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Photo Credit Goes Here

IAPRI-MSU Technical Training

IntrotoAppliedEconometrics:BasictheoryandStataexamples

Trainingmaterialsdevelopedandsessionfacilitatedby

NicoleM.MasonAssistantProfessor,Dept.ofAgricultural,Food,&ResourceEconomics

MichiganStateUniversity

10:00AM–1:00PM,25June2018IndabaAgriculturalPolicyResearchInstitute

Lusaka,Zambia

Why this training?

•  RequestedbyIAPRI

•  Systematicintroductionfornon-economistandrecentlyhiredeconomistteammembers

•  Refresherforinterestedveteraneconomistteammembers

–  Ithasbeenawhilesincewelastcoveredthesetopics(2012-2013)!

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Outline & some questions I hope you’ll be able to answer by the end of today’s session

1.  WhatiseconometricsandwhyisitusefulforIAPRI’swork?2.  Thesimplelinearregressionmodel

a.  Howisitsetup?b.  Whatarethekeyunderlyingassumptions?c.  Howdoweinterpretit?d.  HowdoweestimateitinStata?e.  Howdoweuseittotesthypotheses(ingeneral&Stata)?

3.  (Time-permitting)Anotherappliedeconometricstopicofyourchoice(e.g.,panelestimators,IV/2SLS,etc.)

What is econometrics? •  Whatcomestomindwhenyouheartheword?•  Econometricsistheuseofstatisticalmethodsfor:

– “Estimatingeconomicrelationships”– “Testingeconomictheories”–  Evaluatingpoliciesandprograms

•  Econometricsisstatisticsappliedtoeconomicdata

•  WhyiseconometricsusefulforIAPRI?

Source:Wooldridge(2002,p.1)

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A great resource (this or an earlier version)

Steps in econometric analysis Forthoseofyouthathavedoneapaperthatuseseconometrics,whataresomestepsyouwentthroughingoingfromyourresearchquestion(s)/hypothesis(es)throughtotheanalysisandinference?

1.  Researchquestion(s)2.  Economicmodelorotherconceptual/theoreticalframework3.  Operationalize#2àeconometricmodel4.  Specifyhypothesestobetestedin#35.  Collectandcleandata;createvariables6.  Inspect&summarizedata7.  Estimateeconometricmodel8.  Interpretresults;hypothesistesting&statisticalinference

Source:Wooldridge(2002,p.1)

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Economic model vs. econometric model •  What’sthedifference?•  Economicmodel=“arelationshipderivedfromeconomic

theoryorlessformaleconomicreasoning”–  Examples?

•  Econometricmodel=“anequationrelatingthedependentvariabletoasetofexplanatoryvariablesandunobserveddisturbances,whereunknownpopulationparametersdeterminetheceterisparibuseffectofeachexplanatoryvariable”–  Examples?

Source:Wooldridge(2002,p.794)

Economic model vs. econometric model (cont’d) EX)ModelingdemandforbeefWhatdoeseconomictheorytellusarelikelytobesomecriticalfactorsaffectinganindividual’sdemandforbeef?•  Let:

–  qbeef=beefquantitydemanded–  pbeef=beefprice–  pother=avectorofotherprices(complements,substitutes,etc.)–  income=income–  Z=tastes&preferences(proxiesincaseofeconometricmodel)

•  Howcouldwewritedownageneralfunction(economicmodel)relatingbeefquantitydemandedandthefactorslikelytoaffectit?–  qbeef=f(pbeef,pother,income,Z)

•  Whatwouldthislooklikeitwewerewritingitdownasaneconometricmodel(e.g.,amultiplelinearregressionmodel)?–  qbeef=β0+β1pbeef+potherβ2+β3income+Zβ4+u

We’llgothroughnotation/interpretationinafewminutes

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The simple linear regression model: Motivation •  Letyandxaretwovariablesthatrepresentsomepopulation•  Wewanttoknow:

–  Howdoesychangewhenxchanges?– Whatisthecausaleffect(ceterisparibuseffect)ofxony?

