11. Simultaneous-Equation Models

Transcription

1 11. Simultaneous-Equation Models Up to now: Estimation and inference in single-equation models Now: Modeling and estimation of a system of equations 328

2 Example: [I] Analysis of the impact of advertisement measures on the sales quantity of a pharmaceutical product Data set covering 24 quarters (i = 1,..., 24) of the following variables: sales quantity a i (in 100 g) number of advertisements w i (in double pages) price of the active pharmaceutical ingredient p i (in euros per 100 g) price of advertisement q i (in 1000 euros per double page) 329

3 Example: [II] Conceivable specification of a single equation a i = α + β 1 w i + β 2 p i + u i Problem: bidirectional causality, because (the number of) advertisements w i (has) have an impact on the sales quantity a i the sales quantity a i has an impact on the (number of) advertisments w i 330

4 Quarter Sales Quantity Number of Advertisements Price of Ingredient Price of Advertisement

5 Consequence of bidirectional causality: OLS estimators of the single equations are biased and inconsistent Resort: Modeling of bidirectional causality via simultaneous-equation models (interdependent equation models) 332

6 11.1 Inconsistency of OLS Estimators Consider the following equation system: a i = α + β 1 w i + β 2 p i + u i (1) w i = γ + δ 1 a i + δ 2 q i + v i (2) The error terms u i and v i are assumed to satisfy the following: u i and v i satisfy all #B-Assumptions There is only contemporary correlation between u i and v i : Cov(u i, v i ) = σ for i = 1,..., N Cov(u i, v j ) = 0 for i = j and i, j = 1,..., N 333

7 Consider the following scenario: We assume u i > 0 for any i From Eq. (1) it follows that a i With δ 1 > 0 it follows from Eq. (2) that w i With β 1 > 0 it then follows from Eq. (1) that a i multiplicator process: u i > 0 implies a i and w i 334

8 Analogously: v i > 0 implies w i and a i Obviously: u i > 0 implies w i in Eq. (1) in Eq. (1) w i and u i are positively contemporarily correlated v i > 0 implies a i in Eq. (2) in Eq. (2) a i and v i are positively contemporarily correlated 335

9 Summary: In both equations the error terms are contemporarily correlated with a regressor OLS estimators of the single-equation parameters are biased and inconsistent (cf. Chapter 9, Case #3, Slide 243) 336

10 11.2 Indirect Least Squares Estimation Now: Parameter estimation after an appropriate transformation of the equation system a i = α + β 1 w i + β 2 p i + u i (1) w i = γ + δ 1 a i + δ 2 q i + v i (2) 337

11 Transformation: [I] Inserting w i from Eq. (2) into Eq. (1), we obtain a i = α + β 1 (γ + δ 1 a i + δ 2 q i + v i ) + β 2 p i + u i Rearranging terms, we obtain and (1 β 1 δ 1 )a i = α + β 1 γ + β 2 p i + β 1 δ 2 q i + β 1 v i + u i a i = α + β 1γ 1 β 1 δ 1 + β 2 1 β 1 δ 1 p i + β 1δ 2 1 β 1 δ 1 q i + β 1v i + u i 1 β 1 δ 1 (3) 338

12 Transformation: [II] Analogously, inserting a i from Eq. (1) into Eq. (2) and solving for w i, we obtain w i = γ + δ 1α 1 β 1 δ 1 + β 2δ 1 1 β 1 δ 1 p i + δ 2 1 β 1 δ 1 q i + δ 1u i + v i 1 β 1 δ 1 (4) We define the following terms from Eq. (3) as π 1 α + β 1γ 1 β 1 δ 1, π 2 β 2 1 β 1 δ 1, π 3 β 1δ 2 1 β 1 δ 1, u i β 1v i + u i 1 β 1 δ 1 and, analogously, the terms from Eq. (4) as π 4 γ + δ 1α 1 β 1 δ 1, π 5 β 2δ 1 1 β 1 δ 1, π 6 δ 2 1 β 1 δ 1, v i δ 1u i + v i 1 β 1 δ 1 339

