Topic 1 Simultaneous Equations I: Introduction

Transcription

1 ECONOMETRICS II Topic 1 Simultaneous Equations I: Introduction These slides are copyright 2010 by Tavis Barr. This work is licensed under a Creative Commons Attribution- ShareAlike 3.0 Unported License. See for further information.

2 Topic Outline Introduction Systems of Exogenous Regressors Seemingly Unrelated Regressions Singular Systems Identification Through a Likelihood Function Identification of a Linear System The Rank and Order Conditions Identification Through Error Term Restrictions

3 Introduction So far, we have dealt with systems of the form y = f(x) + ε Here, y is a vector of observations of one variable, whereas X is a matrix of observations of several variables Mostly, this has taken the linear form y = Xβ + ε

4 Introduction In this topic, we will consider systems of the form f m (Y,X,ε m ) = 0, for m = {1,2,...M} Here, Y is a matrix of observations of several variables (M in number) just like X (still k in number) In general, we will have M equations describing the relationship between X and Y

5 Introduction In this topic, we will consider systems of the form f m (Y,X,ε m ) = 0, for m = {1,2,...M} Mostly, this will take the linear form YΓ = XΒ + ε

6 Introduction When we write the system equations in matrix format, we assume that they are stacked and sorted by equation, i.e., X, Y, 1 =0 f1 f 2 X, Y, 2 f M X, Y, M =0

7 Introduction In this topic, we will consider systems of the form f m (Y,X,ε m ) = 0, for m = {1,2,...M} Calling Y endogenous and X exogenous is essentially saying that y m is stochastic given X and Y ~m We will consider ε m as the stochastic component that determines y m (or some deterministic function of it) given the other variables

8 Systems of Exogenous Regressors Suppose we have a system of the form: 1 y1=x1 1 y 2 =X 2 2 M 2 y M =X M M y does not appear on the right-hand side The variables X may or may not be the same in each equation

9 Systems of Exogenous Regressors Suppose we have a system of the form: 1 y1=x1 1 y 2 =X 2 2 M 2 y M =X M M ε has covariance Σ I N, Σ is unrestricted Such a system is called a system of seemingly unrelated regressions

10 Systems of Exogenous Regressors Suppose we have a system of the form: Example: y1=x1 1 1 y 2 =X 2 2 M 2 where y M =X M M E =0 E ' = I N y is industry output for M industries X includes usage of labor, capital, energy, etc. We have N time periods for each industry

11 Systems of Exogenous Regressors Suppose we have a system of the form: y1=x1 1 1 y 2 =X 2 2 M 2 where y M =X M M E =0 E ' = I N Within each equation, the model does not violate the Gauss-Markov assumptions Therefore, OLS works fine

12 Systems of Exogenous Regressors Suppose we have a system of the form: y1=x1 1 1 y 2 =X 2 2 M 2 where y M =X M M E =0 E ' = I N Suppose we want to try to improve on OLS by using a GLS estimator: = X ' I N 1 X 1 X ' I N 1 y

13 Systems of Exogenous Regressors Suppose we want to try to improve on OLS by using a GLS estimator If the exogenous variables in all equations are the same, then this reduces to: [ 11 X ' X 1 12 X ' X 1 1M 1][ X ' X X m=1 21 X ' X 1 22 X ' X 1 2M X ' X M X ' m=1 M1 X ' X 1 M2 X ' X 1 MM X ' M X X ' m=1 ' M 1 1m y m m] 1 2m y m 1 Mm y

14 Systems of Exogenous Regressors If the exogenous variables in all equations are the same, GLS reduces to: [ 11 X ' X 1 12 X ' X 1 1M 1][ X ' X X m=1 21 X ' X 1 22 X ' X 1 2M X ' X M X ' m=1 M1 X ' X 1 M2 X ' X 1 MM X ' M X X ' m=1 Consider the top row of the product: M l=1 1l X ' X 1 M m=1 lm 1 X ' y m ' M 1 1m y m m] 1 2m y m 1 Mm y

15 Systems of Exogenous Regressors Consider the top row of the product: M l=1 1l X ' X 1 M = X ' X 1 X ' l=1 M m=1 M 1l m=1 = X ' X 1 X ' y 1 lm 1 X ' y m lm 1 y m In other words, it's identical to equation-byequation OLS

16 Systems of Exogenous Regressors If exogenous variables are the same in all equations, SUR is identical to equation-byequation OLS If error terms are unrelated across equations, this is obviously also true Evidence suggests that there may be improvements when exogenous variables are less related across equations

17 Systems of Exogenous Regressors Sometimes we have a system of equations where the regressors have to add up to a constant across the equations If a coefficient each equation is a cost share for one input, cost shares have to add up to one Similarly, if coefficient in each equation represents consumer expenditure shares on each good, expenditure shares must sum to one Such a system of equations is called a singular system because Σ will be singular

