Ch2 Exercise Solutions

Chapter 2, Exercise Solutions, Principles of Econometrics, 4e
27
Exercise 2.3 (Continued)
(d)
ˆi yi y ˆ i yi b1 b2 xi , are: The values of the least squares residuals, computed from e
Figure xr2.4(a) Sum of squares for 2
The minimum of this function is approximately 12 and occurs at approximately 2 1.95. The significance of this value is that it is the least-squares estimate. (d) To find the value of that minimizes S () we obtain
190 200 P Z 81 1 P Z 1.1111 0.8667
Chapter 2, Exercise Solutions, Principles of Econometrics, 4e
26
EXERCISE 2.3
(a) The observations on y and x and the estimated least-squares line are graphed in part (b). The line drawn for part (a) will depend on each student’s subjective choice about the position of the line. For this reason, it has been omitted. Preliminary calculations yield:
b1 y b2 x 5.5 1.514286 3.5 10.8
Figure xr2.3 Observations and fitted line
10 2 1 4 6 8
2 y
3 x
4 Fitted values
5
6
(c)
y yi N 33 6 5.5 x xi N 21 6 3.5
3
4
(f) (g) (h) (i)
See figure above. The fitted line passes through the point of the means, x 2,
y 2.4 .
Given b1 5, b2 1.3 and y b1 b2 x , we have y 2.4 b1 b2 x 5 1.3 2 2.4
ˆ y ˆ i N 12 5 2.4 y y
ˆ2
N 2
ˆi2 e

4.3 1.4333 3
ˆ2 1.4333 0.14333 10
(j)
var b2
xi x
2
Chapter 2, Exercise Solutions, Principles of Econometrics, 4e
=0
Chapter 2, Exercise Solutions, Principles of Econometrics, 4e
28
EXERCISE 2.4
(a) If 1 0, the simple linear regression model becomes
yi 2 xi ei
dS 2 xi yi 2 xi2 d
Setting this derivative equal to zero, we haveb xi2 xi yi
or
b
xi yi xi2
Thus, the least-squares estimate is
b2 176 1.9341 91
CHAPTER
2
Exercise Solutions
21
Chapter 2, Exercise Solutions, Principles of Econometrics, 4e
22
EXERCISE 2.1
(a)
x
0 1 2 3 4
y 6 2 3 1 0
xx
-2 -1 0 1 2 0
x x
xi
1 2 3 4 5 6
ˆi 0. e
yi
10 8 5 5 2 3
ˆi e
0.714286 0.228571 −1.257143 0.257143 −1.228571 1.285714
Their sum is (e)
ˆi 1 0.714286 2 0.228571 3 1.257143 4 0.257143 xi e 5 1.228571 6 1.285714
The predicted value for y at x x is
ˆ b1 b2 x 10.8 1.514286 3.5 5.5 y
ˆ b1 b2 x y . That is, the predicted value at the sample mean x is the We observe that y sample mean of the dependent variable y . This implies that the least-squares estimated line passes through the point ( x , y ) . This point is at the intersection of the two dashed lines plotted on the graph in part (b) .
0
x 2,
(b)
y 2.4
b2
x x y y 13 1.3 2 10 x x
b2 is the estimated slope of the fitted line.
b1 y b2 x 2.4 1.3 2 5
which agrees with the approximate value of 1.95 that we obtained geometrically.
215 200 180 200 P Z 100 100 P 2 Z 1.5 0.9104
.4
Figure xr2-2a
0
.1
f(z) .2
.3
-5
-2
0 z
1.5
5
(b)
X 190 y|x$2000 y|x $2000 P X 190 P 2 2 y| x $2000 y| x $2000
ˆi2 e
ˆi xi e
0 1 2 3 4 xi = 10
6 2 3 1 0 yi = 12
5 3.7 2.4 1.1 −0.2 ˆi = y 12
1 −1.7 0.6 −0.1 0.2 ˆi = e 0
1 2.89 0.36 0.01 0.04 ˆi2 = e 4.3
0 −1.7 1.2 −0.3 0.8 ˆi = xi e 0
b1 is the estimated value of E ( y ) when x 0 ; it is the intercept of the fitted line.
(c)
xi2 02 12 22 32 42 30
i 1
5
xi yi 0 6 1 2 2 3 3 1 4 0 11
(b)
xi 21
y 5.5
yi 33
x 3.5
xi x yi y 26.5
xi x
2
17.5
The least squares estimates are:
b2
x x y y 26.5 1.514286 2 17.5 x x
24
EXERCISE 2.2
(a)
180 X y|x$2000 215 y|x$2000 y|x $2000 P 180 X 215 P 2 2 2 y| x $2000 y| x $2000 y|x $2000
4 1 0 1 4
2
yy
3.6 −0.4 0.6 −1.4 −2.4
x x y y
−7.2 0.4 0 −1.4 −4.8 −13
xi =
10
yi
12
xi x xi x
10
2

y y x x y y
(c)
S () ( yi xi )2 ( yi2 2xi yi 2 xi2 ) yi2 2 xi yi 2 xi2
i 1 i 1
N
N
352 2 176 912 352 352 912
40 35 30 SUM_SQ 25 20 15 10 1.6 1.8 BETA 2.0 2.2 2.4
190 200 P Z 100 1 P Z 1 0.8413
Figure xr2-2b
0
.1
f(z) .2
.3
.4
-5
-1
0 z
5
Chapter 2, Exercise Solutions, Principles of Econometrics, 4e