Macroeconometrics. Christophe BOUCHER. Session 2 A brief overview of the classical linear regression model 1

Transcription

1 Macroeconomerics Chrisophe BOUCHER Session 2 A brief overview of he classical linear regression model 1

2 Regression Regression is probably he single mos imporan ool a he economerician s disposal. Bu wha is regression analysis? I is concerned wih describing and evaluaing he relaionship beween a given variable (usually called he dependen variable) and one or more oher variables (usually known as he independen variable(s)).

3 Some Noaion Denoe he dependen variable by y and he independen variable(s) by x 1, x 2,..., x k where here are k independen variables. Some alernaive names for he y and x variables: y x dependen variable independen variables regressand regressors effec variable causal variables explained variable explanaory variable Noe ha here can be many x variables bu we will limi ourselves o he case where here is only one x variable o sar wih. In our se-up, here is only one y variable.

4 Regression is differen from Correlaion If we say y and x are correlaed, i means ha we are reaing y and x in a compleely symmerical way. In regression, we rea he dependen variable (y) and he independen variable(s) (x s) very differenly. The y variable is assumed o be random or sochasic in some way, i.e. o have a probabiliy disribuion. The x variables are, however, assumed o have fixed ( non-sochasic ) values in repeaed samples.

5 Simple Regression For simpliciy, say k=1. This is he siuaion where y depends on only one x variable. Examples of he kind of relaionship ha may be of ineres include: How asse reurns vary wih heir level of marke risk Measuring he long-erm relaionship beween sock prices and dividends. Consrucing an opimal hedge raio Evaluae he relaionship beween invesmen rae and saving rae of counries Ec.

6 Simple Regression: An Example Suppose ha we have he following daa on he excess reurns on a fund manager s porfolio ( fund XXX ) ogeher wih he excess reurns on a marke index: Excess reurn on marke index Year, Excess reurn = r XXX, rf = rm - rf We have some inuiion ha he bea on his fund is posiive, and we herefore wan o find wheher here appears o be a relaionship beween x and y given he daa ha we have. The firs sage would be o form a scaer plo of he wo variables.

7 Graph (Scaer Diagram) Excess reurn on fund XXX Excess reurn on marke porfolio

8 Finding a Line of Bes Fi We can use he general equaion for a sraigh line, y=a+bx o ge he line ha bes fis he daa. However, his equaion (y=a+bx) is compleely deerminisic. Is his realisic? No. So wha we do is o add a random disurbance erm, u ino he equaion. where = 1,2,3,4,5 y = + x + u

9 Why do we include a Disurbance erm? The disurbance erm can capure a number of feaures: - We always leave ou some deerminans of y - There may be errors in he measuremen of y ha canno be modelled. - Random ouside influences on y which we canno model

10 Deermining he Regression Coefficiens So how do we deermine wha and are? Choose and so ha he (verical) disances from he daa poins o he fied lines are minimised (so ha he line fis he daa as closely as possible): y x

11 Ordinary Leas Squares The mos common mehod used o fi a line o he daa is known as OLS (ordinary leas squares). Wha we acually do is ake each disance and square i (i.e. ake he area of each of he squares in he diagram) and minimise he oal sum of he squares (hence leas squares). Tighening up he noaion, le y denoe he acual daa poin ŷ û denoe he fied value from he regression line denoe he residual, y - ŷ

12 Acual and Fied Value y y i û i ŷ i xi x

13 How OLS Works So min. uˆ 1 uˆ 2 uˆ 3 uˆ 4 uˆ 5, or minimise uˆ. This is known 1 as he residual sum of squares. 5 Bu wha was? I was he difference beween he acual poin and he line, y -. ŷ û 2 So minimising y yˆ is equivalen o minimising uˆ 2 wih respec o and.

