Hypothesis Tests and Confident (1)

7/25/2019 Hypothesis Tests and Confident (1)

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Hypothesis Tests andConfdence Intervals in

Multiple Regressors

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Outline

Hypothesis Testing

Joint Hypotheses

Single Restriction Test

Test Score Data

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Hypothesis Tests and ConfdenceIntervals or a Single Coe!icient

is appro"i#ately distri$uted %&'( 1)*&C+T)

Thus hypotheses on can $e tested using the

usual t,statistic( and confdence intervals areconstructed as

So too or

and are generally not independently

distri$uted , so neither are their t,statistics ore on this

later)*

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."a#ple/ The Caliornia class si0edata

The coe!icient on STR in &2) is the e!ect on Test Scoreo a unit change in STR( holding constant thepercentage o .nglish +earners in the district*

Coe!icient on STR alls $y one,hal*

3 confdence interval or coe!icient on STR in &2) is451*1' 6 1*7 '*8-9 : &5 1*( 5 '*27)

The t,statistic or STR : 5 1*1';'*8- : 5 2*8( so


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Tests o Joint Hypotheses

+et ."pn : e"penditures per pupil and consider thepopulation regression #odel

The null hypothesis that >school resources don?t#atter(@ and the alternative that they do(corresponds to


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Ajoint hypothesis specifes a value or two ormore coe!icients( that is( it i#poses a restriction ontco##on sense@ test is to re=ect i either o theindividual t,statistics e"ceeds 1*7 in a$solute value*

But this >co##on sense@ approach doesn?t


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Here?s co##on sense@ test$ased on the tco##on sense@ test is re=ect it1E1*7 and;or t2 E 1*7* Fhat is the pro$a$ility

that this >co##on sense@ test re=ects H'


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ro$a$ility o incorrectly re=ecting the null


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The sizeo a test is the actual re=ection rate under thenull hypothesis*

The si0e o the >co##on sense@ test isn?t 3*

Its si0e actually depends on the correlation $et


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The N,statistic

The N,statistic tests all parts o a =oint hypothesis atonce*

Nor#ula or the special case o the =oint hypothesisand

in a regression


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The N,statistic testing and

The N,statistic is large


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+arge,sa#ple distri$ution o the N,statistic

Consider a special case that t1 and t2 are independent(so * In large sa#ples the or#ula $eco#es

Knder the null( t1 and t2 have standard nor#aldistri$utions that( in this special case( areindependent*

The large,sa#ple distri$ution o the N,statistic is thedistri$ution o the average o t


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The chi,suared distri$ution


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Co#pute p,value using the N,statistic/

p,value : tail pro$a$ility o the distri$ution$eyond the N,statistic actually co#puted*

Implementation in STATAKse the >test@ co##and ater the regression*

."a#ple/ Test the =oint hypothesis that thepopulation coe!icients on STR and e"penditures

per pupil &e"pnPstu) are $oth 0ero( against thealternative that at least one o the populationcoe!icients is non0ero*

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The ho#osedasticity,only N,statistic

To co#pute the ho#osedasticity,only N,statistic

Kse the previous or#ulas( $ut usingho#osedasticity,only standard errors* Or

Run trestricted@ regression) and oneunder the alternative hypothesis &the>unrestricted@ regression)*

The second #ethod gives a si#ple or#ula*

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The >restricted@ and >unrestricted@regressions

."a#ple/ are the coe!icients on STR and ."pn 0eroG

Restricted population regression &that is( under H')/

Knrestricted population regression &under H1)/

The nu#$er o restrictions under H': : 2* The ft


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By ho< #uch #ust the R2increase or the coe!icients on."pn and ct .+ to $e =udged statistically signifcantG

Si#ple or#ula or the ho#osedasticity,only N,statistic


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."a#ple/

Restricted regression/

Knrestricted regression/

2'


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The ho#osedasticity,only N,statistic,su##ary

The ho#osedasticity,only N,statistic re=ects


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Joint confdence sets ctd.


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Confdence set based oninverting the F-statistic


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Su##ary/ testing =oint hypotheses

The >co##on,sense@ approach o re=ecting i eithero the t,statistics e"ceeds 1*7 re=ects #ore than3 o the ti#e under the null &the si0e e"ceeds thedesired signifcance level)*

The heterosedasticity,ro$ust N,statistic is $uilt into STATA &>test@ co##and)* This tests all restrictions at once*

Nor large n( N is distri$uted as *

The ho#osedasticity,only N,statistic is i#portanthistorically &and thus in practice)( and is intuitivelyappealing( $ut invalid


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Testing Single Restrictions on MultipleCoe!icients

Consider the null and alternative hypothesis(

This null i#poses a single restriction & : 1) on#ultiple coe!icients , it is not a =oint hypothesis


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T


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Method 1/ Rearrange &@transor#@) the regression*

Add and su$tract


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&a) Original syste#/

&$) Rearranged &@transor#ed@) syste#/

so

The testing pro$le# is no< a si#ple one/ test


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Method 2/ eror# the test directly

."a#ple/

To test( using STATA(


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Analysis o the Test Score Data

A general approach to varia%le selection and!model specifcation"

Speciy a >$ase@ or >$ench#ar@ #odel*

Speciy a range o plausi$le alternative #odels(


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Is a candidate varia$le statistically signifcantG

Kse =udg#ent( not a #echanical recipe*

And don&t just ma'imize R(.

-2


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Digression about measures oft


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aria$les


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aria$les actually in the Caliorniaclass si0e

data set student,teacher ratio &STR) percent .nglish learners in the district &ct.+)

percent eligi$le or su$sidi0ed;ree lunch

percent on pu$lic inco#e assistance average district inco#e

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A loo at #ore o the Caliornia data

-7


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Digression/ presentation o regressionresults in a ta$le

+isting regressions in >euation@ or# can $e cu#$erso#e


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Su##ary/ Multiple Regression

Multiple regression allo

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Hypothesis Tests and Confident (1)

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