Lecture 4. Classical Linear Regression Model: Overview

Transcription

1 Lecure 4 Classical Linear Regression Model: Overview

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 1,,..., k where here are k independen variables. Some alernaive names for he y and variables: y dependen variable independen variables regressand regressors effec variable causal variables eplained variable eplanaory variable

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

5 Simple Regression For simpliciy, say k=1. This is he siuaion where y depends on only one variable. Eamples 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

6 Simple Regression: An Eample Suppose ha we have he following daa on he ecess reurns on a fund manager s porfolio ( fund XXX ) ogeher wih he ecess reurns on a marke inde: Ecess reurn on marke inde Year, Ecess 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 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) Ecess reurn on fund XXX Ecess reurn on marke porfolio

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

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 ŷ û denoe he acual daa poin 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 i

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

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) () From (1), Bu and. α β α β y β α + = = = y L 0 ) ( β α α = = y L 0 ) ( β α β 0 0 ) ( = = T y y β α β α = y T y = T = = i y y y L ) ( ) ( β α

15 Deriving he OLS Esimaor So we can wrie or (3) From (), (4) From (3), (5) Subsiue ino (4) for from (5), α 0 = y β α = y 0 ) ( β α y β α = = + = + = + T Ty y y y y y ) ( β β β β β β 0 = T T y T β α

16 Deriving he OLS Esimaor Rearranging for β, β (T ) = Ty y So overall we have β = y Ty andα = y T β This mehod of finding he opimum is known as ordinary leas squares.

17 α β Wha do We Use and For? In he CAPM eample used above, plugging he 5 observaions in o make up he formulae given above would lead o he esimaes = and β = We would wrie he fied line as: α y = Quesion: If an analys ells you ha she epecs he marke o yield a reurn 0% higher han he risk-free rae ne year, wha would you epec he reurn on fund XXX o be? Soluion: We can say ha he epeced value of y = * value of, so plug = 0 ino he equaion o ge he epeced value for y: = = y i

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

19 The Populaion and he Sample The populaion is he oal collecion of all objecs or people o be sudied, for eample, 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 α and β). The PRF is y = α + β + u The SRF is y = α + β and we also know ha u = y y. We use he SRF o infer likely values of he PRF. We also wan o know how good our esimaes of α and β are.

21 Lineariy In order o use OLS, we need a model which is linear in he parameers (α and β ). I does no necessarily have o be linear in he variables (y and ). 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 eponenial regression model α β u Y = e X e ln Y = α + β ln X + u Then le y =ln Y and =ln X y = α + β + u

22 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, 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. Var (u ) = σ The variance of he errors is consan and finie over all values of 3. Cov (u i,u j )=0 The errors are saisically independen of one anoher 4. Cov (u, )=0 No relaionship beween he error and corresponding variae

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

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

25 Consisency/Unbiasedness/Efficiency Consisen The leas squares esimaors α and β are consisen. Tha is, he esimaes will converge o heir rue values as he sample size increases o infiniy. Need he assumpions E( u )=0 and Var(u )=σ < o prove his. Consisency implies ha [ ] lim Pr β β > δ = 0 δ > 0 T Unbiased The leas squares esimaes of α and β are unbiased. Tha is E( α )=α and E( β )=β 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 β of parameer β 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 β.

26 Precision and Sandard Errors Any se of regression esimaes of α and β are specific o he sample used in heir esimaion. Recall ha he esimaors of α and β from he sample parameers ( α and β ) are given by y Ty β = and α = y β T Wha we need is some measure of he reliabiliy or precision of he esimaors ( α and β ). 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 SE( ) α = s SE( ) β = s ( where s is he esimaed sandard deviaion of he residuals. T ) 1 ( ) = s = s T T 1 T,

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

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

29 Some Commens on he Sandard Error Esimaors ( ) Consider wha happens if is small or large: y y y y 0 0

30 Some Commens on he Sandard Error Esimaors 3. The larger he sample size, T, he smaller will be he coefficien variances. T appears eplicily in SE( α ) and implicily in SE( β ). T appears implicily since he sum is from = 1 o T. 4. The erm appears in he SE( α ). ( ) The reason is ha measures how far he poins are away from he y-ais.

