Econometrics Analysis

8/18/2019 Econometrics Analysis

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Part 2: Projection and Regression-1/45

Econometrics I

Professor William Greene

Stern School of Business

Department of Economics


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Econometrics I

Part 2 – Projection and

Regression


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Statistical Relationship

Objective: Characterize the ‘relationship’between a variable of interest an a set of!relate! variables

Context: "n inverse eman e#uation$ P % α & β' & γ ($ ( % income) P an ' are

two obviousl* relate ranom variables) Weare intereste in stu*in+ the relationship

between P an ') B* ‘relationship’ we mean ,usuall*- covariation)

,Cause an effect is problematic)-


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Bivariate Distribution - Model for a Relationsi!

Bet"een #"o $ariables We mi+ht posit a bivariate istribution for P an '$ f,P$'-

.ow oes variation in P arise/

With variation in '$ an

0anom variation in its istribution) 1here e2ists a conitional istribution f,P3'- an a

conitional mean function$ E4P3'5) 6ariation in P arisesbecause of

6ariation in the conitional mean$

6ariation aroun the conitional mean$ ,Possibl*- variation in a covariate$ ( which shifts the

conitional istribution


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Conditional Moments

1he conitional mean function is the regressionfunction) P % E4P3'5 & ,P 7 E4P3'5- % E4P3'5 & ε E4ε3'5 % 8 % E4ε5) Proof: ,1he 9aw of iterate

e2pectations- 6ariance of the conitional ranom variable % conitional

variance$ or the scedastic function.

" trivial relationship; ma* be written as P % h,'- & ε$

where the ranom variable ε % P7h,'- has zero mean b*construction) 9oo


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Part 2: Projection and Regression-%/45

Sample Data (Experiment)


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5 !"ser#ations on P and $

Sho%ing &ariation o' P rond E*P+


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&ariation rond E*P,$+

(Conditioning Redces &ariation)


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Means o' P 'or -i#en -rop Means o' $


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nother Conditioning &aria"le


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Conditional Mean .nctions

=o re#uirement that the* be >linear> ,wewill iscuss what we mean b* linear-

Conitional ?ean function: h,@- is thefunction that minimizes E@$(4( A h,@-5

=o restrictions on conitional variances atthis point)



Projections and Regressions

We e2plore the ifference between the linear proection anthe conitional mean function

* an 2 are two ranom variables that have a bivariateistribution$ f,2$*-)

Suppose there e2ists a linear function such that * % α & β2 & ε where E,ε32- % 8 % Cov,2$ε- % 8 1hen$

Cov,2$*- % Cov,2$α- & βCov,2$2- & Cov,2$ε-

% 8 & β 6ar,2- & 8 so$ β % Cov,2$*- 6ar,2- an E,*- % α & βE,2- & E,ε- % α & βE,2- & 8 so$ α % E4*5 7 βE425)



Regression and Projection

Does this mean E4*325 % α & β2/ =o) 1his is the linear projection of * on 2

Ft is true in ever* bivariate istribution$whether or not E4*325 is linear in 2)

* can alwa*s be written * % α & β2 & ε where ε ⊥ 2$ β % Cov,2$*- 6ar,2- etc)

1he conitional mean function is h,2- such that

* % h,2- & v where E4v3h,2-5 % 8) But$ h,2-oes not have to be linear)

1he implication: What is the result of linearl*re+ressin+ * on $; for e2ample usin+ leasts#uares/



Data 'rom a /i#ariate Poplation



0he 1inear Projection Compted

" 1east S3ares


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1inear 1east S3ares Projection

----------------------------------------------------------------------

Ordinary least squares regression ............

LHS=Y Mean = 1.21632

Standard deviation = .3752

!u"#er o$ o#servs. = 1%% Model si&e 'ara"eters = 2

(egrees o$ $reedo" = )

*esiduals Su" o$ squares = .5+

Standard error o$ e = .31)7

,it *-squared = .2))12

dusted *-squared = .2)%)6

--------/-------------------------------------------------------------

0aria#le oe$$iient Standard 4rror t-ratio 't8 Mean o$ 9

--------/-------------------------------------------------------------

onstant .)336)::: .%6)61 12.15% .%%%%

9 .2+51::: .%3%5 6.2) .%%%% 1.556%3

--------/-------------------------------------------------------------


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0he 0re Conditional Mean .nction True Conditional Mean Function E[y|x]

X

.35

.70

1.05

1.40

1.75

.00

1 2 30

E X P E C T D Y


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0he 0re Data -enerating Mechanism

*at does least s+uares ,estiate.


