By T. Wansbeek, E. Meijer
The e-book first discusses extensive numerous facets of the well known inconsistency that arises while explanatory variables in a linear regression version are measured with mistakes. regardless of this inconsistency, the zone the place the real regression coeffecients lies can occasionally be characterised in an invaluable method, specifically whilst bounds are recognized at the dimension blunders variance but in addition while such details is absent. salary discrimination with imperfect productiveness size is mentioned as a big specified case.
Next, it really is proven that the inconsistency isn't unintentional yet primary. because of an identity challenge, no constant estimators might exist in any respect. more information is fascinating. this knowledge will be of assorted forms. One sort is detailed past wisdom approximately capabilities of the parameters. This results in the CALS estimator. one other significant style is within the type of instrumental variables. Many elements of this are mentioned, together with heteroskedasticity, mixture of knowledge from diversified resources, development of tools from the to be had info, and the LIML estimator, that is particularly suitable whilst the tools are weak.
The scope is then widened to an embedding of the regression equation with dimension blunders in a a number of equations atmosphere, resulting in the exploratory issue research (EFA) version. This marks the step from size mistakes to latent variables. Estimation of the EFA version ends up in an eigenvalue challenge. numerous versions is reviewed that contain eignevalue difficulties as their universal characteristic.
EFA is prolonged to confirmatory issue research (CFA) by way of together with regulations at the parameters of the issue research version, and subsequent by means of bearing on the standards to history variables.
These versions are all structural equation types (SEMs), a really basic and demanding category of versions, with the LISREL version as its best-known illustration, encompassing just about all linear equation platforms with latent variables.
Estimation of SEMs should be considered as an software of the generalized approach to moments (GMM). GMM usually and for SEM specifically is mentioned at nice size, together with the generality of GMM, optimum weighting, conditional moments, non-stop updating, simulation estimation, the hyperlink with the strategy of utmost probability, and particularly trying out and version overview for GMM.
The dialogue concludes with nonlinear versions. The emphasis is on polynomial types and types which are nonlinear because of a clear out at the based variables, like discrete selection versions or versions with ordered express variables.
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Extra info for Measurement Error and Latent Variables in Econometrics (Advanced Textbooks in Economics)
18b). The 48 3. 18b) is projected onto the space spanned by V*. The result becomes more insightful when we consider the case where £2 > 0, which implies that there is measurement error in all variables. 17) holds trivially and the first part reduces to or, equivalently, This is an ellipsoid with midpoint \(K 4- K*), passing through K and K* and tangent to the hyperplane (ft — K}'J^XK = 0. 2. 11) on the measurement error variance, the admissible region for ft would be the area between the two parallel hyperplanes, cf.
13) we find Thus, when measurement error is neglected, the explanatory power of the model, as measured conventionally by R2, is underestimated. When the disturbance term is assumed to be normally distributed, the hypothesis ft = 0 can be tested by means of the commonly employed F-statistic. It has a close relationship with R2, In particular, F is a monotonically increasing transformation of R2 and hence it is also biased towards zero. As a result, the null hypothesis will be accepted too often and hence the hypothesis ft = 0 will not be rejected often enough.
The observable variable is a sum of two independent stationary AR(1) processes. /(I - <£2). Clearly, this estimator is inconsistent and its probability limit is the variance-weighted sum of the two AR( 1 )-coefficients. Note that in this example, the data are not independent for different observations, which is a second departure from the standard model. 1, allows for some additional insight when the underlying distribution of E is assumed normal. In addition, we assume that the elements of e and the rows of S and V are independently normally distributed.