18 Systems of Exogenous Regressors In a system of equations where β is constrained, Σ will be singular Example: Translog Cost Function We have cost C(Y,p) = Σ m=1,m p m x(y,p) Shepard's Lemma tells us x i = C/ p i In logs, this derivative is C/ p i = p i x i / C= s i If we have CRS, then this is invariant to Y Suppose we take a Taylor series approximation of C(p), in logs, around p=0

19 Systems of Exogenous Regressors Example: Translog Cost Function Shepard's Lemma says lnc/ lnp i = p i x i / C= s i Suppose we take a Taylor series approximation of C(p), in logs, around p=0: M lnc ln m=1 C ln p m lnp m 1 2 l=1 M M m=1 Here the RHS is a total derivative lnc/ lnp i 2 lnc ln p l ln p ln p ln p l m m We can write this in coefficients as: M M M ln C /Y = m ln p 1 m lm m=1 l=1 m=1 2 ln p ln p l m

20 Systems of Exogenous Regressors Example: Translog Cost Function Shepard's Lemma says lnc/ lnp i = p i x i / C= s i We approximate the total derivative lnc/ lnp i : M ln C /Y = m=1 M M m ln p m l=1 m=1 lm 1 2 ln p l ln p m Another way of stating this: M M s m = m ln p m l=1 m=1 lm 1 2 ln p l This gives us M equations of cost shares

21 Systems of Exogenous Regressors Example: Translog Cost Function Another way of writing the cost function: M M s m = m ln p m l=1 m=1 lm 1 2 ln p l But if these are really cost shares, then: M m=1 M m =1 ; m=1 M lm =0 ; m=1 ml =0 Therefore, system is constrained and covariance matrix is singular

22 Systems of Exogenous Regressors How to solve a singular system? Estimate M-1 equations, infer the last equation's coefficients from other equations Better to do by ML; then we can substitute M 1 M =1 m m=1 directly into the likelihood function for the Mth equation, and we don't have to drop it

23 Systems of Exogenous Regressors The bottom line: Having multiple equations does not in itself create problems for OLS There may be interesting challenges from cross-equation restrictions What makes the remaining material in this topic different is not multiple equations, but multiple endogenous variables

24 Mostly, we will study a system of equations in its linear form YΓ = XΒ + ε We assume that both Β and Γ are nonsingular This implies that there is a unique transformation Γ -1 such that Y = -XΒΓ -1 +εγ -1

25 With a linear system, there will be a unique transformation Γ -1 : Y = -XΒΓ -1 + εγ -1 In other words, there is a unique error term εγ -1 that determines y m given X and Y ~m Therefore, a linear model (and many nonlinear models as well) requires that there are as many equations as endogenous variables

26 Remember the general definition of identification: A model with parameters θ is identified if, given a empirical relationship with a known unique solution f(x, y,β) = 0, there is a one-to-one function g such that θ = g(β) It is underidentified if there exists more than one such g for some admissible values of β

27 Remember the general definition of identification: A model with parameters θ is identified if, given a empirical relationship with a known unique solution f(x, y,β) = 0, there is a one-to-one function g such that θ = g(β) It is overidentified if there exists no such g for some admissible values of β

28 Here, with a system of multiple equations, we would say it is identified if a one-to-one function g: θ = g(β) exists given the system of equations with a known unique solution f m (Y,X,ε m ) = 0, for m = {1,2,...M}

29 Here, with a system of multiple equations, we would say it is identified if a one-to-one function g: c = g(β) exists given the system of equations with a known unique solution f m (Y,X,ε m ) = 0, for m = {1,2,...M} Example: If f is linear, then we have the reduced form y m = XΠ m + ω m, m = {1,2,...M}

30 Example: If f is linear, then we have the reduced form y m = XΠ m + ω m, m = {1,2,...M} The question then becomes: Is there a unique transformation g such that g(y m ) = g(xπ m ) + g(ω m ) Obviously, without further restrictions on g, it is not unique

31 An example of an unidentified model: y 1 = y 2 β 1 x 1 β 2 ε 1 y 2 = x 1 β 3 ε 2 In this case, the first equation would show us: y 1 = x 1 β 3 ε 2 β 1 x 1 β 2 ε 1

32 An example of an unidentified model: y 1 = y 2 β 1 x 1 β 2 ε 1 y 2 = x 1 β 3 ε 2 which implies y 1 = x 1 β 3 ε 2 β 1 x 1 β 2 ε 1 The fact that y 1 contains x 1 is not a problem But y 2 includes ε 2, which might be correlated with ε 1

