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Consider the model

$_$=$_0$+$_1$$_∗$+$_$

In practice we measure $_∗$ by $_$ such that

a) $_$=$_*+3$

b) $_$=$5_*$

What will be the effect of these measurement errors on estimates of true $_0$ $_1$? In terms of biasedness, BLUE, consistency, efficiency?

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    As the question says "measurement error", this should be not as straightforward as claiming that "Running the regression of $Y_i$ with $X_i$ instead of $X_i$* will give a biased estimate of $β_0$, and no change to $β_1$." – 2017-02-27

1 Answers 1

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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't effect the variance and covariance measures, then it will no effect $\hat{\beta}_1$. You can see it if you plug in $$ \hat{\beta}_1 = \frac{\sum (X^*_i - \bar{X}^*)(Y_i - \bar{Y})}{\sum(X^*_i-\bar{X}^*)^2}, $$ $X_i - 3$ instead of $X_i^*$, however, for the $\beta_0$ estimator that measures location, you will get $$ \hat{\beta_0^*} = \bar{Y}_n - \hat{\beta_1}\bar{X}_n^* = \bar{Y}_n + \hat{\beta_1}3-\hat{\beta}_1\bar{X}_n = \hat\beta_0 +3\hat{\beta}_1 . $$ Thus your $\hat{\beta}_0$ estimator will be biased estimator with $$ b(\hat{\beta}_0^*)=E(\hat{\beta}_0^* - \beta_0) = \beta_0+3\beta_1 - \beta_0 = 3\beta_1. $$ As such if $\beta_1 \neq 0$, then your $\hat\beta_0$ will be inconsistent estimator. And as it is biased then there is no sense talking about being BLUE or efficient.