Chapter 1. Economic Analysis and Econometrics. validating theories of behavior of individuals and institutions, and a policy science,

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1 Chaer. Economic Analysis and Economerics. Inroducion The disciline of Economics is boh a ure science, concerned wih develoing and validaing heories of behavior of individuals and insiuions, and a olicy science, concerned wih he forecasing and engineering of behavior. In boh arenas, conces from robabiliy and saisics, and mehods for analyzing and undersanding economic daa, lay an imoran role. In his chaer, we give hree inroducory examles ha illusrae he inerwining of economic behavior, saisical inference, and economeric forecasing. These examles conain some robabiliy and saisics conces ha are exlained in laer chaers. Wha is imoran on firs reading are he general connecions beween economic reasoning, robabiliy, saisics, and economic daa; deails can be osoned.. Pei Verdo s Raional Decision Pei Verdo is a yical rofessional economis, wiy, wise, and unboundedly raional. Pei wors a a universiy in California. Pei s life is filled wih fun and exciemen, he high oin of course being he class she eaches in economerics. To sulemen her modes universiy salary, Pei oeraes a small vineyard in he Naa Valley, and sells her frui o nearby wineries. Pei faces a dilemma. She has o mae a decision on wheher o harves early or lae in he season. If he Fall is dry, hen lae-harvesed frui is imroved by addiional "hang ime", and will fech a remium rice. On he oher hand, if rains come early, hen much of he lae-harvesed frui will be soiled. If Pei harvess early, she avoids he ris, bu also loses he ooruniy for he maximum rofi.

2 McFadden Chaer. Economic Analysis and Economerics Table gives Pei s rofi for each ossible acion she can ae, and each ossible Even of Naure: q==========================================================================================================================================================================================================e Table. Profi Acions Harves Harves Early Lae [ ] Even of Early Rains $30,000 $0,000 Naure Lae Rains $30,000 $40,000 z ==========================================================================================================================================================================================================c $ There is a secialized weaher service, Blue Sy Forecasing, ha sells longrun reciiaion forecass for he Naa Valley. Pei has o choose wheher o subscribe o his service, a a cos of $000 for he year. Table gives he hisorical record over he as 00 years on he join frequency of various forecass and oucomes. q=========================================================================================================================================================================================================================================================================================e Table. Frequency Blue Sy Forecas No Forecas Early Lae Rains Rains TOTAL [ ] Saes of Early Rains Naure Lae Rains ; ========================================================================================================================================================================================================================================================================================= TOTAL z =========================================================================================================================================================================================================================================================================================c $ % The inerreaion of he number 0.3 is ha in 30 ercen of he as 00 years, Blue Sy has forecas early rains and hey do in fac occur. The column oals give he frequencies of he differen Blue Sy forecass. The row oals give he frequencies of he differen Saes of Naure. Thus, Blue Sy forecass early rains half he ime, and he frequency of acual early rains is 0.4. One can also form condiional robabiliies from Table. For examle, he condiional robabiliy of lae rains, condiioned on he even ha lae rains are forecas, equals 0.4/(0.+0.4) = 0.8.

3 McFadden Chaer. Economic Analysis and Economerics 3 Pei wans o maximize execed rofi. In oher words, she wans o mae he robabiliy-weighed average of he ossible rofi oucomes as large as ossible. Pei is no adverse o ris; she figures ha riss will average ou over he years. If Pei does no subscribe o he Blue Sy forecas service, hen all she nows when she has o mae her harvesing decision is he marginal frequency of early rains, 0.4, given in he las column in Table. To calculae execed rofi in his case from a secified acion, Pei mulilies he rofi she will receive in each sae of Naure by he robabiliy of his sae, and sums. If she harvess early, he execed rofi is ($30,000) (0.4) + ($30,000) (0.6) = $30,000. If she harvess lae, he execed rofi is ($0,000) (0.4) + ($40,000) (0.6) = $8,000. Then, if Pei does no subscribe o Blue Sy, she will always choose o harves early. Now suose Pei does subscribe o Blue Sy, and has heir forecas available. In his case, she can do her execed rofi calculaion condiioned on he forecas. In any case, he execed rofi from harvesing early is $30,000, as before. To analyze her oions when she ges he forecas, Pei firs calculaes he condiional robabiliies of early rain, given he forecas: 0.3 Prob(Early Rains Forecas Early Rains) = = Prob(Early Rains Forecas Lae Rains) = = If he forecas is for early rains, he execed rofi from harvesing lae is ($0,000) (0.6) + ($40,000) (0.4) = $,000.

