Investment

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[pic 68]

H0:a1 = 0 (et has unit root → No cointegration) H1: a1 Short-Run Dynamics: ECM

Random Regressors When the stochastic regressor is strictly exogenous E(ui|Xj) = 0 , which implies that E(ui) = 0 and Cov(ui,Xj) = E(uiXj) = 0 for all i and j Lagged dependent variable and autocorrelated errors: yt =1+2xt +3yt-1 + ut with ut = ut−1 + vt where vt ~ i.i.d. (0, ) (identically, independently distributed yt-1 = 1 + 2xt-1+ +3yt-2 + ut-1.) Measurement error Permanent Income Hypothesis Cp = Yp ,Cp = permanent consumption Yp = permanent income. However, Cp and Yp are unobservable. Ct = Cp + CTt, Yt = Yp + YTt , Ct and Yt are actually observed values, and CTt and YTt are transitory components. Ct -CTt = (Yt -YTt), Ct = Yt + ut where ut = CTt- YTt. Clearly Yt and ut are correlated. Two-stage Least Squares (2SLS) Estimator overidentified models Stage 1: Estimate the reduced-form equation using OLS Xi = a1 + a2Zi +a3Wi + vi and then predict the endogenous variable, Xi^. Stage 2: Then, solve the following two moment conditions for the two coefficients. , and Solving it for and is equivalent to estimating the model with the X replaced with Xi^ by OLS yi =1 + 2Xi^ + ei the GMM estimator accounts for heteroscedastic and autocorrelated random errors, while the 2SLS estimator does not. The Hausman-Wu Test for Endogeneity H0:cov(Xi,ui)=0 H1(Xi,ui)0 Procedure Model: yi = 1 + 2Xi + ui (where X is suspected to be endogenous) instrumental variables are Z1 and Z2. Estimate the reduced-form equation by OLS Xi = a0 + a1Z1i + a2Z2i + vi get vi from OLS THEN yi = 1 + 2Xi + vi^ + ui Test H0: 0（no correlation between Xi and ui) H1: 0 (correlation between Xi and ui) using t-statistic. If there are more than one endogenous variable, use an F test. Sargan Test for Instrument ValidityAssume that the model is given by yt = 1 + 2Xi + ui where Xi is an endogenous regressor, and available instruments are Z and W. Then, the number of the coefficients is 2 while the number of the instruments is 3 (1, Z, and W). Compute the 2SLS estimates for 1 and 2, and , using all the three instruments. Obtain the residuals by = yi − − Xi Estimate the auxiliary regression given by = a1 + a2Zi + a3Wi + vi n×R2 ~ χ2(L−K) If the n×R2 statistic is greater than the critical value, the null is rejected implying that at least one of the surplus moment conditions is not valid.异方差Breusch-Pagan (LM) test [pic 105][pic 106][pic 107][pic 108][pic 109][pic 110][pic 69][pic 70][pic 71][pic 72][pic 73][pic 74][pic 75][pic 76][pic 77][pic 78][pic 79][pic 80][pic 81][pic 82][pic 83][pic 84][pic 85][pic 86][pic 87][pic 88][pic 89][pic 90][pic 91][pic 92][pic 93][pic 94][pic 95][pic 96][pic 97][pic 98][pic 99][pic 100][pic 101][pic 102][pic 103][pic 104]

from ols

HET(Koenker robust version) [pic 111][pic 112][pic 113]

white test [pic 114][pic 115][pic 116][pic 117][pic 118][pic 119][pic 120][pic 121][pic 122][pic 123][pic 124][pic 125][pic 127][pic 126]

4.Compute the transformed observations[pic 128][pic 129][pic 130]

Apply least squares