Ecoometrc Methods Revew of Estmato Estmatg the populato mea Radom samplg Pot ad terval estmators Lear estmators Ubased estmators Lear Ubased Estmators (LUEs) Effcecy (mmum varace) ad Best Lear Ubased Estmators (BLUEs) Estmators v estmates Sample statstcs as estmators Sample mea, varace, stadard devato, covarace ad correlato Cofdece tervals as terval estmators Start wth a stadard estmato problem: Estmatg the populato mea a Cosder some populato wth a dstrbuto of, say, heghts, characterzed by the radom varable The mea, μ, ad varace, σ, of are ukow ou wat to estmate μ, the average heght the populato b Radom Samplg - A stadard approach Sample depedetly tmes from ths populato ad use the data to estmate the ukow mea Each of the depedet radom draws from the dstrbuto, s tself a radom varable, Gve the ature of the samplg process, the s are d (depedetly ad detcally dstrbuted) wth dstrbuto A estmator of μ wll be some fucto of the observed values of the s, ad wll accordgly, be a radom varable A pot estmator wll geerate a pot estmate (dvdual value) for each data set A terval estmator wll geerate a rage of estmates, defed by a lower boud ad a upper boud So we essetally have pot estmates for the lower ad upper bouds of the terval estmator Recall that we use upper case letter to deote radom varables ad lower case letters to deote the actual sample values
Ecoometrc Methods Revew of Estmato v We are terested a pot estmate: The samplg process wll geerate partcular observatos { y, y,, y }, ad the actual estmate wll be the value of the pot estmator for the gve set of draw values c Lear Estmators Lear estmators wll have the geeral lear fuctoal form: M = β0 + β + β + + β d Lear Ubased Estmators (LUEs) Ubased: Ths lear estmator wll be ubased f EM ( ) = μ, whch s to say, the expected value of the estmator s the true mea μ Eve though we do t kow the true mea μ, we ca ofte determe whether or ot a estmator of μ s fact based or ot For the actual draw sample { } y, y,, y, the partcular lear estmate ˆm wll be mˆ = β0 + βy + βy + + βy ˆm may or may ot be close to μ but o average, estmates geerated ths fasho wll equal μ f M s a ubased estmator of μ Sce the s are all d wth dstrbuto, each has mea μ, ad so the expected value of M s: EM ( ) = β0 + βμ + βμ + + βμ = β0 + βμ = β0 + μ β v We restrct atteto to lear ubased estmators ad so we oly wat to cosder { β } that satsfy β0 + μ β = μ, rrespectve of what the partcular value of μ happes to be v Sce β0 + μ β = μ for all μ, f you dfferetate both sdes wth respect to μ, you fd that for the lear estmator to always be ubased, β = 0ad β = 0 = (the tercept term must be 0 ad the slope coeffcets must sum to ) v So f we restrct atteto to the set (or class) of lear ubased estmators of μ, we are cosderg oly estmators of the form: M = β+ β + + β, where β = = v Ths defes the class of LUEs (Lear Ubased Estmators) ths problem
Ecoometrc Methods Revew of Estmato e Varace of the LUE Cosder the lear ubased estmator M ust defed Sce the s are parwse depedet, Cov(, ) = 0 for, ad the varace of the sum s the sum of the varaces Ad so, Var( M ) = β Var( ) + β Var( ) + + β Var( ) Ad sce Var( ) = σ for each, ( ) Var M = σ β f Best Lear Ubased Estmator (BLUE) The Best Lear Ubased Estmator wll be the estmator the class of LUEs that has mmum varace So we wat to cosder all lear estmators of the form M = β+ β + + β, where β =, ad fd the partcular set of { } = β that mmzes the varace wth ths group/class of estmators Ths sets up the optmzato problem: m Var( M ) = σ β subect to β = = v Ths s a costraed optmzato problem The optmum wll be defed by: Var( M ) Var( M ) * * = σ β = σ β (for all ad ) ad * β = = Some tuto: If oe partal effect s less tha aother, the take weght from the secod ad add that weght to the frst, ad the overall varace wll decrease So at Var( M ) the mmum varace, all of the partal effects, the s, must be the same * * v Ths mples β = β for all ad, ad sce they sum to, β = for all But the M s ust the Sample Mea: M = = For a partcular sample { } * y, y,, y, mˆ = y = y may or may ot be close to μ but o average, estmates geerated ths fasho wll equal μ, sce M = s a ubased estmator of μ The Sample Mea estmator, M = =, s a radom varable, takg o dfferet values wth dfferet probabltes depedg o the actual draw sample The dstrbuto of M s called a samplg dstrbuto 3