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pplication: Doctor &isits

German Fniviual .ealth Care ata: =%$HI

?oel for number of visits to the octor:

1rue E463Fncome5 % exp,J)KJ 7 )8KLMincome-

9inear re+ression: +M,Fncome-%H)NJ 7 )8OMincome


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Conditional Mean and Projection

Notice the problem with the linear approach. Neati!epre"iction#.


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Representing the Relationship

Conitional mean function: E4* 3 x5 % +,x-

9inear appro2imation to the conitional mean function:9inear 1a*lor series evaluate at x8

1he linear proection ,linear re+ression/-

δ δk

0 K 0 0

k=1 k k k

K 00 k=1 k k

ĝ( ) =g( )! [g | = ](x "x ) = ! (x "x )

$ $ $ $

=γ + Σ γ

γ =

K

k k 0 1 k k

0

"1

g#(x)= (x "E[x ])

E[y]

$ar[ ]% &Co'[ y]%$ $ {


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Representations o' 4


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Smmar

Regression function: E4*325 % +,2-

Projection: +M,*32- % a & b2 whereb % Cov,2$*-6ar,2- an a % E4*57bE425Proection will e#ual E4*325 if E4*325 islinear)

Taylor Series Approximation to +,2-


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Part 2: Projection and Regression-2%/45

0he Classical 1inear Regression Model

1he model is * % f,2J$2$$2Q$βJ$β$βQ- & ε

% a multiple regression moel ,as oppose to

multivariate-) Emphasis on the multiple; aspect of

multiple re+ression) Fmportant e2amples: ?ar+inal cost in a multiple output settin+ Separate a+e an eucation effects in an earnin+s e#uation)

Rorm of the moel A E4*3x5 % a linear function of x)

,0e+ressan vs) re+ressors- ‘Dependent and !independent variables)

Fnepenent of what/ 1hin< in terms of autonomous variation) Can * ust ‘chan+e/’ What ‘causes’ the chan+e/ 6er* careful on the issue of causalit*) Cause vs) association)

?oelin+ causalit* in econometrics


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Part 2: Projection and Regression-2&/45

Model ssmptions: -eneralities

"inearity means linear in the parameters) We’ll return tothis issue shortl*)

#dentifiability) Ft is not possible in the conte2t of themoel for two ifferent sets of parameters to prouce thesame value of E4*3x5 for all x vectors) ,Ft is possible for

some x)- Conditional expected value of t$e deviation of an

observation from the conitional mean function is zero %orm of t$e variance of the ranom variable aroun the

conitional mean is specifie =ature of the process b* which x is observe is not

specifie) 1he assumptions are conitione on theobserve x)

"ssumptions about a specific probabilit* istribution to bemae later)


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Part 2: Projection and Regression-2'/45

1inearit o' the Model

f,2J$2$$2Q$βJ$β$βK - % 2JβJ & 2β & & 2QβQ &otation: 2JβJ & 2β & & 2QβQ % x′β)

Bolface letter inicates a column vector) 2; enotes a

variable$ a function of a variable$ or a function of a setof variables) 1here are Q variables; on the ri+ht han sie of the

conitional mean function); 1he first variable; is usuall* a constant term)

,Wisom: ?oels shoul have a constant term unless

the theor* sa*s the* shoul not)- E4*3x5 % βJMJ & βM2 & & βQM2Q)

,βJMJ % the intercept term-)


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Part 2: Projection and Regression-2(/45

1inearit

Simple linear moel$ E4*3x5%x'

'uaratic moel: E4*3x5% & TJ2 & T2

9o+linear moel$ E4ln*3lnx5% & U< ln2


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Part 2: Projection and Regression-3)/45

1inearit

"inearity means linear in the parameters$not in the variables

E4*3x5 % βJ f J,- & β f ,- & & βQ f Q,-)

f


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ni3eness o' the Conditional Mean

1he conitional mean relationship must hol for an*set of = observations$ i % J$$=. "ssume$ that

= ≥ K ,ustifie later- E4*J3x5 % x(′β E4*3x5 % x)′β

E4*=3x5 % x&′β

"ll n observations at once: E4y*+, % +β - .β

)


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ni3eness o' E*,6+

=ow$ suppose there is a γ ≠ β that prouces thesame e2pecte value$

E 4y*+, % +γ - .γ

/

9et δ % β 7 γ) 1hen$+δ % +β 0 +γ - .