13 Transformation: [III] Thus, the transformed equation system can be written as a i = π 1 + π 2 p i + π 3 q i + u i (5) w i = π 4 + π 5 p i + π 6 q i + v i (6) Remarks: [I] The error terms u i and v i satisfy all #B-Assumptions (why?) Eq. (5) no longer contains the variable w i 340

14 Remarks: [II] Eq. (6) no longer contains the variable a i Eqs. (5) and (6), respectively, contain one endogenous variable (a i or w i ) the genuine exogenous (predetermined) variables p i, q i Via Eqs. (5) and (6), knowledge on π 1,..., π 6 and the exogenous variables p i, q i directly yields E(a i ), E(w i ) (π 1,..., π 6 contains interdependencies between a i and w i ) 341

15 Definition 11.1: (Structural and reduced form) We call the Eqs. (1) and (2) the structural form of the simultaneous-equation system and the transformed Eqs. (5) and (6) its reduced form. Remarks: [I] The structural form contains behavioral equations that typically rest on economic theory The right-hand side of the reduced form only contains genuine exogenous (predetermined) variables (the system has been solved for the endogenous variables) contemporary correlation of the structural form has been removed OLS estimation of π 1,..., π 6 is feasible 342

16 Remarks: [II] Using the definitions of π 1,..., π 6 in the reduced-form Eqs. (3) and (4), we can solve for the structural parameters in the Eqs. (1) and (2): and α = π 1 π 3π 4 π 6, β 1 = π 3 π 6, β 2 = π 2 π 3π 5 π 6 (7) γ = π 4 π 1π 5 π 2, δ 1 = π 5 π 2, δ 2 = π 6 π 3π 5 π 2 (8) (Proof: see Class) via Eqs. (7) and (8), the OLS estimators ˆπ 1,..., ˆπ 6 imply estimators of the structural parameters 343

17 Dependent Variable: SALES_QUANTITY Method: Least Squares Date: 01/07/05 Time: 14:40 Sample: 1997:1 2002:4 Included observations: 24 Variable Coefficient Std. Error t-statistic Prob. C INGREDIENT_PRICE ADVERT_PRICE R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) Dependent Variable: ADVERT_NO Method: Least Squares Date: 01/07/05 Time: 14:41 Sample: 1997:1 2002:4 Included observations: 24 Variable Coefficient Std. Error t-statistic Prob. C INGREDIENT_PRICE ADVERT_PRICE R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic)

18 Definition 11.2: (Indirect least squares estimators) The estimators of the structural parameters, obtained via the OLS estimators ˆπ 1,..., ˆπ 6 and the Eqs. (7) and (8), are called indirect least squares (ILS) estimators. Computation of the ILS estimates: ˆα ILS = , ˆβ ILS 1 = , ˆβ ILS 2 = ˆγ ILS = , ˆδ ILS 1 = , ˆδ ILS 2 =

19 Remarks: The ILS estimators are consistent (due to the consistency of the OLS estimators ˆπ 1,..., ˆπ 6 and the Slutsky-Theorem) Interpretation of the estimates: the ILS estimates ˆβ 1 ILS, ˆβ 2 ILS, ˆδ 1 ILS, ˆδ 2 ILS only reflect primary effects (triggering a multiplicator effect) the entire effects can only be inferred from the OLS estimates ˆπ 2, ˆπ 3, ˆπ 5, ˆπ 6 (estimates of the reduced-form parameters) 346

20 11.3 Identification Up to now: Using the OLS estimators of the reduced-form parameters, we could find unique estimators of all structural parameters Question: Under which conditions is this possible? 347

21 Example 1: [I] (Downsized model) Omitting the advertisement price q i in Eq. (2) yields a i = α + β 1 w i + β 2 p i + u i (9) w i = γ + δ 1 a i + v i (10) (structural form) Reduced form: a i = π 1 + π 2 p i + u i (11) w i = π 3 + π 4 p i + v i (12) 348

