CHAPTER 7:
DEMAND ESTIMATION AND
FORECASTING
I. INTRODUCTION
A. Specification of Empirical Demand
Functions
1. Microeconomic Theory tells us what
variables to include in a demand
model and what the signs of the
parameters should be.
2. Choosing a functional form.
a. linear
b. log - linear
B. Market Determined vs. Management
Determined Prices
1. Estimating Industry demand for
price-taking firms.
2. Estimating Demand for price-setting
firms.
C. Definitions.
1. Endogenous Variable: Variable
whose value is determined by a
system of equations.
2. Exogenous Variable: Variable whose
value is determined outside a system
of equations.
D. Linear Demand Specification.
1. General form
Qx = a + bPx + cM + dPr + eN
Where:
b=
ΔQ x
ΔP
c=
ΔQ x
Δm
d=
ΔQ x
ΔPr
e
ΔQ x
= ΔN
; b < 0 law of demand
; c > 0 normal good,
c < inferior good
Substitutes
; d > 0 Substitutes
d < 0 Complements
;e>0
2. Elasticities
a) E =
ΔQ x
( ΔPx
since b =
Px
)( Q x
)
ΔQ x
ΔPx
P
Eˆ  bˆ. x
Qx
M
Eˆ m  cˆ.
b)
Qx
Pr
Eˆ xr  dˆ.
c)
Qx
3. If the parameter estimates are
significant, the elasticities are
significant.
4. If no particular point on the demand
curve is specified, it is customary to
evaluate the elasticity at the mean
values Px and Qx.
D. Log-linear or constant elasticity
functional form.
1. Qx = aPxbMcPrdNe
2. Transformation:
ln Qx = ln a + b ln Px + c ln M +
d ln Pr + e ln N
3. The expected signs for the
parameters are the same as for the
linear function.
4. bˆ, cˆ, dˆ are also elasticities.
III ESTIMATING INDUSTRY
DEMAND FOR PRICE TAKING
FIRMS.
A. Problem:
Demand: Q = a + bP + cM + Ed
Supply: Q = h + kP + lPI + Es
1. Price is an endogenous variable – as
is the case with all price-taking
firms. Price is therefore correlated
with the random error term which
results in biased estimates.
2. Simultaneous Equation Bias – When
the (OLS) estimation method is used
to estimate the parameter of an
equation for which one or more of
the explanatory variables are
endogenous.
B. Solution: Two-Stage Least Squares
1. Specify the industry supply and demand
equations.
Demand: Q = a + bP + cM + dPr
Supply: Q = h + kP + lPt
2. Check for identification of industry
demand.
In order for demand to be identified, so
that the sample data will trace out the
true demand curve, the specification of
supply must include at least one
exogenous variable that is not in the
demand equation.
3. Process:
a. P, the right hand side variable is
regressed against all of the exogenous
variables in the system.
b. Stage 2: P̂ is now exogenous so that
the following equation is estimated:
Q = a + b P̂ + cM + dPr
c. Example: Estimating world-wide
demand for copper: Data on page 306
1. Equations:
Q = a + bPcopper + cM + dPAluminum
Q = e + fPcopper + gT + hX
2. Is the demand equation identified?
3. Computer printout page 263.
4. Are the parameter estimates
significant at the 5% level?
5. Evaluate the price, income, and
cross-price elasticities.
- Use the last year from the data set.
III. Estimating Demand for price-setting
firm, checkers pizza: Data page 307
A. Linear functional form.
1. Q = a + bP + cM + dPal + cPmac
Where Q = sales of pizza at
Checkers Pizza
P = price of pizza at
checkers.
M = average annual H.H.
income.
Pal = price of pizza at Als.
Pbmac = Price of Big Mac at
McDonalds.
2. Evaluate computer printout.
a) Which parameter estimates are
significant at the 5% level of
significance?
b) As the F-statistic significant at
the 5% level of significance.
c) Write the predictive model.
3. Evaluate elasticities around the values
assigned to the independent variable.
P
Eˆ  bˆ( )
Q
 9.05 
(-213.422)  2784.4 
=
= -0.694
M
Eˆ m  cˆ( )
Q
= (0.9109)
= 0.871
 26,614 


