Econometrics - Lecture 5 Autocorrelation, IV Estimator Contents nAutocorrelation nTests against Autocorrelation nInference under Autocorrelation nOLS Estimator Revisited nCases of Endogenous Regressors nInstrumental Variables (IV) Estimator: The Concept nIV Estimator: The Method nCalculation of the IV Estimator nAn Example nSome Tests nThe GIV Estimator Nov 3, 2017 Hackl, Econometrics, Lecture 4 2 Example: Demand for Ice Cream nVerbeek’s time series dataset “icecream” n30 four weekly observations (1951-1953) nVariables qcons: consumption of ice cream per head (in pints) qincome: average family income per week (in USD, red line) qprice: price of ice cream (in USD per pint, blue line) qtemp: average temperature (in Fahrenheit); tempc: (green, in °C) Nov 3, 2017 Hackl, Econometrics, Lecture 4 3 Demand for Ice Cream, cont’d nTime series plot of consumption of ice cream per head (in pints), cons, over observation periods Nov 3, 2017 Hackl, Econometrics, Lecture 4 4 Demand for Ice Cream, cont’d nConsumption (cons) of ice cream per head (in pints): scatter diagramme of actual values cons over lagged values cons-1 Nov 3, 2017 Hackl, Econometrics, Lecture 4 5 Autocorrelation nTypical for time series data such as consumption, production, investments, etc. nAutocorrelation of error terms is typically observed if qa relevant regressor with trend or seasonal pattern is not included in the model: miss-specified model qthe functional form of a regressor is incorrectly specified qthe dependent variable is correlated in a way that is not appropriately represented in the systematic part of the model nAutocorrelation of the error terms indicates deficiencies of the model specification such as omitted regressors, incorrect functional form, incorrect dynamic nTests for autocorrelation are the most frequently used tool for diagnostic checking the model specification Nov 3, 2017 Hackl, Econometrics, Lecture 4 6 Demand for Ice Cream, cont’d nTime series plot of n Cons: consumption of ice cream per head (in pints); mean: 0.36 qTemp/100: average temperature (in Fahrenheit) qPrice (in USD per pint); mean: 0.275 USD Nov 3, 2017 Hackl, Econometrics, Lecture 4 7 Demand for Ice Cream, cont’d nDemand for ice cream, measured by cons, explained by price, income, and temp n n n n n n n n n n Nov 3, 2017 Hackl, Econometrics, Lecture 4 8 Demand for Ice Cream, cont’d nTime series diagramme of demand for ice cream, actual values (o) and predictions (polygon), based on the model with income and price n Nov 3, 2017 Hackl, Econometrics, Lecture 4 9 Demand for Ice Cream, cont’d nIce cream model: Scatter-plot of residuals et vs et-1 (r = 0.401) n Nov 3, 2017 Hackl, Econometrics, Lecture 4 10 A Model with AR(1) Errors nLinear regression n yt = xt‘b + et 1) n with n et = ret-1 + vt with -1 < r < 1 or |r| < 1 n where vt are uncorrelated random variables with mean zero and constant variance sv2 nFor ρ ¹ 0, the error terms et are correlated; the Gauss-Markov assumption V{e} = se2IN is violated nThe other Gauss-Markov assumptions are assumed to be fulfilled nThe sequence et, t = 0, 1, 2, … which follows et = ret-1 + vt is called an autoregressive process of order 1 or AR(1) process n_____________________ n1) In the context of time series models, variables are indexed by „t“ n n Nov 3, 2017 Hackl, Econometrics, Lecture 4 11 Properties of AR(1) Processes nRepeated substitution of et-1, et-2, etc. results in n et = ret-1 + vt = vt + rvt-1 + r2vt-2 + … nwith vt being uncorrelated and having mean zero and variance sv2: nE{et} = 0 nV{et} = se2 = sv2(1-r2)-1 nThis results from V{et} = sv2 + r2sv2 + r4sv2 + … = sv2(1-r2)-1 for |r|<1; the geometric series 1 + r2 + r4 + … has the sum (1- r2)-1 given that |r| < 1 qfor |r| > 1, V{et} is undefined nCov{et, et-s } = rs sv2 (1-r2)-1 for s > 0 n all error terms are correlated; covariances – and correlations Corr{et, et-s } = rs (1-r2)-1 – decrease with growing distance s in time n n Nov 3, 2017 Hackl, Econometrics, Lecture 4 12 AR(1) Process, cont’d nThe covariance matrix V{e}: n n n n n n n nV{e} has a band structure nDepends only of two parameters: r and sv2 n Nov 3, 2017 Hackl, Econometrics, Lecture 4 13 Consequences of V{e} ¹ s2IT nOLS estimators b for b nare unbiased nare consistent nhave the covariance-matrix n V{b} = s2 (X'X)-1 X'YX (X'X)-1 nare not efficient estimators, not BLUE nfollow – under general conditions – asymptotically the normal distribution nThe estimator s2 = e'e/(T-K) for s2 is biased nFor an AR(1)-process et with r > 0, s.e. from s2 (X'X)-1 underestimates the true s.e. n n n n nand Nov 3, 2017 Hackl, Econometrics, Lecture 4 14 Inference in Case of Autocorrelation nCovariance matrix of b: n V{b} = s2 (X'X)-1 X'YX (X'X)-1 nUse of s2 (X'X)-1 (the standard output of econometric software) instead of V{b} for inference on b may be misleading nIdentification of autocorrelation: nStatistical tests, e.g., Durbin-Watson test nRemedies nUse of correct variances and standard errors nTransformation of the model so that the error terms are uncorrelated Nov 3, 2017 Hackl, Econometrics, Lecture 4 15 Estimation of r nAutocorrelation coefficient r: parameter of the AR(1) process n et = ret-1 + vt nEstimation of ρ nby regressing the OLS residual et on the lagged residual et-1 n n n nestimator is qbiased qbut consistent under weak conditions Nov 3, 2017 Hackl, Econometrics, Lecture 4 16 Autocorrelation Function nAutocorrelation of order s: n n n nAutocorrelation function (ACF) assigns rs to s nCorrelogram: graphical representation of the autocorrelation function nGRETL: Variable => Correlogram n Produces (a) the autocorrelation function (ACF) and (b) the graphical representation of the ACF (and the partial autocorrelation function) n n Nov 3, 2017 Hackl, Econometrics, Lecture 4 17 Example: Ice Cream Demand nAutocorrelation function (ACF) of cons n Nov 3, 2017 Hackl, Econometrics, Lecture 4 18 LAG ACF PACF Q-stat. [p-value] 1 0,6627 *** 0,6627 *** 14,5389 [0,000] 2 0,4283 ** -0,0195 20,8275 [0,000] 3 0,0982 -0,3179 * 21,1706 [0,000] 4 -0,1470 -0,1701 21,9685 [0,000] 5 -0,3968 ** -0,2630 28,0152 [0,000] 6 -0,4623 ** -0,0398 36,5628 [0,000] 7 -0,5145 *** -0,1735 47,6132 [0,000] 8 -0,4068 ** -0,0299 54,8362 [0,000] 9 -0,2271 0,0711 57,1929 [0,000] 10 -0,0156 0,0117 57,2047 [0,000] 11 0,2237 0,1666 59,7335 [0,000] 12 0,3912 ** 0,0645 67,8959 [0,000] Example: Ice Cream Demand nCorrelogram of cons n Nov 3, 2017 Hackl, Econometrics, Lecture 4 19 Contents nAutocorrelation