2.5A model is linear in the parameters; it may or may not be linear in the variables.
2.16aThe scatter plot for male and female verbal scores is as follows:
And the corresponding plot for male and female math scores is as follows:
2.16bOver the years, the male and female verbal scores show a downward trend, whereas after reaching a low in 1980, the math scores for both males and females seem to show an upward trend, of course with year to year variation.
2.16cWe can develop a simple regression model regressing the math score on the verbal score for both sexes.
2.16dThe plot is as follows:
As the graph shows, over time, the two scores have moved in the same direction.
3.2YiXiyixixiyixi2
41-3-399
54-2000
750101
12652104
------
sum2816001914
------
Note:= 7 and = 4
Therefore, ;
3.10Since , that is, the sum of the deviations from mean value is always zero, are also zero. Therefore, . The point here is that if both Y and X are expressed as deviations from their mean values, the regression line will pass through the origin.
, since means of the two variables are zero. This is equation (3.1.6).
3.16aFalse. The covariance can assume any value; its value depends on the units of measurement. The correlation coefficient, on the other hand, is unitless, that is, it is a pure number.
3.16bFalse. See Fig.3.11h. Remember that the correlation coefficient is a measure of linear relationship between two variables. Hence, as Fig.3.11h shows, there is a perfect relationship between Y and X, but that relationship is nonlinear.
3.25aSee figure in Exercise 2.16(d).
3.25bThe regression results are:
= –198.126+1.436Xt
se = (25.211) (0.057)
r2 = 0.966
where Y = female verbal score and X = male verbal score.
3.25cAs pointed out in the text, a statistical relationship, however strong, does not establish causality, which must be established a priori. In this case, there is no reason to suspect causal relationship between two variables.
5.1aTrue. The t test is based on variables with a normal distribution. Since the estimators of β1 and β2are linear combinations of the error ui, which is assumed to be normally distributed under CLRM, these estimators are also normally distributed.
5.1bTrue. So long as E(ui) = 0, the OLS estimators are unbiased. No probabilistic assumptions are required to establish unbiasedness.
5.1cTrue. In this case, Eq.(1) in App. 3A, Sec. 3A.1, will be absent. This topic is discussed more fully in Chap. 6, Sec. 6.1.
5.1eTrue. This follows from Eq.(1) of App. 3A, Sec. 3A.1.
5.1fFalse. All we can say is that the data at hand does not permit us to reject the null hypothesis.
5.1hFalse. A larger σ2may be counterbalanced by a larger . It is only if the latter is held constant, the statement can be true.
5.9a
5.9bPayi = 12129.37 + 3.3076 Spend
se = (1197.351) (0.3117)r2 = 0.6968; RSS = 2.65E+08
5.9cIf the spending per pupil increases by a dollar, the average pay increased by about $3.31. The intercept term has no viable economic meaning.
5.9dThe 95% CI for β2is: 3.3076 ± 2(0.3117) = (2.6842,3.931). Based on this CI you will not reject the null hypothesis that the true slope coefficient is 3.
5.9eThe mean and individual forecast values are the same, namely, 12129.37 + 3.3076(5000) ≈ 28,667. The standard error of the mean forecast value, using Eq.(5.10.2), is 520.5117 (dollars) and the standard error of the individual forecast, using Eq.(5.10.6), is 2382.337. The confidence intervals are:
Mean Prediction:28,667 ± 2(520.5117), that is, ($27,626, $29,708)
Individual Prediction:28,667 + 2(2382.337), that is ($23,902, $33,432)
As expected, the latter interval is wider than the former.
5.9fThe histogram of the residuals can be approximated by a normal curve. The Jarque-Bera statistic is 2.1927 and its p value is about 0.33. So, we do not reject the normality assumption on the basis of this test, assuming the sample size of 51 observations is reasonably large.
6.2a&bIn the first equation an intercept term is included. Since the intercept in the first model is not statistically significant, say at the 5% level, it may be dropped from the model.
6.2cFor each model, a one percentage point increase in the monthly market rate of return leads on average to about 0.76 percentage point increase in the monthly rate of return on Texaco common stock over the sample period.
6.2eNo, the two r2s are not comparable. The r2 of the interceptless model is the raw r2.
6.2fSince we have a reasonably large sample, we could use the Jarque-Bera test of normality. The JB statistic for the two models is about the same, namely, 1.12 and the p value of obtaining such a JB value is about 0.57. Hence do not reject the hypothesis that the error terms follow a normal distribution.