Problem Set #3 Key
Sonoma State University
Economics 494- Seminar in Quantitative Marketing II
Dr. Cuellar
Endogeneity II
We saw from the profit maximizing conditions that estimating the effect of advertising on sales
suffers from endogeneity.
1. Consider the simple model of demand:
S =β0 +β1P + β2I + β2A + u
where S is unit sales, P is the price of output, A is advertising and I is income.
a. Construct an idealized system of equations that will exactly identify the endogenous
variables in the model. Explain fully.
S =β0 +β1P + β2I + β2A + u
P =α0 +α1S +α2Input Price +ε
A = ϒ0 + ϒ1S + ϒ2Price of Advertising + ζ
b. Suppose now that we assume that supply is perfectly elastic and that we have a
conventional downward sloping demand. Using a simple supply and demand paradigm,
show the effects of an exogenous shift in the demand curve caused by an increase in the
error term.
Price
P
D1
S
D0
Q0
Q1
Quantity
c. Can you justify the assumption of a perfectly elastic supply curve? Assume the supply and
demand curves represent the market for wine. Explain fully.
d. What does this say about the endogeneity discussed in part (a) above? Show graphically
and explain fully.
Price is no longer endogenous.
e. Given your answer in (b) above, construct an idealized system of equations that will exactly
identify the endogenous variables in the model. Explain fully.
S =β0 +β1P + β2I + β2A + u
A = ϒ0 + ϒ1S + ϒ2Price of Advertising + ζ
2. Consider now the total revenue function: TR =β0 +β1S + β2I + β2A + u.
a. Construct an idealized system of equations that will exactly identify the endogenous
variables in the model. Explain fully.
TR =β0 +β1S + β2I + β2A + u.
S =α0 +α1P +α2Input Price +ε
A = ϒ0 + ϒ1TR + ϒ2Price of Advertising + ζ
b. Suppose now that we assume that supply is perfectly inelastic and that we have a
conventional downward sloping demand. Using a simple supply and demand paradigm,
show the effects of an exogenous shift in the demand curve caused by an increase in the
error term.
Price
S
P1
D1
P0
D0
Q
Quantity
c. Can you justify the assumption of a perfectly inelastic supply curve? Assume the supply and
demand curves represent the market for a specific brand of wine, say Russian River Valley
pinot noir. Explain fully.
d. What does this say about the endogeneity discussed in part (a) above? Show graphically
and explain fully.
Sales (i.e., quantity) is no longer endogenous.
e. Given your answer in (b) above, construct an idealized system of equations that will exactly
identify the endogenous variables in the model. Explain fully.
TR =β0 +β1S + β2I + β2A + u.
A = ϒ0 + ϒ1S + ϒ2Price of Advertising + ζ
3. Consider the model estimated in question (1) of Problem Set #1:
lnTR=β0 + β1lnQ+ β2lnI+ β3lna+ β4lnA+u
a. Set up a system of structural equations representing sales and advertising based on the
variables in the data set. Unfortunately, the price of advertising is not included in the data
set. The researchers instead used the consumer price index (CPI).
lnTR=β0 + β1lnQ+ β2lnI+ β3lna+ β4lnA+u
lnQ =α0 +α1P +ε
lna = ϒ0 + ϒ1TR + ϒ2CPI( in place of the Price of Advertising) + ζ
b. Are you able to fully identify your model?
There are no exogenous variables excluded from the TR equation that would allow us to identify
demand. Such as:
lnQ =α0 +α1P +α2Input Price +ε
c. What must be true for you to not have an identification problem? Explain fully.
If you assume a perfectly inelastic supply, then Q is exogenous.
d. Use the Hausman test, to test for endogeneity between sales and advertising. Show each
step and discuss your results.