nTests against Autocorrelation nInference under Autocorrelation nOLS Estimator Revisited nCases of Endogenous Regressors nInstrumental Variables (IV) Estimator: The Concept nIV Estimator: The Method nCalculation of the IV Estimator nAn Example nSome Tests nThe GIV Estimator Nov 3, 2017 Hackl, Econometrics, Lecture 4 20 Tests for Autocorrelation of Error Terms nDue to unbiasedness of b, residuals are expected to indicate autocorrelation nGraphical displays, e.g., the correlogram of residuals may give useful hints nResidual-based tests: nDurbin-Watson test nBox-Pierce test nBreusch-Godfrey test n n n n Nov 3, 2017 Hackl, Econometrics, Lecture 4 21 Durbin-Watson Test nTest of H0: r = 0 against H1: r ¹ 0 nTest statistic n n n nFor r > 0, dw is expected to have a value in (0,2) nFor r < 0, dw is expected to have a value in (2,4) ndw close to the value 2 indicates no autocorrelation of error terms nCritical limits of dw qdepend upon xt’s qexact critical value is unknown, but upper and lower bounds can be derived, which depend upon xt’s only via the number of regression coefficients nTest can be inconclusive nH1: r > 0 may be more appropriate than H1: r ¹ 0 n Nov 3, 2017 Hackl, Econometrics, Lecture 4 22 Durbin-Watson Test: Bounds for Critical Limits nDerived by Durbin and Watson nUpper (dU) and lower (dL) bounds for the critical limits and a = 0.05 n n n n n n ndw < dL: reject H0 ndw > dU: do not reject H0 ndL < dw < dU: no decision (inconclusive region) n n Nov 3, 2017 Hackl, Econometrics, Lecture 4 23 T K=2 K=3 K=10 dL dU dL dU dL dU 15 1.08 1.36 0.95 1.54 0.17 3.22 20 1.20 1.41 1.10 1.54 0.42 2.70 100 1.65 1.69 1.63 1.71 1.48 1.87 Durbin-Watson Test: Remarks nDurbin-Watson test gives no indication of causes for the rejection of the null hypothesis and how the model to modify nVarious types of misspecification may cause the rejection of the null hypothesis nDurbin-Watson test is a test against first-order autocorrelation; a test against autocorrelation of other orders may be more suitable, e.g., order four if the model is for quarterly data nUse of tables unwieldy qLimited number of critical bounds (K, T, a) in tables qInconclusive region nGRETL: Standard output of the OLS estimation reports the Durbin-Watson statistic; to see the p-value: qOLS output => Tests => Durbin-Watson p-value Nov 3, 2017 Hackl, Econometrics, Lecture 4 24 Asymptotic Tests nAR(1) process for error terms n et = ret-1 + vt nAuxiliary regression of et on (an intercept,) xt and et-1: produces nRe2 nTest of H0: r = 0 against H1: r > 0 or H1: r ¹ 0 1.Breusch-Godfrey test (GRETL: OLS output => Tests => Autocorr.) qRe2 of the auxiliary regression: close to zero if r = 0 qUnder H0: r = 0, (T-1) Re2 follows approximately the Chi-squared distribution with 1 d.f. qLagrange multiplier F (LMF) statistic: F-test for explanatory power of et-1; follows approximately the F(1, T-K-1) distribution if r = 0 qGeneral case of the Breusch-Godfrey test: Auxiliary regression based on higher order autoregressive process q n n n n Nov 3, 2017 Hackl, Econometrics, Lecture 4 25 Asymptotic Tests, cont’d 2.Similar the Ljung-Box test, based on q QLB = T (T+2) Σsm rs2/(T-s) q with correlations rs between et and et-s; QLB follows the Chi-squared distribution with m d.f. if r = 0 3.Box-Pierce test qThe t-statistic based on the OLS estimate r of r from et = ret-1 + vt, q t = √(T) r q follows approximately the t-distribution, t2 = T r2 the Chi-squared distribution with 1 d.f. if r = 0 qTest based on √(T) r is a special case of the Box-Pierce test which uses the test statistic Qm = T Σsm rs2 n Nov 3, 2017 Hackl, Econometrics, Lecture 4 26 Asymptotic Tests, cont’d nGRETL: qOLS output => Tests => Autocorrelation (shows the Breusch-Godfrey LMF statistic, the Box-Pierce statistic, and the Ljung-Box statistic as well as p-values) qOLS output => Graphs => Residual correlogram (shows – besides the correlogram of the residuals – Ljung-Box statistic and p-value) nRemarks nIf the model of interest contains lagged values of y the auxiliary regression should also include all explanatory variables (just to make sure the distribution of the test is correct) nIf heteroskedasticity is suspected, White standard errors may be used in the auxiliary regression n Nov 3, 2017 Hackl, Econometrics, Lecture 4 27 Demand for Ice Cream, cont’d Hackl, Econometrics, Lecture 4 28 OLS estimated demand function: Output from GRETL Dependent variable : CONS coefficient std. error t-ratio p-value ------------------------------------------------------------- const 0.197315 0.270216 0.7302 0.4718 INCOME 0.00330776 0.00117142 2.824 0.0090 *** PRICE -1.04441 0.834357 -1.252 0.2218 TEMP 0.00345843 0.000445547 7.762 3.10e-08 *** Mean dependent var 0.359433 S.D. dependent var 0,065791 Sum squared resid 0,035273 S.E. of regression 0,036833 R- squared 0,718994 Adjusted R-squared 0,686570 F(2, 129) 22,17489 P-value (F) 2,45e-07 Log-likelihood 58,61944 Akaike criterion -109,2389 Schwarz criterion -103,6341 Hannan-Quinn -107,4459 rho 0,400633 Durbin-Watson 1,021170 Nov 3, 2017 Demand for Ice Cream, cont’d nTest for autocorrelation of error terms nH0: r = 0, H1: r ¹ 0 ndw = 1.02 < 1.21 = dL for T = 30, K = 4; p = 0.0003 (in GRETL: 0.0003025); reject H0 nGRETL also shows the autocorrelation coefficient: r = 0.401 nPlot of actual (o) and fitted (polygon) values Nov 3, 2017 Hackl, Econometrics, Lecture 4 29 Demand for Ice Cream, cont’d nAuxiliary regression et = xt‘b + ret-1 + vt: OLS estimation gives n r = 0.401, R2 = 0.141 nTest of H0: r = 0 against H1: r > 0 1.Breusch-Godfrey test: LMF = 4.11, p-value: 0.053 2.Box-Pierce test: t2 = 4.237, p-value: 0.040 3.Ljung-Box test: QLB = 3.6, p-value: 0.058 nAll three tests reject the null hypothesis Nov 3, 2017 Hackl, Econometrics, Lecture 4 30 Contents nAutocorrelation nTests against Autocorrelation nInference under Autocorrelation nOLS Estimator Revisited nCases of Endogenous Regressors nInstrumental Variables (IV) Estimator: The Concept nIV Estimator: The Method nCalculation of the IV Estimator nAn Example nSome Tests nThe GIV Estimator Nov 3, 2017 Hackl, Econometrics, Lecture 4 31 Inference under Autocorrelation nCovariance matrix of b: n V{b} = s2 (X'X)-1 X'YX (X'X)-1 nUse of s2 (X'X)-1 (the standard output of econometric software) instead of V{b} for inference on b may be misleading nRemedies nUse of correct variances and standard errors nTransformation of the model so that the error terms are uncorrelated Nov 3, 2017 