chapter 2 economie monetara

8/4/2019 Chapter 2 Economie Monetara

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26 Introductory Econometrics for FinanceAppendix: Econometric software package suppliers

Pa ckq gg,.,,',, C,ontafi inform ati nEVicws QMS Sofnvare, Sr"rire 336,4521 Campus Drive #336. hvine. CA 92612-2621, USATel: {+1) 949 8s6 3368; Fax: (+1) g4g 8s6 2044: WeD: \ew,evlews.conlCAtISS Aptech Systcms Inc. pO Box 250, BIack DiaIton.l, WA 98010, USA

Tel: (+l ) 425 432 7855: Fax: (+1) 425 432 j832., Web: lrw.rprech.conrLIMDEP Econometric Softwarc, .15 Gloria l,lace, plainview, Ny 11803, USATel: {+t) 516 938 5254; Fax: (+l) 516 g3B 2441i Wc.b: ww,liDdep.con

MATLAB The lr{arhWorks Inc., 3 Applie Hill Drive, Natick, MA 01760-2098, USAIe1: 1+l) 508 647 7000: Fax: {+l) SO8 647 7OO1: Wc"b: ww.rrarhworl


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28 lnl rodu(tory Economettics Ior Finance

Names for y Names For the rsDependent variable lndependent variablesRegressand RegressorsEffect variable Causal variablesExplained variable Explanatory variabtes

Thele ale various completely interchangeable nanres for. .y and the.rs, and all of these terms will be r.rsed synonyr-rlously in this book (seebox 2.1).2.2 Regression versus correlation

All readers will be au'are of the notion and definition of corrclaLion. Thecolrelation befween rwo variables measures the dcgi'r'e oJ linear.ts.socidrionbenl'een thent. If it is stated that r, and .v are cor.related, it means th;rt r.rnd .r are being treated in a con-rpletely symmetrical way. Thus, it is notimplicd that changes in.t cause changes in r,, or.indeed that cltanges il.rr.cause changes in.r'. Rather, it is simply stated that there is evidencefor a linear relationship beLr,veen the fwo variables, and that movementsin the two are on a\rerage related to an extent given by the correlationcoefficient.In regression. the dependenl variable l.r.r and the indcpendenr vari-able(s) (.ts) are treated very differently. The r, variable is assumed to berandom or'stochastic'in some way, i.e. to have a probability distribution.The .r variables are, however, assumed to have fixed ('non-stochastic') val-ues in repeated sanrples.l Regression as a tool is more flexible and morenowerfirl tha n corr'4]31jgp.

2.3 Simple regressionFol sinrpliciry, suppose for now rhar ir is believed that.r'depends on onlyone .r variable. Again. this is of course a severely lestricted case, but thecase of more explanatory variables n'ill be considered in the next cirap-ter. Three exanrplcs of the kjnd of relarionship rhar rrray be of inrer-cstinclude:r Strictly. the assuntptiotr rhrt the rs are non"srochastic is stroncer thrn required. rnissne thit will be discuss(.d jn ntore detaji in chlpter.4.

A bricf ovewiew o/ the clas'sicai llncnt' r'cglcssiotr nttrrlcl

Fr$fixfScatter Plot of twovariables, Y and x

e How asset rettll'lls var-y with thcir levcl of t'uat'ket t"tsk,* Measuriug the Iong-terltr l'elatiotlsl-tip between stock prlces irdividends

s Constructillg an oPtilllal hedge t'atto'Suppose that a researcher has solDe iciea that there should be a relati,iip o.*.." ftvo variables l and \ ' anci that fin:rncial theory suggtthat an increase tn .t wlll leacl to an increase in 'r" A sensible first stto testing whether there is indeed an associatiot't befween the varialwould be to fornl a scatter plot of them' Suppose that tire otttcoure of tplot is figlrre 2.1In this case, it appeals that there is an approxirllate positive lillrelationship befween t and -r'which rneans that incl'eases in r aLe ltstt'accompanieci by increases in r'' and that the relationsirip befween tl:.on fr. describeci appl'oxilrrately by a straigirt line' It would be possto ciraw by hand onto the grapl-r a line that appears to fit the data'i.'rr.r..p, and slope of the line fittec1 by eye could then be lneasurecl t'tl-re graph. However, rn practice sttch a metltod is lil


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30 Introductory Econometics for Finance

(2.2)where the subscript t(= 1, 2,3, ...) denotes the observation number. Thedisturbance term can capture a number of features (see box 2.2).So how are the appropriate values ofo and p deterrnined? a and I arechosen so that the (verticai) distances from the data points to trre firtedlines are minimised (so that the line fits the data as closely as possible).The parameters are thus chosen to minimise colectivery trre (ver.ticar)distances fi'orr the data poinrs to the fitted line. This courd be crone by'eye-balling' the data and, for each set of variables ' and .r, oue courcrform a scatter plot and draw on a iine that looks as ifit fits the clat;r rvellby hand, as in figur.e 2.2.Note that the ve'ficrri distcrrccs are usually mi'inrsecr r.ather tha, theho.izontal distances or those taken perpendicular to the line. This arises

a Even in the generar case where there is more than one expranatory variabre, somedeterminants of 11 wiil arways in practice be omitted rrom tne moout. rnis ,'ght, t",'example, arise because the number of influences on )j is too large to place in asingle model, or because some determinants of y may be unobservable or notmeasurable.. There may be errors in the way that ) is measured which cannot be modelled.o There are bound to be random outside influences on ;, that agatn cannot bemoderred' For exampre, a terrorist attack, a hurricane or a computer failure courd ailaffect financiar asset returns in a way that cannot be captured In a moder andcannot be forecast reliably. Similarly, many researchers would argue that humanbehaviour has an inherent randomness and unpredictabilityl

to get the line that best 'fits'the data. The rcsearcher. wourcl the. beseeking to find the values of the pal'aneters or coefficieuts, * a'd lJ,which would place the line as close as possible to all of the data pointstal(en together.However, this equation (1, = a + B.r) is an exact one. Assumir1g that thisequation is appropriate, ifthe values ofa and B had been calculated, tl.rengiven a value of ,r, it would be possibre to deternrine with certai'ty whatthe value of ' would be. In'ragine - a moder which says with completecertainry what the value of one variable will be given any value of theother!clearly this nrodel is not rearistic. statisticaily, it wourd corr.espond tothe case where rhe model fitted rhe data perfectiy- that is, ail of trre datapoints lay exactly on a straight line. To make the moclel nore realistic, arandom disturbance te.n, denoted by,, is added to the equation, thus-l'/=a+F.r,*u,

A brief oven'taw of tltt: classical linectr rcg,ression modcl

@Scatter Plot of twovariables wlth a llneof best fit chosen DYeye

as a resttlt of the assunption that r is fi-xecl in repeated samples' so-t)the problem becomes one of detern-rining the appropriate model forgiu"n 1or conditional upon) the observed values of r'This 'eye'balling' procedure nray be acceptable if only indicative resunre required, but of course this method' as well as being tedious' is likto be imprecise. The most con'llrloll method used to flt a line to the dat;known as ordinary least squares (OLS) This approach fonns the wolkhoof ecouonetlic nrodel estimation, and will be discussed in detail in tand subsequent chaPters'Two alternatlve estlmation methods (for deterl-nining the apProprrvalues of the coefficlents cr anci p) are ti,e method of tnourents andmethod of maxinrum likelihood' A generalised versiott of the methodmoments, due to Hansen (1982), is popular' but beyond the scope of tbook. The nethod of maximum likelihood is also widely emplol'ed' 'will be discnssed in detail in chapter 8'

Suppose now fot' ease of exposition' that the satnple of data cont;only-fiu" observations. The metl'rod of OLS entails taking each vertdistance fronl the polnt to the line' squaring it and tilen uriniurtrthe total sum of the areas of squares (l]ence 'least sqtrares )' as sltt]wifigure 2.3. This can be vierved as eqrtivalent to tnininrising the stllll olaieas of the squares dlawn fi'ou the points to the linc'Tightening up the notation, let r', clenote tl're actllal data poirlt forscFr'ation t and let ii cienote the fitted valrte front the regresston linc


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JZ

@ilN4ethod of OLSfitting a line to thedata by minimlsingthe sum of squaredresiduals

EEilPlot of a singleobservati on,together with theline of best fit, theresrdual and thefitted value

I ntrod.uctory Econontetrics for F inance

other words, for the given value of .r'of this observation /, i, is the vah.refor l which the model would have predicted. Note that a hat (^) over avariable or parameter is used to denote a value estimated by a r'oclei.Finally, let ri, denote the residual, which is the difference between theactual value of ' and the vaiue firted by the model for this data point *i.e, (ti * i,). Tliis is shown for just one observation r in figur.e 2.4.\Mrat is done is to nlinimise the sum of the ni. The reason that the sr,ln.lof the squared distances is minimised rather than, for. exa'rple, findingthe sun of ri, that is as close !o zero as possible, is that i' the latter casesome poinrs rvill lie above the line while others lie below it. Then, whenthe sunr to be made as close to zero as possible is for-r'ned, the points

_--- - , -. . , r: I f | | lt

A brief oven'icw oJ tlta classicul lincar rcg,ressiott tttodcl 3:i

above the line wor'tlcl collllt as pol'itivc valttcs' while tl-rose below woulclcount as negatives. So these clist:rnccs will in large part caucel each othetottt'wlriclrwoulcltlleant.hltollecor'rlclfitvir.ttra.Ilyatrylitletothedat:lso long as tlle stllll of the rlistancc's of the l)oillts above the line and tl-tisunr of thc distrlnces of the poit-rts bclow the lille wet'e the sanre ln thacase, thel'e rvottltl tlot ue a r'rrlirlue soltltiott fo| the cstirtra:cd:oeffict:lllIn fact, any fittcd line that goes tlttor'rgl'r thc tttcau of tl-rc olrsetvatton(i,e. i, r)wottlcl sct tllc sttlll Of the ri, to zcro Horvcvet" taking the sqllat("clistances cltsufes tilllt:tll rleviatiorls thilt clltel'the t;tlctrlrtitln :r|e positirancl thet'efot'e do not c:tt'tcc'l ortt'Sorr-rirrirr-risir-rgthcsu]llofsclttat.cclclistatrcesisgivclrbytl-til.rilrrisirlri?] + ri + /ii + tl + rr]t, ol mir.tituisitrg

{t,i;lT.his s.n-r is.l


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JA Intrcductory Econometrics for FinanceTable 2.1 Sample data on fund XXX to motivate OLS estimation

Year, rExcess return on Excess return onfund XXX : rxxx,r - rfr market index = nn, - rf,

12345

L7.839.072.8)a)1.7.2

'13.723.26.9

16.8't2.3

Example 2.1

to use to calcnlate the slope estintate, but the forrrula can also be writtennrole intuitively, as/1 - a(-\-r -.1 )( \'f -.\')L(.r/ - .\')- 12.6)

which is equivalent to the sample covariance befween ,r. and l divided bvthe sample vaLiance of .r.To reiterate, this method of finding the optimurn is known as OLS. Itis also worth noting that it is obvious from the equation for t1, that theregression line wiil go through the mean of the observations - i.e, thatthe point (.i,.y) lies on the regression line.

