The full text on this page is automatically extracted from the file linked above and may contain errors and inconsistencies.
________ Review________ Vol. 68, No. 7 August/September 1986 5 The Discount Rate and Market Interest Rates: Theory and Evidence 22 A M icroeconom ic System-Wide Approach to the Estim ation o f the Demand for Money T h e Review is p u b lis h e d 10 tim e s p e r y e a r by th e R e se a rch a n d P u b lic In fo rm a tio n D e p a rtm e n t o f th e F e d e ra l Reserve R a n k o f St. Lo uis. S in g le-cop y s u b s c rip tio n s a re availab le to th e p u b lic f r e e o f charge. M a il req u ests f o r s u b s c rip tio n s , ba ck issues, o r a d d re s s ch a n g e s to: R e se a rch a n d P u b lic In fo rm a tio n D e p a rtm e n t, F e d e ra l Reserve R a n k o f St. Louis, P.O. Rop 442, St. Lo uis, M is s o u r i 63166. T h e views e x p re sse d a re th o se o f th e in d iv id u a l a u th o rs a n d d o n o t n e ce ssa rily re fle c t o ffic ia l p o s itio n s o f th e F e d e ra l R eserve R a n k o f St. L o u is o r th e F e d e ra l Reserve System. A r t ic le s h e re in m ay be r e p r in te d p r o v id e d th e s o u rc e is cre d ite d . Ple a se p ro v id e th e R ank's R e se a rch a n d P u b lic In fo rm a tio n D e p a rtm e n t w ith a co p y o f r e p r in te d m aterial. Federal Reserve Bank of St. Louis Review August/Septem ber 1986 In This Issue . • . In the first article in this Review, “The D iscount Rate an d Market Interest Rates: Theory and Evidence,” Daniel L. Th orn ton d iscu sses the theoretical links betw een the Federal Reserve’s d iscou n t rate an d market interest rates and p resents som e em pirical evidence on the extent of this link. He finds that, both in theory an d p ractice, the direct relationship betw een the d iscou n t rate an d m oney market rates is extrem ely weak. Consequently, any observed relationship betw een these rates m ust be due to an exp ectation s effect o r to a ch an ge in Federal Reserve behavior. If the latter is co rre ct, however, there should have been a stron ger association betw een the d iscou n t rate and m oney m arket rates following the Federal Reserve’s ch an ge in operating p ro ced u re in the fall of 1982. Th orn ton finds, how ever, that, if anything, this relationship h as b eco m e w eaker. In the seco n d article in this Review, "A M icroecon om ic System -W ide A pproach to the Estim ation of the D em and for M oney,” Salam K. Fayyad d escribes h ow the m icro eco n om ic system -w ide ap p roach to m on ey d em an d differs from th e usual m oney d em an d specifications. Using a neoclassical utility function defined over five expend iture categories, two of w hich are p resu m ed to cap tu re th e flow of “m onetary services,” he d em o n strates how the m icro eco n o m ic system -w ide ap p roach can be im plem ented. Fayyad th en exam ines in-sam ple p red iction s of budget shares for exp en d itu res on m o n etaiy services an d the o th er expen d itu re categories estim ated by this ap p ro ach over the 1/1969— 1/1985 period. His results indicate that the m icro eco n om ic system -w ide ap p ro ach to m o n ey dem an d estim ation yields p red iction s that closely track the actu al behavior of the flow of m on etary services over this period. 3 FEDERAL RESERVE BANK OF ST. LOUIS AUGUST/SEPTEMBER 1986 The Discount Rate and Market Interest Rates: Theory and Evidence Daniel L. Thornton T A . HE relationship betw een the Federal Reserve’s discou n t rate and m oney m arket interest rates c o n tinues to be a topic of m u ch interest and even m ore confusion. A significant nu m b er of m oney m arket an a lysts and som e in public service believe that the dis co u n t rate is an im portant tool through w hich the Federal Reserve exerts its influence over the econ om y — particularly m arket interest rates. This view ap pears to have gathered strength from recen t evidence that discoun t rate changes have a statistically signifi can t effect on m arket interest rates an d from the presu m ed effects of a 1982 change in the Federal Reserve’s operating p ro ced u re.1 Consequently, the long-standing d iscrep an cy betw een w hat eco n om ic th eoiy says about the relationship betw een the dis co u n t rate and market interest rates an d the view am ong m any m oney m arket analysts ap p ears to have b ecom e larger. The purpose of this article is to narrow the gap by pointing out that, both in theoiy and in p ractice, changes in the Federal Reseive’s d iscou n t rate, p e r se, have essentially no effect on m arket inter est rates. At best they “signal’’ chan ges in the Federal Reserve’s use of o th er m ore powerful tools of policy. Any im pact of a discou n t rate ch an ge on market in ter est rates is due to changing exp ectatio n s o r to a change in Federal Reserve operations following the discount rate change. Daniel L. Thornton is a senior economist at the Federal Resen/e Bank of St. Louis. Rosemarie Mueller provided research assistance. 'See Thornton (1982) for a summary of some of the usual sources of confusion; Thornton (1982), Sellon and Seibert (1982) and Smirlock and Yawitz (1985) for empirical estimates of a change in the dis count rate on market interest rates; and Batten and Thornton (1984, 1985) and Hakkio and Pearce (1986) for empirical estimates of an impact of a discount rate change on the foreign exchange market. THE MARKET ANALYST’S VIEW Figure 1 illustrates a com m on ly held view of the relationship betw een a cu t in the d iscou n t rate and the response of m arket interest rates; it show s the hypothetical time path of market interest rates before and after a hypothetical cu t in the Federal Reserve discou n t rate at tim e t„, and it reflects the p ercep tion that a cu t in the d isco u n t rate ca u ses m arket interest rates to be perm an en tly low er than they otherw ise w ould have been. This cau se-and-effect relationship is p u rely qualitative. It is not cle a r w h e th e r a 1 p ercen tage-p oin t cu t in the discou n t rate will low er market rates by 1 p ercen tage point o r only a few basis points, it m erely is asserted that m arket rates will be lower. The view that the discou n t rate is preem inent in the m oney m arket co n trasts sharply with e co n om ic th e ory and the p ercep tion of m any eco n om ists that the discount rate is the least powerful of the Federal Reseive’s tools for influencing the m oney stock and interest rates. Before turning to this analysis in detail, it is instructive to co n sid er som e casual evidence against the idea that the d iscou n t rate is preem inent in the m oney market. Chart 1 show s the th ree-m o n th Treasury bill, federal funds an d d iscou n t rates weekly for the period from O ctober 1982 to Ju n e 1986. W hat do th ese data show about the effect of a d iscou n t rate ch an ge on m arket interest rates? First, in a n u m b er of instances, d iscou n t rate ch an ges are followed closely by a leveling off of m arket interest rates o r by a m ove m ent in the opposite d irection. While this does not rule out the possibility that m arket rates w ould have been higher (lower) if the d iscou n t rate h ad n ot been cu t (raised), it does suggest that the m arket analyst 5 FEDERAL RESERVE BANK OF ST. LOUIS AUGUST/SEPTEMBER 1986 view is not su p p o rted by a simple analysis of interest rate behavior. Second, nearly all discount rate ch an ges follow, rath er than lead, m ovem ents in m arket interest rates in the sam e direction.- It w ould seem that ch an ges in m arket interest rates m otivate d iscoun t rate ch an ges rath er than the reverse. F u rth erm ore, even w hen m ar ket rates declined (increased) following a d iscount rate cu t (increase), it is particularly difficult to d eter m ine w h eth er m arket rates w ould have m oved in the sam e or sim ilar fashion in the ab sen ce of a ch an g e in the d iscou nt rate. While all of this is inconclusive, it provides w eak and often co n trary evidence of a dis co u n t rate/m arket interest rate line of causation, and provides little com fort to those w ho believe the view illustrated by figure 1. F ig u re 1 Hypothetical Response to a Discount Rate Cut THE DISCOUNT RATE AND MARKET RATES IN THEORY B ecau se the interest rate is the price of credit, any im pact of d isco u n t rate ch an ges on m arket interest rates m ust com e via their effect on the supply of o r the d em an d for credit. In this regard, th ree distinct — though not n ecessarily m utually exclusive — effects of a d iscou n t rate ch an ge can be identified. These are illustrated in figure 2. Prior to the d iscou nt rate cut, the credit m arket is in equilibrium at an interest rate of R,,, co rresp on d in g to the intersection of the initial supply and dem and curves, S„ and D„, respectively. The Direct Effect The first effect, called the direct (or substitution) effect, cau ses a shift in the supply of credit. D iscount w indow borrow ing is one m eth od depository institu tions use to adjust their reserve position. Alternatively, they ca n buy federal funds o r sell governm ent secu ri ties directly in the m oney m arket.3 Since these altern a tives are close substitutes, the d em an d for borrow ed reserves d ep en ds on the spread betw een m arket inter est rates, especially the federal funds rate, and the d iscou n t rate. As the federal fu n d s-d iscount rate spread increases, borrow ings from the Federal Re serve tend to increase and vice versa. Thus, the level of d iscount w indow borrow ings usually is exp ressed as: (1) Borr = a(Rf—R(l), a > 0, 2 This is true of other periods as well; see Thornton (1982), p. 14. depository institutions also can call in loans or carry the deficiency over into the next reserve period. They rarely, if ever, use these alternatives, however. Digitized6for FRASER w here Borr den otes the aggregate level of ind eb ted ness of depository institutions to the Federal Reserve and Rf and R(l denote the federal funds an d discount rate, respectively. To illustrate the direct effect of a ch an ge in the discou n t rate on m arket interest rates, assu m e that the discount rate is cut. In response, d ep ositoiy institu tions increase their borrow ings an d red u ce their use of alternative so u rces of reserves. The in crease in borrowings p ro d u ces an increase in the m on etary base and, in turn, the supply of credit — illustrated in figure 2 by a shift from S„ to S,. Thus, a d iscou n t rate cu t has a direct effect, causing m arket interest rates to decline from R„ to R,. The effect of an in crease in the discount rate w ould be sym m etric. The Announcement Effect Additionally, discou n t rate ch an ges ca n have an “an n ou n cem en t effect.” If a change in the discou n t rate is interpreted as a “signal” that the Federal Re serve will alter its policy w ith resp ect to the grow th of reserves and the m oney stock, the m arket m ay react in anticipation of a policy ch an ge. A cu t in the discou n t rate usually is thought to be a signal that the Federal Reserve is going to p u rsu e an easier m on etary policy so the m arket reacts in anticipation of Federal Reserve AUGUST/SEPTEMBER 1986 FEDERAL RESERVE BANK OF ST. LOUIS C h a rt 1 Selected Interest Rates 1982 1983 1984 open m arket operations that will increase the supply of credit.4 Consequently, there is an im m ediate shift in the supply of credit, relative to dem and, in an ticip a tion of further m onetary ease. If the an n ou n cem en t effect o ccu rs, it is over and above the direct effect of a discount rate change, and is illustrated by the shift from S, to S, in figure 2:' The Policy Effect Finally, there could be a “policy effect” if the Federal Reserve actually changes its policy and in creases the "This is not the only possible interpretation for the market. See Batten and Thornton (1984) and Smith (1963) for a discussion of this point. 5 This also could have been illustrated by a reduction in the demand for credit, but was illustrated as a shift in supply to keep the figure simple. 19 8 5 19 8 6 grow th rate of reserves. 1'his also can be illustrated by the shift from S„ to S,. If the m arket co rrectly antici pates the direction and m agnitude of the policy effect, market interest rates will rem ain p erm anently low er at R2. Of course, this requires that the m ark et’s e x p e cta tions be co rrect, both in term s of the actu al ch an ge in Federal Reserve policy and in term s of the im p act of that policy change on the market." As the Federal Reserve p u rch ases m ore securities, sp ecu lators sell off those acquired in anticipation of the policy ch an ge. If the market overanticipates Federal Reserve actions, however, m arket rates first will fall below an d then 6 This brief discussion gives rise to several issues not analyzed in this paper, such as the effectiveness of policy and the credibility of the central bank. For a general discussion of the credibility issue, see Cukierman (1986). 