Econ 423 – Lecture Notes - UMD
Econ 423 – Lecture Notes(These notes are slightly modified versions of lecture notes provided byStock and Watson, 2007. They are for instructional purposes onlyand are not to be distributed outside of the classroom.)12-1
Instrumental Variables RegressionThree important threats to internal validity are: omitted variable bias from a variable that is correlatedwith X but is unobserved, so cannot be included in theregression; simultaneous causality bias (X causes Y, Y causes X); errors-in-variables bias (X is measured with error)Instrumental variables regression can eliminate bias whenE(u X) 0 – using an instrumental variable, Z12-2
IV Regression with One Regressor and One InstrumentYi β0 β1Xi ui IV regression breaks X into two parts: a part that might becorrelated with u, and a part that is not. By isolating thepart that is not correlated with u, it is possible to estimateβ1. This is done using an instrumental variable, Zi, which isuncorrelated with ui. The instrumental variable detects movements in Xi that areuncorrelated with ui, and uses these to estimate β1.12-3
Terminology: endogeneity and exogeneityAn endogenous variable is one that is correlated with uAn exogenous variable is one that is uncorrelated with uHistorical note: “Endogenous” literally means“determined within the system,” that is, a variable that isjointly determined with Y, that is, a variable subject tosimultaneous causality. However, this definition isnarrow and IV regression can be used to address OV biasand errors-in-variable bias, not just to simultaneouscausality bias.12-4
Two conditions for a valid instrumentYi β0 β1Xi uiFor an instrumental variable (an “instrument”) Z to be valid,it must satisfy two conditions:1.2.Instrument relevance: corr(Zi,Xi) 0Instrument exogeneity: corr(Zi,ui) 0Suppose for now that you have such a Zi (we’ll discuss howto find instrumental variables later).How can you use Zi to estimate β1?12-5
The IV Estimator, one X and one ZExplanation #1: Two Stage Least Squares (TSLS)As it sounds, TSLS has two stages – two regressions:(1) First isolates the part of X that is uncorrelated with u:regress X on Z using OLSXi π0 π1Zi vi(1) Because Zi is uncorrelated with ui, π0 π1Zi isuncorrelated with ui. We don’t know π0 or π1 but wehave estimated them, so Compute the predicted values of Xi, Xˆ i , where Xˆ i πˆ0 πˆ1 Zi, i 1, ,n.12-6
Two Stage Least Squares, ctd.(2) Replace Xi by Xˆ i in the regression of interest:regress Y on Xˆ using OLS:iYi β0 β1 Xˆ i ui(2) Because Xˆ i is uncorrelated with ui (if n is large), the firstleast squares assumption holds (if n is large) Thus β1 can be estimated by OLS using regression (2) This argument relies on large samples (so π0 and π1 are wellestimated using regression (1)) This the resulting estimator is called the Two Stage LeastSquares (TSLS) estimator, βˆ TSLS .112-7
Two Stage Least Squares, ctd.Suppose you have a valid instrument, Zi.Stage 1: Regress Xi on Zi, obtain the predicted values Xˆ iStage 2: Regress Yi on Xˆ i ; the coefficient on Xˆ i isthe TSLS estimator, βˆ TSLS .1βˆ1TSLS is a consistent estimator of β1.12-8
The IV Estimator, one X and one Z, ctd.Explanation #2: a little algebra Yi β0 β1Xi uiThus,cov(Yi,Zi) cov(β0 β1Xi ui,Zi) cov(β0,Zi) cov(β1Xi,Zi) cov(ui,Zi) 0 cov(β1Xi,Zi) 0 β1cov(Xi,Zi)where cov(ui,Zi) 0 (instrument exogeneity); thuscov(Yi , Z i )β1 cov( X i , Z i )12-9
The IV Estimator, one X and one Z, ctd.cov(Yi , Z i )β1 cov( X i , Z i )The IV estimator replaces these population covariances withsample covariances:sYZTSLSˆβ1 ,s XZsYZ and sXZ are the sample covariances. This is the TSLSestimator – just a different derivation!12-10
Consistency of the TSLS estimatorsYZTSLSˆβ1 s XZpThe sample covariances are consistent: sYZ cov(Y,Z) andpsXZ cov(X,Z). Thus,βˆ1TSLSsYZ p cov(Y , Z ) β1s XZcov( X , Z ) The instrument relevance condition, cov(X,Z) 0, ensuresthat you don’t divide by zero.12-11
