econometrics analysis
TRANSCRIPT
-
8/18/2019 Econometrics Analysis
1/45
Part 2: Projection and Regression-1/45
Econometrics I
Professor William Greene
Stern School of Business
Department of Economics
-
8/18/2019 Econometrics Analysis
2/45
Part 2: Projection and Regression-2/45
Econometrics I
Part 2 – Projection and
Regression
-
8/18/2019 Econometrics Analysis
3/45
Part 2: Projection and Regression-3/45
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)-
-
8/18/2019 Econometrics Analysis
4/45
Part 2: Projection and Regression-4/45
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
-
8/18/2019 Econometrics Analysis
5/45
Part 2: Projection and Regression-5/45
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
-
8/18/2019 Econometrics Analysis
6/45
Part 2: Projection and Regression-%/45
Sample Data (Experiment)
-
8/18/2019 Econometrics Analysis
7/45Part 2: Projection and Regression-&/45
5 !"ser#ations on P and $
Sho%ing &ariation o' P rond E*P+
-
8/18/2019 Econometrics Analysis
8/45Part 2: Projection and Regression-'/45
&ariation rond E*P,$+
(Conditioning Redces &ariation)
-
8/18/2019 Econometrics Analysis
9/45Part 2: Projection and Regression-(/45
Means o' P 'or -i#en -rop Means o' $
-
8/18/2019 Econometrics Analysis
10/45Part 2: Projection and Regression-1)/45
nother Conditioning &aria"le
-
8/18/2019 Econometrics Analysis
11/45Part 2: Projection and Regression-11/45
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@$(4( A h,@-5
=o restrictions on conitional variances atthis point)
-
8/18/2019 Econometrics Analysis
12/45Part 2: Projection and Regression-12/45
Projections and Regressions
We e2plore the ifference between the linear proection anthe conitional mean function
* an 2 are two ranom variables that have a bivariateistribution$ 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)
-
8/18/2019 Econometrics Analysis
13/45Part 2: Projection and Regression-13/45
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 $; for e2ample usin+ leasts#uares/
-
8/18/2019 Econometrics Analysis
14/45Part 2: Projection and Regression-14/45
Data 'rom a /i#ariate Poplation
-
8/18/2019 Econometrics Analysis
15/45Part 2: Projection and Regression-15/45
0he 1inear Projection Compted
" 1east S3ares
-
8/18/2019 Econometrics Analysis
16/45Part 2: Projection and Regression-1%/45
1inear 1east S3ares Projection
----------------------------------------------------------------------
Ordinary least squares regression ............
LHS=Y Mean = 1.21632
Standard deviation = .3752
!u"#er o$ 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
--------/-------------------------------------------------------------
-
8/18/2019 Econometrics Analysis
17/45Part 2: Projection and Regression-1&/45
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
-
8/18/2019 Econometrics Analysis
18/45Part 2: Projection and Regression-1'/45
0he 0re Data -enerating Mechanism
*at does least s+uares ,estiate.
-
8/18/2019 Econometrics Analysis
19/45Part 2: Projection and Regression-1(/45
-
8/18/2019 Econometrics Analysis
20/45Part 2: Projection and Regression-2)/45
-
8/18/2019 Econometrics Analysis
21/45Part 2: Projection and Regression-21/45
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
-
8/18/2019 Econometrics Analysis
22/45
Part 2: Projection and Regression-22/45
Conditional Mean and Projection
Notice the problem with the linear approach. Neati!epre"iction#.
