Simultaneous. Equation Model. Estimators: Statistical. Properties and. Practical Implications
1 The Linear Simultaneous Equations Model
This chapter deals with the statistical properties of estimators in simultaneous equation models. The discussion covers material that extends from the standard large sample asymptotics and the early works on finite-sample analysis to recent work on exact small-sample properties of IV (instrumental variable) estimators and the behavior of IV estimators when instruments are weak. This section introduces the linear simultaneous equations model and the notation for the rest of the chapter. Sections 2 and 3 then cover limited information and full information estimators of structural parameters. Section 4 moves on to large sample properties of the estimators while Sections 5 and 6 summarize results obtained in finite sample analysis. Section 7 ends the chapter with a summary of practical implications of the finite sample results and alternative asymptotics that involve increasing the number of instruments or reducing the correlation between instruments and endogenous regressors in instrumental variable estimation.
Consider the classical linear simultaneous equations model (SEM) of the form:
Byt + Txt = ut; t = 1, 2,…, T or YB’ + XГ = U, (6.1)
B = G x G matrix of fixed parameters (some of which are unknown),
Г = G x K matrix of fixed parameters (some unknown), yt = G x 1 vector of observations on endogenous variables at "time" t, xt = K x 1 vector of observations on exogenous variables at "time" t, ut = G x 1 vector of structural disturbances in "time" t,
Y = T x G matrix whose tth row is y’t,
X = T x K matrix whose tth row is x’t,
U = T x G matrix whose tth row is u’t,
T = sample size.
The system described in (6.1) consists of G linear equations. Each equation is linear in the components of yt, with a particular row of B and Г containing the coefficients of an equation in the system. Each equation may be stochastic or nonstochastic depending on whether or not the corresponding component of the disturbance vector ut has a nondegenerate probability distribution.
The following assumptions together with (6.1) comprise the classical linear simultaneous equations model:
A1. B is nonsingular. Thus, the model is complete in the sense that we can solve for yt in terms of xt and ut.
A2. X is exogenous. That is, X and U are independently distributed of each other.
A3. X is of full column rank with probability 1.
A4. The disturbances ut, for t = 1, 2,…, T, are uncorrelated and identically distributed with mean zero and positive definite covariance matrix X.
At certain times, we will replace A4 with the stronger assumption,
A4′. The disturbances ut are independent and identically distributed as multivariate normal with mean zero and covariance matrix X.
We refer to the G equations in (6.1) as being structural in that they comprise a simultaneous system which explains the mutual interdependence among G endogenous variables and their relationship to K exogenous variables whose behavior, by assumption A2, is in turn explained by factors outside the system.
This model differs from the standard multivariate linear regression model in the statistics literature to the extent that B is generally not diagonal and Y and U are correlated. Under these conditions, the structural equations are such that, after normalization, some explanatory or right-hand side variables are correlated with the disturbance terms. (One can thus claim that the canonical equivalence is between this structural system and a multivariate linear regression model with measurement errors.)
The structural system in (6.1) also leads to a multivariate linear regression system with coefficient restrictions. Premultiplying (6.1) by B-1, we get the so-called reduced form equations of the model:
yt = – В^Гх t + B-1u t = П x t + vt or Y = – X ГБ’-1 + UB’-1 = X П’ + У (6.2)
where П = – В-1Г, vt = B-1u t, and У = UB’-1.
It follows from A2 and (6.2) that X and У are independently distributed of each other and that, for Q = B^ZB’-1, vt ~ uncorrelated (0, Q), if A4 holds; and vt ~ iid N(0, Q) if A4′ holds. Thus, yt | X ~ uncorrelated (Пxt, Q) under A4 and yt | X ~ independent N(n x t, Q) under A4′.
The standard literature on identification of simultaneous equations models shows that identification of (6.1) through prior restrictions on the structural parameters (B, Г, Z) generally will imply restrictions on the matrix П of reduced form coefficients. For more details on identification of simultaneous equations models, see Hsiao (1983), Bekker and Dijkstra (1990), and Bekker and Wansbeek (chapter 7) in this volume.