Example 5.3
The use of non sample information
Pilar González and Susan Orbe
Dpt. Applied Economics III (Econometrics and Statistics)
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Example 5.3 Restricted Least Squares
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Contents
1
5.3.1. Pizza consumption I.
2
5.3.2. Pizza consumption II.
3
5.3.3. Chicken consumption I.
4
5.3.4. Chicken consumption II.
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Example 5.3 Restricted Least Squares
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Contents
1
5.3.1. Pizza consumption I.
2
5.3.2. Pizza consumption II.
3
5.3.3. Chicken consumption I.
4
5.3.4. Chicken consumption II.
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Example 5.3 Restricted Least Squares
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Example 5.3.1. Pizza consumption I.
Questions.
Load the data file pizza.gdt.
a. Specify a regression model to explain pizza consumption as a linear function of
income and age.
b. It is believed that the marginal effect of income on consumption is equal to the
marginal effect of age on consumption but with opposite sign. Introduce this non
sample information in the regression model specified in item a. Estimate the
restricted model.
b. Comment on the results.
Procedure.
Regression Model: pizzai = β1 + β2 incomei + β3 agei + ui
Restriction: β2 = −β3
Restricted Model: pizzai = β1 + β2 (incomei − agei ) + ui
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Example 5.3.1. Pizza consumption I.
To estimate the restricted model, generate the new regressor (income − age)
and name it IM A.
The estimation results of the restricted model appear in the figure below.
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Example 5.3 Restricted Least Squares
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Example 5.3.1. Pizza consumption I.
Results.
SRF:
p\
izzai = 158.718 + 1.47049 incomei − 1.47049 agei
i = 1, . . . , 40
• Interpretation of the estimated coefficients:
- The estimated consumption of pizza amounts to $158.718 when the variables income
and age take the value zero.
- It is estimated that the consumption of pizza increases by $1.47049 when income
increases by $1000 holding age fixed.
- It is estimated that the consumption of pizza decreases by $1.47049 when age
increases by one year holding income fixed.
• If the restriction imposed on the coefficients is true, the Restricted Least Squares
(RLS) estimator in linear, unbiased and its variance is smaller than the variance of
the OLS estimator. If the restriction is not true, the RLS estimator is biased.
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Example 5.3 Restricted Least Squares
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Contents
1
5.3.1. Pizza consumption I.
2
5.3.2. Pizza consumption II.
3
5.3.3. Chicken consumption I.
4
5.3.4. Chicken consumption II.
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Example 5.3 Restricted Least Squares
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Example 5.3.2. Pizza consumption II.
Questions.
Open the session pizza5.1.1.
a. Estimate a regression of consumption on age and income that allows to
measure the following effect: “The expected change in the consumption of
pizza generated by a unit change in income depends on the age of the client.”
Save the model as an icon to the session.
b. If you knew that β2 = 5, how would you estimate the regression model?
c. Comment on the results.
Procedure.
Regression Model:
pizzai = β1 + β2 incomei + β3 (agei × incomei ) + ui
i = 1, ..., N
Restriction: β2 = 5
Restricted Model: pizzai − 5 incomei = β1 + β3 (agei × incomei ) + ui
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Example 5.3.2. Pizza consumption II.
To estimate the restricted model, generate the new dependent variable
pizzai − 5 × incomei and name it pizza5I. The regressor agei × incomei ,
denoted by AI, was generated in the Example 5.1.2.
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Example 5.3.2. Pizza consumption II.
Results
SRF:
p\
izzai = 99.5067 + 5 incomei − 0.0887108 (agei × incomei )
i = 1, . . . , 40
• Estimated marginal effects:
- Marginal effect of income: It is estimated that the consumption of pizza changes by
(5 − 0.0887108 agei ) dollars when income increases by $1000 holding age constant.
This estimated variation in the expected consumption is not constant throughout the
sample because it depends on the client’s age: the older the client, the smaller the
marginal effect of income on consumption.
- Marginal effect of age: It is estimated that the consumption of pizza decreases by
(0.0887108 incomei ) dollars when age increases by 1 year holding annual income
constant. This estimated variation in the expected consumption is not constant
throughout the sample because it depends on the client’s income: the higher the level
of income, the smaller the marginal effect of age on consumption.