22 Example 1: [II] where π 1 = α + β 1γ 1 β 1 δ 1, π 2 = β 2 1 β 1 δ 1, π 3 = γ + δ 1α 1 β 1 δ 1, π 4 = β 2δ 1 1 β 1 δ 1, u i = β 1v i + u i 1 β 1 δ 1, v i = δ 1u i + v i 1 β 1 δ 1 Number of parameters structural form: 5 (α, β 1, β 2, γ, δ 1 ) reduced form: 4 (π 1, π 2, π 3, π 4 ) computation of all 5 structural parameters impossible 349

23 Example 1: [III] The formulae for π 1,..., π 4 imply γ = π 3 π 1π 4 π 2, δ 1 = π 4 π 2 γ, δ 1 are obtainable (Eq. (10)) α, β 1, β 2 are not obtainable (Eq. (9)) only partially unique ILS estimators 350

24 Example 2: [I] (Extended model) We add the trend variable i in Eq. (1): a i = α + β 1 w i + β 2 p i + β 3 i + u i (13) w i = γ + δ 1 a i + δ 2 q i + v i (14) (structural form) Reduced form: a i = π 1 + π 2 p i + π 3 i + π 4 q i + u i (15) w i = π 5 + π 6 p i + π 7 i + π 8 q i + v i (16) 351

25 Example 2: [II] where π 1 = α + β 1γ 1 β 1 δ 1, π 2 = β 2 1 β 1 δ 1, π 3 = β 3 1 β 1 δ 1, π 4 = β 1δ 2 1 β 1 δ 1, π 5 = γ + δ 1α 1 β 1 δ 1, π 6 = β 2δ 1 1 β 1 δ 1, π 7 = β 3δ 1 1 β 1 δ 1, π 8 = δ 2 1 β 1 δ 1 Number of parameters structural form: 7 (α, β 1, β 2, β 3, γ, δ 1, δ 2 ) reduced form: 8 (π 1,..., π 8 ) some structural parameters are no longer unique 352

26 Example 2: [III] For example, the formulae for π 1,..., π 8 imply δ 1 is not unique δ 1 = π 6 π 2 but also, δ 1 = π 7 π 3 no unique ILS estimation of δ 1 353

27 Summary: In the equation system (1) and (2) all structural parameters are uniquely obtainable In the equation system (9) and (10) only the structural parameters γ, δ 1 are uniquely obtainable In the equation system (13) and (14) only the structural parameters α, β 1, β 2, β 3 are uniquely obtainable 354

28 Definition 11.3: (Identifiability) (a) We call a structural equation exactly identified, if all its parameters are uniquely obtainable. (b) We call a structural equation underidentified, if its parameters are non-obtainable due to a lack of determining equations. (c) We call a structural equation overidentified, if its parameters are not uniquely obtainable due to an oversufficiency of determining equations. 355

29 Obviously: Finding the correct identification type of an equation system appears to be a non-trivial technical task Now: Technical rules for finding the identification type (easy to implement, but not always exact) To this end: Distinction of all model variables between endogenous variables predetermined variables 356

30 Definition 11.4: (Predetermined, endogenous variables) (a) We call a variable predetermined, if its values are determined outside the equation system. (b) We call a variable endogenous, if its values are determined within the equations system. Remarks: [I] We consider the constant term(s) in one or several equations as a single predetermined variable (the column of 1 s in the X-matrix) 357

31 Remarks: [II] Predetermined variables in the extended model: the constant term the variables p i, q i, i Endogenous variables in the extended model: the variables a i, w i Deciding whether a variable is endogenous or predetermined can be a delicate matter in practice: proposition by economic theory determined by the econometrician 358

32 Notation: K total = Number of predetermined variables in the entire equations system K = Number of predetermined variables in the equation under consideration M = Number of endogenous variables in the equation under consideration Now: Necessary conditions of exactly identified and overidentified equations 359

33 Theorem 11.5: (Necessary conditions) If an equation of a system is (a) exactly identified, we then have K total K = M 1, (b) overidentified, we then have K total K > M 1. Remarks: [I] There exist systems with underidentified equations for which we have K total K M 1 However, from the basics of logic we have the following: If an equation satisfies K total K < M 1, then the equation is underidentified (sufficient condition for underidentified equations) 360

