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Introductionto Econometrics
Chapters 1, 2 and 3
The statistical analysis of
economic (and relateddata
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Brief Overview of the Course
! "conomics s#ggests important relationships$ often%ith policy implications$ t virt#ally nevers#ggests '#antitative magnit#des of ca#saleffects.
What is the quantitativeeffect of red#cing class si)e onst#dent achievement*
+o% does another year of ed#cation change earnings*
What is the price elasticity of cigarettes*
What is the effect on o#tp#t gro%th of a 1 percentage
point increase in interest rates &y the ,ed* What is the effect on ho#sing prices of environmental
improvements*
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This course is about using data tomeasure causa effects!
! deally$ %e %o#ld lie an e/periment
What %o#ld &e an e/periment to estimate the effect of classsi)e on standardi)ed test scores*
! #t almost al%ays %e only have o&servational(none/perimental data.
ret#rns to ed#cation
cigarette prices
monetary policy
! ost of the co#rse deals %ith diffic#lties arising from #singo&servational to estimate ca#sal effects
confo#nding effects (omitted factors sim#ltaneo#s ca#sality
correlation does not imply ca#sation3
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! 4earn methods for estimating ca#sal effects #singo&servational data
! 4earn some tools that can &e #sed for other p#rposes5 fore/ample$ forecasting #sing time series data5
! ,oc#s on applications theory is #sed only as needed to#nderstand the %hys of the methods5
! 4earn to eval#ate the regression analysis of others thismeans yo# %ill &e a&le to read6#nderstand empiricaleconomics papers in other econ co#rses5
! 7et some hands-on e/perience %ith regression analysis inyo#r pro&lem sets.
In this course #ou wi$
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! Empirica probem$ Class si)e and ed#cationalo#tp#t
Policy '#estion8 What is the effect on test scores (orsome other o#tcome meas#re of red#cing class si)e &y
one st#dent per class* &y 9 st#dents6class* We m#st #se data to find o#t (is there any %ay to ans%er
this withoutdata*
&eview of 'robabiit# and (tatistics)(* Chapters 2, 3+
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The Caifornia Test (core ata (et
All :-; and :-9 California school districts (n< =20
>aria&les8
! ?PthP
grade test scores (@tanford- achievement test$com&ined math and reading$ district average
! @t#dent-teacher ratio (@TB < no. of st#dents in thedistrict divided &y no. f#ll-time e'#ivalent teachers
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Initia oo/ at the data$(You should already know how to interpret this table)
This ta&le doesnt tell #s anything a&o#t the relationship&et%een test scores and the STR.
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o districts with smaer casses havehigher test scores
(catterpot of test score v. st#dent-teacher ratio
What does this figure show?
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*e need to get some numerica evidence on whetherdistricts with ow (T&s have higher test scores but how
1. Compare average test scores in districts %ith lo% @TBs to
those %ith high @TBs (estimation3
2. Test the n#ll3 hypothesis that the mean test scores in the
t%o types of districts are the same$ against the
alternative3 hypothesis that they differ (hypothesistesting3
D. "stimate an interval for the difference in the mean test
scores$ high v. lo% @TB districts (confidence interval3
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Initia data ana#sis$ Compare districts %ith small3 (@TB E20 and large3 (@TB F 20 class si)es8
1. Estimationof < difference &et%een gro#p means
2. Test the hypothesisthat < 0
3. Constr#ct a confidence intervalfor
Class @i)e Average score
(
@tandard deviation
(sY
n
@mall ;?G.= 1.= 2D9
4arge ;?0.0 1G. 192
Y
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1! Estimation
<
< ;?G.= ;?0.0
< G.=s this a large difference in a real-%orld sense*
@tandard deviation across districts < 1.1
Hifference &et%een ;0PthPand G?PthPpercentiles of test
score distrition is ;;G.; ;?.= < 9.2 This is a &ig eno#gh difference to &e important for school
reform disc#ssions$ for parents$ or for a school
committee*
1n
small
Yi
i=1
nsmall
Ysmall Ylarge
1n
large
Yi
i=1
nlarge
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2! 5#pothesis testing
t=Ys Y
l
ss2
ns
+ sl2
nl
=Ys Y
l
SE(Ys Y
l)
Hifference-in-meanstest8 comp#te the t-statistic$ (remem&er this*
! %here SE( is the standard error3 of $
the s#&scripts sand lrefer to small3 and large3
@TB districts$ and (etc.
Ys Yl Ys Yl
ss
2
=
1
ns 1(Y
iYs
)2
i=1
ns
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Comp#te the difference-of-means t-statistic8
< =.0?
ItI J 1.;$ so reKect (at the ?L significance levelthe n#ll hypothesis that the t%o means are thesame.
@i)e sY n
small ;?G.= 1.= 2D9
large ;?0.0 1G. 192
Y
t=Ys Ylss2
ns
+ sl2
nl
=657.4 650.0
19.42
238+ 17.9
2
182
=7.4
1.83
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3! Confidence interva
A ?L confidence interval for the difference &et%eenthe means is$
( M 1.;NSE(
< G.= M 1.;N1.9D < (D.9$ 11.0
Two equivalent stateents!1. The ?L confidence interval for doesnt incl#de 05
2. The hypothesis that < 0 is reKected at the ?L level.
Yl
Ys
Yl
Ys
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*hat comes ne6t7
! The mechanics of estimation$ hypothesis testing$and confidence intervals sho#ld &e familiar
! These concepts e/tend directly to regression andits variants
! efore t#rning to regression$ ho%ever$ %e %illrevie% some of the #nderlying theory ofestimation$ hypothesis testing$ and confidenceintervals8 Why do these proced#res %or$ and %hy #se these rather
than others*
We %ill revie% the intellect#al fo#ndations of statisticsand econometrics
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&eview of (tatistica Theor#
1. The probabiit# framewor/ for statistica inference
2. "stimation
D. Testing
=. Confidence ntervals
The probabiit# framewor/ for statistica inference
a Pop#lation$ random varia&le$ and distrition
& oments of a distrition (mean$ variance$ standarddeviation$ covariance$ correlation
c Conditional distritions and conditional meansd Histrition of a sample of data dra%n randomly from a
pop#lation8 Y1$ O$ Yn
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)a+ 'opuation, random variabe, anddistribution
Population
! The gro#p or collection of all possi&le entities of interest(school districts
! We %ill thin of pop#lations as infinitely large ( is an
appro/imation to very &ig3
Random variable Y
! Q#merical s#mmary of a random o#tcome (district averagetest score$ district @TB
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Population distribution of Y
! The pro&a&ilities of different val#es of Ythat occ#rin the pop#lation$ for e/. PrRY< ;?0S (%hen Yisdiscrete
! or8 The pro&a&ilities of sets of these val#es$ for e/.PrR;=0 Y ;;0S (%hen Yis contin#o#s.
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)b+ 8oments of a popuation distribution$ mean,variance, standard deviation, covariance,correation
mean< e/pected val#e (e/pectation of Y
< E(Y
< Y
< long-r#n average val#e of Yover repeatedreali)ations of Y
variance < E(Y YP2P