Hackl, Econometrics, Lecture 4 32 HAC-estimator for V{b} nSubstitution of Y in n V{b} = s2 (X'X)-1 X'YX (X'X)-1 n by a suitable estimator nNewey-West: substitution of Sx = s2(X'YX)/T = (StSsstsxtxs‘)/T by n n n n with wj = j/(p+1); p, the truncation lag, is to be chosen suitably nThe estimator n T (X'X)-1 Ŝx (X'X)-1 n for V{b} is called heteroskedasticity and autocorrelation consistent (HAC) estimator, the corresponding standard errors are the HAC s.e. n Nov 3, 2017 Hackl, Econometrics, Lecture 4 33 Demand for Ice Cream, cont’d nDemand for ice cream, measured by cons, explained by price, income, and temp, OLS and HAC standard errors Nov 3, 2017 Hackl, Econometrics, Lecture 4 34 coeff s.e. OLS HAC constant 0.197 0.270 0.288 price -1.044 0.834 0.876 income*10-3 3.308 1.171 1.184 temp*10-3 3.458 0.446 0.411 Cochrane-Orcutt Estimator nGLS estimator nWith transformed variables yt* = yt – ryt-1 and xt* = xt – rxt-1, also called “quasi-differences”, the model yt = xt‘b + et with et = ret-1 + vt can be written as n yt – ryt-1 = yt* = (xt – rxt-1)‘b + vt = xt*‘b + vt (A) nThe model in quasi-differences has error terms which fulfill the Gauss-Markov assumptions nGiven observations for t = 1, …, T, model (A) is defined for t = 2, …, T nEstimation of r using, e.g., the auxiliary regression et = ret-1 + vt gives the estimate r; substitution of r in (A) for r results in FGLS estimators for b nThe FGLS estimator is called Cochrane-Orcutt estimator n Nov 3, 2017 Hackl, Econometrics, Lecture 4 35 Cochrane-Orcutt Estimation nIn following steps 1.OLS estimation of b for b from yt = xt‘b + et, t = 1, …, T 2.Estimation of r for r from the auxiliary regression et = ret-1 + vt 3.Calculation of quasi-differences yt* = yt – ryt-1 and xt* = xt – rxt-1 4.OLS estimation of b from n yt* = xt*‘b + vt, t = 2, …, T n resulting in the Cochrane-Orcutt estimators nSteps 2. to 4. can be repeated in order to improve the estimate r : iterated Cochrane-Orcutt estimator qGRETL provides the iterated Cochrane-Orcutt estimator: q Model => Time series => Autoregressive estimation n Nov 3, 2017 Hackl, Econometrics, Lecture 4 36 Demand for Ice Cream, cont’d nIterated Cochrane-Orcutt estimator n n n n n n n n n n nDurbin-Watson test: dw = 1.55; dL=1.21 < dw < 1.65 = dU Nov 3, 2017 Hackl, Econometrics, Lecture 4 37 Demand for Ice Cream, cont’d nDemand for ice cream, measured by cons, explained by price, income, and temp, OLS and HAC standard errors (se), and Cochrane-Orcutt estimates Nov 3, 2017 Hackl, Econometrics, Lecture 4 38 OLS-estimation Cochrane-Orcutt coeff se HAC coeff se constant 0.197 0.270 0.288 0.157 0.300 price -1.044 0.834 0.881 -0.892 0.830 income 3.308 1.171 1.151 3.203 1.546 temp 3.458 0.446 0.449 3.558 0.555 Demand for Ice Cream, cont’d nModel extended by temp-1 n n n n n n n n n n nDurbin-Watson test: dw = 1.58; dL=1.21 < dw < 1.65 = dU Nov 3, 2017 Hackl, Econometrics, Lecture 4 39 Demand for Ice Cream, cont’d nDemand for ice cream, measured by cons, explained by price, income, and temp, OLS and HAC standard errors, Cochrane-Orcutt estimates, and OLS estimates for the extended model n n n n n n n n n nAdding temp-1 improves the adj R2 from 0.687 to 0.800 Nov 3, 2017 Hackl, Econometrics, Lecture 4 40 OLS Cochrane-Orcutt OLS coeff HAC coeff se coeff se constant 0.197 0.288 0.157 0.300 0.189 0.232 price -1.044 0.881 -0.892 0.830 -0.838 0.688 income 3.308 1.151 3.203 1.546 2.867 1.053 temp 3.458 0.449 3.558 0.555 5.332 0.670 temp-1 -2.204 0.731 General Autocorrelation Structures nGeneralization of model n yt = xt‘b + et n with et = ret-1 + vt nAlternative dependence structures of error terms nAutocorrelation of higher order than 1 nMoving average pattern n Nov 3, 2017 Hackl, Econometrics, Lecture 4 41 Higher Order Autocorrelation nFor quarterly data, error terms may develop according to n et = get-4 + vt n or - more generally - to n et = g1et-1 + … + g4et-4 + vt net follows an AR(4) process, an autoregressive process of order 4 nMore complex structures of correlations between variables with autocorrelation of order 4 are possible than with that of order 1 Nov 3, 2017 Hackl, Econometrics, Lecture 4 42 Moving Average Processes nMoving average process of order 1, MA(1) process n et = vt + avt-1 nεt is correlated with εt-1, but not with εt-2, εt-3, … nGeneralizations to higher orders Nov 3, 2017 Hackl, Econometrics, Lecture 4 43 Remedies against Autocorrelation nChange functional form, e.g., use log(y) instead of y nExtend the model by including additional explanatory variables, e.g., seasonal dummies, or additional lags nUse HAC standard errors for the OLS estimators nReformulate the model in quasi-differences (FGLS) or in differences Nov 3, 2017 Hackl, Econometrics, Lecture 4 44 Contents nAutocorrelation nTests against Autocorrelation nInference under Autocorrelation nOLS Estimator Revisited nCases of Endogenous Regressors nInstrumental Variables (IV) Estimator: The Concept nIV Estimator: The Method nCalculation of the IV Estimator nAn Example nSome Tests nThe GIV Estimator Nov 3, 2017 Hackl, Econometrics, Lecture 4 45 OLS Estimator Nov 3, 2017 Hackl, Econometrics, Lecture 4 46 Linear model for yt yi = xi'β + εi, i = 1, …, N (or y = Xβ + ε) given observations xik, k =1, …, K, of the regressor variables, error term εi OLS estimator b = (Σixi xi’)-1Σixi yi = (X’X)-1X’y From b = (Σixi xi’)-1Σixi yi = (Σixi xi’)-1Σixi xi‘ β + (Σixi xi’)-1Σixi εi = β + (Σixi xi’)-1Σixi εi = β + (X’X)-1 X’ε follows E{b} = (Σixi xi’)-1Σixiyi = (Σixi xi’)-1Σixi xi‘ β + (Σixi xi’)-1Σixi εi = β + (Σixi xi’)-1 E{Σixi εi} = β + (X’X)-1 E{X’ε} OLS Estimator: Properties Nov 3, 2017 Hackl, Econometrics, Lecture 4 47 1.OLS estimator b is unbiased if n(A1) E{ε} = 0 nE{Σixi εi } = E{X’ε} = 0; is fulfilled if (A7) or a stronger assumption is true q(A2) {xi, i =1, …,N} and {εi, i =1, …,N} are independent; is the strongest assumption q(A10) E{ε|X} = 0, i.e., X uninformative about E{εi} for all i (ε is conditional mean independent of X); is implied by (A2) q(A8) xi and εi are independent for all i (no contemporaneous dependence); is less strong than (A2) and (A10) q(A7) E{xi εi} = 0 for all i (no contemporaneous correlation); is even less strong than (A8) OLS Estimator: Properties, cont’d Nov 3, 2017 Hackl, Econometrics, Lecture 4 48 2.OLS estimator b is consistent for β if n(A8) xi and εi are independent for all i n(A6) (1/N)Σi xi xi’ has as limit (N→∞) a non-singular matrix Σxx (A8) can be substituted by (A7) [E{xi εi} = 0 for