DISTURBANCE TERMS. is a scalar and x i

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1 DISTURBANCE TERMS I a feld of research desg, we ofte have the qesto abot whether there s a relatoshp betwee a observed varable (sa, ) ad the other observed varables (sa, x ). To aswer the qesto, we ma costrct the model whch depeds o x. Becase s ot ecessarl explaed ol b x from some reasos dscssed below, however, there alwas exsts the dscrepac betwee the observed vale of ad the predcted vale of obtaed from the model. The dscrepac s take as a dstrbace term or a error term. Sppose that sets of data, x, ), x, ),..., x, ), are observed ( ( ( s a scalar ad x s a vector (sa, k vector). We assme that there s a relatoshp betwee x ad, whch s represeted as the model: = f (x) f (x) s a fcto of x. We sa that s explaed b x or s regressed o x. s called the depedet or explaed varable ad x s a vector of the depedet or explaator varables. Sppose that a vector of the kow parameter (sa, β, whch s a k vector) s clded f (x). Usg the sets of data, we cosder estmatg β f (x). Addg a dstrbace term (sa,, whch s also called a error term), the relatoshp betwee ad x s gve b: = f ( x) +. The dstrbace term dcates the term that caot be explaed b x. Usall, x s assmed to be ostochastc. Note that x s sad to be ostochastc whe t takes a fxed vale. f (x) s determstc, whle s stochastc. f (x) has to be specfed b a researcher. Represetatvel, f (x) s ofte specfed as the lear fcto: f ( x) = xβ. The reasos wh we add the dstrbace term are as follows: () there are some predctable elemets of radomess hma resposes, () a effect of a large mber of omtted varables s cotaed x, () there s a measremet error, (v) a fctoal form of f (x) s ot kow geeral. More detals are as follows. For (), as a example, gross domestc prodct (GDP) data s observed as a reslt of hma behavor, whch s sall predctable ad s thoght of a sorce of radomess. For (), we caot kow all the explaator varables that deped o. Most of the varables are omtted, ad ol the mportat varables eeded for aalss are clded x. The flece of the omtted varables s thoght of a sorce of. For (), some kds of errors are clded almost all the data, ether becase of data collecto dffcltes or becase the explaed varable s heretl measrable ad

2 a prox varable has to be sed ts stead. For (v), covetoall we specf f (x) as: f ( x) = xβ. However, there s o reaso to specf the lear fcto. Exceptoall, we have the case where the fctoal form of f (x) comes from the derlg theoretcal aspect. Eve ths case, however, f (x) s derved from a ver lmted theoretcal aspect, ot ever theoretcal aspect. For smplct, hereafter, cosder the lear regresso model: = x β +, =,,...,. Whe,,..., are assmed to be mtall depedet ad detcall dstrbted wth mea zero ad varace, the sm of sqared resdals, = ( x β ), s mmzed wth respect to β. The, the estmator of β (sa, βˆ ) s: ˆ β = ( = x ) = x, whch s called the ordar least sqares (OLS) estmator. βˆ s kow as the best lear based estmator (BLUE). It s dstrbted = as: N( β, ( x x ) ) der ormalt assmpto o, becase βˆ s rewrtte as: ˆ β = β + ( = x ) = x. Note from the cetral lmt theorem that ( ˆ β β ) s asmptotcall ormall dstrbted wth mea zero ad varace M xx eve whe the dstrbace term s ot ormal we have to assme ) = ( / x M as goes to ft,.e., ( a b dcates that a approaches b ). xx O the dstrbace term, we have assmed as follows: () V ( ) = for all, () Cov (, ) = 0 for all, ad () Cov (, x ) = 0 for all ad. I the followg, we exame βˆ the case where each assmpto s volated. () Volato of the Assmpto: V ( ) = for all Whe the assmpto o varace of s chaged to V ( ) =,.e., heteroscedastc dstrbace term, the OLS estmator βˆ s o loger BLUE. The

3 varace of βˆ s gve b: = x ) ( x x )( V ( ˆ) β = ( x x ). Let b be a solto of mmzato of wth respect to β. The, ( ) / xβ = x / ) b = ( x / ad b ~ N( β,( x x / ) ) are derved = der ormalt assmpto o. We have the reslt that βˆ s ot BLUE becase of V ( b) V ( ˆ) β. The eqalt holds ol whe = for all. For estmato, has to be specfed, e.g., = zγ z represets a vector of the other exogeos varables. () Volato of the Assmpto: Cov (, ) = 0 for all The correlato betwee ad s called the spatal correlato the case of cross-secto data ad the atocorrelato or seral correlato tme seres data. Let ρ be the correlato coeffcet betwee ad ρ = for all = ad ρ = ρ for all. That s, we have Cov(, ) = ρ. The matrx that the (, ) th-elemet s ρ shold be postve defte. I ths stato, the varace of βˆ s: = x ) ( = = x x )( = V ( ˆ) ρ β = ( x x ). Let * b be a solto of the mmzato problem of ρ ( β )( β ) = = x x wth respect to β, where ρ deotes the (, ) th-elemet of the verse matrx of the matrx that the (, ) th-elemet s ρ. The, der ormalt assmpto o, we obta: b * = ( = = = * ρ x x ) ( ρ x ) ad b ~ N( β, ( ρ x x ) ). = = It ca be verfed that we obta the followg: V ( b * ) V ( ˆ β ). The eqalt holds ol whe ρ = for all = ad ρ = 0 for all. For estmato, we eed to

4 specf ρ. For a example, we ma take the followg specfcato: ρ =ρ, whch correspods to the frst-order atocorrelato case (.e., = ρ + ε ε s the depedetl dstrbted error term) tme seres data. For aother example, the spatal correlato model we ma take the form: ρ = whe s the eghborhood of ad ρ = 0 otherwse. () Volato of the Assmpto: Cov (, x ) = 0 for all ad If s correlated wth x for some ad, t s kow that βˆ s ot a based estmator of β,.e., = E ( ˆ) β β, becase of E (( x x ) x ) 0. I order to obta a cosstet estmator of β, we eed the codto: = ( / ) x 0 as. However, we have the fact: ( / ) x / 0 as the case of Cov (, x ) 0. Therefore, βˆ s ot a cosstet estmator of β,.e., ˆ β / β as. To mprove ths cosstec problem, we tlze the strmetal varable (sa, z ), whch satsfes the propertes: ( / ) z 0 ad z x ( / ) = / 0 as. The, t s kow that b = IV ( z x ) z s = a cosstet estmator of β,.e., b IV β as. b IV s called the strmetal varable (IV) estmator. It ca be also show that ( β ) s asmptotcall b IV ormall dstrbted wth mea zero ad varace M M M zx zz xz ) ) ( / z x M zx, / z z M zz = xz zx ( ad (/ ) x z M = M. As a example of z, we ma choose z ˆ = x as xˆ dcates the predcted vale of x whe x s regressed po the other exogeos varables assocated wth x, sg OLS. Hsash Tazak

5 See also Atocorrelato, Cetral Lmt Theorem, Regresso, Seral Correlato, Ubased Estmator. FURTHER READINGS Keed, P. (008). A Gde to Ecoometrcs (6th Ed.), Wle-Blackwell. Maddala, G.S. (00). Itrodcto to Ecoometrcs (3rd Ed.), Joh Wle & Sos Ic.

Lecture 2: The Simple Regression Model

Lecture 2: The Simple Regression Model Lectre Notes o Advaced coometrcs Lectre : The Smple Regresso Model Takash Yamao Fall Semester 5 I ths lectre we revew the smple bvarate lear regresso model. We focs o statstcal assmptos to obta based estmators.