1. Outline 3: Regression Analysis: Estimation of Equations From Economic Theory for Business Management

2. 3.1 Examples of Demand Functions
Outline 3 Regression Analysis: Estimation of Equations from Economic Theory for Business Management  
3.1 Examples of Demand Functions
3.2 Introduction to Regression Analysis (Least Squares)
3.3 Diagnosis of Least Squares Estimation
3.4 Practicum in Regression Using EXCEL

3. 3.1 Examples: Demand for Whole Powdered Milk
Estimated Demand Equation for WPM in Brazil & Argentina: Tables 1a and 2a:

4. 3.1 Examples: Demand for Gasoline in the US
Estimated Demand Equation for Gasoline in the US: See Hughes, et.al. paper; equation (1) and Table 1 for Basic Model

5. 3.2 Introduction to Regression Analysis
Regression is a statistical method for estimating the parameters of an equation (intercept and slopes)
The coefficients (intercept and slopes) are estimated by:
Minimizing the sum of squared distances between the actual values of the dependent variable and the estimated values of the dependent variable

6. 3.2 Introduction to Regression Analysis
Regression can estimate a simple equation with one independent variable, more than one independent variable and more than one dependent variable (simultaneous equation systems)
Y = f(X)
Y = f(X1, X2, …, Xn)
Y = f(X1, X2, …, Xn) and Z = f(Y)
The coefficients (intercept and slopes) are estimated by:
Minimizing the sum of squared distances between the actual values of the dependent variable and the estimated values of the dependent variable

7. 3.2 Introduction to Regression Analysis
Now we add an error term where ε and η are regression error terms. These are stochastic equations as the hypothesized relationship contains randomness:
Y = f(X) + ε
Y = f(X1, X2, …, Xn) + ε
Y = f(X1, X2, …, Xn) + ε (example of PRPM(T))
Z = f(Y) + η
The coefficients (intercept and slopes) are estimated by:
Minimizing the sum of squared distances (squared errors) between the actual values of the dependent variable and the estimated values of the dependent variable

8. 3.2 Introduction to Regression Analysis
The statistical specification of these models are:
Y = a0 + a1 X + ε
Y = a0 + a1 X1 + a2 X2 + … + an Xn + ε
Z = b0 + b1 Y + η (example of PRPM(T) for as system)
The coefficients, i.e., intercept and slopes, or a’s and b’s above are parameters that regression analysis estimates. In the process the regression analysis also estimates the error term(s), namely ε and η.

9. 3.2 Introduction to Regression Analysis
Stochastic v. deterministic graphs and models
Review error term, actual Y versus predicted Y, on stochastic graph for
t = 1, …,4

10. 3.2 Introduction to Regression Analysis
Regression analysis estimates the hypothesized relation between the dependent variable(s) and independent variable(s).
The inference of causation that Y is a function of X comes from the theory that lead to the specification of the model.
Correlation measures the strength of the potential linear relation between two variables.

11. 3.3 Diagnostics of Least Squares Estimation
Comport with expectations from theory
Slope on price in sales model negative?
Statistical significance of the estimated parameters
Statistical significance / fit of the entire estimated model
Distribution of the error term: Normal? Test with Jarque Bera statistic

12. 3.3 Diagnostics of Least Squares Estimation
Use T distribution to test significance of estimated parameters (intercept and slopes):
Y = a0 + a1 X1 + a2 X2 + … + an Xn + ε
T distribution is the ratio of a normal to a chi square distributed variable (e.g., slope / standard error of slope)
T has a zero mean and thicker tails than normal
Tests to see if estimate is different from zero
Choose significance level; usually 5% or less; minimum strength of test from experience is 10%

13. 3.3 Diagnostics of Least Squares Estimation
The Coefficient of Determination: R2
Is a measure of the percent of the variation in the dependent variable that is explained by the independent variable
If a simple regression, R is the correlation coefficient
Ranges from 0 to 3.0
Derivation of R2:
3. Graphically
2. Mathematically

14. 3.3 Diagnostics of Least Squares Estimation
The F Statistic:
A test of the significance of R2 and the significance of the entire regression
Ratio of two variances is F distributed
Note that R2 is a ratio of the explained sum of squares to the total sum of squares

15. 3.5 Practicuum in Regression Using EXCEL
See Class Notes