7 FEDERAL RESERVE BANK OF ST. LOUIS subsequently rise to th eir long-run equilibrium. F u r therm ore, if th e m ark et’s expectation s are in correct an d Federal Reserve policy rem ains u nch an ged , inter est rates will rise back to R, — the only im p act of a d iscou n t rate ch an ge w ould be the d irect effect. AUGUST/SEPTEMBER 1986 F ig u re 2 Three Possible Effects o f a Discount Rate Cut on M arket Intere st Rates DISCOUNT WINDOW BORROWINGS AND THE FED’S OPERATING PROCEDURE Some have argued that the policy effect h as b ecom e m ore im portant sin ce the O ctober 1982 ch an ge in the Federal Reserve’s operating p ro ced u re. At that time, the Board sw itched from a n onborrow ed reserve to a borrow ed reserve operating p ro ced u re. It is now widely believed that the Federal Reserve operates to achieve a certain average level of borrow ed reseives (called the initial borrow ing assum ption) over a given tim e period.7 The m ech an ics of this operating p ro ce dure can be illustrated by tracin g the reaction of the Federal Reserve to an u n exp ected in crease in the d em an d for reserves. O ther things u n chan ged , an in crease in the d em an d for reserves tend s to cau se both borrow ings and the funds rate to rise, as depository institutions attem p t to satisfy their d em an d for re serves in the m on ey m arket and at the Federal Reserve d iscou n t w indow . As borrow ings increase relative to the borrow ing assum ption, the Fed in creases th e su p ply of nonb orrow ed reserves via open m arket p u r ch ases of governm ent secu rities; in respon se, both borrow ing an d the federal funds rate fall. A cu t in the d iscou n t rate, not accom p an ied by a ch an ge in the initial borrow ing assum ption, w orks analogously. If the Federal Reserve cu ts the d iscou n t rate, th e d em an d for borrow ed reseives will increase at all levels of th e federal funds rate, cau sing b orrow ings to in crease relative to the initial borrow ing a s sum ption. If the initial borrow ing assu m p tion is u n ch an g ed , th e F ed m u st in crease th e supply of nonborrow ed reserves through open m arket o p era tions until the federal funds rate h as declined by enough to retu rn borrow ings to th e level of the bor rowing assum ption. The above implies that equation 1 can be w ritten as: (21 Borr* = a(Rf- R d ), w here Borr* d en otes th e Federal Reserve's initial b o r row ing assum ption. Equation 2 implies a co n stan t sp read betw een the federal funds an d d iscou n t rates. 7 For a discussion of this, see Roley (1986), Wallich (1984) and Federal Reserve Bank of New York (1986). 8 Any ch an ge in the d iscou n t rate will be m a tch e d by an equal change in the federal funds rate, providing th ere is no co m p en satory ch an ge in th e borrow ing a s sum ption. It should be em p h asized that it is not the d iscou n t rate change p e r s e that affects m arket interest rates, but the subsequent policy effect if the Fed eral Reserve strictly ad h eres to an operating p ro ced u re th at a t tem p ts to m aintain the level of borrow ings assu m ed by its cu rren t policy directive. If th e m arket perceives this behavior, it cou ld also stren gth en an y a n n o u n ce m ent effect. The Importance o f the Liquidity Effect All of the potential effects of a ch an ge in th e dis co u n t rate on m arket interest rates (but, in particu lar, the policy effect) d ep en d on the so-called "liquidity effect” — the ch an ge in interest rates asso ciated w ith an u n an ticip ated in crease in the grow th rate of the m on ey supply. W hile su ch an effect is w idely tou ted in theoretical discussions, th ere is little em pirical evi d en ce to su p p ort it. Yet, w ithout a liquidity effect o r at least the exp ectatio n of a liquidity effect, ch an g es in the d iscou n t rate cou ld n ot have an im p act on a broad sp ectru m of m arket interest rates* 8 This, of course, ignores the possible effect of changes in expecta tions of inflation on interest rates. See Brown and Santoni (1983), Cagan and Gandolfi (1969) and Melvin (1983) for a review of the direct evidence on the liquidity effect. FEDERAL RESERVE BANK OF ST. LOUIS Which Market Interest Rates? M uch of the d iscussion thu s far has been carried out in term s of the federal funds rate. In reality, there are a large num ber of different rates: the: rates on federal funds, T reasury bills, notes an d bonds, co m m ercial bank loans, m ortgages, etc. H ence, the array of credit m arket assets should be divided into those that are closely related to the d iscou n t rate an d those that are less closely related to it. The m arket for federal funds is one segm ent of the credit m arket that is particularly sensitive to d iscou n t rate ch anges an d to ch anges in Federal Reserve o p era tions. Federal funds are sim ply the reserve assets of one depository institution that are sold (lent) to an o th er for the purp ose of achieving both institutions' desired reserve positions. B ecau se su ch funds are close substitutes for reserves supplied by the Federal Reserve, including those supplied through the dis co u n t w indow, ch an ges in the d iscoun t rate o r F ed eral Reserve policy should initially affect the federal funds rate and subsequently o th er market rates. (See page 10 for a d iscussion of the relationship betw een the discoun t rate and the prim e rate.) Borrowings and the Rate Spread The relationship betw een the d iscount rate and m arket interest rates rests, in one w ay o r an other, on the strength of the relationship betw een borrow ings an d the rate spread. Equations 1 an d 2, however, imply that borrow ings d epend on m ore than the spread betw een the market and d iscount rates. To see this, assum e that there are no im pedim ents to borrow ing so th at dep ository institution s ca n b orrow any am ount they desire at the d iscoun t w indow . If this w ere the case, borrow ings w ould rise w hen ever m ar ket rates w ere above the d iscou n t rate an d fall w h en ever the discou n t rate is above the market rate. If we ab stract from problem s of inflation and inflationary expectation s, the m arket rate w ould always equal the discount ra te :’ But if R, = R(l, however, equation 1 implies that borrow ings w ould be zero. The d ata in ch art 2, w hich show weekly adjustm ent borrowings and the federal funds rate/d isco u n t rate 9 Under this arrangement, one can envision the Federal Reserve pushing down interest rates by lowering the discount rate. As this is done, hovyever, money growth will accelerate and so will inflation. As a result, nominal interest rates will rise and money will grow even faster. Hence, even if the discount window were completely "open,” the Federal Reserve would be unable to control interest rates with the discount rate in anything but the short run. AUGUST/SEPTEMBER 1986 sp read from O ctober 1982 to Ju n e 1986, ind icate that the d iscount an d federal funds rates are seldom equal.1 M oreover, w h en th e rates are equal, borrow " ings are not zero. This is p rim a fa c ie evidence that borrow ing is not explained solely by the interest rate spread. Indeed, Federal Reserve regulations, w h ich set forth the conditions u n d er w h ich dep ository institu tions m ay u se the d iscou n t w indow , m ake it cle a r that borrow ing is a privilege an d explicitly state that it is inappropriate to borrow “to take advantage of a differ ential betw een the d iscou n t rate an d the rate on alternative so u rces of funds.”1 1 A visual inspection of ch a rt 2 show s that th ere is usually a positive relationship betw een borrow ings an d the rate spread, that is, that increases in borrow ings tend to be associated w ith in creases in the spread and vice versa. T here are, how ever, som e m arked d e p artures from this relationship. The m ost obvious of these o ccu rre d w ith the sh arp in crease in borrow ings in M ay-Ju n e 1984 an d November 1985. Both of these events w ere accom p an ied by special circu m stan ces. The form er is associated w ith heavy d iscou n t w indow borrow ings by Continental Bank of Illinois an d the latter with the largest single-day borrow ing from the Federal Reserve w hen th e Bank of New York (BONY) exp erienced a co m p u te r failure on November 21, 1985.1 Even w hen these outliers are ignored, however, 2 there are instances w hen borrow ings and the spread m ove in opposite d irections. M oreover, th ere is co n siderable variation in the relationship betw een the average level of borrow ings an d the average level of the spread. The m ost obvious of these is the period from Ju n e 13, 1984, through O ctober 3, 1984, w hen the sp read averaged over 200 basis points and borrow ings averaged less than a billion dollars, as co m p ared to an average sp read of 70 basis points an d average borrow ings of $.7 billion over the entire p eriod .1 3 The strength of the relationship betw een borrow - '“Borrowing from the Federal Reserve is divided into three categories: adjustment borrowing, seasonal borrowing and extended credit borrowing. The borrowing assumption, however, pertains only to adjustment and seasonal borrowings; see Partian, Hamdani and Camilli (1986). "This is called the “reluctance of banks to borrow from the Federal Reserve,” and at one time there was considerable discussion over whether this reluctance was “inherent” or “induced.” 1 See Federal Reserve Bank of New York (1986) for a discussion of 2 the BONY borrowings. 1 lt could be that depository institutions became more reluctant to 3 borrow from the Federal Reserve in light of the large borrowings by Continental Bank. 9 FEDERAL RESERVE BANK OF ST. LOUIS AUGUST/SEPTEMBER 1986 The Discount Rate and the Prime Rate One possible reason for the hypothesized strong effects of discoun t rate ch an ges on interest rates is the fact that d iscou n t rate ch an ges an d ch an ges in the com m ercial bank prim e rate often o c c u r to geth er and are usually acco m p an ied by a great deal of publicity. Both of th ese rates are adm inistered rates that do not ch an ge daily w ith m arket forces, but change less frequently an d by fairly large am ounts. Because ch anges in the prim e rate often follow on the heels of ch an ges in the d iscou n t rate, it m ay lead som e to co n clu d e incorrectly the latter cau sed the form er. B ecau se both are adm inistered rates, however, they are likely to respo n d similarly but not precisely coterm inously, to market rates. F or exam ple, as market interest rates fall relative to these adm inistered rates, th ese rates b eco m e in creas ingly out of line with the m arket. Hence, there is an incentive for the Federal Beserve to cu t the dis co u n t rate and for com m ercial banks to cu t their Prime rate Date effective prim e rate. If the Federal Beserve cu ts the d iscou n t rate first, banks m ay feel additional p ressu re to cut their prim e rate, but this does n ot im ply th at the form er cau sed the latter. R ather both rates are m erely responding to m arket forces. The table above show s that on four o ccasio n s since O ctober 1982 d iscou n t rate and prim e rate ch an ges w ere effective on the sam e day. In each instance, the an n ou n cem en t of a cu t in the prim e rate followed the an n ou n cem en t of the d iscou n t rate change. F o r the rem aining five ch an g es in the d iscount rate, ch an ges in the prim e rate followed d iscount rate ch an ges by a week o r m ore. Also, there w ere a n u m b er of ch an ges in the prim e rate that w ere n ot even rem otely asso cia te d w ith ch an ges in the discou n t rate. It w ould ap p e a r that changes in m arket interest rates are prim arily re sponsible for ch an ges in both of th ese adm inistered rates. Discount rate Change October 7, 1982 13% to 12% 12% to 11.5% January 11,1983 February 25, 1983 August 8, 1983 March 19, 1984 April 5, 1984 11.5% to 11% 11% to 10.5% 10.5% to 11% 11% to 11.5% 11.5% to 12% May 8, 1984 June 25,1984 September 27,1984 October 16, 1984 October 29, 1984 November 8,1984 12% to 12.5% 12.5% to 13% 13% to 12.75% 12.75% to 12.5% 12.5% to 12% 12% to 11.75% November 28, 1984 December 19,1984 11.75% to 11.25% 11.25% to 10.75% January 15,1985 May 20, 1985 June 18,1985 March 7, 1986 April 21, 1986 10.75% to 10.5% 10.5% to 10% 10% to 9.5% 9.5% to 9% 9% to 8.5% Change 13.5% to 13% October 13,1982 November 22, 1982 Date effective October 12, 1982 November 22, 1982 December 14, 1982 9.5% to 9% 9% to 8.5% April 9, 1984 8.5% to 9% November 21,1984 9% to 8.5% December 24,1984 10 10% to 9.5% 8.5% to 8% May 20, 1985 8% to 7.5% March 7,1986 April 21, 1986 7.5% to 7% 7% to 6.5% FEDERAL RESERVE BANK OF ST. LOUIS AUGUST/SEPTEMBER 1986 C h a rt 2 A d ju s tm e n t plus S easo nal B o rro w in g s from F e d e ra l Reserve an d F ed eral F u n d s -D is c o u n t R ate Spread 1 i— i— i— i— * — O N D J J— i— i— i— i— * * i— — — F M A M J 1982 — * i— '— i— i— i— i— i— i— i— i— — i— i— 1 i— * 1 i— i— * * i— — i— i— i— i — i— i - i — — — — — — J A S O N D J F M A M J J A S O N D J 1 9 83 1984 ings and the sp read can be estim ated statistically by considering the equation: (3) Borr, = a„ + a,(R f—Rd + u,. ) The term u, is a rand om d isturban ce that can be thought of as cap tu ring the effect of all factoi-s oth er than the rate spread that d eterm ine deviations in borrow ing from its average level. From a statistical point of view, the variation in borrow ings can be d e com p osed into two so u rces: the proportion explained by the rate spread and that explained by all o th er factors. (Since the factors that go into u, are not exp lic itly identified, this is called “unexplained variation.’’) Equation 3 is estim ated with ordinary least squares, using the weekly d ata show n in ch art 2. The outliers for the weeks ending May 16 to Ju n e 6, 1984, and N ovember 27, 1985, w ere deleted.'4 The results are ,4lf these outliers are not removed, the R2falls to about .15. F M A M J J A S O N D J 19 8 5 I M A M J 1986 p resen ted in the first row of table 1. The coefficient of determ ination, d en oted R3, m easu res the proportion of the variation in borrow ings explained by the rate spread, and 1-tT is the p roportion of variation e x plained by all o th e r factors. The R3 ind icates that only 35 p ercen t of the variation in borrow ings is a cco u n te d for by the spread, leaving 65 p ercen t to be acco u n te d for bv o th e r factors. T he fit can be im proved by putting in a du m m y variable that takes on the value one for the p eriod from the week ending Ju n e 13, 1984, to O ctober 3, 1984, w hen the sp read w as unusually high, an d zero else w here. The results of including a d u m m y variable are show n in the secon d row of table 1. While including the du m m y variable b oosts the R2 som ew hat, it does n ot explain this anom aly. Nevertheless, even after a c cou n tin g for this ap p aren t shift in the borrow ing fu n c tion, the spread and the d u m m y variable explain only 11 FEDERAL RESERVE BANK OF ST. LOUIS AUGUST/SEPTEMBER 1986 extent of the shift in th e supply of cred it an d the interest sensitivity of the d em an d for credit, so it is possible, in principle, to determ in e the effect of an exogenous ch an ge in the m on ey stock on interest rates. Table 1 Estimates of Equation 3 Intercept Dummy variable .368* (12.21) -.410* (4.03) R2 SE .291* (10.04) .420* (14.74) Spread .35 .28 .419* (9.94) .40 .27 "Indicates the variable is statistically significant at the 5 percent level. 