Econometrics I 8

7/25/2019 Econometrics I 8

1/50

Part 8: Hypothesis Testing-1/50

Econometrics I

Professor William Greene

Stern School of Business

Department of Economics


2/50


Econometrics I

Part 8 Interval Estimation

and Hypothesis Testing


3/50


Interval Estimation

point estimator of

We ac!no"le#ge the sampling varia$ility% Estimate# sampling variance

&sampling variaility ind!"ed y


4/50

Part 8: Hypothesis Testing-#/50

Point estimate is only the $est single guess

'orm an interval( or range of plausi$le values

Plausi$le

li!ely values "ith accepta$le #egreeof pro$a$ility%

To assign pro$a$ilities( "e re)uire a #istri$ution

for the variation of the estimator%

The role of the normality assumption for


5/50


*onfi#ence Interval$! the point estimate

St#%Err+$!, s)r-+./0$$123,!!4 v!

5ssume normality of for no": $!6 7+!(v!/, for the true !% 0$!2!19v! 6 7+(3,

*onsi#er a range of plausi$le values of !given the point

estimate $!% $!sampling error%

;easure# in stan#ar# error units( ?@

Aarger ?@greater pro$a$ility 0confi#enceC1 Given normality( e%g%( ?@ 3% F( ?@3%F Plausi$le range for !then is $! ?@ v!


6/50

Part 8: Hypothesis Testing-%/50

*omputing the *onfi#ence Interval

5ssume normality of for no": $!6 7+!(v!

/, for the true !%

0$!2!19v! 6 7+(3,

v! +./0$&$123,!!is not !no"n $ecause ./ must $eestimate#%

Jsing s/instea# of ./( 0$!2!19Est%0v!1 6 t+72K,%

0Proof: ratio of normal to s)r0chi2s)uare#19#f is pursue# in

your teLt%1

Jse critical values from t #istri$ution instea# of stan#ar#

normal% Will $e the same as normal if 7 M 3%


7/50Part 8: Hypothesis Testing-'/50

*onfi#ence Interval

*ritical t+%NF(/, /%F

*onfi#ence interval $ase# on t: 3%/NOF /%F @ %3F3*onfi#ence interval $ase# on normal: 3%/NOF 3% @ %3F3


8/50Part 8: Hypothesis Testing-8/50

Bootstrap *onfi#ence Interval'or a *oefficient


9/50Part 8: Hypothesis Testing-(/50

Bootstrap *I for Aeast S)uares



Bootstrap *I for Aeast 5$solute Deviations



Testing a Hypothesis Jsing

a *onfi#ence Interval

Given the range of plausi$le values

Testing the hypothesis that a coefficient e)uals

?ero or some other particular value:

Is the hypothesi?e# value in the confi#ence

interval

Is the hypothesi?e# value "ithin the range ofplausi$le values If not( reQect the hypothesis%



Test a Hypothesis 5$out a *oefficient



*lassical Hypothesis Testing

We are intereste# in using the linear regression

to support or cast #ou$t on the vali#ity of a

theory a$out the real "orl# counterpart to our

statistical mo#el% The mo#el is use# to test

hypotheses a$out the un#erlying #atagenerating process%


14/50Part 8: Hypothesis Testing-1#/50

Types of Tests

7este# ;o#els: Restriction on the parameters

of a particular mo#el

y 3& /L & O? & ( O

7onneste# mo#els: E%g%( #ifferent RHS varia$les

yt 3& /Lt& OLt23& tyt 3 & /Lt& Oyt23& "t

Specification tests:6 7+(/, vs% some other #istri$ution



;etho#ology

)ayesian

Prior o##s compares strength of prior $eliefs in t"o states of the

"orl#

Posterior o##s compares revise# $eliefs

Symmetrical treatment of competing i#eas

7ot generally practical to carry out in meaningful situations

*lassi"al

7ullC hypothesis given prominence

Propose to reQectC to"ar# #efault favor of alternativeC 5symmetric treatment of null an# alternative

Huge gain in practical applica$ility


16/50Part 8: Hypothesis Testing-1%/50

7eyman = Pearson ;etho#ology

'ormulate null an# alternative hypotheses

Define ReQectionC region sample evi#ence

that "ill lea# to reQection of the null hypothesis%

Gather evi#ence

5ssess "hether evi#ence falls in reQection region

or not%


17/50Part 8: Hypothesis Testing-1'/50

Inference in the Ainear ;o#el

+orm!lating hypotheses, linear restrictions as a generalframe"or!

