An algorithm for estimating the volatility of the velocity of money

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1 PRA uch Persoal RePEc Archve A algorthm for estmatg the volatlty of the velocty of moey urat Alkhaov ad Leo Taylor KIEP Uversty August 93 Ole at PRA Paper No. 4933, posted 8. August 03 6:06 UTC

2 A algorthm for estmatg the volatlty of the velocty of moey urat Alkhaov ad Leo Taylor Abstract ost macroecoomc models, such as the IS-L, assume equlbrum moey markets. Sce moey demad s a verse fucto of velocty, a accurate estmate of velocty wll lead to errors calculatg the moetary ad geeral equlbra. Ths ote suggests a way to gauge the potetal error estmatg velocty. The algorthm arses from the quatty equato of exchage, whch oe may prefer to a ad hoc model of velocty. Keywords: moetary polcy, smulatos, forecastg trastoal ecoomes, mathematcal statstcs ecoomcs JEL Classfcatos: E47; E5 I. Itroducto Fluctuatos the turover rate of a ut of moey velocty -- complcate the cetral bak s forecast of the ecoomc mpact of a chage moetary polcy. oetarsts coted that whe velocty s stable, a chage moey supply leads to a predctable chage omal come. But whe data are scarce ad ecoomc sttutos are chagg rapdly for example, early the post-sovet trasto to markets come velocty ca be hard to estmate (Ctr, 995). Forecasts of the effects of moetary chages, a gve scearo, may beeft from a way to gauge the magtude of possble errors estmatg velocty. Ths ote suggests such a algorthm. A typcal moetary forecast begs wth the detty that total spedg equals total recepts by the factors of producto. 3 Ths s the quatty equato of exchage, V = PQ. The left-had sde multples the moey supply by velocty V; the rght-had sde Departmet of Ecoomcs, KIEP Uversty; muratsta@gmal.com Departmet of Ecoomcs, KIEP Uversty; ltaylor@kmep.kz. Taylor s the correspodg author. We thak partcpats a semar at the Departmet of Ecoomcs at KIEP the fall of 0 -- especally Gerald Pech, for correctg a error a early draft. 3 The factors clude etrepreeurs.

3 multples the prce P of a typcal budle of goods by the umber of budles Q. Whe V ad Q are costats, a chage duces a proportoal chage P, whch smplfes the bak s forecastg. Ideed, whe the ecoomy produces at full capacty, the Q may be costat; but geeral, a costat V s harder to ustfy. arshall (93) suggested that velocty may be slow to chage because habt determes the share of come that people sped. I truth, velocty s ofte volatle the short ru. For the velocty of the currecy Kazakhsta, the tege, the rato of the stadard devato to the mea vared from to (Table ). 4 The rato was more tha twce as hgh the perod (.97) -- whch cotaed a ecoomc slowdow ad a 5% devaluato of the tege -- as the perod (.076). Table velocty statstcs for the tege Year Stadard Devato Velocty mea Rato Notes: Colum gves the stadard devato of velocty; Colum 3, the mea of velocty, calculated as the aual average of quarterly estmates; ad Colum 4, the rato of the stadard devato to the mea. Appedx B lsts data used the computatos. Source of raw data: The Natoal Bak of Kazakhsta odelg come velocty s ofte dffcult. For example, ts lk to lagged moey volatlty s ot always clear. Fredma (984) argued, effect, that velocty would fall whe ecoomc ucertaty creased, sce people would hold moey as a precauto. 5 4 Research to the tege supply ofte uses or 3. But s more typcal tha these measures are of research to the varace of velocty. 5 For broad perspectves, see Fredma (970, pp. 7-9) -- ad ascaro & eltzer (983), whch develops a geeral equlbrum model whch varablty or rsk affects the choce of portfolos (p. 488).

