Thus, bEZ is an unbiased estimator.
(c) The variance is given by
(d) If , then
Exercise 2.15 (continued)
(e) To convince E.Z. Stuff that var(b2) < var(bEZ), we need to show that
or that
Consider
Thus, we need to show that
or that
or that
This last inequality clearly holds. Thus, is not as good as the least squares estimator. Rather than prove the result directly, as we have done above, we could also refer Professor E.Z. Stuff to the Gauss Markov theorem.
Exercise 2.16
(a) The model is a simple regression model because it can be written as where , , and .
(b) The estimates are in the table below
FirmGEIBMFORDMSFTDISXOM
b1 = −0.000959
(0.00442)0.00605
(0.00483)0.00378
(0.0102)0.00325
(0.00604)0.00105
(0.00468)0.00528
(0.00354)
1.148
(0.0895)0.977
(0.0978)1.662
(0.207)1.202
(0.122)1.012
(0.0946)0.457
(0.0716)
N180180180180180180
Standard errors in parentheses
The stocks Ford, GE, and Microsoft are relatively aggressive with Ford being the most aggressive with a beta value of . The others are relatively defensive with Exxon-Mobil being the most defensive with a beta value of .
(c) All estimates of the are close to zero and are therefore consistent with finance theory. The fitted regression line and data scatter for Microsoft are plotted in Figure xr2.15.
Fig. xr2.15 Scatter plot of Microsoft and market rate
(d) The estimates for given are as follows.
FirmGEIBMFORDMSFTDISXOM
1.147
(0.0891)0.984
(0.0978)1.667
(0.206)1.206
(0.122)1.013
(0.0942)0.463
(0.0717)
Standard errors in parentheses
The restriction aj = 0 has led to small changes in the ; it has not changed the aggressive or defensive nature of the stock.
Exercise 2.17
(a)
Figure xr2.17(a) Price (in $1,000s) against square feet for houses (in 100s)
(b) The fitted linear relationship is
We estimate that an additional 100 square feet of living area will increase the expected home price by $13,402.94 holding all else constant. The estimated intercept −115.4236 would imply that a house with zero square feet has an expected price of $−115,423.60. This estimate is not meaningful in this example. The reason is that there are no data values with a house size near zero.
Figure xr2.17(b) Observations and fitted line
Exercise 2.17 (continued)
(c) The fitted quadratic model is
The marginal effect is . For a house with 2000 square feet of living area the estimated marginal effect is 2(0.1845)20 = 7.3808. We estimate that an additional 100 square feet of living area for a 2000 square foot home will increase the expected home price by $7,380.80 holding all else constant.