Hausman Test
Reduced
Structural
Form
Equation
Equation
w/Error Term
Ln(Adv)
Ln(Revenue)
Ln(Quantity)
1.209
-0.641
[0.00]**
[0.27]
Ln(Income)
-0.508
1.336
[0.02]*
[0.00]**
Ln(Average Adv)
0.261
0.069
[0.00]**
[0.51]
Ln(CPI)
-0.414
[0.02]*
Error
-0.277
[0.57]
Ln(Advertising)
0.43
[0.35]
Constant
6.516
-9.81
[0.00]**
[0.01]**
Observations
50
50
2
Adjusted R
0.92
0.67
F-Statistic
137.52
20.55
Absolute value of t-statistics in brackets
* significant at 5% level; ** significant at 1% level
The coefficient on the error term is not statistically significant indicating the absence of
endogeniety.
e. Re-estimate your equation using two-staged least squares instrumental variable regression.
Question
lnSales
Revenue
(3e)
OLS
2SLS
-0.339
-0.641
[0.16]
[0.25]
lnPrice
lnIncome
lnAdvertising
lnAverage Advertising
Constant
1.147
[0.00]**
0.185
[0.24]
0.119
[0.05]*
-8.201
[0.00]**
50
0.67
Sales
(4e)
OLS
2SLS
-0.309
-0.38
[0.00]** [0.00]**
1.336
0.786
0.832
[0.00]** [0.00]** [0.00]**
0.43
0.363
0.236
[0.33] [0.00]** [0.13]
0.069
0.025
0.08
[0.49]
[0.39]
[0.25]
-9.81
-6.065
-6.292
[0.00]** [0.00]** [0.00]**
50
50
50
0.95
Observations
Adjusted R2
p-values in brackets
* significant at 5% level; ** significant at 1% level
i.
Be sure to test your instrument for relevance.
lnCPI
Constant
lnadv
1.39
[0.00]**
-5.887
[0.00]**
50
0.34
Observations
Adjusted R2
Absolute value of t-statistics in brackets
* significant at 5% level; ** significant at 1% level
Note also, that the F-statistic in the first stage regression is greater than 10.
f.
Do your OLS and IV regression estimates produce different elasticities?
4. Consider the model estimated in question (3) of Problem Set #1:
lnQ=β0 + β1lnP+ β2lnI+ β3lna+ β4lnA+u
a. Set up a system of structural equations representing sales and advertising based on the
variables in the data set. Unfortunately, the price of advertising is not included in the data
set. The researchers instead used the consumer price index (CPI).
lnQ=β0 + β1lnP+ β2lnI+ β3lna+ β4lnA+u
lna = ϒ0 + ϒ1Q + ϒ2CPI( in place of the Price of Advertising) + ζ
b. Are you able to fully identify your model?
No.
c. What must be true for you to not have an identification problem? Explain fully.
If you assume supply is perfectly elastic, then it is exogenous.
d. Use the Hausman test, to test for endogeneity between sales and advertising. Show each
step and discuss your results.
Hausman Test
lnadv
lnqty
lnPrice
-0.571
-0.38
[0.00]**
[0.00]**
lnIncome
0.696
0.832
[0.01]**
[0.00]**
lnAverage Advertising
0.509
0.08
[0.00]**
[0.26]
lnCPI
-0.575
[0.02]*
Residuals
0.143
[0.39]
lnAdvertising
0.236
[0.13]
Constant
-1.609
-6.292
[0.28]
[0.00]**
Observations
50
50
2
Adjusted R
0.83
0.95
Absolute value of t-statistics in brackets
* significant at 5% level; ** significant at 1% level
The coefficient on the error term is not statistically significant indicating the absence of
endogeniety.
e. Re-estimate your equation using two-staged least squares instrumental variable regression.
i. Be sure to test your instrument for relevance.
Tested in 3e(i).
f.
Do your OLS and IV regression estimates produce different elasticities?
5. Can parts 3(c) and 4(c) both be true?
No, supply cannot be perfectly inelastic and perfectly elastic at the same time. However, you could
assume a perfectly inelastic supply in the short run and a perfectly elastic supply in the long run.