14 Deriving he OLS Esimaor Bu, so le Wan o minimise L wih respec o (w.r..) and, so differeniae L w.r.. and (1) (2) From (1), Bu and. x y ˆ ˆ ˆ y x L 0 ) ˆ ˆ ( 2 ˆ x y x L 0 ) ˆ ˆ ( 2 ˆ 0 ˆ ˆ 0 ) ˆ ˆ ( x T y x y y T y x T x i x y y y L 2 2 ) ˆ ˆ ( ) ˆ (

15 Deriving he OLS Esimaor (con d) So we can wrie or (3) From (2), (4) From (3), (5) Subsiue ino (4) for from (5), 0 ˆ ˆ x y x y x 0 ) ˆ ˆ ( y x ˆ ˆ x Tx Tyx y x x x x x y y x x x y y x 0 ˆ ˆ 0 ˆ ˆ 0 ) ˆ ˆ ( ˆ ˆ x T T y T

16 Deriving he OLS Esimaor (con d) Rearranging for, ˆ 2 2 (T x ) x Tyx x y So overall we have ˆ x y Txy andˆ y 2 2 x Tx ˆ x This mehod of finding he opimum is known as ordinary leas squares.

17 Wha do We Use and For? In he CAPM example used above, plugging he 5 observaions in o make up he formulae given above would lead o he esimaes = and = We would wrie he fied line as: yˆ x Quesion: If an analys ells you ha she expecs he marke o yield a reurn 20% higher han he risk-free rae nex year, wha would you expec he reurn on fund XXX o be? Soluion: We can say ha he expeced value of y = * value of x, so plug x = 20 ino he equaion o ge he expeced value for y: ˆ y i

18 Accuracy of Inercep Esimae Care needs o be exercised when considering he inercep esimae, paricularly if here are no or few observaions close o he y-axis: y 0 x

19 The Populaion and he Sample The populaion is he oal collecion of all objecs or people o be sudied, for example, Ineresed in Populaion of ineres predicing oucome he enire elecorae of an elecion A sample is a selecion of jus some iems from he populaion. A random sample is a sample in which each individual iem in he populaion is equally likely o be drawn.

20 The DGP and he PRF The populaion regression funcion (PRF) is a descripion of he model ha is hough o be generaing he acual daa and he rue relaionship beween he variables (i.e. he rue values of and ). The PRF is y x u The SRF is yˆ ˆ ˆ x and we also know ha uˆ y yˆ. We use he SRF o infer likely values of he PRF. We also wan o know how good our esimaes of and are.

21 Lineariy In order o use OLS, we need a model which is linear in he parameers ( and ). I does no necessarily have o be linear in he variables (y and x). Linear in he parameers means ha he parameers are no muliplied ogeher, divided, squared or cubed ec. Some models can be ransformed o linear ones by a suiable subsiuion or manipulaion, e.g. he exponenial regression model u Y e X e ln Y ln X u Then le y =ln Y and x =ln X y x u

22 Linear and Non-linear Models This is known as he exponenial regression model. Here, he coefficiens can be inerpreed as elasiciies. Similarly, if heory suggess ha y and x should be inversely relaed: y u x hen he regression can be esimaed using OLS by subsiuing 1 z x Bu some models are inrinsically non-linear, e.g. y x u

23 Esimaor or Esimae? Esimaors are he formulae used o calculae he coefficiens Esimaes are he acual numerical values for he coefficiens.

24 Simple linear regression : esimaion of an opimal hedge raio (1) Objecive : an invesor whishes o hedge a long posiion in he S&P 500 using shor posiion in fuures conracs Minimise he variance of he hedged porfolio reurns The appropriae hedge raio will be he slope esimae ( of spo reurns on fuures reurns ˆ ) in a regression The hedge raio = number of unis of he fuures asse o sell per uni of he spo asse held Excel files: SandPhedge.xls monhly daa for he S&P 500 index and S&P 500 fuures

25 Simple linear regression : esimaion of an opimal hedge raio (2) 1. Creaing a workfile and imporing daa workfile hedge m 2002:2 2007:7 cd C:\Users\Chrisophe\Deskop\Econo_SerTemp\daa1 read(b2,s=sandphedge) SandPhedge.xls 2 2. Transform he level of he 2 series ino percenage reurns Genr rfuures=100*dlog(fuures) Genr rspo=100*dlog(spo) 3. Descripive saisics and correlaions his rfuures his rspo cor rfuures rspo 4. Regress on saionary series equaion hedgereg.ls rspo c rfuures 5. Regress on non-saionary series equaion hedgereg_level.ls spo c fuures save hedge.wf1

