3 Endogeneity Instrumental variables.pdf

Applied Econometrics Endogeneity & Instrumental Variables Lecture 3 Autumn 2023 D.Sc. (Econ.) Elias Oikarinen Professor (Associate) of Economics University of Oulu, Oulu Business School Today (…and continues tomorrow) • Endogeneity in regression models • Instrumental Variables regression as a...

Applied Econometrics Endogeneity & Instrumental Variables Lecture 3 Autumn 2023 D.Sc. (Econ.) Elias Oikarinen Professor (Associate) of Economics University of Oulu, Oulu Business School Today (…and continues tomorrow) • Endogeneity in regression models • Instrumental Variables regression as a solution to endogeneity: Two Stage Least Squares • Basics of Instrumental Variables (IV) regression • Extra reading/information marked with * (i.e. slides that we do not go through during the lecture, but students should consider on their own) • Handout largely based on Stock and Watson, ”Introduction to Econometrics” • Discussion on endogeneity in Brooks, Ch 7 & Gujarati, Ch 19 – worth reading through! • Eviews 10 User Guide, Ch 21 2 Chapter 12 Instrumental Variables Regression © Pearson Education Limited 2015 3 Outline 1. Terminology 2. IV Regression: Why and What; Two Stage Least Squares 3. The General IV Regression Model 4. Checking Instrument Validity a) Weak and strong instruments b) Instrument exogeneity 5. Application: Demand for cigarettes 6. Examples: Where Do Instruments Come From? 4 Terminology: Endogeneity and Exogeneity An endogenous variable is one that is correlated with u An exogenous variable is one that is uncorrelated with u In IV regression, we focus on the case that X is endogenous and there is an instrument, Z, which is exogenous. Digression on terminology: “Endogenous” literally means “determined within the system.” If X is jointly determined with Y, then a regression of Y on X is subject to simultaneous causality bias. But this definition of endogeneity is too narrow because IV regression can be used to address omittedvariable bias and errors-in-variable bias. Thus we use the broader definition of endogeneity above. © Pearson Education Limited 2015 5 Instrumental Variables (IV) Regression – Why? • A fundamental assumption of regression analysis is that the right-hand side variables are uncorrelated with the disturbance term. If this assumption is violated, both OLS and weighted LS are biased and inconsistent. • Three possible sources of violation to this assumption are, i.e. problems that results in E(u|X) ≠ 0: • Simultaneous causality bias (X causes Y, Y causes X); • Omitted variable bias from a variable that is correlated with X but is unobserved (so cannot be included in the regression) and for which there are inadequate control variables; • Errors-in-variables bias (X is measured with error) • Instrumental variables regression can eliminate bias when E(u|X) ≠ 0 – using an instrumental variable (IV), Z. 6 Example of Endogeneity due to Simultaneous Causality • Housing market equilibrium equations • In equilibrium, demand = supply Dt = f(Yt,Ht,Pt) = 0 + 1Yt + 2Ht – 3Pt St* = g(Pt,ct) = 0 + 1Pt – 2ct  Pt = h(Dt,St) = β0 + β1Yt + β2Ht – β3St • If we estimate the price function, we face simultaneity problem  OLS is not a proper way to estimate the price regression • What about estimating the demand function or the supply function? (POLL) P = housing price level; Y = disposable income per household; H = number of households; c = construction costs • More examples to come later on in the handout 7 The IV Estimator with a Single Regressor and a Single Instrument (SW Section 12.1) Yi = β0 + β1Xi + ui , where Xi is endogenous • IV regression breaks X into two parts: a part that might be correlated with u, and a part that is not. By isolating the part that is not correlated with u, it is possible to estimate β1. • This is done using an instrumental variable, Zi, which is correlated with Xi but uncorrelated with ui. Instrumental variables (instruments) can be used to replace endogenous variables on the right hand side of a regression equation. The instruments are correlated with the variables they replace but not with the error term in the regression. © Pearson Education Limited 2015 8 Two Conditions for a Valid Instrument Yi = β0 + β1Xi + ui For an instrumental variable (an “instrument”) Z to be valid, it must satisfy two conditions: 1. Instrument relevance: corr(Zi,Xi) ≠ 0 2. Instrument exogeneity: corr(Zi,ui) = 0 Suppose for now that you have such a Zi (we’ll discuss how to find instrumental variables later). How can you use Zi to estimate β1? © Pearson Education Limited 2015 9 The IV estimator with one X and one Z Explanation #1: Two Stage Least Squares (TSLS) As it sounds, TSLS has two stages – two regressions: (1) Isolate the part of X that is uncorrelated with u by regressing X on Z using OLS: Xi = π0 + π1Zi + vi (1) • Because Zi is uncorrelated with ui, π0 + π1Zi is uncorrelated with ui. We don’t know π0 or π1 but we have estimated them, so… • Compute the predicted values of Xi, , where 𝑋෠𝑖 = 𝜋ො 0 + 𝜋ො1 𝑍𝑖 , i = 1,…,n. © Pearson Education Limited 