40 p ercen t of the total variation in borrow ings, leaving the bulk of the variation to be explained by o th er facto rs.1 5 IN SEARCH OF THE DIRECT EFFECT: SOME EMPIRICAL ESTIMATES Separating the three possible effects of d isco u n t rate ch an ges on m arket interest rates — the direct, policy an d a n n o u n cem en t effects — is difficult. The results in table 1, how ever, provide a basis for estim ating the likely direct effect of a d isco u n t rate ch an ge on interest rates. From the seco n d row of table 1, w e see that a 1 p ercen tage-p oin t (100 basis-point) decline in th e dis co u n t rate will cau se borrow ings to in crease by $.419 billion. All o th er things th e sam e, this will in crease the m on etary base (in the form of borrow ed reserves) by the sam e am ou nt. Given an M l-m o n etary base m ulti plier of 2.7, this will p ro d u ce a $1.13 billion in crease in M l.1 Such ch an ges in the m on ey stock shift the supply 6 of cred it to the right, cau sing m arket interest rates to fall. The effect of this on m arket rates d ep en ds on the 1 Because borrowings fluctuate with market interest rates, they can 5 be a source of cyclical variation in the money stock. Because of this, some have suggested that the discount rate be tied to some market interest rate. Opponents of this view have argued that no single interest rate adequately represents the appropriate opportunity cost for all institutions. If this were true, rates other than the federal funds rate might explain borrowings. To test this, the second equation on table 1 was reestimated with the difference between the threemonth Treasury bill and federal funds rates added as a separate regressor. The coefficient on the difference between these rates was not statistically significantly different from zero at the 5 percent level (t-ratio = 1.26). Hence, it appears that the federal funds rate is the primary interest rate on which borrowing depends. 1 The M1 multiplier averaged much less than this during all of the 6 period under consideration, i.e., 2.7 is approximately its current level. 12 The largest estim ates of this liquidity effect co m e from estim ated sh o rt-ru n m on ey d em an d equations. F o r exam ple, usual estim ates suggest that a $1.13 billion ch an ge in M l w ould p ro d u ce a 67 basis-point initial ch an ge in the th ree-m o n th T reasu ry bill rate, but only a six basis-point effect in the lon g-ru n equilib rium ra te .1 It is well known, how ever, th at su ch eq u a 7 tions have u n reasonably large estim ates of th e liquid ity effect.1 O ther studies, w hich attem p t to estim ate " the liquidity effect directly, sh ow only sm all an d tra n sient effects of u n an ticip ated ch an ges in m o n ey on interest rates. Using th ese estim ates, a $1.13 billion ch an ge in the m oney stock w ould p ro d u ce about a one basis-point ch an ge in the T-bill rate initially, w ith no long-run effect w hatsoever.I U Put into an o th er perspective, sin ce O ctob er 1982 the average, absolute w eekly ch an ge in M l h as b een $1.77 billion, m ore than one an d one-half tim es th e esti m ated $1.13 billion ch an ge in M l associated w ith a full 1 p ercen tage-p oin t ch an ge in th e d isco u n t rate. Thus, the direct effect of a ch an ge in the d isco u n t rate on m arket interest rates, all o th er things co n stan t, is likely to be small. Technical Fs. Nontechnical Changes in the Discount Rate Alternatively, estim ates of the m agnitude of th e di rect effect can be obtained by classifying d iscou n t rate changes accord in g to th e reaso n they w ere m ade. Some d iscou n t rate ch an ges are m ad e solely as te ch n i cal adjustm ents, designed to align the d iscou n t rate with market interest rates. Other’s are m ad e for- policyre la te d re a so n s. T h ese a re called n o n te ch n ica l changes. 1These estimates are based on current levels of M1 and interest 7 rates. Using a short-run interest elasticity estimate from the “nominal-adjustment” specification of the short-run demand for money of - .015 and a money stock of $670 billion, the percentage change in the interest rate would be about 11 percent. A T-bill rate of 6 percent translates into a 67 basis-point change in market interest rates. The long-run effect was calculated under the assumption of a long-run elasticity of about - .14 ( - .015/.11). These estimates are in line with the results from Thornton (1985). 1 See Carr and Darby (1981). 8 1 See Brown and Santoni (1983). Similar estimates would be ob 9 tained from Cagan and Gandolfi (1969) and Melvin (1983). FEDERAL RESERVE BANK OF ST. LOUIS Since the response of borrow ings to a d iscou n t rate change should be the sam e regardless of the reason for the change, ce teris paribus, the direct effect of a discou nt rate change on m arket interest rates should be the sam e for all ch an ges in the d iscou n t rate.-" Fu rtherm ore, there should be no ch an ge in the m ar ket’s p ercep tion of policy w hen d iscou n t rate changes are purely tech n ical ad ju stm ents. F or n on tech n ical changes, however, not only is there a direct effect due to th e im p act on borrow ings and th e supply of credit, but a potential an n ou n cem en t effect, w hich m ay or m ay not be validated by subsequent Federal Reseive actions. If the d iscoun t rate ch anges th at are m ade purely as tech n ical ad ju stm ents do n ot affect m arket interest rates, this is fu rth er evidence that there is essentially no direct effect of d iscou n t rate ch anges. Any interest rate effects co m e through an an n o u n ce m ent effect o r subsequent policy chan ges. It should be noted that the fact that the Federal Reserve changes the d isco u n t rate from tim e to time solely to bring it in line w ith m arket interest rates is itself p rim a fa c ie evidence that the link betw een bor rowings an d the federal funds/discount rate spread is not the sole d eterm inant of depository institution borrowing. If it w ere, the Federal Reserve should never have to make su ch tech n ical adjustm ents, but this is not the case. Of the nine d iscou n t rate ch an ges from O ctober 1982 to Ju n e 1986 listed in table 2, three w ere stated to have been m ade solely for technical reasons and three of the rem aining six m entioned tech n ical co n ce rn s as one of the reasons for the ch an ge.2 1 Recent em pirical work provides strong evidence that only d iscount rate ch an ges m ade for policy rea sons affect m arket interest rates.2- This work is u p dated here by estim ating the equation: 10 (4) AR, = a„ + X a,A R ,, + 0ADR, + u„ i= 1 “ This discussion assumes that the Federal Reserve is not trying to control the money stock, and in particular, it is not using a monetary base or total reserves target. If it were, any change in the discount rate would have no direct effect on interest rates because the effect of such a change would be neutralized by compensatory open market operations. 2 The classification used is based upon the Federal Reserve's an 1 nounced statement of intentions as used by Thornton (1982) and Batten and Thornton (1984, 1985). Smirlock and Yawitz (1985) investigate alternative schemes, but find that the one employed here works best. Their results are supported by Hakkio and Pearce (1986). “ See Thornton (1982), Batten and Thornton (1984, 1985), Smirlock and Yawitz (1985) and Hakkio and Pearce (1986). AUGUST/SEPTEMBER 1986 w here AR den otes the o ne-d ay ch an ge in a m arket interest rate, and ADR d en otes the ch an ge in the d iscou n t rate.23 This equation w as estim ated using daily d ata from O ctober 1 ,1 9 8 2 , to Ju n e 1 1 ,198G, using both the federal funds and th ree-m o n th T reasury bill rates. The T-bill rate w as selected to rep resen t m arket interest rates in general. E stim ates of th e coefficient on ADR and som e sum m ary statistics are p resen ted in table 3.2J The results indicate that a ch an ge in the d iscou n t rate has a positive, significant effect on both the federal funds and T-bill rates on the n ext m arket day. The effect on the federal funds rate is roughly 2.5 tim es that on the T-bill rate. W hen the d iscou n t rate ch an ges are partitioned into those m ade for tech n ical reason s (ADRT) and those m ade for n on tech n ical reason s (ADRNT), the results indicate th at d iscou n t rate ch an ges m ad e solely for tech n ical reason s h ad no significant effect on the federal funds rate. The results for the T-bill rate are less clear. The coefficient on d isco u n t rate ch an ges m ad e solely for tech n ical reason s is sm aller th an that for policy-related reason s, but is statistically signifi can t at the 5 p ercen t level. A clo ser look, how ever, reveals that only o ne of the th ree d iscou n t rate ch an ges m ad e solely for tech n ical reason s is a sso ci ated with m ovem ent in the T-bill rate in the exp e cte d d irection. The h alf-percent decline in the d isco u n t rate on O ctober 1 2 ,1 9 8 2 , is associated w ith a 37 basispoint decline in the T-bill rate. In co n trast, the half p ercen t increase on April 9 ,1 9 8 4 , is asso ciated w ith a 9 basis-point decline in the T-bill rate an d th e half p ercen t d ecrease on April 21, 1986, is associated with no change in the T-bill rate. W hen d iscou n t rate ch an ges m ad e for p urely te ch nical reason s are partitioned into the one m ad e on O ctober 12, 1982 (ADRTO), and the o th e r tw o (ADRT), the results indicate that significance of tech n ical ch an ges on th e th ree-m o n th T reasu ry bill rate is due to the change on O ctober 12. Fu rth erm ore, the effect on the federal funds rate is significant at th e 10 p e r ?3 ADR takes on the value of the discount rate change on the day that the change became effective. The one exception is the change that was announced on November 21, 1984, effective immediately. Since the announcement was made at 4:15 p.m. EST after the market closed, the ADR takes on a value on November 23. (The federal funds rate declined by 35 basis points between November 21 and 23 and increased by 4 basis points between November 20 and 21). 2 The coefficients on the distributed lag of the dependent variable are 4 not reported because they are intended only to capture the effect of all previously known information on these interest rates and are not of importance themselves. 