26 The Assumpions Underlying he Classical Linear Regression Model (CLRM) The model which we have used is known as he classical linear regression model. We observe daa for x, bu since y also depends on u, we mus be specific abou how he u are generaed. We usually make he following se of assumpions abou he u s (he unobservable error erms): Technical Noaion Inerpreaion 1. E(u ) = 0 The errors have zero mean 2. Var (u ) = 2 The variance of he errors is consan and finie over all values of x 3. Cov (u i,u j )=0 The errors are saisically independen of one anoher 4. Cov (u,x )=0 No relaionship beween he error and corresponding x

27 The Assumpions Underlying he CLRM Again An alernaive assumpion o 4., which is slighly sronger, is ha he x s are non-sochasic or fixed in repeaed samples. A fifh assumpion is required if we wan o make inferences abou he populaion parameers (he acual and ) from he sample parameers ( and ) Addiional Assumpion 5. u is normally disribued

28 Properies of he OLS Esimaor If assumpions 1. hrough 4. hold, hen he esimaors and deermined by OLS are known as Bes Linear Unbiased Esimaors (BLUE). Wha does he acronym sand for? Esimaor - is an esimaor of he rue value of. Linear - is a linear esimaor Unbiased - On average, he acual value of he and s will be equal o he rue values. Bes - means ha he OLS esimaor has minimum variance among he class of linear unbiased esimaors. The Gauss-Markov heorem proves ha he OLS esimaor is bes.

29 Consisency/Unbiasedness/Efficiency Consisen The leas squares esimaors and are consisen. Tha is, he esimaes will converge o heir rue values as he sample size increases o infiniy. Need he assumpions E(x u )=0 and Var(u )= 2 < o prove his. Consisency implies ha lim Pr ˆ 0 0 T Unbiased The leas squares esimaes of and are unbiased. Tha is E( )= and E( )= Thus on average he esimaed value will be equal o he rue values. To prove his also requires he assumpion ha E(u )=0. Unbiasedness is a sronger condiion han consisency. Efficiency An esimaor of parameer is said o be efficien if i is unbiased and no oher unbiased esimaor has a smaller variance. If he esimaor is efficien, we are minimising he probabiliy ha i is a long way off from he rue value of.

30 Precision and Sandard Errors Any se of regression esimaes of and are specific o he sample used in heir esimaion. Recall ha he esimaors of and from he sample parameers ( and ) are given by ˆ x y Txy andˆ y ˆ x 2 2 x Tx Wha we need is some measure of he reliabiliy or precision of he esimaors ( and ). The precision of he esimae is given by is sandard error. Given assumpions 1-4 above, hen he sandard errors can be shown o be given by where s is he esimaed sandard deviaion of he residuals. SE( ˆ) s SE( ˆ) s T ( x 2 x) 1 ( x x) x 2 2 s s T 2 x 2 x 2 T 1 x Tx 2 2 x 2,

31 Esimaing he Variance of he Disurbance Term The variance of he random variable u is given by Var(u ) = E[(u )-E(u )] 2 which reduces o Var(u ) = E(u 2 ) We could esimae his using he average of : Unforunaely his is no workable since u is no observable. We can use he sample counerpar o u, which is : Bu his esimaor is a biased esimaor of 2. s 2 1 T û 2 u 2 u s 2 1 T ˆ 2 u

32 Esimaing he Variance of he Disurbance Term (con d) An unbiased esimaor of is given by s T ˆ 2 u 2 where uˆ 2 is he residual sum of squares and T is he sample size. Some Commens on he Sandard Error Esimaors 1. Boh SE( ) and SE( ) depend on s 2 (or s). The greaer he variance s 2, hen he more dispersed he errors are abou heir mean value and herefore he more dispersed y will be abou is mean value. 2. The sum of he squares of x abou heir mean appears in boh formulae. The larger he sum of squares, he smaller he coefficien variances.