Ecoometrc Methods Revew of Estmato v So: Estmators are radom varables (pot) Estmates are umbers (the value of the estmator for the partcular sample observed) g Sce the Sample Mea s ubased ad has mmum varace the class of LUE's, t s a BLUE the best (mmum varace) estmator the class of lear ubased estmators (LUE s) Notce that ths result holds, eve f we do't kow the actual value of σ Also otce that ths holds for ay dstrbuto (we have't yet sad aythg about the partcular dstrbuto of ) More geerally a Estmators: Suppose that you have a d radom sample { },, from the dstrbuto,, ad you wat to use ths data to estmate some ukow parameter of the dstrbuto, θ A estmator wll be a fucto of the ' s, say W = h(,, ), ad wll tself be a radom varable (takg o dfferet values wth dfferet probabltes) b Ubasedess: A estmator s ubased f, o average, t s rght or more formally, f t s expected value s the true value of the parameter EW ( ) = θ We ve show above that the Sample Mea s a ubased estmator of the true mea, μ The bas of the estmator s the dfferece betwee ts expectato ad the true value of the ukow parameter: Bas( W ) = E( W ) θ 3 Some commo estmators a Mea: The sample mea, =, s a ubased estmator ( fact, t s BLUE) of the mea of, μ (see above) E ( ) = E ( ) = μ The partcular estmated sample mea s y = y b Varace: The sample varace, S = ( ), s a ubased estmator of the varace of, σ, whe the mea of s ukow: ES ( ) = Var ( ) = σ Sometmes we wrte ths estmator as S or S We dvde by - to geerate a ubased estmator There are crcumstaces uder whch you mght wat to dvde by, or eve + but wth large samples, the cosequetal dffereces are small 4
Ecoometrc Methods Revew of Estmato The partcular estmated sample varace s s = syy = ( y y) c Stadard devato: The sample stadard devato s the square root of the sample varace, S = S = ( ) It s geerally a based estmator of the Stadard Devato of, σ, sce geeral, the expected value of the square root of somethg s ot equal to the square root of the expected value of that somethg But that fact does t stop us from usg t The partcular estmated stadard devato s s = sy = y y ( ) d Covarace: The sample covarace, S = ( X X)( ), s a ubased estmator of the Covarace of X ad, ES ( ) = Cov (, ) = σ, whe the meas of X ad ( μ X ad μ ) are ukow The partcular estmated sample covarace s sxy = ( x x)( y y) S e Correlato Estmator: The sample correlato estmator, ˆ ρ =, s geerally a based SXS σ estmator of the correlato of X ad, corr( X, ) = ρ = σ σ 4 Effcecy of (ubased) estmators a Cosder two ubased estmators of θ, W ad W The W s more effcet tha W f for every parameter value θ, Var( W) Var( W), ad for at least oe value of θ, Var( W) < Var( W) (so W ever has hgher varace) b Why restrct to ubased estmators? Otherwse, t s easy to fd (really bad) estmators wth zero varace, as was show class 5 Cofdece tervals as terval estmators a Cofdece tervals are the most commo terval estmators b Cosder the pot estmator L l(,, ) terval ad U u( ) = for the lower boud of the cofdece =,, for ts upper boud c The radomly geerated cofdece terval [L,U] wll be a terval estmator, where the specfc values of the terval edpots wll vary wth the observed data sample d Note that f t s the case that 95% of the tme, tervals geerated ths fasho cota the true mea μ, the we have a 95% cofdece terval (estmator) for μ X 5