4 McFadden Chaer. Economic Analysis and Economerics 4 This is less han $30,000, so Pei will definiely harves early if early rain is forecas. If he forecas is for lae rain, hen he execed rofi from harvesing lae is ($0,000) (0.) + ($40,000) (0.8) = $34,000. This is greaer han $30,000, so Pei will harves lae if he forecas is for lae rain. Is subscribing o Blue Sy worh while? If Pei does no, hen she will always harves early and her execed rofi is $30,000. If Pei does subscribe, hen her execed rofi in he even of an early rain forecas is $30,000 and in he even of a lae rain forecas is $34,000. Since he frequency of an early rain forecas is 0.5, Pei s overall execed rofi if she subscribes is ($30,000) (0.5) + ($34,000) (0.5) = $3,000. This is $000 more han he execed rofi if Pei does no subscribe, so ha he value of he informaion rovided by he subscriion is $000. This is more han he $000 cos of he informaion, so Pei will choose o subscribe and earn an overall execed rofi, ne of he subscriion cos, of $3,000. Pei Verdo s decision roblem is a yical one for an economic agen facing uncerainy. She has a crierion (execed rofi) o be oimized, a "model" of he robabiliies of various oucomes, he ossibiliy of collecing daa (he forecas) o refine her robabiliy model, and acions ha will be based on he daa colleced. An economerician facing he roblem of saisical inference is in a similar siuaion: There is a "model" or "hyohesis" for an economic henomenon, daa ha rovides informaion ha can be used o refine he model, and a crierion o be used in deermining an acion in resonse o his informaion. The acions of he economerician, o declare a hyohesis "rue" or "false", or o mae a forecas,

5 McFadden Chaer. Economic Analysis and Economerics 5 are similar in siri o Pei s choice. Furher, he soluion o he econmerician s inference roblem will be similar o Pei s soluion. A exboo definiion of economerics is he alicaion of he rinciles of saisical inference o economic daa and hyoheses. However, Pei s roblem suggess a deeer connecion beween economeric analysis and economic behavior. The decision roblems faced by raional economic agens require saisical inference, and hus are "economeric" in naure. The soluions o hese roblems require he same logic and echniques ha mus be brough o bear more formally in scienific inference. Thus, all raional economic agens are informal woring economericians, and he sudy of formal economerics can iself rovide models of economic behavior. Turned around, economerics is simly a codificaion of he "fol" echniques used by economic agens o solve heir decision roblems. Thus, he sudy of economerics rovides no only he body of ools needed in emirical and alied economics for daa analysis, forecasing, and inference, bu also he conces needed o exlain economic behavior. 3. Soc Mare Efficiency The hyohesis is ofen advanced ha he soc mare is efficien. Among he ossible meanings of his erm is he idea ha arbiragers are sufficienly acive and ervasive so ha oenial windows of ooruniy for excess rofi are immediaely closed. Consider a broad-based soc mare index, such as he S&P 500 Mare Index. The rofi o be made by aing a dollar ou of a "ris-free" channel (usually defined o be 30-day U.S. Treasury Bills), buying a dollar s worh of a Mare Index Fund, and selling i one day laer is R = [M + d M - (+i )M ]/M, m