Hypothesis Testing linear restrictions

5nalytical frame"or!: y $

&Hypothesis: 2 . 0(

!stantive restri"tions, What is a testa$le hypothesis

Su$stantive restriction on parameters

Re#uces #imension of parameter spaceImposition of restriction #egra#es estimation criterion



Testa$le Implications of a Theory Investors care a$out interest rates an# eLpecte#

inflation:

I $3& $/r & $O#p & e

Investors care a$out real interest rates

I c3& c/0r2#p1 & cO#p & e

7o testa$le restrictions implie#%

c3 $3( c/$/2$O( cO$O% Investors care onlya$out real interest rates

I f3& f/0r2#p1 & fO#p & e% fO


19/50Part 8: Hypothesis Testing-1(/50

The eneral inear Hypothesis, H0, - . 0

!niying depart!re point, egardless o the hypothesis4 least s.!aresis !niased

E+,

The hypothesis ma6es a "laim ao!t the pop!lation

. 0 Then4 i the hypothesis is tr!e4 E7 . 9 0

The sample statisti"4 . :ill not e.!al ;ero

T:o possiilities, . is small eno!gh to attri!te to sampling variaility . is too large e"tion region



5pproaches to Defining the ReQection Region

031 Imposing the restrictions lea#s to a loss of fit% R/must go #o"n% Does it go #o"n a lotC 0I%e%( significantly1%Ru

/ unrestricte# mo#el( Rr/ restricte# mo#el fit%

F - 0Ru/= Rr

/19 4 9 +03 = Ru/19072K1, F+(72K,%

0/1 Is - .close to 0 Basing the test on the #iscrepancyvector: m - .Jsing the Wald criterion: m0Uar+m,123mhas a chi2s)uare# #istri$ution "ith J #egrees of free#om

But( Uar+m, +/0$&$123,%

If "e use our estimate of /( "e get an '+(72K,( instea#% 07ote( this is $ase# on using ee90N2K1 to estimate /%1

These are the same for the linear mo#el



Testing 'un#amentals 2 I

I?E o a test Pro$a$ility it "ill incorrectly

reQect a trueC null hypothesis%

This is the pro$a$ility of a Type I error%

Under the null hypothesis, F(3,100) has

an F distribution with (3,100) degrees

of freedom. Even if the null is true, F

will be larger than the ! "riti"al value

of #.$ about ! of the time.


22/50


Testing Proce#ures

Ho: to determine i the statisti" is @large@7ee# a Vnull #istri$ution%V

If the hypothesis is true( then the statistic "ill have a certain#istri$ution% This tells you ho" li!ely certain values are(an# in particular( if the hypothesis is true( then VlargevaluesV "ill $e unli!ely%

If the o$serve# statistic is too large( conclu#e that theassume# #istri$ution must $e incorrect an# thehypothesis shoul# $e reQecte#%

'or the linear regression mo#el "ith normally #istri$ute##istur$ances( the #istri$ution of the relevant statistic is '"ith an# 72K #egrees of free#om%


23/50


Distri$ution Jn#er the 7ullDensity of F[3,100]

X

.250

.500

.750

.000

1 2 3 40

FDENSITY


24/50

Part 8: Hypothesis Testing-2#/50

5 Simulation ELperiment

sample ; 1 - 100 $matrix ; fvalues=init(1000,1,0)$proc$create ; fakelogc = rnn(-.319!,1."#3)$ %oefficients all = 0regress ; &uietl' ; ls = fakelogc %ompute regression ; rs=one,log&,logplpf,logpkpf$calc ; fstat = (rs&r*+3)+((1-rs&r*)+(n-"))$ %ompute matrix ; fvalues(i)=fstat$ ave 1000 sen*procexecute ; i= 1,1000 $ 1000 replications

istogram ; rs = fvalues ; title= tatistic for 0/#=3="=0$


25/50


Simulation Results

%& out"omes to the right of

#.$ in this run of thee'periment.