31 Eample: How o Calculae he Parameers and Sandard Errors Assume we have he following daa calculaed from a regression of y on a single variable and a consan over observaions. Daa: Calculaions: y = 83010, T = , β = =, = RSS = , ( * * ) *( ) y = 86.65, = 035. We wrie α = * = y = α + β y =

32 Eample SE(regression), s = T u = =.55 SE( α) =.55* SE( β ) =.55* ( ) ( ) ( ) = 3.35 = We now wrie he resuls as y = (3.35) (0.0079)

33 An Inroducion o Saisical Inference We wan o make inferences abou he likely populaion values from he regression parameers. Eample: Suppose we have he following regression resuls: y = (14.38) (0.561) β = is a single (poin) esimae of he unknown populaion parameer, β. How reliable is his esimae? The reliabiliy of he poin esimae is measured by he coefficien s sandard error.

34 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 eample, suppose given he regression resuls above, we are ineresed in he hypohesis ha he rue value of β is in fac 0.5. We would use he noaion H 0 : β = 0.5 H 1 : β 0.5 This would be known as a wo sided es.

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

36 The Probabiliy Disribuion of he Leas Squares Esimaors We assume ha u N(0,σ ) Since he leas squares esimaors are linear combinaions of he random variables i.e. β = wy The weighed sum of normal random variables is also normally disribued, so α N(α, Var(α)) β N(β, Var(β)) 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.

37 The Probabiliy Disribuion of he Leas Squares Esimaors Sandard normal variaes can be consruced from α and β : α α var ( α ) ~ N ( 0,1) and β β var ( β ) ~ N ( 0,1) Bu var(α) and var(β) are unknown, so α α ~ SE( ) α T and β β ~ SE( ) β T

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

39 The Tes of Significance Approach 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- 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 α. 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 eplanaion is ha we would only epec a resul as ereme as his or more ereme 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.

40 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 -sided es: f().5% rejecion region 95% non-rejecion i.5% rejecion region

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

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

43 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.

44 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 variae 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 -).

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

46 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, ) Eamples 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 σ, he variance of he disurbances.

47 The Confidence Inerval Approach o Hypohesis Tesing An eample 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 β. Confidence inervals are almos invariably wo-sided, alhough in heory a one-sided inerval can be consruced.

48 How o Carry ou a Hypohesis Tes Using Confidence Inervals 1. Calculae, β and SE( α ), SE( β ) as before. α. Choose a significance level, α, (again he convenion is 5%). This is equivalen o choosing a (1-α) 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- degrees of freedom. 4. The confidence inerval is given by ( β SE( ), β β + SE( )) β cri cri 5. Perform he es: If he hypohesised value of β (β*) lies ouside he confidence inerval, hen rejec he null hypohesis ha β = β*, oherwise do no rejec he null.

49 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 β = β* if he es saisic lies wihin he non-rejecion region, i.e. if β β * cri + cri SE( β ) Rearranging, we would no rejec if SE( ) β β β* + SE( ) β cri cri β SE( ) β β* cri β + cri SE( ) β Bu his is jus he rule under he confidence inerval approach.

50 Consrucing Tess of Significance and Confidence Inervals: An Eample Using he regression resuls above, y = , T= (14.38) (0.561) Using boh he es of significance and confidence inerval approaches, es he hypohesis ha β =1 agains a wo-sided alernaive. The firs sep is o obain he criical value. We wan cri = 0;5%

51 Deermining he Rejecion Region f().5% rejecion region.5% rejecion region

52 Performing he Tes The hypoheses are: H 0 : β = 1 H 1 : β 1 Tes of significance approach β β * es sa = SE( β ) = = Do no rejec H 0 since es sa lies wihin Confidence inerval approach β ± SE( ) β cri = ± = ( 0.051,1.0433) Since 1 lies wihin he confidence inerval, non-rejecion region do no rejec H 0

53 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 : β = 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 eample, say we waned o use a 10% size of es. Using he es of significance approach, β β * es sa = SE( β ) = = as above. The only hing ha changes is he criical -value.

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

55 Changing he Size of he Tes: The Conclusion 0;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).

56 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.. 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 = α H 0 is false Type II error = β

57 The Trade-off Beween Type I and Type II Errors The probabiliy of a ype I error is jus α, 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 ereme as his could have occurred purely by chance. 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 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.

58 The Eac 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 6 = The p-value = 0.1. Do we rejec a he 5% level?...no Do we rejec a he 10% level?...no Do we rejec a he 0% level?...yes

59 Generalising he Simple Model o Muliple Linear Regression Before, we have used he model y = α + β + u = 1,,...,T Bu wha if our dependen (y) variable depends on more han one independen variable? For eample he number of cars sold migh plausibly depend on 1. he price of cars. he price of public ranspor 3. he price of perol 4. he een of he public s concern abou global warming Similarly, sock reurns migh depend on several facors. Having jus one independen variable is no good in his case - we wan o have more han one variable. I is very easy o generalise he simple model o one wih k-1 regressors (independen variables).