4 3 oey volatlty would relate drectly to come velocty. 6 Hall & Noble (987) tested for Grager causalty Uted States data ad cocluded that the log of velocty was caused partly by ts ow lags ad by lags of the volatlty of moey growth. Other studes dcated that these results mght vary wth the perod studed, sce the moetary evromet evolves over tme due to chages such factors as regulato ad flato (Brocato & Smth, 989; ehra, 987 ad 989). The results ehra (989) were also sestve to specfcato of the equato e.g., levels or frst dffereces. I addto, Grager-causalty estmates ofte deped o the legths of the lags specfed, cocluded Thorto & Batte (985). Thorto (995) tured up evdece supportg Fredma s hypothess for three of e dustral coutres studed, but oly certa tme perods. Thorto cocluded that the Fredma hypothess would appear to have lttle geeral applcablty (p.90). II. Aalyss For a way to estmate velocty that s ot ad hoc, beg wth the quatty equato of exchage: 7 PQ V =. (II.) At tmes, we may have ucerta estmates for the three rght-had varables Equato II.. For example, we may lack relable mothly data for these varables ( partcular, for Q) whe estmatg mothly velocty. Or the aalyst may base her predcto of velocty o assumed values of the depedet varables assumptos that may ot come true. I ether evet, Equato II. may estmate V mprecsely. It would be useful to have a estmate of V s volatlty. Cosder P, Q ad as radom varables. The a Taylor seres ad a well-kow property of varace (Larse & arx, 006, p.38; Appedx A below) gve a frst-order approxmato of the varace of V: 6 From ths perspectve, the creased volatlty of the tege the perod of 009- may have reflected ucertaty about the optmal amout of moey to hold followg the global facal crss of The velocty of moey creases omal come (PQ), sce people wll sped a gve moey supply faster whe they ca afford more purchases; ad velocty falls wth a crease moey, sce there are more tege ow to face aggregate purchases of a gve sze.

5 4 Q P P Q var( V ) var( P) + var( Q) + 4 PQ P Q cov( P, ) cov(, Q). 3 3 QP var( ). + cov( P, Q) (II.) I some short-ru cases, P, Q ad may be depedet of oe aother -- each subect to radom factors, such as measuremet error, whch eed ot affect the other two varables. The covaraces the are zero, ad the last three terms Equato II. wll dsappear. Equato II. apples what we wll call the Larse-arx algorthm. (The two mathematcas had developed t to help terpret detal X-rays (Larse & arx, 006).) Gve the log-ru varaces of P, Q ad, the equato ca forecast the varace of velocty a scearo specfyg the former three varables. For example, suppose that the Natoal Bak of Kazakhsta cosders a crease the moey supply equal to the forecasted aual rate of growth Q, 7.5%. The Bak assumes that P would ot chage. I addto to the levels of P, Q, ad, oe mght assume for these varables ther average aual varaces for the perod (Table ). By Equato II., the predcted value of velocty s 7.7. By Equato II., the predcted stadard devato of velocty s.39, or.8 of the mea. Ths rato s 70% hgher tha the average for 000-0, so the Bak may wsh to act o ts scearo forecast wth cauto. If velocty follows a ormal dstrbuto, the the 95% cofdece terval that s mpled for t s about (4.9, 0.5). Table Forecastg example Varables Level Varace Stadard devato Prce level 8.7, Output 8, ,9, ,90.9 moey 3,89,483.9,30,845,69,97.7,40,985.0 Velocty The covaraces amog, P ad Q play a crtcal role. Keyesa moetary polcy assumes that the short-ru correlato betwee prces ad moey s low eough to permt a fuso of moey to affect real GDP rather tha the prce level. But Kazakhsta, usg aual data for 000 through 0, the smple correlatos of the Cosumer Prce Idex, the moey supply (0 or ), ad output Kazakhsta all exceed.97. For mothly data, the correlato betwee the CPI ad also exceeds.97. The three varables may each relate to a tme tred, or they may be cotegrated; but the pot s that ther covaraces caot be gored.

