EconS 451: Lecture # 8
Empirical Estimation Review
• Describe in general terms what we are attempting to solve with
empirical estimation.
• Understand why Ordinary Least Squares has been a very popular
estimation technique.
• Understand the five assumptions of the Classical Linear
Regression Model.
• Understand how to choose the appropriate functional form.
• Understand the applications associated with Indicator (Dummy)
Variables.
• Estimation Examples.
What is our goal ?
•
Our economic understanding about how
certain variables interact…….leads us to
develop a functional specification.
Dependent Variable = F (Explanatory Variables)
yt  0  1 xt   t
yt   0  1 xt1  .......   k xtk   t
How would we define a relationship ?
Income and Food Expenditure
Food Expenditure
300
250
200
150
100
50
0
0
200
400
600
800
Income
1,000
1,200
1,400
We can be more specific!
•
Ordinary Least Squares:
•
Minimizes the sum of the squared errors to
produce a line that best fits the data.
n
Min  
i 1
2
i


n
 (Y  Y )
i
i 1
i
2
How would we define a relationship ?
OLS Estimation
300
Food Expenditure
250
y = 0.1282x + 40.824
200
150
100
Y
Predicted Y
Linear (Predicted Y)
50
0
0
200
400
600
Income
800
1,000
1,200
1,400
Assumptions of
Classical Linear Regression Model
Assumption: 1
• Dependent variable is a linear function of a specific set
of independent variables, plus a disturbance.
yt  0  1 xt   t
Violations
• Wrong regressors.
• Nonlinearity.
• Changing parameters.
Assumptions of
Classical Linear Regression Model
Assumption: 2
• Expected value of disturbance term is zero.
E ( t )  0
Violations
• Biased intercept.
Assumptions of
Classical Linear Regression Model
Assumption: 3
• Disturbances (error term) have uniform variances and
are uncorrelated.
E ( t r )   , t  r
2
E ( t r )  0 , t  r
Violations
• Heteroskedasticity.
• Autocorrelated errors.
Assumptions of
Classical Linear Regression Model
Assumption: 4
• Observations on independent variables can be
considered fixed in repeated samples.
X  fixed in repeated samples
Violation
• Autoregression.
Assumptions of
Classical Linear Regression Model
Assumption: 5
• No exact linear relationship between independent
variables.
T
 (x
t
 x)  0
2
i 1
Violation
• Multicollinearity.
Interpreting Results
Regression Statistics
Multiple R
0.562956
R Square
0.316919
Adjusted R Square
0.298943
Standard Error
37.80161
Observations
40
ANOVA
df
SS
MS
F
1
25193.03
25193.03
17.6303
Residual
38
54300.56
1428.962
Total
39
79493.58
Regression
Significance F
0.000155854
Interpreting Results
Variable
Coefficients
Standard
Error
t Stat
P-value
Intercept
40.82
22.13
1.842
0.072962
Income
0.128
0.030
4.195
0.000156
Choosing a Functional Form
• Linear
• Quadratic
• Hyperbola
yt  0  1 xt   t
yt   0  1 xt   2 xt2   t
yt   0 
1
xt
 t
• Semi-Log
yt  0  1 ln xt   t
• Double Log
ln yt  0  1 ln xt   t
• Log-Inverse
ln yt   0 
1
xt
 t
Choosing a Functional Form
• Use economic theory.
• Plot the independent variable against the
dependent variable to discern pattern.
• First without any transformation.
• Then make the different transformations that you
may be interested to see and plot them against the
dependent variable.
Using Indicator Variables (Dummies)
Expenditure
Income
Structural
Dummy
52.50
258.30
0
58.32
343.10
0
81.79
425.00
0
119.90
467.50
0
125.80
482.90
0
100.46
487.70
1
121.51
496.50
1
100.08
519.40
1
127.75
543.30
1
104.94
548.70
1
• Capture Structural
Change
• Some unusual
occurrence that
isn’t capture
elsewhere in the
other variables
! Estimation Demo Using Excel !
See Example
Summary Questions
• What are the five assumptions of the classical linear regression
model?
• Describe in words, how Ordinary Least Squares works.
• What is measured by the R-Square term?
• How can you determine if a variable is statistically significant?
• What steps do you take to determine the appropriate functional
form for estimating an equation?
• When would you ever utilize an indicator (dummy) variable in your
estimation…..and how would you do it?