ECONOMICS 35* -- NOTE ECON 35* -- NOTE Specfcaton -- Aumpton of the Smple Clacal Lnear Regreon Model (CLRM). Introducton CLRM tand for the Clacal Lnear Regreon Model. The CLRM alo known a the tandard lnear regreon model. Three et of aumpton defne the CLRM.. Aumpton repectng the formulaton of the populaton regreon equaton, or PRE. Aumpton A. Aumpton repectng the tattcal properte of the random error term and the dependent varable. Aumpton A-A4 3. Aumpton repectng the properte of the ample data. Aumpton A5-A8 ECON 35* -- Note : Specfcaton of the Smple CLRM Page of 6 page
ECONOMICS 35* -- NOTE Fgure. Plot of Populaton Data Pont, Condtonal Mean E(Y X), and the Populaton Regreon Functon PRF Y Ftted value Weekly conumpton expendture, $ 75 5 5 75 PRF = β + β X 5 6 8 4 6 8 4 6 Weekly ncome, $ Recall that the old lne n Fgure. the populaton regreon functon, whch take the form f (X ) = E(Y X ) = β + β X. For each populaton value X of X, there a condtonal dtrbuton of populaton Y value and a correpondng condtonal dtrbuton of populaton random error u, where () each populaton value of u for X = X and u X = Y E(Y X ) = Y β β X, () each populaton value of Y for X = X Y X = E(Y X ) + u = β + β X + u. ECON 35* -- Note : Specfcaton of the Smple CLRM Page of 6 page
ECONOMICS 35* -- NOTE. Formulaton of the Populaton Regreon Equaton (PRE) Aumpton A: The populaton regreon equaton, or PRE, take the form Y = β X + β + u or Y = β + βx + u (A) The PRE (A) gve the value of the regreand (dependent varable) Y for each value of the regreor (ndependent varable) X. The ubcrpt on Y and X are ued to denote ndvdual populaton or ample value of the dependent varable Y and the ndependent varable X. The PRE (A) tate that each value Y of the dependent varable Y can be wrtten a the um of two part.. A lnear functon of the ndependent varable X that called the populaton regreon functon (or PRF). The PRF for Y take the form f(x ) = β + β X where β and β are regreon coeffcent (or parameter), the true populaton value of whch are unknown, and X the value of the regreor X correpondng to the value Y of Y.. A random error term u (alo called a tochatc error term). Each random error term u the dfference between the oberved Y value and the value of the populaton regreon functon for the correpondng value X of the regreor X: u = Y f(x ) = Y ( β + β X ) = Y β β X The random error term are unobervable becaue the true populaton value of the regreon coeffcent β and β are unknown. ECON 35* -- Note : Specfcaton of the Smple CLRM Page 3 of 6 page
ECONOMICS 35* -- NOTE The PRE (A) ncorporate three dtnct aumpton. Y = β + β X + u (A) A.: Aumpton of an Addtve Random Error Term. The random error term u enter the PRE addtvely. Techncally, th aumpton mean that the partal dervatve of Y wth repect to u equal :.e., Y u = for all ( ). A.: Aumpton of Lnearty-n-Parameter or Lnearty-n-Coeffcent. The PRE lnear n the populaton regreon coeffcent β j (j =, ). Th aumpton mean that the partal dervatve of Y wth repect to each of the regreon coeffcent a functon only of known contant and/or the regreor X ; t not a functon of any unknown parameter. Y β j = f ( X ) j =, where f ( X ) contan no unknown parameter. j j A.3: Aumpton of Parameter or Coeffcent Contancy. The populaton regreon coeffcent β j (j =, ) are (unknown) contant that do not vary acro obervaton. Th aumpton mean that the regreon coeffcent β and β do not vary acro obervaton --.e., do not vary wth the obervaton ubcrpt. Symbolcally, f β j the value of the j-th regreon coeffcent for obervaton, then aumpton A.3 tate that β j = β j = a contant (j =, ). ECON 35* -- Note : Specfcaton of the Smple CLRM Page 4 of 6 page
ECONOMICS 35* -- NOTE 3. Properte of the Random Error Term Aumpton A: The Aumpton of Zero Condtonal Mean Error The condtonal mean, or condtonal expectaton, of the random error term u for any gven value X of the regreor X equal to zero: E ( u X) = or E( u X ) Th aumpton ay two thng: = (for all X value) (A). the condtonal mean of the random error term u the ame for all populaton value of X --.e., t doe not depend, ether lnearly or nonlnearly, on X;. the common condtonal populaton mean of u for all value of X zero. Implcaton of Aumpton A Implcaton of A. Aumpton A mple that the uncondtonal mean of the populaton value of the random error term u equal zero: or ( u X) E = ( u) E ( u X ) = ( u ) E = (A-) E =. (A-) Th mplcaton follow from the o-called law of terated expectaton, whch E E u X = E u. Snce E ( u X) = by A, t follow that tate that [ ( )] ( ) E ( u) E[ E( u X) ] = E[ ] = =. The logc of (A-) traghtforward: If the condtonal mean of u for each and every populaton value of X equal zero, then the mean of thee zero condtonal mean mut alo be zero. ECON 35* -- Note : Specfcaton of the Smple CLRM Page 5 of 6 page
