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1 Chapter - The Smple Lnear Regresson Model The lnear regresson equaton s: where + β + β e for,..., and are observable varables e s a random error How can an estmaton rule be constructed for the unknown parameters β and β? The least squares prncple, also known as the method of ordnar least squares (OLS, s to fnd a soluton that mnmzes the sum of squared errors: S e ( β β ote: negatve and postve errors get equal weght. Econ 36 - Chapter
2 Ths s a mnmzaton problem. Soluton s a calculus eercse. Mnmzng ponts b and b are obtaned as the soluton to the frst order condtons: S 0 β S and 0 β Dfferentaton gves: S β ( β β β β and S β ( β β β β Econ 36 - Chapter
3 To obtan the mnmzng pont set the above equatons to 0 and evaluate at b and b. Ths gves: and b b 0 b b 0 ow dvde through b and rearrange terms to get the normal equatons: b + b ( b + b ( The normal equatons can be solved. From equaton ( rearrange terms to get: b b (3 b where and are the sample means. That s Equaton (3 s the ntercept estmator. 3 Econ 36 - Chapter
4 4 Econ 36 - Chapter Substtute (3 nto ( to get: + b b ( Rearrange to get: b ote, the above used the result: ow solve to get the slope estmator: b (4 Equatons (3 and (4 are called the ordnar least squares (OLS estmators for β and β.
5 The estmators b and b gve an estmaton rule. How are these results used? Collect a numerc data set for an applcaton of nterest. and are now the numerc observatons. Do the calculatons n equatons (3 and (4. Ths gves the numerc estmates denoted b b and b. These numbers are called ordnar least squares (OLS estmates. Statstcal ote A numerc data set can be vewed as one sample from the populaton. Suppose repeated samplng from the populaton was possble. A dfferent sample wll have a dfferent set of numerc data. Ths means that dfferent samples wll eld dfferent least squares pont estmates. 5 Econ 36 - Chapter
6 It s useful to note that the slope estmator can be epressed n a number of equvalent was. Equaton (4 can be wrtten as: b (4a Another equvalent formula s: ( ( b (4b ( 6 Econ 36 - Chapter
7 7 Econ 36 - Chapter Equaton (4b can be used to get Equaton (4 b usng the results: ( ( ( + + and ( (
8 Another statement for the slope estmator s: b cov(, var( where the sample varance s: var( ( and the sample covarance s: cov(, ( ( ote the dvsor s. 8 Econ 36 - Chapter
9 The least squares ftted values or predcted values are: ŷ b + b for,..., The least squares resduals are: ê ŷ b b for,..., Propertes of least squares estmaton The least squares ftted lne passes through the sample means, The average value of ŷ s the sample mean. That s, ŷ The sum of the resduals s zero. That s, ê 0 ote: the above propertes requre that an equaton ntercept ncluded n the lnear regresson equaton. β s 9 Econ 36 - Chapter
10 The ntercept estmate b Interpretng the Estmates The ntercept estmate ma not have a meanngful economc nterpretaton f the sample observatons do not have values around 0. Eample For the household ependture functon s weekl household ependture on food (n dollars and s weekl ncome (n $00. The estmated lnear regresson equaton s: ŷ for,..., 40 As a frst guess, ou ma sa that a household wth zero ncome (0 wll spend about $83.4 each week on food. Thnk agan. An nspecton of the data set shows that, for the sample of 40 households, the mnmum weekl household ncome s 3.69 ( $369 and the mamum ncome s ($3,340. The ftted regresson lne ma not be useful for predctng food ependture at levels of ncome below the mnmum observed value or eceedng the mamum level n the data set. 0 Econ 36 - Chapter
11 The slope estmate b Ths gve a margnal mpact the estmated ncrease n the mean of the dependent varable for a one unt ncrease n the eplanator varable. Eample For the estmated food ependture equaton the slope estmate s 0.. The economc nterpretaton s: for a tpcal household, weekl ependture on food ncreases b about $0. for an addtonal $00 n ncome. For reportng purposes, values can be rescaled. For eample, an equvalent statement s: a $0 ncrease n weekl household ncome leads to an ncrease n weekl ependture on food b about $.04. Econ 36 - Chapter
12 Elastct Economsts are famlar wth the measure of elastct defned as the percentage ncrease n for a one percent ncrease n. An elastct can be obtaned as: ε dln( dln( d d β Ths wll var at ever sample observaton. How can a summar measure be found? A fast method s to evaluate the elastct at the sample means (,. Ths gves an elastct estmate calculated as: ε ˆ b For the household food ependture eample: 9.60 ε ˆ b Ths sas that a % ncrease n household ncome wll lead, on average, to a 0.7% ncrease n weekl food ependture. An estmated ncome elastct less than one suggests that food s a necesst rather than a luur good. Econ 36 - Chapter
13 There can be more than one method for gettng results. Here s another method for estmatng an elastct from the least squares estmaton results. Defne the observatons: z for,..., The sample mean s: An elastct estmate s: z z ε ˆ Z z b For the food ependture estmaton results: Z ε ˆ z b ( Ths method gves a dfferent numercal answer compared to the prevous calculaton. 3 Econ 36 - Chapter
