Topic 2: Endogeneity and IV TSLS

Dr Pham Thi Bich Ngoc Hoa Sen University ngoc.phamthibich@hoasen.edu.vn  Endogeneity refers to the fact that an independent variable (IV) included in the model is a choice variable (not exogenous) Structure y = Xβ + ε E[ε|X ] = g(X)  Regression y = Xβ + g(X) + [ε - g(X)] = E[y|X] + u, E[u|X ]=0 Projection ε = Xθ + w "Regression of y on X" y = X(β + θ) + w The problem : NOI SINH: U TUONG TAC VOI X X is not exogenous Exogeneity: E[it | x it ]  (current period) Strict Exogeneity: E[it | x i1 , x i2 , , x iT ]  (all periods) (We assume no correlation across individuals.) (W Green, 2012)    Omitted variable bias Measurement error Simultaneity Omitting a variable (X2) creates a bias only if: X2 is an explanator of Y (so, when omitted, it becomes a component of the error term) X2 is correlated with X1 (so that X2 creates a correlation between X1 and the error term) LWAGE  1  2EDUC( X, Ability, Motivation, )  3EXP  4EXP   ε(Ability, Motivation) Increased Ability is associated with increases in EDUC( X, Ability, Motivation, ) and ε(Ability, Motivation) What looks like an effect due to increase in EDUC may be an increase in Ability The estimate of 3 picks up the effect of EDUC and the hidden effect of Ability   Measurement error also induces a correlation between our included explanator and the error term Instead of observing Xi , we observe X* y   x * + ν, E[x*]=0, E[v|x*]=0 x = x* + u, E[u|x*]=0, E[v|u]=0 y = x  (v  u) = x   E[y | x]  x  E[(v  u) | x] =  x -  E[u | x]  Cov[u, x]  = x -   x   Var[x]   Var[u]   = x -   x   Var[x*]  Var[u]  Var[x*]   =  x  x, || < ||   Var[x*]  Var[u]    An independent variable (IV) included in the model is a choice variable, potentially affected by the dependent variable (DV) Examples: ◦ IV = Exports; DV = GDP ◦ IV = education; DV = income  Given: both X and Y are jointly determined Because X and Y are determined simultaneously, X can adjust in response to shocks to Y () Thus X will be correlated with       The classic example of simultaneous causality in economics is supply and demand Both prices and quantities adjust until supply and demand are in equilibrium A shock to demand or supply causes BOTH prices and quantities to move Thus, any attempt to estimate the relationship between prices and quantities (say, to estimate a demand elasticity) suffers from SIMULTANEITY BIAS Econometricians have a frequent interest in estimating elasticities resulting from such an equilibrium process Simultaneity bias is a MAJOR problem Endogeneity bias      Suppose that Y2it is an endogenous explanatory variable: ◦ Y1it = a0 + a1 Y2it + a2 Xit + uit (1) ◦ Y2it = b0 + b1 Xit + b2 Zit + vit (2) Equations (1) and (2) have a “triangular” structure Given this triangular structure, the OLS estimate of a1 in equation (1) is unbiased only if vit is uncorrelated with uit If vit is correlated with uit, then Y2it is correlated with uit which means that the OLS estimate of a1 would be biased To avoid this bias, we must estimate equation (1) “instrumental variables” (IV) regression rather than OLS 10   The most-up-to-date implementation of ivreg2 requires Stata version 11 or later ivreg2 may be used with time-series or panel data varlist1 are the exogenous regressors or "included instruments" varlist_iv are the exogenous variables excluded from the regression or "excluded instruments" varlist2 the endogenous regressors that are being "instrumented" 18 Used for panel data:  ssc install xtivreg2 xtivreg2 depvar [varlist1] (varlist2=varlist_iv) [weight] [if exp] [in range] , {fe | fd} [options] 19     IV estimation can be extended to the multiple regression case Call the model we are interested in estimating the structural model Our problem is that one or more of the variables are endogenous We need an instrument for each endogenous variable 20    If there is just one instrument for our endogenous variable, we can’t test whether the instrument is uncorrelated with the error We say the model is just identified If we have multiple instruments, it is possible to test the overidentifying restrictions – to see if some of the instruments are correlated with the error 21     In the instrumental variable regression, if we have multiple endogenous regressors x1, …, xk and multiple instruments z1, …, zm, the coefficients on the endogenous regressors are said to be: Exactly identified if m = k Overidentified if m > k Underidentified if m < k  can not identify the coefficients 22  The Sargan-Hansen test is a test of overidentifying restrictions  The joint null hypothesis (H0) is that the instruments are valid instruments, i.e., uncorrelated with the error term, and that the excluded instruments are correctly excluded from the estimated equation  Under the null, the test statistic is distributed as chisquared in the number of (L-K) overidentifying restrictions  A rejection casts doubt on the validity of the instruments