• If the restriction imposed on the coefficients is true, the RLS estimator in linear,
unbiased and its variance is smaller than the variance of the OLS estimator. If the
restriction is not true, the RLS estimator is biased.
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Example 5.3 Restricted Least Squares
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Contents
1
5.3.1. Pizza consumption I.
2
5.3.2. Pizza consumption II.
3
5.3.3. Chicken consumption I.
4
5.3.4. Chicken consumption II.
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Example 5.3 Restricted Least Squares
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Example 5.3.3. Chicken consumption I.
Questions.
Load the data file chicken.gdt.
a. Estimate a regression of consumption on income, price of chicken and price of
pork. Save the results as an icon to the session.
b. Assume that β3 + β4 = 1. Estimate the model specified in the previous item
subject to this non sample information.
c. Comment on the results and save the session.
Procedure.
Regression Model: Yt = β1 + β2 X2t + β3 X3t + β4 X4t + ut
Restriction: β3 + β4 = 1
t = 1, ...T
Restricted Model: Yt − X4t = β1 + β2 X2t + β3 (X3t − X4t ) + ut
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Example 5.3 Restricted Least Squares
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Example 5.3.3. Chicken consumption I.
To estimate the restricted, generate the new dependent variable Y − X4 and
name it Y mX4. You have to generate as well the new regressor X3 − X4 and
name it X3mX4.
The results of estimating the restricted model appear in the figure below.
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Example 5.3 Restricted Least Squares
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Example 5.3.3. Chicken consumption I.
Results.
SRF:
Ybt = 26.4943 + 0.00777726 X2t − 0.889705 X3t + 1.889705 X4t
•
t = 1990, . . . , 2012
Estimated marginal effects of the explanatory variables:
- It is estimated that the consumption of chicken increases by 0.00777726 kg when the disposable income
increases by e1 and the prices of chicken and pork remain constant.
- It is estimated that the consumption of chicken decreases by 0.889705 kg when the price of chicken
increases by e1 and the explanatory variables income and price of pork remain constant.
- It is estimated that the consumption of chicken increases by 1.889705 kg when the price of pork increases
by e1 and the explanatory variables income and price of chicken remain constant.
•
If the restriction imposed on the coefficients is true, the RLS estimator in linear,
unbiased and its variance is smaller than the variance of the OLS estimator. If the
restriction is not true, the RLS estimator is biased.
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Example 5.3 Restricted Least Squares
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Contents
1
5.3.1. Pizza consumption I.
2
5.3.2. Pizza consumption II.
3
5.3.3. Chicken consumption I.
4
5.3.4. Chicken consumption II.
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Example 5.3 Restricted Least Squares
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Example 5.3.4. Chicken consumption II.
Questions.
a. Load the data file chicken.gdt and estimate a regression model to determine
the consumption of chicken as a quadratic function of its price and a linear
trend. Save the results as an icon to the session.
b. Write down the restricted model that results from including the information
“the explanatory variable price of chicken is not relevant” in the model
specified in item a.
c. Estimate the restricted model and comment on the results.
Procedure.
Regression Model: Yt = β1 + β2 X3t + β3 X32t + β4 timet + ut
Restriction: β2 = β3 = 0
Restricted model: Yt = β1 + β4 timet + ui
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Example 5.3 Restricted Least Squares
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Example 5.3.4. Chicken consumption II.
The estimation results of the restricted model appear in the figure below.
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Example 5.3.4. Chicken consumption II.
Results.
SRF:
Ybt = 26.9083 + 1.06344 timet
t = 1990, . . . , 2012
• Interpretation of the estimated coefficients:
- 26.9083 kg: the estimated consumption of chicken in 1989.
- 1.06344 kg: the estimated annual variation in the consumption of chicken.
• If the restriction imposed on the coefficients is true, the RLS estimator in linear,
unbiased and its variance is smaller than the variance of the OLS estimator. If the
restriction is not true, the RLS estimator is biased.
Pilar González and Susan Orbe | OCW 2014
Example 5.3 Restricted Least Squares
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