Instrumental Variables and Two-Stage Least Squares

Transcription

1 Instrumental Variables and Two-Stage Least Squares Generalised Least Squares Professor Menelaos Karanasos December 2011

2 Generalised Least Squares: Assume that the postulated model is y = Xb + e, (1) where e N(0, σ 2 Ω), where Ω is a positive de nite matrix-this implies that Ω 1 is also a positive de nite matrix. Thus it is possible to nd a nonsingular matrix P such that Ω 1 = P 0 P. or Ω = (P 0 P) 1 = P 1 (P 0 ) 1.

3 Premultiply the linear model in equation (1) by P, to obtain Py = PXb + Pe (2) Denote Pe by u. Then Var(u) = E (uu 0 ) = E (Pee 0 P 0 ) = PE (ee 0 ) P 0 = σ 2 P {z } {z} Ω P 0 σ 2 Ω P 1 (P 0 ) 1 = σ 2 PP 1 (P 0 ) 1 P 0 = σ 2 I. Thus the transformed variables in equation (2) satisfy the conditions under which OLS is BLUE.

4 The coe cient estimated vector from OLS regression of Py on PX is the generalized least squares (GLS) estimator: From the OLS theory it follows that β G = [(PX ) 0 PX ] 1 (PX ) 0 Py = (X 0 P 0 P {z} Ω 1 X ) 1 X 0 P 0 Py = (X 0 Ω 1 X ) 1 X 0 Ω 1 y. Var(β G ) = σ 2 [(PX ) 0 PX ] 1 = σ 2 (X 0 P 0 P {z} Ω 1 X ) 1 = σ 2 (X 0 Ω 1 X ) 1.

5 An unbiased estimator of the unknown σ 2 is readily obtained from the application of OLS to the transformed model. It is s 2 = bu 0 bu/n k where β G = (X 0 Ω 1 X ) 1 X 0 Ω 1 y. = (Py PX β G ) 0 (Py PX β G )/N k = (y X β G ) 0 P 0 P {z} Ω 1 (y X β G )/N k = (y X β G ) 0 Ω 1 (y X β G )/N k,

6 Note that the procedures outlined so far imply knowledge of Ω. In practice Ω is unknown, and it is important to develop feasible generalized least squares (FGLS). Finally, note that if σ 2 Ω = V, where V is a positive de nite variance-covariance matrix. Then, it follows directly that and β G = (X 0 {z} Ω 1 X ) 1 X 0 {z} Ω 1 y = σ 2 V 1 σ 2 V 1 = (X 0 V 1 X ) 1 X 0 V 1 y, Var(β G ) = σ 2 (X 0 {z} Ω 1 X ) 1 σ 2 V 1 = σ 2 (X 0 V 1 X ) 1.

7 Feasible GLS procedure: Let V = Further, we hypothesize that σ σ 2 N σ 2 i = a 0 + a 1 z a 2 i, i = 1,..., N, where z i is a single variable, possibly one of the regressors, thought to determine the heteroscedasticity.

8 Because the OLS residuals be = y can run the nonlinear regression X β are consistent estimates of e one be 2 i = a 0 + a 1 z a 2 i + v i. Estimates of the disturbance variances are then bσ 2 i = α 0 + α 1 z α 2 i, i = 1,..., N. These estimates give the bv matrix and a feasible GLS procedure.

9 Instrumental variables (IV) estimators: Consider the relation y i = bx i + e i, (3) where for simplicity the constant term has been dropped. Suppose however, that the observed value x i can be represented as the sum of the true value ex i and a random measurement error v i, that is x i = ex i + v i. In this case the apropriate relation may be y i = bex i + e i (4)

10 If we assume that equation (4) is the maintained speci cation but that observations are only available on x i and not on ex i, what happens if we use OLS? The OLS slope is β = y x {z} bex i +e i x(bex + e) x 2 = x 2 = b xex x 2 + xe x 2 ) E (β) = b xex 6= b. x 2 Thus OLS is biased. This is an example of speci cation error.

11 In this case one should make use of instrumental variables which are also commonly referred to as instruments. Suppose that it is possible to nd a data matrix Z of order N l (l k), which posseses two vital properties: 1. The variables in Z are correlated with those of X 2. The variables in Z are (in the limit) uncorrelated with the disturbance term e. Premultiplying the general relation by Z 0 gives with Z 0 y = {z} {z} Z 0 X b + {z} Z 0 e, y X e Var(e ) = (Z 0 e) = E [(Z 0 e)(z 0 e) 0 ] = E (Z 0 ee 0 Z ) = Z 0 E (ee 0 )Z = σ 2 Z 0 Z.