•  Examplesofyandx(ingeneralorinyourresearch)?

y xBeefdemand BeefpriceMaizeyield Qtyoffertilizerused

Childnutrition HHreceivessocialcashtransferCropdiversification HHreceivesFISPe-voucherForestconservation Communityforestmgmt.program

Anatomy of a simple linear regression model y=β0+β1x+u

•  uistheerrortermordisturbance– ufor“unobserved”– Representsallfactorsotherthanxthataffecty– Someuseεinsteadofu

•  Terminologyforyandx?y x

Dependentvariable IndependentvariableExplainedvariable ExplanatoryvariableResponsevariable ControlvariablePredictedvariable Predictorvariable

Regressand RegressorOutcomevariable Covariate

β0(intercept)andβ1(slope)arethe

populationparameterstobeestimated

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To use data to get unbiased estimates of β0 and β1, we have to make some assumptions about the relationship b/w x and u

y = β0 + β1x + u 1.  E(u) = 0 (not restrictive if have an intercept, β0) 2.  *** E(u|x) = E(u) (i.e. the average value of u does

not depend on the value of x) #1 & #2 à E(u|x) = E(u) = 0 (zero conditional mean) •  If this holds, x is “exogenous”; but if x is

correlated with u, x is “endogenous” (we’ll come back to this later)

What does this assumption imply below? •  yield = β0 + β1fertilizer + u, where u is unobserved land quality

(inter alia) •  wage = β0 + β1educ + u, where u is unobserved ability (inter alia)

When is this assumption reasonable?

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Spurious correlations http://www.tylervigen.com/spurious-correlations

What is E(y|x) if we assume E(u|x)=0? y = β0 + β1x + u

What is ∂E( y | x)∂x

and how

do we interpret this result?

∂E( y | x)∂x

= β1

What is the interpretation of β0?

Interpretation: β0 is the expected value of y when x = 0 (intercept)

E(y|x) = β0 + β1x

Hint: Apply the rules for conditional expectations.

Interpretation: β1 is the expectedchange in y given a one unit increase in x, ceteris paribus (slope)

y

x1

2

3

0β

1β slope

intercept

E( y | x) = β0 + β1x

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Why is it called linear regression? y = β0 + β1x + u

•  Linear in parameters, β0 and β1

•  Does NOT limit us to linear relationships between x and y

•  But rules out models that are non-linear in parameters, e.g.:

y = 1β0 + β1x

+ u

y = Φ β0 + β1x( ) + u

y =β0

β1

x + u

Estimating β0 and β1 y = β0 + β1x + u

•  Suppose we have a random sample of size N from the population of interest. Then can write:

yi = β0 + β1xi + ui , i = 1, 2, 3, …, N We don’t know β0 and β1 but want to estimate them.

How does linear regression use the data in our sample to estimate β0 and β1?

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(Ordinary) least squares (OLS) approach •  The estimated values of β0 and β1 are the values

that minimize the sum of squared residuals •  “Fitted” values of y and residuals:

Fitted (estimated, predicted) values of y: yi = β0 + β1xi

Source: Wooldridge (2002)

Residuals:ui = yi − yi

= yi − β0 − β1xi

OLS: Choose β0 and β1 to minimize:

ui2

i=1

N∑ = yi − β0 − β1xi( )2

i=1

N∑

yi = β0 + β1xi + ui

The OLS estimators for β0 and β1

β1 =(xi − x )( yi − y)

i=1

N∑

(xi − x )2

i=1

N∑

β0 = y − β1x

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Ifxis2,whatisyourestimateofy(i.e.,whatisthefittedvaluefory,y-hat)?

Fitted (estimated, predicted) values of y: yi = β0 + β1xi

Basic Stata commands –  regressyx Linearregressionofyonx

•  EX)regresswageeduc

–  predictnewvar1,xb Computefittedvalues

•  EX)predictwagehat,xb(Ijustmadeupthenamewagehat)

–  predictnewvar2,residComputeresiduals

•  EX)predictuhat,resid(Ijustmadeupthenameuhat)

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Obtaining OLS estimates – example (Stata) Wooldridge (2002) Example 2.4: Wage and education Use Stata to run the simple linear regression of wage (y) on educ (x).