13 AUGUST/SEPTEMBER 1986 FEDERAL RESERVE BANK OF ST. LOUIS Table 2 Discount Rate Changes, October 1982 to June 1986 Date effective Change Classification Reason October 12,1982 10% to 9.5% T Action taken to bring the discount rate into closer alignment with short term market interest rates November 22,1982 9.5% to 9% P Action taken against the background of continued progress toward greater price stability and indications of continued sluggishness in business activity and relatively strong demand for liquidity December 14,1982 9% to 8.5% P Action taken in light of current business conditions, strong competitive pressures on prices and further moderation of cost increases, a slowing of private credit demands and present indications of some tapering off in growth of the broader monetary aggregates April 9, 1984 8.5% to 9% T Action taken to bring discount rate into closer alignment with short-term interest rates November 21,1984 9% to 8.5% P Action taken in view of slow growth of M1 and M2 and the moderate pace of business expansion, relatively stable prices and a continued strong dollar internationally December 24,1984 8.5% to 8% P Essentially the same as before plus to bring the discount rate into more appropriate alignment with short-term market interest rates May 20,1985 8% to 7.5% P Action taken in the light of relatively unchanged output in the industry sector stemming from rising imports and a strong dollar. Rate reduction is consistent with declining trend in market interest rates March 7,1986 7.5% to 7% P Action taken in context of similar action by other important industrial countries and for closer alignment with market interest rates. A further consideration was a sharp decline in oil prices April 21, 1986 7% to 6.5% T Action taken to bring discount rate into closer alignment with prevailing levels of market rates P = policy related T = technical Source: Federal Reserve Bulletin, paraphrased from statem ents in various issues, and the Wall Street Journal. cen t level w hen th ese data are partitioned in this way. This is the only instance w hen a tech n ical ch an ge in the d iscou n t rate had a significant effect on m arket rates.2’ The p rep on d eran ce of evidence suggests that d iscou n t rate ch an ges m ade solely for tech n ical re a sons have no statistically significant effect on market interest rates.2 This result is co n sisten t w ith o u r pre6 2 This change was announced two days after the Federal Reserve de 5 emphasized M1 as a monetary target. (See Thornton (1983) for a discussion of this period.) While there was no immediate announce ment of the decision to de-emphasize M1, there were leaks to this effect, so the market may have interpreted the October 12 decrease in the discount rate as an indication that the Federal Resen/e would move toward an easier policy. There were leaks to the press on October 7 that the Federal Reserve would pay more attention to interest rates and less to M1 growth. See BNA’s Daily Report for Executives, October 8,1982. “ This finding has been reiterated by Thornton (1982), Smirlock and Yawitz (1985) and the results presented in table 5 for the money 14 vious finding that there is little, if any, direct effect of a d iscount rate change on m arket interest rates. It cou ld be, how ever, that d iscou n t rate ch an ges m ade solely for tech n ical reason s are m ore readily an ticip ated than those m ad e for policy reason s.2 If 7 this w ere the case, and if the m arket perceived the effect of the corresp on d in g ch an ge in th e m o n ey su p ply on interest rates, m arket rates w ould ch an ge p rior to the ch an ge in the d iscou n t rate so th ere w ould be no statistically significant effect following th e a n n ou n cem en t of a d iscou n t rate ch an ge. Hakkio and Pearce (1986) report that d iscou n t rate ch an g es m ade for tech n ical reason s are n o m ore readily fo recasted than those m ade for n on tech n ical reason s. H ence, this market, and by Batten and Thornton (1984, 1985) and Hakkio and Pearce (1986) for the foreign exchange market. 2 This conjecture is offered by Batten and Thornton (1984). 7 AUGUST/SEPTEMBER 1986 FEDERAL RESERVE BANK OF ST. LOUIS Table 3 Estimates of Equation 4 for Technical and Nontechnical Discount Rate Changes Constant ADR ADRNT ADRT ADRT0 R2 SE .20 .35 .20 .35 .20 .35 .03 .08 .03 .08 .04 .08 Federal funds rate -.011 (0.94) .690* (2.95) (0.91) .827* (2.90) .412 (1.02) -.010 (0.87) .829* (2.91) -.009 (0.02) -.010 1.289 (1.81) Treasury bill rate -.000 (0.12) .267* (4.74) -.000 (0.10) .299* (4.32) .204* (2.08) .000 (0.02) .297* (4.33) -.066 (0.56) .789* (4.55) 'Indicates statistical significance at the 5 percent level. alternative interp retation ap pears to have little m erit The Discount Rate jIs a Penalty Rate A nother w ay of estim ating the direct effect of a d iscou n t rate change on m arket interest rates conies from noting that dep ository institutions have little incentive to borrow from the Federal Reserve w hen the d iscou n t rate is a “penalty rate,” that is, w hen it is above th e federal funds rate. D epository institutions th at borrow from the Federal Reserve w hen the d is cou n t rate is a penalty rate are assu m ed to do so for reason s o th er than to m inim ize the explicit co st of obtaining reserve-adjustm ent funds. C hanges in the discou n t rate that co m e w hen the d iscou n t rate is a penalty rate — especially ch an ges that leave the dis co u n t rate at penalty levels — should have no effect on borrow ing and, h en ce, no direct effect on m arket in terest rates.-0 If estim ates indicate that d iscou n t rate 2 Their “forecasts," however, are based on in-sampie results and are 8 not true ex ante forecasts. ^While this idea is common in the literature, e.g., Broaddus and Cook (1983) and Sellon and Seibert (1982), it is sometimes presented in such a way that it appears that the only effect is the direct effect. In this case, any finding of a significant effect of a discount rate change on market interest rates implies that it is produced via the direct effect. We have shown, however, this is not the case. ch an ges m ad e w hen the d iscou n t rate is n ot a penalty rate do not have an effect on m arket rates, while those m ad e w hen the d isco u n t rate is a penalty rate do have a significant effect, this w ould be fu rth er evidence that there is no direct effect of a d isco u n t rate ch an ge on m arket interest rates. To test this hypothesis, d iscou n t rate ch an g es w ere partitioned into those w hen the d iscou n t rate w as a penalty rate (ADRP) p rior to th e an n o u n cem en t and those w hen the d iscou n t rate w as not a penalty rate (ADRNP).3 The results, rep orted in table 4, indicate '1 that ch an ges m ade w hen the d isco u n t rate w as a penalty rate are statistically significant.3 F u rth erm ore, 1 “ The partition used was based upon whether the discount rate was a penalty rate with respect to the federal funds rate. There was only one instance when the discount rate was a penalty with respect to the T-bill rate. Such a partition is of little interest, however, since the evidence in footnote 15 indicates that the federal funds rate is the relevant opportunity cost variable. 3 Sellon and Seibert (1982) performed a similar analysis on data for 1 the period from February 1980 to August 1982 and found that discount rate changes made when the discount rate was a penalty rate had no statistically significant effect on market interest rates or borrowings. During this period, however, such discount rate changes were primarily those made for technical reasons; thus it appears that the Sellon and Seibert result is due to this fact and not to the fact that the discount rate was at a penalty level at the time of the change. See Thornton (1982) for the technical vs. nontechnical results over a similar period. 15 FEDERAL RESERVE BANK OF ST. LOUIS ch an ges m ad e w h en the d iscou n t rate w as not a penalty rate w ere not statistically significant. These results are precisely the opposite of those th at should have been obtained if the effect of a d iscou n t rate change, rep orted in table 3, w ere due to a d irect effect. Evidence on the Announcement and Policy Effects The evidence indicates that d iscou n t rate changes do n ot directly affect m arket interest rates. C on se quently, the effect on m arket rates indicated in table 3 m u st be due to an a n n o u n cem en t effect, a policy effect o r both. B ecau se the effect m easu red in table 3 o ccu rs on the day following the an n o u n cem en t of a ch an g e in the d iscou nt rate an d ch anges m ad e for tech n ical reasons have no effect on m arket rates, this strongly suggests that it is, at least in part, an an n o u n cem en t effect, ft is im possible to determ ine, how ever, w heth er the exp ectation s w ere subsequently validated by ch an ges in the rate at w hich the Federal Reserve supplied reserves.3 2 Attem pts m ade to test directly for a policy respon se following a d iscoun t rate ch an ge w ere incon clu sive.1 1 Nevertheless, som e evidence b ears on the policy ef fect, at least in term s of its im plications for th e period following the O ctober 1982 change in the Federal Beserve’s operating p roced u re. First, if the Fed's new operating p ro ced u re attem p ts to m aintain a co n stan t sp read betw een the federal funds and d iscoun t rate, borrow ings always should be close to their assu m ed level. Chart 3 plots the actu al level of ad ju stm ent plus seasonal borrow ings an d th eir assu m ed level for weekly d ata from O ctober 6, 1982, through D ecem ber 1985. As the ch art shows, the actual level of borrow ing often deviates from the initial borrow ing assum ption, AUGUST/SEPTEMBER 1986 Table 4 Estimates of Equation 4 for Penalty and Non-Penalty Discount Rate Changes Constant ADRP ADRNP R2 SE .20 .35 .03 .08 Federal funds rate -.010 (0.93) .741* (2.58) .588 (1.46) Treasury bill rate -.000 (0.04) .372* (5.41) .060 (0.62) ‘ Indicates statistical significance at the 5 percent level. som etim es by a considerable m agnitude. Two of the m ost notable deviations, of co u rse, o ccu rre d in mid1984 and November 1985. Even w h en th ese unusual periods are ignored, the average absolute deviation of borrow ings from the initial borrow ing assu m p tion is $226 million, over 40 p ercen t of th e average level of the initial borrow ing assu m p tion during the period. Fu rtherm ore, there is a ten d en cy for the initial b or row ing assu m p tio n to follow, ra th e r th a n lead, changes in actu al borrow ings. It ap p ears th at the federal funds/discount rate sp read is m aintained w hen the borrow ing assu m p tion ch an g es; the d e m and for borrow ed reserves is not forced to conform to the borrow ing assum ption. “ Alternatively, Smirlock and Yawitz (1985) allow for the change in the discount rate to impact market interest rates with a lag of up to five days. Because they cannot reject the hypothesis that effects past the initial day are significant, they conclude that the rapid adjustment is consistent with market efficiency. Because the market rates nearly always return to levels prior to discount rate changes, how ever, it is possible to find no statistically significant long-run effect simply by making the lag “long enough” or a permanent effect (as they found) by making it “short enough.” Second, if the policy effect is strong, the resp o n se of m arket interest rates, especially th e federal funds rate, to a change in the d iscou n t rate should be larger since the O ctober 1982 change in the o peratin g p ro ced u re. To test this, equation 4 w as reestim ated for the period from O ctober 1, 1979, to Ju n e 11, 1986, and the re sp on se of m arket interest rates to n on tech n ical changes in the d iscou n t rate w as allow ed to be differ ent for the periods O ctober 1, 1979, to O ctober 5 ,1 9 8 2 , and O ctober 6, 1982, to Ju n e 11, 1986. The results are rep orted in table 5 w ith the coefficients for th e prea n d p o s t - O c t o b e r 1 9 8 2 p e r i o d s d e n o t e d by ADBNTPBE82 and ADBNTPOST82, respectively.M “ Several attempts to directly test various hypotheses were con ducted, but the results were unsatisfactory. For example, discount rate changes that indicate a change in policy — regardless of the reason given for the change — should be followed by a sharp change in the growth of nonborrowed reserves. Hence, statistical tests of nonborrowed reserve growth before and after discount rate changes were undertaken. Because the nonborrowed reserve data only are available biweekly, the tests were also done using weekly M1 data. The results indicated no statistically significant change in the growth rate of either nonborrowed reserves or M1; however, the data were highly variable and the observations few. Hence, these tests should be considered inconclusive. ^Because of the differences in the variation of the dependent vari ables between the periods, the equation was estimated adjusting for heteroskedasticity. Also, the pre-October 1982 period includes a surcharge variable because Thornton (1982) has shown the results are sensitive to this modification. While not reported here, the surcharge coefficient is nearly identical to that reported by Thornton. The coefficient on ADRNTPRE82 differs from that reported by Thornton primarily because of a difference in the sample period; however, all of the qualitative conclusions are the same. 16 FEDERAL RESERVE BANK OF ST. LOUIS AUGUST/SEPTEMBER 1986 C h a rt 3 A d ju s tm e n t plus S easo n al B o rro w in g s from F e d eral Reserve a n d In itia l B o rro w in g A ssum ption Billioas of dollars Billiots of dollars 5 ,------------- --------------- 5 A c tu a l b orrow in gs The results sh ow that th e responsiveness of the federal funds rate to ch an g es in th e d iscou n t rate w as essentially the sam e during the two periods. Indeed, an F -test of the equality o f the two coefficients does not reject the hypothesis that the respon se w as the sam e. There is a statistically significant difference in the responsiveness of th e T-bill rate; however, it has b ecom e less, not m ore, responsive to ch an ges in the d iscount rate. The evidence suggests th at the shift in the F ed ’s operating p ro ced u re has not increased the initial response of m arket interest rates to d iscou n t rate ch an ges; if anything it appeal's to have low ered it. Finally, there is one additional p iece of evidence on the an n ou n cem en t vs. policy effect of a d iscou n t rate ch an ge. The effect of th e d iscou n t rate o n m arket interest rates, especially the policy effect, im plies c a u sality running from the federal funds rate to o th er m arket interest rates. In o rd er to investigate this, tests of “G ranger cau sality” w ere co n d u cte d using both daily an d w eekly d ata for th e federal funds an d threem o n th Treasu ry bill rates. T h ese tests are designed to determ in e w h eth er ch an g es in o ne rate p reced e or follow ch an ges in th e oth er. (Details an d results are p resen ted in the appendix.) The results using th e daily d ata ind icate th at ch an g es in th e T-bill rate p r e c e d e ch an ges in the federal funds, the reverse of w h at the policy-effect hyp oth esis w ould m ost strongly imply. T h e results using w eekly d ata are less definitive, indi catin g that at tim es eith er rate p re ce d e s th e other. While this result is not