1 Outline. 1. Motivation. 2. SUR model. 3. Simultaneous equations. 4. Estimation

Transcription

1 1 Outline. 1. Motivation 2. SUR model 3. Simultaneous equations 4. Estimation

2 2 Motivation. In this chapter, we will study simultaneous systems of econometric equations. Systems of simultaneous equations can be derived from equilibrium models. In the text, the following very simple example is considered from a model of the macroeconomy: C t = α + α 1 Y t + α 2 r t + ε 1t I t = γ + γ 1 r t + ε 2t Y t = C t + I t + G t C t consumption, Y t income, I t investment (endogenous), r t interest, G t spending (exogenous).

3 The top two equations are behavioral equations and the bottom equation is an accounting identity. It is convenient to write this with the endogenous variables on the lhs and the exogenous variables on the rhs: C t = β C1 + β C2 G t + β C3 r t + v Ct I t = β I1 + β I2 G t + β I3 r t + v It Y t = β Y 1 + β Y 2 G t + β Y 3 r t + v Yt The β s will be functions of the α s that are somewhat tedious to derive, e.g. β C1 = α + α 1 γ 1 1 α 1, β C2 = α 1 1 α 1, β C3 = α 1γ 1 + α 2 1 α 1

4 The first set of equations is called the structural model and the second set of equations is called the reduced form representation of the model. One way to view the problem of identification is to ask whether the reduced form coefficients uniquely identify the structural parameters. We will begin our study of simultaneous equations with the seemingly unrelated regressions model. SUR could be used to estimate our reduced form model since SUR requires the endogenous variables to be excluded from the r.h.s. The error terms in the reduced form model will have a positive covariance because the reduced form v Ct, v It,v Yt will involve both ε 1t and ε 2t.

5 Then we will move to a models where endogenous variables are not necessarily excluded from other regression equations. 3 SUR Model We begin by consider a system of several individual regression equations, like the reduced form of our simple macro model above. There are j =1,...,J equations and t =1,...,T time periods: y tj = X t β j + ε tj

6 We shall assume that the variance has the following structure: Var[y j X] = w jj I T Cov[y i,y j X] = w ij I T For a single equation, the error terms are spherically distributed. However, we allow for the covariance to be positive across observations within a single time period. No dependence between observations across time periods.

7 Note that since we have assumed away the possibility of time series dependence in our data, the variances and covariances can be summarized by the J by J matrix: Ω =[w ij ; i, j =1,...,J] Define y v =[y 1,y 2,...,y J ] whereweabusenotation that y j =[y tj, t=1,...,t] Define X V similarly to result in conformable multiplication:

8 E[y V X] = Xβ 1 Xβ 2 Xβ 3. Xβ J X V = diag(x; j =1,...,J)= β = β 1 β 2 β 3. β J = X V β X X.. X X Given that the endogenous y j s only appear on the lhs of our equations, we could estimate our model using only ols.

9 b β OLS = ³ X V X V 1 X V y V However, this would exploit a possible source of efficiency, namely that the covariances between our equations are positive. Therefore, a GLS procedure will lead to more efficient estimates and give us the correct matrix of standard errors for hypothesis testing, etc... The variance matrix in our model can be written as: Var[y v X V ] = Ω I T w 11 I T w 12 I T... w 1J I T w = 21 I T w 22 I T... w 2J I T.... w J1 I T w J2 I T... w JJ I T

10 In the above, denotes a Kronecker product. As is standard in GLS, we want to reweight our observations using the square root of our variance matrix. This leads to the GLS formula: bβ GLS = ³ X V (Ω I) 1 X V 1 X V (Ω I) 1 y V Inthiscase,itturnsoutthatGLSandOLSare equivalent! To prove this, we use properties of Kronecker products. For comformable matrices, it turns out that:

11 (A B) 1 = A 1 B 1 (A B)(C D) = (AC BD) Therefore, it follows that: (Ω I T ) 1 X v = ³ Ω 1 I T (IJ X) = ³ Ω 1 X = (I J X) ³ Ω 1 X = X v (Ω I T ) 1 Next, applying this result to our GLS estimator implies that:

12 bβ GLS = ³ X V (Ω I) 1 X V 1 X V (Ω I) 1 y V = ³ (Ω I) 1 XV X 1 V (Ω I) 1 XV y V = ³ XV X 1 V (Ω I)(Ω I) 1 XV y V = ³ XV X 1 V X V y V = β b OLS However, it turns out that this equivalence fails in the important case of linear restrictions imposes on the β s. For example, it can be shown that certain linear restrictions are imposed in the simple macro model discussed earlier. In this case, there would be an increase in efficiency from a FGLS procedure.

13 The analysis of this model is similar to the analysis of OLS with linear restrictions. 4 Simultaneous Equations. IntheSURmodel,therewerenoendogenousvariablesontherhsofourregressions. We will write our more general simultaneous equations model as: ytγ + x tb = ε t Γ =[γ ij ]andb =[β kj ] (1 J) (J J)+(1 K) (K J) =(1 J)

14 In the above, y t denotes the endogenous variables, x t the exogenous variables, and the equations come from behavioral relationships or identities. The simple macro model we considered at the start of this lecture can obviously be formed in this way. If Γ is a nonsingular matrix, then we can write: yt =( x tb + ε t)γ 1 yt = x tb Γ 1 + ε tγ 1 = x tπ + v t Thus, we could view the reduced form of our system as a SUR model.

