文档介绍：E c o n om i cs 5 2 8 I n C h oi
H K U S T Sp r ing 20 04
R e v i ew P r o b l em s f or M id t erm
1 L e a s t S q u ar es
1 . In t he l inea r regression mod el
y = X β+ ε,
there is a need f or changing the u nit of measurement for the dependent v ariab le
y. So y ∗= c y ( c is a constant ) is now used as a dependent variable.
( a) D oes this practice change R 2 ?
(b) W hat happens to R 2 if the unit of measurement is changed only for the
regressor?
2. Consider the linear regression model
′ 2
y i = α+ β X i + ε i , ε i ∼ i id
, σ,
= 0
(a) Is the O L S estimator of β aff ected by the nonz ero mean of ε i ?
(b) Can the least sq uares estimator of α estimate it accurately?
3 . P rove that the OLS estimators of β 1 in the following linear models are identical
y t = x t β 1 + t β 2 + ε t
∗∗
y t = x t β 1 + ε t
∗∗
where y t and x t are de— trended y t and x t obtained by regressing y t and x t on t
∗∗
and setting y t and x t equal to the respective residuals.
4. Discuss the validity of the following statements.
(a) Sum of residuals is always zero.
(b) If a regression produces R 2 greater than 0 . 5 , the regression is a reliable
one.
2
( c ) I n a r e g res si o n m od el
y i = α x i + ε i ,
sw it ch ing the indep endent and dependent v ariab les and ru nning a least
sq uares provide a valid estimator of 1 .
α