e views session 2

8/10/2019 e Views Session 2

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Principles of Econometrics class of October 14th FEUNLNotes by Jos Mrio Lopes1

1 !eteros"e#asticity in a cross$section frame%or" &e'amples from chapter ()

Heteroskedasticity happens whenever Var(ui|x1,x2,) is not constant for allobservations

!ast class, you"ve seen how to robustify your standard errors when you suspect to be in

the presence of heteroskedasticity #n the $ultiple re%ression $odel,

uxxxy kk +++++= 2211&

'ou would have to co$pute

2

1

2

2(

((

x

n

i i

ij

jSST

r

Var ==

here ri*denotes the ithresidual fro$ re%ressin% x*on all other independent variables

(see section +2)

#n Views, this can be done by choosin% on the -.ptions/ 0enu, the hite standard

errors

1#f you find any typo in these notes, please e$ail $e so # can correct it

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obust standard errors and t statistics are appropriate as the sa$ple si3es increases e

don"t always use these robust standard errors because, in s$all sa$ples, the robust t

statistics can depart a lot fro$ the t distribution

Hence, it isi$portant to know whether there is or there isn"t heteroskedasticity in our

sa$ple !et"s perfor$ a few exa$ples pickin% exa$ples fro$ the book 4ake theexa$ple on the de$and for ci%arettes, fro$ chapter + Open the corresponding workfile

e wish to esti$ate the de$and for ci%arettes $easured by the nu$ber of ci%arettes

s$oked per day as a function of inco$e, the price of a pack of ci%arettes, education,

a%e, s5uared a%e and the presence of a ban on restaurants fro$ the state the person

surveyed lives

e %et the followin% results6

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neither inco$e nor ci%arette price is si%nificant and their i$pacts would be s$allanyway (e%, if inco$e increases by 1&7, ci%s increases by (&++&81&&)91&:&&++

ci%arettes per day);

education reduces s$okin%;

s$okin% increases with a%e up until approx fter that, it falls

*+t no%, a -ery important .+estion/ is there heteros"e#asticity0 f so, the +s+al

stan#ar# errors an# t statistics %ill be %ron2 an# OL3 %ill not be efficient e %ill

perform 5+st a co+ple tests to chec" for heteros"e#asticity 3ee other tests a-ailable

on E6ie%s

First, let7s r+n the *re+sch$Pa2an test for heteros"e#asticity6

1) sti$ate the $odel by .!?, keep the s5uared .!? esti$ated residuals

2) un an auxiliary re%ression of the s5uared .!? esti$ated residuals on the

independent variables @eep the s5uared fro$ this re%ression

=) Aor$ either the A (followin% a A(k,nk1)) or the !0 (followin% a chis5uare with k

de%rees of freedo$) #f the pvalue is %reater than B7, we do not re*ect the null

of ho$oskedasticity

#n Views, this is very easy to do

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Cehold how $any options you have for runnin% a heteroskedasticity testD


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Aor a CE test, we %et

Coth the A test and the !0 (obs9s5uared of the auxiliary re%ression) conclude for the

re*ection of the null of ho$oskedasticity

'ou should check that Views is doin% this ri%ht HowF Generate the residuals yourself

and perfor$ the re%ression as usual (ew .b*ect85uation, etc) 'ou will %et the sa$eoutput as above

hite test for heteros"e#asticitytakes into account the possibility that the variance

structure $i%ht be richer 4he s5uares and crossproducts of the independent variables

are also included in the ri%hthand side >lternatively, whenever you have too $any

independent variables, you can use the fitted values of the dependent variable and the

s5uared fitted values of the independent variable

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#n our case, you %etHeteroskedasticity Test: White

F-statistic 2.159258 Prob. F(25,781) .9

!bs"#-s$%ared 52.172&5 Prob. 'hi-$%are(25) .11

caed e*+aied 11.81 Prob. 'hi-$%are(25) .

4his $eans that the null of ho$oskedasticity is re*ectedAro$ this point, we can correct the standard errors usin% the hite robust standard

errors

.r, we can transfor$ the $odel and run .!? on this transfor$ed $odel HowF

Feasible 8enerali9e# Least 3.+aresprocedure6

%enerate the esti$ated s5uared residuals (the residuals fro$ the $odeluxxy kko ++++= 11 ;

re%ress the lo% of the esti$ated s5uared residuals on the independent variables (why

the lo%F), obtain the fitted values of this re%ression(

g

exponentiate the fitted values to %et )exp(

gh =

esti$ate the e5uation uxxy kko ++++= 11 by L3, usin%(

81 has wei%hts

?ince we have to esti$ate h, AG!? will not be unbiased but it is consistent and

asy$ptotically $ore efficient than .!?