particularly surprising, the fact th at the stro n ger (m ost statistically significant) effect is from the T-bill rate to th e federal funds rate is in co n sistent w ith a stron g policy effect. While th ese results are disquieting to those w ho su p p ort the policy effect, th ey are n ot conclusive. The im p ortan ce o ne assigns to the a n n o u n ce m e n t o r pol 17 AUGUST/SEPTEMBER 1986 FEDERAL RESERVE BANK OF ST. LOUIS Table 5 Estimates of Equation 4 with the Discount Rate Partitioned into the Pre- and Post-October 1982 Periods Dependent Variable Federal funds rate Treasury bill rate Constant ADRT ADRNTPRE82 ADRNTPOST82 -.006 (0.54) (139) .824* (2.85) .779* (2.71) -.001 (0.23) .129 (1.68) .686* (6.57) .292* (4.43) .382 F-Test' .013 10.206* R2 SE .14 1.01 .04 1.00 ’ Indicates statistical significance at the 5 percent level. ’Test of the hypothesis that the coefficients on ADRNTPRE82 and ADRNTPOST82 are equal. icy effects d ep en ds on th e interpretation of a d isco u n t rate ch an ge. If it is believed th at d iscou n t rate ch an ges are prim arily signals that th e Federal Reserve is going to con tin u e its p resen t policy of ease o r restraint, the policy effect should be nil. If, on th e o th er hand, d iscou n t rate ch an ges typically signal a ch an ge in the rate at w hich the Federal Reserve is going to supply reserves to the system , the extent to w h ich o ne b e lieves this ch an ge will affect m arket interest rates d ep en d s on o n e’s view of th e liquidity effect. If the liquidity effect is believed to be w eak an d transient — as m ost em pirical w ork suggests — the resp o n se of th e m arket to su ch ch an ges is essentially noise, w ith no real significance for the future co u rse (or level) of m arket interest rates. In su ch instances, d iscoun t rate cu ts th at are followed by m o re exp an sion ary m o n e tary policy ultim ately m ight be followed by higher, not lower, interest rates if su ch a policy ch an ge gives rise to exp ectatio n s of h igher inflation. On the o th er hand, if one believes th at th e liquidity effect is stron g an d lasting, ch an ges in the d isco u n t rate will be thought to have p erm an en t effects on m arket interest rates, but only if followed by a ch ange in Federal Reserve policy. the a n n o u n ce d ch an ge in th e d isco u n t rate; an d (3) the "policy effect,” the im p act of a subsequent ch an g e in Federal Reserve activity on th e m arket. Special a t tention w as given to th e hyp oth esis th at th e im p act of d iscou n t rate ch an ges on m arket interest rates b e cam e stro n ger following the Federal Reserve’s sw itch from a n onborrow ed reserve to a b orrow ed reserve operating p ro ced u re in O ctob er 1982. The evidence sh ow ed a statistically significant effect of a ch an ge in the discou n t rate on both the federal funds an d Treasu ry bill rates im m ediately following the d iscou n t rate ch an ge. A series of tests provided evidence, con sisten t with the theory, that th e direct effect of a discou n t rate ch an ge is nil. Consequently, the im pact of a d iscou n t rate ch an ge on m arket rates is due to an an n ou n cem en t effect, a policy effect o r both. The rapidity with w hich m arket rates resp o n d to the d iscou n t rate ch an g e suggests th at th e an n o u n cem en t effect is operative. F u rth erm ore, som e ind irect tests of th e policy effect p ro d u ced results th at are in con sis tent w ith it, suggesting th at d iscou n t rate ch an ges have h ad n o p erm an en t effect on m arket interest rates. CONCLUSIONS REFERENCES This article w as intend ed to clarify th e relationship betw een th e Federal Reserve’s d isco u n t rate an d m ar ket interest rates. T hree distinct, thou gh n ot m utually exclusive, potential effects of a d isco u n t rate ch an ge on m arket interest rates w ere outlined: (1) th e “direct, ce te ris paribus, effect,” w hich ab stracts from market reaction s to the d iscoun t rate ch an ge an d any su b se quent ch ange in Federal Reserve operation s; (2) the "an n o u n cem en t effect,” w h ich reflects the changing exp ectatio n s of th e Federal Reserve’s activity b ased on 18 Batten, Dallas S., and Daniel L. Thornton. “Discount Rate Changes and the Foreign Exchange Market,” Journal of International Money and Finance (December 1984), pp. 279-92. _________“The Discount Rate, Interest Rates and Foreign Ex change Rates: An Analysis with Daily Data,” this Review (February 1985), pp. 22-30. Broaddus, Alfred, and Timothy Cook. “The Relationship Between the Discount Rate and the Federal Funds Rate Under the Federal Reserve’s Post-October 6, 1979 Operating Procedure,” Federal Reserve Bank of Richmond Economic Review (January/February 1983), pp. 12-15. AUGUST/SEPTEMBER 1986 FEDERAL RESERVE BANK OF ST. LOUIS Brown, W. W., and G. J. Santoni. “Monetary Growth and the Timing of Interest Rate Movements,” this Review (August/September 1983), pp. 16-25. Cagan, Phillip, and Arthur Gandolfi. “The Lag in Monetary Policy as Implied by the Time Pattern of Monetary Effects on Interest Rates,” American Economic Review (May 1969), pp. 277-84. Carr, Jack, and Michael R. Darby. “The Role of Money Supply Shocks in the Short-Run Demand for Money,” Journal of Monetary Economics (September 1981), pp. 183-99. Cukierman, Alex. “Central Bank Behavior and Credibility: Some Recent Theoretical Developments,” this Review (May 1986), pp. 5-17. Daily Report for Executives, The Bureau of National Affairs, Inc., Washington (October 8,1982), L12. Federal Reserve Bank of New York. “Monetary Policy and Open Market Operations in 1985,” Quarterly Review (Spring 1986), pp. 34-53. Hakkio, Craig S., and Douglas K. Pearce. “Exchange Rates and Discount Rate Changes,” Federal Reserve Bank of Kansas City working paper 86-06 (1986). Melvin, Michael. “The Vanishing Liquidity Effect of Money on Inter est: Analysis and Implications for Policy,” Economic Inquiry (April 1983), pp. 188-202. Partian, John C., Kausar Hamdani, and Kathleen Camilli. "Re serves Forecasting for Open Market Operations,” Federal Re serve Bank of New York Quarterly Review (Spring 1986), pp. 1933. Roley, V. Vance. “Market Perceptions of U.S. Monetary Policy Since 1982,” Federal Reserve Bank of Kansas City Economic Review (May 1986), pp. 27— 40. Sellon, Gordon H., and Diane Seibert. “The Discount Rate: Experi ence Under Reserve Targeting,” Federal Reserve Bank of Kansas City Economic Review (September-October 1982), pp. 3-18. Smirlock, Michael, and Jess Yawitz. “Asset Returns, Discount Rate Changes, and Market Efficiency,” The Journal of Finance (Septem ber 1985), pp. 1,141-158. Smith, Warren L. “The Instruments of General Monetary Control,” National Bank Review (September 1963), pp. 47-76. Thornton, Daniel L. “The Discount Rate and Market Interest Rates: What’s the Connection?” this Review (June/July 1982), pp. 3-14. _________“The FOMC in 1982: De-emphasizing M1,” this Review (June/July 1983) pp. 26-35. ________ . “Money Demand Dynamics: Some New Evidence,” this Review (March 1985), pp. 14-23. Thornton, Daniel L., and Dallas S. Batten. “Lag-Length Selection and Tests of Granger Causality Between Money and Income,” Journal of Money, Credit and Banking (May 1985), pp. 164-78. Wallich, Henry C. “Recent Techniques of Monetary Policy,” Fed eral Reserve Bank of Kansas City Economic Review (May 1984), pp. 21-30. (See appendix on next page.) 19 FEDERAL RESERVE BANK OF ST. LOUIS AUGUST/SEPTEMBER 1986 Appendix w here A d en otes the first difference o perator, i.e., ARfl = R„ — Rn ,, an d H, an d RTd en ote the federal funds an d three-m on th Treasury bill rates, respectively. T h e p ro ced u re co n sists of testing the hypothesis th at p., = |x, = . . . = |xK = 0. If this hyp oth esis is rejected , it is said the “causality” runs from the federal funds rate (R() to the th ree-m on th Treasury bill rate (R,.). To test for causality running from the Treasu ry bill rate to the federal funds rate, the equation Tests o f Granger Causality T ests of “G ranger cau sality” are really tests of tem poral ordering of time series. The test of causality running from the federal funds rate to the T reasury bill rate is perform ed by estim ating, using ordinary least squares (OLS), the equation K K K AR,, = a. + X 8,AK„ , + 2 (jl,AR„ ,, i= 1 K ARf, = 0„ + 2 XAR„. + 2 eiARn , i= 1 i= 1 i= 1 Table A.1 Granger Causality Results for ARf and ART: Daily Data Tests of jjl’s = 0 Lags of ARr Lags of AR, 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12 .342 .624 .570 .678 .775 .707 .718 .494 .339 .267 .238 .211 .355 .610 .524 .647 .739 .682 .706 .480 .325 .242 .227 .208 .358 .616 .481 .617 .707 .666 .696 .457 .305 .223 .217 .205 .382 .646 .519 .677 .741 .713 .751 .492 .311 .223 .220 .218 .377 .646 .526 .681 .715 .709 .754 .473 .291 .197 .198 .199 .378 .647 .527 .682 .716 .704 .746 .471 .292 .198 .199 .200 .346 .583 .486 .651 .672 .695 .650 .462 .317 .250 .237 .217 .346 .585 .497 .662 .685 .704 .653 .436 .284 .216 .213 .199 .338 .575 .498 .660 .683 .694 .634 .439 .246 .171 .178 .176 .342 .586 .512 .675 .675 .686 .645 .428 .218 .185 .178 .168 .342 .587 .513 .675 .676 .686 .646 .429 .218 .184 .172 .159 .341 .585 .512 .675 .675 .684 .639 .423 .219 .186 .172 .163 8 9 10 11 12 .059 .031* .068 .084 .145 .174 .166 .220 .299 .381 .435 .424 .061 .032* .070 .087 .149 .178 .168 .222 .301 .382 .434 .425 .057 .026* .058 .071 .125 .146 .135 .177 .246 .317 .381 .384 .060 .026* .055 .064 .114 .132 .116 .152 Tests Of e’S = 0 Lags of AR, Lags of ART 1 2 3 4 5 6 7 1 2 3 4 5 6 7 8 9 10 11 12 .681 .837 .597 .640 .524 .625 .673 .686 .770 .792 .850 .787 .379 .581 .372 .409 .288 .382 .476 .526 .620 .634 .714 .633 .304 .419 .469 .462 .302 .385 .480 .552 .648 .654 .734 .649 .195 .249 .386 .540 .293 .360 .466 .540 .641 .659 .740 .628 .173 .167 .300 .453 .437 .474 .590 .676 .765 .799 .859 .745 .107 .097 .198 .310 .397 .524 .639 .733 .809 .835 .878 .777 .102 .054 .117 .166 .256 .338 .377 .482 .563 .638 .707 .658 'Indicates significance at the 5 percent level. 20 .098 .050* .106 .147 .235 .303 .315 .408 .477 .560 .627 .579 .213 .276 .343 .319 AUGUST/SEPTEMBER 1986 FEDERAL RESERVE BANK OF ST. LOUIS Table A.2 Granger Causality Results for ARf and ART: Weekly Data Tests of ( ’s = 0 x Lags of ART Lags of AR, 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12 .269 .538 .291 .325 .209 .025* .028* .038* .061 .074 .092 .099 .258 .505 .337 .374 .248 .031* .034* .047* .074 .088 .109 .117 .242 .493 .423 .512 .352 .053 .056 .077 .115 .136 .161 .170 .314 .580 .466 .602 .501 .086 .084 .113 .162 .169 .205 .215 .319 .570 .477 .613 .535 .086 .085 .112 .161 .170 .210 .209 .312 .564 .453 .584 .531 .051 .066 .088 .130 .147 .192 .207 .361 .615 .501 .617 .549 .066 .092 .108 .158 .185 .241 .265 .415 .673 .536 .648 .590 .077 .108 .131 .187 .216 .275 .294 .352 .649 .567 .677 .648 .058 .085 .117 .162 .225 .293 .346 .339 .632 .482 .607 .593 .056 .085 .116 .164 .225 .297 .356 .341 .635 .485 .606 .596 .057 .087 .118 .166 .228 .299 .361 .434 .736 .525 .632 .544 .058 .087 .118 .161 .221 .292 .370 Tests Of e ’S = 0 Lags o f AR, Lags of ART 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12 .073 .181 .043* .027* .045* .040* .045* .027* .044* .062 .044* .063 .031* .097 .029* .016* .024* .022* .027* .017* .027* .036* .028* .041* .022* .070 .008* .005* .007* .009* .012* .007* .013* .014* .013* .020* .039* .115 .021* .015* .018* .021* .027* .017* .028* .030* .026* .038* .041* .123 .024* .018* .021* .024* .031* .018* .030* .033* .027* .041* .038* .115 .020* .015* .015* .021* .028* .016* .027* .030* .025* .037* .046* .134 .024* .017* .017* .024* .032* .019* .030* .034* .028* .042* .041* .123 .029* .022* .022* .030* .044* .029* .045* .049* .041* .059 .046* .137 .080 .069 .063 .071 .103 .101 .149 .144 .115 .150 .045* .136 .078 .071 .065 .073 .107 .104 .153 .141 .109 .143 .042* .128 .077 .065 .066 .067 .103 .107 .158 .157 .110 .146 .054 .157 .087 .071 .111 .167 .161 .103 .138 .108 .146 .183 'Ind ica te s significance at the 5 percent level. is estim ated and the hypothesis that e, = e., = . . . = eK = 0 is tested. If the hypothesis is rejected , the cau sal ity runs from the T reasu ry bill rate to the federal funds rate. If the hypotheses co n cern in g the jjl ’ s an d the e's are both rejected, there is said to be bidirectional causality betw een the rates. If n eith er is rejected, the series are said to be independent. The tests w ere perform ed using both daily and weekly data. B ecause the test results are quite sen si tive to the ord er of the lag, K, the tests w ere perform ed on all orders up to K = 12.' The significance levels 1 For a discussion of this procedure, see Thornton and Batten (1985). corresp on d in g to the F -statistics for all ord ers are presen ted in tables A.l and A.2 for the daily and weekly data, respectively. The tests using daily d ata show unidirectional c a u sality from R, to R,, the opposite of w hat is required for policy actions to be tran sm itted from the federal funds rate to o th e r m arket interest rates. It should be noted that the daily federal funds rate series exhibits co n sid erably m ore variability than the T-bill rate series. W hen these data are sm ooth ed by averaging over a week, the tests indicate bidirectional causality; how ever, the stron ger relationship ap p ears to be running from the T-bill rate to the federal funds rate. 21 FEDERAL RESERVE BANK OF ST. LOUIS AUGUST/SEPTEMBER 1986 A Microeconomic System-Wide Approach to the Estimation of the Demand for Money Salam K. Fayyad A STABLE d em and-for-m oney relationship is a n ecessary condition for the viability of a m on etary policy based on the use of m on etary aggregates as interm ediate policy targets. In recen t years, stan d ard m oney-dem and form ulations have exhibited large shifts that rem ain largely u nexplained today despite extensive research efforts devoted to determ ining the reasons for these shifts. This p ap er p resen ts an alternative to the stan d ard single-equation m ethod of estim ating the d em and for m oney. The alternative, called the m icro eco n om ic system -w ide ap p roach to dem an d analysis, differs in several fundam ental wavs from the usual m onevd em an d specification. The p u rp ose of this article is to show how the system -w ide ap p roach can be applied to estim ating the d em an d for m oney. The results indi cate that in-sam ple prediction s m ade using this ap p ro ach closely track the actu al data over the 1969-85 period. THE STANDARD MONEY DEMAND FORMULATION: A BRIEF REVIEW Over the past three d ecad es, m ost dem and-form oney studies have em ployed sim ilar specifications. Typically they use incom e (as a tran saction variable) an d one o r m ore (typically two) interest rates (to c a p Salam K. Fayyad is an assistant professor at Yarmouk University (Jordan) and a former visiting scholar at the Federal Reserve Bank of St. Louis. The author is intellectually indebted to his teacher, William A. Barnett. Laura A. Prives provided research assistance. 