1. 5  ECONOMETRICS CHAPTER Yi = B1 + B2 ln(Xi2) + ui
Functional Forms and Regression Models
The X and Y variables do not have to be linear.
(The model must be linear in parameters.) 
Yi = B1 + B2 ln(Xi2) + ui
This is okay, even though it is not linear in the
explanatory variable.

2. Y = bX  10,000 = 104 logbY = X  log1010,000 = 4 100 = e4.6051
Review of Logarithms
Y = bX  10,000 = 104
logbY = X  log1010,000 = 4
100 = e4.6051
loge100 = ln(100) =
ln(AB) = ln(A) + ln(B)
ln(Ak) = k ln(A)
ln(1) = 0

3. ln(Yi) = ln(A) + B2ln(Xi)
The Log-Linear Model
Yi = AXiB2
We ignore the error term, ui, for the time being.
We can take the natural log of both sides:
ln(Yi) = ln(AXiB2)
ln(Yi) = ln(A) + B2ln(Xi)
ln(Yi) = B1 + B2ln(Xi)

4. Demand curve w/ constant slope
Price and Quantity 20 40 60 80
100
120 25 50 75
125
150
Price ($)
Quantity (100s)
Demand curve w/ constant slope
Q = 150 – 1.25 P
slope = -1.25
% ΔQD
% ΔP
Price elasticity
of demand =
= slope x P/Q
Elasticity = x 40/100 =
Elasticity = x 100/25 =

5. Constant price elasticity of demand20 40 60 80
100
120
Price ($)
Quantity (100s)
Constant price elasticity of demand
Q = 64,000 P-2
slope = -128,000 P-3
Price elasticity
of demand
= slope x P/Q
Elasticity = x 80/10 =
Note:
ln(Q) = ln(64,000) – 2 ln(P)
ln(Q) = ln(P)
Elasticity = -2 x 40/40
= -2

6. Qi = B1 + B2 Pi elasticity = B2 x Pi / Qi ln(Qi) = B1 + B2ln(Pi)
Price and Quantity
Linear (constant slope) model:
If B2 = 2,150…
$1 increase in price
2,150 more units.
Qi = B1 + B2 Pi
elasticity = B2 x Pi / Qi
Log-linear (constant elasticity) model:
If B2 = 0.45…
10% increase in price
4.5% increase in units
ln(Qi) = B1 + B2ln(Pi)
elasticity = B2

7. ln(Yi) = B1 + B2ln(X2i) + B3ln(X3i) + ui
Multiple Log-Linear Models
ln(Yi) = B1 + B2ln(X2i) + B3ln(X3i) + ui
Cobb-Douglas Production Function
Output = A (LaborB1)(CapitalB2)
ln(Yi) = B1 + B2ln(Labori) + B3ln(Capitali) + ui

8. Double labor or capital, output increases but does not double.
Constant
returns to
scale
Double labor or capital, output increases but does not double.
Double both labor and capital, output doubles.

9. ln(Yi) = B1 + B2ln(Labori) + B3ln(Capitali) + ui
Cobb-Douglas Production Function
ln(Yi) = B1 + B2ln(Labori) + B3ln(Capitali) + ui
Constant returns to scale: B2 + B3 = 1
Decreasing returns to scale: B2 + B3 < 1

10. ln(Yi) = B1 + B2X2i + B3X3i + ui The Semilog Model
Log variable(s) only on one side of the equation.
Most useful for variables that grow exponentially.
Pop1 = 100
Pop2 = 100 x 1.04 = 104
Pop3 = 104 x 1.04 =
Pop4 = x 1.04 =

11. ln(Popi) = 4.63 + .0385 ti + ui A one year increase in time
results in a 3.85% change in population