all i, no contemporaneous correlation] 3.OLS estimator b is asymptotically normally distributed if (A6), (A8) and n(A11) εi ~ IID(0,σ²) are true; nfor large N, b follows approximately the normal distribution b ~a N{β, σ2(Σi xi xi’ )-1} nUse White and Newey-West estimators for V{b} in case of heteroskedasticity and autocorrelation of error terms, respectively n Hackl, Econometrics, Lecture 4 49 Assumption (A7): E{xi εi} = 0 for all i nImplication of (A7): for all i, each of the regressors is uncorrelated with the current error term, no contemporaneous correlation n(A7) guaranties unbiasedness and consistency of the OLS estimator nStronger assumptions – (A2), (A10), (A8) – have same consequences nIn reality, (A7) is not always true: alternative estimation procedures are required for ascertaining consistency and unbiasedness nExamples of situations with E{xi εi} ≠ 0 (see the following slides): nRegressors with measurement errors nRegression on the lagged dependent variable with autocorrelated error terms (dynamic regression) nUnobserved heterogeneity nEndogeneity of regressors, simultaneity n n n Nov 3, 2017 Contents nAutocorrelation nTests against Autocorrelation nInference under Autocorrelation nOLS Estimator Revisited nCases of Endogenous Regressors nInstrumental Variables (IV) Estimator: The Concept nIV Estimator: The Method nCalculation of the IV Estimator nAn Example nSome Tests nThe GIV Estimator Nov 3, 2017 Hackl, Econometrics, Lecture 4 50 Hackl, Econometrics, Lecture 4 51 Regressor with Measurement Error n yi = β1 + β2wi + vi nwith white noise vi, V{vi} = σv², and E{vi|wi} = 0; conditional expectation of yi given wi : E{yi|wi} = β1 + β2wi nExample: yi: household savings , wi: household income nMeasurement process: reported household income xi may deviate from household income wi n xi = wi + ui n where ui is (i) white noise with V{ui} = σu², (ii) independent of vi, and (iii) independent of wi nThe model to be analyzed is n yi = β1 + β2xi + εi with εi = vi - β2ui nE{xi εi} = - β2 σu² ≠ 0: requirement for consistency and unbiasedness of OLS estimates is violated nxi and εi are negatively (positively) correlated if β2 > 0 (β2 < 0) n n Nov 3, 2017 Hackl, Econometrics, Lecture 4 52 Consequences of Measurement Errors nInconsistency of b2 = sxy/sx2 n plim b2 = β2 + (plim sxε)/(plim sx2) = β2 + E{xi εi} / V{xi} n n n n β2 is underestimated nInconsistency of b1 = n plim (b1 - β1) = - plim (b2 - β2) E{xi} n given E{xi} > 0 for the reported income: β1 is overestimated; inconsistency of b2 “carries over” nThe model does not correspond to the conditional expectation of yi given xi: n E{yi|xi} = β1 + β2xi - β2 E{ui|xi} ≠ β1 + β2xi n as E{ui|xi} ≠ 0 n Nov 3, 2017 Hackl, Econometrics, Lecture 4 53 Dynamic Regression nAllows modelling dynamic effects of changes of x on y: n yt = β1 + β2xt + β3yt-1 + εt n with εt following the AR(1) model n εt = ρεt-1 + vt n vt white noise with σv² nFrom yt = β1 + β2xt + β3yt-1 + ρεt-1 + vt follows n E{yt-1εt} = β3 E{yt-2εt} + ρ²σv²(1 - ρ²)-1 n i.e., yt-1 is correlated with εt n Remember: E{et, et-s } = rs sv2 (1-r2)-1 for s > 0 nOLS estimators not consistent if ρ ≠ 0 nThe model does not correspond to the conditional expectation of yt given the regressors xt and yt-1: n E{yt|xt, yt-1} = β1 + β2xt + β3yt-1 + E{εt |xt, yt-1} Nov 3, 2017 Hackl, Econometrics, Lecture 4 54 Omission of Relevant Regressors nTwo models: n yi = xi‘β + zi’γ + εi (A) n yi = xi‘β + vi (B) nTrue model (A), fitted model (B) nOLS estimates bB of β from (B) n n nOmitted variable bias: E{(Σi xi xi’)-1 Σi xi zi’}γ = E{(X’X)-1 X’Z}γ nNo bias if (a) γ = 0, i.e., model (A) is correct, or if (b) variables in xi and zi are uncorrelated (orthogonal) nOLS estimators are biased, if relevant regressors are omitted that are correlated with regressors in xi Nov 3, 2017 Hackl, Econometrics, Lecture 4 55 Unobserved Heterogeneity nExample: Wage equation with yi: log wage, x1i: personal characteristics, x2i: years of schooling, ui: abilities (unobservable) n yi = x1i‘β1 + x2iβ2 + uiγ + vi nModel for analysis (unobserved ui covered in error term) n yi = xi‘β + εi n with xi = (x1i‘, x2i)’, β = (β1‘, β2)’, εi = uiγ + vi nGiven E{xi vi} = 0 n plim b = β + Σxx-1 E{xi ui} γ nOLS estimators b are not consistent if xi and ui are correlated (γ ≠ 0), e.g., if higher abilities induce more years at school: estimator for β2 might be overestimated, hence effects of years at school etc. are overestimated: “ability bias” nUnobserved heterogeneity: observational units differ in other aspects than ones that are observable n Nov 3, 2017 Hackl, Econometrics, Lecture 4 56 Endogenous Regressors nRegressors in X which are correlated with error term, E{X‘ε} ≠ 0, are called endogenous nOLS estimators b = β + (X‘X)-1X‘ε qE{b} ≠ β, b is biased; bias E{(X‘X)-1X‘ε} difficult to assess qplim b = β + Σxx-1q with q = plim(N-1X‘ε) nFor q = 0 (regressors and error term asymptotically uncorrelated), OLS estimators b are consistent also in case of endogenous regressors nFor q ≠ 0 (error term and at least one regressor asymptotically correlated): plim b ≠ β, the OLS estimators b are not consistent nEndogeneity bias nRelevant for many economic applications nExogenous regressors: with error term uncorrelated, all regressors that are not endogenous Nov 3, 2017 Hackl, Econometrics, Lecture 4 57 Consumption Function nAWM data base, 1970:1-2003:4 nC: private consumption (PCR), growth rate p.y. nY: disposable income of households (PYR), growth rate p.y. n Ct = β1 + β2Yt + εt (A) n β2: marginal propensity to consume, 0 < β2 < 1 nOLS estimates: n Ĉt = 0.011 + 0.718 Yt n with t = 15.55, R2 = 0.65, DW = 0.50 nIt: per capita investment (exogenous, E{It εt} = 0) n Yt = Ct + It (B) nBoth Yt and Ct are endogenous: E{Ct εi} = E{Yt εi} = σε²(1 – β2)-1 nThe regressor Yt has an impact on Ct; at the same time Ct has an impact on Yt Nov 3, 2017 Hackl, Econometrics, Lecture 4 58 Simultaneous Equation Models nIllustrated by the preceding consumption function: n Ct = β1 + β2Yt + εt (A) n Yt = Ct + It (B) nVariables Yt and Ct are simultaneously determined by equations (A) and (B) nEquations (A) and (B) are the structural equations or the structural form of the simultaneous equation model that describes both Yt and Ct nThe coefficients β1 and β2 are behavioural parameters nReduced form of the model: one equation for each of the endogenous variables Ct and Yt, with only the exogenous variable It as regressor nThe OLS estimators are biased and