Suppose that some data have been coilected on the excess returns on afund manager's portfolio ('fund XXX'1 rogether with the excess returns ona market index as shown in table 2.1,Tire fund manager has some intuition that the beta (in the CAPMframework) on this fund is positive, and she therefore wants to findwhether there appears to be a relationship between x and y given the data.Again, the first stage could be to form a scatter plot of the fwo variables(figure 2,5).Clearly, there appears to be a positive, approximately linear relation-ship berween .r and .r,, although there is not much data on which to basethis conclusionl Plugging the five observations in to rnake up the for-mulae given in Q.a) and (2.5) would lead to the estirnates d = -1.74 andE = L6a. The fitted line would be rvritten as

.i, = -1.74* I.64.r, (2.7)where .r', is the excess return of the market portfolio over tlte r.isk fi.eerate (i.e. rnt * tf1, also known as the n?arket risk preniunt.

IT'?'rr9ilScatter plot ofexcess returns onfund XXX versusexcess returns onthe n.rarket Portfolio

A brieJ oveniew oJ the clossical lincar rcgrcssitm model

o

oooo

l5

fund XXX to be?The exPected value of \' :'-7'74 +into (2.7)

ir = -1,14 + 1.64 x 20 : 3l'061.64 x vah,re of .{', so Plug t =

'1{}

xl0* ll

o ttl

Excess return on market Portfolio

2,3.7 What are A and fi used for?This question is plobably best ausu'ercc1 by posing another questioll lf '",r"lyit tells you that she expects the lnarket to yield a return 20% higl'than the r.isk-free rate next yeaf, what would yoll expect the leturn '

Tlrus, fbr a glven expecred marl(et risl< premium of 2a%' and givenriskiness, ftrnd XXX would be expected to earn an excess over the rjfree rate of approximately 31%' h'r this setup' the regression beta is :tthe CAPM beta, so that fund XXX has an estimated beta of 1 64' sgesting that the fund is rather rislqr' In this case' the residual suniqu"ro t'eaches its minimum value of 30'33 with these OLS coeffici'values.Although it n-ray be obvior'rs, it is wolth stating that it is not actvlsirto conduct a regressiot't analysis using only five obsewationsl Thnsresultspresentedherecanbeconsiclereclindicativeandforillustrattotthe techniqr-re only. Son-re further discussions on appropriate saniple s:for regression analysis are given in chapter 4'

The cocfficient estlnlate of 1.64 for 19 is iuterpreted as saying that.r'increases by 1 ul'rit. r'rvrll be expectcci' everythitrg else being cqto increase by 1.64 urrits'. Of coLlrse' if i had beell negative' a nse Itvoukl on a\/erage car.ise a fall in r" a, the intelcePt coefficient estilllat{


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E@ilNo observationsclose to the y"axis36 lntroductory Econonrctdcs for Flnonce

lntel'preted as the value that wor-rld be tal(en by the dependent variable 'if the independent variable.r took a value of zero.'Units'her.e refer to theunits of measurement of .r, and r,,. So, for example, suppose that lj : t.O+,,r is neasured in per cent and .t' is measured in t}rousands of US dollars.Then it would be said that if .r rises by 1%, .r.will be expected to rise onaverage by $1.64 thousand (or $1,640). Note rhat changing the scale ofr.or .r' will make no difference to the ovel'all results since the coefficientestimates will change by an off-setting factor to leave the ovelall r.elation-sl.rip berween l and .r unchanged (see Gujar.ati, 2003, pp. 1,69-773 for aproofl. Thus, if tl're units of measur.ement of l. were hundreds of clollar.sinstead of thousands, and everything else reurains unchanged, the slopecoefficient estimate would be 16.4, so that a 1% increase in _r.wouid leadto an increase in r.of $16.4 hundreds (or $1,640) as befor.e. All oti.rer.prop_erties of the oLS estimator discussed below are also invariant to chanscsir-r the scalir-rg of the data.

A word of caution is, however, in or.der concerning the leliability ofestinates of the constarlt teilr. Although the strict interpretation r:f theintelcept is ir-rdeed as stated abovc', ir-r practice, it is oftei-r the case thatthele are no values of.r' close to zero in the san'rple. in such i'stances,estimates of the value of the intercept will be unreliable. For. examplc,consider' figr.rle 2.6, which demonstrates a situation where no points areclose to the r,-axis.

*..---.,rrlll

A brief ovtn'iew o/ thc cias.sical lit'nar rcgrcssion modcl 'Ir-r such cases, one coulcl not e?pcct to obtain robust estimates of tl-lvalueoffwhetl'flszeroasalloftlrc-infortr-ratiorrintlresarnpleperrail

to th case whet'e t is consicle|ably h|ger thall zcro'A similar- cilt-ttiolt should bc exct'cisccl whetr prodttcing pledlctious 1'l using vrlr.res of .f that are a long way outside the l'allge ol values rthe satnple. ln cxaltlplc 21, r tal(es valtrcs between 7% and 23% in tlavnilable clata. So, it would not be rclvisilble to ttse this Droclel to detet'ntitthecxpectedcxccsslctuf]lorlthefirrldiftlreexllectcdcxcessfetLlfn(thc lnarl(et werc, say l%' ot'30"/" or' -5% (i c' the nlat'l


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Intraduct1ry Econont ettics Jor Finance

it would srill in generai not be possible to obtain a perfect fit of the lineto the data. In some textbooks, a distinction is clrawn befween the pRF(the underlytng true relationship befrveen ), and r) and the DGp (theprocess describing the way that the actual obseryations on )i corne about),although in tl-ris book, the nvo ternrs will be r.rsed synonynrously.The sample regression function, SRF, is the relationship that has beenestimated using the santple observations, and is often written ast-,:a*f.r, (2.10)

Notice that there is no error or resiclual term in (2.10); all this equatior-rstates is rhat given a parricular value of.r, niultiplying it by f and aclclingd will give the model fitred or expected value for r,, denotecl i. It is alsopossible to rvrite-ri=d+lJ.r,*ir, (2.11)

Equation (2.11) splits the observed value of .\, iltto two cornponents: thefitted vaiue from the model. and a residual term.The SRF is used to infer likely values of the pRF. That is, the estimates& and B are constructed, for the sample of data at hand, but what is reallyof interest is the true relationship befween .r and r, - in other words,the PRF is r,vhat is really wanted, but all that is ever available is the SRFiHowever, what can be said is how likely it is, given the figures calculatedfor d and f, that the corresponding population paraneters take on certainvalues.

2,4.3 Linearity and possible forms for the regression functionIn order to use OLS, a model that is linear is required, This means that,in tlre simnle hirr3ri3lg case, the relationship between.r and y must be_' -__- -".,r'_capable of being expressed diagranratically using a straight line. Morespecifically, the model must be linear in the parameters (a and p), but itdoes not necessarily have to be linear in the variables (.t. and r). By ,linear.in the parameters'. it is nreant that rhe parametel.s are not multipliedtogether, divided, squared, or cubed, etc.Models that are not linear in the variables can often be made to tai(ea linear form by applying a sLritable transformation or rnanipulation. For'example, consider the follorting exponential regression ntodel

A hrief overtiew of Lhe classtcal ltnror regrcssiott tttodel,Taking logarithnrs ofboth sides, applying the laws oflogs and rearran!the right-hand side (RHS)

ln /, = In(A )-l f,ln X, * rt, l';where A anci I ale Par:rllletcrs to bc cstillliitcd. Now lct rr : )rt(A)' r', =aud -1, : ln X,

.\i = cr '1- p.r, * rt, {'This is lcnown as all cxponsrllinl r-cg|e.ssion rrrodcl since f vat'ies acco-to sotne exponeut (power.) firnction nf X. In fact, wltcn :r fcgl'essloll cLtion is expressed in'clouble logalithruic for"nr" rvhicl-r nlealls that Itl-re clcpendent ancl the independent variables are uatut'al logarithms.coefficient estin-lates arc irltcfPfetecl as elasticitics (stlictly, they al'echanges on a logaritl-ruric scale). Thus a coefficient estitll)ate of 1.2 for(2'13)or(2.14)isirlterpreteclasstatil]gthat.afiseirrXofl%willlea.average, everytlling else being equal, to a |ise in Y of 1'2%" Converselr.yand'tin]evelslatl.lerthanlogaritlrrrricfor.nr(e.g.(2.9)),tlrecoefficjdenote unit chatlges as described above'Sinrilarly, if theor.y sl"tggests tlrat .r slrorrld be invet.sely related tocording to a model of the fornr

,p,., (,li=(1+-+l/r.l/the reglession can be estinated usir-rg OLS by setting

I.lr

and regressing .r on a constant and i. Clearly, then' a surprisingly vrarrayofmodelscanbeestinateclusingoLSbynrakingsuitabletrat',nations to tl-re variables. on the otl-rer hand, some nodels are intrin-(n.on-linear, e.g.