34 Remarks: [II] A sufficient condition of all identification types is provided by the so-called rank condition (cf. Section ) An identification rule similar to the rank condition is the socalled order condition 361

35 Definition 11.6: (Order condition of identifiability) The order condition classifies an equation of a simultaneous system as (a) underidentified, if K total K < M 1, (b) exactly identified, if K total K = M 1, (c) overidentified, if K total K > M 1. Remarks: [I] The order condition does not always yield exact results (cf. Remark #1 after Theorem 11.5, Slide 360) 362

36 Remarks: [II] Cases, in which the order condition erroneously classifies an actually underidentified equation as exactly or overidentified, rarely occur in practice in practice, we often use the easy-to-check order condition instead of the more complicated rank condition (being aware of a small error risk) 363

37 Remarks: [III] K total K is the number of missing predetermined variables in the equation under consideration M 1 is the number of endogenous variables appearing on the right-hand side of the equation the order condition compares the number of excluded predetermined variables in the considered equation with the number of endogenous variables on the right-hand side of the equation 364

38 Examples: [I] Consider the original model Eq. (1): K total = 3, K = 2, M = 2 K total K = 3 2 = 1 = M 1 Eq. (1) is exactly identified Eq. (2): K total = 3, K = 2, M = 2 K total K = 3 2 = 1 = M 1 Eq. (2) is exactly identified 365

39 Examples: [II] Downsized model Eq. (9): K total = 2, K = 2, M = 2 K total K = 2 2 = 0 < 1 = M 1 Eq. (9) is underidentifed Eq. (10): K total = 2, K = 1, M = 2 K total K = 2 1 = 1 = M 1 Eq. (10) is exactly identified 366

40 Examples: [III] Extended model Eq. (13): K total = 4, K = 3, M = 2 K total K = 4 3 = 1 = M 1 Eq. (13) is exactly identified Eq. (14): K total = 4, K = 2, M = 2 K total K = 4 2 = 2 > 1 = M 1 Eq. (10) is overidentified 367

41 11.4 Two-Stage Least Squares Estimation Summary: If a single equation of a system is exactly identified, we can consistently estimated all structual parameters of the equation via the ILS method underidentified, we cannot consistently estimate the structural parameters of the equation overidentified, the ILS method does not provide unique estimators of the structural parameters of the equation 368

42 Now: There is a general estimation procedure, the so-called twostage LS method (TSLS), that provides the same results as the ILS method for exactly identified equations unique and consistent estimators of overidentified equations Remark: Cf. the relationship to the general IV estimation in single equations in Chapter 9, Slides

43 Stage #1 of TSLS estimation: [I] Consider the extended model a i = α + β 1 w i + β 2 p i + β 3 i + u i (17) w i = γ + δ 1 a i + δ 2 q i + v i (18) with the overidentified Eq. (18) a i and v i are contemporarily correlated OLS estimators are inconsistent apply the generalized IV estimation procedure 370

44 Stage #1 of TSLS estimation: [II] Conceivable instrumental variable for a i : linear combination of all predetermined variables (justification: predetermined variables are independent of v i ) Predetermined variables: p i, i, q i, constant term instrumental variable has the form z i = µ 1 + µ 2 p i + µ 3 i + µ 4 q i with optimal parameters µ 1,..., µ 4 (chosen so that the correlation with a i becomes maximal) 371

45 Stage #1 of TSLS estimation: [III] Find the optimal linear combination via OLS estimation of the reduced form (15) for a i a i = µ 1 + µ 2 p i + µ 3 i + µ 4 q i + u i Optimal parameter values are µ 1 = ˆπ 1,..., µ 4 = ˆπ 4 the desired instrumental variable z i is given by z i = ˆπ 1 + ˆπ 2 p i + ˆπ 3 i + ˆπ 4 q i or, more compactly written, z i = â i where â i = ˆπ 1 + ˆπ 2 p i + ˆπ 3 i + ˆπ 4 q i 372

