Description
Problem 1 [35 points]: Suppose the following regression model is fitted to a data set with
observations {(xi, yi), i = 1, 2, …, n}:
, ~ (0, )
2 2
y xi
e e N
iid
i
i i
(a) [11 points] Based on the least squares method and the fact that RSS/
2
2
~ n1
(df = n-1
since df = n from the data and df = 1 from estimating ), compute the least squares
estimates for
and
2
.
(b) [5 points] Is
ˆ
an unbiased estimator for β? Verify.
(c) [4 points] Show that the regression line passes through the point
n
i
i i
n
i
i
x y
n
x
n
x x y
1
2
1/ 2
1
2 4
1/ 2
4 1
,
1
, , but does NOT pass through the point
(x, y) .
(d) [7 points] Derive the maximum likelihood estimates (MLE)
~
and
~2
.
(e) [8 points] Suppose (x1, x2, x3, x4, x5) = (-2, -1, 0, 1, 2) and (y1, y2, y3, y4, y5)=(8, 1, 1, 3, 9).
What the values of the least squares estimates
ˆ
and
2
ˆ
? Does the sum of residuals
equal to zero?
Problem 2 [10 points]: Consider the residuals {
i
e
ˆ
} from a simple linear regression
i i i i i
e y y y x 0 1
ˆ ˆ
ˆ
ˆ
, i = 1, 2, …, n
where
x
ˆ
y –
ˆ
SXY/SXX and
ˆ
1
0
1
are the OLS estimates for β1 and β0.
Show that {
i
e
ˆ , i=1,2,…n} are uncorrelated with the explanatory variables {xi, i= 1,2,…n}.
That is,
( )(ˆ ˆ) 0
1
1
ˆ( , ˆ)
1
n
i
i i
x x e e
n
x e .
Page 2/2
Problem 3 (R problem) [20 points]: The R library ‘alr3’ contains the “brains” data, which
provided the average body weight (in kg) and average brain weight (in gram) for 62 species
of mammals. (https://rdrr.io/rforge/alr3/man/brains.html)
Suppose that we are interested in how the log of average brain weight
(y=log(brains$BrainWt)) is affected by the log of average body weight (x=log(brains$BodyWt)).
(a) [10 points] Based on the R codes similar to those from Ch2 page 23, obtain the OLS
estimates
0
ˆ
, 1
ˆ
and
2
ˆ .
(b) [6 points] Based on the plot and the abline functions as in Ch1 page 27, generate the
scatterplot of the data, and add the regression line obtained in part (a) to the plot.
(c) [4 points] Suppose an outlier is defined as observation (xi, yi) with
| ˆ | 2
ˆ
i
e
. Do you
think there is outlier in the data set? Verify.
(Note: A more precise definition of outlier will be introduced in Chapter 7, which
removes the impact of the outlier (xi, yi) itself when estimating
ˆ
).
Problem 4 [35 points]: Suppose we want to fit the our data {(xi, yi), i=1, 2, …n} based on the
following simple linear regression model:
, with E( ) 0, Var( ) and , 1,…, are uncorrelated 2
yi
0
1
xi
ei
ei
ei
ei
i n
Given that
11, 73.14545, 3.95455, 60961.94, 202.25, 3373.75
1 1
2
1
2
n
i
i i
n
i
i
n
i
i n x y x y x y .
(a) [8 points] Compute SXY, SXX and SYY.
(b) [6 points] Show that the OLS estimates
1
ˆ
= 0.09100. What are the OLS estimates for
0
and
2
?
(c) [4 points] Compute
| )
ˆ
Var(
ˆ
0 X
and
| )
ˆ
Var(
ˆ
1 X .
(d) [5 points] Suppose (x, y) = (74.5, 2.0) is an observation in the data set. Based on the
definition of outlier as in Problem 3(c), do you think the observation is an outlier?
Explain.
[part(e) and (f)]] Suppose the point (50.3, 3.0) is added to the data set, and the new OLS
estimates
*
0
ˆ
,
*
1
ˆ
and
2 *
ˆ
are obtained from the 12 observations.
(e) [8 points] Show that
*
1
ˆ
= 0.08190.
(f) [4 points] What are the values of OLS estimate
*
0
ˆ
and
2 *
ˆ
?
– End of the Assignment –
