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Basic Econometrics, Gujarati and Porter CHAPTER 19: THE IDENTIFICATION PROBLEM 19.1 Using the definitions of M, m, K, and k, and letting R equal the number of variables (endogenous as well as predetermined) excluded from a given equation, then, by Definition 19.1, R = ( M − m) + ( K − k ) ≥ ( M − 1) Subtracting ( M − m) from each side, we obtain ( K − k ) ≥ m − 1, which is Definition 19.2. 19.2 The structural coefficients are: β 0 = π 3 − β1π 0 α 0 = π 3 − α1π 0 β1 = π4 π1 β2 = π 5 − 19.3 α1 = π 2π 4 π1 π5 π2 α2 = π 4 − π 1π 5 π2 (a) The reduced form equations are: Yt = π 0 + π 1 I t + wt (1) Ct = π 2 + π 3 I t + wt (2) For this system M = 2 (C,Y) and K =1 (I). The order condition applied to (2) shows that it is exactly identified. The income identity is identified by definition. (b) The reduced form equations are: Wt = π 0 + π 1UN t + π 2 M t + wt P = π + π UN + π R + π M (1) (2) 6 t 7 t For this system, M = 2 ( W , P ) and K = 3 ( UN , R , M ) . By the order condition, Eq. (1) overidentified, but Eq. (2) is just identified. t 4 5 t (c) This problem is designed to show the tedious nature of developing reduced form equations. The solution is left to the reader. 19.4 See Exercise 19.3. The rank condition test provides the same result. 19.5 The reason that the supply equation is overidentified is that the demand equation contains two predetermined variables, I and R . If it contained just one, the supply equation would be just identified. Thus, if α 2 = 0 or α 3 = 0 , the supply equation would be just identified. 207 Basic Econometrics, Gujarati and Porter 19.6 (a) For this system, M = 2 (Y1, Y2) and K = 2 (X1, X2). By the order condition, Y1 and Y2 are both exactly identified. (b) In this case Y1 is identified, but not Y2. 19.7 (a) Following the system (19.2.12) and (19.2.22), it can be shown that: πˆ πˆ βˆ10 = [πˆ 20 − 22 πˆ10 ] = −3; βˆ12 = 22 = 1.25 πˆ12 πˆ12 πˆ πˆ βˆ20 = [πˆ 20 − 21 πˆ10 ] = −6; βˆ21 = 21 = 2 πˆ11 πˆ11 πˆ πˆ πˆ πˆ γˆ11 = [πˆ 21 − 11 21 ] = 2.25; γˆ12 = [πˆ 22 − 12 21 ] = −6 πˆ11 πˆ11 (b) To test this hypothesis, we need the standard error of γˆ11 . But as you can see from (a), γˆ11 is a nonlinear function of the πˆ coefficients and it is not easy to estimate its standard error. 19.8 (a) In this example, Y1 is not identified but Y2 is. This system is similar to the system (19.2.12) and (19.2.13). Thus, πˆ βˆ21 = 3 = 1.5; βˆ20 = (πˆ 2 − βˆ21πˆ 0 ) = −4 πˆ1 The other structural coefficients cannot be identified. (b) In this case both Y1 and Y2 are identified. 19.9 In this system M = 4 and K = 5. Here all the equations are overidentified. 19.10 Here M = 4 and K = 4. By the order condition, Y1 and Y2 are not identified, but Y3 and Y4 are just identified. 19.11 Here M = 5 and K = 4. By the order condition, Y1 , Y2 , and Y5 are just identified, Y3 is not identified and Y4 is overidentified. To show how the rank condition works, consider the first equation. It excludes variables Y3 , Y5 , X 2 and X 3 . For this equation to be identified, there must be at least one 4 x 4 non-zero determinant from coefficients of the variables excluded from this equation but included in the remaining equations. One such determinant is: 208 Basic Econometrics, Gujarati and Porter β 23 0 γ 22 γ 23 1 β 35 0 λ33 ≠0 0 0 0 γ 43 0 1 γ 52 γ 53 Thus by the rank condition also the first equation is identified. Follow a similar procedure for the other equations. 19.12 For this model, M = 4 and K = 2. By the order condition, the equation for C is identified, and those for I and T are overidentified. With r treated as exogenous, M = 4 and K = 3. By the order condition now the equations for C, T, and I are all overidentified. 