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Unformatted text preview: Chapter 9 Simultaneous equations estimation Overview Until this point the analysis has been confined to the fitting of a single regression equation on its own. In practice, most economic relationships interact with others in a system of simultaneous equations, and when this is the case the application of ordinary least squares (OLS) to a single relationship in isolation yields biased estimates. Having defined what is meant by an endogenous variable, an exogenous variable, a structural equation, and a reduced form equation, the first objective of this chapter is to demonstrate this. The second is to show how it may be possible to use instrumental variables (IV) estimation, with exogenous variables acting as instruments for endogenous ones, to obtain consistent estimates of the coefficients of a relationship. The conditions for exact identification, underidentification, and overidentification are discussed. In the case of overidentification, it is shown how two-stage least squares can be used to obtain estimates that are more efficient than those obtained with simple IV estimation. The chapter concludes with a discussion of the problem of unobserved heterogeneity and the use of the Durbin–Wu–Hausman test in the context of simultaneous equations estimation. Further material Note : This item relates to the fitting of a simultaneous equations model using time series data. You should therefore skip it when you first read this chapter and return to it when you have read Chapter 11 of the text. Simultaneous equations estimation in time series models Time series models very often consist of a set of simultaneous relationships, macroeconomic models and market- clearing models being obvious examples. One difference in the fitting of simultaneous equations models with time series data, by contrast with cross-sectional data, is the possibility of using lagged endogenous variables as instruments. For instance, consider the model Y t = β 1 + β 2 X t + β 3 Y t –1 + u t X t = α 1 + α 2 Y t + v t Both X t and Y t are endogenous, and yet the second equation is identified because Y t –1 can act as an instrument for Y t , provided that the disturbance terms u t and v t satisfy the regression model assumptions. Y t –1 is described as a predetermined variable because at time t it is fixed and therefore not correlated with v t , despite not being strictly exogenous. It is important that the regression model assumptions are satisfied and, in particular, that v t is not subject to autocorrelation. If v t were subject to autocorrelation, Y t –1 would not be independent of it. Y t –1 is partly determined by X t –1 in the first equation. X t –1 is partly determined by v t –1 in the second. Hence Y t –1 is partly determined by v t –1 . If v t is subject to autocorrelation, it will also be partly determined by v t –1 , and so Y t –1 will be correlated with it....
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- Spring '10