.\i:a+f3-t! *utSuch nrodels cannot be estimated using oLS, but n-right be estir.Ilable ta nou-linear estilration metl-tod (see chapter 8)

2.1.4 Estimator or estimate?Estinrators are the fbr-mulae used to calculate tlrc coefficienls - for exi,the expressior.rs given in (2.4) and (2,5) above, rvhile tl'Ie estinratcthe other hancl, are the actual nunrcrical values for thc coelficient-s tli,obtained fi'on'r the samPle., :,{ \:'r"' (2.121


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40 Itltroductory Ecouontetrics for Finance

2.5 Simple linear regression in EViews - estimationof an optimal hedge ratioThis section shows how to run a bivariate regression usi'g EViews. Tlieexample considers the situation wherc an investor wisires to hc.clge a l


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E@@[ilEquation estimationwindowI nI,odu(tory Econotnctrics far Finoncc

Equdtion specrfrcationDependent variable follovied by list of regressofs includrng AR[lAand PDL terns, OR an explicit equation like Y=c(1)+c(2)-X.

fspot c rfutures

A brief ovcrttew 0J lhc ri0sslcal linror regression ntodel

Prob(F-statistic) 0 358093

Dependent Variable RSPOThlethod Least SquaresDate.08/09/07 Tinre 10 17Sanrple (adjusted) 20021'103 2007 tll07lncluded observations. 65 after adJustnrentsCoeflicient Std Enor t-statistic Probc 0363302 0444369 0817569 041S7RFUTURES oiiiaoo 0133790 0925781 03581

i-rquur.O O 013422 llean dependent var 0 421 203Adltrsted R-squareo u 002238 S D dependent var 3 542992S E of regressrorr , , ;;695' Akaike info crrterion 5 400342Sunr squarec resro , J2.4960 Schtvarz criterion 5 467246Log likelihood -'r i3 5i i i Hannan'Ournn criler 5 426740F-statistic u 857070 Durbin-Watson stat 2 1 16689

r7fttf,f{il:rrr-Estimation resu|rs

Estinration settingsNlethod: I t5 - Least Squares (i',lLS and ARMA)sampte, i zoozr,roz ioozuoz -..1

the short run relationship berween the tlvo series By contrast' the 'parameter tn a regr-ession using the raw spot and futures indices (olog of the spot serles and the 1og of the futures series) can be interpas nreasuring the long run relationship beween them' Tl'ris isstte oiong and short runs will be discr-rssed in detail in chapter 4 For now'Quick/Estimate Equation and enter tl.re variables spot c futures rreluation Specification dialog box' click OK' tiren name the regrerresults 'levelreg'. The intelcePt estimate (d) in this regressiot] isand the slope estrnate (P) is 0'9S' The intercept can be conside-r'ed iproxitnate the cost ofcarry, while as expected' the long-ternr relatiobenveen spot and futures prices is altnost 1:1 - see chapter 7 fol ftrdiscussionoftheestilllationandilrterpretatiolrofthislong.tel.tlrrel.ship. Finally, click the Save button to save tl're rvhole wolkfile'

2.6 The assumptions undellying the classical linear regression nThe rlroclel .\'r :.r * pt, * rr, that has been cierivecl allove' togetltetthelssrrtrlptiollsllsteC]beiow,isknownastheIl(]ssicdllirlcrtt.tt'.c'l.t'.sstott

l--o-_lfc.^*awill be presented with a dialog box, which, when it has been completed,will look like screenshot 2.2.In the 'Equation Specification'window you insert the list ofvariablesto be used, with the dependent variabie (l') first, and including a constant(c), so type rspot c rfutures. Note that it would have been possible to wlitethis in an equation fornat as rspot = c(1) * c(2)-rfutures, but this is uor-ecuntbersome.In the 'Estimation settings'box, the default estimation method is OLSand the defauit sample is the whole sample, and these need not be modi'fied. Click OK and the regression results will appear, as in screenshot 2.3.

The parameter estimates for the intercept (d) and slope tBt are 0.36 and0.12 respectively. Name the regression results returnreg, and it will uowappear as a new object in the 1ist. A large number of other statistics arealso presented in the regression output - the purpose and interPretatronof these will be discussed later in this and subsequent chapters.Now estimate a regression for the levels of the series rather thanthe returns (i.e. run a regression of spot on a constant and flltures) alldexamine tl'le paranleter estimates. The return reglession slope paratle-ter estilnated above measures the optimal hedge r-atio and also ureasurcs

,. | rrt I |lt If ]1


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(CLRM). Data for'"r, is obsenrable, but since r', also depends on r/, it is neces-sary to be specific about ho"v the rr, are generated. T'he set of assumptionsshown in box 2.3 are usually made concerning the rr,s, the unobservableerror or distulbance terms. Note that no assumptions are made concern-ing their observable counterparts, the estimated rnodel's residuals.

As long as assumption I holds, assurlption 4 can be equivaler-rtly writtellE(.r,rr,) = 0. Both fonnulations imply that the regressor is orthogoral to(i.e. unrelated to) the error term. An alternative assunption to 4, whichis slightly stronger, is that the -rr are noll-stochastic or fixed in lepeatedsarnples. This means that there is no sampling variation in.r,. and thatits value is determined outside the model.

A fifth assurnption is required to make valid inferences about the pop-ulation paraneters (the actual n and B) from the sample paran'leters (dand 13) estin'rated using a finite anount of data:(5)u, - N(0, o21-i.e. that u, is nornally distributed

2.7 Properties of the OLS estimatorIf assumptions l-4 hold. then the cstimrtols d, and p detcrminccl by OLSwill have a number of desirable propelties, and ale l


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Illtroductlry Eccnometics for Finance2"7.3 Efficiency

An estimator p of a parameter fl is said to be efficient if no other estrmr-tor has a smaller variance. Broadly speaking, if the estimator is efficient,it will be minirnising the p.obabiliry that it is a lorg way off from thetrue value of p. In other words, if the estirnator is 'best', the uncertai'ryassociated with estimation will be minimised fol the class of linear un.biased estimators. A technical way to state this would be to say that anefficient estlmator would have a plobabiliry distribution that is nar.r.owlydispersed around the true value.

2.8 Precision and standard errorsArly set of regression estinrates ri and 13 a'e specific to the sarnple nsecin their estimation. In other words, if a different sa'rple of data wasselected fron-r within the population, the data points (the.r, ancl r,,)rviilbe different, leading to different values of the OLS estinates.Recall thar the oLS esrinators ld and d) are given bv (2,a) and (2.5). Itwould be desirable to have an idea of how'gooci'these estimates of a a'dB are in the sense ofhaving sonle measure ofthe reliabilifv or precision ofthe estimators (ri' and 79;. tt is thus useful to k'ow whether one ."n hou.confidence in the estinrates, and whether they are likely to vary muchfrom one sample to another sample within the given popuiation. An ideaof the sampling variability and hence of the precision of the estimatescan be calculated using only the sampie of data available. This estimare isgiven by its standard error. Given assumptions 1-4 above, valid estimator.sof the standard errors can be shown to be given by

,S6(d) = .r

sr(r) : .,(2.20)

12.211whe'e s is the estimated standard deviation of the residuals (see below).These folnulae are derived rn the appendix to this chapter.It is wolth noting that the standard errors give only a general indicationof the likely accllracy of the regression parameters. They clo not shorvhow accurate a palticular set of coefficient estirnates is. If the standar.derfors are small, it shows that the coefficients are likely to be preciseon average, not how precise they are for this particulai.sample. Thr.rsstandard errors give a measure of the degree 0f u'rcertainty in the estinrated

A ilriclcrvcrvieu, of lhc cla'ssicol lirtt'or rt:grcsston ntodtl

valnes {or. t5e coe{ficicnts. It t'an bc scctr that they arc a fttncticithe, actual obscr.vrtior.rs on thc cxirlaDatoty varirble, .f, thc saurple7., ancl anothcf terlr, i. Thc l;rst of thcsc is uD cstitnate of tlie varjof the disturl;;ulce fcl'nt.'l-hc uctual vat-iatrcc or thc disttll'ballcc tclusualiy detrotc'cl by ol. llow tlttt ittt cstitt.titte of ol bt' obtaillcd?

2.8.7 Esttntating the variance of the error term (o:)Front cler.r.rcr.rtitfy stalistics, the vrr-i;tuce of a t'atrclotl'l valiable ir, is giv,

var(tt,) : I',1(rr, ) - [:(rr1 )lr (Assr.tnrptiou l of thc CLRM was that thc cxpectcd or average value oerrols is zct'o. Under this rssr"rnlption, l2-22) above t'educes to

var(rr, ) = EIrri]So what is requir.cd is an cstinrate of thc'average value of rrl, wl'rich tbe calculated as

'ls.- _ - \7'u 'Unfoltunately (224) is tlot woll


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48Hr-

Introductory Econonxtrics Jor Finance

this is left to the appendix to this chapter. Sorne general intuition is n.:-.,given as to why the folrnulae for the standar.d eLrors given by (2.20) :rld(2.21) contain the telns that they clo and in the form ttrat they clo. l'irepresentation offered in box 2.4 loosely follows that of Hill, Gr.iffitl-rs and.fudge (1997), wl.rich is the clearesr that this aLrrhor h.rs scen,

(1) The larger the sample size, I, the smaller will be the coefllcient standard errors.fappears explicitly in .SE(d) and implicitly jn SE(lq). fappears impliciily since thesum f (-r, *i)r is from r = | Io T. The reason for this is simply that, at least fornow, it ls assumed that every observation on a series represents a piece of usefulinformation which can be used to help determine the coefficient estimates. So theIarger the size of the sample, the more information wrll have been used in estimationofthe parameters, and hence the more confidence will be placed jn those estlmates.(2) Both St(&) and .t6(p) depend on s2 (or sy. Recall from above that sr is the estimateof the error variance, The larger this quantity is, the more dispersed are the residuals,and so the greater is the uncertainty in the model. lf s2 is large, the cjata points arecollectively a long way away from the line.(3) Thesumofthesquaresofthex,abouttheirmeanaDpearsinbothformulae-sinceI (r, - i)2 appears in the denominators. The larger the sum of squares, the smallerthe coefficient variances. Consider what happens if f (.r, - i)2 is smalj or large, asshown in figures 2.7 and 2.8, respectively.

