Lecture 2: The Simple Regression Model
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1 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. A good derstadg o the assmptos of the sgle lear regresso model wll help o to derstad the mportace of each assmpto of the mltvarate regresso model. Defe the smple lear regresso SLR model: where s a depedet varable, s a depedet varable or a covarate, ad s the error term or dstrbace. We wat to obta kow two parameters ad. These two parameters are called tre vales that ca ot be observed. What we ca do s to estmate these two parameters. We deote the estmated parameters as ad. We ma call them as estmated coeffcets or estmators. There are several was to estmate the two parameters. The most commo method s the least sqared method. Least Sqared Resdals Method: Defe ad : kow parameters or coeffcets ad : Least Sqared estmates b ad b : coeffcets that are sed to obta ad b ad b are two varables that cold take a vales. We se these two varables whle we are searchg for ad. Whe the sm of sqared resdals s mmzed, b ad b wold be eqal to ad, respectvel. Defe b b s the resdal.
2 Mmzg Problem: Fd a par of b ad b that mmzes the sm of sqared resdals: m b b Frst order codtos F.O.C. wth respect to w.r.t b ad b are w.r.t. b w.r.t. b Notce that b ad b are replaced b ad becase the frst order dervatves are set to be zero ad. From, we have From, B sg, we have Ε b
3 3 We ca orgaze ths to fd, Ths, the least sqared estmates are 3 4 Ubasedess of SLR I ths sb-secto, we eame the baasedess of the smple ler regresso SLR estmates. To show the baasedess, we eed for assmptos, whch are specfed below. We eed to derstad wh we eed each assmpto ad how the SLR estmates wll be based whe oe of the for assmptos s volated. Assmptos: SLR Lear parameters: SLR Radom samplg ad are radom SLR 3 Zero codtoal mea SLR 4 Sample varato Ε - >
4 4 From 4, we have Here, we have sed SLR. 5 At ths pot, we ca see that: f the epected vale of the secod term s zero, the the epected vale of wll be eqal to, the kow tre vale. B takg the epectato of the both sdes of 5, we have 6 Net, we have mportat steps: becase s jst a costat. B the defto, we kow that Cov.
5 Uder SLR3, we assme Cov, ths we have. Ths, we have Therefore der assmptos SLR-4, the least sqared estmate s based. Whe s based, s also based. However, SLR 3 s a ver strog assmpto especall a sgle lear regresso model becase the dstrbace,, cldes so ma mportat omtted varables. Whe SLR 3 s volated, we have a omtted varable problem. We wll dscss abot the omtted varables problem the et lectre. For ow, we smpl assme that the for assmptos are satsfed. Varaces of OLS stmators For the OLS estmates to be the most effcet estmates amog ma other estmates, we eed to add oe more assmpto: SLR 5 Homoskedastct: Var Φ The homoskedastct assmpto dcates that the sze of the varace of s costat or does ot deped o. Whe ths assmpto s volated, the we sa the error term ehbts heteroskedastct. Note that eve der the heteroskedastct, the least sqared estmates are ot based as we have show prevosl. For ow, we assme the homoskedastct ad obta the varaces of estmates. B sg 4, we have Var From SLR5, Var Φ Var [ ] 5
6 6 Var 7 Note: the larger the Φ, the larger the Var, ad the larger the varace, the smaller the Var. Check the tetbook page 56 for the varace of the estmated costat term, Var. The Stadard rror of We ca get the stadard devato of b takg the postve sqared root of the varace: sd Var. However, Φ s kow. Ths, we have to estmate Φ. The based estmator of Φ s SSR. Ths s adjsted b the - degrees of freedom. The degree of freedom s - becase we have observatos wth two estmators, whch clde a tercept. Ths the stadard error of s / SSR se
7 The stadard error for ca be obtaed b replacg the tet. wth eqato.58 R-sqared The R-sqared s the fracto of the sample varato that s eplaed b. R SS / SST SSR / SST SS: the eplaed sm of sqares; SSR: the resdal sm of sqares; SST: the total sm of sqares. SST SS SSR Regresso Throgh the Org Sppose we have a model sch as. Least Sqared Approach: m b Frst order codtos F.O.C. wth respect to w.r.t b ad b are ~ w.r.t. b ~ 8 7
8 Method of Momets Approach To take the method of momet approach, we start wth two assmptos. The frst assmpto s ot strog. Bt the secod assmpto s ver strog. For the least sqared estmates to be based, however, we stll eed to assme the same secod assmpto. Ths, these assmptos are also eeded for least sqared estmates to be based. Both approaches reach the same estmates. Two assmptos are: 9 Ths we have two restrctos ad two kow parameters ad. Ths we are able to solve these two eqatos. These are eactl the same as ad. Ths we obta the same estmates: 3 ad 4. 8
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