Command: regress wage educ (or: reg wage educ)

β1

β0

wagei = β0 + β1educi + ui

What are β0 and β1 below?

Practice time! 1.  Openthedataset“WAGE1.DTA”inStata

a.  Typethecommand“describe”toseewhatvariablesareinthedataset

b.  Estimatethemodelonthepreviousslide(regwageeduc)c.  Findandinterpret(putinasentence!)theestimatesofβ0andβ1in

theregressionoutput2.  Openthedataset“RALS1215_training.dta”inStata

a.  Use“describe”toseewhatvariablesareinthedatasetb.  Regressthevariableforgrossvalueofcropproductiononthe

variableforlandholdingsizec.  Findandinterpret(putinasentence!)theestimatesofβ0andβ1in

theregressionoutput

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Total, explained, & residual sum of squares, R2

SST ≡ ( yi − y)2

i=1

N∑Total sum of squares:

SSE ≡ ( yi − y)2

i=1

N∑Explained sum of squares:

SSR ≡ ui

2

i=1

N∑Residual sum of squares:

SST = SSE+ SSRCoefficient of determination or R2:

R2 = SSE / SST = 1− (SSR / SST )

Proof is on p. 39 of Wooldridge (2002)

Interpretation? The proportion of the sample variation in y that is explained by x

SST (total SS), SSE (explained SS), SSR (residual SS), and R2 in Stata

R2 = SSE / SST = 1− (SSR / SST ) SST = SSE+ SSR

SSE

SSR

SST

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Practice time! LookattheoutputfromyourRALS-basedregression:1.  WhatistheSSE?2.  WhatistheSSR?3.  WhatistheSST?4.  WhatistheR-squared?(Findtheactualnumberandthencheckthatit

equalsSSE/SST)5.  InterprettheR-squared(putitinasentence)

Why is it called R2? •  Letter R sometimes used to refer to correlation

coefficient (others use ρ, which is “rho”)

R2 is the squared sample correlation coefficient between yi and yi

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Practice time! ImmediatelyafteryourRALSregressioncommand,usethepreviousslideasaguideand:1.  Computethepredictedvaluesofgvharv,callingthemgvharvhat2.  Computethecorrelationcoefficientbetweengvharvandgvharvhat3.  Squarethiscorrelationcoefficient(usingtheStata“display”command)4.  ComparetheR-squaredyoujustcomputed“byhand”totheStata-

generatedR-squaredintheregressionoutput.Dotheymatch?

“My R-squared it too low!”

DoesalowR2meantheregressionresultsareuseless?Whyorwhynot?

y = β0 + β1x

β1 may still be good (unbiased) estimate of ceteris paribus (causal) effect of x on y even if R2 is low

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Unbiasedness & assumptions needed for it •  Whatdoesitmeanforanestimatortobeunbiased?

•  WhatassumptionsdoweneedtomakeinorderfortheOLSestimatortobeunbiased?(Hint:wetalkedaboutthekeyassumptionearliertoday.)

E(β1) = β1 and E(β0 ) = β0

Unbiasedness of OLS (simple linear regression) If the following 4 assumptions hold, then OLS is unbiased. (OLS is also consistent under these assumptions, and under slightly weaker assumptions à AFRE 835.)

SLR.1. Linear in parameters:

SLR.2. Random sampling

**SLR.3. Zero conditional mean (exogeneity):

SLR.4. Sample variation in x

y = β0 + β1x + u

E(u | x) = E(u) = 0

β1 =(xi − x )( yi − y)

i=1

N∑

(xi − x )2

i=1

N∑Why necessary? Hint:

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Can’t estimate slope parameter if no variation in x

Source:Wooldridge(2002)

OLS estimators for β0 and β1 are unbiased under SLR.1-SLR.4

•  What does unbiasedness mean in plain language?

•  The key assumption is E(u|x) = E(u) = 0 (zero conditional mean/exogeneity) – SLR.3 •  Under SLR.1-SLR.4, OLS estimate of β1 is the

causal effect (ceteris paribus effect) of x on y •  If E(u|x) ≠ E(u), then x is endogenous to y à OLS

estimates biased à need to take other measures to deal with this (IV/2SLS, panel data methods, etc.)