ln figure 2.7, the data are close together so that f (x, - i)2 is small. ln this ftrstcase, it is more difficult to determine with any degree of certainty exac|y where theline should be. 0n the other hand, in figure 2.8, the points are widely djspersed

A hrfu'f ovovie tv of lhc rla.ssirrri lincrtr- rt'gt'c.ssiott ntorii:l

across a long sect;on of the line, so that one could hold more confldence irestinrates in this case.(4) Thetermf.rf affectsonlytheinterceptstandarderrorandnottheslopestanerror. The reason is that f .r,2 measures how far the points are away from the y''Consider figures 2.9 and 2,IO.

ln figure 2.9, all of the points are bunched a long way from the y"axis, which mit more difficult to accurately estimate the point at which the estimated line crothe _y'axis (the intercept). ln figure 2.10, the points collectively are closer to

@EilEffect on thestandard errors oftho coefficientestimates when(.xr - i) are narrowlydispersed

@Effect on tnestandard errors ofthe coefflcientestimates when(.1, - i) are widelydrspersed

EmuEIEffect on thestandard errors ofrf large

O


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50

BtrfTTIEffect on theslandard errors of.r; small

Example 2.2

Introductory Econornctrics for Finance

xthe y-axis and hence it will be easier to determine where the line actually crossesthe axis. Note that this intuition will work only in the case wnere all of the r, arepositivel

Assume that the following data have been calculated from a regrssion of.r'on a single variable.r and a constant over 22 observations\-,. ,, - prnrnr r/2.-r.rt - ' = 22. .i = 416.5. .r-'= 86.65.\-': = iQrs6sl PSS = Ilo,6/-^t *Determine the appropriate values of the coefficient estinrates and theirstandard errors.

This question can sirnply be ansrvered by plugging the appropriate nunt-bers into the formulae given above. The calculations are, 830101 - Q2 t 416.5 > 86.65 r- _iq 1q654 * 12 x (116.5 )l

r? : 86.65 - 0.3.5 x 416.5 : -59.12The sample regression function would be written as

l.)i:(}+trl.\/i, : _-59. tl * 0.35.r,

A brit.f ovtniew of the clossicnl linaar rrgression models

Now, tr,r|ning to the standrl'd crr"ot'calculations, it is necessary to o)an cstinrate, r, of the cl'ror vanancc/>-ai i'iltui.tEtrt'r'ii'rrl,'itt. r -,/ .- -, - : 255" , / -l v l(), / rq tqo-s+(/./,;' - I \s ! ll r.tu t,loSi - 12 z J 1r,.5: r - 1 t{

'\Arlir: 155 ' \/..ur,urs+ - rr -lrn+ = r)rrlt7gWith tl're staudard eI|o|s calculated, the results are written asii = -59,l2+0,.15.r/(,1.35) (0.0079)The standald error estin)ates arc r.tsual)y placed in paretttheses uncltlclcvanI cocffi cicnt c:.til]l.rtcs.

2.9 An introduction to statistical inferenceOften, financial theory will suggest that certain coefficients sl-toulclon prrt jcular valucs, or valrres witltin a given range. It is thus of irrto deternrine whether the relationships expected from financial tlare r-rpl'reld by the data to hand or not. Estinrates of n and li haveobtained froni tl-re sauple, but these values are not of any pal'ticulterest: the population valr"res that describe the true relationship betthe variables would be of lnore interest, but are never available. II't:inferences are made concerning the likely population values from tlgression paralneters that have been estimated from tl're sample o1to hand. In doing this, the aim is to determine whether the differ-betlveen the coefficient estimates th:it are actually obtained, and exltions arising frorn financial theory, are a long $'ay frot.tr one altothcstatistical sense.

Example 2.3 Sr-rppose the following regression results have been calculated:i, =20.3*0.-5091.r,

( 14.38) 10.256I ),l = O.SOSf is a single (point) estimate of the unknowt.t popr.rlation peter, /4. As stated above, the reliability of the point estitttate is ttrer


14/31

Introductory Econlmetrics Jar Firlance

by the coefficient's standard error. The information fronr one or nrore ofthe sample coefficients and their stanclard errol.s cau be used to 'rakenferences about the population para',eters. so the estimate of the slopecoefficient is f = o.sort, but it is obviors that this 'umber is rikely to\fary to sonre degree from one sanrple to the next. It might be of inter.estto answer the question,'ls it plausible, given tliis estil:rate, that the tluepopulation paraneter, B, could be 0.5? Is it plausible that f could be t?,,etc. Answers to these quesfions can be obtainecl tl-rrough hypothc.sls rc.sfi rrg.2,9,1- Hypothesis testing: so/re concepts

In the hypothesis testing framelvork, the'e are always two l-rypotheses thatgo together, l 0.5 are subsumed under the alternative hypothesis.Sometimes. sone prior information may be availabre, suggesting for.example that p > 0.5 would be expected rather than p < O.S.ln this case,f '< 0.5 is no longer of interest to us, and hence a one-sided test would beconducted:Ho:f=0.-sHr tf>0.5

Here the null hypotrresis that the true value of B is 0.5 is beirrg testedagalnst a one-sided alternative fhat , is nrore than 0.5.on the other hand, one colrld envisage a situation where there is pr.ror.i.formation that 19 < 0.5 is expected. For example, srrppose that an ,t-vestment bank bor"rght a piece of new risk managenrent soffwafe thar isrntended to better track the riskiness inher-ent in its traclers'books anclfhat , is sone measure of the risk that pr.evior-rsly tool< the vrlue O.S.clearly, it would'ot nake sense to expect the'isk to have r-ise', ancl so

A bricJ ovct-victu o/ thc r/rrs.sical littt'rt t r cgt-cssion ntoricll r 0.5, coil'esPoncling to an iltcrcilse in t'isl


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54

rj|.5?.errrThe normaldi stribution

lntroductory Ecotrometrics for Finance

viao ' N(0, l) and

eu *ct-!/,1

rn,.JJI](A'

A brief oventiew o.f Lltc classical lintru' rcgression nlodellr

fable 2.2 Critical values from the standard normal versust-distributionSignificance level (%) N(0,1) trc50%5"/o2.5%0.5%

1.641.962.57

001.68 2.13').0) 2.782.70 4.60@fEgThe t'distributionversus lng normal

.T

Standard'ormal variables can be constructed from d and, fl by subtract.ing the mean and dividing by the square root of the variancecr-cl fl*Br r\\r! I /,/ var( fi\The square roots ofthe coefficientvariances are the standard error.s. unfor-tunately' the standard errors of the true coefficient values u'der. the pRFare never known - au that is availabre are their sample counterparts, trrecalculated standard errors of the coefficient estimates, Sf(a) ana Sffi4).4Replacing the true values of the standard errors with the sample es_timated versions induces another source of uncertainty, and arso meansthat the standardised sratistics follow a r-distributio'with r - 2 degreesof freedom (defined below) rather than a normal distribution. so

A normal variate can be scaled to have zero rnean and unit vatian,by subtracting its mean and dividing by its standard deviatlon. There isspecific relationship befween the r- and the standard normal distributioand the t-distribution has another parameter, its degrees of freedom.

\&rtrat does the t-distlibution look like? It looks similar to a nornldistribution, br.rt with fatter tails, and a smaller peak at the mean,shown in figure 2.12.Sone exanrples of the percentiles from the normal and l-distributio;taken from the statistical tables are given in table 2.2. \Mren used in tlcontext of a hypothesis test, these pelcentiles become critical values. Tlvalues presented in table 2.2 would be those clitical values appropriafor a onc-sidcd tcst of thc given significancc lcvcl.it can be scen that as the number of deglees of freedom for thedistribution incLeases frour 4 to 40. the clitical values fall substantialIrr figule 2.12. lhis is rcprescnted by a gradrral incrcasc irr thc hcigltttl-re distribution at the centre and a reduction in the fatness of the tailsthe number of degrees of freedonr increases. In the limit, a l-distributic'with an infinite number of desrees of freedorn is a standard nornral, i

R_R+ltrrsE(p) ''-zThis result is not formally proved here. For a fornal proof, see Hril,Criffiths and Judge (1997, pp. BS_90).

2.9.3 A note on the t and the normal distributionsTire normal distribution, srrown in flgure 2.11, sr.rouid be famiriar to read_ers. Note its characteristic 'beit'shape and its symmetry arou'd the rlean(of zero for a standard normal distributioni.a strjctly' these are the estinrated standard euors condjfional on the paramcter estirrnres.and so should be denoted S6(A) and .rE(B). but the additional Iayer ofhats \vill beontitted lter since the nteaning should be obvious front the context.


16/31

56 Introductory Econometrics for I:inancer- : N(0, l), so the norntal distribution can be viewed as a sDecial case ofthe t.

Putting the limit case, /a, aside, the critical values for the r-distributionare larger in absolute value than those from the stanclar.d nor.mal. Thisarises from the increased uncertainfy associatecl with the situation wherethe errol variance nlust be estimatcd. so'ow the l-distr.ibution is used,and fora given statistic to constitute tlre same amount of re'liable evidenceagainst tlre nr.rll, it has to be bigger in absolute value fhan in circuurstanceswhere the nomtal is applicablc.Ther-e are broadly two approaches to testiltg hypotheses uncler regles-sion analysis: the test of significance apploacl'r ancl the conficlcnce inter-valapploacl'r. Each of these will now be consider.ed ir.r tnr.n.

2,9.4 The test of significance approachAssulDe the regression equation is given by ti : u I p.t., * tt,. r :1.2..... f. The steps involved in doing a test of significar-rce ar.e shownin box 2.5.

