Consider the model 𝑋 𝑖 ∗In practice we measure 𝑋 𝑖 ∗ by 𝑋 𝑖 such thata) 𝑋 𝑖 ∗ + 3b) 5 𝑋 𝑖 ∗What will be the effect of these measurement errors on estimates of true 𝛽 0 and 𝛽 1 ? In terms of biasedness, BLUE, consistency, efficiency?

Question

Consider the model            𝑋              𝑖        ∗In practice we measure             𝑋              𝑖        ∗ by             𝑋              𝑖       such thata)             𝑋              𝑖        ∗  +  3b)   5            𝑋              𝑖        ∗What will be the effect of these measurement errors on estimates of true             𝛽        0   and             𝛽        1  ? In terms of biasedness, BLUE, consistency, efficiency?

Lilliana Burton · Accepted Answer

Let us take a look at the first case, the fact that we measure       X    i    =      X    i    ∗    +  3 instead of       X    i    ∗   it won&#039;t effect the variance and covariance measures, then it will no effect                     β        ^              1  . You can see it if you plug in                    β        ^              1    =            ∑      (              X        i        ∗            −                                    X            ¯                          ∗            )      (              Y        i            −                        Y          ¯                    )              ∑      (              X        i        ∗            −                                    X            ¯                          ∗                    )        2              ,      X    i    −  3 instead of       X    i    ∗  , however, for the       β    0   estimator that measures location, you will get                    β        0        ∗            ^        =                    Y        ¯              n    −                    β        1            ^                          X        ¯              n    ∗    =                    Y        ¯              n    +                    β        1            ^        3  −                    β        ^              1                      X        ¯              n    =                    β        ^              0    +  3                    β        ^              1    .Thus your                     β        ^              0   estimator will be biased estimator with  b  (                    β        ^              0    ∗    )  =  E  (                    β        ^              0    ∗    −      β    0    )  =      β    0    +  3      β    1    −      β    0    =  3      β    1    .As such if       β    1    ≠  0, then your                     β        ^              0   will be inconsistent estimator. And as it is biased then there is no sense talking about being BLUE or efficient.

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