26/50

Part 8: Hypothesis Testing-2%/50

Testing 'un#amentals 2 II

PABE o a test the pro$a$ility that it "ill

correctly reQect a false nullC hypothesis This is 3 = the pro$a$ility of a Type II error%

The po"er of a test #epen#s on the specific

alternative%


27/50

Part 8: Hypothesis Testing-2'/50

Po"er of a Test

ull *ean + 0. e-e"t if observed mean /1. or 2 1..4rob(e-e"t null5mean+0) + 0.0

4rob(e-e"t null5mean+.)+0.0$0#

4rob(e-e"t null5mean+1)+0.1$00. 6n"reases as the (alternative) mean rises.

BE TA

.084

.168

.251

.335

.419

.000

-3 -2 -1 0 1 2 3 4 5-4

N2 N0 N1

V

a

ria

b

le


28/50


Test Statistic

'or the fit measures( use a normali?e# measure of

the loss of fit:

( )( )

( )

( )

r rr

r r

r r

0 sin"e

and

0 sin"e

/ /

u r / /u r/

u

/ /u uu

yy yy

u u

u u

u u

R 2R 9 '+(n 2K, R R

32R 9 072K1

ften useful

R 32 R 32S S

Insert these in ' an# it $ecomes

9 '+(n2K,

9072K1

e e e e

e e e ee e e e

e e


29/50

Part 8: Hypothesis Testing-2(/50

Test Statistics

+orming test statisti"s,

'or #istance measures use Wal# type of #istance

measure( W m+Est%Uar0m1,23m

n important relationship et:een t and +

'or a single restriction( mr&2 q% The variance

is rX0Uar+,1r

The #istance measure is m9 stan#ar# error of m%


30/50


n important relationship et:een t and +

'or a single restriction( '+3(72K, is the s)uare of the tratio%

#

7hi 89uared:;< = ;F

7hi s9uared: >< = ( >)

where the two "hi/s9uared variables are independent.

6f ; + 1, i.e., testing a single restri"tion,

7hi 89uared:1< =1F7hi s9uared: >< = ( >)

(:0,1< = ( >)

:0,1)

=


31/50


5pplication

Time series regression(

AogG 3& /logY & OlogPG

& logP7* & FlogPJ* & logPPT

& NlogP7 & 8logPD & logPS &

Perio# 3 2 3F% 7ote that all coefficients in themo#el are elasticities%


32/50


'ull ;o#el----------------------------------------------------------------------r*inar' least s&uares regression ............2=2 4ean = .39#99 tan*ar* *eviation = .#"5!5 6umer of oservs. = 34o*el si7e 8arameters = 9

egrees of free*om = #!:esi*uals um of s&uares = .005


33/50


Test 5$out ne ParameterIs the price of pu$lic transportation really relevant H: %

*onfi#ence interval: $ t0%F(/N1 Stan#ar# error

%33FN3 /%F/0%N8F1

%33FN3 %33/N 02%FFFN (%/N81

*ontains %% Do not reQect hypothesis

Regression fit if #rop Without APPT( R2s)uare# %FNO

*ompare R/( "as %F(

'03(/N1 +0%F 2 %FNO193,9+032%F190O21,

/%38N 3%N// 0"ith some roun#ing #ifference1


34/50

Part 8: Hypothesis Testing-3#/50

Hypothesis Test: Sum of *oefficients 3----------------------------------------------------------------------r*inar' least s&uares regression ............2=2 4ean = .39#99 tan*ar* *eviation = .#"5!5 6umer of oservs. = 34o*el si7e 8arameters = 9

egrees of free*om = #!:esi*uals um of s&uares = .005


35/50


Imposing the Restriction----------------------------------------------------------------------2inearl' restricte* regression2=2 4ean = .39#959 tan*ar* *eviation = .#"5!!9" 6umer of oservs. = 34o*el si7e 8arameters = 5


36/50

Part 8: Hypothesis Testing-3%/50

oint HypothesesJoint hypothesis: Income elasticity = +1, Own price elasticity = -1.