46 Stage #2 of TSLS estimation: It is now possible to perform classical IV estimation (cf. Definition 9.7, Slide 291) we obtain ˆγ CIV, ˆδ CIV 1, ˆδ CIV 2 However, in simultaneous equation systems it is true that from the regression CIV-estimator = OLS-estimator i.e. ˆγ CIV = ˆγ OLS, w i = γ + δ 1 â i + δ 2 q i + v i (19) ˆδ 1 CIV = ˆδ 1 OLS, ˆδ 2 CIV = ˆδ 2 OLS 373

47 Definition 11.7: (Two-stage least squares (TSLS) estimator) The OLS estimator of Eq. (19) is called the two-stage least squares (TSLS) estimator. Remark: Since the TSLS estimator and the classical IV estimator coincide, the TSLS estimator is consistent 374

48 Summarizing example: [I] Extended model in structural form a i = α + β 1 w i + β 2 p i + β 3 i + u i (20) w i = γ + δ 1 a i + δ 2 q i + v i (21) and in reduced form a i = π 1 + π 2 p i + π 3 i + π 4 q i + ui (22) w i = π 5 + π 6 p i + π 7 i + π 8 q i + vi (23) (cf. Slides ) 375

49 Summarizing example: [II] Owing to the order condition we know that Eq. (20) is exactly identified Eq. (21) is overidentified Estimation of the overidentified Equation (21) via the TSLS method 376

50 Dependent Variable: SALES_QUANTITY Method: Least Squares Date: 01/14/05 Time: 18:54 Sample: 1997:1 2002:4 Included observations: 24 Variable Coefficient Std. Error t-statistic Prob. C INGREDIENT_PRICE I ADVERT_PRICE R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) Summarizing example: [III] Stage #1: OLS estimation of Eq. (22) yields the instrument variable z i = â i = p i i q i 377

51 Summarizing example: [IV] Stage #2: OLS estimation of w i = γ + δ 1 â i + δ 2 q i + v i yields the (consistent) TSLS estimator ˆγ TSLS = , (cf. EViews-Output on Slide 379) ˆδ TSLS 1 = , ˆδ TSLS 2 =

52 Dependent Variable: ADVERT_NO Method: Least Squares Date: 01/14/05 Time: 19:00 Sample: 1997:1 2002:4 Included observations: 24 Variable Coefficient Std. Error t-statistic Prob. C Z ADVERT_PRICE R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic)

53 11.5 Examples Example #1: (System with lagged variables) [I] Consider the preceding model Conjecture: the number of advertisements of the previous period w i 1 has an impact on the current sales quantity a i sales quantity of the previous period a i 1 has an impact on the current number of advertisements w i model: a i = α + β 1 w i + β 2 p i + β 3 w i 1 + u i (24) w i = γ + δ 1 a i + δ 2 q i + δ 3 a i 1 + v i (25) 380

54 Example #1: [II] a i 1 and w i 1 are predetermined at date i endogenous variables: a i, w i predetermined variables: constant term, p i, q i, a i 1, w i 1 Order condition: For both equations we have: K total = 5, K = 3, M = 2 K total K = 2 > 1 = M 1 both equations are overidentified parameter estimation via TSLS method (see Class) 381

55 Example #2: (Keynesian macro-model) [I] For i = 1,..., N we consider the equations C i = α + β Y i + u i (26) Y i = C i + I i (27) where C = consumption expenditure, Y = income, I = investment (cf. Slide 297) Special feature: Equation (27) does not include an error term any parameters 382

56 Example #2: [II] We know from Chapter 9, Slides : separate OLS estimation of Eq. (26) is inconsistent Identification (order condition) Eq. (27) does not contain any parameters For Eq. (26) we have: K total = 2, K = 1, M = 2 K total K = 1 = M 1 Eq. (26) is exactly identified parameter estimation via ILS or TSLS 383

57 11.6 Matrix Representation Now: Formal matrix representation of simultaneous equations Structural Form Notation: [I] Consider a simultaneous system with M equations Notation of the M endogenous variables: y 1,..., y M or y m with m = 1,..., M 384