Example #1: Supply and demand for butterIV regression was originally developed to estimate demandelasticities for agricultural goods, for example butter:ln(Qibutter ) β0 β1ln( Pi butter ) ui β1 price elasticity of butter percent change in quantityfor a 1% change in price (recall log-log specificationdiscussion) Data: observations on price and quantity of butter fordifferent years The OLS regression of ln(Qibutter ) on ln( Pi butter ) suffers fromsimultaneous causality bias (why?)12-12
Simultaneous causality bias in the OLS regression ofln(Qibutter ) on ln( Pi butter ) arises because price and quantity aredetermined by the interaction of demand and supply12-13
This interaction of demand and supply produces Would a regression using these data produce the demandcurve?12-14
But what would you get if only supply shifted? TSLS estimates the demand curve by isolating shifts inprice and quantity that arise from shifts in supply. Z is a variable that shifts supply but not demand.12-15
TSLS in the supply-demand example:ln(Qibutter ) β0 β1ln( Pi butter ) uiLet Z rainfall in dairy-producing regions.Is Z a valid instrument?(1) Exogenous? corr(raini,ui) 0?Plausibly: whether it rains in dairy-producing regionsshouldn’t affect demand(2) Relevant? corr(raini,ln( Pi butter )) 0?Plausibly: insufficient rainfall means less grazingmeans less butter12-16
TSLS in the supply-demand example, ctd.ln(Qibutter ) β0 β1ln( Pi butter ) uiZi raini rainfall in dairy-producing regions.· P butter )Stage 1: regress ln( Pi butter ) on rain, get ln(i· P butter ) isolates changes in log price that arise fromln(isupply (part of supply, at least)· P butter )Stage 2: regress ln(Qibutter ) on ln(iThe regression counterpart of using shifts in the supplycurve to trace out the demand curve.12-17
Example #2: Test scores and class size The California regressions still could have OV bias (e.g.parental involvement). This bias could be eliminated by using IV regression(TSLS). IV regression requires a valid instrument, that is, aninstrument that is:(1) relevant: corr(Zi,STRi) 0(2) exogenous: corr(Zi,ui) 012-18
Example #2: Test scores and class size, ctd.Here is a (hypothetical) instrument: some districts, randomly hit by an earthquake, “double up”classrooms:Zi Quakei 1 if hit by quake, 0 otherwise Do the two conditions for a valid instrument hold? The earthquake makes it as if the districts were in a randomassignment experiment. Thus the variation in STR arisingfrom the earthquake is exogenous. The first stage of TSLS regresses STR against Quake,thereby isolating the part of STR that is exogenous (the partthat is “as if” randomly assigned)12-19
Inference using TSLS In large samples, the sampling distribution of the TSLSestimator is normal Inference (hypothesis tests, confidence intervals) proceedsin the usual way, e.g. 1.96SE The idea behind the large-sample normal distribution of theTSLS estimator is that – like all the other estimators wehave considered – it involves an average of mean zero i.i.d.random variables, to which we can apply the CLT. Here is a sketch of the math (see SW App. 12.3 for thedetails).12-20
sYZTSLSˆβ1 s XZ1 n(Yi Y )( Z i Z ) n 1 i 1 1 n( X i X )( Z i Z ) n 1 i 1n Y ( Zi Z)ii 1n X (Zii Z)i 1Substitute in Yi β0 β1Xi ui and simplify:nβˆ1TSLS nβ1 X i ( Z i Z ) ui ( Z i Z )i 1i 1n X (Zii Z)i 1so 12-21
nβˆ1TSLS β1 u (Zi Z)ii 1n. X (Zii Z)i 1nsoβˆ1TSLS – β1 u (Zi Z)ii 1n X (Zii Z)i 1Multiply through by n :n ( βˆ1TSLS1 n( Z i Z )ui n i 1– β1) 1 nX i ( Zi Z ) n i 112-22
n ( βˆ1TSLS1 n( Z i Z )ui n i 1– β1) 1 nX i ( Zi Z ) n i 1p1 n1 n X i ( Z i Z ) ( X i X )( Z i Z ) cov(X,Z) 0n i 1n i 1 n1(Z ni Z )ui is dist’d N(0,var[(Z–µZ)u]) (CLT)i 1so:whereβˆ1TSLS is approx. distributed N(β1,σ β2ˆTSLS1σ2βˆ TSLS1),1 var[( Z i µZ )ui ] .2n [cov( Z i , X i )]where cov(X,Z) 0 because the instrument is relevant12-23