22 ture the effect of the opp ortun ity co st of holding money) as explanatory variables; the d ep en d en t vari able is generally the stock of real M l b alances. The w ide a cce p ta n ce of the stan d ard m on ey d e m and specification is understandable. It em bodies a proposition, w hich, sin ce K eynes’ G eneral Theory, has co n stitu ted a key tenet of the received w isdom o n the d em an d for m oney: the desire to hold m on ey balan ces is directly related to th e n eed to co n d u ct tran sactio n s and inversely related to the opp ortun ity co st of h old ing m oney b alances. In addition, it perform ed rem ark ably well in a statistical sense. The coefficients of the explanatory variables h ad “sensible” signs an d m agni tudes, and the estim ated m odel fit the d ata very well. The disquietude accom p an yin g Goldfeld’s (1976) d iscovery th at his sta n d a rd fo rm u lation of the d em and-for-m oney function began in 1973 to system atically overpredict th e real m on ey b alan ces u n d e r scores the im p ortan ce th at h as been a tta ch e d to the stability, and h en ce predictability, of the d em an d for m oney. It is not surprising th at the rep o rted shift in Goldfeld’s specification, o r w hat, after 1976, b ecam e generally known as the “ca se of the m issing m o n ey,” instigated a seem ingly tireless search for a verifiable explanation of w hat hap p ened .' A review of the vast literature devoted to finding the reason s for the shift in m oney d em an d reveals that these studies are largely u n su ccessfu l in a cco u n tin g 1 recent study suggests that the demand-for-money function has A undergone shifts in the periods 111/1962, IV/1973, IV/1979, and I/ 1980. See Mizrach and Santomero (1986). FEDERAL RESERVE BANK OF ST. LOUIS for it. F o r exam ple, Laidler has co m m en ted that: T h e first th in g to be said . . . is w h atev er e lse th e y do, th e y d o n o t re s c u e th e d e m a n d fo r M , fu n ctio n from th e su s p ic io n o f in stab ility . . . . [T]he o ften u n s a tis fa c to ry re su lts . . . in d ic a te th at fu rth e r w o rk is req u ired , ra th e r th a n th a t th e lin e o f inqu iry th at th e y re p re s e n t sh o u ld be a b a n d o n e d .2 The inconclusiveness of the evidence on w hat cau sed the stan d ard m on ey-d em and specification to shift in 1973 ca n be viewed as an indication that exam ining alternative ap p roach es to the dem an d -for-m on ey for m ulation might be useful. The alternative offered h ere is derived from a m icro eco n o m ic system -w ide a p p ro ach to d em an d analysis. A NEW APPROACH TO MODELING MONEY DEMAND: THE MICROECONOMIC SYSTEM-WIDE APPROACH The basic prem ise w hich underlies any m icro th eo retic ap p ro ach to co n su m e r d em an d analysis is that the co n su m er m axim izes a n eoclassical utility fu n c tion subject to a budget co n strain t.1 A m odel co n sis tent with both the principles of m icro eco n om ic theory and aggregation theory yields specific behavioral im plications w hich can be tested using available data aggregated over goods and co n su m ers. (Some criti cism s of including m oney in this ap p roach ap p ear on page 24.) This study uses a neoclassical utility function defined over five expen d itu re categories, two of w hich are presu m ed to cap tu re the “m on etary services” in the U.S. econ om y. By restricting the analysis to five expenditure categories, this study assu m es the exist en ce of a m acro utility function that is weakly sep ara ble in these categories.4 T he solution to the co n su m er ch o ice problem w hen the utility function is defined over five goods is a system of five d em an d equations. In each equation, the quantity d em an d ed of a specific good is expressed as a function of the total am ou nt available for' sp en d 2 See Judd and Scadding (1982), p. 1014. 3 neoclassical utility function is one that is continuously twice A differentiable and quasiconcave with positive marginal utility every where. 4 neoclassical utility function is weakly separable in a block of goods A if and only if the marginal rate of substitution between any two goods inside the block is independent of consumption outside that block. While this separability assumption may seem overly restrictive, it is actually less restrictive than that maintained by studies in which money is considered to be the sole argument in the utility function. See, for example, Ewis and Fisher (1984). AUGUST/SEPTEMBER 1986 ing on all five goods, and (in the general case) of their prices. Naturally, th e exact specification of these d e m an d equations will d ep en d on the specific form of the utility function ch o sen . This study, how ever, uses a general d em an d system co n sisten t with the m axim i zation of an arbitrary neoclassical utility function. Thus, while the system is subject to all the restrictions that eco n o m ic theory implies, the results are invariant to the functional form of the utility function being m axim ized. This ch o ice avoids the loss of generality w hich m ay result if a p articu lar functional form is specified an d perm its testing of h yp oth eses about the stru ctu re of the utility function itself. The m icro eco n om ic system -w ide ap p roach to d e m and analysis deals with the allocation of total sp en d ing am ong the individual goods co n sid ered . Thus, for the specific set of goods ch osen , the exp lan atory vari ables in the d em an d system are th e am ou nt available for spending and the p rices of these goods. This a p p roach provides a convenient m ean s for acquiring detailed inform ation about utility-based attributes of goods; this inform ation is readily available by in sp e ct ing the signs (and, of cou rse, the statistical an d e co nom ic significance) of estim ated incom e, own- and cross-p rice param eters. The d em an d m odel u sed in this study, th e absolute p rice version of the R otterdam m odel, w as ch o sen primarily b ecau se the theoretical restriction s are readily exp ressed in term s of the m o d el’s p aram eters. This makes it relatively easy to im pose and to test the validity of these restrictions.3 A nother attractive fea ture of the Rotterdam m odel is that it ca n be used to provide predictions of the value (budget) sh ares of the goods included in the analysis. These p red iction s can be u sed to co m p u te m easures of inform ational in a c cu racies useful in assessing the p erfo rm an ce o f the dem and system as a w hole and the individual d e m and equations as well. The m on etary variables u sed are the real flow of m on etary services provided by various m on etary a s sets, not the sim ple sum of the real stocks of m on etary aggregates generally u sed in stan d ard m on ey-d em an d analysis. A m easure of the m onetary-service flows w as obtained by evaluating the stocks of m on etary assets at th eir corresp on d in g u ser-co st p rices (see the dis cussion of the data below). T he u ser-cost price of each m on etary asset is the difference betw een the interest 5 the aggregated-over-consumers version of the model derived by ln Barnett (1979), the macro parameters are subject to the same restrictions as their micro counterparts. (See footnote 9.) 23 FEDERAL RESERVE BANK OF ST. LOUIS AUGUST/SEPTEMBER 1986 Can Money Be Included in a Microeconomic System-Wide Demand Model? The applicability of the theoretical restrictions in the general case of d em and for m oney and goods h as been questioned. In fact, if, as in the Samuelsonian tradition, the utility-based analysis of the dem an d for m oney is handled by putting m oney an d prices in the utility function, then, by the strong results p ro d u ced by Sam uelson and Sato (1984), the restrictions in question, as they pertain to the d em a n d f o r go o d s, are sim ply unattainable. While potentially disquieting, the Sam uelson-Sato results are not unqualifiedly binding. Indeed, these results are founded in th e view, long esp oused by Sam uelson, that, in co n n ectio n w ith the inclusion of m on ey in the preferen ce stru ctu res, m oney is w anted solely for the p u rp oses of facilitating tran s actions. As Sam uelson (1983), p. 117, states: In th is c o n n e c tio n , I have re fe re n c e to n o n e o f th e te n u o u s c o n c e p ts o f m o n ey , a s a n u m e ra ire c o m m od ity, o r as a c o m p o s ite co m m o d ity , b u t to m o n ey p ro p er, th e d istin g u ish in g fe a tu re s o f w h ic h a re its in d ire c t u se fu ln e ss, n o t fo r its o w n sake but fo r w h at it c a n buy, its a c ce p ta b ility , its n o t b e in g "u s e d u p ” b y u se , etc., e tc . This is the rationale behind S am uelson’s inclusion of p rices and m oney in the utility function specified to be h om ogeneous of degree zero in both m oney an d p rices. It is precisely this form ulation to w hich the Sam uelson-Sato results pertain. One could argue that S am uelson’s view of w hat m oney is w anted for is unduly restrictive. In fact, of the assets currently regarded as potential so u rces of m on etary services in the U.S. eco n om y (see table 1), only a few are “generally accep tab le in e x ch an g e.”1 Fu rtherm ore, the supposition, based on Sam uelson’s view, that m oney can n o t properly be treated like o th er com m od ities can also be ques- 'In this study, the Federal Reserve’s definition of monetary assets was taken as given. The use of the Fed’s definitions does not mean that the list of assets which appears in table 1 includes all assets that provide monetary services in the U.S. economy or, for that matter, that all assets included in the list provide such services. Digitized 24 FRASER for tioned. H ouseholds co n su m e the services provided by various expen d itu re categories ostensibly be cau se of the utility they derive from these exp en d i ture categories: in general, little, if any, effort is d irected tow ard d eciphering the n atu re of the util ity involved in those cases. By the sam e token, it can be m aintained that m oney is held b ecau se of the utility it provides, w ithout having to sp ecu late as to w h eth er that utility derives from m oney's ‘general acceptability in exch an g e,” the serenity its h olders exp erience by holding it, o r from any o th er knowable o r even unknow able attribute.2 Indeed, if m oney ca n be treated like o th er goods, then it can be includ ed in the utility function in precisely the sam e m an n er as any o th e r good. In that case, the Sam uelson-Sato results w ould not apply, an d one cou ld thus im pose o r test for any of the restriction s implied by eco n om ic theory. Interestingly, even w ithin the Sam uelson-Sato framework, the theoretical restriction s w ould not be unattainable if the utility function w ere weakly separable in the block of goods (see Sam uelson an d Sato (1984), pp. 5 9 2 -9 5 ). Hence, it is legitim ate to im pose o r test for any of the restriction s im plied by th eoiy if one assu m es, or, even better, tests for an d (where applicable) im p oses blockw ise w eak sep ara bility in goods. The la tte r w as d on e in this study, since weak separability in the block of goods could not be statistically rejected .3 2 course, this amounts to suggesting that money is held for the Of “moneyness” of it. While tautological, this statement can be made operational by hypothesizing that, on the margin, the extent to which income is forgone when monetary assets are held is a measure of the moneyness that these assets possess. The gain that is realized by adopting this hypothesis is considerable; not only does it play a key role in the measurement of the flow of monetary services in terms of readily observable data, it also inherently captures the various degrees of moneyness provided by various monetary assets. 