33 An example of an unidentified model: y 1 = y 2 β 1 x 1 β 2 ε 1 y 2 = x 1 β 3 ε 2 which implies y 1 = x 1 β 3 ε 2 β 1 x 1 β 2 ε 1 Here, y 2 includes ε 2, which might be correlated with ε 1 In general, identifying a system means distinguishing the error processes that generate each variable

34 Another example of an unidentified model: q = β 0 p β 1 ε 1 q = β 2 p β 3 ε 2 This is meant to represent a supply and demand system, with one equation representing supply and the other representing demand Using OLS, both equations will result in the same coefficients

35 Another example of an unidentified model: q = β 0 p β 1 ε 1 q = β 2 p β 3 ε 2 Using OLS, both equations will result in the same coefficients To identify which is which, we need to separate the supply shocks from the demand shocks

36 Identification via the likelihood function is more straight-forward to understand, though usually more difficult to implement Let f m (Y,X,ε m ) = 0, for m = {1,2,...M} be a system of equations Let L(θ;Y,X) be the likelihood function for the parameters of the model Then, the system is identified if, at the maximum likelihood L *, there is a unique θ such that L(θ;Y,X) = L *

37 Let L(θ;Y,X) be the likelihood function for the parameters of the model Then, the system is identified if, at the maximum likelihood L *, there is a unique θ such that L(θ;Y,X) = L * This generalizes to any pseudo-likelihood function L whereby the maximum of L(θ;Y,X) over possible values of θ is a consistent estimator of θ

38 A system is identified if, at the maximum likelihood L *, there is a unique θ such that L(θ;Y,X) = L * Notice that the likelihood function, given Y and X, is derived from the probability of observing the error terms given the parameters of the model So this requirement in one sense states that the error terms, as specified by the likelihood function, are clarified enough to identify one model from the other

39 Identification of a linear system: We have a non-identified reduced form model: y m = XΠ m + ω m We then have a (hopefully) identified, structural form model: y m = YΓ + XΒ + ε m

40 Identification of a linear system: We then have a (hopefully) identified, structural form model: y m = YΓ + XΒ + ε m We will sometimes use the equivalent model YΓ * = XΒ + ε m where Γ * = I - Γ

41 Identification of a linear system: We are attempting to transform an observable reduced form model into an identified structural form model: y m = XΠ m + ω m Y m Γ * m = X m Β m + ε m This implies that there must be a unique transformation Γ from the reduced form to the structural form such that Β m = ΠΓ * m

42 We are attempting to transform an observable reduced form model into an identified structural form model: y m = XΠ m + ω m y m = YΓ + XΒ + ε m Note the implicit assumption that γ mm = 0 and that y m appears with a coefficient of 1 in the mth equation We can dispense with this assumption, but not without imposing an equivalently strong normalization to replace it

43 We need a unique transformation Γ that gets us from the reduced form to the structural form: y m = XΠ m + ω m y m = YΓ + XΒ + ε m This is an M x M matrix with M(M-1) unknown terms We can identify it either through restrictions on Γ and Β, or indirectly through restrictions on Σ = ΓΩΓ, the structural covariance matrix

44 Identification of a linear system through exclusion (γ mo, β mj = 0 ): We need a unique transformation Γ that gets us from the reduced form to the structural form y m = XΠ m + ω m y m = YΓ + XΒ + ε m or YΓ * + XΠΓ * + ωγ * Consider the mth row of Γ. Suppose we declare that for a set of M * m variables y* m, the coefficient γ mo = 0

45 Identification of a linear system through exclusion (γ mo, β mj = 0 ): We need a unique transformation Γ that gets us from the reduced form to the structural form y m = XΠ m + ω m y m = YΓ + XΒ + ε m or y m = YΓ * + XΠΓ * + ωγ * Declaring β mj = 0 helps by creating restrictions on linear combinations of elements of Π How many exclusions do we need to identify the model?

46 Identification of a linear system through exclusion (γ mo, β mj = 0 ): Some notation: Label Description No. Y m Endogenous variables included M m in the mth equation Y * m Endogenous variables excluded M* m from the mth equation X m Exogenous variables included k m in the mth equation X * m Endogenous variables excluded k* m from the mth equation

47 Consider the matrix cross-product ΠΓ * for the mth equation when the coefficient on the jth exogenous variable is set to zero: * γ m o o: y 0 Y m π o j =β m j =0 Here each oth element of Γ * is being m multiplied by the jth variable in each oth equation; the sum equals the coefficient of the jth variable in Β m

48 Consider the matrix cross-product ΠΓ * for the mth equation when the coefficient on the jth exogenous variable is set to zero: * γ m o o: y 0 Y m π o j =β m j =0 This is an equation with M m unknowns, one for each non-excluded endogenous variable To solve for these unknowns, we need to exclude M m exogenous variables from the mth equation

