ECONOMICS 2123B
Assignment 1
Due on June 24, 2015
Note: The assignment is due at the beginning of class. Late assignments will be given a
mark of zero. For full credit you need to show your work.
Question 1
Consider the simple linear regression model Yi =
Sept 27, 2016
Economics 2122A, Fall 2016
Sample Midterm Exam 1
Answer the questions in the spaces provided, showing the steps you used to
arrive at your answer. If you run out of room, clearly label where you continue
your answer.
Name:
1. About 20% of th
MULTIPLE REGRESSION WITH TWO EXPLANATORY VARIABLES: EXAMPLE
EARNINGS = b 1 + b 2S + b 3EXP + u
b1
EARNINGS
EXP
S
This sequence provides a geometrical interpretation of a multiple regression model with two
explanatory variables.
1
MULTIPLE REGRESSION WITH
POSSIBLE DIRECT MEASURES FOR ALLEVIATING MULTICOLLINEARITY
2
2
u
u
1
1
2
2
2
X 2 i X 2 1 rX 2 , X 3 nMSD( X 2 ) 1 rX 2 , X 3
2
b2
What can you do about multicollinearity if you encounter it? We will discuss some possible measures,
looking at the model
POSSIBLE INDIRECT MEASURES FOR ALLEVIATING
MULTICOLLINEARITY
2
2
u
u
1
1
2
2
2
X 2 i X 2 1 rX 2 , X 3 nMSD( X 2 ) 1 rX 2 , X 3
2
b2
(1)
Combine the correlated variables.
In this sequence, we look at four possible indirect methods for alleviating a prob
PROPERTIES OF THE MULTIPLE REGRESSION COEFFICIENTS
ASSUMPTIONS FOR MODEL A
A.1
The model is linear in parameters and correctly
specified.
Y X . X u
1
A.2
2
2
k
k
There does not exist an exact linear relationship among the
regressors in the sample.
A.3 The
F TEST OF GOODNESS OF FIT
Model
Y = b 1 + b 2X + u
(Y Y )2 (Y Y )2 e 2
TSS ESS RSS
In an earlier sequence it was demonstrated that the sum of the squares of the actual values of Y (TSS:
total sum of squares) could be decomposed into the sum of the squares
RANDOM COMPONENTS, UNBIASEDNESS OF THE REGRESSION COEFFICIENTS
True model
Y 1 2 X u
Fitted model
Y b1 b2 X
The regression coefficients are special types of random variable. We will demonstrate this using the
simple regression model in which Y depends on X
PRECISION OF THE MULTIPLE REGRESSION COEFFICIENTS
True model
Y 1 2 X 2 3 X 3 u
Fitted model
Y b1 b2 X 2 b3 X 3
2
2
u
u
1
1
2
b2
2
2
2
X 2 i X 2 1 rX 2 , X 3 nMSD X 2 1 rX 2 , X 3
This sequence investigates the variances and standard errors of the slope
MULTICOLLINEARITY
Y 2 3 X 2 X 3
X 3 2 X 2 1
X2
X3
Y
10
19
51
11
21
56
12
23
61
13
25
66
14
27
71
15
29
76
Suppose that Y = 2 + 3X2 + X3 and that X3 = 2X2 1. There is no disturbance term in the equation for
Y, but that is not important. Suppose that we hav
HEDONIC PRICING
k
Pi 1 j X ji ui
j 2
Hedonic pricing supposes that a good or service has a number of characteristics that individually give
it value to the purchaser. The market price of the good is then assumed to be a function, typically a
linear combin
F TEST OF GOODNESS OF FIT FOR THE WHOLE EQUATION
Y 1 2 X 2 . k X k u
H 0 : 2 . k 0
H 1 : at least one 0
This sequence describes two F tests of goodness of fit in a multiple regression model. The first relates
to the goodness of fit of the equation as a wh
F TESTS RELATING TO GROUPS OF EXPLANATORY VARIABLES
Y 1 2 X 2 u
Y 1 2 X 2 3 X 3 4 X 4 u
We now come to more general F tests of goodness of fit. This is a test of the joint explanatory power of
a group of variables when they are added to a regression model
PREDICTION
k
True model
Pi 1 j X ji ui
j 2
k
Fitted model
Pi b1 b j X ji
j 2
In the previous sequence, we saw how to predict the price of a good or asset given the composition of
its characteristics. In this sequence, we discuss the properties of such pre
GRAPHING A RELATIONSHIP IN A MULTIPLE REGRESSION MODEL
. reg EARNINGS S EXP
Source |
SS
df
MS
-+-Model | 22513.6473
2 11256.8237
Residual | 89496.5838
537 166.660305
-+-Total | 112010.231
539 207.811189
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-sq
WESTERN UNIVERSITY
G. Stirling
ECONOMICS 2123B
March, 2015
Sample Question
11.
Suppose you are given the following information:
Model 1
lnYt = 4.99 +23.2D1 + 36.5D2 + .732 lnX2 2.798 D1lnX2 + 4.251 D2ln X2 .371 X3
+.405 D1lnX3 .236 D2lnX3
Model 2
adjusted
A MONTE CARLO EXPERIMENT
True model
Fitted model
Y b1 b2 X
Y b 1 b 2 X u
b2
X X Y Y b a u
X X
i
i
2
2
t
i
i
Xi X
ai
X j X 2
In the previous slideshow, we saw that the error term is responsible for the variations of b2 around its
fixed component b 2. We
ONE-SIDED t TESTS OF HYPOTHESES RELATING TO REGRESSION COEFFICIENTS
True model
Y 1 2 X u
Fitted model
Y b1 b2 X
Null hypothesis
H 0 : 2 20 ,
Alternative hypothesis
0
H1 : 2 2
Test statistic
b2 20
t
s.e. b2
Reject H0 if
t t crit
In the previous sequence,
Econ 2122, Fall 2016
Practice final exam questions
Note: This may sound a little longer than a usuall final exam. Some extra parts are added for the
matter of more practice.
1. Let the random variable M represent the duration of a major Hollywood movie. S
SIMPLE REGRESSION MODEL
Y
Y b 1 b 2 X
b
1
X1
X2
X3
X4
X
Suppose that a variable Y is a linear function of another variable X, with unknown parameters b 1 and
b 2 that we wish to estimate.
1
SIMPLE REGRESSION MODEL
Y
Y b 1 b 2 X
b
1
X1
X2
X3
X4
X
Suppose t
GOODNESS OF FIT
Two useful results:
e 0
X e
i i
0
This sequence explains measures of goodness of fit in regression analysis. It is convenient to start by
demonstrating two useful results. The first is that the mean value of the residuals must be zero.
1
G
DERIVING LINEAR REGRESSION COEFFICIENTS
True model
Y 1 2 X u
Y
6
Y3
Y2
5
4
3
Y1
2
1
0
0
1
2
3
X
This sequence shows how the regression coefficients for a simple regression model are
derived, using the least squares criterion (OLS, for ordinary least squar
CHANGES IN THE UNITS OF MEASUREMENT
Yi 1 2 X i ui
Yi b1 b2 X i
b2
X X Y Y
X X
i
i
2
i
Suppose that the units of measurement of Y or X are changed. How will this affect the regression
results? Intuitively, we would anticipate that nothing of substance wi