58 Notation: [II] Notation of the K total = K + 1 predetermined variables: x 0, x 1,..., x K or x k with k = 0, 1,..., K where x 0 = (column of 1 s of the X-matrix)

59 Notation: [III] Eq. m of the system: where y m = α m x 0 + β 1m x 1 + β 2m x β Km x K +γ 1m y γ m 1m y m 1 + γ m+1m y m γ Mm y M + u m (28) u m : error vector of Eq. m α m : constant term of Eq. m β km : coefficient of the kth predetermined variable in Eq. m γ jm : coefficient of the jth endogenous variable in Eq. m 386

60 Notation: [IV] Interchanging both sides and subtracting y m from both sides of Eq. (28) yields (where 0 0 N 1 and γ mm = 1) γ 1m y 1 + γ 2m y γ Mm y M + α m x 0 +β 1m x 1 + β 2m x β Km x K + u m = 0 (29) All M equations yield the structural form: γ 11 y γ M1 y M + α 1 x 0 + β 11 x β K1 x K + u 1 = 0 γ 12 y γ M2 y M + α 2 x 0 + β 12 x β K2 x K + u 2 =. 0 γ 1M y γ MM y M + α M x 0 + β 1M x β KM x K + u M = 0 387

61 Assumption on the error-term vectors u 1,..., u M : u 1,..., u M satisfy all four #B-Assumptions We permit contemporary correlation among the error-term vectors of different equations (cf. Slide 333) formal representation: Cov(u m ) = E(u m u m ) = σ2 m I N (m = 1,..., M) E(u m u n) = σ mn I N (m n, m, n = 1,..., M) 388

62 Matrix representation: [I] We collect the M equations of the structural form from Slide 387 in matrix form We define the (N M) and (N K + 1) matrices Y [ y 1 y 2 y M ] = X [ x 0 x 1 x 2 x K ] = y 11 y 12 y 1M y 21 y 22 y 2M.... y N1 y N2 y NM 1 x 11 x 21 x K1 1 x 12 x 22 x K x 1N x 2N x KN 389

63 Matrix representation: [II] For Eq. m we define the (M 1) and (K + 1 1) parameter vectors (m = 1,..., M) γ m γ 1m γ 2m. γ Mm and β m structural form of the equation system: α m β 1m β 2m. β Km Yγ 1 + Xβ 1 + u 1 = 0 N 1 Yγ 2 + Xβ 2 + u 2 =. 0 N 1 (30) Yγ M + Xβ M + u M = 0 N 1 390

64 Matrix representation: [III] More compact representation of the structural form: Y [ γ 1 γ 2 γ M ] + X [ β1 β 2 β M ] + [ u 1 u 2 u M ] = 0 N M (31) (side-by-side collocation of the M equations from (30)) 391

65 Matrix representation: [V] More compact notation: Γ [ γ 1 γ 2 γ M ] = γ 11 γ 12 γ 1M γ 21 γ 22 γ 2M.... γ M1 γ M2 γ MM (32) B [ β 1 β 2 β M ] = α 1 α 2 α M β 11 β 12 β 1M β 21. β 22.. β 2M. β K1 β K2 β KM (33) 392

66 Matrix representation: [VI] u 11 u 12 u 1M U [ ] u u 1 u 2 u M = 21 u 22 u 2M.... u N1 u N2 u NM (34) most compact representation of the structural form: YΓ + XB + U = 0 N M (35) 393

67 Example: [I] Extended sales-quantity model (i = 1,..., N) a i = α + β 1 w i + β 2 p i + β 3 i + u i w i = γ + δ 1 a i + δ 2 q i + v i Matrices for the representation in the form (35): Y = a 1 w 1 a 2. w 2. a N w N, Γ = [ ] 1 δ1 β 1 1, 394

68 Example: [II] 1 p 1 1 q 1 1 p X = 2 2 q p N N q N, B = α γ β 2 0 β δ 2 u 1 v 1 u, U = 2 v 2.. u N v N (see Class) 395

69 Reduced Form Now: Rearrangement of the structural form (35) so that the endogenous variables are on the left-hand side we only have predetermined variables on the right-hand side 396

