A COMPANION TO Theoretical Econometrics

Latent Variable Models

Consider again the bivariate measurement error model (8.7), where the unobservable random variables En, є n, and vn are assumed to be mutually inde­pendent with expectation zero. The variables yn and xn are observable. Now let us assume that there is a third observable variable, zn, say, that is linearly related to E n in the same way as yn is:

Zn = lEn + Un, (8.18)

where un is independent of E n, є n, and vn, and has also mean zero. From this extended model, we obtain the following equations for the variances and covariances of the observable variables (the "covariance" equations):

о y = о EP2 + о2 cyx = о IP cyz = о EPy о 2x = о E + о v ох2 = о Ey

о Z = о Ey 2 + о2.

This system of six equations in six unknown parameters can be solved uniquely for the unknown parameters. Since the left-hand variables are consistently estim­ated by their sample counterparts consistent estimators for the parameters follow immediately. For example, the estimator of P is S = cyz/cxz = y'z/x'z, which is equivalent to the IV estimator with z as instrumental variable.

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A COMPANION TO Theoretical Econometrics

Normality tests

Let us now consider the fundamental problem of testing disturbance normality in the context of the linear regression model: Y = Xp + u, (23.12) where Y = (y1, ..., …

Univariate Forecasts

Univariate forecasts are made solely using past observations on the series being forecast. Even if economic theory suggests additional variables that should be useful in forecasting a particular variable, univariate …

Further Research on Cointegration

Although the discussion in the previous sections has been confined to the pos­sibility of cointegration arising from linear combinations of I(1) variables, the literature is currently proceeding in several interesting …

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