2015 10 Two Stage Least Squares, ctd. (2) Replace Xi by Xˆ in the regression of interest: i regress Y on Xˆ i using OLS: Yi = β0 + β1 Xˆ i + ui (2) • Because Xˆ i is uncorrelated with ui, the exogeneity assumption holds for regression (2). (This requires n to be large so that π0 and π1 are precisely estimated.) • Thus, in large samples, β1 can be estimated by OLS using regression (2) • The resulting estimator is called the Two Stage Least TSLS Squares (TSLS) estimator, ˆ1 . © Pearson Education Limited 2015 11 Two Stage Least Squares: Summary Suppose Zi, satisfies the two conditions for a valid instrument: 1. Instrument relevance: corr(Zi,Xi) ≠ 0 2. Instrument exogeneity: corr(Zi,ui) = 0 Two-stage least squares: Stage 1: Regress Xi on Zi (including an intercept), obtain the predicted values Xˆ i Stage 2: Regress Yi on Xˆ i (including an intercept); the coefficient on Xˆ i is the TSLS estimator, ˆ TSLS . 1 ˆ1TSLS is a consistent estimator of β1. © Pearson Education Limited 2015 12 * The IV Estimator, one X and one Z, ctd. Explanation #2: A direct algebraic derivation Yi = β0 + β1Xi + ui Thus, 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 by instrument exogeneity; thus cov(Yi , Z i ) β1 = cov( X i , Z i ) © Pearson Education Limited 2015 13 * 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 with sample covariances: sYZ TSLS ˆ = , 1 s XZ sYZ and sXZ are the sample covariances. This is the TSLS estimator – just a different derivation! © Pearson Education Limited 2015 14 * The IV Estimator, one X and one Z, ctd. Explanation #3: Derivation from the “reduced form” The “reduced form” relates Y to Z and X to Z: Xi = π0 + π1Zi + vi Yi = γ0 + γ1Zi + wi where wi is an error term. Because Z is exogenous, Z is uncorrelated with both vi and wi. The idea: A unit change in Zi results in a change in Xi of π1 and a change in Yi of γ1. Because that change in Xi arises from the exogenous change in Zi, that change in Xi is exogenous. Thus an exogenous change in Xi of π1 units is associated with a change in Yi of γ1 units – so the effect on Y of an exogenous change in X is β1 = γ1/π1 units. © Pearson Education Limited 2015 15 * The IV estimator from the reduced form, ctd. The math: Xi = π0 + π1Zi + vi Yi = γ0 + γ1Zi + wi Solve the X equation for Z: Zi = –π0/π1 + (1/π1)Xi – (1/π1)vi Substitute this into the Y equation and collect terms: Yi = γ0 + γ1Zi + wi = γ0 + γ1[–π0/π1 + (1/π1)Xi – (1/π1)vi] + wi = [γ0 – π0γ1 /π1] + (γ1/π1)Xi + [wi – (γ1/π1)vi] = β0 + β1Xi + ui, where β0 = γ0 – π0γ1 /π1, β1 = γ1/π1, and ui = wi – (γ1/π1)vi. © Pearson Education Limited 2015 16 * The IV estimator from the reduced form, ctd. Xi = π0 + π1Zi + vi Yi = γ0 + γ1Zi + wi yields Yi = β0 + β1Xi + ui, where β1 = γ1/π1 Interpretation: An exogenous change in Xi of π1 units is associated with a change in Yi of γ1 units – so the effect on Y of an exogenous unit change in X is β1 = γ1/π1. © Pearson Education Limited 2015 17 Example #1: Supply and demand for butter IV regression was first developed to estimate demand elasticities for agricultural goods, for example, butter: ln( • Qibutter ) = β0 + β1ln( Pi butter ) + ui β1 = price elasticity of butter demand = percent change in quantity for a 1% change in price (recall log-log specification discussion) • Data: observations on price and quantity of butter for different years – time series data • butter butter Q The OLS regression of ln( i ) on ln( Pi ) suffers from simultaneous causality bias (why?) © Pearson Education Limited 2015 18 Simultaneous causality bias in the OLS regression of ln( Qibutter ) on ln( Pi butter ) arises because price and quantity are determined by the interaction of demand and supply: © Pearson Education Limited 2015 19 This interaction of demand and supply produces data like… Note: This is a time series model. Hence, the points in the graph reflect different quantity-price pairs observed in different periods of time. Would a regression using these data produce the demand curve? © Pearson Education Limited 2015 20 But…what would you get if only supply shifted? Note: In order to be able to estimate the slope of demand curve, and thus the price elasticity of demand, we need to keep the demand curve fixed – that is, we want remove the changes in quantity that have taken place due to demand curve shifts. • TSLS estimates the demand curve by isolating shifts in price and quantity that arise from shifts in supply. • Z is a variable that shifts supply but not demand. © Pearson Education Limited 2015 21 TSLS in the supply-demand example: ln(Qibutter ) = β0 + β1ln(Pi butter ) + ui Let Z = rainfall in dairy-producing regions. Is Z a valid instrument? (1) Relevant? corr(raini,ln( Pi )) ≠ 0? Plausibly: insufficient rainfall means less grazing means less butter means higher prices butter (2) Exogenous? corr(raini,ui) = 0? Plausibly: whether it rains in dairy-producing regions shouldn’t affect demand for butter © Pearson Education Limited 2015 22 TSLS in the supply-demand example, ctd. ln( Q butter) = β0 + β1ln(P butter) + ui i i Zi = raini = rainfall in dairy-producing