E(β1) = β1 and E(β0 ) = β0

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If make one more assumption - homoskedasticity (SLR.5) - then OLS is “BLUE”

LetV(u)=σ2 SLR.5. Homoskedasticity (constant variance):

V (u | x) =V (u) =σ 2

Whichimplies:

V ( y | x) =V (u | x) =σ 2

Homoskedasticity

Source:Wooldridge(2002)

Heteroskedasticity

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If SLR.1 through SLR.5 hold, then OLS is “BLUE”

•  Best(mostefficient,i.e.,smallestvariance)

•  Linear(linearfunctionoftheyi)

•  Unbiased

•  Estimator

Also,ifhomoskedastic,thenthe“regular”varianceformulasforOLSestimatorsarecorrect(i.e.,areunbiasedestimatorsfortruevariances).(Ifheteroskedastic,thentheseformulasandtheregularstandarderrorsreportedbyStatabiasedàtoosmall.)Whyisthisaproblem?

In Stata

V (βi ), i = 0,1

σ2

σ

σ

β ja.k.a.

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Practice time! LocatethestandarderrorsforyourestimatesintheRALS-basedregressionQuestion:Whydoweneedthesestandarderrors?Howwillweusethem?

The sampling distributions of the OLS estimators •  By the Central Limit Theorem, under

assumptions SLR.1-SLR.5, the OLS estimators are asymptotically (i.e., as Nà∞) normally distributed

•  If we add one more assumption, then we can obtain the sampling distribution of the OLS estimators in finite samples

SLR.6. Normality: The population error, u, is independent of x and is normally distributed with E(u)=0 and V(u)=σ2, i.e.:

u ~ Normal(0,σ 2 )

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SLR.1-SLR.6 = “classical linear model assumptions”

•  CLM = SLR.1-SLR.6

•  CLM assumptions imply y | x ~ Normal(β0 + β1x, σ 2)

Source:Wooldridge(2002)

ByaddingSLR.6,wecandohypothesistestingusingt-statisticseveninfinitesamples

**tdistributionconvergestostandardnormalasd.f.à∞

T =β j − β j

V (β j )~ t with N − 2 d.f.

T-statistic refresher UnderSLR.1-SLR.6(simplelinearregressioncase):

σ

β j

Replaceβjwiththevalueunderthenullhypothesis

e.g., H0 :β j = 0, H1 :β j ≠ 0

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P-value refresher Whatisap-value?•  “Thesmallestsignificancelevelatwhichthenullhypothesiscan

berejected”(Wooldridge2002,p.800)•  Significancelevel=P(TypeIerror)

=P(rejectthenullwhenthenullisactuallytrue)•  Supposeyouareconductingyourhypothesistestusingthe10%

significancelevelasyourcut-offforstatisticalsignificance•  Whatdoyouconcludeifp-value>0.10?Doyourejectthenull

hypothesisoffailtorejectit?•  Whatdoyouconcludeifp-value<0.10?

Type I vs. Type II error refresher

TypeIerror:rejectH0whenH0istrueProbability:α(significancelevel)

TypeIIerror:failtorejectH0whenH0isfalseProbability:β(1-β=powerofthetest)–differentβ!

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T-statistics and p-values in Stata output

The p-values reported by Stata are for H0 :β j = 0 vs. H1 :β j ≠ 0

Practice time! UsingyourRALS-basedregressionoutput:1.  Whatisthevalueofthet-statisticthatyouwouldusetotest

H0:βland=0vs.H1:βland≠0?Finditintheregressionoutputandalsocalculateitusingtheformulaafewslidesback.