60 Muliple Regression and he Consan Term Now we wrie y = β + β + β β + u k k, =1,,...,T Where is 1? I is he consan erm. In fac he consan erm is usually represened by a column of ones of lengh T: 1 = β 1 is he coefficien aached o he consan erm (which we called α before).

61 Differen Ways of Epressing he Muliple Linear Regression Model We could wrie ou a separae equaion for every value of : We can wrie his in mari form y = Xβ +u where y is T 1 X is T k β is k 1 u is T 1 T kt k T T T k k k k u y u y u y = = = β β β β β β β β β β β β

62 Inside he Marices of he Muliple Linear Regression Model e.g. if k is, we have regressors, one of which is a column of ones: T 1 T 1 T 1 Noice ha he marices wrien in his way are conformable. + = T T T u u u y y y β β

63 How Do We Calculae he Parameers (he β ) in his Generalised Case? Previously, we ook he residual sum of squares, and minimised i w.r.. α and β. In he mari noaion, we have The RSS would be given by = u T u u u 1 [ ] = = = ' T T T u u u u u u u u u u u u

64 The OLS Esimaor for he Muliple Regression Model In order o obain he parameer esimaes, β 1, β,..., β k, we would minimise he RSS wih respec o all he βs. I can be shown ha y X X X k = = 1 1 ) ( β β β β

65 Calculaing he Sandard Errors for he Muliple Regression Model Check he dimensions: β is k 1 as required. Bu how do we calculae he sandard errors of he coefficien esimaes? Previously, o esimae he variance of he errors, σ, we used u s =. T uu Now using he mari noaion, we use s ' = T k where k = number of regressors. I can be proved ha he OLS esimaor of he variance of β is given by he diagonal elemens of s( X' X) 1, so ha he variance of β 1 is he firs elemen, he variance of β is he second elemen, and, and he variance of is he k h diagonal elemen. β k

66 Calculaing Parameer and Sandard Error Esimaes for Muliple Regression Models: An Eample Eample:The following model wih k=3 is esimaed over 15 observaions: y β + β + + = 1 β 3 3 and he following daa have been calculaed from he original X s ( X' X) 1 = ,( X' y) =., u ' u = Calculae he coefficien esimaes and heir sandard errors. To calculae he coefficiens, jus muliply he mari by he vecor o obain 1 ( X ' X ) X ' y. To calculae he sandard errors, we need an esimae of σ. u RSS s = 1096 T k = = 091.

67 Calculaing Parameer and Sandard Error Esimaes for Muliple Regression Models: An Eample (con d) The variance-covariance mari of β is given by s ( X' X) 1 = 091. ( X' X) 1 = The variances are on he leading diagonal: Var( β ) = 183. SE( β ) = Var( β ) = 091. SE( β ) = 096. Var( β ) = 393. SE( β ) = We wrie: y = ( 1.35) ( 0.96) ( 1.98)

68 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 β i β * i es saisic = SE( β i ) If he es is H 0 : β i = 0 H 1 : β 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 β i * = 0, es sa = β i SE( β ) i The raio of he coefficien o is SE is known as he -raio or -saisic.

69 The -raio: An Eample In he las eample above: Coefficien SE raio Compare his wih a cri wih 15-3 = 1 d.f. (½% in each ail for a 5% es) =.179 5% = % Do we rejec H 0 : β 1 = 0? (No) H 0 : β = 0? (Yes) H 0 : β 3 = 0? (Yes)

70 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 eplain 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

71 An Eample 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 R = α + β ( R R ) + The model: for j = 1,, 115 j We are ineresed in he significance of α j. f j j m f u j The null hypohesis is H 0 : α j = 0.

72 Frequency Disribuion of -raios of Muual Fund Alphas (gross of ransacions coss)

73 Frequency Disribuion of -raios of Muual Fund Alphas (ne of ransacions coss)

74 Can UK Uni Trus Managers Bea he Marke? We now perform a varian on Jensen s es in he cone of he UK marke, considering monhly reurns on 76 equiy uni russ. The daa cover he period January 1979 May 000 (57 observaions for each fund). Some summary saisics for he funds are: Mean Minimum Maimum 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 000 R R = α + β ( R R ) + ε j f j j m f j

75 Can UK Uni Trus Managers Bea he Marke? : Resuls Esimaes of Mean Minimum Maimum Median α -0.0% -0.54% 0.33% -0.03% β raio on α 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.