(1) Estimate A, p and SE@), SE(.b in the usuat way.(2) Calculate the test statistic. This is given by the formulaB-8,ICSL.loltsltC: ------;- /? ?nlsL(p)

where p- is the value of p under the null hypothesis. The null hypothesis is Ho : 19= ,0- and the alternative hypothesis is Ht : fl * p- (for a two_sided test).(3) A tabulated distribution with which to compare the estimated test statistics is re-quired. Test statistics derived in this way can be shown to follow a /-distribution withI - 2 degrees of freedom.(4) choose a 'significance level', often denoted d (not the sanre as the regressionintercept coefficient). lt is conventional to use a significance level of 5%.(5) Given a significance level, a rejection region and non-rejection region can be de.termined. If a 5% significance level is employed, this means that 5% of the totaldistribution (5% of the area under the curve) wiil be in the rejection region. Trratrejection region can either be split in half (for a two-sided test) or it can ail fall onone side ofthe .r-axis, as is the case for a one_sided testFor a two-sided test, the 5% rejection region is split equally between the two tairs,as shown in Iigure 2.13.

For a one-sided test, the 5% rejection region is rocated sorery in one tair of thedistribution, as shown in figures 2.!4 and 2.1s, for a test where tlre alternaIveis of the 'less tl'ran' form, and where the alternative is of the 'greater than' form,respectively.

A lrrrrt' overliew rrl fhe rirrs-sical lnaar rt'g,rcssion ntodcl

tffir|grFfRejection regions fora twosided 5%hvoothesis te$t

@Rejection region fora one-sidedhypolhesis test ofthe formh: li = F-'H':li B-

5"/.ieclion region

\/IC

.f(x)

2.5Yoreiection region 95% non-reieclion region

n.)

95% non-rei{iction region

/ (x)

95% non-reieclion region


17/31

Introductory Econometrics for Finonce

(6) use the /-tables to obtain a critical value or values with which to compare the teststatistic, Ttle critical value will be that varue of r that puts 5% into the rejectionregion.(7) Finally pedorm the test. If the test statistic ries in the rejection region then rejectthe null hypothesis (H6), else do not reiect H".

Steps 2-7 require further comment. In step 2, trre estimatecl var'e of p rscornpared with the value that is sr"rbject to test undef the null hypothesis,but this difference is 'normalised'or scaled by the standard error.of thecoefficient estimate. Tire standard error is a measure of how conflclentone is in the coefficient estimate obtained in the first stage. If a standarderror is small. the value of the test statistic will be large relative to thecase where the standard error is large. For a srrall standard error, it wouldnot Iequire the estin-rated and hypothesised values to be far away from oneanother for the null hypothesis to be rejected. Dividing by rhe standarderror also ensures that, under the flve GLRM assumptions, the test statisticfollows a tabuiated distribution.In this context, the number of degrees of freedonr can be interpretedas the number of pieces of additio'al infor'ration beyond the rninimun.rrequirement. If rwo parameters are estimated (a and p - the interceptand the slope of the iine, respectively), a nrinimum of fwo observations isrequired to fit this line to the data. As the number of degrees of freedomincreases, the critical values in the tables decrease in absolute terms, sinceless caution is required and one can be more confident that the resultsare approprrate.

The significance level is also so'retimes called the size of the test (notethat this is compietely different from the size of the sample) and it de-fermines the region where the null hypothesis under test will be rejectedor not rejected. Remember that the distributions in figures 2.13-2.15 arefor a random variable. purely by chance, a random variable rvill take onextreme values (either large and positive values or large and negative val-ues) occasionally. More specifically, a significance level of 5?6 means thata result as extreme as this or more extreme would be expected oniy 5%of the time as a consequence of chance alone. To give one illustratio', ifthe 5% critical value for a one-sided test is 1.68, this irnplies that the reststatistic would be expected to be greater than this only 5g,o of the tirrre bychance alone. There is nothing magical about the test - ail that is done isto speci8/ an arbitrary cutoff value for the test statistic tl-rat deternrineswhether the null hypothesis would be rejected or not. It is conventionalto use a 5% size of test, but l0% and 1% are also commonly usecl.

A brieJ ovewiew of thc classical lineor rcgression mod.els

However, one potential problem with the ttse of a fixed (e'g. 5o ) srzof test is that if tl'te sample size is sr,rfficiently large, any null hypothesican be rejected. This is palticr"rlar-ly wolt'isoltle in finance, where tens t'thousancls of observations or lllore are often available, What happens ithat the standard errors |eclrrce as the santple size increases, thUs leadinto an iucrease in the value ofall r-test statistics.This plobiem is frequentloverlooked in enrpirical wot'k, but sollle econometlicians have suggestcthat a lower size of test (e.g. 1%) should be r-rsed for large samples (see' ft'example, Leanter', 1978, for a discussion of these issues).Note also thc r.rsc of tcr-nrinology in counection with hypothesis test'it is saicl that the null hypothesis is either rejected or not rejected. It tincorrect to state that ifthc null hypothesis is not rejected, it is'acceptet(although this error is fi'equently made in practice), and it is never saithat the altet-native hypothesis is accePted or rejected Oue reason wlit is not sensible to say that the null hypotl]esis is'accepted'is thatis impossible to know whether the nnll is actually true or not! In angiven situation, mauy null hypotheses will not be rejected. For exatlpl,suPPose that Hn : p : 0.5 atld Ho : d : 1 a|e seParately tested against tlrrelevant two-sided alternatives and neither null is rejected. Clearly thenwoulcl not tlake sense to say that'H1) : p :0 5 is accepted'and'H6 : p =is accepted', since the true (but ur-rknown) value of ,4 cannot be botlt 0ancl 1. So, to suntnarise, the trull hypothesis is either rejected or ntrejected on the basis of the available evideuce.

2.9.5 The confidence interval approach to hypothesis testing (box 2'6)To give an example of its usage, one n'rigirt estimate a palameter, say p'be 0.93, and a '95% confidence interval'to be (0.77' 1.09). This means th'in many repeated samples, 95% of the time, the trtte value of p will 1contained rvithin this interual. confldence intervals are almost invariabestimated in a fwo-sided folm, although in theory a one-sided inten'can be constructed. constructing a 95% confidence inten'al is equivalelto using the 5% level in a test of sigr-rificance'

2.9.6 The test of srgnlflcance and confidence interval approaches alwaysgive the same conc/usionUncier the test of significrllce rpirt'oach' the nttil hypothcsis thnt p =u,ill not bc rejected if the test statisfic lies rvithirl tltc trorl'tejcctiott t'r'gtc'i.c. if the foilorving cor.rditior.r holds

,4 * rl.-t,,tr '-

-.*t,,',)Lll )


18/31

60 lntroductory F,conometrics fot' Finance

(1) Calculate A, B and SE(d), SEG) as before.(2) Choose a significance rever, c (again the convention is 5%). This is equivarent tochoosing a (l _ a)tjoo/" conf,dence intervali.e. 57o significance level = 957o confidence interyal(3) Use the r{ables to find the approprlate critjcal value, whjch will again have Z*2degrees of freedom,(4) The confidence intervat for B js given by(B - Gir. stfl, B +/-ir. s6(l))

Note that a centre dot (.) is sometimes used jnstead of a cross (x) to denote whentwo quantities are multiplied together.(5) Perform the test: if the hypothesised varue of p (i.e. B.) rres oulside the confrdenceintervar, then reject the nuil hypothesis that B = B-, otherwise do not reiect the nuil.

Rearranging, the null l.rypothesis would not be rejectecl if-r.,.;,. SErf,t = i * f' 1- | ,,t. SErf tr.e. one would not reject ifB - t,,,,. Sf(p) S f. = t,+ t,,.it. SE(B)

But this is just the rure for non-rejection under. trre confidence intervalapproach. so it will arways be rhe case that, for a given signiflcance revel,the test of significance and confidence intervar approaches will proviclethe same conclusion by construction. one testing approach is sir.nply a'algebraic rearrangement of the other.Example 2.+

Given the regression results above,r;:zO.l*0.5091.r,

( I 4.38) (0.256 I ) (2.3'1)using both the test ofsignificance and confidence interval approaches, testthe h5pothesis that p : 1 against a two-sided alternative. This hypothesismight be of interest, for a unit coefficient on the explanatory variabreimplies a 1:1 relationship berween movenrents in.r.and .,-'o'e,'ents i' r,.The null and alter.native hlpotheses are respectively:

H11 :fl-1Ht:fi*l

A brief ottetlittu of fhc cia.ssicol lineor reg,rcssictn ntodel 6i

Iest of srgnlfcance appraachi p.tt.sr slol =tzS E(fr)0.5091 -l ,.,,- -'.''10.2-56 |

Find t,.,.,, = ll rr = *2.086Do not reject ll0 since test statistic

lies within non-rejection reglon

Co nfi d e n ce i nte r v a I a pproa ch

Find t,,.;, = ho.sq : *2.086p L t,,t, S E( B)= 0.5091 i 2.086 0.2561= (-0.025r,1.0.133)Do not reject Ho since I lieswithin the confidence interval

'fhe rcsults of the test accor.cling to each approach are shown in box 2.'A couple of coniltlents are in ot.der.. First, the critical value from thl-distribution that is r.eqr,rir.ed is for 20 degrees of fi'eedom and at the 5'level. This ntealts that 5% of the total clistribution rvill be in the |eje,tion region, and since this is a rwo-sided test, 2.5% of the distributioris requirecl to be containecl in each tail. From the symmerly of theciistribution aroltnd zero, the critical values in the upper and lower tawill be equal in rnagnitude, but oPposite in sign, as shown in figure 2 1rwhat if instead the researcher wanted to test H0:d = 0 or Ho:f = -In order to test these hypotheses using the test of significance approacJthe test statistic would have to be reconstructed in each case, although tl:cr.itical value would be the same. On the other hand, no additional u'otwould be required if the confldence interval approach had been adopter

@EilCritical values andrejection regions for

2,SYorejeclion region

\2 'SYorejection Iegion

/(.r)

95% non-reiection region

-2.086 +2.086


19/31

52 lntroductory Econometrics for Financesince it effectively permits the testing of an infinite number of hypotheses.So for example, suppose that the researcher wanted to tesr

H1; :f=0versus

H1 :ft'0and

Hr:lJ=2versus

H1 : fi lZIn the first case, the null hypothesis (that f = 0) would nor be rejectedstnce 0 lies within the 95% confidence interval. By the same argument, thesecond null hypothesis (that f =2) would be rejected since 2 lies outsidethe estimated conf,dence interval.On the other hand, note that this book has so far considered only tl.relesults under a5% size of test. In marginal cases (e,g. H6 :19 = 1, where thetest statistic and critical value are close together), a completely differentanswer may arise if a different size of test was used. This is where the testof significance approach is preferable to the construction of a confidenceinterval.