β

0 .γ

- 1)

Fs this possible/ + is an =×Q matri2 ,= rows$ Qcolumns-) What oes +δ % 1 mean/ We

assume this is not possible) 1his is the ‘fullrank’ assumption A it is an ‘ientifiabilit*’assumption) ltimatel*$ it will impl* that we can ‘estimate’ β) ,We have *et to evelop this)-1his re#uires = ≥ Q .


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1inear Dependence

E2ample: from *our te2t:

x % 4J $ =onlabor income$ 9abor income$ 1otal income5 ?ore formal statement of the uni#ueness conition:

&o linear dependencies: =o variable 2


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n Endring rt Mster

*0 do larger!aintings coand

iger !rices.

#e Persistence of

Meor0 alvador

Dali 1(31

#e Persistenceof conoetrics

reene 2)11

ra!ics so" relative

si6es of te t"o "or7s


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8n 9nidentified :But $alid;

#eor0 of 8rt 8!!reciationnanced Monet 8rea ffect Model< =eigt

and *idt ffects

>og:Price; ? @ A 1 log 8rea A

2 log 8s!ect Ratio A

3 log =eigt A

4 ignature A

C

:8s!ect Ratio ? =eigt/*idt; #is is a!erfectl0 res!ectable teor0 of art !rices

=o"ever it is not !ossible to learn about

te !araeters fro data on !rices areas

as!ect ratios eigts and signatures


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Part 2: Projection and Regression-3%/45

7otationDe%ne col&mn !ector# o' n ob#er!ation# on y an" the K

!ariable#.

= Xβ ( εThe a##&mption mean# that the ran) o' the matri$ X i# K .No linear "epen"encie# *+ ,- C/-0N 12N3 o' the matri$ X.

1 11 12 1 11

2 21 22 2 22

N1 N2 NK

εβ εβ = = × +

εβ

y

M M M M MM

K

K

K

N N K

y x x x

y x x x

y x x x


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Part 2: Projection and Regression-3&/45

!ected $alues of Deviations fro te

Eonditional Mean

Xbserve y will e#ual E4*3x5 & ranom variation)* % E4y 3x5 & ε ,isturbance-

Fs there an* information about ε in x/ 1hat is$ oesmovement in x provie useful information aboutmovement in ε/ Ff so$ then we have not full* specifiethe conitional mean$ an this function we are callin+ ‘E 4*3x5’ is not the conitional mean ,re+ression-

1here ma* be information about ε in other variables)But$ not in x) Ff E4ε3x5 ≠ 8 then it follows thatCov4ε$x5 ≠ 8) 1his violates the ,as *et still not full*efine- ‘inepenence’ assumption


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Part 2: Projection and Regression-3'/45

8ero Conditional Mean o' 9

E4ε3all ata in +5 % 8

E4ε3+5 % 1 is stron+er than E4εi 3 xi5 % 8

1he secon sa*s that


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Part 2: Projection and Regression-4)/45

Eonditional =ooscedasticit0 and

Gonautocorrelation

Disturbances provie no information about each other$whether in the presence of + or not)

6ar4ε3+5 % σ

#) Does this impl* that 6ar4ε5 % σ#/ (es:

Proof: 6ar4ε

5 % E46ar4ε

3+55 & 6ar4E4ε

3+55)

Fnsert the pieces above) What oes this mean/ Ft is anaitional assumption$ part of the moel) We’ll chan+e

it later) Ror now$ it is a useful simplification


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Goral Distribution o' 9

An assumption of limitedusefulness

se to facilitate finite sample erivations of certaintest statistics)

1emporar*)


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The 1inear Model

y % +β23$ = observations$ Q columns in +$ incluin+ acolumn of ones)

Stanar assumptions about +

Stanar assumptions about 3*+ .43*+,-15 .43,-1 and Cov435x,-1

0e+ression/

Ff E4y3+5 % +β then E4*3x5 is also the proection)


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Corn%ell and Rpert Panel DataCornwell an" 1&pert 1et&rn# to 4choolin Data 565 n"i!i"&al# 7 Year#8ariable# in the %le are

E*+ = ,ork ex-erience.K/ = ,eek0 ,orkedCC = occu-ation 1 i2 3lue collar

456 = 1 i2 7anu2acturing indu0try/8T9 = 1 i2 re0ide0 in 0out:/M/; = 1 i2 re0ide0 in a city (/M/;)M/ = 1 i2 7arriedFEM = 1 i2 2e7ale8545 = 1 i2 ,age 0et 3y union contractE6 = year0 o2 education

.;


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Speci'ication: $adratic E''ect o' Experience


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Econometrics Analysis

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