Consider the one-variable regression model Yi = β0 + β1Xi + ui and suppose that it satisfies the least squares assumptions .The regressor Xi is missing, but data on a related variable, Zi, are available, and the value of Xi is estimated usingX ̃i = E(Xi | Zi). Let wi = X ̃ i - Xi.a. Show that X ̃i is the minimum mean square error estimator of Xi using Zi. That is, let X^i = g(Zi) be some other guess of Xi based on Zi, and show that E[(X^i - Xi)2] ≥ E[(X ̃i - Xi)2].b. Show that E(wi | X ̃i) = 0.c. Suppose that E(ui | Zi) = 0 and that X ̃i is used as the regressor in place of Xi. Show that β^1 is consistent. Is β^0 consistent?
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Mean, Median, Mode
It is a descriptive summary of a data set. It can be defined by using some of the measures. The central tendencies do not provide information regarding individual data from the dataset. However, they give a summary of the data set. The central tendency or measure of central tendency is a central or typical value for a probability distribution.
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A z-score is a unit of measurement used in statistics to describe the position of a raw score in terms of its distance from the mean, measured with reference to standard deviation from the mean. Z-scores are useful in statistics because they allow comparison between two scores that belong to different normal distributions.
Consider the one-variable regression model Yi = β0 + β1Xi + ui and suppose that it satisfies the least squares assumptions .The regressor Xi is missing, but data on a related variable, Zi, are available, and the value of Xi is estimated usingX ̃i = E(Xi | Zi). Let wi = X ̃ i - Xi.
a. Show that X ̃i is the minimum mean square error estimator of Xi using Zi. That is, let X^i = g(Zi) be some other guess of Xi based on Zi, and show that E[(X^i - Xi)2] ≥ E[(X ̃i - Xi)2].
b. Show that E(wi | X ̃i) = 0.
c. Suppose that E(ui | Zi) = 0 and that X ̃i is used as the regressor in place of Xi. Show that β^1 is consistent. Is β^0 consistent?
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