14 Assessng the Least Squares Estmators The model of economc behavour s epressed as the lnear regresson equaton: where + β + β e for,,..., and are observable varables s the dependent varable s the eplanator varable β and β are unknown parameters (coeffcents β s the ntercept coeffcent β s the slope coeffcent e s a random error The method of ordnar least squares (OLS fnds an estmaton rule for β and β to mnmze the sum of squared errors: S e ( β β 4 Econ 36 - Chapter
15 Soluton gves the least squares (OLS estmators b and b. The predcted or ftted values are ŷ b + b for,..., The resduals are ê ŷ b b for,..., The estmators b and b are functons of the and. e s vewed as a random varable. Therefore s a random varable and b and b are also random varables and ther statstcal propertes can be analzed. 5 Econ 36 - Chapter
16 To establsh some statstcal results a number of assumptons are requred. The standard assumptons are: ( The lnear regresson equaton s correctl specfed as: β + β + e ( E(e 0 for all (,,..., Ths sas the random errors have zero mean. That s, an omtted varables that are captured n e do not sstematcall affect the mean value of. 6 Econ 36 - Chapter
17 (3 var( e σ (sgma-squared for all Ths sas equal error varance for all observatons. Ths s called homoskedastct (equal spread. ote that assumpton ( mples var(e E [( e E(e ] ( Ee for all (4 e,e 0 cov( j for all j Ths sas the covarance between an two errors s zero. ote that assumpton ( mples cov(e,e j E E(e [( e E(e ( e E(e ] e j The correlaton between two errors s defned as: cov(e var(e,e j var(e j Ths shows zero covarance s equvalent to uncorrelated errors. j j 7 Econ 36 - Chapter
18 (5 must have at least dfferent values. That s, var( > 0 (5* s non-random or non-stochastc. That s, the values are fed n repeated samplng. Ths means cov(e, E E(e 0 [( e E(e ( E( ] ( Ths sas the error s uncorrelated wth the eplanator varable. The above assumptons can now be used to establsh the statstcal propertes of the least squares (OLS estmator. b. The focus of the presentaton wll be the slope estmator Smlar results can be obtaned for the ntercept estmator b. 8 Econ 36 - Chapter
19 If the standard assumptons are satsfed then b s an unbased estmator of β. That s, E(b β Ths result can be shown. Introduce w for,,..., ( A result s ( 0 Therefore w 0 Also w w ( ( ( 9 Econ 36 - Chapter
20 That s, the w have the propertes w 0 and w ow state the slope estmator as: b ( ( w ( w w ( w Ths shows that b s a lnear functon of the a weghted average of the wth the w as weghts. Use assumpton ( to substtute for the to get b β w ( β w + β + β + e w + we β + we 0 Econ 36 - Chapter
21 Take epectatons to fnd E(b β + E w e β + we(e use assumpton (5* β use assumpton ( E(e 0 for all Ths sas the slope estmator b s an unbased estmator of β. What does ths mean? Wth a sample of numerc data a slope estmate can be calculated. Ths estmate wll be smaller or larger than the true unknown populaton value β. Another sample of observatons wll eld a dfferent slope estmate that agan wll be smaller or larger than the true parameter. In repeated samplng, the average of all the calculated slope estmates wll equal β. Econ 36 - Chapter
22 The varance of the slope estmator can be found as follows. var( b var( β + we var( we w var(e β s a constant use assumptons (5* and (4 σ w use assumpton (3 var( e σ σ ( ( substtute for w σ ( Econ 36 - Chapter
23 The varance gves a measure of the precson of the estmator. Inspecton of the varance formula shows the followng: an ncrease n sample sze generall leads to lower varance. Ths holds snce ( ncreases as ncreases. Ths gves ncreased precson of the estmator. the greater the varablt n the more precse s the estmator. Ths holds snce the varance of the slope estmator can be epressed as: var( b σ ( var( the smaller the varablt n (as reflected n more precse s the estmator. σ the 3 Econ 36 - Chapter
24 The Gauss-Markov Theorem It has been shown that the least squares (OLS estmator b s a lnear unbased estmator of β. Here, lnear means that b s a weghted average of the. The Gauss-Markov theorem sas: If the standard set of assumptons s satsfed, then the least squares estmator has mnmum varance n the class of lnear unbased estmators. That s, the least squares estmator s BLUE (Best Lnear Unbased Estmator. Best means mnmum varance. 4 Econ 36 - Chapter
25 What does ths mean? Suppose * b s another lnear unbased estmator of β. That s, b * k where the k are some weghts that are dfferent from the w. Also E (b * β The Gauss-Markov theorem sas * > var( b var(b If an of the standard assumptons are volated then the least squares method ma not be the best. There ma be an estmator wth lower varance but t s ether based or not lnear n. 5 Econ 36 - Chapter
26 Estmatng the Varance of the Error Term ow take another look at the varance of the slope estmator: var( b σ ( The error varance σ s unknown. An estmator for σ s needed. The sum of squared resduals s: SSE ê ( b b An unbased estmator for σ s: σˆ ê SSE The dvsor s the number of degrees of freedom n the sum of squares. The degrees of freedom (df s the number of ndependent peces of nformaton used to comple the sum of squares from observatons. For ths applcaton, two degrees of freedom are lost for the two estmated parameters (the ntercept and the slope. 6 Econ 36 - Chapter
27 ow replace the unknown error varance var( b as: v âr(b σˆ ( The standard error of the slope estmator b s: se(b vâr(b σ to get an estmator for 7 Econ 36 - Chapter
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