates that including an explicit measure of the rigidity of the prime makes lagged exogenous rates redundant. The estimated change in the prime under the friction model is given by E(aP,)-- Y, , I'F( +
,, ( : )I + ( , f ( -
(,2),
(7)
where f(*) and F(*) are the normal density and cumulative density functions respectively. Fig. 3 is a graph of both the estimated and actual changes in the prime for each interval. The graph provides further evidence of the ability of the friction model to capture ~he rigidity of this rate.
134
S.M. Forbes and L.S. Mayne, Friction model of the prime
5. Conclusion The friction model of the prime presented in this study appears to be appropriate in periods of both rising and falling rates when the prime exhibits its characteristic rigidity. Upper and lower thresholds are identified which, when exceeded, are associated with changes in the prime. Moreover, the estimated parameters of the model provide evidence that contemporaneous rather than lagged exogenous rates are important in forecasting changes in the prime.
Appendix A: Likelihood and log-i~kelihood functions Let y,=Ap,, (/)Xlt= Ylt, ~x2t= Y2I. The likelihood function for a sample of changes in the prime which has p positive, r negative, and q zero observations is written: P
I
L = 1-]
g e-(l/2a2}O,.-4,x~.)2
n"l q
× H
[F(~)X2m,~y2)-F(f~Xtm,cY2)"]
m=l
-ff-.,ff
× |l
- -
e --( l / 2a2)(yk -- ~x2k)2
k= Jt ~
where ~xj,
1
In the following log-likelihood function, F(dpxj,, ~r2) will be written as F~t: q
log(L)= ~ log(F2,,,-Fl,n) .=1 P
n=I
(/~+r-----)log27ta2 2 i"
k=l
References Axilrod, A.H., 1985, U.S. ~q~,,netary policy in recent years: An overview, Federal Reserve B-Iletin 71, no. 1, 14-24. Brady, T.F., 1985, Loan pricing and busines:~ lending at commercial banks, Federal Reserve Bulletin 71, no. 1, 1-13.
S.M. Forbes and L.S. Mayne, Friction model of the prime
135
Fischer, G.C., 1982, The prime: Myth and reality (School of Business Administration, Temple University, Philadelphia, PA). Gilbert, R.A., 1985, Operating procedures for conducting monetary policy, Federal Reserve Bank of St. Louis Review, Feb., 13-21. Goldberg, M.A., 1982, The pricing of the prime rate, Journal of Banking and Finance 6, 277-296. Goldberg, M.A., 1984, The sensitivity of the prime rate to money market conditions, The Journal of Financial Research VII, no. 4, 269-280. Hadjimichalakis, K.G., 1981, Symmetric versus asymmetric interest rate adjustment mechanisms: The prime rate, below-prime lending and the commercial paper rate, Economics Letters 7, 257-264. Judge, G.C., W.E. Griffiths, R.C. Hill and T.C. Lee, 1982, Introduction to the theory and practice of econometrics (Wiley, New York). Mayer, T., 1987, Disclosing monetary policy, Monograph Series in Finan~ and Economics, no. 1 (Salomon Brothers Center for the Study of Financial Institutions, Graduate School of Business Administration, New York University, New York). Rosett, R.N., 1959, A statistical model of friction in economics, Econometrica 26, 263-267. Stigler, G.J., 1968, The organization of industry (Richard D. Irwin, Homewood, IL). Wallich, H.C., 1984, Recent techniques of monetary policy, Federal Reserve Bank of Kansas City Economic Review, May, 21-30.