Problems in Applying the Linear Regression Model Appendix

Problems in Applying the Linear Regression Model Appendix 4 A • • 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 1. 2. 3. 4. 5. 6. Autocorrelation Heteroscedasticity Specification and Measurement Error Multicollinearity Simultaneous equation relationships and the identification problem Nonlinearities Slide 1

1. Autocorrelation • Problem: also known as serial correlation » 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: 1. Find more variables that explain these patterns 2. Take first differences of data: Q = a + b • P Slide 2

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

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: 1. Transform data, e. g. , transform them into logs 2. Take averages of each sub-sample and use weighted least squares Slide 4

Scatter of Error Terms Heteroscedasticity Height alternative log Ht = a + b • AGE 1 2 5 8 AGE Slide 5

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 Slide 6

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 R 2) and makes economic sense. Slide 7

4. Multicollinearity • Sometimes independent variables aren’t independent. EXAMPLE: Let Q = Eggs sold Q = a + b P d + c Pg where Pd is price for a dozen eggs and Pg is the price for a gross of eggs. • Regression Output Q = 22 - 7. 8 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. Slide 8

Multicollinearity • Problem: » Coefficients are unbiased » The t-values are small, often insignificant • Symptoms: » High R-squares but low t-values • Cures: 1. Drop a variable. Usually the remaining variable becomes significant. 2. Do nothing if forecasting, since the added Rsquare of more variables is worthwhile Slide 9

1. 5. Identification Problem and the Simultaneity Problem • 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 Slide 10

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. P S 1 S 2 S 3 Demand quantity Slide 11

Suppose instead that SUPPLY is Fixed P • 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 D 3 D 2 D 1 quantity Slide 12

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. P S 2 S 1 ? D 2 D 1 quantity Slide 13

Simultaneous Systems 1. Demand is Qd = a + b P + c Y + e 1 2. Supply is Qs = d + e P + f W + e 2 Ø Where P is price, Y is income, W is the wage rate, and each has an error term. Ø Notice that P is in both of the demand supply function. P is “endogenously” determined by both demand supply. Ø The simultaneity problem is that price is not independent, as it is determined by the whole system Ø 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”. Slide 14

6. Nonlinear Forms • Semi-logarithmic transformations. Sometimes taking the logarithm of the dependent variable or an independent variable improves the R 2. Examples are: • log Y = + ß·X. Y Ln Y =. 01 +. 05 X X » Here, Y grows exponentially at rate ß in X; that is, ß percent growth period. • Y = + ß·log X. Here, Y doubles each time X increases by the square of X. Slide 15

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) Y E. g. , Y = 500 + 2 ( 1/X) • shapes can be: » declining slowly • if beta positive » rising slowly • if beta negative X Slide 16

• Quadratic, cubic, and higher degree polynomial relationships are common in business and economics. w Profit and revenue are cubic functions of output. w Average cost is a quadratic function, as it is Ushaped w Total cost is a cubic function, as it is S-shaped • TC = ·Q + ß·Q 2 + ·Q 3 is a cubic total cost function. • If higher order polynomials improve the R-square, then the added complexity may be worth it. Slide 17

Multiplicative or Double Log • With the double log form, the coefficients are elasticities • Q = A • P b • Yc • P s 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. Slide 18

Soft Drink Case, pp. 167 -168 a cross section of 50 states Linear Specification Cans = 515 - 242 Price + 1. 19 Income + 2. 91 Temp Predictor Constant Price Income Temp R-Sq = 69. 8% Coeff 514. 8 -241. 80 1. 195 2. 9136 St. Dev 113. 2 43. 65 1. 688 0. 7071 T P 4. 55 0. 000 -5. 54 0. 000 0. 71 0. 483 4. 12 0. 000 R-Sq(adj) = 67. 7% The Price elasticity in Wyoming is = ( Q/ P)(P/Q) = -241. 8(2. 31/102)= -5. 476 Slide 19

Double Log Soft Drink Case Ln Cans = 2. 47 - 3. 17 Ln Price + 0. 202 Ln Income + 1. 12 Ln Temp Predictor Constant Ln Price Ln Income Ln Temp Coef Std Dev T P 2. 466 1. 385 1. 78 0. 082 -3. 1695 0. 6485 -4. 89 0. 000 0. 2020 0. 1834 1. 10 0. 277 1. 1196 0. 2611 4. 29 0. 000 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? Slide 20