regions. Stage 1: regress ln(Pi butter ෠ ) on rain, get In(𝑃 𝑏𝑢𝑡𝑡𝑒𝑟 ෠ ) (= “𝑋”) 𝑏𝑢𝑡𝑡𝑒𝑟 In(𝑃෠ ) isolates changes in log price that arise from supply (part of supply, at least) butter Stage 2: regress ln(Qi ) on In(𝑃෠ 𝑏𝑢𝑡𝑡𝑒𝑟 ) The regression counterpart of using shifts in the supply curve to trace out the demand curve. POLL © Pearson Education Limited 2015 23 Inference using TSLS • In large samples, the sampling distribution of the TSLS estimator is normal • Inference (hypothesis tests, confidence intervals) proceeds in the usual way, e.g. ± 1.96SE (with 95% intervals) • (see SW Appendix 12.3 for the details of the math...) © Pearson Education Limited 2015 24 Inference using TSLS, ctd. ˆ1TSLS is approx. distributed N(β1,  ˆ TSLS ), 2 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’ll discuss what happens if they aren’t valid shortly. • Important note on standard errors: – The OLS standard errors from the second stage regression aren’t right – they don’t take into account the estimation in the first stage ( Xˆ is estimated); The computation of the correct ones in i “rather involved” – Instead, use a single specialized command (in a software package) that computes the TSLS estimator and the correct SEs. – As usual, use heteroskedasticity-robust SEs if necessary © Pearson Education Limited 2015 25 Example #2: Demand for Cigarettes cigarettes ln( Qi ) = β0 + β1ln( Pi cigarettes ) + ui Why is the OLS estimator of β1 likely to be biased? • Data set: Cross-section data on annual cigarette consumption and average prices paid (including tax), by state, for the 48 continental US states, 1995. • Proposed instrumental variable: • Zi = general sales tax per pack in the state = SalesTaxi • Do you think this instrument is plausibly valid? 1. Relevant? corr(SalesTaxi, ln( P cigarettes )) ≠ 0? i 2. Exogenous? corr(SalesTaxi,ui) = 0? © Pearson Education Limited 2015 26 Cigarette demand, ctd. First stage OLS regression: In(𝑃෠ icigarettes)= 4.63 + .031SalesTaxi, n = 48 Second stage OLS regression: ln(Qicigarettes)= 9.72 – 1.08 In(𝑃෠ icigarettes), n = 48 Combined TSLS regression with correct, heteroskedasticityrobust standard errors: ln(Qicigarettes)= 9.72 – 1.08 In(Picigarettes), n = 48 (1.53) (0.32) © Pearson Education Limited 2015 27 EViews Example: Cigarette demand, First stage Instrument = Z = rtaxso = general sales tax (real $/pack) (data etc. included in Moodle to reproduce the results…) X Z LS(COV=WHITE) LRAVGPRS C RTAXSO Dependent Variable: LRAVGPRS Method: Least Squares Date: 09/13/19 Time: 18:21 Sample: 1 48 Included observations: 48 Huber-White-Hinkley (HC1) heteroskedasticity consistent standard errors and covariance Variable Coefficient Std. Error t-Statistic Prob. C RTAXSO 4.616546 0.030729 0.028918 0.004835 159.6444 6.354935 0.0000 0.0000 R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood F-statistic Prob(F-statistic) Prob(Wald F-statistic) 0.470996 0.459496 0.093945 0.405979 46.43466 40.95588 0.000000 0.000000 Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion Hannan-Quinn criter. Durbin-Watson stat Wald F-statistic 4.781380 0.127783 -1.851444 -1.773478 -1.821981 2.008192 40.38520 X-hat: Forecast → lravgprsf Now we have the predicted values from the 1st stage © Pearson Education Limited 2015 28 Second stage Y X-hat LS(COV=WHITE) LPACKPC C LRAVGPRSF Dependent Variable: LPACKPC Method: Least Squares Date: 09/13/19 Time: 18:24 Sample: 1 48 Included observations: 48 Huber-White-Hinkley (HC1) heteroskedasticity consistent standard errors and covariance Variable Coefficient Std. Error t-Statistic Prob. C LRAVGPRSF 9.719877 -1.083587 1.597119 0.333695 6.085882 -3.247237 0.0000 0.0022 R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood F-statistic Prob(F-statistic) Prob(Wald F-statistic) 0.152490 0.134066 0.226447 2.358809 4.204011 8.276648 0.006069 0.002178 Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion Hannan-Quinn criter. Durbin-Watson stat Wald F-statistic 4.538837 0.243346 -0.091834 -0.013867 -0.062370 2.099150 10.54455 • These coefficients are the TSLS estimates • The standard errors are wrong because they ignore the fact that the first stage was estimated → Running IV as a single command yields the correct SEs → Next slide © Pearson Education Limited 2015 29 Combined into a single command: Y X Z TSLS(COV=WHITE) LPACKPC C LRAVGPRS @ RTAXSO Dependent Variable: LPACKPC Method: Two-Stage Least Squares Date: 09/13/19 Time: 18:27 Sample: 1 48 Included observations: 48 White heteroskedasticity-consistent standard errors & covariance Instrument specification: RTAXSO Constant added to instrument list Variable Coefficient Std. Error t-Statistic Prob. C LRAVGPRS 9.719877 -1.083587 1.528322 0.318918 6.359835 -3.397693 0.0000 0.0014 R-squared Adjusted R-squared S.E. of regression F-statistic Prob(F-statistic) J-statistic 0.401129 0.388110 0.190354 11.71294 0.001313 0.000000 Mean dependent var S.D. dependent var Sum squared resid Durbin-Watson stat Second-Stage SSR Instrument rank 4.538837 0.243346 1.666792 1.992587 2.358809 2 Estimated cigarette demand equation: ln(Qicigarettes) = 9.72 – 1.08 In(Picigarettes), n = 48 (1.53) (0.31) © Pearson Education Limited 2015 30 R2 in TSLS Regression Although R2 is presented in the TSLS estimations statistics, it does not have the same interpretation as in the classical linear regression model and sometimes it can actually be negative. The reported R2 in IV regressions should be taken cautiously. 