ECONOMICS 35* -- NOTE Implcaton of A: the Orthogonalty Condton. Aumpton A alo mple that the populaton value X of the regreor X and u of the random error term u have zero covarance --.e., the populaton value of X and u are uncorrelated: or ( u X) E = Cov ( X,u) E( Xu) = E ( u X ) = Cov( X u ) E( X u ) = (A-), = = (A-). The equalty ( X,u ) E( X u ) Cov = n (A-) follow from the defnton of the covarance between X and u, and from aumpton (A): Cov ( X,u ) E{ [ X E(X )][ u E(u )]} = E{ [ X E(X )] u} = E[ X u E(X )u ] = E(X u ) E(X )E(u ) = E(X u ) by defnton nce E(u ) = by A and A - nce E(X ) a contant nce E(u ) = by A and A -.. Implcaton (A-) tate that the populaton random error term u have zero covarance wth, or are uncorrelated wth, the correpondng populaton regreor value X. Th aumpton mean that there ext no lnear aocaton between u and X. Note that zero covarance between X and u mple zero correlaton between X and u, nce the mple correlaton coeffcent between X and u, denoted a ρ X, u ), defned a ( Cov(X, u ) Cov(X, u ) ρ (X, u ) =. (X )(u ) d(x )d(u ) From th defnton of ρ X, u ( ),.e., = ( X, u ρ ) Cov( X, u ), t obvou that = mple that Cov ( X, u ) = ( X, u ) = ρ. ECON 35* -- Note : Specfcaton of the Smple CLRM Page 6 of 6 page
ECONOMICS 35* -- NOTE Implcaton 3 of A. Aumpton A mple that the condtonal mean of the populaton Y value correpondng to a gven value X of the regreor X equal f X = β + β X : the populaton regreon functon (PRF), ( ) E ( u X) = E( Y X) f ( X) = β + βx E( u X ) = E( Y X ) f ( X ) = β + βx = (A-3) =. (A-3) Proof: Take the condtonal expectaton of the PRE (A) for ome gven X : Y = β + β X + u ( =,, N) (A) E(Y = β X ) = E( β = E( β + β X X ) + E(u X ) + β X X ) + β X nce E(u X ) = by aumpton A nce E( β + β X X ) = β + β X. Meanng of the Zero Condtonal Mean Error Aumpton A Each value X of X dentfe a egment or ubet of the relevant populaton. For each of thee populaton egment or ubet, aumpton A ay that the mean of the random error u zero. In other word, for each populaton egment, potve and negatve value of u "cancel out" o that the average value of the random error u for each populaton value X of X equal zero. Aumpton A rule out both lnear dependence and nonlnear dependence between X and u; that, t requre that X and u be tattcally ndependent. The abence of lnear dependence between X and u mean that X and u are uncorrelated, or equvalently that X and u have zero covarance. But lnear ndependence between X and u not uffcent to guarantee the atfacton of aumpton A. It poble for X and u to be both uncorrelated, or lnearly ndependent, and nonlnearly related. Aumpton A therefore alo requre that there be no nonlnear relatonhp between X and u. ECON 35* -- Note : Specfcaton of the Smple CLRM Page 7 of 6 page
ECONOMICS 35* -- NOTE Volaton of the Zero Condtonal Mean Error Aumpton A The random error term u repreent all the unknown, unobervable and unmeaured varable other than the regreor X that determne the populaton value of the dependent varable Y. Anythng that caue the random error u to be correlated wth the regreor X wll volate aumpton A: Cov( X,u) or ( X,u) ρ ( u X) If X and u are correlated, then ( u X) Note that the convere not true: Cov ( X,u) = or ( X,u) = E. E mut depend on X and o cannot be zero. ρ doe not mply that ( u X) E =. Reaon: Cov ( X, u) meaure only lnear dependence between u and X. But any nonlnear dependence between u and X wll alo caue E ( u X) to depend on X, and hence to dffer from zero. So Cov ( X,u) = not enough to nure that aumpton A atfed. Common caue of correlaton or dependence between X and u --.e., common caue of volaton of aumpton A.. Incorrect pecfcaton of the functonal form of the relatonhp between Y and X. Example: Ung Y a the dependent varable when the true model ha ln(y) a the dependent varable. Or ung X a the ndependent varable when the true model ha ln(x) a the ndependent varable.. Omon of relevant varable that are correlated wth X. 3. Meaurement error n X. 4. Jont determnaton of X and Y. ECON 35* -- Note : Specfcaton of the Smple CLRM Page 8 of 6 page