Inference using TSLS, ctd.βˆ1TSLS is approx. distributed N(β1,σ β2ˆ TSLS ),1 Statistical inference proceeds in the usual way. The justification is (as usual) based on large samples This all assumes that the instruments are valid – we’lldiscuss what happens if they aren’t valid shortly. Important note on standard errors:o The OLS standard errors from the second stageregression aren’t right – they don’t take into account theestimation in the first stage ( Xˆ is estimated).io Instead, use a single specialized command that computesthe TSLS estimator and the correct SEs.o as usual, use heteroskedasticity-robust SEs12-24
Example: Cigarette demand, ctd.ln(Qicigarettes ) β0 β1ln( Pi cigarettes ) uiPanel data: Annual cigarette consumption and average prices paid(including tax) 48 continental US states, 1985-1995Proposed instrumental variable: Zi general sales tax per pack in the state SalesTaxi Is this a valid instrument?(1) Relevant? corr(SalesTaxi, ln( Pi cigarettes )) 0?(2) Exogenous? corr(SalesTaxi,ui) 0?12-25
Cigarette demand, ctd.For now, use data from 1995 only.First stage OLS regression:·P cigarettes ) 4.63 .031SalesTax , n 48ln(iiSecond stage OLS regression:·Q cigarettes ) 9.72 – 1.08 ln(·P cigarettes ) , n 48ln(iiCombined regression with correct, heteroskedasticity-robuststandard errors:·Q cigarettes ) 9.72 – 1.08 ln(·P cigarettes ) , n 48ln(ii(1.53) (0.32)12-26
STATA Example: Cigarette demand, First stageInstrument Z rtaxso general sales tax (real /pack)XZ. reg lravgprs rtaxso if year 1995, r;Regression with robust standard errorsNumber of obs F( 1,46) Prob F R-squared Root MSE -- Robustlravgprs Coef.Std. Err.tP t [95% Conf. Interval]------------- -------------rtaxso .0307289.00483546.350.000.0209956.0404621cons --------------------X-hat. predict lravphat;Now we have the predicted values from the 1st stage12-27
Second stageYX-hat. reg lpackpc lravphat if year 1995, r;Regression with robust standard errorsNumber of obs F( 1,46) Prob F R-squared Root MSE -- Robustlpackpc Coef.Std. Err.tP t [95% Conf. Interval]------------- ravphat -1.083586cons ------------------- These coefficients are the TSLS estimates The standard errors are wrong because they ignore the factthat the first stage was estimated12-28
Combined into a single command:YXZ. ivreg lpackpc (lravgprs rtaxso) if year 1995, r;IV (2SLS) regression with robust standard errorsNumber of obs F( 1,46) Prob F R-squared Root MSE -- Robustlpackpc Coef.Std. Err.tP t [95% Conf. Interval]------------- ravgprs -1.083587cons ------------------Instrumented: lravgprsThis is the endogenous regressorInstruments:rtaxsoThis is the instrumental ----------------------------------OK, the change in the SEs was small this time.but not always!·Q cigarettes ) 9.72 – 1.08 ln(·P cigarettes ) , n 48ln(ii(1.53) (0.32)12-29
Summary of IV Regression with a Single X and Z A valid instrument Z must satisfy two conditions:(1) relevance: corr(Zi,Xi) 0(2) exogeneity: corr(Zi,ui) 0 TSLS proceeds by first regressing X on Z to get X̂ , thenregressing Y on X̂ . The key idea is that the first stage isolates part of thevariation in X that is uncorrelated with u If the instrument is valid, then the large-sample samplingdistribution of the TSLS estimator is normal, so inferenceproceeds as usual12-30
The General IV Regression Model So far we have considered IV regression with a singleendogenous regressor (X) and a single instrument (Z). We need to extend this to:o multiple endogenous regressors (X1, ,Xk)o multiple included exogenous variables (W1, ,Wr)These need to be included for the usual OV reasono multiple instrumental variables (Z1, ,Zm)More (relevant) instruments can produce a smallervariance of TSLS: the R2 of the first stage increases,so you have more variation in X̂ . Terminology: identification & overidentification12-31
Identification In general, a parameter is said to be identified if differentvalues of the parameter would produce differentdistributions of the data. In IV regression, whether the coefficients are identifieddepends on the relation between the number of instruments(m) and the number of endogenous regressors (k) Intuitively, if there are fewer instruments than endogenousregressors, we can’t estimate β1, ,βko For example, suppose k 1 but m 0 (no instruments)!12-32