31 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 regressing Y on X̂ X̂ , then • The key idea is that the first stage isolates part of the variation in X that is uncorrelated with u • If the instrument is valid, then the large-sample sampling distribution of the TSLS estimator is normal, so inference proceeds as usual © Pearson Education Limited 2015 32 The General IV Regression Model (SW Section 12.2) • So far we have considered IV regression with a single endogenous regressor (X) and a single instrument (Z). • We need to extend this to: – multiple endogenous regressors (X1,…,Xk) – multiple included exogenous variables (W1,…,Wr) or control variables, which need to be included for the usual OV reason – multiple instrumental variables (Z1,…,Zm). More (relevant) instruments can produce a smaller variance of TSLS: the R2 of the first stage increases, so you have more variation in X̂ . • New terminology: identification & overidentification © Pearson Education Limited 2015 33 Housing Market Example cont’d • Housing market equilibrium equilibrium • In equilibrium, demand = supply Dt = f(Yt,Ht,Pt) = 0 + 1Yt + 2Ht – 3Pt St* = g(Pt,ct) = 0 + 1Pt – 2ct  Pt = h(Dt,St) = β0 + β1Yt + β2Ht – β3St • How many potentially endogenous regressors are there in the price model? (POLL) 34 The General IV Regression Model: Summary of Jargon Yi = β0 + β1X1i + … + βkXki + βk+1W1i + … + βk+rWri + ui • Yi is the dependent variable • X1i,…, Xki are the endogenous regressors (potentially correlated with ui) • W1i,…,Wri are the included exogenous regressors (uncorrelated with ui) or control variables (included so that Zi is uncorrelated with ui, once the W’s are included) • β0, β1,…, βk+r are the unknown regression coefficients • Z1i,…,Zmi are the m instrumental variables (the excluded exogenous variables) • The coefficients are overidentified if m > k; exactly identified if m = k; and underidentified if m < k. © Pearson Education Limited 2015 35 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 estimate β1,…, βk. If so, you can test whether the instruments are valid (a test of the “overidentifying restrictions”) – we’ll return to this later • underidentified if m < k. There are too few instruments to estimate β1,…, βk. If so, you need to get more instruments! © Pearson Education Limited 2015 36 TSLS with a Single Endogenous Regressor Yi = β0 + β1X1i + β2W1i + … + β1+rWri + ui • m instruments: Z1i,…, Zm • First stage – Regress X1 on all the exogenous regressors: regress X1 on W1,…,Wr, Z1,…, Zm, and an intercept, by OLS – Compute predicted values Xˆ 1i , i = 1,…,n • Second stage – Regress Y on Xˆ 1i , W1,…, Wr, and an intercept, by OLS – The coefficients from this second stage regression are the TSLS estimators, but SEs are wrong • To get correct SEs, do this in a single step in your regression software © Pearson Education Limited 2015 37 Example #2: Demand for cigarettes, ctd. Suppose income is exogenous, and we also want to estimate the income elasticity: ln(Qicigarettes)= β0 + β1ln(Picigarettes) + β2ln(Incomei) + ui We actually have two instruments: Z1i = general sales taxi Z2i = cigarette-specific taxi • Endogenous variable: ln(Picigarettes) (“one X”) • Included exogenous variable: ln(Incomei) (“one W”) • Instruments (excluded endogenous variables): general sales tax, cigarette-specific tax (“two Zs”) • Is β1 over–, under–, or exactly identified? (POLL) © Pearson Education Limited 2015 38 Example: Cigarette demand, one instrument IV: rtaxso = real overall sales tax in state Y X W Z TSLS(COV=WHITE) LPACKPC C LRAVGPRS LPERINC @ RTAXSO LPERINC Dependent Variable: LPACKPC Method: Two-Stage Least Squares Date: 09/13/19 Time: 18:30 Sample: 1 48 Included observations: 48 White heteroskedasticity-consistent standard errors & covariance Instrument specification: RTAXSO LPERINC Constant added to instrument list Variable Coefficient Std. Error t-Statistic Prob. C LRAVGPRS LPERINC 9.430658 -1.143375 0.214515 1.259393 0.372303 0.311747 7.488260 -3.071090 0.688107 0.0000 0.0036 0.4949 R-squared Adjusted R-squared S.E. of regression F-statistic Prob(F-statistic) J-statistic Instrumented: Instruments: lravgprs rtaxso lperinc © Pearson Education Limited 2015 0.418934 0.393109 0.189575 6.533672 0.003227 0.000000 Mean dependent var S.D. dependent var Sum squared resid Durbin-Watson stat Second-Stage SSR Instrument rank 4.538837 0.243346 1.617235 1.960299 2.313601 3 EViews lists ALL the exogenous regressors as instruments – slightly different terminology than we have been using 39 Example: Cigarette demand, two instruments Y X W Z1 TSLS(COV=WHITE) LPACKPC C LRAVGPRS LPERINC @ RTAXSO Z2 RTAX LPERINC Dependent Variable: LPACKPC Method: Two-Stage Least Squares Date: 09/13/19 Time: 