33 Some Commens on he Sandard Error Esimaors Consider wha happens if x 2 x is small or large: y y y y 0 x x 0 x x

34 Some Commens on he Sandard Error Esimaors (con d) 3. The larger he sample size, T, he smaller will be he coefficien variances. T appears explicily in SE( ) and implicily in SE( ). T appears implicily since he sum x is from = 1 o T. x 2 4. The erm appears in he SE( ). The reason is ha y-axis. x 2 2 x measures how far he poins are away from he

35 Example: How o Calculae he Parameers and Sandard Errors Assume we have he following daa calculaed from a regression of y on a single variable x and a consan over 22 observaions. Daa: x x 2 Calculaions: y , T , RSS 22, , ( 22 * * ) x *( ) 2 y 86.65, We wrie * yˆ ˆ ˆ x yˆ x

36 Example (con d) SE(regression), s T ˆ 2 u SE( ) 2.55* SE( ) 2.55* We now wrie he resuls as yˆ x (3.35) (0.0079)

37 An Inroducion o Saisical Inference We wan o make inferences abou he likely populaion values from he regression parameers. Example: Suppose we have he following regression resuls: yˆ x (14.38) (0.2561) is a single (poin) esimae of he unknown populaion parameer,. How reliable is his esimae? The reliabiliy of he poin esimae is measured by he coefficien s sandard error.

38 Hypohesis Tesing: Some Conceps We can use he informaion in he sample o make inferences abou he populaion. We will always have wo hypoheses ha go ogeher, he null hypohesis (denoed H 0 ) and he alernaive hypohesis (denoed H 1 ). The null hypohesis is he saemen or he saisical hypohesis ha is acually being esed. The alernaive hypohesis represens he remaining oucomes of ineres. For example, suppose given he regression resuls above, we are ineresed in he hypohesis ha he rue value of is in fac 0.5. We would use he noaion H 0 : = 0.5 H 1 : 0.5 This would be known as a wo sided es.

39 One-Sided Hypohesis Tess Someimes we may have some prior informaion ha, for example, we would expec > 0.5 raher han < 0.5. In his case, we would do a one-sided es: H 0 : = 0.5 H 1 : > 0.5 or we could have had H 0 : = 0.5 H 1 : < 0.5 There are wo ways o conduc a hypohesis es: via he es of significance approach or via he confidence inerval approach.

40 The Probabiliy Disribuion of he Leas Squares Esimaors We assume ha u N(0, 2 ) Since he leas squares esimaors are linear combinaions of he random variables i.e. w y The weighed sum of normal random variables is also normally disribued, so N(, Var()) N(, Var()) Wha if he errors are no normally disribued? Will he parameer esimaes sill be normally disribued? Yes, if he oher assumpions of he CLRM hold, and he sample size is sufficienly large.

41 The Probabiliy Disribuion of he Leas Squares Esimaors (con d) Sandard normal variaes can be consruced from and : ˆ var ~ N 0,1 and ˆ var ~ N 0,1 Bu var() and var() are unknown, so ˆ SE( ˆ) ~ T 2 and ˆ SE( ˆ) ~ T 2

42 Tesing Hypoheses: The Tes of Significance Approach Assume he regression equaion is given by, y x u for =1,2,...,T The seps involved in doing a es of significance are: 1. Esimae, and SE( ), SE( ) in he usual way 2. Calculae he es saisic. This is given by he formula * es saisic SE( ) where * is he value of under he null hypohesis.

43 The Tes of Significance Approach (con d) 3. We need some abulaed disribuion wih which o compare he esimaed es saisics. Tes saisics derived in his way can be shown o follow a - disribuion wih T-2 degrees of freedom. As he number of degrees of freedom increases, we need o be less cauious in our approach since we can be more sure ha our resuls are robus. 4. We need o choose a significance level, ofen denoed. This is also someimes called he size of he es and i deermines he region where we will rejec or no rejec he null hypohesis ha we are esing. I is convenional o use a significance level of 5%. Inuiive explanaion is ha we would only expec a resul as exreme as his or more exreme 5% of he ime as a consequence of chance alone. Convenional o use a 5% size of es, bu 10% and 1% are also commonly used.