19.13 From Eq. (19.1.2), the reduced form of the income equation is: Yt = π 0 + π 1 I t + ut The OLS results are: Yˆt = 10.0000 + 5.0000 I t t = ( 8.458) (12.503) R 2 = 0.897 From Eq. (19.1.4), the reduced form for consumption is: Ct = π 2 + π 3 I t + wt The OLS results are: Cˆt = 10.000 + 4.000 I t t = (8.458) (10.002) r 2 = 0.848 For this model, M = 2 and K = 1. By the order condition, the consumption function is just identified. The estimates of the structural coefficients are: πˆ πˆ βˆ0 = 0 = 2; βˆ1 = 3 = 0.8 πˆ1 πˆ 2 19.14 See Exercise 19.1. From Eq. (19.3.1), with Definition 19.2, K − k ≥ m −1 Add k to each side. This yields K ≥ m + k −1 19.15 (a) The reduced form equations are: 209 Basic Econometrics, Gujarati and Porter Y Pt = π 1 + π 2 t + π 3 Ft + π 4Wt + π 5Ct −1 + π 6Tt −1 + π 7 Pt −1 + vt Nt Y Qt = π 8 + π 9 t + π 10 Ft + π 11Wt −1 + π 12Ct −1 + π 13Tt −1 + π 14 Pt −1 + wt Nt (b) Here M =2, K = 7. By the order condition, both equations are overidentified. Empirical Exercises 19.16 (a) & (b) Here M = 2 and K = 2. By the order condition, the demand function is not identified and the supply function is overidentified. (c) The reduced form equations are: Yt = π 0 + π 1 Rt + π 2 Pt + vt M t = π 3 + π 4 Rt + π 5 Pt + wt (d) To test for simultaneity in the supply function, (1) Estimate the reduced form for Yt and obtain the residuals, vˆt (2) Regress Mt on Yt and vˆt (3) The null hypothesis is that there is no simultaneity, i.e., the coefficient of vˆt in step (2) is not statistically significant. The Stata results of this exercise are as follows: Source | SS df MS -------------+-----------------------------Model | 120078361 2 60039180.3 Residual | 2071202.91 34 60917.7326 -------------+-----------------------------Total | 122149564 36 3393043.43 Number of obs = 37 F( 2, 34) = 985.58 Prob > F = 0.0000 R-squared = 0.9830 Adj R-squared = 0.9820 Root MSE = 246.82 d-statistic = 0.2538 -----------------------------------------------------------------------------m2 | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------Y | 0.8055752 0.0184182 43.74 0.000 .7681449 .8430054 vˆt | -0.02237 0.1043517 -0.21 0.832 -.2344382 .1896982 _cons | -2517.945 134.307 -18.75 0.000 -2790.889 -2245 -----------------------------------------------------------------------------Since the coefficient of the residual term is not statistically significant, do not reject the null hypothesis that there is no 210 Basic Econometrics, Gujarati and Porter simultaneity. Therefore, we can assume there is no simultaneity present. (e) Here we use the exogeneity test discussed in the chapter. We estimate the following regression: M t = β1 + β 2Yt + β 3Yˆt + ut (1) where Yˆ is obtained from the regression of the reduced form for Y t given in (c). If the estimated β3 is statistically different from zero, reject the hypothesis that Yt is exogenous. Source | SS df MS -------------+-----------------------------Model | 120078361 2 60039180.3 Residual | 2071202.9 34 60917.7323 -------------+-----------------------------Total | 122149564 36 3393043.43 Number of obs = 37 F( 2, 34) = 985.58 Prob > F = 0.0000 R-squared = 0.9830 Adj R-squared = 0.9820 Root MSE = 246.82 -----------------------------------------------------------------------------m2 | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------Y | 0.7832051 0.1027135 7.63 0.000 0.5744663 0.991944 Y_hat | 0.0223701 0.1043517 0.21 0.832 -0.1896982 0.2344383 _cons | -2517.945 134.307 -18.75 0.000 -2790.889 -2245 -----------------------------------------------------------------------------Dependent Variable: M2 Variable C Yt Yˆt Coefficient -2295.7898 0.3292 0.4791 R2 =0.9928 d = 0.5946 Std. Error 78.9873 0.06825 0.0695 t-Statistic -29.0652 4.8238 4 6.8929 As these results show, the coefficient of Yˆt is not statistically significant, leading to the conclusion that Y is exogenous. 19.17 (a) To test this, we can apply the techniques from Section 8.7 for testing the equality of two regression coefficients using the restricted least squares F test. The implications of this are left to the reader to explore. 211 Basic Econometrics, Gujarati and Porter (b) To see if νˆt is correlated with u2t , we can treat νˆt as a regressor in the Qt model (19.4.7) and, after the equation substitution, perform another restricted least squares F test. 212