6 McFadden Chaer. Economic Analysis and Economerics 6 where M equals he mare index, and is by definiion he rice of one share of he mare index fund, d is he dividend rae, and i is he ineres rae, and boh d and i are assumed o be nown in eriod. If he arbirager faces a higher ineres rae han he ris-free rae, hen i should be reinerreed as he relevan rae. The argumen is ha he execaion of R m, as of eriod, mus be zero, since if i were osiive, he arbirager would buy he fund unil he rice M is bid u o eliminae he rofi ooruniy. Similarly, if he execaion of R m in eriod were negaive, hen he arbirager would sell he fund and bid down he rice M unil he rofi ooruniy is again eliminaed. The condiion of a zero execaion of R m given he informaion available in eriod imlies ha M-M M +d -i =ε, where ε is a disurbance wih mean zero given eriod informaion. The robabiliy ha he mare index decreases beween and hen equals he robabiliy ha ε < d - i. Assume ha d - i is a consan λ, and ha ε has a disribuion ha is saionary over ime. (These assumions are quesionable, and he conclusions below may be alered if hey are false.) Then he robabiliy ha he mare index decreases on any day is Q Prob(ε < λ), and he robabiliy ha i increases on any day is P Prob(ε > λ). Noe ha Q + P, wih equaliy holding if he robabiliy of no change in he index is zero. Define a osiive run of lengh n o be a sequence of n successive days on which he index increases, wih a day on each end on which he index does no increase. Similarly, define a negaive run of lengh n o be a sequence of n successive days on which he index decreases, braceed by days where rice does no decrease. Noe

7 McFadden Chaer. Economic Analysis and Economerics 7 ha runs in his definiion can be erminaed by no-change days as well as by reversals in he direcion of change. The able below gives he number of osiive and negaive runs in he S&P 500 Mare Index beween July 3, 96 and December 3, 990. Only days on which he mares are oen are couned; a oal of 765 index changes are observed over his eriod, of which 3753 are increases, 3366 are decreases, and 46 are no change. Then, esimaes from he samle frequencies are P ^ = 3753/765 = and Q ^ = 3366/765 = q====================================================================================================================================================================================================================e RUN NUMBER OF NUMBER OF LENGTH OBSERVED OBSERVED POSITIVE RUNSNEGATIVE RUNS [ ] [ ] TOTAL z ====================================================================================================================================================================================================================c $ $ From robabiliy heory, if he robabiliy P of a rice increase is consan, indeenden of he as hisory of he rocess, hen he robabiliy of a osiive run of lengh n is P n- (-P). Similarly, he robabiliy of a negaive run of lengh n is Q n- (-Q). Using he esimaes P ^ for P and Q ^ for Q, we can hen calculae he execed numbers of runs of each lengh. For examle, he execed number of osiive runs of lengh 3 is given by he oal number of osiive runs (585) imes ^ he robabiliy P (-P) of a run of lengh 3, evaluaed a he esimae P =

8 McFadden Chaer. Economic Analysis and Economerics 8 Then, he observed and execed numbers of runs can be comared. This is done in he able below. Because he numbers of longer runs are small, we combine all he runs of eigh or more days in lengh. q==================================================================================================================================================================================================================================================================================================================e RUN NUMBER OFNUMBER OFNUMBER OFNUMBER OF LENGTHOBSERVED EXPECTED OBSERVED EXPECTED POSITIVE POSITIVE NEGATIVE NEGATIVE [ ] z ==================================================================================================================================================================================================================================================================================================================c $ $ $ $ The able shows ha here are many fewer runs of lengh one han are execed, and many more runs of hree days or more han are execed. This suggess ha eiher he efficien mare hyohesis is wrong, or ha faciliaing assumions we made along he way on he behavior of d - i and ε are wrong. Of course, one canno readily rule ou by visual insecion he ossibiliy ha he hyoheses are rue, and he differences in he able are he resul of he saisical noise in he daa. However, we can use a "goodness-of-fi" es o mae a saemen abou he robabiliy ha he aern of deviaions beween observed and execed oucomes in he able above could arise by chance. In his case, he goodness-of-fi es esablishes ha i is exremely unliely ha he aerns in he able arose by chance. This examle shows how an economic hyohesis can be formulaed as a condiion on a robabiliy model of he daa generaion rocess, and how saisical ools can be used o judge wheher he economic hyohesis is rue. In his case, he evidence

