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**Unformatted text preview: **7.7 (a) No. An r-value cannot exceed 1 in absolute value. Plugging
the given data in Eq. (7.11.2), the reader can should verify that:
r113 = 2.295, which is logically impossible. (b) Yes. Following the same procedure as in (a), the reader will
find that r113 = 0.392, which is possible. (c) Yes, again it can be shown that r113 = 0.880, which is possible. 7.10 (o) 3: (b) If you multiply X; by 2, you can verify from Equations
(7.4.7) and (7.4.8), that the slopes remain unaffected. On the other
hand, if you multiply Y by 2, the slopes as well as the intercept
coefﬁcients and their standard errors are all multiplied by 2. Always
keep in mind the units in which the regressand and regressors are
measured. 7.11 From (7.11.5) we knowr that R: = all + a: - 2mm _
1— I";
Therefore? when r23 = D, that is, no correlation between variables
X2 and X3,
11'?2 = r212 + r213, that is, the multiple coefﬁcient of
determination is the sum of the coefﬁcients of determination in the regression of I" on X2 and that of Y on X3. 8.1 (a) In the ﬁrst model, where sales is a linear function of time, the
rate of change of sales, (dedt) is postulated to be a constant, equal to {31, regardless of time I. In the second model the rate of change is
not constant because (dedt) = or, + 2052:, which depends on time I. (b) The simplest thing to do is plot Y against time. If the resulting
graph looks parabolic, perhaps the quadratic model is appropriate. (c) This model might be appropriate to depict the earnings proﬁle
of a person. Typically, when someone enters the labor market, the
entry-level earnings are low. Overtime, because of accumulated
experience, earnings increase, but after a certain age they start
declining. (d) Look up the web sites of several car manufacturers, or Motor
Magazine, or the American Automobile Association for the data. 8.6 Start with equation (8.5.11) and write it as: —k R1
F = m ., which can be rewritten as:
k —1 R?-
F E 3%)) = (1 R2) . after further algebraic manipulation, we
n _ _
obtain
R2 FUc _1) , which is the desired result. _ F(k—1)+(n—k)
For regression (8.2.1), n=64, k = 3. Therefore,
F @3152} = 3.15.. approx. (Note use 60 df in place of 62 df). Therefore, putting these values in the preceding R2 formula,
we obtain: RZ- 2(3.15) =6.3c=m,}936 _ 2(3.15)+61 67.3 This is the critical R2 value at the 5% level of signiﬁcance. Since
the observed of R2 of 0.7077 in (8.2.1) far exceeds the critical value,
we reject the null hypothesis that the true li’2 value is zero. 7.24 (a)
CI = —20.6327 + 0.7 340 Yd + 0.0360Wea3rh — 5.5212 Interest r= (—1.6085) (53.3762) (14.4882) (2.306?) a2 = 0.9994 (b) The three independent variables are statistically signiﬁcant at the
5% level. It seems that increases in Income (Yd) and Wealth are
related to increases in Ccnsumptien, whereas an increase in the Interest rate corresponds tc a decrease in the Ccnsumpticn level.
This makes sense. Dependent Variable: C01
Method: Least Squares Date: 02703709 Time: 17:14
Sample: 1947 2000
Included observations: 54 WEALTH
INTEREST R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
F-statistic
Prob(F—statistic) Coefﬁcient 0.734028
0.035976 —5.521224
—20.63274 0.999402
0.999366 37.79877
71437.35 —270.6877
27838.41
0.000000 Std. Error t-Statistic 0.013752 53.37622 0.002483 14.48822 2.306675 —2.393 585 12.82699 —1.608541
Mean dependent var SD. dependent var
Akaike info criterion
Schwarz criterion Harman—Quinn criter.
Durbin—Watson stat Prob. 0.0000
0.0000 0.0205
0.1140 2888.356
1500.903
10.17362
10.32095
10.23044
1.310554 ...

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- Spring '13