44 Deermining he Rejecion Region for a Tes of Significance 5. Given a significance level, we can deermine a rejecion region and nonrejecion region. For a 2-sided es: f(x) 2.5% rejecion region 95% non-rejecion region 2.5% rejecion region

45 The Rejecion Region for a 1-Sided Tes (Upper Tail) f(x) 95% non-rejecion 5% rejecion region

46 The Rejecion Region for a 1-Sided Tes (Lower Tail) f(x) 5% rejecion region 95% non-rejecion region

47 The Tes of Significance Approach: Drawing Conclusions 6. Use he -ables o obain a criical value or values wih which o compare he es saisic. 7. Finally perform he es. If he es saisic lies in he rejecion region hen rejec he null hypohesis (H 0 ), else do no rejec H 0.

48 A Noe on he and he Normal Disribuion You should all be familiar wih he normal disribuion and is characerisic bell shape. We can scale a normal variable o have zero mean and uni variance by subracing is mean and dividing by is sandard deviaion. There is, however, a specific relaionship beween he - and he sandard normal disribuion. Boh are symmerical and cenred on zero. The -disribuion has anoher parameer, is degrees of freedom. We will always know his (for he ime being from he number of observaions -2).

49 Wha Does he -Disribuion Look Like? normal disribuion -disribuion

50 Comparing he and he Normal Disribuion In he limi, a -disribuion wih an infinie number of degrees of freedom is a sandard normal, i.e. ( ) N( 01, ) Examples from saisical ables: Significance level N(0,1) (40) (4) 50% % % % The reason for using he -disribuion raher han he sandard normal is ha we had o esimae 2, he variance of he disurbances.

51 The Confidence Inerval Approach o Hypohesis Tesing An example of is usage: We esimae a parameer, say o be 0.93, and a 95% confidence inerval o be (0.77,1.09). This means ha we are 95% confiden ha he inerval conaining he rue (bu unknown) value of. Confidence inervals are almos invariably wo-sided, alhough in heory a one-sided inerval can be consruced.

52 How o Carry ou a Hypohesis Tes Using Confidence Inervals 1. Calculae, and SE( ), SE( ) as before. 2. Choose a significance level,, (again he convenion is 5%). This is equivalen o choosing a (1-)100% confidence inerval, i.e. 5% significance level = 95% confidence inerval 3. Use he -ables o find he appropriae criical value, which will again have T-2 degrees of freedom. 4. The confidence inerval is given by ( ˆ SE( ˆ), ˆ SE( ˆ)) cri cri 5. Perform he es: If he hypohesised value of (*) lies ouside he confidence inerval, hen rejec he null hypohesis ha = *, oherwise do no rejec he null.

53 Confidence Inervals Versus Tess of Significance Noe ha he Tes of Significance and Confidence Inerval approaches always give he same answer. Under he es of significance approach, we would no rejec H 0 ha = * if he es saisic lies wihin he non-rejecion region, i.e. if * cri cri SE( ) Rearranging, we would no rejec if Bu his is jus he rule under he confidence inerval approach. cri ˆ SE( ˆ) ˆ * SE( ˆ) cri SE( ˆ) cri * ˆ cri SE( ˆ)

54 Consrucing Tess of Significance and Confidence Inervals: An Example Using he regression resuls above, yˆ x (14.38) (0.2561), T=22 Using boh he es of significance and confidence inerval approaches, es he hypohesis ha =1 agains a wo-sided alernaive. The firs sep is o obain he criical value. We wan cri = 20;5%

55 Deermining he Rejecion Region f(x) 2.5% rejecion region 2.5% rejecion region

56 Performing he Tes The hypoheses are: H 0 : = 1 H 1 : 1 Tes of significance approach * es sa SE( ) Do no rejec H 0 since es sa lies wihin Confidence inerval approach ˆ SE( ˆ) cri ( ,1.0433) Since 1 lies wihin he confidence inerval, non-rejecion region do no rejec H 0

57 Tesing oher Hypoheses Wha if we waned o es H 0 : = 0 or H 0 : = 2? Noe ha we can es hese wih he confidence inerval approach. For ineres (!), es H 0 : = 0 vs. H 1 : 0 H 0 : = 2 vs. H 1 : 2

58 Size of a Tes The size of a es, ofen called significance level, is he probabiliy of commiing a Type I error. A Type I error occurs when a null hypohesis is rejeced when i is rue. This es size is denoed by α (alpha). The 1- α is called he confidence level, which is used in he form of he (1- α)*100 percen confidence inerval of a parameer. Type I error is he false rejecion of he null hypohesis and ype II error is he false accepance of he null hypohesis. As an aid memoir: hink ha our cynical sociey rejecs before i acceps.