70 Structural form (35): with the YΓ + XB + U = 0 N M (N M) matrix Y (contains endogenous variables) (N K + 1) matrix X (contains predetermined variables) (M M) parameter matrix Γ (K + 1 M) parameter matrix B (N M) error-term matrix U 397

71 Note: The (M M) matrix Γ is quadratic and regular the inverse matrix Γ 1 exists Right-hand-side multiplication of the structural form (35) with Γ 1 and rearranging terms yields Y = XBΓ 1 UΓ 1 (36) 398

72 Notation: [I] We define the (K + 1 M) matrix Π BΓ 1 = π 01 π 02 π 0M π 11. π 12.. π 1M. π K1 π K2 π KM (Π is a pure parameter matrix) = [ π 1 π 2 π M ] We define the (N M)-matrix v 11 v 12 v 1M V UΓ 1 v = 21 v 22 v 2M.... v N1 v N2 v NM (V is the error-term matrix) = [ v 1 v 2 v M ] 399

73 Notation: [II] the form (36) can be written as Y = XΠ + V (37) or, expressed in terms of the M equations, as y 1 = Xπ 1 + v 1 (38) y 2 = Xπ 2 + v 2 (39). y M = Xπ M + v M (40) 400

74 Obviously: On the left-hand sides of the Eqs. (37), (38) (40) we only find endogenous variables On the right-hand side of the Eqs. (37), (38) (40) we only find predetermined variables and the error terms (37), (38) (40) constitutes the reduced form Remark: The reduced-form parameters of Equation m are included in the parameter vector π m 401

75 Estimation: The M equations (38) (40) can be estimated separately via OLS For Equation m (m = 1,..., M) the OLS estimator is given by π m = (X X) 1 X y m (41) 402

76 More compact representation: Side-by-side collocation of the M estimators from Eq. (41) yields [ π 1 π 2 π M ] = (X X) 1 X [ y 1 y 2 y M ] Defining Π [ π 1 π 2 π M ] we obtain the compact formula Π = (X X) 1 X Y (42) 403

77 Identification of an Equation Question: Is the Equation m of the structural system identified? Formal consideration: [I] Consider the following term from Slide 399: Π = BΓ 1 (43) where, according to the definitions given in the Eqs. (32) and (33) on Slide 392, B = [ β 1 β 2 β M ] and Γ = [ γ 1 γ 2 γ M ] 404

78 Formal consideration: [II] Right-hand-side multiplication of Eq. (43) with Γ yields Π [ γ 1 γ 2 γ M ] = [ β1 β 2 β M ] (M equations) the form of Eq. m is given by Πγ m = β m (44) Eq. (44) constitutes a relationship between the structural parameters of Eq. m (in γ m, β m ) and all reduced parameters (in Π) 405

79 Problem of identification: Is it possible to conclude from the known matrix Π on the unknown vectors γ m and β m? Recall: The order condition (see Definition 11.6, Slide 362) classifies Eq. m of the system as underidentified, exactly identified, overidentified on the basis of the variables numbers K total, K m, M m The order condition is not always exact Exact of identifiability: combination of a (simple) rank condition and the order condition 406

80 Simple rank condition: [I] We collect all structural parameters in the matrix [ Γ B ] = γ 11 γ 21 γ M1 α 1 β 11 β K1 γ 12. γ 22.. γ M2. α 2. β 12.. β K2. γ 1M γ 2M γ MM α M β 1M β KM Row m of the matrix [Γ B ] contains all structural parameters of Eq. m 407

81 Simple rank condition: [II] We genrate the matrix R from [Γ B ]: we consider row m of [Γ B ] we cancel all those columns of [Γ B ], for which there is a non-zero entry in row m we cancel row m of [Γ B ] this yields the (M 1) (K total K m + M M m )-matrix R The simple rank condition considers the rank of the matrix R [rank(r)] and compares rank(r) with M 1 408