15 We will estimate the model using a GMM strategy. We will make the following assumptions: 1. The (ε t,x t ) t =1,..., T are iid. 2. E(ε t x t )= 3. Γ is nonsingular. There are JK orthogonality conditions for GMM estimation. E x t ε t = E xt E ε t x t = by iterated expectations and orthogonality. Similar to the theory of ols, a LLN implies that:

16 E T h xt ε ti p (K J) WeshallalsoassumethatE T xt x t p D = E[x t x t ] Hence it follows that: E T h xt ε t i = E T h xt ³ y t Γ + x tb i h = E T xt (ε t x tb )Γ 1 Γ + x i tb since yt = ( x tb + ε t)γ 1 hh = E T xt ε i t Γ 1 Γ + x t x t (ΠΓ + B ) i p D (ΠΓ + B )= If there is a unique solution in terms of B and Γ to the above equations, then we are identified.

17 If the matrix D is nonsingular, this boils down to proving that ΠΓ +B = is uniquely determined by Γ and B (recall that Π = B Γ 1 ) A first problem for identification is going to be that if we multiply everything through by a constant, α (ΠΓ + B )=. Thus it is customary to consider problems where γ ij =1forsomecoefficient in our Γ matrix. Another problem is that we have J 2 + JK free parameters in Γ and B but we have only JK moment equations Thus we have violated the order condition for GMM estimation.

18 It is customary to impose exclusion restrictions in our model. An example is to exclude some demand shifter from the supply equation and vise versa. In algebraic examples, by shifting the supply equation, we could learn the shape of the demand equation. Note that Porter excludes Great Lakes open dummy from his supply equation for example. Next, we describe conditions required for the identification of the parameters of equation j. Let s make K + J M j restrictions on our parameters, e.g.

19 R γj γ j + R βj β j = r j ³³ K + J Mj J (J 1)+ ³³ K + J Mj K (K 1) = ³³ K + J Mj 1 Combining this with our moment restrictions on the parameters, we have that: " Π I K #" γj # " # R γj R βj β j = r j The dimension of the first matrix is (2K + J M j ) (K + J) Obviously, it is only possible to solve this system if there are more rows (restrictions) than columns (free parameters), that is:

20 2K + J M j K + J K M j As in our study of GMM, this is an order condition. Similar to the study of GMM, we must put in a rank condition that rules out colinearity between our restrictions. By algebra, we can rewrite the first matrix in the above equation as: " " Π I K R γj R βj B # = I K R γj Γ + R βj B " IK I J #" Γ 1 I K R γj # #

21 The two outer matrices on the rhs of the above equation have full rank. The last K columns obviously have rank K. We need the first set of columns to have rank J. This happens if and only if rank(r γj Γ +R βj B )= J. The rank condition can be generalized beyond the identification of just a single equation to the identification of multiple equations. This theorem is similar to the one described above andisstatedinthetext.

22 5 Estimation. One approach to estimation would be ols estimation of the reduced form and then translate back to the structural model. However, this approach is not efficient and would be problematic if the model was overidentified. The theory of estimation is fairly standard and fits into our GMM framework. A first alternative estimator would be to estimate each equation, one at a time. For example, we might estimate our demand equation using 2SLS.

23 An instrument for price would be a supply shifter. More generally, we can use the exogenous variables that enter into all the equations as potential instruments. This is called a limited information approach. We could then use our first stage 2SLS estimates to construct an estimate of the variance matrix if we rule out autocorrelation and heteroskedasticity across observations. This is called 3SLS. Maximum likelihood is also an option if we are willing to specify a parametric distribution on the error terms.

24 6 Application Revisted. The econometric model can be summarized as: By t = Γx t + I t + U t Ã! 1 log Qt y t = x log p t = L t t S t Ã! 1 α1 B =, = β 1 1 Ã! α α and Γ = 2 β β 2 u t = à β 3! à u1t u 2t! Error terms are normally distributed, no serial dependence or heteroskedasticity. 2by2VariancematrixΣ for joint distribution of à u1t u 2t!.

25 Note that we have made a number of exclusion restrictions which is required to identify the model. Conditional on I t, the likelihood function for y t is: h(y t I t ) = (2π) 1 Σ 1/2 exp( 1/2(By t Γx t + I t ) Σ 1 (By t Γx t + I t )) A problem for maximization is that I t is unobserved. This is called a switching regression model. In order to deal with this, the EM algorithm is required (we will talk about this later in the course).

26 The basic idea is that we start out with an initial set of probabilities for I t =1,callitw 1,...,w T (say from the railway news) We then obtain an estimate of λ = E T wt sample frequency of the w s. as the This is the E (expectation) step of the EM algorithm. Next, we maximize the likelihood function given w 1,..., w T. Call our MLE (B, Γ, ) This is the M (maximization) step of the EM algorithm.

27 We then apply Bayes Theorem to update our values of w, i.e. w 1 t = λ h(y t I t =1,B, Γ, ) λ h(y t I t =1,B, Γ, )+ ³ 1 λ h(y t I t =,B, Γ, ) Iterate this procedure until convergence is acheived. A likelihood ratio test is done to ask whether =, which is soundly rejected.