59 Power of a Tes The power of a saisical es is he probabiliy ha i will correcly lead o he rejecion of a false null hypohesis Type II error, denoed by ß, is he probabiliy of failing o rejec he null hypohesis when i is false. The power of a es is equal o 1 - ß

60 The Errors Tha We Can Make Using Hypohesis Tess We usually rejec H 0 if he es saisic is saisically significan a a chosen significance level. There are wo possible errors we could make: 1. Rejecing H 0 when i was really rue. This is called a ype I error. 2. No rejecing H 0 when i was in fac false. This is called a ype II error. Resul of Tes Significan (rejec H 0 ) Insignifican ( do no rejec H 0 ) Realiy H 0 is rue Type I error = H 0 is false Type II error =

61 The Trade-off Beween Type I and Type II Errors The probabiliy of a ype I error is jus, he significance level or size of es we chose. To see his, recall wha we said significance a he 5% level mean: i is only 5% likely ha a resul as or more exreme as his could have occurred purely by chance. Noe ha here is no chance for a free lunch here! Wha happens if we reduce he size of he es (e.g. from a 5% es o a 1% es)? We reduce he chances of making a ype I error... bu we also reduce he probabiliy ha we will rejec he null hypohesis a all, so we increase he probabiliy of a ype II error: less likely o falsely rejec Reduce size more sric rejec null of es crierion for hypohesis more likely o rejecion less ofen incorrecly no Less ofen Type I error More ofen Type II error rejec So here is always a rade off beween ype I and ype II errors when choosing a significance level. The only way we can reduce he chances of boh is o increase he sample size.

62 A Special Type of Hypohesis Tes: The -raio Recall ha he formula for a es of significance approach o hypohesis esing using a -es was i * i es saisic SE i If he es is H 0 : i = 0 H 1 : i 0 i.e. a es ha he populaion coefficien is zero agains a wo-sided alernaive, his is known as a -raio es: Since i * = 0, es sa i SE( ) i The raio of he coefficien o is SE is known as he -raio or -saisic.

63 Changing he Size of he Tes Bu noe ha we looked a only a 5% size of es. In marginal cases (e.g. H 0 : = 1), we may ge a compleely differen answer if we use a differen size of es. This is where he es of significance approach is beer han a confidence inerval. For example, say we waned o use a 10% size of es. Using he es of significance approach, * es sa SE( ) as above. The only hing ha changes is he criical -value.

64 Changing he Size of he Tes: The New Rejecion Regions f(x) 5% rejecion region 5% rejecion region

65 Changing he Size of he Tes: The Conclusion 20;10% = So now, as he es saisic lies in he rejecion region, we would rejec H 0. Cauion should herefore be used when placing emphasis on or making decisions in marginal cases (i.e. in cases where we only jus rejec or no rejec).

66 Some More Terminology If we rejec he null hypohesis a he 5% level, we say ha he resul of he es is saisically significan. Noe ha a saisically significan resul may be of no pracical significance. E.g. if a shipmen of cans of beans is expeced o weigh 450g per in, bu he acual mean weigh of some ins is 449g, he resul may be highly saisically significan bu presumably nobody would care abou 1g of beans.

67 The -raio: An Example Suppose ha we have he following parameer esimaes, sandard errors and -raios for an inercep and slope respecively. Coefficien SE raio Compare his wih a cri wih 15-3 = 12 d.f. (2½% in each ail for a 5% es) = % = % Do we rejec H 0 : 1 = 0? (No) H 0 : 2 = 0? (Yes)

68 Wha Does he -raio ell us? If we rejec H 0, we say ha he resul is significan. If he coefficien is no significan (e.g. he inercep coefficien in he las regression above), hen i means ha he variable is no helping o explain variaions in y. Variables ha are no significan are usually removed from he regression model. In pracice here are good saisical reasons for always having a consan even if i is no significan. Look a wha happens if no inercep is included: y x

69 An Example of he Use of a Simple -es o Tes a Theory in Finance Tesing for he presence and significance of abnormal reurns ( Jensen s alpha - Jensen, 1968). The Daa: Annual Reurns on he porfolios of 115 muual funds from R The model: for j = 1,, 115 j R ( R R ) We are ineresed in he significance of j. f j j m f u j The null hypohesis is H 0 : j = 0.