If p-value >5%  instruments are valid 23  For the 2SLS estimator, the test statistic is Sargan's statistic, typically calculated as N*R-squared from a regression of the IV residuals on the full set of instruments  Under the assumption of conditional homoskedasticity, Hansen's J statistic becomes Sargan's statistic The J statistic is consistent in the presence of heteroskedasticity and (for HAC-consistent estimation) autocorrelation;  Sargan's statistic is consistent if the disturbance is homoskedastic and (for AC-consistent estimation) if it is also autocorrelated With robust, bw and/or cluster, Hansen's J statistic is reported 24  Endogeneity tests of one or more endogenous regressors can implemented using the endog option  Under the null hypothesis (H0) that the specified endogenous regressors can actually be treated as exogenous, the test statistic is distributed as chi-squared with degrees of freedom equal to the number of regressors tested  Unlike the Durbin-Wu-Hausman tests reported by ivendog, the endog option of ivreg2 can report test statistics that are robust to various violations of conditional homoskedasticity  If p-value [...]... (2) - Stage 2: Regress eq (4) Y1it = a0 + a1 (b0^ + b1^ Xit + b2^ Zit ) + a2 Xit + uit (4)  The models have a triangular structure 16  Using the ivregress command ◦ The models can be estimated using two-stage least squares (2SLS), limited-information maximum likelihood (LIML) or generalized method of moments (GMM)  help ivregress: ivregress 2sls depvar [varlist1] (varlist2 = varlist _iv) [if] [in]... varlist _iv) [if] [in] [weight] [, options]  ssc install ivreg2 ivreg2 depvar [varlist1] (varlist2=varlist _iv) [weight] [if exp] [in range] [, options] 17   The most-up-to-date implementation of ivreg2 requires Stata version 11 or later ivreg2 may be used with time-series or panel data varlist1 are the exogenous regressors or "included instruments" varlist _iv are the exogenous variables excluded from the... regression or "excluded instruments" varlist2 the endogenous regressors that are being "instrumented" 18 Used for panel data:  ssc install xtivreg2 xtivreg2 depvar [varlist1] (varlist2=varlist _iv) [weight] [if exp] [in range] , {fe | fd} [options] 19     IV estimation can be extended to the multiple regression case Call the model we are interested in estimating the structural model Our problem... + a2 Xit + uit (3) ◦ All the explanatory variables (Xit and Zit) are exogenous  The basic idea underlying IV regression is to remove vit from the Y1it model so that our estimate of a1 is unbiased 13 Instrumental Variable:  Note that vit is removed from the Y1it model if we use the predicted rather than the actual values of Y2it on the right hand side ◦ Y1it = a0 + a1 (b0^ + b1^ Xit + b2^ Zit ) + a2...    Instrumental Variables (IV) estimation is used when your model has endogenous x’s That is, whenever Cov(x,u) ≠ 0 Thus, IV can be used to address the problem of omitted variable bias Additionally, IV can be used to solve the classic errors-in-variables problem 11 A Triangle Relationship:  Suppose that Y2it is... N*R-squared from a regression of the IV residuals on the full set of instruments  Under the assumption of conditional homoskedasticity, Hansen's J statistic becomes Sargan's statistic The J statistic is consistent in the presence of heteroskedasticity and (for HAC-consistent estimation) autocorrelation;  Sargan's statistic is consistent if the disturbance is homoskedastic and (for AC-consistent estimation)... estimation) autocorrelation;  Sargan's statistic is consistent if the disturbance is homoskedastic and (for AC-consistent estimation) if it is also autocorrelated With robust, bw and/ or cluster, Hansen's J statistic is reported 24  Endogeneity tests of one or more endogenous regressors can implemented using the endog option  Under the null hypothesis (H0) that the specified endogenous regressors can actually... be treated as exogenous, the test statistic is distributed as chi-squared with degrees of freedom equal to the number of regressors tested  Unlike the Durbin-Wu-Hausman tests reported by ivendog, the endog option of ivreg2 can report test statistics that are robust to various violations of conditional homoskedasticity  If p-value k Underidentified if m < k  can not identify the coefficients ... causality in economics is supply and demand Both prices and quantities adjust until supply and demand are in equilibrium A shock to demand or supply causes BOTH prices and quantities to move Thus,... moments (GMM)  help ivregress: ivregress 2sls depvar [varlist1] (varlist2 = varlist _iv) [if] [in] [weight] [, options]  ssc install ivreg2 ivreg2 depvar [varlist1] (varlist2=varlist _iv) [weight] [if... variable (IV) included in the model is a choice variable, potentially affected by the dependent variable (DV) Examples: ◦ IV = Exports; DV = GDP ◦ IV = education; DV = income  Given: both X and Y