2.  Conductthishypothesistestatthe10%level(usingthep-valuereportedinStata).Whatdoyouconclude?

3.  Whatdoesthismeaninpractice?

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95% confidence intervals in Stata output

Interpretationof95%confidenceinterval(CI):”Ifrandomsampleswereobtainedoverandoveragain,withβjLandβjUcomputedeachtime,thenthe(unknown)populationvalueβjwouldlieintheinterval[βjL,βjU]for95%ofthesamples.Unfortunately,forthesinglesamplethatweusetoconstructtheCI,wedonotknowwhetherβjisactuallycontainedintheinterval.Wehopewehaveobtainedasamplethatisoneofthe95%ofallsampleswheretheintervalestimatecontainsβj,butwehavenoguarantee.”(Wooldridge2002,p.134)

Confidence interval interpretation

βj

Twenty(20)95%CIsforβj

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95% confidence intervals in Stata output

95%CIsareusefulfortestinghypothesesatthe5%sig.levelaboutvaluesofβjunderthenullotherthanzero(againstthecorresponding2-sidedalternativehypothesis)Ifthevalueofβjunderthenullfallswithinthe95%CI,whatwouldyouconclude?•  Failtorejectthenullinfavorofthealternativeatthe5%levelIfthevalueofβjunderthenullisoutsideofthe95%CI,whatwouldyouconclude?•  Rejectthenullinfavorofthealternativeatthe5%level

Practice time! Usethe95%CIinyourRALS-basedregressionoutputtotestanullhypothesisofyourchoiceforβj(againstitscorresponding2-sidedalternativehypothesis).Conductyourhypothesistestatthe5%significancelevel.Whatdoyouconclude?

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(Time-permitting) What else do you want to cover?

Thank you for your attention & participation! NicoleMason(masonn@msu.edu)AssistantProfessorDepartmentofAgricultural,Food,&ResourceEconomics(AFRE)MichiganStateUniversity(MSU)

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Main reference •  Wooldridge,J.(2002).Introductoryeconometrics:Amodernapproach(2ndedition).Cincinnati,OH:South-WesternCollegePub.

EXTRA SLIDES

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Aside: NPR “Hidden Brain” example of a natural experiment, and when it might be reasonable to assume E(u|x)=E(u) •  Listen for the following:

•  What is the dependent variable? •  What is the main explanatory variable of interest? •  Why might it be reasonable to assume E(u|x)=E(u) here? •  What is a natural experiment?

•  Dependent variable: cognitive function of elderly •  Main explanatory variable: wealth •  E(u|x)=E(u) might be reasonable – Congress

computational mistake – people in one cohort got higher benefits that next cohort (level of benefits shouldn’t be correlated with unobservables)

Aside: Natural experiments Anaturalexperimentoccurswhensomeexogenousevent—often a change in government policy—changes theenvironment in which individuals, families, firms, or citiesoperate.Anaturalexperimentalwayshasacontrolgroup,whichisnotaffectedbythepolicychange,andatreatmentgroup,whichisthoughttobeaffectedbythepolicychange.Unlikewithatrueexperiment,wheretreatmentandcontrolgroupsarerandomlyandexplicitlychosen, thecontrolandtreatment groups in natural experiments arise from theparticularpolicychange.(Wooldridge,2002:417)

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Putting it all together: simple linear regression

β1 =(xi − x )( yi − y)

i=1

N∑

(xi − x )2

i=1

N∑

β0 = y − β1x y = β0 + β1x + u

OLSestimatorsforβ0andβ1:

Expectedvalues(underSLR.1-SLR.4): E(β1) = β1 and E(β0 ) = β0

V (β1) = σ 2

(xi − x )2

i=1

N∑

V (β0 ) =σ 2N −1 xi

2

i=1

N∑

(xi − x )2

i=1

N∑

Samplevariances(underSLR.1-SLR.5):

where σ 2 = 1

N − 2ui

2

i=1

N∑ = SSR

N − 2

σ is the standard errorof the regression

A useful cheat-sheet for interpreting models with logged variables

Source:Wooldridge(2002)

β1 =

ΔyΔx

β1

100= Δy

%Δx

100β1 =

%ΔyΔx

β1 =

%Δy%Δx

y = β0 + β1x + u

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Acknowledgements

ThistrainingwasmadepossiblebythegeneroussupportoftheAmericanPeopleprovidedtotheFeedtheFutureInnovation Lab for Food Security Policy [grant numberAID-OAA-L-13-00001] through the United States AgencyforInternationalDevelopment(USAID).The contents are the responsibility of the trainingmaterial authoranddonotnecessarily reflect theviewsofUSAIDortheUnitedStatesGovernment.

www.feedthefuture.gov