ECONOMICS 35* -- NOTE Aumpton A3: The Aumpton of Contant Error ance The Aumpton of Homokedatc Error The Aumpton of Homokedatcty The condtonal varance of the random error term u are dentcal for all obervaton (.e., for all populaton value X of X), and equal the ame fnte potve contant σ for all : or ( u X) E( u X) = σ > = (A3) ( u X ) E( u X ) = σ = (for all X value) (A3) > where σ a fnte potve (unknown) contant. The frt equalty n (A3) follow from the defnton of the condtonal varance of u and aumpton (A): { X} { } ( u X ) E [ u E(u X ) ] = E [ u ] X = E( u X ). by defnton becaue E(u X ) = by aumpton A Implcaton of A3: Aumpton A3 mple that the uncondtonal varance of the random error u alo equal to σ : [ ] = E( u ) = σ ( u ) E ( u E(u )) where ( u ) E( ) =. u = becaue E(u ) = by A-. By aumpton A and A3, E( u X) σ By the law or terated expectaton, [ E( u X )] E( u ) Thu, =. E =. ( u ) E( u ) = E[ E( u X )] = E[ σ ] = σ =. ECON 35* -- Note : Specfcaton of the Smple CLRM Page 9 of 6 page
ECONOMICS 35* -- NOTE Implcaton of A3: Aumpton A3 mple that the condtonal varance of the populaton value Y of the regreand Y correpondng to any gven value X of the regreor X equal the contant condtonal error varance σ : ( u X ) = σ ( Y X ) σ = (A3-) Proof: Start wth the defnton of the condtonal varance of Y for ome gven X : { X} { X } ( Y X ) E [ Y E(Y X ) ] = E [ Y β βx ] = E( u X ) = σ by defnton nce E(Y X ) = β nce u nce E u + β X β X by A by A ( X ) = σ by aumpton A3. = Y β Meanng of the Homokedatcty Aumpton A3 For each populaton value X of X, there a condtonal dtrbuton of random error, and a correpondng condtonal dtrbuton of populaton Y value. Aumpton A3 ay that the varance of the random error for X = X the ame a the varance of the random error for any other regreor value X = X (for all X X ). That, the varance of the condtonal random error dtrbuton correpondng to each populaton value of X are all equal to the ame fnte potve contant σ. ( u X ) ( u X ) = σ = for all X X. > Implcaton A3- ay that the varance of the populaton Y value for X = X the ame a the varance of the populaton Y value for any other regreor value X = X (for all X X ). The condtonal dtrbuton of the populaton Y value around the PRF have the ame contant varance σ for all populaton value of X. ( Y X ) ( Y X ) = σ = for all X X. > ECON 35* -- Note : Specfcaton of the Smple CLRM Page of 6 page
ECONOMICS 35* -- NOTE Aumpton A4: The Aumpton of Zero Error Covarance The Aumpton of Nonautoregreve Error The Aumpton of Nonautocorrelated Error Conder the random error term u and u ( ) correpondng to two dfferent populaton value X and X of the regreor X, where X X. Th aumpton tate that u and u have zero condtonal covarance: Cov( u, u X, X ) = Euu ( X, X) = (for all X X ) (A4) The frt equalty n (A4) follow from the defnton of the condtonal covarance of u and u and aumpton (A): Cov ( u, u X, X ) E{ [ u E(u X )][ u E(u X )] X, X } = ( u u X, X ) E(u X ) E(u by defnton E nce = X ) by A. = The econd equalty n (A4) tate the aumpton that all par of error term correpondng to dfferent value of X have zero covarance. Implcaton of A4: Aumpton A4 mple that the condtonal covarance between the populaton value Y of Y when X = X and the populaton value Y of Y when X = X where X X equal to zero: (,, ) = Cov( Y Y X X ) Cov u u X X,, = X X. ECON 35* -- Note : Specfcaton of the Smple CLRM Page of 6 page
ECONOMICS 35* -- NOTE Proof: () Begn wth the defnton of the condtonal covarance for Y and Y for gven X and X value: nce (,, ) {[ ( ) ][ ( ) ], } = Euu ( X, X) Cov Y Y X X E Y E Y X Y E Y X X X Y E(Y X ) = Y β β X = u by aumpton A and A, and mlarly Y E(Y X ) = Y β β X = u by aumpton A and A. () Therefore (,, ) (, ) Cov Y Y X X = E u u X X = by aumpton A4. Meanng of A4: Aumpton A4 tate that the random error term u correpondng to X = X have zero covarance wth, or are uncorrelated wth, the random error term u correpondng to any other regreor value X = X, where X X. Equvalently, aumpton A4 tate that the populaton value Y of Y correpondng to X = X have zero covarance wth, or are uncorrelated wth, the populaton value Y of Y correpondng to X = X for any dtnct par of regreor value X X. Th mean there no ytematc lnear dependence or aocaton between u and u, or between Y and Y, where and correpond to dfferent obervaton (that, to dfferent regreor value X X ). The aumpton of zero covarance, or zero correlaton, between each par of dtnct obervaton weaker than the aumpton of ndependent random amplng A5 from an underlyng populaton. The aumpton of ndependent random amplng mple that the ample obervaton are tattcally ndependent. The aumpton of tattcally ndependent obervaton uffcent for the aumpton of zero covarance between obervaton, but tronger than neceary. ECON 35* -- Note : Specfcaton of the Smple CLRM Page of 6 page