Identification, ctd.The coefficients β1, , βk are said to be: exactly identified if m k.There are just enough instruments to estimate β1, ,βk. overidentified if m k.There are more than enough instruments to estimateβ1, ,βk. If so, you can test whether the instruments arevalid (a test of the “overidentifying restrictions”) – we’llreturn to this later underidentified if m k.There are too few instruments to estimate β1, ,βk. If so,you need to get more instruments!12-33
The general IV regression model: Summary of jargonYi β0 β1X1i βkXki βk 1W1i βk rWri ui Yi is the dependent variable X1i, , Xki are the endogenous regressors (potentiallycorrelated with ui) W1i, ,Wri are the included exogenous variables orincluded exogenous regressors (uncorrelated with ui) β0, β1, , βk r are the unknown regression coefficients Z1i, ,Zmi are the m instrumental variables (the excludedexogenous variables) The coefficients are overidentified if m k; exactlyidentified if m k; and underidentified if m k.12-34
TSLS with a single endogenous regressorYi β0 β1X1i β2W1i β1 rWri ui m instruments: Z1i, , Zm First stageo Regress X1 on all the exogenous regressors: regress X1on W1, ,Wr, Z1, , Zm by OLSo Compute predicted values Xˆ , i 1, ,n1i Second stageo Regress Y on X̂ 1, W1, , Wr by OLSo The coefficients from this second stage regression arethe TSLS estimators, but SEs are wrong To get correct SEs, do this in a single step12-35
Example: Demand for cigarettesln(Qicigarettes ) β0 β1ln( Pi cigarettes ) β2ln(Incomei) uiZ1i general sales taxiZ2i cigarette-specific taxi Endogenous variable: ln( Pi cigarettes ) (“one X”) Included exogenous variable: ln(Incomei) (“one W”) Instruments (excluded endogenous variables): general salestax, cigarette-specific tax (“two Zs”) Is the demand elasticity β1 overidentified, exactly identified,or underidentified?12-36
Example: Cigarette demand, one instrumentYWXZ. ivreg lpackpc lperinc (lravgprs rtaxso) if year 1995, r;IV (2SLS) regression with robust standard errorsNumber of obs F( 2,45) Prob F R-squared Root MSE - Robustlpackpc Coef.Std. Err.tP t [95% Conf. Interval]------------- -------------lravgprs nc .214515.31174670.690.495-.413375.842405cons -----------------Instrumented: lravgprsInstruments:lperinc rtaxsoSTATA lists ALL the exogenous regressorsas instruments – slightly differentterminology than we have been --------------------------------- Running IV as a single command yields correct SEs Use , r for heteroskedasticity-robust SEs12-37
Example: Cigarette demand, two instrumentsYWXZ1Z2. ivreg lpackpc lperinc (lravgprs rtaxso rtax) if year 1995, r;IV (2SLS) regression with robust standard errorsNumber of obs F( 2,45) Prob F R-squared Root MSE -- Robustlpackpc Coef.Std. Err.tP t [95% Conf. Interval]------------- -------------lravgprs nc .2804045.25388941.100.275-.230955.7917641cons -------------------Instrumented: lravgprsInstruments:lperinc rtaxso rtaxSTATA lists ALL the exogenous regressorsas “instruments” – slightly differentterminology than we have been ---------------------------------12-38
TSLS estimates, Z sales tax (m 1)·Q cigarettes ) 9.43 – 1.14 ln(·P cigarettes ) 0.21ln(Income )ln(iii(0.31)(1.26) (0.37)TSLS estimates, Z sales tax, cig-only tax (m 2)·Q cigarettes ) 9.89 – 1.28 ln(·P cigarettes ) 0.28ln(Income )ln(iii(0.96) (0.25)(0.25) Smaller SEs for m 2. Using 2 instruments gives moreinformation – more “as-if random variation”. Low income elasticity (not a luxury good); income elasticitynot statistically significantly different from 0 Surprisingly high price elasticity12-39
The General Instrument Validity AssumptionsYi β0 β1X1i βkXki βk 1W1i βk rWri ui(1) Instrument exogeneity: corr(Z1i,ui) 0, , corr(Zmi,ui) 0(2) Instrument relevance: General case, multiple X’sSuppose the second stage regression could be run usingthe predicted values from the population first stageregression. Then: there is no perfect multicollinearity inthis (infeasible) second stage regression. Multicollinearity interpretation Special case of one X: the general assumption isequivalent to (a) at least one instrument must enter thepopulation counterpart of the first stage regression, and(b) the W’s are not perfectly multicollinear.12-40