#f ci%sIresids5 stands for the esti$ated h, we have to divide the $odel by

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18s5uare root(h) hyF ?ee book (there is a univariate exa$ple there for savin%s, start

fro$ there)

e will %et

e+edet /ariabe: '03#('04#03F)

6ethod: east $%aresate: 1129 Tie: 17:

a+e: 1 87

0c%ded obseratios: 87

'oe;;iciet td. rror t-tatistic Prob.

13#('04#03F) 5.5&71 17.81& .15&& .7517

!(0?23#('04#03F) -.527 .99 -5.9897 .#T>=#


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'ou can actually see several %raphs at the sa$e ti$e if you select a 8ro+pof variables

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!et"s esti$ate a si$ple $odel, now

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=he lo2 of the price seems to be si2nificant >o+ may thin" this is O?, b+t it is not

*oth -ariables are tren#in2 thro+2ho+t the sample

f yo+ ta"e a loo" at the resi#+als, yo+ can see if %hat yo+7re #oin2 ma"es sense or

not

4hey are not stationary (there are for$al tests to see this, na$ely unit root tests like the

Dickey-Fuller or Phillips-Perrontests and you can always look at the correlo%ra$ of

the residuals) 4his $eans we should rethink your specification .ur previous re%ression

wassp+rio+s

e no% a## a linear tren# to ta"e acco+nt of the tren#in2 beha-io+r of LN6P@

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!E#M does not co$e si%nificant any$ore e conclude that there are other factors

beyond the price that are captured by the linear trend that see$ to be i$portant2

Notice that these other factors are not mo#elle# 5+st by a##in2 a linear tren#

Moreo-er, the fact that a linear tren# appears to be informati-e sho+l#n7t promptyo+ to 2et carrie# a%ay an# start obsessi-ely a##in2 a h+2e train of tren# terms

&linear, .+a#ratic,A)

hat we just did !adding a linear trend" has a detrending interpretation# it is

e$ui%alent to regressing all %aria&les o%er a constant and a linear trend' sa%ing the

residuals and regressing the residuals of the dependent %aria&le regression o%er the

residuals of the independent %aria&les regressions !see &ook"(

:: mportant ass+mptions an# problems in a =ime 3eries frame%or"/

=he 8a+ss$Mar"o- theorem re.+ires both homos"e#asticity an# absence of

serially correlate# errors Other%ise, the OL3 estimator %ill not be *LUE an# the

+s+al stan#ar# errors an# t$statistics %ill no lon2er be -ali#

!o% #o %e test for the presence of serial correlation0

!et"s see a few possibilities available in Views

'ou can take a look at the Nurbinatson statistic, that appears at the botto$ of the

results

e+edet /ariabe: 0d@%sted #-s$%ared .959 .. de+edet ar .1725&

.. o; reAressio .1&&1 >kaike i;o criterio -.97&252

% s$%ared resid .8&75 chBarC criterio -.851

oA ikeihood 2.&59 F-statistic 1.797

Durbin-Watson stat 1.048727 Prob(F-statistic) .29

4he B+rbin$atson test, valid under classical assu$ptions, is based on the .!?

residuals and one can show that N is approxi$ately 2(1) where is the firstorder

2'ou should always test the residuals to see if they"re wellbehaved #n this case, they are stillnonstationary #n a practical work, you should keep on lookin% for a correct specification

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correlation coefficient between residuals at tand residuals at t-) #f the Nurbinatson

statistic is near 2, the correlation coefficient will be near & Hence, we are lookin% for a

value si%nificantly below 2 (for a positive correlation coefficient) and si%nificantly

above 2 (for a ne%ative correlation coefficient) #$a%ine you were testin% if was close

to 3ero (N close to 2) a%ainst an alternative hypothesis that was bi%%er than 3ero

(N s$aller than 2) 4here are two critical values, d!and dO, tabled by ?avin and hite(1LKK), dependin% on the nu$ber of observations and the nu$ber of re%ressors 4his

$eans that, if N falls between d!and dO, the results are inconclusive

>fter you esti$ate your $odel, you have a ?erial correlation $enu under esidual

4ests

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4his is the *re+sch$8o#frey test 4he ull is of absence of autocorrelation Here, we

re*ect this ull6 there is evidence to say there is autocorrelation e basically are

keepin% the residuals of the re%ression, and re%ressin% u tover ut1, ut2, and the

re%ressors #f the A statistic re*ects this $odel, we conclude that there is no

autocorrelation=

.nce you find out that here is firstorder serial correlation, you can transfor$ the $odel

to take this into account61 P esti$ate the ori%inal $odel and take the esti$ated residuals

2 P run the re%ression of Qt over Qt1 to co$pute the correlation coefficient

= P Aor everyvariable xt(and for the dependent variable), co$pute the 5uasi

differenced variable xtxt1pply .!? to the e5uation with the 5uasidifferenced variables 4he usual standard

errors, t statistics and A are asy$ptotically valid

e views session 2

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