8. Instrumental variables regression

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1 8. Instrumental variables regression Recall: In Section 5 we analyzed five sources of estimation bias arising because the regressor is correlated with the error term Violation of the first OLS assumption These threats to internal validity are Omitted variable bias Misspecification of the functional form Measurement error Sample selection bias Simultaneous causality 213

2 Now: General technique that helps to obtain a consistent estimator of the unknown coefficients when the regressor X is correlated with the error term u Instrumental variables (IV) regression Basic idea: Think of the variation in X as having two parts: one part that is correlated with u (the problematic part) a second part that is uncorrelated with u (the unproblematic part which can be used for estimation) 214

3 Issues of this section: How can we isolate the problematic from the unproblematic parts in the variations of X? By the use of instrumental variables (instruments) What are good instruments and how can we find them? 215

4 8.1. The IV estimator with a single regressor and a single instrument IV model and assumptions: We consider the single-regressor model Y i = β 0 + β 1 X i + u i, i = 1,..., n, (8.1) X i and u i are assumed to be correlated, that is Corr(X i, u i ) 0 We use the additional instrumental variable Z to isolate that part of X i that is uncorrelated with u i 216

5 Terminology: We call variables correlated with the error term endogenous We call variables uncorrelated with the error term exogenous Two conditions for a valid instrument Z: 1. Instrument relevance condition: Corr(Z i, X i ) 0 (variation in the instrument Z i is related to variation in X i ) 2. Instrument exogeneity condition: Corr(Z i, u i ) = 0 (that part of the variation in X i captured by Z i is exogenous) 217

6 Implication of these conditions: The relevant and exogenous instrument Z can capture movements in X that are exogenous This exogenous part of X can be used to consistently estimate β 1 Formalization of this concept: Two stage least squares estimation (TSLS) First stage: Decomposition of X into the problematic and the problemfree components Second stage: Use the problem-free component to estimate β 1 218

7 Two stage least squares estimator: 1. Consider the regression equation X i = π 0 + π 1 Z i }{{} Part #1 + v i }{{} Part #2 (8.2) Part #1 is that part of X i that can be predicted by Z i Since Z i is exogenous it follows that Corr(π 0 + π 1 Z i, u i ) = π 1 π 1 Corr(Z i, u i ) = 0 (Part #1 is the problem-free part) Part #2 is v i for which we have Corr(v i, u i ) 0 (Part #2 is the problematic part) We apply OLS to Eq. (8.2) to obtain ˆπ 0 and ˆπ 1 219

8 Two stage least squares estimator: [continued] 2. We use the predicted values ˆX i = ˆπ 0 + ˆπ 1 Z i and consider the regression equation Y i = β 0 + β 1 ˆX i + u i (8.3) We apply OLS to Eq. (8.3) and obtain the TSLS estimators β0 TSLS of β 0 and β1 TSLS of β 1 220

9 Example: Estimation of the demand curve for butter based on data on the quantity of butter consumed (Q butter i ) and butter prices (Pi butter ) sampled over n years (i = 1,..., n) We aim at estimating the butter demand curve where Y i = β 0 + β 1 X i + u i, Y i = ln(q butter i ) X i = ln(pi butter ) β 1 = price elasticity of butter demand 221

10 Example: [continued] We have a simultaneous causality bias here since there are causal links from ln(pi butter ) to ln(q butter i ), but also from ln(q butter i ) to ln(pi butter ) via the interaction between the demand for and the supply of butter It follows from Section (Slides ) that the regressor ln(pi butter ) is likely to be correlated with the error term OLS estimator of β 1 will be inconsistent 222

11 Equilibrium price and quantity data 223

12 Equilibrium price and quantity data [continued] 224

13 Equilibrium price and quantity data [continued] 225

14 Example: [continued] To circumvent this problem we need an instrumental variable Z i which shifts the supply curve but leaves the demand curve unaffected Such an instrument Z i could be the the variable RAINFALL in the butter-producing region Relevance condition: Below average rainfall reduces cattle-grazing and thus reduces butter production at a given price: Corr(RAINFALL i, ln(p butter i )) = 0 Exogeneity condition: Demand for butter does not depend on the rainfall: Corr(RAINFALL i, u i ) = 0 226

15 Example: [continued] TSLS estimation: Stage 1: Regress ln(pi butter ) on RAINFALL i and compute ln(p butter i ) (Isolation of price changes due to shifts in the supply curve) Stage 2: Regress ln(q butter i ) on ln(p butter i ) 227

16 Statistical inference for TSLS: It can be shown that the TSLS estimator ˆβ TSLS 1 is consistent and, in large samples, approximately normally distributed: where σ 2ˆβ TSLS 1 ˆβ TSLS 1 N(β 1, σ = 1 n 2ˆβ TSLS), 1 Var {[Z i E(Z)] u i } [Cov(Z i, X i )] 2 (8.4) The standard error of ˆβ TSLS 1 can be estimated by estimating the variance and covariance terms appearing on the righthand side of Eq. (8.4) and taking the square root of the estimate of σ 2ˆβ TSLS 1 228