76 Jan Performance of UK Uni Truss Jan-80 Jan-81 Jan-8 Jan-83 Jan-84 Jan-85 Jan-86 Jan-87 Jan-88 Jan-89 Jan-90 Jan-91 Jan-9 Jan-93 Jan-94 Jan-95 Jan-96 Jan-97 Jan-98 Jan-99 Jan-79

77 Tesing Muliple Hypoheses: The F-es We used he -es o es single hypoheses, i.e. hypoheses involving only one coefficien. Bu wha if we wan o es more han one coefficien simulaneously? We do his using he F-es. The F-es involves esimaing regressions. The unresriced regression is he one in which he coefficiens are freely deermined by he daa, as we have done before. The resriced regression is he one in which he coefficiens are resriced, i.e. he resricions are imposed on some βs.

78 The F-es: Resriced and Unresriced Regressions Eample The general regression is y = β 1 + β + β β u (1) We wan o es he resricion ha β 3 +β 4 = 1 (we have some hypohesis from heory which suggess ha his would be an ineresing hypohesis o sudy). The unresriced regression is (1) above, bu wha is he resriced regression? y = β 1 + β + β β u s.. β 3 +β 4 = 1 We subsiue he resricion (β 3 +β 4 = 1) ino he regression so ha i is auomaically imposed on he daa. β 3 +β 4 = 1 β 4 = 1- β 3

79 The F-es: Forming he Resriced Regression y = β 1 + β + β (1-β 3 ) 4 + u y = β 1 + β + β β u Gaher erms in β s ogeher and rearrange (y - 4 ) = β 1 + β + β 3 ( 3-4 ) + u This is he resriced regression. We acually esimae i by creaing wo new variables, call hem, say, P and Q. P = y - 4 Q = 3-4 so P = β 1 + β + β 3 Q + u is he resriced regression we acually esimae.

80 Calculaing he F-Tes Saisic The es saisic is given by es saisic = RRSS URSS URSS T k m where URSS RRSS m T = RSS from unresriced regression = RSS from resriced regression = number of resricions = number of observaions k = number of regressors in unresriced regression including a consan in he unresriced regression (or he oal number of parameers o be esimaed).

81 The F-Disribuion The es saisic follows he F-disribuion, which has d.f. parameers. The value of he degrees of freedom parameers are m and (T-k) respecively (he order of he d.f. parameers is imporan). The appropriae criical value will be in column m, row (T-k). The F-disribuion has only posiive values and is no symmerical. We herefore only rejec he null if he es saisic > criical F-value.

82 Eamples : Deermining he Number of Resricions in an F-es H 0 : hypohesis β 1 + β = 1 β = 1 and β 3 = -1 β = 0, β 3 = 0 and β 4 = 0 3 If he model is y = β 1 + β + β β u, hen he null hypohesis No. of resricions, m H 0 : β = 0, and β 3 = 0 and β 4 = 0 is esed by he regression F-saisic. I ess he null hypohesis ha all of he coefficiens ecep he inercep coefficien are zero. Noe he form of he alernaive hypohesis for all ess when more han one resricion is involved: H 1 : β 0, or β 3 0 or β 4 0

83 Wha we Canno Tes wih Eiher an F or a -es We canno es using his framework hypoheses which are no linear or which are muliplicaive, e.g. H 0 : β β 3 = or H 0 : β = 1 canno be esed.

84 The Relaionship beween he and he F- Disribuions Any hypohesis which could be esed wih a -es could have been esed using an F-es, bu no he oher way around. For eample, consider he hypohesis H 0 : β = 0.5 H 1 : β 0.5 We could have esed his using he usual -es: β 05. es sa = or i could be esed in he framework above for he F-es. SE( β) Noe ha he wo ess always give he same resul since he -disribuion is jus a special case of he F-disribuion. For eample, if we have some random variable Z, and Z (T-k) hen also Z F(1,T-k)

85 F-es Eample Quesion: Suppose a researcher wans o es wheher he reurns on a company sock (y) show uni sensiiviy o wo facors (facor and facor 3 ) among hree considered. The regression is carried ou on 144 monhly observaions. The regression is y = β 1 + β + β β u - Wha are he resriced and unresriced regressions? - If he wo RSS are and 397. respecively, perform he es. Soluion: Uni sensiiviy implies H 0 :β =1 and β 3 =1. The unresriced regression is he one in he quesion. The resriced regression is (y )= β 1 + β 4 4 +u or leing z =y - - 3, he resriced regression is z = β 1 + β 4 4 +u In he F-es formula, T=144, k=4, m=, RRSS=436.1, URSS=397. F-es saisic = Criical value is an F(,140) = 3.07 (5%) and 4.79 (1%). Conclusion: Rejec H 0.