49 Consider the matrix cross-product ΠΓ m, partitioned between excluded and nonexcluded exogenous variables: Π :x X j j m * Π j :x j X m = Γ * m B j 0 The bottom half of the system has k * m equations and M m unknowns Once this system of equations is solved for Γ * m, we can use the top half to derive Β m

50 Consider the matrix cross-product ΠΓ m, partitioned between excluded and nonexcluded exogenous variables: Π :x X j j m * Π j :x j X m = Γ * m B j 0 First condition for identification: k* m M m, i.e., the number of exogenous variables excluded is at least as great as the number of endogenous variables included This is known as the order condition

51 Consider the matrix cross-product ΠΓ m, partitioned between excluded and nonexcluded exogenous variables: Π :x X j j m * Π j :x j X m = Γ * m B j 0 Second condition for identification: The portion of Π corresponding to variables in X * has to have full rank m This is known as the rank condition

52 Considering just exclusion restrictions, we have the following three cases: 1. k* m < M m or the rank condition is not met. The equation is unidentified and the structural form cannot be estimated. 2. k* m = M m and the rank condition is met. The equation is exactly identified. 3.k* m > M m and the rank condition is met. The equation is overidentified and some of the restrictions can be tested.

53 We have just seen how a linear system can be identified based on exclusions from Β and Γ Obviously, linear restrictions such as β mj = 1 will have the same effect since they turn β mj into a known element and provide an equation in Γ * m We can also use non-linear restrictions; in this case, the rank and order conditions do not necessarily apply since we don t necessarily need M equations to solve for M unknowns

54 Additionally, we can identify based on restrictions on Σ Remember that Σ = Γ* ΩΓ * and that Ω is estimable A restriction on an element of Σ therefore acts as an equation in Γ * : σ ij = Γ * i Ω i Γ* j

55 Example: A triangular system. Suppose that: y1 y 2 y X y X y 1 32 y X y M = 0 M1 y 1 M2 y 2 M3 y M X M = M Note that this system is not identified because there are no excluded exogenous variables

56 Triangular systems appear in macroeconomics from time to time. Someone will argue that some variables are not structurally affected by others Example: A money (m) demand equation, as a function of inflation (π) and interest rates (r): t 1 r t m t t = 0 21 t 1 31 t 1 X 1 2 X 2 32 m t 3 X 3 41 t 1 42 m t 43 r t 1 4 X = 4

57 Example: A triangular system. Suppose that y1 y 2 y X1 21 y X y 1 32 y X y M = 0 M1 y 1 M2 y 2 M3 y M X M = 1 M Now, suppose that we declare that Σ is diagonal Consider a transformation of the above system by a matrix F into another admissible system It must be that F is upper triangular, since it has to produce 0's above the diagonal for the y's

58 Example: A triangular system. Suppose that y1 y 2 y X1 21 y X y 1 32 y X y M = 0 M1 y 1 M2 y 2 M3 y M X M = 1 M Now, the transformed cross-equation covariance matrix, Σ, also has to be diagonal Because Σij =0 for i j, and Σ 12 =F 2 'ΣF 1 = σ 11 f 12 = 0, it must be that f 12 = 0

59 Example: A triangular system. Suppose that y1 y 2 y X1 21 y X y 1 32 y X 3 0 y M = 0 M1 y 1 M2 y 2 M3 y M X M = 1 M 3 Now, the transformed cross-equation covariance matrix, Σ, also has to be diagonal Recursively, we can show that if F j is diagonal, then F j+1 is also diagonal

60 Example: A triangular system. Suppose that y1 y 2 y X1 21 y X y 1 32 y X 3 0 y M = 0 M1 y 1 M2 y 2 M3 y M X M = 1 M 3 Recursively, we can show that if F j is diagonal, then F j+1 is also diagonal But the diagonal elements have to be one to produce y m by itself in the mth equation, so F must be the identity matrix

61 Another way to think about identification: There are M m endogenous variables in the mth equation, each of which is potentially correlated with the error term For each one, we need an instrumental variable that is correlated with the endogenous variable, but not correlated with the error term By excluding a variable from each equation, we are assuming it is uncorrelated with the error term

62 Another way to think about identification: There are M m endogenous variables in the mth equation, each of which is potentially correlated with the error term For each one, we need an instrumental variable that is correlated with the endogenous variable, but not correlated with the error term The rank condition implies that the excluded variables will be correlated with the endogenous variables

63 Back to the supply/demand example: Suppose we are looking at the foreign demand for Ethiopian coffee We hypothesize that the supply is affected by the weather in Kaffa (w), but the demand is not: q s = β 11 β 12 p β 13 w ε 1 q d = β 21 β 22 p ε 2 In this case, the demand equation is identified, but the supply equation is not We are effectively using the weather as an instrumental variable for the price