not consistent n n n Nov 3, 2017 Hackl, Econometrics, Lecture 4 59 Consumption Function, cont’d nReduced form of the model: n n n n n nOLS estimator b2 from (A) is inconsistent; E{Yt εt} ≠ 0 n plim b2 = β2 + Cov{Yt εt} / V{Yt} = β2 + (1 – β2) σε²(V{It} + σε²)-1 n for 0 < β2 < 1, b2 overestimates β2 nThe OLS estimator b1 is also inconsistent n Nov 3, 2017 Contents nAutocorrelation nTests against Autocorrelation nInference under Autocorrelation nOLS Estimator Revisited nCases of Endogenous Regressors nInstrumental Variables (IV) Estimator: The Concept nIV Estimator: The Method nCalculation of the IV Estimator nAn Example nSome Tests nThe GIV Estimator Nov 3, 2017 Hackl, Econometrics, Lecture 4 60 Hackl, Econometrics, Lecture 4 61 An Alternative Estimator nModel n yi = β1 + β2 xi + εi n with E{ εi xi } ≠ 0, i.e., endogenous regressor xi : OLS estimators are biased and inconsistent nInstrumental variable zi satisfying 1.Exogeneity: E{εi zi} = 0: is uncorrelated with error term 2.Relevance: Cov{xi , zi} ≠ 0: is correlated with endogenous regressor nTransformation of model equation n Cov{yi , zi } = β2 Cov{xi , zi} + Cov{εi , zi} n gives n Nov 3, 2017 Hackl, Econometrics, Lecture 4 62 IV Estimator for β2 nSubstitution of sample moments for covariances gives the instrumental variables (IV) estimator n n n nConsistent estimator for β2 given that the instrumental variable zi is valid , i.e., it is qExogenous, i.e. E{εi zi} = 0 qRelevant, i.e. Cov{xi , zi} ≠ 0 nTypically, nothing can be said about the bias of an IV estimator; small sample properties are unknown nCoincides with OLS estimator for zi = xi Nov 3, 2017 Hackl, Econometrics, Lecture 4 63 Consumption Function, cont’d nAlternative model: Ct = β1 + β2Yt-1 + εt nYt-1 and εt are certainly uncorrelated; avoids risk of inconsistency due to correlated Yt and εt nYt-1 is certainly highly correlated with Yt, is almost as good as regressor as Yt nFitted model: n Ĉ = 0.012 + 0.660 Y-1 n with t = 12.86, R2 = 0.56, DW = 0.79 (instead of n Ĉ = 0.011 + 0.718 Y n with t = 15.55, R2 = 0.65, DW = 0.50) nDeterioration of t-statistic and R2 are price for improvement of the estimator Nov 3, 2017 Hackl, Econometrics, Lecture 4 64 IV Estimator: The Concept nAlternative to OLS estimator nAvoids inconsistency in case of endogenous regressors nIdea of the IV estimator: qReplace regressors which are correlated with error terms by regressors which are nuncorrelated with the error terms n(highly) correlated with the regressors that are to be replaced q and use OLS estimation nThe hope is that the IV estimator is consistent (and less biased than the OLS estimator) nPrice: IV estimator is less efficient; deteriorated model fit as measured by, e.g., t-statistic, R2 Nov 3, 2017 Contents nAutocorrelation nTests against Autocorrelation nInference under Autocorrelation nOLS Estimator Revisited nCases of Endogenous Regressors nInstrumental Variables (IV) Estimator: The Concept nIV Estimator: The Method nCalculation of the IV Estimator nAn Example nSome Tests nThe GIV Estimator Nov 3, 2017 Hackl, Econometrics, Lecture 4 65 Hackl, Econometrics, Lecture 4 66 IV Estimator: General Case nThe model is n yi = xi‘β + εi n with V{εi} = σε² and n E{εi xi} ≠ 0 nat least one component of xi is correlated with the error term nThe vector of instruments zi (with the same dimension as xi) fulfils n E{εi zi} = 0 n Cov{xi , zi} ≠ 0 nIV estimator based on the instruments zi n n n Nov 3, 2017 Hackl, Econometrics, Lecture 4 67 IV Estimator: Distribution nThe (asymptotic) covariance matrix of the IV estimator is given by n n nIn the estimated covariance matrix , σ² is substituted by n n n which is based on the IV residuals nThe asymptotic distribution of IV estimators, given IID(0, σε²) error terms, leads to the approximate distribution n n with the estimated covariance matrix n Nov 3, 2017 Hackl, Econometrics, Lecture 4 68 Derivation of the IV Estimator nThe model is n yi = xi‘β + εt = x0i‘β0 + βKxKi + εi n with x0i = (x1i, …, xK-1,i)’ containing the first K-1 components of xi, and E{εi x0i} = 0 nK-th component is endogenous: E{εi xKi} ≠ 0 nThe instrumental variable zKi fulfils n E{εi zKi} = 0 nMoment conditions: K conditions to be satisfied by the coefficients, the K-th condition with zKi instead of xKi: n E{εi x0i} = E{(yi – x0i‘β0 – βKxKi) x0i} = 0 (K-1 conditions) n E{εi zi} = E{(yi – x0i‘β0 – βKxKi) zKi} = 0 nNumber of conditions – and of corresponding linear equations – equals the number of coefficients to be estimated Nov 3, 2017 Hackl, Econometrics, Lecture 4 69 Derivation of the IV Estimator, cont’d nThe system of linear equations for the K coefficients β to be estimated can be uniquely solved for the coefficients β: the coefficients β are said “to be identified” nTo derive the IV estimators from the moment conditions, the expectations are replaced by sample averages n n n nThe solution of the linear equation system – with zi’ = (x0i‘, zKi) – is n n nIdentification requires that the KxK matrix Σi zi xi’ is finite and invertible; instrument zKi is relevant when this is fulfilled Nov 3, 2017 Contents nAutocorrelation nTests against Autocorrelation nInference under Autocorrelation nOLS Estimator Revisited nCases of Endogenous Regressors nInstrumental Variables (IV) Estimator: The Concept nIV Estimator: The Method nCalculation of the IV Estimator nAn Example nSome Tests nThe GIV Estimator Nov 3, 2017 Hackl, Econometrics, Lecture 4 70 Hackl, Econometrics, Lecture 4 71 Calculation of IV Estimators nThe model in matrix notation n y = Xβ + ε nThe IV estimator n n with zi obtained from xi by substituting instrumental variable(s) for all endogenous regressors nCalculation in two steps: 1.Reduced form: Regression of the explanatory variables x1, …, xK – including the endogenous ones – on the columns of Z: fitted values 2. 