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ECO391 Lecture Handout Over 15 .3

Sp rin g 2 00 3 , G. H oy t

I. ! at i" #! e $e t! o d o% Le a" t S& u ar e " ' O r din a ry Le a" t S &u a re " (

). #!eory

*. +oru-aC. )pp-icat io n

II. Standard Error o% t!e E"tiate

I. Ordinary Lea"t S&uare" 'OLS( 'a- "o ca- -ed #!e $et!od o% Lea"t S&uare" (

). #!e #!eory

Ordi na ry leas t squa r es is a st ati sti ca l techni que that us es s ampl e dat a to e sti ma t e thetrue populat ion relationship between two var iables.

Recall that :1) E' i/ i( o 1 /i is the p op u-a t io n re gr e " " io n -in e

2) i'!at( o 1/ i is the " a p- e regr e " "i o n e& ua ti o n

OLS allows us to find o an d 1 .

onsider the followin! sca tt e r plo t dia!ram the shows the actual " observed da ta point s ina sample:

#

$

%any lines could fit throu!h these data point s. &e want to determine the line with the'best fit.'

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!a t do e" it ea n to "a y a -in e %it " t! e dat a t! e e "t

Rec all th at e i(hat) " the residu al" repre se nt s the dista nc e betw ee n the sa mpl ere!r ess ion line and the obse rved dat a poin t" ($ i"#i). he line t hat mini mi*es thesum of these dis tances is the one tha t !ives us the bes t fit.

+owever " some of the values of the residuals a re ne!at ive in si!n while o ther s a re posi tive. ,f we su m th e re siduals" posi tive values will ca ncel ou t ne!ative values sothe sum will not accurately reflect the total amount of error .

o solve thi s problem we square the residuals before we add them to!ether .

#! e $e t! o d o% -e a" t " &u ar e " 4 'OLS( produces a line that mini mi*es the sum of thesquared ver ti ca l dis tances from the line to the observed data poin ts .

i.e. it minimi*es Σ e i2 - e 1

2 e 22 e /

2 . .. .. .. .. e n2 " where n is the sample si*e

(hats over all of the e 0s)he sum of t he resi dua ls (uns qua r ed ) is eact ly *er o. (Lat er " you can us e this bit ofinformation to chec your wor.)

*. + or u- a " Ho6 do e " OLS g e t e "t i a t e " o% t! e co e% %i ci e n t "

e i2 i" a-" o ca-- ed t! e re "i du a- "u o% " &u ar e " 'SSE (. #!i" i" t !e a ou nt

t! at 6 e 6 an t t o in i i7 e .

SSE e i'!at(2 '1(

' i ' i'!at( ( 2 '2( ' i o 1/ i(

2 '3(

3ow consi der equation (/). , am !oin ! to as you to try to re memb er a littl e calculu s.&e can consider (/) as a mathema t ical function of b o an d b 1" f(b o" b 1)

&e want to minimi* e the su m of the squ ar ed error ter ms so we wan t to minimi* eequa ti on (/). ,n t er ms of calcul us this means we want to find the critical poin ts of afunction. &e want to find the values of b o and b 1 that minimi*e the function. o do thiswe t ae a firs t der iva tive of function (/) with r espect to b o a n d set it e qu al to *ero an dsolve for b o. &hen we do this we !et the function:

n

X b-Y =b ii

o

ΣΣ1

'8(

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(4)

,f we t ae the firs t der iva tive of (/) with r espect to b 1 a nd se t it eq ual to *ero a nd th ensolve for b 1 we !et the followin! equation:

(5)

6qua ti ons (4) and (5) !ive us the for mulas that we need to find the val ues of b o a nd b 1

th at es ti ma t e th e tru e po pul ati on rela tio ns hi p b et we e n $ an d #. ,f we plu! (5)" th eformula for b 1" into (4) " the formula for b o" we may al so writ e b o as follows:

(3ot e that Σ$2≠ (Σ$)2) 6quations (4) 7 (8) !ive us b o and b 1 solely in terms or $ and #

(sample data .)

) X ( - X n

Y X -Y X n=b

2

ii

iiii

ΣΣ

ΣΣΣ

21 '5(

) X ( - X n

Y X X -Y X =b

2

ii

iiiiio

ΣΣ

ΣΣΣΣ

2

2

'(

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. 6$9%L6:Let0 s t ry an eample. onsider the followin! dependen t and independ ent variables :

$i - the number of children in a family#i - th e nu mb e r of loav es of br ea d con su m e d by a fa mily in a !iv en thr ee we e

period

;amily<" n - 5

$, #i $i#i $i2 #i(hat) e i(hat) e i(hat)

2

1 2 4

2 / =

/ 1 /

4 5 >

5 > 1=

n - 5 Σ$i - Σ#i - Σ$i#i - Σ$i2 - Σe i(hat)

-Σe i(hat) 2

-

1( +ind o4

2(+ind 1 4

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/) &rite out the full sample re!ression line? ,nterpret the coefficients " b o and b 1.

8( :redict ion4

@iven that $i - 8" p redi ct the value that we epect #i to t ae !iven our sample r e!ress ionline. (i.e. find #i(hat) .) omplete the sith column of the table.

5( Ca-cu-ate t!e re" idua-" 4

Recall e i(hat) - #i 7 #i(hat) (;ill in the seventh column of the table.)

hec: he sum of the res iduals should be approimately *ero" Σe i(hat) - A.

8) +ind e i'!at(2 or SSE : (omplete the ei!hth column of the table.)

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)-ternative +oru-a"4

he formulas for the estimated coeffici ent s can be manipula ted and writ ten in a vari ety ofways. +ere are a f ew other alt erna t ives . One se t of alt erna t ives a re the formulas !ivenin the tet .

( )( )[ ]( ) 2

1

X X

Y Y X X b

i

ii

−Σ

−−Σ=

n

X X and

X n X

Y X nY X b

iii Σ

=−Σ

−Σ= "

221

n

Y Y and X bY b

i

o

Σ=−= "1

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15./ #! e S ta n d a r d Error o% t !e E"t i a t e ' S e (

2

B2

−

Σ=

n

eS i

e

#!i" e a" ur e appr o;i a t e " t!e avera g e di"t an ce o% t!e rea- dat a point " %ro t!ee "t i a t e d re gr e " " i o n -in e.

i) S e i " ea " ur e d in t !e "a e uni t o% e a " ur e a" t! e vari a -e . So if the # variable is

m ea s u r ed in d oll ar s a nd S e - C>. 2/ " on av er a !e " our a ct ua l da ta poi nt s va ry fro m th ei r

est imated values by about C>.2/.

i i) S e c an e u" ed a" a ea " ur e o% t! e &ua-i t y o% %it o% t!e "a p -e re gr e" " io n -ine .

#! e " a -- e r t! e S e , t! e et t e r t! e %it .

iii) )n a -t e rn a t i v e % or u - a % or S e :

2

1

2

−

Σ−Σ−Σ=

n

Y X bY bY S iiioi

e

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