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

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

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

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

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

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

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

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.

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

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.

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

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%

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

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

3.5 Practicuum in Regression Using EXCEL See Class Notes