Homework 1
STAT 331
Fall 2008
1. (5 points) Prove that covariance between residuals and predictor
variable is 0.
Proof:
)
,
cov(
)
,
cov(
*
1
*
0
x
y
x
x
β
ε


=
)
,
cov(
)
,
cov(
)
,
cov(
*
1
*
0
x
x
x
y
x


=
)
,
cov(
)
,
cov(
)
,
cov(
*
1
*
1
x
x
x
y
x
y
x



=
)
,
cov(
)
,
cov(
)
,
cov(
0
)
,
cov(
x
x
x
x
y
x
y
x


=
= 0
2. (5 points)
Show that for SLR squared tstatistic for a slope is
equal to Fstatistic.
For the SLR
t
t
t
X
Y
+
+
=
1
0
, the tstatistic for the slope is
2
*
1
*
1
~
)
.(
.

=
n
t
e
s
T
under the null hypothesis
0
1
=
, where
*
1
denotes the
least square estimator of
1
,
)
.(
.
*
1
e
s
denotes the standard error of
*
1
,
and n represents the total number of observations.
[
]
2
*
1
2
*
1
*
0
*
1
*
0
1
2
1
2
)
(
)
(
)
2
/(
)
ˆ
(
)
1
2
/(
)
ˆ
(
)
/(
)
1
/(
σ
∑
∑
∑
=
=
=
+

+
=




=


=
n
t
n
t
n
t
n
t
n
t
n
t
X
X
n
Y
Y
Y
Y
k
n
SSE
k
SSR
F
[
]
[
]
)
2
,
1
(
2
2
1
*
1
*
1
2
2
*
2
1
*
2
*
1
2
2
1
*
~
)
.(
.
)
(
/
)
(

=
=
=
=

=

=
∑
∑
n
n
t
n
t
n
t
n
t
F
T
e
s
X
X
X
X
Therefore, the squared tstatistic for a slope is equal to Fstatistic.
Note that this implies that
)
(
)
(
)


(
)
2
,
1
(
2
2
2
F
F
P
F
t
P
T
t
P
n
n
n
≤
=
≤
=
≤



where
F
T
=
2
.
3. (40 points)
DESCRIPTIVE ABSTRACT:
The datafile contains 11 years of quarterly sales for four kinds of
retail establishments, along with nonagricultural employment and wage
and salary disbursements. The current task is to develop a model for
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Documentpredicting general merchandise dealer sales (GMER) based on national
income wage and salary disbursements (WASA).
These data are published monthly in the statistical section of the
Survey of Current Business.
VARIABLE DESCRIPTIONS:
1. TIME: Quarter, from 1st quarter 1979 to 4th quarter 1989
2. WASA: National income wage and salary disbursements ($ billions)
3. EMPL: Employees on payrolls of nonagricultural establishments
(thousands)
4. BLDG: Building material dealer sales ($ millions)
5. AUTO: Automotive dealer sales ($ millions)
6. FURN: Furniture and home furnishings dealer sales ($ millions)
7. GMER: General merchandise dealer sales ($ millions)
Values are aligned and delimited by blanks.
YOUR TASK:
To forecast general merchandise dealer sales for 14 quarters of 1989
using national income wage and salary disbursements. The training
window is from the 1
st
quarter of 1979 to the 4
th
quarter of 1988. You
should
a) construct a simple linear regression of general merchandise
dealer sales and national income wage and salary disbursements,
include the summary output of the linear regression, the
scatterplot, correlation, covariance and Rsquared of GMER vs.
WASA;
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 YuliaGel
 Covariance, Regression Analysis, Variance, residuals, national income wage, general merchandise dealer, merchandise dealer sales

Click to edit the document details