2.Regression of y on the fitted explanatory variables: 3. Nov 3, 2017 Hackl, Econometrics, Lecture 4 72 Calculation of IV Estimators: Remarks nThe KxK matrix Z’X = Σi zixi’ is required to be finite and invertible nFrom n n n it is obvious that the estimator obtained in the second step is the IV estimator nHowever, the estimator obtained in the second step is more general; see below nIn GRETL: The sequence „Model > Instrumental variables > Two-Stage Least Squares…“ leads to the specification window with boxes (i) for the regressors and (ii) for the instruments Nov 3, 2017 Hackl, Econometrics, Lecture 4 73 Choice of Instrumental Variables nInstrumental variable are required to be nexogenous, i.e., uncorrelated with the error terms nrelevant, i.e., correlated with the endogenous regressors nInstruments nmust be based on subject matter arguments, e.g., arguments from economic theory nshould be explained and motivated nmust show a significant effect in explaining an endogenous regressor nChoice of instruments often not easy nRegression of endogenous variables on instruments nBest linear approximation of endogenous variables nEconomic interpretation not of importance and interest n n n Nov 3, 2017 Contents nAutocorrelation nTests against Autocorrelation nInference under Autocorrelation nOLS Estimator Revisited nCases of Endogenous Regressors nInstrumental Variables (IV) Estimator: The Concept nIV Estimator: The Method nCalculation of the IV Estimator nAn Example nSome Tests nThe GIV Estimator Nov 3, 2017 Hackl, Econometrics, Lecture 4 74 Hackl, Econometrics, Lecture 4 75 Returns to Schooling: Causality? nHuman capital earnings function: n wi = β1 + β2Si + β3Ei + β4Ei2 + εi n with wi: log of individual earnings, Si: years of schooling, Ei: years of experience (Ei = agei – Si – 6) nEmpirically, more education implies higher income nQuestion: Is this effect causal? nIf yes, one year more at school increases wage by β2 (Theory A) nAlternatively, personal abilities of an individual causes higher income and also more years at school; more years at school do not necessarily increase wage (Theory B) nIssue of substantial attention in literature Nov 3, 2017 Hackl, Econometrics, Lecture 4 76 Returns to Schooling: Endogenous Regressors nWage equation: besides Si and Ei, additional explanatory variables like gender, regional, racial dummies, family background nModel for analysis: n wi = β1 + zi‘γ + β2Si + β3Ei + β4Ei2 + εi n zi: observable variables besides Ei, Si nzi is assumed to be exogenous, i.e., E{zi εi} = 0 nSi may be endogenous, i.e., E{Si εi} ≠ 0 qAbility bias: unobservable factors like intelligence, family background, etc. enable to more schooling and higher earnings qMeasurement error in measuring schooling qEtc. nWith Si, also Ei = agei – Si – 6 and Ei2 are endogenous nOLS estimators may be inconsistent n Nov 3, 2017 Hackl, Econometrics, Lecture 4 77 Returns to Schooling: Data nVerbeek‘s data set “schooling” nNational Longitudinal Survey of Young Men (Card, 1995) nData from 3010 males, survey 1976 nIndividual characteristics, incl. experience, race, region, family background, etc. nHuman capital earnings or wage function q log(wagei) = β1 + β2 edi + β3 expi + β3 expi² + εi n with edi: years of schooling (Si), expi: years of experience (Ei) nVariables: wage76 (wage in 1976, raw, cents p.h.), ed76 (years at school in 1976), exp76 (experience in 1976), exp762 (exp76 squared) nFurther explanatory variables: black: dummy for afro-american, smsa: dummy for living in metropolitan area, south: dummy for living in the south Nov 3, 2017 OLS Estimation Nov 3, 2017 Hackl, Econometrics, Lecture 4 78 OLS estimated wage function Model 2: OLS, using observations 1-3010 Dependent variable: l_WAGE76 coefficient std. error t-ratio p-value ---------------------------------------------------------- const 4.73366 0.0676026 70.02 0.0000 *** ED76 0.0740090 0.00350544 21.11 2.28e-092 *** EXP76 0.0835958 0.00664779 12.57 2.22e-035 *** EXP762 -0.00224088 0.000317840 -7.050 2.21e-012 *** BLACK -0.189632 0.0176266 -10.76 1.64e-026 *** SMSA76 0.161423 0.0155733 10.37 9.27e-025 *** SOUTH76 -0.124862 0.0151182 -8.259 2.18e-016 *** Mean dependent var 6.261832 S.D. dependent var 0.443798 Sum squared resid 420.4760 S.E. of regression 0.374191 R-squared 0.290505 Adjusted R-squared 0.289088 F(6, 3003) 204.9318 P-value(F) 1.5e-219 Log-likelihood -1308.702 Akaike criterion 2631.403 Schwarz criterion 2673.471 Hannan-Quinn 2646.532 Hackl, Econometrics, Lecture 4 79 Instruments for Si, Ei, Ei2 nPotential instrumental variables nFactors which affect schooling but are uncorrelated with error terms, in particular with unobserved abilities that are determining wage nFor years of schooling (Si) qCosts of schooling, e.g., distance to school (lived near college), number of siblings qParents’ education nFor years of experience (Ei, Ei2): age is natural candidate n Nov 3, 2017 Step 1 of IV Estimation Nov 3, 2017 Hackl, Econometrics, Lecture 4 80 Reduced form for schooling (ed76), gives predicted values ed76_h, Model 3: OLS, using observations 1-3010 Dependent variable: ED76 coefficient std. error t-ratio p-value ---------------------------------------------------------- const -1.81870 4.28974 -0.4240 0.6716 AGE76 1.05881 0.300843 3.519 0.0004 *** sq_AGE76 -0.0187266 0.00522162 -3.586 0.0003 *** BLACK -1.46842 0.115245 -12.74 2.96e-036 *** SMSA76 0.841142 0.105841 7.947 2.67e-015 *** SOUTH76 -0.429925 0.102575 -4.191 2.85e-05 *** NEARC4A 0.441082 0.0966588 4.563 5.24e-06 *** Mean dependent var 13.26346 S.D. dependent var 2.676913 Sum squared resid 18941.85 S.E. of regression 2.511502 R-squared 0.121520 Adjusted R-squared 0.119765 F(6, 3003) 69.23419 P-value(F) 5.49e-81 Log-likelihood -7039.353 Akaike criterion 14092.71 Schwarz criterion 14134.77 Hannan-Quinn 14107.83 Step 2 of IV Estimation Nov 3, 2017 Hackl, Econometrics, Lecture 4 81 Wage equation, estimated by IV with instruments age, age2, and nearc4a Model 4: OLS, using observations 1-3010 Dependent variable: l_WAGE76 coefficient std. error t-ratio p-value ---------------------------------------------------------- const 3.69771 0.435332 8.494 3.09e-017 *** ED76_h 0.164248 0.036887 4.453 8.79e-06 *** EXP76_h 0.044588 0.022502 1.981 0.0476 ** EXP762_h -0.000195 0.001152 -0.169 0.8655 BLACK -0.057333 0.056772 -1.010 0.3126 SMSA76 0.079372 0. 037116 2.138 0.0326 ** SOUTH76 -0.083698 0.022985 -3.641 0.0003 *** Mean dependent var 6.261832 S.D. dependent var 0.443798 Sum squared resid 446.8056 S.E. of regression 0.385728 R-squared 0.246078 Adjusted R-squared 0.244572 F(6, 3003) 163.3618 P-value(F) 4.4e-180 Log-likelihood -1516.471 Akaike criterion 3046.943 Schwarz criterion 3089.011 Hannan-Quinn 3062.072 Hackl, Econometrics, Lecture 4 82 Returns to Schooling: Summary of Estimates nEstimated regression coefficients and t-statistics n 1) 1) 1) 1) 1) 1) 1) 1) 1) n 1) The model differs from that used by Verbeek 1) Nov 3, 2017 OLS IV1) TSLS1) IV (M.V.) ed76 0.0740 0.1642 0.1642 0.1329 21.11 4.45 3.92 2.59 exp76 0.0836 0.0445 0.0446 0.0560 12.75 1.98 1.74 2.15 exp762 -0.0022 -0.0002 -0.0002 -0.0008 -7.05 -0.17 -0.15 -0.59 black -0.1896 -0. 