18:39 Sample: 1 48 Included observations: 48 White heteroskedasticity-consistent standard errors & covariance Instrument specification: RTAXSO RTAX LPERINC Constant added to instrument list Variable Coefficient Std. Error t-Statistic Prob. C LRAVGPRS LPERINC 9.894956 -1.277424 0.280405 0.959217 0.249610 0.253890 10.31566 -5.117680 1.104436 0.0000 0.0000 0.2753 R-squared Adjusted R-squared S.E. of regression F-statistic Prob(F-statistic) J-statistic Prob(J-statistic) Instrumented: lravgprs Instruments: rtaxso rtax lperinc © Pearson Education Limited 2015 0.429422 0.404063 0.187856 13.28079 0.000029 0.311833 0.576557 Mean dependent var S.D. dependent var Sum squared resid Durbin-Watson stat Second-Stage SSR Instrument rank 4.538837 0.243346 1.588044 1.946351 1.845868 4 EViews lists ALL the exogenous regressors as “instruments” – slightly different terminology than we have been using 40 TSLS estimates, Z = sales tax (m = 1) In(Qicigarettes)= 9.43 – 1.14 In(Picigarettes) + 0.21ln(Incomei) (1.26) (0.37) (0.31) TSLS estimates, Z = sales tax & cig-only tax (m = 2) In(Qicigarettes)= 9.89 – 1.28 In(Picigarettes) + 0.28ln(Incomei) (0.96) (0.25) (0.25) • Smaller SEs for m = 2. Using 2 instruments gives more information – more “as-if random variation.” • Low income elasticity (not a luxury good); income elasticity not statistically significantly different from 0 • Surprisingly high price elasticity (why surprising?) © Pearson Education Limited 2015 41 * The General Instrument Validity Assumptions Yi = 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’s Suppose the second stage regression could be run using the predicted values from the population first stage regression. Then: there is no perfect multicollinearity in this (infeasible) second stage regression. • Special case of one X: the general assumption is equivalent to (a) at least one instrument must enter the population counterpart of the first stage regression, and (b) the W’s are not perfectly multicollinear. © Pearson Education Limited 2015 42 * The IV Regression Assumptions Yi = β0 + β1X1i + … + βkXki + βk+1W1i + … + βk+rWri + ui 1. 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 new 3. The X’s, W’s, Z’s, and Y have nonzero, finite 4th moments • #3 is not new 4. 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 © Pearson Education Limited 2015 43 Checking Instrument Validity (SW Section 12.3) Recall the two requirements for valid instruments: 1. Relevance (special case of one X) At least one instrument must enter the population counterpart of the first stage regression. 2. Exogeneity All the instruments must be uncorrelated with the error term: corr(Z1i,ui) = 0,…, corr(Zmi,ui) = 0 What 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? © Pearson Education Limited 2015 44 Checking Assumption #1: Instrument Relevance We will focus on a single included endogenous regressor: Yi = β0 + β1Xi + β2W1i + … + β1+rWri + ui First stage regression: Xi = π0 + π1Z1i +…+ πmZmi + πm+1W1i +…+ πm+kWki + ui • The instruments are relevant if at least one of π1,…, πm are nonzero. • The instruments are said to be weak if all the π1,…, πm are either zero or nearly zero. • Weak instruments explain very little of the variation in X, beyond that explained by the W’s • If instruments are weak, the sampling distribution of TSLS and its t-statistic are not (at all) normal, even with n large. © Pearson Education Limited 2015 45 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 on Z1,…,Zm are zero. • The first-stage F-statistic tests the hypothesis that Z1,…,Zm do not enter the first stage regression. • Weak instruments imply a small first stage F-statistic. © Pearson Education Limited 2015 46 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 than 10, then the set of instruments is weak. • If so, the TSLS estimator will be biased, and statistical inferences (standard errors, hypothesis tests, confidence intervals) can be misleading. © Pearson Education Limited 2015 47 * Checking for Weak Instruments with a Single X, ctd. • Why compare the first-stage F to 10? • Simply rejecting the null hypothesis that the coefficients on the Z’s are zero isn’t enough – you need substantial predictive content for the normal approximation to be a good one. • Comparing the first-stage F to 10 tests for whether the bias of TSLS, relative to OLS, is less than 10%. If F is smaller than 10, the relative bias exceeds 10%—that is, TSLS can have substantial bias (see SW App. 12.5). © Pearson Education Limited 2015 48 What to do if you have weak instruments • Get better instruments (often easier said than done!) • If you have many instruments, some are probably weaker than others and it’s a good idea to drop the weaker ones (dropping an irrelevant instrument will increase the firststage F) • If you only have a few instruments, and all are weak, then you need to do some IV analysis other than TSLS… – Separate the problem of estimation of β1 and construction of confidence intervals – This seems odd, but if TSLS isn’t normally distributed, it makes sense (right?) © Pearson