9 McFadden Chaer. Economic Analysis and Economerics 9 is ha eiher he efficien mares hyohesis does no hold, or ha here is a roblem wih one of he assumions ha we made along he way o faciliae he analysis. A more careful sudy of he ime-series of soc mare rices ends o suor one asec of he efficien mares hyohesis, ha execed rofi a ime from an arbirage o be comleed he following eriod is zero. Thus, he elemenary economic idea ha arbiragers disciline he mare is suored. However, here do aear o be longer-run ime deendencies in he mare, as well as heerogeneiy, ha are inconsisen wih some sronger versions of he efficien mares hyohesis. 4. The Caial Asse Pricing Model The reurn ha an invesor can earn from a soc is a random variable, deending on evens ha iminge on he firm and on he economy. By selecing he socs ha hey hold, invesors can rade off beween average reurn and ris. A basic, and influenial, heory of raional orfolio selecion is he Caial Asse Pricing (CAP) model. This heory concludes ha if invesors are concerned only wih he mean and variance of he reurn from heir orfolio, hen here is a single orfolio of socs ha is oimal, and ha every raional invesor will hold his orfolio in some mix wih a risless asse o achieve he desired balance of mean and variance. Since every invesor, no maer wha her aiudes o ris, holds he same soc orfolio, his orfolio will simly be a share of he oal mare; ha is, all invesors simly hold a mare index fund. This is a owerful conclusion, and one ha aears o be easily refuable by examining he orfolios of individuals. This suggess ha oher facors, such as irraionaliy, ransacions coss, or heerogeneiy in informaion, are influencing behavior. Neverheless, he

10 McFadden Chaer. Economic Analysis and Economerics 0 CAP model is ofen useful as a normaive guide o oimal invesmen behavior, and as a ool for undersanding he benefis of diversificaion. To exlain he CAP model, consider a mare wih socs, indexed =,,...,. Le P be he rice of soc a he end of monh ; his is a random variable when considered before he end of monh, and afer ha i is a number ha is a realizaion of his random variable. Suose an invesor can wihdraw or deosi funds in an accoun ha holds U.S. Treasury 30-Day Bills ha ay an ineres rae I during monh. (The invesor is assumed o be able o borrow money a his rae if necessary.) Convenionally, he T-Bill ineres rae is assumed o be ris-free and nown o he invesor in advance. The rofi, or excess reurn, ha he invesor can mae from wihdrawing a dollar from her T-Bill accoun and buying a dollar s worh of soc is given by P + D - P R =, P -I,, where D is he dividend aid by he soc a he end of monh. The excess reurn R is again a random variable. Le r denoe he mean of R. Le σ denoe is variance, and le σ denoe he covariance of R and R. Noe ha σ and σ are j j wo differen noaions for he same variance. The square roo of he variance, σ, is called he sandard deviaion. Now consider a orfolio in which A dollars are invesed in socs, wih a fracion θ of each dollar allocaed o shares of soc, for =,...,. The excess reurn o his orfolio is hen AR, where R = s θ R =