For example, supPose that now a 70% size of test is usecl for the nuliirypothesis given in example 2.4. Using the test of significance approach,te.tr .rtari.tric: t {SE(P)

0.5091 - I= lfGi_ : _1.e17as above' The only thing that changes is the critical r-value. At the 10%level (so that 5% of the total distribution is placed in each of tl.re tailsfor this fwo.sided test), the required critical value is t211111"y, = *1.72-5. Sollow as the test statistic lies in the rejection region, H6 woulcl be r.ejectecl.In order to use a 109'o test under the confidence interval appr.oach, theinterual itself would have to have been re-estilnated since the cntical valueis entbedded in the calcr-rlation of the confidence interval.So the test ofsignificance and confldence interval approaches both l-ravetheir relative merits. The testing of a number of clifferent l1lpotSeses iseaster under the confidence interval approach, while a copsiclelation of

A brief ovct,rtit:ru ctf the classical lInacn" rcgrcssion ntodelfrthe effect of tlie size of the tcst on the conclrtsiou is easier to addres

r-rndet' the tcst of significatrce appl'oach'caution shoulcl rher.efor-e bc usccl when placing emphasis on or makitrclecisions i11 the coutext of ntarginal cases (i.c. in cases where the nuis only.just lejcctcd or lrot rc.iectccl). In this sitrtation, the appropIirtconclusion to dt';tw is th:lt thc rcsults lit-c tnat'gitlal rnd that no strotlg irlerencc caD lre nt:rclc ouc wilv Or-thc othcr. A thoroLlgh eu'rpi|ical anaiysshoulcl involve concluctiDg a scnsitir.'ity:rrlalysis otl the resttlts to detcntine whctitcr.usir-rg a cliflircrlt sizc of tcst alters the conclusiorls ltwofth st;ttiltg again that it is convcntronal to considel' sizes of test of 10'5,% ancl 1{x,. If thc conclusion (i.c. 'r'e.iect'or 'clo trot |eject') is robnst tchanges ir-r the sizc of the tcst, then One can be tnole confident that tl,conclusions a|e approp|i;lte. If the oLltcollle oF tlre test is qualitatively:tered when thc sizc of the test is n'rodifiecl, the cotrclusion n-ltlst be tlt,there is t'to conclusion olle way ol' the other!It is also \vor.th uoting that if a given r-rr.rl1 hypothesis is rejected using1% significance level, it will also auton.ratically be rejected at the 5% Iev(so that thele is no need to acttlally state the latter' Dougherry (199p. 100), gives the anaiogy of a high jumper. If the high jumper cau cle2 metres, it is obvious that the juntper could also clear 1.5 netles. Tl1% signiflcance level is a l-righer hurdle than the 5% sigDificance levtSimilarly, if the null is Dot rejected at the 5% level of significance, lt wrautonratically not be rejectecl at any strongel level of significance (e.g. 1')In this case, if the jumper carlnot clear 1,5 metres, there is no way s/1will be able to clear 2 ll-retres.

2.9.7 Some more terminologYIf the null l.rypothesis is rejected at the 5% level, it would be said that tlresult of the test is 'statistically significant'. If the nr-rll l'rypothesis is n'rejected, itwould be said that the result of the test is'not significant" 'thatitis.insignificant''Finally,ifthenullh}'pothesisisrejectedattj1% level, the result is terned 'liighly statistically significant"Note that a statistically significant result may be of no practical snificance. For example, if the estin'rated beta for a stocl< under a CAI'regression is 1.05, and a null hypothesis that fJ:1is rejected' the I'esrwill be statistically significant. But it may be the case that a slightly highbcta will r.nake no differcnce to an investor's ciroice as to whethcl to i)the stocl( or Ilot. In that case, one rvould say that the lesult of the t'was strtistically significant but financially ol' plactically insignificant.


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lntroductory Ecotlunetrics fot I:inanceTable 2.3 Crassifying hypothesis testing errors and correcr conclusrons

RealiryHs i! true H6 is false

SignificantResult of test (reject H1y1Insignifican t(do nor reject Ho)Typelcrlttr':a ,// -- -_I lyl)('Il (llol .- I

2'9'8 Crassifyingthe errors that can be made using hypothe.srs testsH1, is usually rejected if the test statistic is statistically sienifica't at achoser significance rever. Trrere are rr,vo possible errol.s that co'rd be urade:(1) Rejecting H11 when it was reaily true; this is called atylte I error.(2) Not 'ejecting Hrr whe' it was in ract false; this is cairecr a 4,pe II crror.Tlie possible scenarios can be summar-isec-l in table 2.3.T'e probabiliry of a type I error is just cr, the siguifica'ce level or srzeoftest chosen. To see this, recall what is meant by,significance'at the s%level: it is only 5% likely that a result as or lrore extrerne as this coulcihave occurred purely by chance. Or, to put this another way, it is only 5%likely that this null would be rejected when it was in fact true.Note that there is no chance for a free lunch (i.e. a costless gain) here!\Mrat happe's if ti-re size of the test is reduced (e.g. fi-om a 5% fest to a

19/o test)? The chances of making a rype i error wourd be reduced. .. but sou'ould the probabiliry that the null hypothesis r.r'ould be r.ejecteci at ail,so i'creasing the probabiriry of a rype II error'. Trre two cornpeting effecrsof reducing the size of the test can be shown rn box 2.g.So there always exists, therefore, a dir-ect trade-off between wDe rand rype II errors wrren crroosing a significance lever. T)re oury..v to

Less likely Lowerto falsely +chance ofReduce size+More strict +Reject null/re)ec| type lerrorof test (e.g. criterion for hypothesjs\5% ta I%) rejection less often N4ore likely to Higherincorrectly +chance ofnot reject type ll error

A lrritf trvLrritrv n/thc 'Lissi.rrl Iincar rLyrcssion nodtl*

reduce the chances of both is to incrcase the sample size or to seleia samplc with more valirtiotr, thus inclcasing the antount of infornr'tion upon which the lesults of thc hypothcsis test are based. In practicrllp to a ccIt;rit'r levcl, typc Iclrors:rre usr"rally consitlerecl t.note seriotirncl hence a stuall sizc'ol lcst is usually choscn (5% or 1% are the nlo'conl lllon ).Thc plobability of a rypc I ct'r-ot' js tJrc plobability of incot'rcctly rejcting a collcct nrrll hypothcsis, which is rlso the size of tl're test. Alloth(inrportrnt pie'ce of tet'nrittolctgy in thrs ;ilca is the pot't,er 0f .l lc.sl. The pow,ofa test is dcfincd as thc plobability of (apploprirtely) r'ejecting atr itrcc,lect null hypothcsis. The powcr of the test is also eqr-ral to one lttlltus tllprobabiliry of a type II c'r't'or'.An optinral tcst wouid be otre with an actual test size that ttlatchtthe nominal size ancl which hacl as high a power as possible. Such a tewould imply, fbl ex:rntple. thet r-rsi:rg a 5'./, significatrce level wouid resr.iin the nr.rll be ing lcjected cxactly 5",' of the tin-re by chattce alone, atrthat an iucoLLect nr-ri1 hypothcsis rn'ould be rejected close to 100% of thtime.

2.LO A special type of hypothesis test: the l-ratioRecall that the folnrula under a test ofsignilicance approach to hypothesresting using a r-test for thc slope parameter was

fi-f'.le.\l slalt.rlt(: '-......'.-..:Tsr(p) (2.3with the obvious acl.tustlnents to test a hypothesis about the intercept.the test is

H1y:f:0H1 :Bl0

i.e. a test that thc population parameter is zero r1;aiust a fwo-sided alttnative, tl-ris is knor.vn as a r-ratio tcst. Since p- :0, the cxpressiou in (2.3collapses to

11l(.\l.\l(1lt\lt( :-.s[(n ) 12.:l

Thus the ratio of the coefficient to lts stalldard cn'or'. given by thcxplcssion, is l


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Example 2.5Introductory Econometrics for Finance

Suppose that we have calculated the estimates for the inter.cept ancl theslope (1.10 and -19.88 respectively) and tl-reir corresponding standard er-rors (1.35 and 1.98 respectively). The r-r'atios associated with each of theintercept and slope coefficients rvould be given byn7Cocfficient I.l0 -t9,88.tt I..15 t.98r-r'atio 0,8i - 10.0JNote that if a coefficient is negative, its r-ratio will also be 'egarive. Inorder to test (separately) tl're null hypotl.reses that tr : 0 and I = 0, thetest statistics would be compared rvith the appropriate critical value frorna l-distribution. In this case, the nuntber of degrees of fi.eedorn, given byf - I, is equal to 15-3=12. The 5% cr.itical value for this fwo-sided test(renember, 25% in each tail for a 5% rest) is 2.179, while rhe 1% rwo_sidedcritical value (0.5% in each tail) is 3.055. civen these /-ratios and criticalvalues, would the following null hypotheses be rejected?H0:cr:0?Hs: B-0? (No)( les)

If H6 is rejected, it would be said that the test statistic is signr/icnnf. If thevariable is not'signiflcant', it means that while the estinated value of thecoefllcient is not exactly zero (e,g. 1,10 in the exantple above), the coeffl-cient is indistinguishable statistically from zero. If a zelo were placed i'the fltted equation instead of the estimated value, this would lnean thatwhatever happened to the value of that explanatory variable, the depen-dent variable would be unaffected, This would then be taken to nean tltatthe variable is not helping to explain variations in ,r, and that it couldtherefore be removed from the regression equation. For example, if the r-ratio associated with.r had been -1.04 rather than -10.04 (assuning thatthe standard error stayed the same), the variable would be classed as in-significant (i,e. not statistically different from zero), The only insignificantterm in the above regression is the intercept. There are gooci statisticalreasons for always retaining the constant, even if it is not significant: seechapter 4.It is worth noting that, for degrees of freedom g.eater thar al.or.ircl 2s,the 5% rwo'sided critical value is apploximately *2. so, as a rule of thunrb(i.e. a rough guidel, the null hypothesis would be r.ejected if the r-statisticexceeds 2 in absointe value.