70 Frequency Disribuion of -raios of Muual Fund Alphas (gross of ransacions coss) Source Jensen (1968). Reprined wih he permission of Blackwell publishers.

71 Frequency Disribuion of -raios of Muual Fund Alphas (ne of ransacions coss) Source Jensen (1968). Reprined wih he permission of Blackwell publishers.

72 Can UK Uni Trus Managers Bea he Marke? We now perform a varian on Jensen s es in he conex of he UK marke, considering monhly reurns on 76 equiy uni russ. The daa cover he period January 1979 May 2000 (257 observaions for each fund). Some summary saisics for he funds are: Mean Minimum Maximum Median Average monhly reurn, % 0.6% 1.4% 1.0% Sandard deviaion of reurns over ime 5.1% 4.3% 6.9% 5.0% Jensen Regression Resuls for UK Uni Trus Reurns, January 1979-May 2000 R R ( R R ) j f j j m f j

73 Can UK Uni Trus Managers Bea he Marke? : Resuls Esimaes of Mean Minimum Maximum Median -0.02% -0.54% 0.33% -0.03% raio on In fac, gross of ransacions coss, 9 funds of he sample of 76 were able o significanly ou-perform he marke by providing a significan posiive alpha, while 7 funds yielded significan negaive alphas.

74 The Overreacion Hypohesis and he UK Sock Marke Moivaion Two sudies by DeBond and Thaler (1985, 1987) showed ha socks which experience a poor performance over a 3 o 5 year period end o ouperform socks which had previously performed relaively well. How Can This be Explained? 2 suggesions 1. A manifesaion of he size effec DeBond & Thaler did no believe his a sufficien explanaion, bu Zarowin (1990) found ha allowing for firm size did reduce he subsequen reurn on he losers. 2. Reversals reflec changes in equilibrium required reurns Ball & Kohari (1989) find he CAPM bea of losers o be considerably higher han ha of winners.

75 The Overreacion Hypohesis and he UK Sock Marke (con d) Anoher ineresing anomaly: he January effec. Anoher possible reason for he superior subsequen performance of losers. Zarowin (1990) finds ha 80% of he exra reurn available from holding he losers accrues o invesors in January. Example sudy: Clare and Thomas (1995) Daa: Monhly UK sock reurns from January 1955 o 1990 on all firms raded on he London Sock exchange.

76 Mehodology Calculae he monhly excess reurn of he sock over he marke over a 12, 24 or 36 monh period for each sock i: U i = R i - R m n = 12, 24 or 36 monhs Calculae he average monhly reurn for he sock i over he firs 12, 24, or 36 monh period: R i 1 n n 1 U i

77 Porfolio Formaion Then rank he socks from highes average reurn o lowes and from 5 porfolios: Porfolio 1: Bes performing 20% of firms Porfolio 2: Nex 20% Porfolio 3: Nex 20% Porfolio 4: Nex 20% Porfolio 5: Wors performing 20% of firms. Use he same sample lengh n o monior he performance of each porfolio.

78 Porfolio Formaion and Porfolio Tracking Periods How many samples of lengh n have we go? n = 1, 2, or 3 years. If n = 1year: Esimae for year 1 Monior porfolios for year 2 Esimae for year 3 Monior porfolios for year 36 So if n = 1, we have 18 INDEPENDENT (non-overlapping) observaion / racking periods.

79 Consrucing Winner and Loser Reurns Similarly, n = 2 gives 9 independen periods and n = 3 gives 6 independen periods. Calculae monhly porfolio reurns assuming an equal weighing of socks in each porfolio. Denoe he mean reurn for each monh over he 18, 9 or 6 periods for he W L winner and loser porfolios respecively as R p and R p respecively. Define he difference beween hese as = -. R D R p L R p W Then perform he regression R D = 1 + (Tes 1) Look a he significance of 1.