3 The manner in which weak separability was tested for is dis cussed at length in Fayyad (1986), chapter 4. A preliminary draft of a paper on this subject is available on request. AUGUST/SEPTEMBER 1986 FEDERAL RESERVE BANK OF ST. LOUIS rate paid on that asset and the m axim u m available holding-period yield.0 THE MODEL F o r n goods, the (discrete-tim e) absolute p rice ver sion of the Rotterdam m odel is given by n (1) w^Dx,, = ^D m ,* + 2 Tr^Dp,, + eM i= i i = l , ..., n.7 The x ’s and p ’s denote the quantities and p rices of the various goods, respectively, an d the subscript t in dexes time. D is the log-change operator; thu s Dxh = Adog x„). W; den otes the expend iture (value) share of the ith good, w* = (w, + w h)/2 is that g o o d ’s average value share over two su ccessive tim e periods. Thus, the d ependent variables in the m odel are the (average) share-w eighted grow th rates of the quantities of goods. The explanatory variables are the grow th rates of real incom e (mt ) and prices.8 The last term in the * system of dem and equation 1, e„ d en otes the dem an d d isturbance. The p roperties of this term are discu ssed in appendix A in con ju n ction w ith the estim ation p ro ced u re em ployed in this study. The p aram eters of th e m od el are (xf ( = p, an d tt m . is the m arginal budget sh are of the ith PiPi dxi good. T ( = ---- 1 — , (j = r.j ° ^ 1 m dpi i. 1, ..., n ) ) is the ith good s p rice coefficient. U nder the stan d ard assu m p tions of eco n o m ic theory, if the co n su m er m axim izes a n eo classical utility function subject to a budget co n straint, then the above p aram eters satisfy the follow ing co n strain ts: (2) E |x, = 1 , i (3) 2 ttm = 0 , i 6 The maximum available holding-period yield is the highest yield of those available either on the monetary assets or on Baa-rated bonds. 7 detailed discussion of the model’s derivation and applications can A be found in Theil (1971, 1975, 1976, 1980), Barten (1969), and Barnett (1979,1981). 8 this study, the terms “expenditure” and “income” are used ln interchangeably. When the latter is used, however, it means “full income,” that is, income augmented by expenditure on the mone tary assets that are included in this study. In the estimation proce dure employed in this study, Dm* is replaced by Dx„ where Dx, = 2 wj Dx„. See Theil (1971), pp. 331-32. (4) [T ji) is sym m etric an d negative sem idefinite.8 T In the estim ation p ro ced u re em ployed in this study, the co n strain ts = 1, 2,iTn = 0, an d [Tri(] = [iTn] w ere im posed. The negative sem idefiniteness of th e [irM ] m atrix w as not im posed; how ever, it w as ch eck ed for. THE DATA The d ata con sist of U.S. quarterly tim e series of exp en d itu res on, an d p rices of, food, nondurables, services and two blocks of m on etaiy assets for the period 1/1969-1/1985. Together, the two blocks of m o n etary assets, M l and ABM1, com p rise the 27 assets that the Federal Reserve Board cu rren tly recognizes as potential so u rce s of m o n etaiy services in the U.S. econ om y. M l is the n arrow m o n etaiy aggregate, co n sisting of cu rre n cy and total checkable deposits. ABM1 co n sists of the n on -M l m o n etaiy assets show n in table 1. Data on the first three com m od ity grou p s (food, nondurables an d services) w ere obtained as follows: A tim e series on the p rice of e a ch co m m o d ity grou p (ph) w as gen erated from available tim e series on cu rren tdollar an d (1972) co n stan t-d ollar co n su m p tio n ex pen d itu res (ph q„ an d p i7, qi| respectively) an d the ( identity (ph ii7 = (p,, qT i72 q„). P er-cap ita co n stan t/p J 1 /p dollar exp en d itu res in ea ch q u arter w ere th en ob tained by dividing th e aggregate co n stan t-d ollar e x p e n d itu re s by th e c o r r e s p o n d in g m id -q u a rte r population size (N,). Thus, in term s of the variables 1 w hich ap p ear in the estim ated system , x„ = — pl 7, q„. One ca n gen erate the d ata on th e quantities and p rices of M l an d ABM1 m on etary services as follows: (1) Convert the nom inal balan ces of m on etaiy assets into real b alan ces by deflating the form er by the "tru e cost-of-living ind ex.” In this study, this index w as the geom etric m ean of the C on su m er Price Index and the C om m erce D ep artm en t's implicit 9 question may arise as to whether these restrictions are applicable, A given that the data are aggregated over both goods and consumers. Insofar as goods are concerned, Hicks’ composite commodity theo rem can be used, assuming that each of the commodity groups is an elementary good. Resolving the more formidable issue of aggrega tion over consumers requires using the aggregation results pro duced by Barnett (1979). In his aggregated-over-consumers abso lute price version of the Rotterdam model, Barnett treats the macrocoefficients, n and it, as population versions of weighted average microcoefficients, with the weights proportional to corre sponding incomes. He then shows that the macrocoefficients have the same properties as their micro counterparts, p and -nir ., 25 FEDERAL RESERVE BANK OF ST. LOUIS AUGUST/SEPTEMBER 1986 dexes for M l an d ABM1 for the entire sam ple period. Table 1 Potential Monetary Assets Asset Description Component 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 Currency and traveler's checks Demand deposits held by households Demand deposits held by business firms Other checkable deposits less Super NOW accounts Super NOW accounts at commercial banks Super NOW accounts at thrifts Overnight repurchase agreements Overnight Eurodollars Money market mutual fund shares Money market demand deposit accounts at commercial banks Money market demand deposit accounts at thrifts Savings deposits less MMDAs at commercial banks Savings deposits less MMDAs at savings and loans Savings deposits less MMDAs at mutual savings banks Savings deposits less MMDAs at credit unions Small time deposits and retail repurchase agreements at commercial banks Small time deposits and retail repurchase agreements at thrifts Small time deposits at credit unions Large time deposits at commercial banks Large time deposits at thrifts Institutional money market mutual funds Term repurchase agreements at commercial banks and thrifts Term Eurodollars Savings bonds Short-term Treasury securities Banker’s acceptances Commercial paper p rice d eflator for p erso n al co n su m p tio n p enditures. ex (2) Evaluate the real balan ces of each m o n etary asset in the base period at its real u ser-cost p rice to obtain the real exp en d iture on th at asset during the base period."' (3) Sum the exp en d itu res th u s obtained over the co m pon en ts of M l an d ABM1. (4) C om pute the Tornqvist-Theil Divisia quantity in- '°The user-cost price of money was derived by Barnett (1978). See also Barnett (1986) and (1981), chapter 7. 26 (5) Set the b ase-period exp en d itu res obtained in (3) equal to the respective quantity ind exes co m p u te d in (4) an d interpolate to acquire co m p lete series on the real exp en d itu res on th e m o n e ta iy services provided by M l and ABM1. (6) C on stru ct Tornqvist-Theil Divisia p rice ind exes of the u ser-cost p rices of the respective co m p o n en ts of M l and ABM1. THE ESTIMATION RESULTS The m axim um likelihood estim ates of the p aram e ters of th e absolute p rice version of the R otterdam m odel and the associated incom e and price elastici ties are rep orted in table 2 ." E stim ates of the incom e coefficients ((A are all positive an d statistically signifi ,) can t at usual significance levels, indicating that the five com m od ity groups includ ed in this study are norm al goods. The incom e elasticity of d em an d for M l show n in table 2 (0.53) is sim ilar to those rep o rted in o th er studies. M oreover, it co rresp o n d s closely to its th e o retical value of 0.50 im plied by the Baum ol (19 5 2 )Tobin (1956) inventory — th eo retic m odel of the tra n s action s dem an d for m oney. The own- and cross-p rice coefficients (ttm are g en er ) ally estim ated w ith less p recision than the incom e coefficients. C onsistent w ith the stan d ard a ssu m p tions of eco n o m ic theory, estim ates of the (Slutsky) ow n-price coefficients are negative, although n ot all are statistically significantly different from zero at usual significance levels.1 3 "The income elasticity of demand for the ith commodity group, (x ,, is given by | , = jl This result can be verified by a simple manipu lation of the definition ix = , dm On the other hand, the Hicks-Alien price elasticity of demand for the ith group, (i,,, is given by jjl, , = which can also be verified by a simple manipulation of the definition , 1 1 = m apj' 1 Negativity of the own-price coefficients, the diagonal elements in the 2 [iiij matrix, is a necessary, but not sufficient, condition for it to be negative semidefinite; a matrix is negative semidefinite if and only if all of its characteristic roots are nonpositive, and at least one root is zero. This property, which was not imposed in this study, was examined by computing the characteristic roots of the estimates of the [tt,,] matrix in table 2. The computed characteristic roots are (0.0000,0.0000, -0.0022, -0.0097, -0.1743); thus, the negativ ity condition is satisfied. FEDERAL RESERVE BANK OF ST. LOUIS AUGUST/SEPTEMBER 1986 Table 2 Maximum Likelihood Estimates of the Absolute Price Version of the Rotterdam Model Equation M i Food Nondurables ABM1 Services M 1 Food 0.187083 (6.682878) -0.116241 (-7.380222) 0.058199 (4.867328) 0.057593 (3.167629) 0.000352 (2.913345) 0.000097 (1.779196) Nondurables 0.291246 (9.685506) 0.058199 (4.867328) -0.034326 (-1.918331) -0.024572 ( - 1.228305) 0.000549 (2.129760) 0.000150 (1.319010) 0.467938 (10.304847) 0.057593 (3.167629) -0.024572 (-1.228305) -0.034144 (-1.135707) 0.000882 (1.596066) 0.000242 (1.686502) ABM1 0.042172 (2.090867) 0.000352 (2.913345) 0.000549 (2.129760) 0.000882 ( 1.596066) 0.001399 (-2.558969) 0.000384 ( -1.832747) M1 0.011561 (3.043011) 0.000097 (1.779196) 0.000150 (1.319010) 0.000242 (1.686502) - 0.000384 (-1.832747) -0.000105 -0.276950) Services Note: t-ratios are in parentheses. Average Income Elasticities Food Nondurables Services ABM1 M1 0.828044 1.390072 0.942341 1.027321 0.525813 Average Hicks-Allen Elasticities Food Food Nondurables -0.514493 0.257394 0.254911 Services ABM1 M1 0.001560 0.000428 0.000718 Nondurables 0.277775 -0.163834 -0.117278 0.002619 Services 0.115982 -0.049483 -0.068760 0.001775 0.000487 ABM1 0.008585 0.013365 0.021474 -0.034071 -0.009354 M1 0.004394 0.006841 0.010991 -0.017464 -0.004763 The Model’s In-Sample Predictive Performance As stated earlier, the R otterdam m odel can tie used to provide predictions of the value (budget) shares. The m odel's implied prediction of the sh are of the ith good at time t is given by Wi,1+1 = w h - eH, w here w„ is the actual value share of the ith good, and eh is the residual of the ith dem an d equation at tim e t. The in-sam ple predictive p erform ance of the R otter d am m o d el c a n be ev alu ated in te rm s of its inform ation-theory results; a general discussion of this m eth od of assessing prediction a ccu ra cy is p re sented in appendix B. C om puted m easu res of infor m ation (prediction) in accu racies from the m odel are rep orted in table 3, along w ith inform ation -in accu racy m easures for a naive (no-change) extrapolation of the value shares. The rep orted m easu res show substantial red u ction s in inform ation in accu racies w hen the m o d e l re s u lts a re c o m p a r e d w ith th e n aiv e p red iction s.1 3 F u rth er insight into the m o d el’s in-sam ple p red ic tions m ay be gained by plotting the; actual and p re dicted shares; this is done in ch arts 1 -5 . An inspection of these ch arts reveals that the m o d el’s in-sam ple predictions track the data extrem ely well; this is esp e cially true for the M l equation despite considerable variability in the actual sh ares of M l. These results suggest that the dem an d for M l, as derived in this 1 ln fact, in view of the greater variability of the shares of M1 and 3 ABM1 relative to the shares of the other goods and services shown in charts 1-5, it is not surprising that predictions from the money equations “beat” the naive model by a larger margin than predic tions from the other equations. In the presence of high period-toperiod variation in the actual shares, the no-change naive model will always perform poorly. 