Education Limited 2015 49 Checking Assumption #2: Instrument Exogeneity • Instrument exogeneity: All the instruments are uncorrelated with the error term: corr(Z1i, ui) = 0,…, corr(Zmi, ui) = 0 • If the instruments are correlated with the error term, the first stage of TSLS cannot isolate a component of X that is uncorrelated with the error term, so X̂ is correlated with u and TSLS is inconsistent. • If there are more instruments than endogenous regressors, it is possible to test – partially – for instrument exogeneity. © Pearson Education Limited 2015 50 Testing Overidentifying Restrictions Consider 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 different from each other, then something must be wrong: one or the other (or both) of the instruments must be invalid. • The J-test of overidentifying restrictions makes this comparison in a statistically precise way. • This can only be done if #Z’s > #X’s (overidentified). © Pearson Education Limited 2015 51 * The J-test of Overidentifying Restrictions Suppose #instruments = m > # X’s = k (overidentified) Yi = β0 + β1X1i + … + βkXki + βk+1W1i + … + βk+rWri + ui The J-test is the Anderson-Rubin test, using the TSLS estimator instead of the hypothesized value β1,0. The recipe: 1. First estimate the equation of interest using TSLS and all m instruments; compute the predicted values Yˆi , using the actual X’s (not the X̂ ’s used to estimate the second stage) uˆi = Yi –Yˆi 2. Compute the residuals 3. Regress against Z1i,…,Zmi, W1i,…,Wri 4. Compute the F-statistic testing the hypothesis that the coefficients on Z1i,…,Zmi are all zero; 5. The J-statistic is J = mF © Pearson Education Limited 2015 52 * The J-test, ctd J = mF, where F = the F-statistic testing the coefficients on Z1i,…,Zmi in a regression of the TSLS residuals against Z1i,…,Zmi, W1i,…,Wri. Distribution of the J-statistic • Under the null hypothesis that all the instruments are exogeneous, J has a chi-squared distribution with m–k degrees of freedom • If m = k, J = 0 • If some instruments are exogenous and others are endogenous, the J statistic will be large, and the null hypothesis that all instruments are exogenous will be rejected. © Pearson Education Limited 2015 53 * Checking Instrument Validity: Summary This summary considers the case of a single X. The two requirements for valid instruments are: 1. Relevance • At least one instrument must enter the population counterpart of the first stage regression. • If instruments are weak, then the TSLS estimator is biased and the and t-statistic has a non-normal distribution • To check for weak instruments with a single included endogenous regressor, check the first-stage F – If F>10, instruments are strong – use TSLS – If F<10, weak instruments – take some action. © Pearson Education Limited 2015 54 * Checking Instrument Validity: Summary 2. Exogeneity • All the instruments must be uncorrelated with the error term: corr(Z1i,ui) = 0,…, corr(Zmi,ui) = 0 • We can partially test for exogeneity: if m>1, we can test the null hypothesis that all the instruments are exogenous, against the alternative that as many as m–1 are endogenous (correlated with u) • The test is the J-test, which is constructed using the TSLS residuals. • If the J-test rejects, then at least some of your instruments are endogenous – so you must make a difficult decision and jettison some (or all) of your instruments. © Pearson Education Limited 2015 55 * Application to the Demand for Cigarettes Why are we interested in knowing the elasticity of demand for cigarettes? • Theory of optimal taxation. The optimal tax rate is inversely related to the price elasticity: the greater the elasticity, the less quantity is affected by a given percentage tax, so the smaller is the change in consumption and deadweight loss. • Externalities of smoking – role for government intervention to discourage smoking – health effects of second-hand smoke? (non-monetary) – monetary externalities © Pearson Education Limited 2015 56 Model of cigarette demand ln( Q cigarettes ) = β0 + β1ln( P cigarettes ) + β2ln(Incomeit) + uit i i Estimation strategy • corr(ln( P cigarettes ), uit) is plausibly nonzero because of i supply/demand interactions • We need to use IV estimation methods to handle the simultaneous causality bias that arises from the interaction of supply and demand. © Pearson Education Limited 2015 57 Use TSLS to estimate the demand elasticity Y TSLS(COV=WHITE) LPACKPC C W LRAVGPRS X Z PERINC @ RTAXSO LPERINC Dependent Variable: LPACKPC Method: Two-Stage Least Squares Date: 09/13/19 Time: 19:12 Sample: 1 48 Included observations: 48 White heteroskedasticity-consistent standard errors & covariance Instrument specification: RTAXSO LPERINC Constant added to instrument list Variable Coefficient Std. Error t-Statistic Prob. C LRAVGPRS LPERINC 9.430658 -1.143375 0.214515 1.259393 0.372303 0.311747 7.488260 -3.071090 0.688107 0.0000 0.0036 0.4949 R-squared Adjusted R-squared S.E. of regression F-statistic Prob(F-statistic) J-statistic 0.418934 0.393109 0.189575 6.533672 0.003227 0.000000 Mean dependent