The IV Regression AssumptionsYi β0 β1X1i βkXki βk 1W1i βk rWri ui1. E(ui W1i, ,Wri) 0 #1 says “the exogenous regressors are exogenous.”2. (Yi,X1i, ,Xki,W1i, ,Wri,Z1i, ,Zmi) are i.i.d. #2 is not new3. The X’s, W’s, Z’s, and Y have nonzero, finite 4th moments #3 is not new4. The instruments (Z1i, ,Zmi) are valid. We have discussed this Under 1-4, TSLS and its t-statistic are normally distributed The critical requirement is that the instruments be valid 12-41
Checking Instrument ValidityRecall the two requirements for valid instruments:1. Relevance (special case of one X)At least one instrument must enter the populationcounterpart of the first stage regression.2. ExogeneityAll the instruments must be uncorrelated with the errorterm: corr(Z1i,ui) 0, , corr(Zmi,ui) 0What happens if one of these requirements isn’t satisfied?How can you check? What do you do?If you have multiple instruments, which should you use?12-42
Checking Assumption #1: Instrument RelevanceWe will focus on a single included endogenous regressor:Yi β0 β1Xi β2W1i β1 rWri uiFirst stage regression:Xi π0 π1Z1i πmZmi πm 1W1i πm kWki ui The instruments are relevant if at least one of π1, ,πm arenonzero. The instruments are said to be weak if all the π1, ,πm areeither zero or nearly zero. Weak instruments explain very little of the variation in X,beyond that explained by the W’s12-43
What are the consequences of weak instruments?If instruments are weak, the sampling distribution of TSLSand its t-statistic are not (at all) normal, even with n large.Consider the simplest case:Yi β0 β1Xi uiXi π0 π1Zi uisYZTSLSˆ The IV estimator is β1 s XZ If cov(X,Z) is zero or small, then sXZ will be small: Withweak instruments, the denominator is nearly zero. If so, the sampling distribution of βˆ TSLS (and its t-statistic) is1not well approximated by its large-n normalapproximation 12-44
An example: the sampling distribution of the TSLSt-statistic with weak instrumentsDark line irrelevant instrumentsDashed light line strong instruments12-45
Why does our trusty normal approximation fail us?sYZTSLSˆβ1 s XZ If cov(X,Z) is small, small changes in sXZ (from one sampleto the next) can induce big changes in βˆ TSLS1 Suppose in one sample you calculate sXZ .00001. Thus the large-n normal approximation is a poorapproximation to the sampling distribution of βˆ TSLS1 A better approximation is that βˆ1TSLS is distributed as theratio of two correlated normal random variables (see SWApp. 12.4) If instruments are weak, the usual methods of inference areunreliable – potentially very unreliable.12-46
Measuring the strength of instruments in practice:The first-stage F-statistic The first stage regression (one X):Regress X on Z1,.,Zm,W1, ,Wk. Totally irrelevant instruments all the coefficients onZ1, ,Zm are zero. The first-stage F-statistic tests the hypothesis that Z1, ,Zmdo not enter the first stage regression. Weak instruments imply a small first stage F-statistic.12-47
Checking for weak instruments with a single X Compute the first-stage F-statistic.Rule-of-thumb: If the first stage F-statistic is less than10, then the set of instruments is weak. If so, the TSLS estimator will be biased, and statisticalinferences (standard errors, hypothesis tests, confidenceintervals) can be misleading. Note that simply rejecting the null hypothesis that thecoefficients on the Z’s are zero isn’t enough – you actuallyneed substantial predictive content for the normalapproximation to be a good one. There are more sophisticated things to do than just compareF to 10 but they are beyond this course.12-48
What to do if you have weak instruments? Get better instruments (!) If you have many instruments, some are probably weakerthan others and it’s a good idea to drop the weaker ones(dropping an irrelevant instrument will increase the firststage F)12-49