F'"F'.FII

I

A brief oveniew oJ tlte classical lincar rtgrcssiort modelSome aurhors placc the r-ratils itt patctrthcscs below the correspondin;

coefficient estimates rather than tl're standard errors. One thus needs t(check wl'rich convention is being r.tsed itr each particular application, anralso to state this clearly whcn pt'esenting estirlatior-r I'esults.

Thele will now follow two finance casc studies that involve only threstimation of bivariate lineal reglession models and the constructlon an(interpretatiorr of r-rlttios,

2,tl An example of the use of a simple l-test to test a theory infinance: can US mutual funds beat the market?Jensen (1968) was the first to systelnatically test the peLforutattce of nutt'llrfunds, and in particular exatttine whether any'beat the market', He useta sample of anuual returlls on the poltfolios of 115 nrutuai frtnds fron1945-64. Each of the 115 fi-rnds was sub.jected to a seParate OLS time sertercgression of the forrtt

Rl, - R1 = aj * lJi(R,,, - Rx) * ttlrwhere Ru is the return on pol'tfolio / at time t, R7i is the return onrisk-free proxy (a l-year government bond), R,,,, is the return on a ]nalket portfolio pl'ox!, rr;1 is an error term, and a1, p.i arc parameters to b'estimated. The quantity of interest is the significance of (IJ, since thrparameter defines whether the fund outPerForms or underperforms thmarket index. Thus the null hypothesis is given by: H6 : ai :0. A positiv'and significant a, for a given fund would suggest that the fund is ablto earn significant abnormal retufns in excess of the market-required r'turn fol a fund of this given riskiness' This coefficient has becone knowras'Jensen's alpha'. Sorne su[Ilrlary statistics across the 115 funds for thestinated regression results for (2.52) are given in table 2.4.

Table 2.4 Summary statistics for the estimated regression results for (2.52)Extremal values

Mean value ' Median value Minimum Maximumri, -0.011l! 0.840Sample size 17

-0.0090,84819

- 0.0800.21910

0.0581.405

20Sorrric: Jcnsen (1968). RePrinted with the pcrnrissioll of Blaclcvc-ll Ptrblishc'ts

of the classicul li:iear rcgrcssiott rnodal


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68

ESEftNFreq uencydistribution oft-ratios of mutualur ru orPr rd> \Brus5of transact onscosts) source:Jensen (1968),Reprinted with tlrepermission ofBlackwell Pub ishers

EEilFreq uencydlstribution oft-ratios of nrutualfund alphas (net oft/dl5duL u r> LU>t5 jSource: Jer.rsen(1968). Reprlntedwrth the pernTissionof BlackwePublis he rs

Intrldu.t()ry Econometrics t'ot' [:inance

j-i

. {)

li

-lt-ratio

o l:

10

.r -l -: I 0 I : l,-ratioAs tabie 2.4 shows, the average (defined as eithel the nrean or the nre-dian) fund was unable to 'beat the market', recording a negative alphain both cases. There weLe, however', some funds that did ntanage to per-fornl significantly better than expected given their level of risk, with tl.rebest fund of all yielding an alpha of 0.058. htterestingly, the aver.age fundhad a beta estimate of ar-ound 0.85, indicating that, in the CAPM context,most filnds wcre iess lislq'than the malket index. Ti-ris resuit utay beattributable to the fr.rnds investing predon-rinantly in (mrtule) bJue chip

stocl


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70 Introductory Econometrics lar FinanceTable 2.6 c,APlr4 regression resuits ior unit trust returns, Januarv 1,979-Mav 2000

Eslimates of Mean Minimum Maximum Medianu(Vc)Pt-Iatio On a

*0.020.91*0.07

- 0.540.56 ,0.030.91- 0.2s0.331.093.11

Ill?[EEfilPerformance of UKunrt !rusts.197 9-2000 30002500200015001000500

rud*s*JoP*Sg*q**tr"C.*tr"*f,"rJ{,*.*rygf*tr"C*Jc],.t*wide variation in the performances of the funcls. The worst-perforrringfund yields an average return of 0.6% per month over the 20-year- pe_riod, while the best would give 1.4% per nonth. This variabiiiry is furtherdemonstrated in figure 2.19, which plots over tin.re the value of 100 in-vested in each of the funds in January 1979.A regression of the form (2.52) is applied to the UI( data, and the sun_mary results presented in table 2,6. A number of features of the regressionresults are worthy of further comment. First, rnost of the funds have esti-mated betas less than one again, perhaps suggesting that the fund man-agers have historically been risk-averse or in'esting disproportionately inblue chip conpanies in mature sectors. second, gross oftransactions costs,'ine funds of the sample of 76 were able to significantly outperforrn themarket by ploviding a significant positive alpha, while seven funcls yieldedsignificalrt negatirre alphas, The average fund (r.vhere 'average' js measuredusing either the mean ol the median) is not able to ear. any excess retul.nover the required rate given its level of risk.

A lu'icf ot,tr-victv oJ thc classical lirtcor rcgrc.s.siott rnodcl

(1) Ihatthe'overreacttoneffect'isjustanothermanifestationofthe'sizeeffect',Thesizeeffect is the tendency of small firms to generate on average, superior returns to largefirms. The argument would follow that the losers were small firms and that thesesmall firms would subsequently outperform the large firms, DeBondt and Thaler didnot believe this a sufficient explanation, but Zarowin (1990) found that allowing forfirm size did reduce the subsequent return on the losers'(2\ Thatthereversalsoffortunereflectchangesinequilibriumrequiredreturns.Thelosersare argued to be likely to have considerably higher CAPM betas, reflecting investors'perceptions that they are more risl{y. Of course, betas can change over time, and asubstantial fall in the flrms' share prices (forthe losers) would lead to a rise in theirleverage ratios, leading in all likelihood to an increase in their perceived risklnessTherefore, the required rate of return on the losers will be larger, and their ex postperformance better. Ball and Kothari (1989) find the cAPM betas of losers to beconsiderably higher than those of winners.

2.13 The overreaction hypothesis and the UK stock market2.73.7 Motivation

Trvo studies by DeBondt and Thaler'(1985, 1937) showed that stocks exp(liencing a poor perfornrance over a 3-s-year period subsequently tend 1outperform stocks that had previously perforrred lelatively well. This in'plies that, on average, stocks which are 'losers'in terms of their returnsubsequently become 'winners', and vice versa. This chapter now exallines a paper by Clare and Thomas (1995) that conducts a sinrilar studusing montl.rly UK stock returns from January 1955 to 1990 (36 years) orall firms traded on the London Stock exchange.

This phenonrenon seenls at first blush to be inconsistent with the ef1cient marl


24/31

for

Pottfolio RankingPortfolio 1 Best pedorming 20% of lirmsPortfolio 2 Next 20%Porttolio 3 Next 20%Portfolio 4 Next 20%Portfolio 5 Worst pedorming 2O% of firms

Estimate R; for year 1Monitor portfolio s f or year 2Estimate i, for year 3:Monitor portfolios for year 36

Then the a\/erage monthly return over each stock i for frre nrst 12-, z4-, ol36-month peliod is calcr:lated;(2.s4)

txp - '\-l"r -..- / vtt|t:1

The stoci(s are then ranl


25/31

winners at the t\,vo'veai l-rorizon, and 1.55% higher return at the three-yelihcrizon. Recall that the estimated value of tl-re coefficient in a regr.essionofa variable on a constant only is equal to the average value ofthat var.!-able. It can also be seeu that the estimated coefficients on the coltstanttern'rs for each horizon are exactly equal to thc differ.ences between tirelcturns of the losers and the wiurtcrs. This coefficient is statistically siglif-icant at the two-year horizon, and marginally significant at the three-yearhorizon.Iu the second test regression, I represents the differ.ence behveen themarket betas of the winuer and loser portfolios. None of t5e beta coeffi-cient estimates a1'e even close to being significant, ancl the ilclusiol ofthe risk term tnakes virtttally no difference to the coefficielt values 6rsignificances of the intercept terns.Removal of the January returns from the samples reduces the subsc-quent degree of overperformance of the loser portfolios, and the signiFicances of the ri1 tel'lrrs is somewhat reducecl. It is concluded, thelefore,that only a Part of the overreaction phenomenon occuls in Ja11aly. Clareand Tiromas then proceed to examine whetlter the overreaction effect isrelated to firm size, although the results are not presented here.2.13.3 Conclusions

The main conclusions from Clare and Thomas'study are:(1) There appears to be evidence of overreactions in ul( stock returns. asfound in previous US studies.(2) These over-reactions are unrelated to the CAPM beta.(3) Losers that subsequently become winners tend to be srnar, so thatmost of the overreaction in the ul( can be attributed to the size effect.2.I4 The exact significance level

The exact signiflcance level is also commonly known as the p-value. Itgives the ntarginal significance lelel where one would be indiffer.ent befweenrejecting and not rejecting the null hypothesis. Ifthe test statistic is ,large'in absolute value, the p-value will be small, and vice versa. Fo' example,consider a test statistic tl'rat is distributed as a /(,2 ar.rd takes a valne of 1..47.wo'ld the null hypothesis be rejected? It would depend on the size of thelest. Now, suppose that the p-vaiue for this test is calc'rated to be 0.12:*. Is the null lejected at the 5% level? Nos Is the nr.rll rejected at the i07; level? Noe Is the nr.rll rejected at the 20% level? ]e_s

II

A brief ovcrvitt^,t o.f the classical lintar rL:y,ression ntorlel$

Table 2.8 Part of the EViews regression output revisitedCoefficient Std. Error t-Statistic Prob.c 0.363302RFiJTURIiS O, 123860 0.4443690. r 33790 0.817569 0,41670.L)257 81. 0.3 581

