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| When a linear model is constructed as ''Y,,t,, = β,,0,, + β,,1,,x,,t,, + e,,t,,'' but estimated as ''Y,,t,, = β,,0,, + β,,1,,X,,t,, + e,,t,, = β,,0,, + β,,1,,(x,,t,, + u,,t,,) + e,,t,,'', then the expected values are characterized by: | When a linear model is constructed as ''Y,,t,, = β,,0,, + β,,1,,x,,t,, + e,,t,,'' but estimated as ''Y,,t,, = βˆ,,0,, + βˆ,,1,,X,,t,, = βˆ,,0,, + βˆ,,1,,(x,,t,, + u,,t,,)'', then the expected values are characterized by: |
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| And the covariance matrix is specified as: | And the [[Statistics/CovarianceMatrices|covariance matrix]] is specified as: |
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We assume that ''x,,t,,'', ''e,,t,,'', and ''u,,t,,'' are all independent. The regression coefficient that has been attenuated (biased towards 0) by measurement error is represented as ''γ''. {{attachment:gamma.svg}} The relationship between ''γ'' and ''β,,1,,'' is characterized by ''κ''. {{attachment:kappa1.svg}} In more plain terms, this is: {{attachment:kappa2.svg}} ''κ'' is the '''reliability ratio'''. |
Measurement Error Models
Measurement Error Models (ISBN: 9780470316665) was written by Wayne A. Fuller in 1987.
A measurement Xt is decomposed into the true value xt and the measurement error ut.
When a linear model is constructed as Yt = β0 + β1xt + et but estimated as Yt = βˆ0 + βˆ1Xt = βˆ0 + βˆ1(xt + ut), then the expected values are characterized by:
And the covariance matrix is specified as:
We assume that xt, et, and ut are all independent.
The regression coefficient that has been attenuated (biased towards 0) by measurement error is represented as γ.
The relationship between γ and β1 is characterized by κ.
In more plain terms, this is:
κ is the reliability ratio.