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University of California, Davis
Managerial Economic(ARE) 106, Summer 2009
Instructor: John H. Constantine
KEY
—Problem Set #2: Due MONDAY, August 17, 2009
Problem 1
(A table is provided at the end of the assignment to calculate the numbers)
:
The following data give the daily price (x, ($/unit)) and consumption (y) of some commodity Z; the data
were collected at nine shops in Davis that sell Z for a one week period.
The objective is to estimate the
demand for Z.
Obs
x
y
1
51
0
2
31
5
3
15
8
4
91
2
5
14
6
6
41
8
7
32
2
8
61
4
9
81
1
(a)
Plot the data.
(b)
The PRF for this model is:
y
i
=
β
1
+
β
2
x
i
+ e
i
.
Use the method of leastsquares to estimate (by hand) the sample regression function.
You must
provide numeric estimates for the following variables:
(i)
β
1
(ii)
β
2
(iii)
σ
2
(iv)
var(
β
2
)
(v)
s.e.(
β
2
)
(c)
State the SRF for the given sample above.
Plug in the parameter estimates from part (b) into this
equation
.
y
∧
= 19.61 – 0.90
x
.
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(d)
What is the economic interpretation of both b
1
and b
2
, the sample estimates of
β
1
and
β
2
,
respectively.
The equation gives an estimate for the underlying model that generates the given data.
“b
1
” is the
intercept of the estimated equation. It is also the predicted value when
x
i
is equal to 0.
“b
2
”
is the slope of the estimated equation. It is the average marginal response of
y
i
to the change of
x
i
. i.e., if
x
i
increases one unit,
y
i
will ON AVERAGE increase (b
1
> 0) or decrease (b
1
> 0) by
b
1
units.
Our estimate of
β
1
and
β
2
1
2
= – 0.90, respectively.
The important parameter estimate is b
2
.
We expect that, on average, for each additional one dollar
increase in the price of Z that quantity demanded will fall by 0.90 units.
(d)
Clearly distinguish the terms var(b
2
) and
σ
2
and
state what your estimates mean.
Var(b
2
) is the variance of b
2
.
b
2
is random since different samples will give us different estimates of
b
2
, i.e., we get a different b
2
from each sample.
Var(b
2
) is estimated as 0.83.
σ
2
is the variance of the error term (e).
It is a measure of dispersion of the data around the
estimated regression line.
^
2
σ
= 134.05.
(e)
Plot the data points given in the table and draw in your estimated regression function.
Chart is given on the following page.
(f)
Suppose x is 9.
What is the predicted y (i.e.,
∧
y
), as determined by your regression equation?
Does this coincide with observation 4, where x = 9 and y = 12?
Why or why not?
y
∧
= 19.61 – 0.91(9) = 11.42.
The actual y value for observation 4 is 12.
No, the prediction does not coincide with observation 4.
The predicted values are the prediction of E(
yx
), not the actual datum
y
i
.
(g)
The estimated linear regression line always passes through the mean values x and y (assuming we
do not suppress the intercept, which we have not done here.)
Verify that this estimated regression
line passes through the orgin.
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This note was uploaded on 11/19/2009 for the course ARE ARE106 taught by Professor Constantine during the Spring '09 term at UC Davis.
 Spring '09
 CONSTANTINE

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