# CHAPTER 7

# MULTIPLE REGRESSION MODEL

## Outline

**7.1** Model Specification

**7.2** Shrinkage Estimators and Testing

**7.3** Bias and Risk Expressions

**7.4** Comparison

**7.5** Problems

In this chapter, we discuss the multiple regression model in detail, beginning with the estimation of the model parameters and test of hypothesis. In the continuation, we introduce improved estimators of the regression coefficients with their dominance properties. In this case, we derive the characteristic properties based on the so-called symmetric balanced loss function.

# 7.1 Model Specification

The multiple regression model is arguably the most widely used statistical tool applied in almost every discipline of the modern era. The estimation of parameters of the multiple regression model is of interest to many users. To deal with a common multiple regression equation, consider the linear model

(7.1.1)

where *y* = (*y*_{1}, …, *y*_{n})′ is an (*n* × 1) vector of observations, *X* = (*x′*_{1}, …, *x′*_{n})′ is a nonstochastic (*n* × *p*) matrix of full rank *p*, **β** is a (*p* × 1) vector of unknown regression coefficients, and ε is an (*n* × 1) random error-vector distributed as *M*^{(n)}_{t} (**0**, σ^{2}**V**_{n}, γ_{o}) (in this case).

# 7.2 Shrinkage Estimators and Testing

Using standard conditions, it is well known that the LSE of **β** is (see Ravishanker and Dey, 2001 for details)

(7.2.1)

According to Theorem 2.6.3, its distribution is *M*^{(p)}_{t} (**β**, σ^{2}*C*^{−1}, γ_{o}).

From Corollary 2.6.3.1, ...