11 McFadden Chaer. Economic Analysis and Economerics is he excess reurn o he one dollar soc orfolio characerized by he shares (θ ). The excess reurn R is again a random variable, wih mean r = s s s θ r and variance σ = θθσ. j j This imlies ha he orfolio wih A = = j= dollars invesed has an excess reurn wih mean Ar and variance A σ (or sandard deviaion Aσ ). The covariance of R and R is given by σ cov(r,r ) = s θσ. j j Define he bea of soc (wih resec o he orfolio ) by he j= formula β = σ /σ, and noe ha ( ) s s s s σ = θ θσ θ σ θ β σ, j j = j= = = 9 0 s and hence ha θβ =. = Now consider he raional invesor s roblem of choosing he level of invesmen A and he orfolio mix θ,θ. Assume ha he invesor cares only abou he mean and sandard deviaion of he excess reurn from her orfolio, and always refers a higher mean and a lower sandard deviaion. The invesor s ases for ris will deermine how mean and sandard deviaion are raded off. The figure below shows he alernaives available o he invesor. For a secified mix (θ ), he locus of orfolios obainable by varying A is a sraigh line hrough he origin. Anoher mix (θ ) will give anoher sraigh line hrough he origin. The locaion of hese lines reflecs he oeraion of diversificaion o reduce ris; i.e., by holding a mix of socs, some of which are liely o go u when ohers are

12 McFadden Chaer. Economic Analysis and Economerics going down, one may be able o reduce sandard deviaion for a given level of he * * mean. There will be an oimal mix (θ ) ha gives a line ha is roaed as far o he norhwes as ossible in he figure. Differen invesors will locae a differen oins along his oimal line, by icing differen A levels, o maximize heir various references for mean versus sandard deviaion. However, all invesors * * will choose exacly he same mix (θ ) for he socs ha hey hold. Bu if every invesor holds socs in he same roorions, hen hese mus also be he * * roorions ha revail in he mare as a whole. Then (θ ) will be he shares by value of all he socs in he mare. Such a orfolio is called a mare index fund. The CAP model hen concludes ha raional invesors who care only abou he mean and sandard deviaion of excess reurn will hold only he mare index fund, wih he levels of invesmen reflecing heir heerogeneous ases for ris. q mean θ θ r * θ z====================================================================================================================================================================== σ sandard deviaion * * The roblem of deermining he oimal orfolio mix (θ ) is mos easily solved by considering a closely relaed roblem. An invesor s orfolio is characerized by A and (θ ). Given a choice among all he orfolios ha achieve a secified level of mean reurn, he invesor would wan o choose he one ha minimizes variance. In he figure, his corresonds o geing as far o he

13 McFadden Chaer. Economic Analysis and Economerics 3 lef as ossible when consrained o he feasible orfolios on a secified horizonal line. From he revious discussion, he soluion o his roblem will be a orfolio wih he oimal mix of socs, and he only difference beween his roblem and he one of maximizing references will be in deermining he overall invesmen level A. The roblem of minimizing variance for a given mean is one of consrained minimizaion: s s Choose A,θ 0 o minimize A θθσ, j j = j= s subjec o A θ r = c (a consan) and θ =. = = The firs-order (uhn-tucer) condiions for his roblem are s * * () A θσ λar + µ, wih equaliy unless θ = 0, for =,..., j j j= s s * * s * () A θθσ = λ θ r, j j = j= = * where he scalars λ and µ are Lagrange muliliers. Mulily () by θ and sum over o obain he resul ( ) s s * * s * s * AA θθσ - λ θ r = µ θ. j j = j= = = 9 0 * Using (), his imlies µ = 0. Then, () imlies ha he oimal θ saisfy s

14 McFadden Chaer. Economic Analysis and Economerics 4 s * * (3) θσ γr, wih equaliy unless θ = 0, for =,...,, j j j= * where γ is a scalar defined so ha he θ sum o one. Since by he earlier commens his mix is simly he mix in he oal mare, equaliy will hold for all socs ha are in he mare and have osiive value. Assume now ha r = s * s s * * θ r and σ = θθσ j j refer o he oimal = = j= orfolio. The lef-hand-side of (3) equals σ β σ, so his condiion can be rewrien * (4) βσ γr, wih equaliy unless θ = 0, for =,...,. Mulilying boh sides of his inequaliy by θ and summing yields he condiion s * s γr γ θr = σ θ β = σ, = = or r = σ /γ. Subsiuing his ino (4) gives us he final form of a main resul of he CAP model, q========================================================================================================================================================================================================================================================================================e r βr, wih equaliy if he soc is held, for =,..., z========================================================================================================================================================================================================================================================================================c * The mean reurns are no observed direcly, bu he realizaions of monhly reurns on individual socs and he mare are observed. Wrie an observed reurn as he sum of is mean and a deviaion from he mean, R = r + ε and R = r +