The hypothesis implies that logG = 1+ logY log!g + "log!#$ + ...

%trategy: &egress logG logY + log!g on the other 'aria(les an)

$ompare the s*ms o s*ares

Kit tMo restrictions impose*:esi*uals um of s&uares = .0#55!!it :-s&uare* = .99!900Inrestricte*:esi*uals um of s&uares = .00531it :-s&uare* = .9901

= ./0233 - .//44516706 7 .//4451752-866 = 51.338841

The critical or 849 with 0,03 )egrees o ree)om is 5.54".

The hypothesis is reecte).


37/50

Part 8: Hypothesis Testing-3'/50

Basing the Test on R/

5fter $uil#ing the restrictions into the mo#el an# computing

restricte# an# unrestricte# regressions: Base# on R/s(

' 00%F3F 2 %N19/190032%F3F190O211

2O%FN33 0Z1

WhatVs "rong The unrestricte# mo#el use# AHS logG%

The restricte# one use# logG2logY% The calculation is safeusing the sums of s)uare# resi#uals%


38/50


Wal# Distance ;easure

Testing more generally a$out a single parameter%

Sample estimate is $!

Hypothesi?e# value is !

Ho" far is !from $! If too far( the hypothesis is

inconsistent "ith the sample evi#ence%

;easure #istance in stan#ar# error units

t 9


39/50

Part 8: Hypothesis Testing-3(/50

The Wal# Statistic

-1

Most test statistics are Wald distance measures

W = (random vector - hyothesi!ed value"# times

[$ariance of di%erence] times

(random vector - hyothesi!ed value"

0 0 -1 0

= &ormali!ed distance measure

= ( - " [$ar( - "] ( - "

Distri'uted as chi-suared() " if (1" the distance isnormally distri'uted and (*" the variance matri+ is

the true one, not the esti

!

mate


40/50

Part 8: Hypothesis Testing-#0/50

Ro$ust Tests

The Wal# test generally "ill 0"hen properly

constructe#1 $e more ro$ust to failures of the

narro" mo#el assumptions than the t or '

Reason: Base# on ro$ustC variance estimators

an# asymptotic results that hol# in a "i#e range

of circumstances%

5nalysis: Aater in the course = after #evelopingasymptotics%


41/50


Particular *asesSome particular cases:ne coefficient e)uals a particular value: ' +0$ 2 value1 9 Stan#ar# error of $ ,/ s)uare of familiar t ratio%

Relationship is ' + 3( #%f%, t/+#%f%,5 linear function of coefficients e)uals a particular value

0linear function of coefficients 2 value1/

' 2222222222222222222222222222222222222222222222222222 Uariance of linear function

7ote s)uare of #istance in numeratorSuppose linear function is !"!$!Uariance is !l"!"l*ov+$!($l,

This is the Wal# statistic% 5lso the s)uare of the some"hatfamiliar t statistic%

Several linear functions% Jse Wal# or '% Aoss of fit measures may $e easier tocompute%


42/50


Hypothesis Test: Sum of *oefficients

?o the three aggregate pri"e elasti"ities sum to @eroA

B0C$ C& C + 0

&+ :0, 0, 0, 0, 0, 0, 1, 1, 1


43/50


Wal# Test


44/50

Part 8: Hypothesis Testing-##/50

Jsing the Wal# Statistic--E 4atrix ; : = C0,1,0,0,0,0,0,0,0 +

0,0,1,0,0,0,0,0,0F$--E 4atrix ; & = C1+-1F$--E 4atrix ; list ; m = :