var S.D. dependent var Sum squared resid Durbin-Watson stat Second-Stage SSR Instrument rank 4.538837 0.243346 1.617235 1.960299 2.313601 3 NOTE: - Estimated elasticity = –1.14 (SE = 0.37) – surprisingly elastic! - Income elasticity small, not statistically different from zero - Must check whether the instrument is relevant… 58 © Pearson Education Limited 2015 Check instrument relevance: compute first-stage F LS(COV=WHITE) LRAVGPRS C RTAXSO LPERINC Dependent Variable: LRAVGPRS Method: Least Squares Date: 09/13/19 Time: 19:13 Sample: 1 48 Included observations: 48 Huber-White-Hinkley (HC1) heteroskedasticity consistent standard errors and covariance Variable Coefficient Std. Error t-Statistic C RTAXSO LPERINC 3.590811 0.027395 0.389283 0.172727 0.004096 0.065391 20.78890 6.688088 5.953118 R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood F-statistic Prob(F-statistic) Prob(Wald F-statistic) 0.638896 0.622847 0.078475 0.277126 55.59861 39.80900 0.000000 0.000000 Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion Hannan-Quinn criter. Durbin-Watson stat Wald F-statistic Wald Test: Equation: Untitled Prob. Test Statistic Value 0.0000 0.0000 t-statistic 6.688088 0.0000 F-statistic 44.73053 Chi-square 44.73053 4.781380 0.127783 -2.191609 Null Hypothesis: C(2)=0 -2.074659 -2.147413 2.312244 48.98071 df Probability 45 (1, 45) 1 0.0000 0.0000 0.0000 We didn’t need to run “test” here! With m=1, the F-stat is the square of the t-stat: 6.69^2 = 44.7 First stage F = 44.7 > 10, so instrument is not weak Can we check instrument exogeneity? No: m = k © Pearson Education Limited 2015 59 Cigarette demand – 2 IVs Y TSLS(COV=WHITE) LPACKPC C W LRAVGPRS X Z1 PERINC @ RTAXSO Z2 RTAX LPERINC Dependent Variable: LPACKPC Method: Two-Stage Least Squares Date: 09/13/19 Time: 19:15 Sample: 1 48 Included observations: 48 White heteroskedasticity-consistent standard errors & covariance Instrument specification: RTAXSO RTAX LPERINC Constant added to instrument list Variable Coefficient Std. Error t-Statistic Prob. C LRAVGPRS LPERINC 9.894956 -1.277424 0.280405 0.959217 0.249610 0.253890 10.31566 -5.117680 1.104436 0.0000 0.0000 0.2753 R-squared Adjusted R-squared S.E. of regression F-statistic Prob(F-statistic) J-statistic Prob(J-statistic) 0.429422 0.404063 0.187856 13.28079 0.000029 0.311833 0.576557 Mean dependent var S.D. dependent var Sum squared resid Durbin-Watson stat Second-Stage SSR Instrument rank 4.538837 0.243346 1.588044 1.946351 1.845868 4 rtaxso = general sales tax only rtax = cigarette-specific tax only Estimated elasticity is -1.28, even more elastic than using general sales tax only! © Pearson Education Limited 2015 60 First-stage F – both instruments X Z1 LS(COV=WHITE) LRAVGPRS C RTAXSO Z2 W RTAX LPERINC Dependent Variable: LRAVGPRS Method: Least Squares Date: 09/13/19 Time: 19:05 Sample: 1 48 Included observations: 48 Huber-White-Hinkley (HC1) heteroskedasticity consistent standard errors and covariance Variable Coefficient Std. Error t-Statistic Prob. C RTAXSO RTAX LPERINC 4.103034 0.010890 0.009352 0.108345 0.088380 0.002137 0.000870 0.039653 46.42477 5.096820 10.75151 2.732358 0.0000 0.0000 0.0000 0.0090 R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood F-statistic Prob(F-statistic) Prob(Wald F-statistic) 0.940328 0.936260 0.032261 0.045794 98.80605 231.1234 0.000000 0.000000 Null hypothesis: F(2, 44) = 209.7 © Pearson Education Limited 2015 Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion Hannan-Quinn criter. Durbin-Watson stat Wald F-statistic Wald Test: Equation: Untitled Test Statistic F-statistic Chi-square Value df Probability 209.6763 419.3525 (2, 44) 2 0.0000 0.0000 Null Hypothesis: C(2)=C(3)=0 4.781380 0.127783 -3.950252 -3.794319 -3.891325 1.604605 263.1230 drtaxso = 0 and drtax = 0 > 10 so instruments aren’t weak P = 0.0000 61 Test the overidentifying restrictions From slide 60 With m>k, we can test the overidentifying restrictions… POLL Dependent Variable: LPACKPC Method: Two-Stage Least Squares Date: 09/13/19 Time: 19:15 Sample: 1 48 Included observations: 48 White heteroskedasticity-consistent standard errors & covariance Instrument specification: RTAXSO RTAX LPERINC Constant added to instrument list Variable Coefficient Std. Error t-Statistic Prob. C LRAVGPRS LPERINC 9.894956 -1.277424 0.280405 0.959217 0.249610 0.253890 10.31566 -5.117680 1.104436 0.0000 0.0000 0.2753 R-squared Adjusted R-squared S.E. of regression F-statistic Prob(F-statistic) J-statistic Prob(J-statistic) 0.429422 0.404063 0.187856 13.28079 0.000029 0.311833 0.576557 Mean dependent var S.D. dependent var Sum squared resid Durbin-Watson stat Second-Stage SSR Instrument rank 4.538837 0.243346 1.588044 1.946351 1.845868 4 • Under the null hypothesis that all the instruments are exogeneous, J has a chi-squared distribution with m–k degrees of freedom • Here, J = 0.31, distributed chi-squared with d.f. = 1; the 5% critical value is 3.84, so do not reject the null hypothesis that both the instruments are exogenous (exact p-value=0.58) © Pearson Education Limited 2015 62 How should we interpret