82 Remarks: For the concept of the rank of a matrix, cf. Econometrics I, Section 2.3 Computation of rank(r) via appropriate software (e.g. EViews, Mathematica) Now: Combination of the simple rank condition and the order condition yields an exact identification rule 409

83 Definition 11.8: (Rank condition of identifiability) The Eq. m of a simultaneous system is (a) underidentified, if and only if rank(r) < M 1. (b) exactly identified, if and only if rank(r) = M 1 and K total K m = M m 1. (c) overidentified, if and only if rank(r) = M 1 and K total K m > M m

84 Indirect Estimation Assumption: Let Eq. m of the structural system be exactly identified Yγ m + Xβ m + u m = 0 N 1 411

85 Then: [I] We obtain the reduced form of the system Y = XΠ + V or, written as M single equations, y 1 = Xπ 1 + v 1 y 2 = Xπ 2 + v 2. y M = Xπ M + v M 412

86 Then: [II] OLS estimation yields estimates of all reduced parameters in the matrix Π Computation of the estimates of the structural parameter vectors γ m and β m from the estimates Π via Eq. (44) on Slide 405: Πγ m = β m 413

87 Remarks: The ILS method is consistent The ILS method does not provide estimates of underidentified and overidentified equations 414

88 Two-Stage Least Squares Estimation Assumption: Let Eq. m of the structural system be overidentified Yγ m + Xβ m + u m = 0 N 1 (45) 415

89 Preliminary notation: [I] We rearrange the matrix Y so that where Y m = [ y m Y (1) m Y (2) m ] y m : endogenous variables of Eq. m to be estimated Y (1) m : other endogenous variables contained in Eq. m Y (2) m : endogenous variables not contained in Eq. m 416

90 Preliminary notation: [II] Analogous rearrangement of the parameter vector γ m so that γ m = [ 1 γ (1) m 0 ] where 1 = γ mm : parameter for y m γ (1) m : parameters for endogenous variables in Y (1) m 0: (null)parameter for missing endogenous variables in m 417

91 Preliminary notation: [III] we write Eq. (45) on Slide 415 as [ y m Y m (1) Y (2) m ] 1 γ (1) m 0 + Xβ m + u m = 0 N 1 Multiplying terms and solving for y m, we obtain y m = Y (1) m γ (1) m + Xβ m + u m = [ Y (1) m X ] [ γ (1) m β m ] + u m (46) 418

92 Stage #1 of TSLS estimation: [I] Finding the optimal linear combination of the instrumental variables via OLS estimation of the reduced form (cf. Slide 372) y 1 = Xπ 1 + v 1 y 2 = Xπ 2 + v 2. y M = Xπ M + v M We collect the OLS estimates in the matrix Π = [ π 1 π 2 π M ] 419

93 Stage #1 of TSLS estimation: [II] We rearrange Π in an analogous way as we rearranged Y: where Π m = [ π m Π (1) m Π (2) m π m : estimates of the reduced form of Eq. m Π (1) m : estimates of the other endogenous variables included in Eq. m of the structural form Π (2) m : estimates of all endogenous variables not included in Eq. m of the structural form ] 420

94 Stage #1 of TSLS estimation: [III] Computation of the estimates of Y (1) m : Ŷ (1) m = X Π (1) m Stage #2 of TSLS estimation: [I] We replace Y (1) m obtain in the structural form (46) by Ŷ(1) m y m = [ Ŷ(1) m X ] [ γ (1) m β m ] and + u m (47) 421

95 Stage #2 of TSLS estimation: [II] OLS estimation of Eq. (47) yields the TSLS estimators of the parameters of Eq. m of the structural form (46): γ (1),TSLS m β TSLS m = [ [ ] [ Ŷ(1) m X Ŷ (1) m X ] ] 1 [ Ŷ(1) m X ] ym = Ŷ(1) m Ŷ(1) m Ŷ(1) m X Ŷ(1) m X X X 1 [ Ŷ(1) m y m X y m ] (48) Remark: The TSLS estimators given in Eq. (48) are consistent 422