17 Statistical inference for TSLS: [continued] These standard errors are routinely computed by economometric software packages like EViews Because ˆβ TSLS 1 is normally distributed in large samples, hypothesis tests and confidence intervals about β 1 can be conducted in the usual way Attention: The ususal OLS standard errors of Stage 2 are not identical to the TSLS standard errors described above and thus are invalid (since these ignore the prediction errors of the ˆX i ) One should use the special TSLS routines implemented in the software packages 229

18 8.2. The general IV regression model Now: Generalization of the IV regression model to multiple regressors and instruments Four types of variables: The dependent variable Y Problematic endogenous regressors Included exogenous regressors Instrumental variables 230

19 Definition 8.1: (General IV regression model) The general IV regression model is Y i = β 0 +β 1 X 1i +...+β k X ki +β k+1 W 1i +...+β k+r W ri +u i, (8.5) i = 1,..., n, where Y i is the dependent variable, β 0, β 1,..., β k+r are unknown regression coefficients, X 1i,..., X ki are k endogenous regressors potentially correlated with u i, W 1i,..., W ri are r included exogenous regressors which are uncorrelated with u i or are control variables, u i is the error term, Z 1i,..., Z mi are m instrumental variables. 231

20 Definition 8.1: (General IV regression model) [continued] The coefficients are overidentified if there are more instruments than endogenous regressors (m > k), they are underidentified if m < k, and they are exactly identified if m = k. Estimation of the IV regression model requires exact identification or overidentification. Now: Adaption of the TSLS principle to the general IV model described in Definition

21 TSLS in the general IV model: Consider the general IV regression model (8.5) from Slide First-stage regression(s): Regress X 1i on the instrumental variables (Z 1i,..., Z mi ) and the included exogenous variables (W 1i,..., W ri ) using OLS, that is estimate the following equation via OLS: X 1i = π 0 + π 1 Z 1i π m Z mi + π m+1 W 1i π m+r W ri + v i (8.6) Compute the predicted values ˆX 1i from this regression Repeat this for all endogenous regressors X 2i,..., X ki, thereby computing the predicted values ˆX 2i,..., ˆX ki 233

22 TSLS in the general IV model: [continued] 2. Second-stage regression Regress Y i on the predicted values of the endogenous variables ˆX 1i,..., ˆX ki and the included exogenous variables (W 1i,..., W ri ), that is estimate the following equation via OLS: Y 1i = β 0 + β 1 ˆX 1i β k ˆX ki + β k+1 W 1i β k+r W ri + u i (8.7) The TSLS estimators β0 TSLS,..., βk+r TSLS are the OLS estimators from the second-stage regression (8.7) Remark: The two stages are done automatically within TSLS estimation commands in EViews 234

23 Now: Adaption of the conditions for a valid instrument Z from Slide 217 (relevance and exogeneity) to the general IV regression model Intuitively: When there are multiple included endogenous variables, the condition for instrument relevance must be formulated in a way that it rules out multicollinearity in the second-stage regression should reflect that the instruments provide enough information about the exogenous movements in the endogenous variables to sort out their seperate effects on Y 235

24 Definition 8.2: (Conditions for valid instruments) A set of m instruments Z 1i,..., Z mi must satisfy the following two conditions to be valid: 1. Instrument relevance: In general, let ˆX 1i be the predicted value of X 1i from the regression of X 1i on the instruments Z 1i,..., Z mi and the included exogenous regressors W 1i,..., W ri and let ˆX 2i,..., ˆX ki be analogously defined. Furthermore, let 1 denote the n-dimensional vector 1 (1,..., 1). Then ( ˆX 1,..., ˆX k, W 1,..., W r, 1) are not perfectly multicollinear. 236

25 Definition 8.2: (Conditions for valid instruments) [continued] 1. Instrument relevance: [continued] If there is only one endogenous regressor X i, then for the previous condition to hold, at least one instrument Z ji, (j = 1,..., m), must have a non-zero coefficient in the regression equation X i = π 0 + π 1 Z 1i π m Z mi + π m+1 W 1i π m+r W ri + v i. 2. Instrument exogeneity: All instruments are uncorrelated with the error term: Corr(Z 1i, u i ) = 0,..., Corr(Z mi, u i ) =

26 Next: Under which conditions are the TSLS estimators consistent and do have a sampling distribution that is normal in large samples? If we can specify conditions under which this is the case, then the principles of statistical inference for TSLS in the single-regressor case as described on Sildes carry over to the general case of multiple instruments and multiple endogenous variables (t-statistics, F -statistics, confidence intervals) 238