12 This suggests the use of the GLS. The resultant estimator is β GLS = β IV = [X {z} 0 Z (Z 0 Z ) 1 {z} Z 0 X ] 1 {z} X 0 Z (Z 0 Z ) 1 Z 0 y (5) {z} (X ) 0 X (X ) 0 y = (X 0 P z X ) 1 X 0 P z y, where P z = Z (Z 0 Z ) 1 Z 0. The variance-covariance matrix is Var(β IV ) = σ 2 [X 0 Z (Z 0 Z ) 1 Z 0 X ] 1 = σ 2 (X 0 P z X ) 1 and the distrurbance variance maybe estimated consistenly from bσ 2 = (y X β IV ) 0 (y X β IV )/N. Note that the use of N or N k or N l in the divisor does not matter asymptotically.

13 Special case: When l = k, that is, when Z contains the same number of columns as X, we have a special case of the foregoing results. Now X 0 Z is k k and nonsingular. This implies that [(X {z} 0 Z (Z 0 Z ) 1 (Z {z} 0 X )] 1 = (X ) 0 X (Z 0 X ) 1 (Z 0 Z )(X 0 Z ) 1. Thus the estimator in equation (5) reduces to β IV = (Z 0 X ) 1 (Z 0 Z )(X 0 Z ) 1 (X 0 Z )(Z 0 Z ) 1 Z 0 y = (Z 0 X ) 1 Z 0 y Moreover, Var(β IV ) simpli es to Var(β IV ) = σ 2 [X 0 Z (Z 0 Z ) 1 Z 0 X ] 1 = σ 2 (Z 0 X ) 1 (Z 0 Z )(X 0 Z ) 1.

14 Two-stage least square (2SLS) The IV estimator may also be seen as the result of a double application of least squares: Stage (i): Regress each variable in the X matrix on Z (X = Zd + u) to obtain a matrix of tted values X bx = Z {z} δ (Z 0 Z ) 1 Z 0 X = Z (Z 0 Z ) 1 Z 0 {z } P Z X = P Z X.

15 Stage (ii): Regress y on bx to obtain the 2SLS estimated β vector β 2SLS = ( bx 0 bx ) 1 bx 0 y = (X 0 PZ 0 P {z } Z X ) {z} 1 X 0 PZ 0 y {z } bx 0 bx 0 bx = (X 0 P Z X )X 0 P Z y = β GLS, since P 0 Z P Z = [Z (Z 0 Z ) 1 Z 0 ] 0 = Z (Z 0 Z ) 1 Z 0 = P Z and P 0 Z P Z = P 2 Z = Z (Z 0 Z ) 1 Z 0 Z (Z 0 Z ) 1 Z 0 = Z (Z 0 Z ) 1 Z 0 = P Z.

16 Thus the IV estimator can be obtained by a two-stage least-squares procedure. The variance-covariance matrix and the estimated disturbance term are given by Var(β IV ) = σ 2 [X 0 Z (Z 0 Z ) 1 Z 0 X ] 1 = σ 2 (X 0 P z X ) 1 and bσ 2 = (y X β IV ) 0 (y X β IV )/N.

17 Choice of instruments: The crucial question is, where do we nd the instruments? Some of them are often variables from X matrix itself. Any variables that are thought to be exogenous and indepedent of the disturbance are retained to serve in the Z matrix. When some of the X variables are used as instruments, we may particion X and Z as X = [X 1 X 2 ], Z = [X 1 Z 1 ], where X 1 is of order N r (r < k), X 2 is N (k r), and Z 1 is N (l r).

18 It can be shown that bx, the matrix of regressors in the second-stage regression, is then bx = [X 1 bx 2 ], and bx 2 = Z (Z 0 Z ) 1 Z 0 X 2, {z } P Z that is bx 2 are the tted values of X 2 obtained from the regression of X 2 on the full set of instruments: X 2 = Z γ + v and bx 2 = Z bγ, bγ = (Z 0 Z ) 1 Z 0 X 2.

19 There still remains the question of how many instruments to use. The minimum number is k. The asymptotic e ciency increases with the number of isntruments. However, the small sample bias also increases with the number of instruments. If, in fact, we select N instruments, it is simple to show that P Z = I in which case the IV estimator is simply the OLS which is biased and inconsistent. If on the other hand, we use the minimum or close to the minimum, number of instruments, the results may also be poor. It has been shown that the mth moment of the 2SLS estimator exists if and only if m < l k + 1. Thus, if there are just as many instruments as explanatory variables, the 2SLS estimator will not have a mean. With one more instrument there will be a mean but not variance, and so forth.