ECONOMICS 35* -- NOTE 4. Properte of the Sample Data Aumpton A5: Random Samplng or Independent Random Samplng The ample data cont of N randomly elected obervaton on the regreand Y and the regreor X, the two obervable varable n the PRE decrbed by A. In other word, the ample obervaton are randomly elected from the underlyng populaton; they are a random ubet of the populaton data pont. Thee N randomly elected ample obervaton on Y and X are wrtten a the N par Sample data [(Y, X ), (Y, X ),... (Y N, X N )] (Y, X ) =,..., N. Implcaton of the Random Samplng Aumpton A5 The aumpton of random amplng mple that the ample obervaton are tattcally ndependent.. It thu mean that the error term u and u are tattcally ndependent, and hence have zero covarance, for any two obervaton and. Cov = ( u, ) Random amplng ( u, u X, X ) Cov =.. It alo mean that the dependent varable value Y and Y are tattcally ndependent, and hence have zero covarance, for any two obervaton and. Random amplng ( Y,Y X, X ) u Cov = ( Y, ) Cov =. The aumpton of random amplng therefore uffcent for aumpton A4 of zero covarance between obervaton, but tronger than neceary for A4. Y ECON 35* -- Note : Specfcaton of the Smple CLRM Page 3 of 6 page
ECONOMICS 35* -- NOTE When the Random Samplng Aumpton A5 Approprate? The random amplng aumpton uually approprate for cro-ectonal regreon model,.e., for regreon model formulated for cro ecton data. Defnton: A cro-ectonal data et cont of a ample of obervaton on ndvdual economc agent or other unt taken at a ngle pont n tme or over a ngle perod of tme. A dtnguhng charactertc of any cro-ectonal data et that the ndvdual obervaton have no natural orderng. A common, almot unveral charactertc of cro-ectonal data et that they uually are contructed by random amplng from underlyng populaton. The random amplng aumpton hardly ever approprate for tme-ere regreon model,.e., for regreon model formulated for tme ere data. Defnton: A tme-ere data et cont of a ample of obervaton on one or more varable over everal ucceve perod or nterval of tme. A dtnguhng charactertc of any tme-ere data et that the obervaton have a natural orderng -- pecfcally a chronologcal orderng. A common, almot unveral charactertc of tme-ere data et that the ample obervaton exhbt a hgh degree of tme dependence, and therefore cannot be aumed to be generated by random amplng. ECON 35* -- Note : Specfcaton of the Smple CLRM Page 4 of 6 page
ECONOMICS 35* -- NOTE Aumpton A6: The number of ample obervaton N greater than the number of unknown parameter K: number of ample obervaton > number of unknown parameter N > K. (A6) Meanng of A6: Unle th aumpton atfed, t not poble to compute from a gven ample of N obervaton etmate of all the unknown parameter n the model. Aumpton A7: Noncontant Regreor The ample value X of the regreor X n the ample (and hence n the populaton) are not all the ame;.e., they are not contant: X c =,..., N where c a contant. (A7) Techncal Form of A7: Aumpton A7 requre that the ample varance of the regreor value X ( =,..., N) mut be a fnte potve number for any ample ze N;.e., (X X) ample varance of X ( X ) = = X >, N where X > a fnte potve number. Meanng of A7: Aumpton A7 requre that the regreor ample value X take at leat two dfferent value n any gven ample. Unle th aumpton atfed, t not poble to compute from the ample data an etmate of the effect on the regreand Y of change n the value of the regreor X. In other word, to calculate the effect of change n X on Y, the ample value X of the regreor X mut vary acro obervaton n any gven ample. ECON 35* -- Note : Specfcaton of the Smple CLRM Page 5 of 6 page
ECONOMICS 35* -- NOTE Aumpton A8: No Perfect Multcollnearty The ample value of the regreor n a multple regreon model do not exhbt perfect multcollnearty. Th aumpton relevant only n multple regreon model that contan two or more non-contant regreor. It nature examned later n the context of multple regreon model. ECON 35* -- Note : Specfcaton of the Smple CLRM Page 6 of 6 page