27 The IV regression assumptions: The variables and errors in the IV regression model in Eq. (8.5) should satisfy the following conditions: 1. E(u i W 1i,..., W ri ) = 0 2. (X 1i,..., X ki, W 1i,..., W ri, Z 1i,..., Z mi, Y i ) are i.i.d. draws from their joint distribution 3. Large outliers are unlikely: X s, W s, Z s, and Y variables have nonzero finite fourth moments 4. The two conditions for valid instruments stated in Definition 8.2 hold 239

28 Remarks: The calculation of TSLS standard errors is done automatically by software packages like EViews One should use heteroskedasticity-robust standard errors for the same reasoning as in the conventional multiple linear regression model 240

29 8.3. Checking instrument validity Important question: Is a given set of instruments valid in a particular application? Meaning of instrument relevance : Instrumental relevance plays a role akin to the sample size A more relevant instrument produces a more accurate estimator, just as a large sample size produces a more accurate estimator The more relevant is the instrument, the better is the normal approximation to the sampling distribution of the TSLS estimator and its t- and F -statistics 241

30 Problems with weak instruments: If the instruments are weak, then the TSLS estimator can be badly biased and the normal distribution is a poor approximation to the sampling distribution of the TSLS estimator No justification for performing statistical inference as described even when the sampling size is large TSLS is no longer reliable Checking for weak instruments: How relevant must instruments be for the normal distribution to provide a good approximation in practice? Complicated answer in the general IV model Simple rule of thumb in the practically most relevant case of a single endogenous regressor 242

31 Rule of thumb 8.3: (Checking for weak instruments) Consider the first-stage F -statistic testing the hypothesis that the coefficients on the instruments Z 1i,..., Z mi in the first-stage regression (8.6) on Slide 233 are all simultaneously equal to zero: H 0 : π 1 = π 2 =... = π m = 0 vs. H 1 : At least one π j 0 (j = 1,..., m). When there is a single endogenous regressor, a first-stage F - statistic less than 10 indicates that the instruments are weak. In this case the TSLS estimator is biased (even in large samples) and the TSLS t-statistics and confidence intervals are unreliable. 243

32 Meaning of instrument exogeneity : If the instruments are not exogenous, then the TSLS is inconsistent TSLS estimation and inference based on it are unreliable Statistical tests for exogenous instruments: No statistical tests are available when the coefficients are exactly identified (that is when m = k in the IV model (8.5) on Slide 231) If the coefficients are overidentified, that is when m > k in Eq. (8.5), it is possible to test the hypothesis that the extra instruments are exogenous under the maintained assumption that there are enough valid instruments to identify the coefficients of interest 244

33 Theorem 8.4: (The overidentifying restrictions test) Let û TSLS i be the residuals from TSLS estimation of Eq. (8.5) from Slide 231. Use OLS to estimate the regression coefficients in û TSLS i = δ 0 + δ 1 Z 1i δ m Z mi + δ m+1 W 1i δ m+r W ri + e i, (8.8) where e i is the regression error term. Let F denote the homoskedasticity-only F -statistic testing the null hypothesis H 0 : δ 1 =... = δ m = 0. The overidentifying restrictions test statistic is (The J-test.) J = m F. 245

34 Theorem 8.4: (The J-test) [continued] Under the null hypothesis that all instruments are exogenous (suggesting that the instruments should approximately be uncorrelated with ûi TSLS ), and if e i is homoskedastic, in large samples J is distributed χ 2 m k, where m k is the degree of overidentification, that is, the number of instruments minus the number of endogenous regressors. Remark: An application of the J-test is provided in the case study The demand for cigarettes See class for details 246

35 8.4. Where do valid instruments come from? Important question: How can we find instrumental variables for a given application that are both relevant and exogenous? Two main approaches: 1. Use economic theory to suggest instruments 2. Find an exogenous source of variation in X arising from a random phenomenon that induces shifts in the endogenous regressor 247

36 Example of Approach #1: Consider the butter demand example from Section 8.1. Understanding of the economics of agricultural markets leads us to look for an instrument that shifts the supply curve but not the demand curve This leads us to consider weather conditions in agricultural regions Instrument variable: RAINFALL in agricultural regions 248

37 Example of Approach #2: Consider the effect on test scores of class size The regressor CLASS SIZE may be correlated with the error term because of omitted variable bias In some districts, however, earthquake damages may increase the average class size This variation in class size may be unrelated to potentially omitted variables that affect student achievement Instrument variable: that portion of CLASS SIZE that acrrues to earthquake damage 249

38 Case studies: Three examples of how researchers use their expert knowledge of their empirical problem to find adequate instrumental variables: Does putting criminals in jail reduce crime? Does cutting class sizes increase test scores? Does aggressive treatment of heart attacks prolong lives? (see class for a thorough discussion) 250

Econometrics Problem Set 11

Econometrics Problem Set 11 Econometrics Problem Set WISE, Xiamen University Spring 207 Conceptual Questions. (SW 2.) This question refers to the panel data regressions summarized in the following table: Dependent variable: ln(q