We would like to obtain an estimator of β1 from the following regression model with only one independent regressor:

yi = β0 + β1x1i + ui . (1)

however, there is another variable x2i , which is missing from the model and potentially correlated with x1i . that is, the true model would be

yi = β0 + β1x1i + β2x2i + vi (2)

where vi is an observation error, which satisfies e (vi |x1i , x2i) = 0.

(a) show that the ols estimator of β1 obtained from model (1) is biased.

(b) when is this bias equal to 0?

(c) derive the ols estimators of β1 and β2 from model (2).

(d) show that, when the sample covariance between x1i and x2i is equal to 0, then the ols estimator of β1 derived in (c) is the same as the ols estimator of β1 derived in (a).

In δabc shown below, ∠bac is congruent to ∠bca: triangle abc, where angles a and c are congruent given: base ∠bac and ∠acb are congruent. prove: δabc is an isosceles triangle. when completed (fill in the blanks), the following paragraph proves that line segment ab is congruent to line segment bc making δabc an isosceles triangle. (4 points) construct a perpendicular bisector from point b to line segment ac . label the point of intersection between this perpendicular bisector and line segment ac as point d: m∠bda and m∠bdc is 90° by the definition of a perpendicular bisector. ∠bda is congruent to ∠bdc by the definition of congruent angles. line segment ad is congruent to line segment dc by by the definition of a perpendicular bisector. δbad is congruent to δbcd by the line segment ab is congruent to line segment bc because consequently, δabc is isosceles by definition of an isosceles triangle. 1. corresponding parts of congruent triangles are congruent (cpctc) 2. the definition of a perpendicular bisector 1. the definition of a perpendicular bisector 2. the definition of congruent angles 1. the definition of congruent angles 2. the definition of a perpendicular bisector 1. angle-side-angle (asa) postulate 2. corresponding parts of congruent triangles are congruent (cpctc)