In flact, the uLrll would hlrve becn lc'jccted at the 12% Ievel or hightTo see this, consider conducting a selies of tests with size 0.1%,0.2'0.3y", 0.4%, . .. 1%, . .., 5%, .. 10%, . . . Eventually, the critical value and te:statistic will n-reet and this will be the p"value. p-values ale alnrost ahvalprovided aLrtonlatically by softrva|e packages. Note horv usefttl they ar,They provicle all of the information recluired to conduct a hypothesis te:withor.rt requiring ofthe IesealcheI the need to calculate a test statistic c'to find a critical value from a t;rble - both ofthese steps have already beet;rken by rhe p.tcl


26/31

i' the seco'd. The fact that these test statistics ar.e both very sn-ralr is in_dicative that neithcr ofthese'ut hypotheses is lir


27/31

se.r'ies. For exampleX2=y12 creares a new variable called X2 that is lialfofxXSq=X^2 creates a new va|iable XSqthat is X squaredLX:LOG(X) creates a new varirble LX that is thelog of XIACX=X(- l) creates a new var-iable LAGX containing Xlagged by one per.iodLACX2=X(*2) creates a new variable LAGX2 containing Xlagged by two periods

Other functions include:d(X) flrst difference of Xd(X.n) rrth order difference of Xdlog1x) first difference of the logarithm of Xdlog(X,n) lth order differ-ence of the logarithnr of Xabs(X) absoiute valr-te of X

If, in the transformation, the new series is given the sanle narne as tl.reold series, then the old series will be overwritten, Note that the retllrr-isfor the S&P index could have been constructed using a sin-rpler conmaudin the 'Genr'window such asRSANDP= 100- DLOG(SANDP)as we used in chapter 1. Before we can transform the returns iDto ex-cess returns, we need to be slightty careful because the stock returnsare monthly, but the Treasury bill yields are annualised. We could runthe whole analysis using monthly data or using annualised data and itshould not matter which rve use, but the two series rnust be rneasuredconsistently. So, to turn the T-bill yields into monthly figures and to writeover the original series, press the Genr button again and typeUSTB3lr{=USTB3M/12Now to compute the excess returns, click Genr again ancl typeERSANDP=RSANDP-USTB3 l\{where 'ERSANDP'will be used to denote the excess r.eturns, so that theoriginal raw returns series will remain in the workfile. Tl-re Ford returnscan simiiarly be transformed into a set of excess returns.Now tirat the excess returns have been obtained for. the fwo series.before rr,rnning the regression, plot the data to examine visually whethe|

the series appear to trtove togcthct'. l'o do this, cl'cate a new object b'clicking orl the Object/New Object ncnLr on thc mcllu bar-. Select Graphprovide a nalte (call thc grapli Graphl) and thcrt in the new windovprovide the nanres of the scrics to l)lot. Itt this :rew willdow typeI]RSANDP I]RFORDThen prcss OI( and scLce nshot 2.4 will uppear.

- ERSANDP - ERFORD

This is a time-series plot of the fivo variables, but a scatter plot may Lrmore informative. To examiue a scatter plot, Click Options, choose thType tab, then select Scatte[ from the list and click OI(. There appears tbe a weak association befween ERF-|AS and ERFORD. Close tl're windorv tthe graph and return to the workfile window.To estimate the CAPM eqr-ration, click on Object/New Objects' In thnerv window, select Equation and nante the object CAPM, Click otl Olln thc window. spccifu the reglessiott eqr.ration. The lcg:'cssiott cqrt.trictakes the form

(R-.;ur - r'1), : tt -l ll(Rv - r'1)1 * tr1

l1"I/

,1,10*!,,+-ui11.,]\

lntrodLtctory F.cotlometrics fot' Iiinance A Ltrtef otet'vitt't ry' tlrc ciassirril littcrl' tt'grcsston ttloricl 81


28/31

Since the data have already been transfolnrecl to obtain the excess returns,in order to specifo this regression equ;rtion, type in the equation windowERFORD C ERSANDPTo Lrse all the observatior-rs in the sanrple and to estirnate fhe regressionusing LS - Leasf Squarcs (NLS and ARMA), click on OK. The lesu]ts scLeenappears as in the followiug table. Mal


29/31

The next step is to rearrange (2A.2) and (2A.3) in order to obtain expres_sions for d and 1i. From (2A.2)T,r',-a-flt,1=o eA.4)

Expandi'g the parentheses and .ecallirg that the sun runs fron 1 to rso that there rvill be f ternrs in d\-'.-r;-i\--,/_: tu-t)/_\1=tl

But f .r', : ha and Ir., = It, so it is possible to write (2A.5) as-;-.\'- te _ /f.\. = il (2A.6;of

s2A.2 Derivation of the OLS standard error estimators for the intercept andslope in the bivariate case

Recall tltat the varirncc of the t';rtltlonl vat'irble ri can bc wt'itten rsvar-(r?) = t'.ltt -' I:([t)))

and since the OLS cstitrtatot'is uttbirscdv;rr'(n) : I:(u - u)J {24.1(

By sinrilal algulttcltts, tltc var.iattce of tl-re slope estittratot'cru be wt'tttcas

var{ft: E(i-fllt (2A.1:wolking first with (2A'17)' lcflliirrg fJ with tllc folnlr-tla for it given l'the OLS c'stillator

- lfr.t,-.irrr, ir \'r:rrrFr=L(--l) (2A.t'' 1-"' Ir IReplacing.r,, with u * f,.t1{ tr,, atrd r"eplacing r with d + lli in (2A 18)

(24.1'

(24.2l

\24.2

(2,4.s)

(2A.t)

(24.8)

i-d-p.i:0From (2A.3)

I t,(.lr - ri - 1ir,) = oFrom (2A.7)

A-a:)._pirSubstituting into (2A.8) for r? fiom (2A.9)

f.r,.r', - i'+ i9.i - d',.r = 0I.,1, - r- f .., + di f r, - d )-ri : o L.tTr,,, - Tr\' + BT.t:- lf"f = o

Rearranging for 79,B(r.tt- f.:)=ril-\--,,.\Lt| /2^t.'1

Dividing both sides of (2A.13) Uy (ri2 * f rj) gives\-",,-r-l=Fl_r, Lincl d=i-/lL/, l

(24.e)

(2,4.10)(2/'.11,)

12A.12l

(24.13)

/\-r.r,-.tt(rr r Il:,.tt, -,r -d.r, \-v;rrtpr= u[ -=.: J

-p) ij;\.i'\ L{.\,-\)- ft.r', -.tlrCancelling cr attd tntrltiplyirtg t)rc 1;rst F relm in t24.19) by-1-\'\t - \ |

.. ,.1 I,. - trf r, -t- rt, - 51.t,- lLrt, .it:\' ,r^ r,r ',rt il) -"''' - "\ f {r, * r)r IRearrangtng

"t''u(p) : t\Now the f terms in (2A.22) will cattcel to give

I 5- rr,t r, - .i t\:,,,'.ri,_,,(E*; )(24.14) \24.2

lntroLlucLory Econonutrics Jor Finance A bricf ctveniew rrf [/rc clossicol lineor rcgrassfun ntrfiel a)


30/31

The denornirator of l2A.24l can be tal


31/31

(c) Why are the squares of the vertical distances taken rather than theabsolute values?2. Exprain, with the use of equations, the difference between the sampreregression function and the population regression functron.3. What is an estimator? rs the oLS estimator superior to ail otherestimators? Why or why not?4. What five assunrptions are usually rnade about the unobservable errorterms In the classical linear regression model (CLRM)? Briefly explainthe meaning of each. Why are these assurlptions made?5. Which of the following models can be estimated (followrng a suitablerearrangenler'rt if necessary) using ordinary least squares (oLS), whereX, :, Z are variables and ct, p, ), are parameters to be estimated?(Htnt: the nrodels need to be linear in the paranreters.)

market, at the 5% tevel. Wriie down the null and alternative hypOthesisWhat do you conclude? Are the analyst's claims empirically verified?7. The analyst also tells you that shares in Chris Mining PLC have nosystematic risk, in other words that the returns on its shares arecompletely unrelatecl to movements in the market. The value of betaand its standard error are calculated to be 0,214 and 0.186'respectively. The model is estimated over 38 quarterly observatl0ns'Write down the null and alternative hypotheses Test this nulihypothesis against a two-sided alternative.

8. Forrn and interpret a 95% and a 99% confidence Interval for beta usintrthe figures given in questiorr 7.9. Are hypotheses tested concerning the actual vaiues of the coefficients(i.e. 0) or their estimated values (i.e. /l r and why?10. Using EViews, select one of the other stock series from the 'capm wklfiIeandestimateaCAPMbetaforthatstock.Testthenu||hypothesisthat the true beta ls one and also test the null hypothesis that the tru(alpha (intercept) is zero. What are your conclusions?

)1=t}+f1.\',-.',:-t : g" x!' e"'.ri:o+py.r, 1-,,,ln(J/):a*fht(.tr)*u,-)i:a+8.r,:,-u1

6. The capital asset pricjng model (CApM) can be written asE(R,): Rr + piLE(R.) - Ril

(2.57)(2.58)(2.5e)(2.60)(2.6r)

(2.62)usrng the standard notation.

The first step in using the cApM is to estimate the stock's beta usingthe market model. The market model can be written asR,, = di * B; R,,,, + u,, (2.63)

where R;, is the excess return for security i at time r, R,,,, is the excessreturn on a proxy for the market portfolio at time t , and u , is an iidrandom disturbance ternr. The cofficient beta in this case is arso theCAPM beta for security 1,suppose that you had estimated (2.63) and found that the estimatedvalue of beta for a stock, f was 7.L47. The standard error associatedwith this coefficient SE{p) is estjlnated to be O.O548.A city analyst has told you that this security closely follows themarket, but that it is no more risky, on average, than the nrarket. Thiscan be tested by the nuli hypotheses that the value of ileta is one. Themodel is estimated over 62 daily observations. Test this hypothesisagainst a one-sided aiternative that the security is more risky than the

chapter 2 economie monetara

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