The world is not a straight line, but we may be able to approximate economic relationships by a straight line
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- Winfred Potter
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1 Lectre 3 What do we know now? The world s not a straght lne, t we may e ale to approxmate economc relatonshps y a straght lne If so then can se the dea of Ordnary Least Sqares (OLS whch gves the est straght lne (the est ft to the data y N mnmsng the sm of sqared resdals If we do ths then the eqatons that gve the OLS estmate of the ntercept and slope of the straght lne are Cov(, Var(
2 Today Go over how to nterpret the meanng of an OLS estmate Look at some algera that gves some more ntton aot what OLS s dong Come p wth a smmary statstc that tells s how closely the OLS straght lne captres the real world varaton Fnd ot why OLS has sch a good reptaton as an estmaton technqe
3 So Cov(, Var( are how the compter determnes the sze of the ntercept and the slope respectvely n an OLS regresson The OLS eqatons gve a nce, clear nttve meanng aot the nflence of the varale on the sze of the slope, snce t shows that: the greater the covarance etween and, the larger the (asolte vale of the slope the smaller the varance of, the larger the (asolte vale of the slope
4 It s eqally mportant to e ale to nterpret the effect of an estmated regresson coeffcent Gven OLS essentally passes a straght lne throgh the data, then gven d d So the OLS estmate of the slope wll gve an estmate of the nt change n the dependent varale y followng a nt change n the level of the explanatory varale (so yo need to e aware of the nts of measrement of yor varales n order to e ale to nterpret what the OLS coeffcent s tellng yo
5 Rememer every tme yo estmate an O<LS regresson there wll e a predcted vale (forecast y and a resdal for each oservaton
6 Shold get sed to smmarsng and plottng the predcted and resdal vales
7 Resdals total nmer of semnars attended Ftted vales fnal QM mark 5 5 total nmer of semnars attended Becase hardly any the vales le exactly on the predcted straght lne, there are oth nder and over predctons Ths s reflected n the graph of resdals whch are scattered aove and elow zero (Note oth the the forecast vales and the resdal vales are measred n terms of marks ganed
8 Let s play darts No. Let s do some more algera
9 PROPERTIES OF OLS 6 Usng the fact that for any ndvdal oservaton,, the ols resdal s the dfference etween the actal and predcted vale S. n So that and dvdng y N N N N Smmng over all N oservatons n the data set
10 6 and snce _ ( _ N N N So the mean vale of the OLS resdals s zero (as any resdal shold e, snce random and npredctale y defnton Snce the sm of any seres dvded y the sample sze gves the mean, can wrte _
11 The nd sefl property of OLS s that the mean of the OLS predcted vales eqals the mean of the actal vales n the data (so OLS predcts average ehavor n the data set another sefl property Ths also means that the OLS regresson lne passes throgh the mean of the dependent varale
12 Proof: smmng Dvdng y N n n n _ We know from aove that _ so
13 The 3 rd sefl reslt s that Cov(, the covarance etween the ftted vales of and the resdals mst e zero Proof: See Prolem Set
14 GOODNESS OF FIT Now we know how to smmarse the relatonshps n the data sng the OLS method, we next need a smmary measre of how well the estmated OLS lne fts the data Thnk of the dsperson of all possle y vales (the varaton n eng represented y a crcle And smlarly the dsperson n the range of x vales
15 The more the crcles overlap the more the varaton n the data explans the varaton n y Lttle overlap n vales so not explan mch of varaton n Large overlap n vales so varale explans mch of varaton n
16 To derve a statstcal measre whch does mch the same thng rememer that ( Usng the rles on covarances (see prolem set we know that Var( Var( Var( Var( Cov(, Var( Var( So the varaton n the varale of nterest, var(, s explaned y ether the varaton n the varales nclded n the OLS model, Var( or y varaton n the resdal Var( 9
17 So we se the rato var( R var( As a measre of how well the model fts the data. (R s also known as the coeffcent of determnaton So R measres the % of varaton n the dependent varale explaned y the model. If the model explans all the varaton n y then the rato eqals If the model explans none of the varaton then the rato = So the closer the rato s to one the etter the ft.
18 9 It s more common however to se one frther algerac adjstment. The /n s common to oth sdes, so can cancel ot and sng the reslts that _ ( ( ( n n n _ Gven ( says that Var( Var( ( Var Can wrte ths as ( ( Then we have It follows that
19 The left sde of the eqaton s the sm of the sqared devatons of aot ts sample mean. Ths s called the Total Sm of Sqares. The rght hand sde conssts of the sm of sqared devatons of the predctons arond the sample mean (the Explaned Sm of Sqares ( ( and the Resdal Sm of Sqares TSS ESS RSS From ths can have an alternatve defnton of the goodness of ft R _ ( y y _ ( y y ESS TSS RSS TSS 9
20 Can see from aove that t mst hold that <= R <= when ESS =, then R = (and model explans none of the varaton n the dependent varale when ESS = TSS, then R = (and model explans all of the varaton n the dependent varale In general the R les etween these two extremes. o wll fnd that for cross-secton data (e samples of ndvdals, frms etc the R are typcally n the regon of. for tme-seres data (e samples of aggregate (whole economy data measred at dfferent ponts n tme the R are typcally n the regon of.9
21 Look at the R n the two regressons. The frst ses tme seres data, the second ses cross-secton data
22 GOODNESS OF FIT So the R measres the proporton of varance n the dependent varale explaned y the model Another sefl nterpretaton of the R s that t eqals the sqare of the correlaton coeffcent etween the actal and predcted vales of Proof: We know the formla for the correlaton coeffcent r, S. In for Cov(, Var( Var( (actal vale = predcted vale pls resdal r, Cov([ ], Var( Var( 43
23 GOODNESS OF FIT Expand the covarance terms Cov([ ], Var( Var( Cov(, Cov(, Var( Var( Var( Var( Var( (snce already proved Cov(, And can always wrte any varance term as sqare root of the prodct Var( Var( Var( Var( 43
24 GOODNESS OF FIT Cancellng terms Var( Var( Var( Var( Var( Var( so rxy R Ths the correlaton coeffcent s the sqare root of R. Eg R =.5 mples correlaton coeffcent etween varale & varale (or etween and predcted vales =.5 =.5 43
25 So whle we wold lke the R to e as hgh as possle yo can only compare R n models wth the SAME dependent ( varale
26 Var( Var(, Cov(, Cov(, Cov(, Cov(, Cov(, Cov(, Cov( ],[ Cov(, Cov(, Cov(, Cov( ], Cov([, Cov( e e e e e the covarance etween the ftted vales of and the resdals mst e zero., Cov( The 3 rd sefl reslt s that
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