15 McFadden Chaer. Economic Analysis and Economerics 5 ε. Noe ha R = s * s * θ R, so hen ε = θε. For all socs held in he = = mare, he CAP model imlies r = β r. Use he form R = r + ε o rewrie he equaion r = β r as R - ε = β (R - ε ). Define ν = ε - β ε. Then, he equaion becomes (5) R = β R + ν. This equaion can be inerreed as a relaion beween mare ris, embodied in he mare excess reurn R, and he ris of soc, embodied in R. The disurbance ν in his equaion is someimes called he secific ris in soc, he roorion of he oal ris in his soc ha is no resonsive o mare flucuaions. This disurbance has he following roeries: (a) Eν =0; (b) Eν = E(ε - β ε ) = σ + β σ -βσ σ - β σ (c) Eν R = E(ε - βε )ε = σ - βσ =0. Equaion (5) is a linear regression formula. Proeries (a)-(c) are called he Gauss-Marov condiions. We will see ha hey imly ha an esimae of β wih desirable saisical roeries can be obained by using he mehod of leas squares. Then, he CAP model s assumions on behavior imly an economeric model ha can hen be fied o rovide esimaes of he mare beas, ey arameers in he CAP analysis. The mare bea s of individual socs are ofen used by orfolio managers o assess he meris of adding or deleing socs o heir orfolios. (Of course, he

16 McFadden Chaer. Economic Analysis and Economerics 6 CAP model says ha here is no need o consider holding orfolios differen han he mare index fund, and herefore no need for orfolio managers. Tha hese hings exis is iself evidence ha here is some deficiency in he CAP model, erhas due o failures of raionaliy, he resence of ransacions cos, or he abiliy o mimic he excess reurn of he mare using various subses of all he socs on he mare because he oimal orfolio is no unique.) Furher, saisical analysis of he validiy of he assumions (a)-(c) can be used o es he validiy of he CAP model. The β s in formula (5) convey informaion on he relaionshi beween he excess reurn on an individual soc and he excess reurn in he mare. Subrac means in (5), square boh sides, and ae he execaion o ge he formula σ = β σ + δ, where δ = Eν is he variance of he disurbance. This equaion says ha he ris of soc equals he mare ris, amlified by β, lus he secific ris. Soc will have high ris if i has large secific ris, or a β ha is large in magniude, or boh. A osiive β imlies ha evens ha influence he mare end o influence soc in he same way; i.e., he soc is ro-cyclic. A negaive β imlies ha he soc ends o move in a direcion oosie ha of he mare, so ha i is couner-cyclic. Socs ha have small or negaive β are defensive, aiding diversificaion and maing an imoran conribuion o reducing ris in he mare orfolio. The CAP model described in his examle is a widely used ool in economics and finance. I illusraes a siuaion in which economic axioms on behavior lead o a widely used saisical model for he daa generaion rocess, he linear regression

17 McFadden Chaer. Economic Analysis and Economerics 7 model wih errors saisfying Gauss-Marov assumions. An imoran feaure of his examle is ha hese saisical assumions are imlied by he economic axioms, no aached ad hoc o faciliae daa analysis. I is a hay, bu rare, circumsance in which an economic heory and is economeric analysis are fully inegraed. To see such harmonies, heoriss need o draw ou and mae recise he emirical imlicaions of heir wor, and economericians need o develo models and mehods ha minimize he need for exra faciliaing assumions.