Error-in-variables regression

One assumption until now has been that the dependent and independent variables are measured without error. The impact of measurement error on the regression parameter estimates depends on whether the error affects the dependent or independent variable. When Y has measurement error, the effect on the regression model is not problematic if the measurement errors are uncorrelated and unbiased. In this case the linear model becomes 

Before proceeding, a distinction needs to be made between X being simply a random variable and X being random due to random measurement error. This distinction is important and the question is sometimes asked, what s the difference If X is random but measured accurately, the experimenter has no control over its measurement, and its value may vary from study to study. An example of this might be the weight of subjects in a clinical study. If this random variable X is measured without error, then an exact, accurate measurement of X can be obtained only for that study. If, however, X is random due to measurement error, then repeated measurement of X within the same study will result in differing values of X each time X is measured and a misleading relationship between X and Y will be obtained. 

One other distinction needs to be made between random X and X with random measurement error. Neither implies that X is biased. Bias implies a constant effect across all measurements. For example, if a weight scale is not calibrated properly and when no one is standing on it, the scale records a measure of 1 kg, then when any person is measured the weight will be biased high by 1 kg. This is not the type of measurement error that is being discussed here because any constant bias in a measuring instrument will be reflected in the 

If both X and Y are random variables and X is measured without random error, then all the theory presented for the case of fixed x is still applicable if the following conditions are true  

The conditional distribution for each of the Yj given Xj is independent and normally distributed with conditional mean 0o + 0iXj and conditional variance a2, and 

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