6 5 III. Coclusos The Larse-arx algorthm may be most useful whe appled to short-ru moetary relatoshps, sce these may be harder to estmate tha log-ru oes. Ylmaz, Oskebayev & Kaat (00) fd that a model of demad Kazakhsta, based o a output proxy, the terest rate, ad o foreg exchage rates, has the expected coeffcet sgs the log ru but ot the short. The algorthm may dcate how severe the msspecfcato short-ru estmates may be. IV. Refereces Brocato, J. ad Smth, K. L., 989. Velocty ad the Varablty of oey Growth: Evdece from Grager-Causalty Tests: Commet. Joural of oey, Credt, ad Bakg, (), pp Ctr, D., 995. Polcy Expereces ad Issues the Baltcs, Russa, ad Other Coutres of the Former Sovet Uo. Iteratoal oetary Fud. Occasoal Paper #33. Avalable at: [Accessed ay 03] Fredma,., 970. A Theoretcal Framework for oetary Aalyss. Joural of Poltcal Ecoomy, 78(), pp Fredma,., 984. Lessos From the oetary Polcy Expermet. The Amerca Ecoomc Revew, 74(), pp Hall, T. E. ad Noble, N. R., 987. Velocty ad the Varablty of oey Growth: Evdece from Grager-Causalty Tests: Note. Joural of oey, Credt, ad Bakg, 9(), pp.-6. Larse, R. J. ad arx,. L., 006. A Itroducto to athematcal Statstcs ad Its Applcatos. NJ: Pearso Pretce Hall. arshall, A., 003 [93]. oey, credt ad commerce. NY: Prometheus Books. ascaro, A. ad eltzer, A. H., 983. Log- ad Short-Term Iterest Rates a Rsky World. Joural of oetary Ecoomcs,, pp ehra, Y. P., 987. oey Growth Volatlty ad Hgh Nomal Iterest Rates. Ecoomc Revew, November/December, pp.0-9. ehra, Y. P., 989. Velocty ad the Varablty of oey Growth: Evdece From Grager-Causalty Tests: Commet. Joural of oey, Credt, ad Bakg, (), pp.6-6. Natoal Bak of Kazakhsta, 03. Varous macroecoomc statstcs. Avalable at [Accessed ay 03] Thorto, D. L. ad Batte, D. S., 985. Lag-Legth Selecto ad Tests of Grager- Causalty Betwee oey ad Icome. Joural of oey, Credt, ad Bakg, 7(), pp

7 6 Thorto, J., 995. Fredma s oey Supply Volatlty Hypothess: Some Iteratoal Evdece. Joural of oey, Credt, ad Bakg, 7(), pp Ylmaz,., Oskebayev, Y. ad Abdulla, K., 00. Demad for oey Kazakhsta: Romaa Joural of Ecoomc Forecastg, 3(), pp.8-9. V. Appedx A V. The Case of Idepedet Radom Varables I a Taylor seres, a frst-order expaso approxmates a fucto g aroud some pot (μ, μ,,μ ): g g( W μ, W,..., W ) g( μ, μ,..., μ ) + ( W ). = W where the dervatves are evaluated at the pot (μ, μ,,μ ). For velocty, such a Taylor seres would be V V V V (, P, Q) V ( μ m, μ p, μ q ) + ( μ m ) + ( P μ p ) + ( Q μ q ), P Q (V..) where μ m, μ p ad μ q are arbtrary costats. Note that V μ, μ, μ ) s also a costat. ( m p q A well-kow result cocerg the varace of a lear sum of depedet radom varables W wth fte meas s that Var( = a W ) = = a Var( W ) (V..) where a s a costat. Applyg Equato V.. to Equato V.. gves us

8 7 V V V Var( V ) Var ( μm) + ( P μp) + ( Q μq) P Q V V V = Var( μm) + Var( P μp) + Var( Q μ q ) P Q V V V = Var( ) + Var( P) + Var( Q), P Q (V..3) where the last le uses Equato V.. aga: Var ( X μ ) = Var( X ) + Var( μ) = Var( X ), sce μ s a costat. Note that the - coeffcet of the varace of μ s squared. V. The Geeral Case Whe covaraces are ot zero, the the geeral verso of Equato V.. s = Var ( a W ) = a Var( W ) + a a Cov( W, W ), = < k k k (V..) where we have used the result Cov( a W, a W = a a ) = E( a W a W ) E( a W ) E( a W [ E( W W ) E( W ) E( W )] = a a Cov( W, W ). ) = a a E( W W ) a E( W ) a E( W ) Equato II. specfes Equato V...

9 8 VI. Appedx B Table 3 Dataset for smulato Tme P Q , ,844. 6, ,477.7,4.3 0, ,04. 35,65.4 8, ,045. 9,6.3 35, , , , , , , , ,37.0,05, , ,89.9,50, , ,43.5,7, , ,508.7,340, ,88.4,05,448.,854, ,03.9,96,04.8 3,553,008.3 P s the Cosumer Prce Idex. It s averaged from mothly data. Q s real output. It equals omal gross domestc product dvded by the CPI. Each aual estmate of Q sums the four quarterly estmates. The moey supples 0 ad are measured mllos of tege. They are averaged from mothly data. Source of raw data: The Natoal Bak of Kazakhsta

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