【正文】 xcluded variables 2, the eqn is not identified Equation 14: Not identified Equation 15: Just identified Equation 16: Overidentified 如果模型中每個結(jié)構(gòu)方程都是可識別的，則稱結(jié)構(gòu)型聯(lián)立方程組模型是可識別的。 Example of the order condition 169。 Chris Brooks 2021, 陳磊 2021 654 5 外生性的定義 Leamer（ 1985）： p310 變量 X對變量 Y是外生的，如果變量 Y關(guān)于 X的條件分布不隨產(chǎn)生 X的過程的變化而改變。 外生性的兩種形式： ? 前定變量：與方程中的當(dāng)前和未來誤差項獨立。 ? 嚴(yán)格外生變量：與方程中任何時期的誤差項獨立。 前定變量的通常定義：包括外生變量和滯后的內(nèi)生變量 169。 Chris Brooks 2021, 陳磊 2021 655 ? How do we tell whether variables really need to be treated as endogenous or not? ? Consider again equations (14)(16). Equation (14) contains Y2 and Y3 but do we really need equations for them? ? We can formally test this using a Hausman test as follows: 1. Obtain the reduced form equations corresponding to (14)(16). The reduced forms turn out to be: (17)(19) Estimate the reduced form equations (17)(19) using OLS, and obtain the fitted values, 5 Tests for Exogeneity Y X X vY X vY X v1 10 11 1 12 2 12 20 21 1 23 30 31 1 3? ? ? ?? ? ?? ? ?? ? ?? ?? ?? , ? , ?Y Y Y1 2 3169。 Chris Brooks 2021, 陳磊 2021 656 2. Run the regression corresponding to equation (14). 3. Run the regression (14) again, but now also including the fitted values as additional regressors: (20) 4. Use an Ftest to test the joint restriction that ?2 = 0, and ?3 = 0. If the null hypothesis is rejected, Y2 and Y3 should be treated as endogenous. Tests for Exogeneity ?,?Y Y2 311331222514332101 ?? uYYXXYYY ???????? ???????169。 Chris Brooks 2021, 陳磊 2021 657 ? Consider the following system of equations: (2123) ? Assume that the error terms are not correlated with each other. Can we estimate the equations individually using OLS? ? Equation 21: Contains no endogenous variables, so X1 and X2 are not correlated with u1. So we can use OLS on (21). ? Equation 22: Contains endogenous Y1 together with exogenous X1 and X2. We can use OLS on (22) if all the RHS variables in (22) are uncorrelated with that equation’s error term. In fact, Y1 is not correlated with u2 because there is no Y2 term in equation (21). So we can use OLS on (22). 6 Recursive Systems Y X X uY Y X X uY Y Y X X u1 10 11 1 12 2 12 20 21 1 21 1 22 2 23 30 31 1 32 2 31 1 32 2 3? ? ? ?? ? ? ? ?? ? ? ? ? ?? ? ?? ? ? ?? ? ? ? ?169。 Chris Brooks 2021, 陳磊 2021 658 ? Equation 23: Contains both Y1 and Y2。 we require these to be uncorrelated with u3. By similar arguments to the above, equations (21) and (22) do not contain Y3, so we can use OLS on (23). ? This is known as a RECURSIVE or TRIANGULAR system. We do not have a simultaneity problem here. ? But in practice not many systems of equations will be recursive... Recursive Systems 169。 Chris Brooks 2021, 陳磊 2021 659 Indirect Least Squares (ILS) ? Cannot use OLS on structural equations, but we can validly apply it to the reduced form equations. ? If the system is just identified, ILS involves estimating the reduced form equations using OLS, and then using them to substitute back to obtain the structural parameters. ? However, ILS is not used much because 1. Solving back to get the structural parameters can be tedious. 2. Most simultaneous equations systems are overidentified. 7 Estimation procedures for Systems 169。 Chris Brooks 2021, 陳磊 2021 660 ? In fact, we can use this technique for justidentified and overidentified systems. ? Two stage least squares (2SLS or TSLS) is done in two stages: Stage 1: ? Obtain and estimate the reduced form equations using OLS. Save the fitted values for the dependent variables. Stage 2: ? Estimate the structural equations, but replace any RHS endogenous variables with their stage 1 fitted values. Estimation of Systems Using TwoStage Least Squares 169。 Chris Brooks 2021, 陳磊 2021 661 Example: Say equations (14)(16) are required. Stage 1: ? Estimate the reduced form equations (17)(19) individually by OLS and obtain the fitted values, . Stage 2: ? Replace the RHS endogenous variables with their stage 1 estimated values: (24)(26) ? Now and will not be correlated with u1, will not be correlated with u2 , and will not be correlated with u3 . Estimation of Systems Using TwoStage Least Squares ?, ? , ?Y Y Y1 2 3Y Y Y X X uY Y X uY Y u1 0 1 2 3 3 4 1 5 2 12 0 1 3 2 1 23 0 1 2 3? ? ? ? ? ?? ? ? ?? ? ?? ? ? ? ?? ? ?? ?? ????Y2?Y3?Y3?Y2169。 Chris Brooks 2021, 陳磊 2021 662 ? TSLS是比較經(jīng)濟 、 易用的方法 。 ? 如果在第一階段估計時所得到的 R2非常高 ， 那么古典 OLS估計量與TSLS估計量將非常接近；如果在第一階段估計時所得到的 R2非常低 ， TSLS估計量將沒有太大的實際意義 。 ? TSLS估計量是有偏估計量 ， 但卻是一致估計量 。 ? It is still of concern in the context of simultaneous systems whether the CLRM assumptions are supported by the data. ? If the disturbances in the structural equations are autocorrelated, the 2SLS estimator is not even consistent. ? The standard error estimates also need to be modified pared with their OLS counterparts, but once this has been done, we can use the usual t and Ftests to test hypotheses about the structural form coefficients. Estimation of Systems Using TwoStage Least Squares 169。 Chris Brooks 2021, 陳磊 2021 663 ? Recall that the reason we cannot use OLS directly on the structural equations is that the endogenous variables are correlated with the errors. ? One solution to this would be not to use Y2 or Y3 , but rather to use some other variables instead. ? We want these other variables to be (highly) correlated with Y2 and Y3, but not correlated with the errors they are called INSTRUMENTS. ? Say we found suitable ins