1. Problems in Applying the Linear Regression Model Appendix 4A
The assumptions of the linear regression model don’t always hold in the real world
We now examine statistical problems, which is the central focus of the economic sub-field called econometrics
Autocorrelation
Heteroscedasticity
Specification and Measurement Error
Multicollinearity
Simultaneous equation relationships and the identification problem
Nonlinearities

2. Autocorrelation also known as serial correlation
Problem:
Coefficients are unbiased
but t-values are unreliable
Symptoms:
look at a scatter of the error terms to see if there is a pattern, or
see if Durbin Watson statistic is far from 2.
Cures:
Find more variables that explain these patterns
Take first differences of data: Q = a + b•P 20

3. Scatter of Error Terms Positive Autocorrelation Figure 4A.1 page 171Y 1 2 6 3 4 5 7 8 X 21

4. 2. Heteroscedasticity Problem: Coefficients are unbiased
t-values are unreliable
Symptoms:
different variances for different sub-samples
scatter of error terms shows increasing or decreasing dispersion
Cures:
Transform data, e.g., transform them into logs
Take averages of each sub-sample and use weighted least squares 22

5. Scatter of Error Terms Heteroscedasticity
Height
alternative
log Ht = a + b•AGE
AGE 23

6. 3. Specification & Measurement Error
Salary = a + b (Strike Outs) in baseball
b is positive !!!
Why?
Omitted variable which is the number of Hits
Salary = c + d (Strike Outs) + e ( Hits )
here d is negative and e is positive

7. Specification & Measurement Error
Problem:
Coefficients are biased – we can even have the wrong sign as in the baseball example
Even adding more observations will not cure this bias
Symptoms:
The results don’t make economic sense
Cure:
Think through the relationships and find the missing variables in the specification
See if the new specification improves the fit (higher R2) and makes economic sense. 22

8. 4. Multicollinearity Regression Output
Q = Pd -.9 Pg
(1.2) (1.45)
R-square = .87
(t-values in parentheses)
N = 100 observations
Notice that:
R-square is 87%
But that neither coefficient is statistically significant.
Sometimes independent variables aren’t independent.
EXAMPLE: Let Q = Eggs sold
Q = a + b Pd + c Pg
where Pd is price for a dozen eggs
and Pg is the price for a gross of eggs. 19

9. Multicollinearity Problem: Coefficients are unbiased
The t-values are small, often insignificant
Symptoms:
High R-squares but low t-values
Cures:
Drop a variable. Usually the remaining variable becomes significant.
Do nothing if forecasting, since the added R-square of more variables is worthwhile 22

10. 5. Identification Problem and the Simultaneity Problem
Coefficients are biased
Symptom:
Independent variables are known to be part of a system of equations
Cure:
Use as many independent variables as possible 18

11. Graphical Explanation of the Identification Problem
Suppose we estimate the following demand curve Q = a + b P.
Suppose Supply varies and Demand is FIXED.
All points lie on the demand curve
The demand curve is said to be identified. S1 S2 S3
Demand
|____________________________Quantity
quantity 4

12. Suppose instead that SUPPLY is Fixed
Let DEMAND shift and supply is fixed on doesn’t change.
All Points are on the SUPPLY curve.
We say that the SUPPLY curve is identified.
Supply D3 D2 D1
quantity 5

13. When both Supply and Demand Vary
Often both supply and demand vary.
Equilibrium points are in shaded region.
A regression of Q = a + b P will be neither a demand nor a supply curve. S2 S1 ? D2 D1
quantity 6

14. Simultaneous Systems Demand is Qd = a + b P + c Y + e1
Supply is Qs = d + e P + f W + e2
Where P is price, Y is income, W is the wage rate, and each has an error term.
Notice that P is in both of the demand and supply function. P is “endogenously” determined by both demand and supply.
The simultaneity problem is that price is not independent, as it is determined by the whole system
The cure for this problem is usually to have as many independent variables as possible in the demand regression to make demand act like it is “fixed”.

15. 6. Nonlinear Forms
Semi-logarithmic transformations. Sometimes taking the logarithm of the dependent variable or an independent variable improves the R Examples are:
log Y =  + ß·X.
Here, Y grows exponentially at rate ß in X; that is, ß percent growth per period.
Y =  + ß·log X. Here, Y doubles each time X increases by the square of X.
Ln Y = X Y X

16. Reciprocal Transformations
The relationship between variables may be inverse. Sometimes taking the reciprocal of a variable improves the fit of the regression as in the example:
Y =  + ß·(1/X)
shapes can be:
declining slowly
if beta positive
rising slowly
if beta negative Y
E.g., Y = ( 1/X) X

17. Polynomials
Quadratic, cubic, and higher degree polynomial relationships are common in business and economics.
Profit and revenue are cubic functions of output.
Average cost is a quadratic function, as it is U-shaped
Total cost is a cubic function, as it is S-shaped
TC = ·Q + ß·Q2 + ·Q3 is a cubic total cost function.
If higher order polynomials improve the R-square, then the added complexity may be worth it.

18. Multiplicative or Double Log
With the double log form, the coefficients are elasticities
Q = A • P b • Yc • Ps d
multiplicative functional form
So: Ln Q = a + b•Ln P + c•Ln Y+ d•Ln Ps
Transform all variables into natural logs
Called the double log, since logs are on the left and the right hand sides. Ln and Log are used interchangeably. We use only natural logs. 16

19. Soft Drink Case, pp. 167-168 a cross section of 50 states
Linear Specification
Cans = Price Income Temp
Predictor Coeff StDev T P
Constant
Price
Income
Temp
R-Sq = 69.8% R-Sq(adj) = 67.7%
The Price elasticity in Wyoming is = (DQ/DP)(P/Q) = (2.31/102)=

20. Double Log Soft Drink Case
Ln Cans = Ln Price Ln Income Ln Temp
Predictor Coef Std Dev T P
Constant
Ln Price
Ln Income
Ln Temp
R-Sq = 67.4% R-Sq(adj) = 65.1%
Characterize the demand for soft drinks in the US.
Are soft drinks inelastic? Are they luxuries?
Which specification fits the data better?