-
8/18/2019 Econometrics Analysis
23/45
Part 2: Projection and Regression-23/45
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]%$ $ {
-
8/18/2019 Econometrics Analysis
24/45
Part 2: Projection and Regression-24/45
Representations o' 4
-
8/18/2019 Econometrics Analysis
25/45
Part 2: Projection and Regression-25/45
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-
-
8/18/2019 Econometrics Analysis
26/45
Part 2: Projection and Regression-2%/45
0he Classical 1inear Regression Model
1he model is * % f,2J$2$$2Q$βJ$β$βQ- & ε
% a multiple regression moel ,as oppose to
multivariate-) Emphasis on the multiple; aspect of
multiple re+ression) Fmportant e2amples: ?ar+inal cost in a multiple output settin+ Separate a+e an eucation effects in an earnin+s e#uation)
Rorm of the moel A E4*3x5 % a linear function of x)
,0e+ressan vs) re+ressors- ‘Dependent and !independent variables)
Fnepenent of what/ 1hin< in terms of autonomous variation) Can * ust ‘chan+e/’ What ‘causes’ the chan+e/ 6er* careful on the issue of causalit*) Cause vs) association)
?oelin+ causalit* in econometrics
-
8/18/2019 Econometrics Analysis
27/45
Part 2: Projection and Regression-2&/45
Model ssmptions: -eneralities
"inearity means linear in the parameters) We’ll return tothis issue shortl*)
#dentifiability) Ft is not possible in the conte2t of themoel for two ifferent sets of parameters to prouce thesame value of E4*3x5 for all x vectors) ,Ft is possible for
some x)- Conditional expected value of t$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)
-
8/18/2019 Econometrics Analysis
28/45
Part 2: Projection and Regression-2'/45
1inearit o' the Model
f,2J$2$$2Q$βJ$β$βK - % 2JβJ & 2β & & 2QβQ &otation: 2JβJ & 2β & & 2QβQ % x′β)
Bolface letter inicates a column vector) 2; enotes a
variable$ a function of a variable$ or a function of a setof variables) 1here are Q variables; on the ri+ht han sie of the
conitional mean function); 1he first variable; is usuall* a constant term)
,Wisom: ?oels shoul have a constant term unless
the theor* sa*s the* shoul not)- E4*3x5 % βJMJ & βM2 & & βQM2Q)
,βJMJ % the intercept term-)
-
8/18/2019 Econometrics Analysis
29/45
Part 2: Projection and Regression-2(/45
1inearit
Simple linear moel$ E4*3x5%x'
'uaratic moel: E4*3x5% & TJ2 & T2
9o+linear moel$ E4ln*3lnx5% & U< ln2
-
8/18/2019 Econometrics Analysis
30/45
Part 2: Projection and Regression-3)/45
1inearit
"inearity means linear in the parameters$not in the variables
E4*3x5 % βJ f J,- & β f ,- & & βQ f Q,-)
f
-
8/18/2019 Econometrics Analysis
31/45
Part 2: Projection and Regression-31/45
ni3eness o' the Conditional Mean
1he conitional mean relationship must hol for an*set of = observations$ i % J$$=. "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
)
-
8/18/2019 Econometrics Analysis
32/45
Part 2: Projection and Regression-32/45
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 .
-
8/18/2019 Econometrics Analysis
33/45
Part 2: Projection and Regression-33/45
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
-
8/18/2019 Econometrics Analysis
34/45
Part 2: Projection and Regression-34/45
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
-
8/18/2019 Econometrics Analysis
35/45
Part 2: Projection and Regression-35/45
8n 9nidentified :But $alid;
#eor0 of 8rt 8!!reciationnanced Monet 8rea ffect Model< =eigt
and *idt ffects
>og:Price; ? @ A 1 log 8rea A
2 log 8s!ect Ratio A
3 log =eigt A
4 ignature A
C
:8s!ect Ratio ? =eigt/*idt; #is is a!erfectl0 res!ectable teor0 of art !rices
=o"ever it is not !ossible to learn about
te !araeters fro data on !rices areas
as!ect ratios eigts and signatures
-
8/18/2019 Econometrics Analysis
36/45
Part 2: Projection and Regression-3%/45
7otationDe%ne col&mn !ector# o' n ob#er!ation# on y an" the K
!ariable#.
= Xβ ( εThe a##&mption mean# that the ran) o' the matri$ 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
-
8/18/2019 Econometrics Analysis
37/45
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$x5 ≠ 8) 1his violates the ,as *et still not full*efine- ‘inepenence’ assumption
-
8/18/2019 Econometrics Analysis
38/45
Part 2: Projection and Regression-3'/45
8ero Conditional Mean o' 9
E4ε3all ata in +5 % 8
E4ε3+5 % 1 is stron+er than E4εi 3 xi5 % 8
1he secon sa*s that
-
8/18/2019 Econometrics Analysis
39/45
-
8/18/2019 Econometrics Analysis
40/45
Part 2: Projection and Regression-4)/45
Eonditional =ooscedasticit0 and
Gonautocorrelation
Disturbances provie no information about each other$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
-
8/18/2019 Econometrics Analysis
41/45
Part 2: Projection and Regression-41/45
Goral Distribution o' 9
An assumption of limitedusefulness
se to facilitate finite sample erivations of certaintest statistics)
1emporar*)
-
8/18/2019 Econometrics Analysis
42/45
Part 2: Projection and Regression-42/45
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)
-
8/18/2019 Econometrics Analysis
43/45
Part 2: Projection and Regression-43/45
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
.;
-
8/18/2019 Econometrics Analysis
44/45
Part 2: Projection and Regression-44/45
Speci'ication: $adratic E''ect o' Experience
-
8/18/2019 Econometrics Analysis
45/45