27 FEDERAL RESERVE BANK OF ST. LOUIS AUGUST/SEPTEMBER 1986 Table 3 Average Information Inaccuracies1 Rotterdam Model Naive System Results Uncorrected information inaccuracy Information inaccuracy with d.f. correction Percent reduction from naive 14.40 14.82 96.73% 452.60 3.505 71.22% 4.141 65.35% 5.943 82.79% 5.083 98.67% 0.646 98.74% 12.18 Single Equation Results Food information inaccuracy Percent reduction from naive Nondurables information inaccuracy Percent reduction from naive Services information inaccuracy Percent reduction from naive ABM1 information inaccuracy Percent reduction from naive M1 information inaccuracy Percent reduction from naive 'The information inaccuracies are to be multiplied by 10-4. C h a rt 1 Actual vs. Predicted V alue Shares of Food 28 11.95 34.54 381.30 51.30 FEDERAL RESERVE BANK OF ST. LOUIS AUGUST/SEPTEMBER 1986 C h a rt 2 Actual vs. Predicted Value Shares of Nondurables C h a rt 3 Actual vs. Predicted Value Shares of Services 29 FEDERAL RESERVE BANK OF ST. LOUIS Actual vs. Predicted Value Shares of ABM1 C h a rt 5 Actual vs. Predicted Value Shares of M l 30 AUGUST/SEPTEMBER 1986 AUGUST/SEPTEMBER 1986 FEDERAL RESERVE BANK OF ST. LOUIS study, was m ore stable than the d em an d for o th er goods, services and financial assets over the sam ple period. CONCLUDING REMARKS This study has d iscu ssed an ap p roach to the esti m ation of the dem an d for m oney that relies on a m ethodology m arkedly different from that em ployed in the conventional m oney d em an d analysis. The a p p roach is explicitly derived from the principles of m icro eco n om ic th eoiy and em p h asizes the im p or tance of interaction am ong goods. Th e m odeling p ro cess is not influenced by a search for “goodn ess of fit”; instead the em phasis is p laced on the m od el’s co n sist en cy with explicit utility-m axim izing conditions. The em pirical results p ro d u ced in this study show that it is possible to specify a m odel of m oney dem and that closely tracks the actual behavior of the flow of M l’s m on etary services despite its considerable varia bility over this period. Thus, there seem s to be nothing m ysterious about that variability; it can be explained in term s of ch an ges in relevant eco n o m ic variables. T hese results indicate that m oney dem and has been considerably m ore stable over the past two d ecad es than stan d ard m oney d em an d analysis has suggested. ________and Kenneth Singleton. “The Microeconomic Theory of Monetary Aggregation,” New Approaches to Monetary Economics (Cambridge University Press, forthcoming). Barten, A. P. “Maximum Likelihood Estimation of a Complete De mand System,” European Economic Review, Vol. 1 (1969), pp. 773. Baumol, William J. “The Transactions Demand for Cash: An Inven tory Theoretic Approach,” Quarterly Journal of Economics (Novem ber 1952), pp. 545-56. Ewis, Nabil A. and Douglas Fisher. "The Translog Utility Function and the Demand for Money in the United States,” Journal of Money, Credit, and Banking (February 1984), pp. 34-52. Fayyad, Salam K. “Monetary Asset Component Grouping and Ag gregation: An Inquiry into the Definition of Money” (Ph.D. disserta tion, The University of Texas at Austin, 1986). Goldfeld, Stephen M. “The Demand for Money Revisited," Brook ings Papers on Economic Activity (3:1973), pp. 577-646. _________ “The Case of the Missing Money,” Brookings Papers on Economic Activity (3:1976), pp. 683-739. Judd, John P. and John L. Scadding. “The Search for a Stable Money Demand Function: A Survey of the Post-1973 Literature,” Journal of Economic Literature, Vol. 20 (1982), pp. 993-1023. Mizrach, Bruce and Anthony M. Santomero. “The Stability of Money Demand and Forecasting through Changes in Regimes,” The Review of Economics and Statistics (May 1986), pp. 324-28. Samuelson, Paul A. Foundations of Economic Analysis (Harvard University Press, 1983). ________ and Ryuzo Sato. "Unattainability of Integrability and Definiteness Conditions in the General Case of Demand for Money and Goods,” American Economic Review (September 1984), pp. 588-604. Theil, Henri. Economics and Information Theory (North-Holland, Amsterdam, 1967). _________Principles of Econometrics (Wiley, 1971). _________ Theory and Measurement of Consumer Demand, Vol. 1 (North-Holland, Amsterdam, 1975). REFERENCES Barnett, William A. “The User Cost of Money," Economics Letters, Vol. 1 (1978), pp. 145-49. _________ “Theoretical Foundations for the Rotterdam Model,” Review of Economic Studies, Vol. 46 (1979), pp. 109-30. _________ Consumer Demand and Labor Supply (North-Holland, Amsterdam, 1981). _________ Theory and Measurement of Consumer Demand, Vol. 2 (North-Holland, Amsterdam, 1976). _________The System-Wide Approach to Microeconomics (Uni versity of Chicago Press, 1980). Tobin, James. “ The Interest Elasticity of Transactions Dem and for Cash,” Review of Economics and Statistics (August 1956), pp. 241-47. (See app en dixes A and 1 on following pages) 5 31 FEDERAL RESERVE BANK OF ST. LOUIS AUGUST/SEPTEMBER 1986 Appendix A In o rd e r to e stim a te th e fu n ctio n a l form o f th e d em an d eq u atio n s, a s to c h a s tic v ersio n o f th a t form sh o u ld b e s p e c ified a n d th e d is tu rb a n ce te rm s in te rp re te d . T h e sy ste m o f d em a n d e q u a tio n s c a n b e w ritte n as follow s, 5 2 (1) X„ = m x , + t1 t,, P| + e„, t /I , " 't w h e re X„ ( = w*, Dx,,) is a T -d im e n sio n a l v e cto r o f o b serv a tio n s o n th e le ft-h a n d -sid e v ariables o f th e ith co m m o d ity group, PM = D ph) is a T -d im e n sio n a l v e cto r o f th e lo g -ch a n g e ( in th e p rice o f th e ith co m m o d ity grou p , |x, is th e m arginal b u d g et s h a re o f th e ith co m m o d ity gro u p , I t t J is a 5 x 5 5 matrix of the price coefficients, and X, ( = 2 w * Dxh) is a i= t T -d im e n sio n a l v e cto r o f th e (bu d get-sh are) w eig h ted su m o f th e lo g -ch an g e in e x p e n d itu re s on th e five co m m o d ity grou p s. T h e last term in e q u a tio n 1, e m, is th e d is tu rb a n ce term o f th e ith d em an d e q u atio n . T h e d is tu rb a n c e te rm s, [ e j , are a ssu m e d to c a p tu re th e ran d o m e ffe cts o f all v ariab les o th e r th a n in c o m e a n d all p ric e s . T h e d is tu rb a n c e te rm s are fu rth e r a ssu m e d to be n o rm ally d istrib u te d w ith m e a n ze ro a n d a v a ria n ce-co v a ria n ce m atrix 2 (x) I„ s u c h th at e m ) = a u (3) a n d E (eis, eit) = 0 f o r s = t, fo rs^ t, w h ere (x) is th e K ro n e ck e r p ro d u c t, I, is a T x T id en tity m atrix, a n d cr*, is th e i, jth e le m e n t o f th e 5 x 5 m atrix, S . A n o th er p ro p e rty o f th e d e m a n d -d is iu rb a n c e te rm s is th at th e ir su m v an ish es w ith u n it p ro b ab ility (see B arten (1969), p. 16 an d T h e il (1971), p. 333). A p o te n tia lly tro u b le so m e im p lica tio n o f th is p ro p erty is th at 2(7^ = E(eit(eM+ ... + e j ) = 0. i Thus, th e c o v a ria n ce m atrix, 2 , is sin g u lar, a n d as s u c h , c a n n o t have a rank th at is larg er th a n n — 1. In w h at follow s, it is a ssu m e d th at th e ran k o f 2 is e x a ctly n —1. In o rd e r to circu m v e n t th e c o m p lic a tio n s p o se d by th is sin g u larity p ro b lem , o n e e q u a tio n o f th e sy stem (1) is d ele te d . T h e leg itim acy o f th is p ro ced u r e c a n be v erified easily by s u m m in g over i an y fo u r o f th e five e q u a tio n s o f th e sy stem and u sin g th e p ro p e rtie s o f th at sy ste m in o rd e r to re co v e r the d ele te d e q u atio n . In fact, a m a jo r ad van tage o f th e e s tim a tio n m e th o d u se d in th is study, full in fo rm atio n m ax im u m lik elih o o d (FIML), is th at th e p a ra m e te r e stim a te s it p ro d u c e s are invariant to th e e q u a tio n d e le te d (see B arten (1969), pp. 2 5 -2 7 ). 32 For notational convenience, the system of demand equa tions (II may be written as follows, (4) y, = g(x„0) + e ,, i= i (2) E(£is, Formulation o f the Likelihood Function where y, are the vectors of the left-hand-side variables of (1), x, are the vectors of the right-hand-side variables, e, are the vectors of the demand-disturbance terms, and 0 is the vector of the parameters |x and tt^. Since the additive distur , bance vectors e , = ( e ,„ ..., e 41), t = l, ..., T, are assumed to be independently normally distributed with mean 0 and variance-covariance matrix 2, it follows that the vectors y, must also be independently normally distributed with mean g(x„ 0 ) and variance-covariance matrix 2 . In arriving at the vector-valued function g, it is assumed for notational convenience that prior restriction on the parameters has already been eliminated by substitution. Given the observed data on y = (y„ ...,yr)a n d x = (x„ ...,x.r), the log-likelihood function on 0 and 2 is given by (5) L (0, 2 ;y , x) = —(T(n —1)/2) log 2 it - 1 /2 l2 l T 1 2 f(y, ~ g (x„ 0 ) ) ' 2 (y, —g (x, 2))]. t= i This function is to be maximized with respect to the ele ments of the parameter vector 0 and the elements of the variance-covariance matrix, 2 . For computational conven ience, however, and since the asymptotic distribution of 2 is not at all needed, a stepwise-optimization procedure is used. This procedure involves first maximizing the loglikelihood function (5) with respect to the elements of 2, for a given value of 0 , to obtain an expression for 2 in terms of the elements of 0 . Thus, for 0 = 0 *, the value of 2 that maximizes (5) is given by 1 1 (6) 2 * (0 *;y , x) = — 2 (y ,-g (x„ 0 ) )(y ,-g (x„ 0 ) )'. t= l Substitution of 2* into (5) yields 1 (7) L (0;y, x) = - (T(n - l)/2) log 2 tt l2 (0 ;y , x) I - “ ( T ( n - l) ), which is the concentrated likelihood function. The second step in the optimization procedure becomes immediately clear when one recognizes that maximizing the log-likelihood function (5) is equivalent to minimizing the determinant of 2 in (7). The latter is accomplished by searching the feasible parameter space for the value of 0 at which I2 I is minimized. The values of the elements of © thus obtained,©, are the maximum likelihood estimates of FEDERAL RESERVE BANK OF ST. LOUIS the system (1). T h e a sy m p to tic co v arian ce m atrix o f© is o b tained by invertin g th e m atrix - [ ff- L ], w h ic h is n u m e ri- d&d&' AUGUST/SEPTEMBER 1986 m atrix p e rta in o n ly to th e e stim a te d p a ra m e ters. T h e a s y m p to tic co v a ria n ce m atrix o f th e e n tire v e cto r o f (e sti m ated as w ell as co m p u ted ) p a ra m e te r e stim a te s is derived in Fayyad (1986). cally ev alu ated at (n = (n N aturally, th e e le m e n ts o f that ) ). Appendix B Available resu lts from in fo rm a tio n th e o iv ca n lie u se d to develop m e a su res by w h ic h th e p e rfo rm a n c e o f e a c h o f the estim a te d e q u a tio n s as w ell as th at o f th e sy ste m as a w h o le can be gau ged .' C o n sid e r an in fin ite sim a l c h a n g e in th e bu d get sh a re o f th e ith co m m o d ity ' w = p.x/m l: xi , Pi , PA . dw: = - dp, + — dx, — — r dm , m 1 m nv from w h ic h it follow s that (1) dw, = w, d log f), + w, d log x, — w, d log m . T h e fin ite -ch a n g e an alo g o f e q u a tio n 1 is given by (2) Aw,, = w*, I)p,, + w*t Dx,, - w*, Dm,. S in ce Dm, = Dx, + Dp,, it follow s th at e q u a tio n 2 c a n be rew ritten as (3) Aw,, = w* Dx,, + w*, (Dp,, — Dp,) — w*, Dx,. O bseiv e th at the first te rm on the rig h t-h a n d -sid e o f e q u a tion 3, to b e in te rp re te d as th e q u a n tity c o m p o n e n t o f the ch a n g e in th e bu d get sh a re o f th e ith good, is th e d e p e n d e n t variable o f th e ith d em a n d e q u a tio n o f th e e stim a te s sy stem . T h u s, given the lo g -ch a n g es in real in c o m e a n d relative p rices, th e R o tterd am m o d el c a n be u sed to provide c o n d i tion al fo re c a sts o f w*. Dx,, and , th ro u g h e q u a tio n 3 o f A w,,. S in ce th e p re d ictio n o f wf, Dx,, is equ al to th e rig h t-h an d sid e o f th e ith d em an d e q u a tio n w ith th e d is tu rb a n c e term d eleted , it follow s th at w lil+, = w„ - e ,, , w h ere w lil+, is th e im p lied p re d ictio n o f w i t , a n d e ,, is th e re sid u al o f th e ith d em an d e q u a tio n in p e rio d t. In view o f the fact th at th e b u d g e t sh a re s are p o sitive an d ad d up to unity, th e y m ay be view ed as p ro b a b ilitie s. A m e a su re o f the m o d e l fit c a n be a c q u ire d b y d ete rm in in g the e x p e c te d gain in in fo rm a tio n from th e a c tu a l sh a re s, w h ic h ca n be view ed as p o s te rio r p ro b a b ilitie s, w h e n th e 'See Theil (1967), pp. 1-48; (1971), pp. 646-50, and Barnett (1981), pp. 149-54. im p lied p re d ic tio n s (or th e fitted values) o f th e s e s h a re s are view ed as p rio r p ro b a b ilitie s. T h a t m e a su re is given by (4) f, = ■ n 2 . i= t w Mlog Y V w ,, , " w h e re I, is the in fo rm a tio n in a c c u ra c y o f th e p re d ictio n s prov id ed by th e sy stem o f d e m a n d e q u a tio n s. It is to be n o te d th a t not o n ly is th is m e a su re o f in fo rm a tio n in a c c u racy ad ditive over g o o d s, as is in d ic a te d by th e e x p re ssio n in (4), but it is also ad ditive o v er tim e. T h u s, it is p o ssib le to c o n s tr u c t an average in d e x o f in fo rm a tio n in a c cu ra cy , 1, over th e p erio d from t, to t, by u sin g th e follow ing form u la (5) I = 2 I, . ______!_____ t. - ». + 1 t = t, O b seiv e th at the in fo rm a tio n -in a c cu ra c y m e a s u re s p re se n te d above p e rta in to th e p re d ic tio n s w h ic h are prov id ed by th e sy stem o f d e m a n d e q u a tio n s as a w h o le. It is p o ssib le to a c q u ire a sin g le -e q u a tio n m e a su re o f in fo rm a tio n in a c c u ra c y by u sin g th e fo rm u la (6) I,, = w it log ^ + (1 —w„) log J_ ^1!, _ w„ 1 —w„ w h e re 1 —w Mis th e c o m b in e d b u d g et s h a re o f all c o m m o d i ties o th e r th a n th e ith. As b efo re, th e average (over tim e) in d ex o f a sin g le -e q u a tio n in fo rm a tio n in a c c u ra c y c a n be o b ta in e d by u sin g fo rm u la 5. In o rd e r to provide co m p a ra b ility o f the in fo rm atio n in a c c u ra c y a c ro ss m o d e ls, a c o r re c tio n (ad ju stm e n t) facto r sh o u ld be a p p lied to th e s e m e a s u re s (see T h e il (1971), pp. t>.r l - 5 2 , a n d B a rn e tt (1981), p. 150). A d ju stm e n t w as i ach iev e d in th is stu d y by m u ltip lyin g th e in fo rm a tio n -in a c cu ra c y m e a su re in e a c h c a s e by a fa c to r o f ML/1ML-K), w h e re M is th e n u m b e r o f jo in tly e stim a te d e q u a tio n s, L is th e n u m b e r o f tim e p e rio d s (qu arters), a n d K is th e n u m b e r o f u n re s tr ic te d p a ra m e te rs. Clearly, th is p ro c e d u re is clo s e ly akin to the d eg re e s-o f-fre e d o m a d ju s tm e n t o f the c o rre la tio n c o e fficie n t. 33