0573 -0.0573 -0.1031 -10.76 -1.01 -0.89 -1.33 R2 0.291 0.246 F-test 204.9 163.4 Hackl, Econometrics, Lecture 4 83 Some Comments nInstrumental variables (age, age2, nearc4a) nare relevant, i.e., have explanatory power for ed76, exp76, exp762 nWhether they are exogenous, i.e., uncorrelated with the error terms, is not answered nTest for exogeneity of regressors: Wu-Hausman test nEstimates of ed76-coefficient: nIV estimate: 0.16 (0.13), i.e., 16% higher wage for one additional year of schooling; more than the double of the OLS estimate (0.07); not in line with “ability bias” argument! ns.e. of IV estimate (0.04) much higher than s.e. of OLS estimate (0.004) nLoss of efficiency especially in case of weak instruments: R2 of model for ed76: 0.12; Corr{ed76, ed76_h} = 0.35 Nov 3, 2017 GRETL’s TSLS Estimation Nov 3, 2017 Hackl, Econometrics, Lecture 4 84 Wage equation, estimated by GRETL’s TSLS Model 8: TSLS, using observations 1-3010 Dependent variable: l_WAGE76 Instrumented: ED76 EXP76 EXP762 Instruments: const AGE76 sq_AGE76 BLACK SMSA76 SOUTH76 NEARC4A coefficient std. error t-ratio p-value ---------------------------------------------------------- const 3.69771 0.495136 7.468 8.14e-014 *** ED76 0.164248 0.0419547 3.915 9.04e-05 *** EXP76 0.0445878 0.0255932 1.742 0.0815 * EXP762 -0.00019526 0.0013110 -0.1489 0.8816 BLACK -0.0573333 0.0645713 -0.8879 0.3746 SMSA76 0.0793715 0.0422150 1.880 0.0601 * SOUTH76 -0.0836975 0.0261426 -3.202 0.0014 *** Mean dependent var 6.261832 S.D. dependent var 0.443798 Sum squared resid 577.9991 S.E. of regression 0.438718 R-squared 0.195884 Adjusted R-squared 0.194277 F(6, 3003) 126.2821 P-value(F) 8.9e-143 Hackl, Econometrics, Lecture 4 85 Returns to Schooling: Summary of Estimates nEstimated regression coefficients and t-statistics n 1) 1) 1) 1) 1) 1) 1) 1) 1) n 1) The model differs from that used by Verbeek 1) Nov 3, 2017 OLS IV1) TSLS1) IV (M.V.) ed76 0.0740 0.1642 0.1642 0.1329 21.11 4.45 3.92 2.59 exp76 0.0836 0.0445 0.0446 0.0560 12.75 1.98 1.74 2.15 exp762 -0.0022 -0.0002 -0.0002 -0.0008 -7.05 -0.17 -0.15 -0.59 black -0.1896 -0. 0573 -0.0573 -0.1031 -10.76 -1.01 -0.89 -1.33 R2 0.291 0.246 0.196 F-test 204.9 163.4 126.3 Hackl, Econometrics, Lecture 4 86 Some Comments nVerbeek‘s IV estimates nDeviate from GRETL results nNo report of R2; definition of R2 does not apply to IV estimated models nIV estimates of coefficients nare smaller than the OLS estimates; exception is ed76 nhave higher s.e. than OLS estimates, smaller t-statistics nQuestions nRobustness of IV estimates to changes in the specification nExogeneity of instruments nWeak instruments Nov 3, 2017 Contents nAutocorrelation nTests against Autocorrelation nInference under Autocorrelation nOLS Estimator Revisited nCases of Endogenous Regressors nInstrumental Variables (IV) Estimator: The Concept nIV Estimator: The Method nCalculation of the IV Estimator nAn Example nSome Tests nThe GIV Estimator Nov 3, 2017 Hackl, Econometrics, Lecture 4 87 Hackl, Econometrics, Lecture 4 88 Some Tests nQuestions of interest 1.Is it necessary to use IV estimation, must violation of exogeneity be expected? To be tested: the null hypothesis of exogeneity of suspected variables 2.If IV estimation is used: Are the chosen instruments valid (relevant)? nFor testing nexogeneity of regressors: Wu-Hausman test, also called Durbin-Wu-Hausman test, in GRETL: Hausman test nrelevance of potential instrumental variables: Sargan test or over-identifying restrictions test nWeak instruments, i.e., only weak correlation between endogenous regressor and instrument: Cragg-Donald test Nov 3, 2017 Hackl, Econometrics, Lecture 4 89 Wu-Hausman Test nFor testing whether one or more regressors xi are endogenous (correlated with the error term); H0: E{εi xi} = 0 nIf the null hypothesis qis true, OLS estimates are more efficient than IV estimates qis not true, OLS estimates are inefficient, the less efficient but consistent IV estimates to be used nBased on the assumption that the instrumental variables are valid, i.e., given that E{εi zi} = 0, the null hypothesis E{εi xi} = 0 can be tested against the alternative E{εi xi} ≠ 0 nThe idea of the test: nUnder the null hypothesis, both the OLS and IV estimator are consistent; they should differ by sampling errors only nRejection of the null hypothesis indicates inconsistency of the OLS estimator Nov 3, 2017 Hackl, Econometrics, Lecture 4 90 Wu-Hausman Test, cont’d nBased on the differences between OLS- and IV-estimators; various versions of the Wu-Hausman test nAdded variable interpretation of the Wu-Hausman test: checks whether the residuals vi from the reduced form equation of potentially endogenous regressors contribute to explaining n yi = x1i’b1 + x2i’b2 + vi’γ + εi nx2: potentially endogenous regressors nvi: residuals from reduced form equation for x2 (predicted values for x2: x2 + v) nH0: γ = 0; corresponds to: x2 is exogenous nFor testing H0: use of nt-test, if γ has one component, x2 is just one regressor nF-test, if more than 1 regressors are tested for exogeneity Nov 3, 2017 Hackl, Econometrics, Lecture 4 91 Hausman Test Statistic nBased on the quadratic form of differences between OLS- estimators bLS and IV-estimators bIV nH0: both bLS and bIV are consistent, bLS is efficient relative to bIV nH1: bIV is consistent, bLS is inconsistent nHausman test statistic n H = (bIV – bLS)’ V (bIV – bLS) n with estimated covariance matrix V of bIV – bLS follows the approximate Chi-square distribution with J d.f. Nov 3, 2017 Hackl, Econometrics, Lecture 4 92 Wu-Hausman Test: Remarks nRemarks nTest requires valid instruments nTest has little power if instruments are weak or invalid nVarious versions of the test, all based on differences between OLS- and IV-estimators nIn GRETL: Whenever the TSLS estimation is used, GRETL produces automatically the Hausman test statistic Nov 3, 2017 Hackl, Econometrics, Lecture 4 93 Sargan Test nFor testing whether the instruments are valid nThe validity of the instruments zi requires that all moment conditions are fulfilled; for the R-vector zi, the R sums n n n must be close to zero nTest statistic n n has, under the null hypothesis, an asymptotic Chi-squared distribution with R-K df nCalculation of ξ: ξ = NRe2 using Re2 from the auxiliary regression of IV residuals ei = on the instruments zi Nov 3, 2017 Hackl, Econometrics, Lecture 4 94 Sargan Test: Remarks nRemarks nIn case of an identified model (R = K), all R moment conditions are fulfilled, ξ = 0 nOver-identified model: R > K; the Sargan test is also called over-identifying restrictions test nRejection implies: the joint validity of all moment conditions and hence of all instruments is not acceptable nThe Sargan test gives no indication of invalid instruments nIn GRETL: Whenever the TSLS estimation is used and R > K, GRETL produces automatically the Sargan test statistic n Nov 3, 2017 Hackl, Econometrics, Lecture 4 95 Cragg-Donald Test nWeak (only marginally valid) instruments, i.e., only weak correlation between endogenous regressor and instrument : nBiased IV estimates nInconsistent IV estimates nInappropriate large-sample approximations to the finite-sample distributions even for large N nDefinition of weak instruments: estimates are biased to an extent that is unacceptably large nNull hypothesis: instruments are weak, i.e., can lead to an asymptotic relative bias greater than some value b n Nov 3, 2017 Hackl, Econometrics, Lecture 4 96 Cragg-Donald Test, cont’d nTest procedure nRegression of the endogenous regressor on all instruments, both external, i.e., ones not included among the regressors, and internal nF-test of the null hypothesis that the coefficients of all external instruments are zero nIf F-statistic is less a not too large value, e.g., 10: consider the instruments as weak n Nov 3, 2017 Contents nAutocorrelation nTests against Autocorrelation nInference under Autocorrelation nOLS Estimator Revisited nCases of Endogenous Regressors nInstrumental Variables (IV) Estimator: The Concept nIV Estimator: The Method nCalculation of the IV Estimator nAn Example nSome Tests nThe GIV Estimator Nov 3, 2017 Hackl, Econometrics, Lecture 4 97 From OLS to IV Estimation nLinear model yi = xi‘β + εi nOLS estimator: solution of the K normal equations n 1/N Σi(yi – xi‘b) xi = 0 nCorresponding moment conditions n E{εi xi} = E{(yi – xi‘β) xi} = 0 nIV estimator given R instrumental variables zi which may overlap with xi: based on the R moment conditions n E{εi zi} = E{(yi – xi‘β) zi} = 0 nIV estimator: solution of corresponding sample moment conditions n Nov 3, 2017 Hackl, Econometrics, Lecture 4 98 Number of Instruments nMoment conditions n E{εi zi} = E{(yi – xi‘β) zi} = 0 n one equation for each component of zi nzi possibly overlapping with xi nGeneral case: R moment conditions nSubstitution of expectations by sample averages gives R equations n n 1.R = K: one unique solution, the IV estimator; identified model 2. 2.R < K: infinite number of solutions, not enough instruments for a unique solution; under-identified or not identified model Nov 3, 2017 Hackl, Econometrics, Lecture 4 99 The GIV Estimator 3.R > K: more instruments than necessary for identification; over-identified model nFor R > K, in general, no unique solution of all R sample moment conditions can be obtained; instead: nthe weighted quadratic form in the sample moments n n with a RxR positive definite weighting matrix WN is minimized ngives the generalized instrumental variable (GIV) estimator n n n n Nov 3, 2017 Hackl, Econometrics, Lecture 4 100 The weighting matrix WN nWN: positive definite, order RxR nDifferent weighting matrices result in different consistent GIV estimators with different covariance matrices nOptimal choice for WN? nFor R = K, the matrix Z’X is square and invertible; the IV estimator is (Z’X)-1Z’y for any WN n Nov 3, 2017 Hackl, Econometrics, Lecture 4 101 GIV and TSLS Estimator nOptimal weighting matrix: WNopt = [1/N(Z’Z)]-1; corresponds to the most efficient IV estimator n nIf the error terms are heteroskedastic or autocorrelated, the optimal weighting matrix has to be adapted nRegression of each regressor, i.e., each column of X, on Z, i.e., on the R column of Z, results in and n nThis explains why the GIV estimator is also called “two stage least squares” (TSLS) estimator: 1.First step: regress each column of X on Z 2.Second step: regress y on predictions of X n n n Nov 3, 2017 Hackl, Econometrics, Lecture 4 102 GIV Estimator and Properties nGIV estimator is consistent nThe asymptotic distribution of the GIV estimator, given IID(0, σε²) error terms, leads to n n which is used as approximate distribution in case of finite N nThe (asymptotic) covariance matrix of the GIV estimator is given by n n nIn the estimated covariance matrix, σ² is substituted by n n the estimate based on the IV residuals Nov 3, 2017 Hackl, Econometrics, Lecture 4 103 Your Homework 1.Use the data set “icecream” of Verbeek for the following analyses: a)Estimate the model where cons is explained by price and temp; show a diagramme of the residuals which may indicate autocorrelation of the error terms. b)Use the Durbin-Watson and the Breusch-Godfrey test against autocorrelation; state suitably H0 and H1. c)Compare (i) the OLS and (ii) the HAC standard errors of the estimated coefficients. d)Repeat a), using (i) the iterative Cochrane-Orcutt estimation and (ii) OLS estimation of the model in differences; compare and interpret the results. 2.For the Durbin-Watson test: (a) show that dw ≈ 2 – 2r; (b) can you agree with the statement “The Durbin-Watson test is a misspecification test”. Nov 3, 2017 Hackl, Econometrics, Lecture 4 104 Your Homework, cont’d 3.Use the data set “schooling” of Verbeek for the following analyses based on the wage equation n log(wage76) = b1 + b2 ed76 + b3 exp76 + b4 exp762 n + b5 black + b6 momed + b7 smsa76 + e a)Assuming that ed76 is endogenous, (i) estimate the reduced form for ed76, including external instruments smsa66, sinmom14, south66, and mar76; (ii) assess the validity of the potential instruments; what indicate the correlation coefficients? b)Estimate, by means of the GRETL Instrumental variables (Two-Stage Least Squares …) procedure, the wage equation, using the external instruments black, momed, sinmom14, smsa66, south76, mar76, and age76. Interpret the results including the Hausman and the Sargan test. c)Compare the estimates for b2 (i) from the model in b), (ii) from the model with instruments black, momed, smsa66, south76, mar76, and age76, and (iii) with the OLS estimates. Nov 3, 2017 Hackl, Econometrics, Lecture 4 105 Your Homework, cont’d 4.The model for consumption and income consists of two equations: n Ct = β1 + β2Yt + εt n Yt = Ct + It a.Show that both Ct and Yt are endogenous: q E{Ci εi} = E{Yi εi} = σε²(1 – β2)-1 b.Derive the reduced form of the model Nov 3, 2017 Hackl, Econometrics, Lecture 4 106