1. 建立如下的双变量PRF ：
模型I ：Y i =β1+β2X i +u i
模型II ：Y i =α1+α2(X i ?X
?)+u i a. 求β1和α1的估计量。它们是否相同？它们的方差是否相同？ b. 求β2和α2的估计量。它们是否相同？它们的方差是否相同？ c. 如果模型II 比模型I 好，好在哪里？
2. 令β
?YX 和β?XY 分别为Y 对X 回归和X 对Y 回归中的斜率。说明 β
?YX β?XY =r 2 其中r 为X 和Y 之间的相关系数。
3. The following table contains the ACT scores and the GPA (grade point average) for eight
college students. Grade point average is based on a four-point scale and has been rounded to one digit after the decimal.
slope estimates in the equation
GPA ?=β0?+β1?ACT Comment on the direction of the relationship. Does the intercept have a useful interpretation here?
Explain. How much higher is the GPA predicted to be if the ACT score is increased by five points? (ii) Compute the fitted values and residuals for each observation, and verify that the residuals (approximately) sum to zero.
(iii) What is the predicted value of GPA when ACT = 20?
4. In the linear consumption function
cons ?=β0?+β1?inc
the (estimated) marginal propensity to consume (MPC) out of income is simply the slope, β1?, while the average propensity to consume (APC) is
cons ?/inc = β0?/inc + β1?
Using observations for 100 families on annual income and consumption (both measured in
dollars), the following equation is obtained:
cons ?=?124.84+0.853inc
n = 100, R 2= 0.692.
(i) Interpret the intercept in this equation, and comment on its sign and magnitude. (ii) What is the predicted consumption when family income is $30,000? (iii) With inc on the x -axis, draw a graph of the estimated MPC and APC.