the J-test rejection? • What if J-test would have rejected the null hypothesis that both the instruments are exogenous? • That would mean that either rtaxso is endogenous, or rtax is endogenous, or both! • The J-test doesn’t tell us which! In these kind of situations, You must exercise judgment… • Why might e.g. cig-only tax be endogenous? – Political forces: history of smoking or lots of smokers; political pressure for low cigarette taxes – If so, cig-only tax is endogenous → (if j-test had rejected) © Pearson Education Limited 2015 use just one instrument, the general sales tax 63 * The Demand for Cigarettes: Summary of Empirical Results • Use the estimated elasticity based on TSLS with the general sales tax as the only instrument: Elasticity = -1.28, SE = 0.25 • This elasticity is surprisingly large (not inelastic) – a 1% increase in prices reduces cigarette sales by nearly 1.3%. This is much more elastic than conventional wisdom in the health economics literature. • This is a short-run elasticity. What would you expect a longrun (ten-year change) elasticity to be – more or less elastic? © Pearson Education Limited 2015 64 * Assess the Validity of the Study Remaining threats to internal validity? 1. Omitted variable bias? – Unobserved factors that vary across states 2. 3. 4. 5. 6. Functional form mis-specification? (could check this) Remaining simultaneous causality bias? Errors-in-variables bias? Selection bias? (no, we have all the states) An additional threat to internal validity of IV regression studies is whether the instrument is (1) relevant and (2) exogenous. How significant are these threats in the cigarette elasticity application? © Pearson Education Limited 2015 65 * Assess the Validity of the Study, ctd. External validity? • We have estimated a short-run elasticity – can it be generalized to a long-run elasticity? Why or why not? • Suppose we want to use the estimated elasticity of -1.28 to guide policy today. Here are two changes since the period covered by the data (1995) – do these changes pose a threat to external validity (generalization from 1995 to today)? – Levels of smoking today are lower than in 1995 – Cultural attitudes toward smoking have changed against smoking since 1995 © Pearson Education Limited 2015 66 * Where Do Valid Instruments Come From? (SW Section 12.5) General comments The hard part of IV analysis is finding valid instruments • Method #1: “variables in another equation” (e.g. supply shifters that do not affect demand) • Method #2: look for exogenous variation (Z) that is “as if” randomly assigned (does not directly affect Y) but affects X. • These two methods are different ways to think about the same issues – see the link… – Rainfall shifts the supply curve for butter but not the demand curve; rainfall is “as if” randomly assigned – Sales tax shifts the supply curve for cigarettes but not the demand curve; sales taxes are “as if” randomly assigned © Pearson Education Limited 2015 67 * Conclusion (SW Section 12.6) • A valid instrument lets us isolate a part of X that is uncorrelated with u, and that part can be used to estimate the effect of a change in X on Y • IV regression hinges on having valid instruments: 1. Relevance: Check via first-stage F 2. Exogeneity: Test overidentifying restrictions via the Jstatistic • A valid instrument isolates variation in X that is “as if” randomly assigned. • The critical requirement of at least m valid instruments cannot be tested – you must use your head. © Pearson Education Limited 2015 68 * Some IV FAQs 1. When might I want to use IV regression? Any time that X is correlated with u and you have a valid instrument. The primary reasons for correlation between X and u could be: • Omitted variable(s) that lead to OV bias – Ex: ability bias in returns to education • Measurement error – Ex: measurement error in years of education • Selection bias – Patients select treatment • Simultaneous causality bias – Ex: supply and demand for butter, cigarettes © Pearson Education Limited 2015 69 * 2. What are the threats to the internal validity of an IV regression? • The main threat to the internal validity of IV is the failure of the assumption of valid instruments. Given a set of control variables W, instruments are valid if they are relevant and exogenous. – Instrument relevance can be assessed by checking if instruments are weak or strong: Is the first-stage F-statistic > 10? – Instrument exogeneity can be checked using the J-statistic – as long as you have m exogenous instruments to start with! In general, instrument exogeneity must be assessed using expert knowledge of the application. © Pearson Education Limited 2015 70 Other Options to Handle Simultaneity (in time series models) • Vector autoregressive (VAR) & Vector error-correction models (VECM) (multiple equation time series models) • Estimators that are designed for estimating single-equation models with endogeneity issues; in time series regressions e.g. FMOLS (Fully-Modified OLS) & DOLS (Dynamic OLS) that are designed for non-stationary but cointegrated data 71