80 Allowing for Differences in he Riskiness of he Winner and Loser Porfolios Problem: Significan and posiive 1 could be due o higher reurn being required on loser socks due o loser socks being more risky. Soluion: Allow for risk differences by regressing agains he marke risk premium: R D = 2 + (R m -R f ) + (Tes 2) where R m R f is he reurn on he FTA All-share is he reurn on a UK governmen 3 monh -bill.

81 Is here an Overreacion Effec in he UK Sock Marke? Resuls Panel A: All Monhs n = 12 n = 24 n =36 Reurn on Loser Reurn on Winner Implied annualised reurn difference -0.37% 1.68% 1.56% Coefficien for (3.47): ˆ (0.29) ** (2.01) (1.55) Coefficiens for (3.48): ˆ (-0.30) ˆ (-0.25) Panel B: All Monhs Excep January Coefficien for (3.47): ˆ (-0.72) ** (2.01) (0.21) * (1.63) * (1.41) (-0.06) (1.05) Noes: -raios in parenheses; * and ** denoe significance a he 10% and 5% levels respecively. Source: Clare and Thomas (1995). Reprined wih he permission of Blackwell Publishers.

82 Tesing for Seasonal Effecs in Overreacions Is here evidence ha losers ou-perform winners more a one ime of he year han anoher? To es his, calculae he difference beween he winner & loser porfolios as previously,, and regress his on 12 monh-of-he-year dummies: R D R M D i i i1 Significan ou-performance of losers over winners in, June (for he 24-monh horizon), and January, April and Ocober (for he 36-monh horizon) winners appear o say significanly as winners in March (for he 12-monh horizon). 12

83 Conclusions Evidence of overreacions in sock reurns. Losers end o be small so we can aribue mos of he overreacion in he UK o he size effec. Commens Small samples No diagnosic checks of model adequacy

84 The Exac Significance Level or p-value This is equivalen o choosing an infinie number of criical -values from ables. I gives us he marginal significance level where we would be indifferen beween rejecing and no rejecing he null hypohesis. If he es saisic is large in absolue value, he p-value will be small, and vice versa. The p-value gives he plausibiliy of he null hypohesis. e.g. a es saisic is disribued as a 62 = The p-value = Do we rejec a he 5% level?...no Do we rejec a he 10% level?...no Do we rejec a he 20% level?...yes

85 Hypohesis esing : hedging revisied Reload he hedge.wf1 workfile creaed above Re-examine he resuls able from reurns regression We wan o es he null hypohesis ha H 0 : = 1 vs H 1 : = 0 hedgereg.wald c(2)=1 hedgereg_level.wald c(2)=1

86 Esimaion and hypohesis esing: he CAPM 1. Creaing a workfile and imporing daa workfile CAPM m 2002:1 2007:4 cd C:\Users\Chrisophe\Deskop\Econo_SerTemp\daa1 read(b2,s=able) capm.xls 6 2. Transform he level of he 5 series ino percenage reurns and consider monhly T- Bill yields Genr rsandp=100*dlog(sandp) Genr rford=100*dlog(ford) Genr rgm=100*dlog(gm) Genr rmicrosof=100*dlog(microsof) Genr rsun=100*dlog(sun) Genr USTB3M=USTB3M/12 3. Compue he 5 excess reurns Genr ersandp=rsandp - USTB3M Genr erford=rford - USTB3M Genr ermicrosof=rmicrosof - USTB3M Genr ersun=rsun - USTB3M Genr ergm=rgm - USTB3M

87 Esimaion and hypohesis esing: he CAPM (2) 4. Plo he daa o examine in which measure he individual reurns move ogeher wih he index (line graph hen scaer plo) Plo ersandp erford Sca ersandp erford (..) 5. Esimae he CAPM : R Ford r ( R r ) u f M f equaion ford_capm.ls erford c ersandp 6. Tes if he CAPM bea of Ford sock is 1 ford_capm.wald c(2)=1 save capm.wf1