Narrow Results

Fuzzy regression has been applied to marketing, management, and sales forecasting for many years. In this paper, two types of forecasting methods within the framework of fuzzy regression analysis are discussed. The first type is a conventional forecasting method, which intends to find the value of a dependent variable when the given values of independent variables are beyond the range covered by historical data. The second type considers forecasting in which the necessary values of independent variables are desired when the given value of a dependent variable does not fall within the range of historical data. While the first type can be applied in planning, the second one is useful for control and management. Numerical examples are provided for illustration.

The following sections are included:

• INTRODUCTION

• POPULATION PARAMETERS AND THE REGRESSION MODELS
  • Data Description
  • Building the Population Regression Model
  • Sample Versus Population Regression Model

• THE LEAST-SQUARES ESTIMATION OF α AND β
  • Scatter Diagram
  • The Method of Least Squares
  • Estimation of Intercept and Slope

• STANDARD ASSUMPTIONS FOR LINEAR REGRESSION

• THE STANDARD ERROR OF ESTIMATE AND THE COEFFICIENT OF DETERMINATION
  • Variance Decomposition
  • Standard Error of Residuals (Estimate)
  • The Coefficient of Determination

• THE BIVARIATE NORMAL DISTRIBUTION AND CORRELATION ANALYSIS
  • The Sample Correlation Coefficient
  • The Relationship Between r and b
  • The Relationship Between r and R²

• Summary

• Appendix 13A Derivation of Normal Equations and Optimal Portfolio Weights

• Appendix 13B The Derivation of Equation 13.16

• Appendix 13C The Bivariate Normal Density Function
  • Using a Mathematics Aptitude Test to Predict Grade in Statistics

• Appendix 13D American Call Option and the Bivariate Normal CDF
  • Valuating American Option

• Questions and Problems

In this chapter, we shall proceed one step further to investigate how a conditional mean as mentioned in Chap. 2 could be estimated using linear statistical relationship between the two variables. The two-variable linear regression is studied and an application on financial futures hedging will be investigated later in the chapter.