2784
.
4


PAL
ˆ
ˆ
EAL  d (
)
Q
= (101.303)
= 0.368
 10.12 


2784
.
4


PBMAC
Eˆ BMAC  eˆ(
)
Q
= (71.8448)
= 0.030
 1.15 


 2784.4 
B. Log linear functional form.
1. Q = aPbMcPALdPBmace
2. Transform
ln Q = ln a + b ln P + c ln M
+ d ln Pal + e ln PBmac
3. Evaluate computer printout on page
269.
a) Which parameter estimates are
significant at the 5% level of
significance?
b) Is the F-statistic significant at the
5% level of significance?
c) Write the predictive equation in
exponential form.
4. What are the elasticities?
III. TIME-SERIES FORECASTING.
A. Linear trend forecasting.
1. Common business applications:
Sales over time.
2. Hypothesize a linear trend between
sales and time.
Qt = a + bt
where:
Qt = sales in dollars
t = time.
Qˆ t  aˆ bˆt
3. If there is a statistically significant
trend, you can use the model to
forecast by extrapolating into future
time periods.
B. Example of linear trend forecasting:
Terminator Pest Control.
1. Hypothetical data: figure 7.4
computer print-out on page 274.
2. Test for significance at the 0.01 level
of significance.
tcritical =
3. Forecast future sales.
Qˆ t  aˆ bˆt
Q̂
= 46.57 + 4.53t
April 2000
May 2000
June 2000
= 46.57 + 4.53(16)
= 119.
Q̂ 17 = 46.57 + 4.53(17)
= 123.6
Q̂ 18 = 46.57 + 4.53(18)
= 128.1
Q̂ 16
C. Example: Lumber sales for Georgia
Lumber Products
Observations: 2001 (III) – 2003 (II)
1. Computer printout – page 273.
2. Test for significance at the 0.05 level
t critical =
3. Forecast future price of lumber for
next two quarters.
P̂ = 2066 + 25t
P̂ 2003(III) = 2066 + 25 (9)
= $ 2,291 per ton
P̂ 2003(IV) = 2066 + 25 (10)
= $ 2,316 per ton
IV. SEASONAL OR CYCLICAL
VARIATION.
A. Example: Retail sales and Christmas
season.
B. Graphical depiction (figure 7.5)
1. There is an upward shift of the
trend line in the fourth quarter.
2. To take this into account:
Estimate: Qˆ t = a + bt + c
Where c = 0 for quarters 1, 2, 3 and
1 for quarter 4.
C. Correcting for seasonal variation
using a dummy variable
1. A dummy variable is a variable that
takes on values of 0 or 1.
2. Process:
a. D = 0 for first three quarters and 1
for fourth quarter (Table 7.1)
b. Estimate:
Qt  a  bt cD
c. To forecast sales in the first three
quarters
Qˆ t  aˆ bˆt
To forecast sales in the fourth
quarter
ˆ
Qˆ t  (aˆ  cˆ)  bt
D. Multiple Dummy Variables
1. Seasonal Variation in every quarter
Specify:
Qt = a + bt + c1D1 + c2D2 + c3D3
Where:
D1, D2, D3 are dummy variables for
quarters 1, 2, 3
2. Hypothetical Example: Sales data
for statewide trucking company.
a. Data: Table 7.2
b. Computer Printout: Pg. 278
1. Test for significance at the 0.05
level of significance.
2. Interpret meaning of:
a. Slope Parameters
b. Intercept Parameter
3. Predicted Sales:
Q̂ 2005, I =139,625–69,788+2737.5 (17)
Q̂ 2005, II =139,625–58,775+2737.5 (18)
Q̂ 2005, III =139,625–62,013+2737.5 (19)
Q̂ 2005, IV =139,625+2737.5 (20)
Chapter 7 Assignment
Technical Problems: 1, 2, 3, 7, 8, 9, 10
Applied Problems: 1, 4
Not Responsible for Section 7.7