45/50


5pplication: *ost 'unction


46/50

Part 8: Hypothesis Testing-#%/50

Regression Results----------------------------------------------------------------------r*inar' least s&uares regression ............2=% 4ean = 3.0!1# tan*ar* *eviation = 1."#!3 6umer of oservs. = 154o*el si7e 8arameters = 9 egrees of free*om = 1"9:esi*uals um of s&uares = #.313 tan*ar* error of e = .1311

it :-s&uare* = .9931" *>uste* :-s&uare* = .99#!!--------?-------------------------------------------------------------@arialeA %oefficient tan*ar* Brror t-ratio 8CADAEtF 4ean of G--------?-------------------------------------------------------------%onstantA .##53 3.150#" 1."" .10#3 OA -.#10 .#1!! -.95 .33"5 ".#09 2A -.533


47/50

Part 8: Hypothesis Testing-#'/50

Price Homogeneity: nly Price Ratios ;atter

/& O& 3% N& 8& %----------------------------------------------------------------------2inearl' restricte* regression....................2=% 4ean = 3.0!1# tan*ar* *eviation = 1."#!3 6umer of oservs. = 154o*el si7e 8arameters = ! egrees of free*om = 11:esi*uals um of s&uares = #.5#

tan*ar* error of e = .13!3it :-s&uare* = .99#3:estrictns. C #, 1"9F (pro) = 5.(.0003)6ot using 2 or no constant. :s&r* L ma' e 0--------?-------------------------------------------------------------@arialeA %oefficient tan*ar* Brror t-ratio 8CADAEtF 4ean of G--------?-------------------------------------------------------------%onstantA -!.#"0!5


48/50


Imposing the RestrictionsAl"er#a"i$el%& '()*+"e ",e re-"ri'"e re/re--i(# b% '(#$er"i#/ "( *ri'e ra"i(-a#I)*(-i#/ ",e re-"ri'"i(#- ire'"l%. T,i- i- a re/re--i(# ( l(/'* (# l(/**&l(/*l* e"'.----------------------------------------------------------------------r*inar' least s&uares regression ............2=2%8 4ean = -.319 tan*ar* *eviation = 1."#3

6umer of oservs. = 154o*el si7e 8arameters = ! egrees of free*om = 11:esi*uals um of s&uares = #.5# (restricte*):esi*uals um of s&uares = #.313 (unrestricte*)--------?-------------------------------------------------------------@arialeA %oefficient tan*ar* Brror t-ratio 8CADAEtF 4ean of G--------?-------------------------------------------------------------%onstantA -!.#"0!5


49/50

Part 8: Hypothesis Testing-#(/50

Wal# Test of the Restrictions

*hi s)uare# @'Mal* ; fn1 = k ? l ? f - 1 ; fn# = &k ? &l ? &f J 0 $-----------------------------------------------------------K2 proce*ure. Bstimates an* stan*ar* errorsfor nonlinear functions an* >oint test ofnonlinear restrictions.Kal* tatistic = 1!.039!58ro. from %i-s&uare*C #F = .000#0--------?--------------------------------------------------@arialeA %oefficient tan*ar* Brror +t.Br. 8CARAE7F--------?--------------------------------------------------ncn(1)A -1.5!"


50/50

Test of Homotheticity

*ross Pro#uct Terms omotetic 8ro*uction/ T!= T5= T9= 0.

Kit linearit' omogeneit' in prices alrea*' impose*, tis is T= T!= 0.

-----------------------------------------------------------K2 proce*ure. Bstimates an* stan*ar* errors

for nonlinear functions an* >oint test ofnonlinear restrictions.Kal* tatistic = #.!1508ro. from %i-s&uare*C #F = .#!"0--------?--------------------------------------------------@arialeA %oefficient tan*ar* Brror +t.Br. 8CARAE7F--------?--------------------------------------------------ncn(1)A -.0#9" .0#1# -1.390 .1""ncn(#)A -.00"# .0#3" -.19 .5"1

--------?--------------------------------------------------

De test Moul* pro*uce = ((#.90"90 J #.5#)+1)+(#.5#+(15-!))= 1.#5

Documents

Econometrics I 8