Estimation with weak instruments There are no consistent estimators if instruments are weakor irrelevant. However, some estimators have a distribution morecentered around β1 than does TSLS One such estimator is the limited information maximumlikelihood estimator (LIML) The LIML estimatoro can be derived as a maximum likelihood estimator12-50
Checking Assumption #2: Instrument Exogeneity Instrument exogeneity: All the instruments areuncorrelated with the error term: corr(Z1i,ui) 0, ,corr(Zmi,ui) 0 If the instruments are correlated with the error term, thefirst stage of TSLS doesn’t successfully isolate acomponent of X that is uncorrelated with the error term, soX̂ is correlated with u and TSLS is inconsistent. If there are more instruments than endogenous regressors,it is possible to test – partially – for instrumentexogeneity.12-51
Testing overidentifying restrictionsConsider the simplest case:Yi β0 β1Xi ui, Suppose there are two valid instruments: Z1i, Z2i Then you could compute two separate TSLS estimates. Intuitively, if these 2 TSLS estimates are very differentfrom each other, then something must be wrong: one or theother (or both) of the instruments must be invalid. The J-test of overidentifying restrictions makes thiscomparison in a statistically precise way. This can only be done if #Z’s #X’s (overidentified).12-52
Suppose #instruments m # X’s k (overidentified)Yi β0 β1X1i βkXki βk 1W1i βk rWri uiThe J-test of overidentifying restrictionsThe J-test is the Anderson-Rubin test, using the TSLSestimator instead of the hypothesized value β1,0. The recipe:1. First estimate the equation of interest using TSLS and allm instruments; compute the predicted values Yˆ , using theiactual X’s (not the X̂ ’s used to estimate the second stage)2. Compute the residuals uˆi Yi – Yˆi3. Regress uˆi against Z1i, ,Zmi, W1i, ,Wri4. Compute the F-statistic testing the hypothesis that thecoefficients on Z1i, ,Zmi are all zero;5. The J-statistic is J mF12-53
J mF, where F the F-statistic testing the coefficientson Z1i, ,Zmi in a regression of the TSLS residuals againstZ1i, ,Zmi, W1i, ,Wri.Distribution of the J-statistic Under the null hypothesis that all the instruments areexogeneous, J has a chi-squared distribution with m–kdegrees of freedom If m k, J 0 (does this make sense?) If some instruments are exogenous and others areendogenous, the J statistic will be large, and the nullhypothesis that all instruments are exogenous will berejected.12-54
Checking Instrument Validity: SummaryThe two requirements for valid instruments:1. Relevance (special case of one X) At least one instrument must enter the populationcounterpart of the first stage regression. If instruments are weak, then the TSLS estimator is biasedand the and t-statistic has a non-normal distribution To check for weak instruments with a single includedendogenous regressor, check the first-stage Fo If F 10, instruments are strong – use TSLSo If F 10, weak instruments – take some action12-55
2. Exogeneity All the instruments must be uncorrelated with the errorterm: corr(Z1i,ui) 0, , corr(Zmi,ui) 0 We can partially test for exogeneity: if m 1, we can testthe hypothesis that all are exogenous, against thealternative that as many as m–1 are endogenous(correlated with u) The test is the J-test, constructed using the TSLSresiduals.12-56
Econ 423 – Lecture Notes (These notes are slightly modified versions of lecture notes provided by Stock and Watson, 2007. They are for instructional purposes only and are not to be distributed outside of the classroom.) . where cov(X,
Introduction of Chemical Reaction Engineering Introduction about Chemical Engineering 0:31:15 0:31:09. Lecture 14 Lecture 15 Lecture 16 Lecture 17 Lecture 18 Lecture 19 Lecture 20 Lecture 21 Lecture 22 Lecture 23 Lecture 24 Lecture 25 Lecture 26 Lecture 27 Lecture 28 Lecture
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