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Chapter 4 - Further development and analysis of the classical linear regression model. In this chapter, you will learn how to: Construct models with more than one explanatory variable, test multiple hypotheses using an F-test, determine how well a model fits the data, form a restricted regression, derive the OLS parameter and standard error estimators using matrix algebra, estimate multiple regression models and test multiple hypotheses in EViews. | ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 Chapter 4 Further development and analysis of the classical linear regression model ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 Generalising the Simple Model to Multiple Linear Regression Before, we have used the model t = 1,2,.,T But what if our dependent (y) variable depends on more than one independent variable? For example the number of cars sold might plausibly depend on 1. the price of cars 2. the price of public transport 3. the price of petrol 4. the extent of the public’s concern about global warming Similarly, stock returns might depend on several factors. Having just one independent variable is no good in this case - we want to have more than one x variable. It is very easy to generalise the simple model to one with k-1 regressors (independent variables). ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 Multiple Regression and the Constant Term Now we write , . | ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 Chapter 4 Further development and analysis of the classical linear regression model ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 Generalising the Simple Model to Multiple Linear Regression Before, we have used the model t = 1,2,.,T But what if our dependent (y) variable depends on more than one independent variable? For example the number of cars sold might plausibly depend on 1. the price of cars 2. the price of public transport 3. the price of petrol 4. the extent of the public’s concern about global warming Similarly, stock returns might depend on several factors. Having just one independent variable is no good in this case - we want to have more than one x variable. It is very easy to generalise the simple model to one with k-1 regressors (independent variables). ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 Multiple Regression and the Constant Term Now we write , t=1,2,.,T Where is x1? It is the constant term. In fact the constant term is usually represented by a column of ones of length T: 1 is the coefficient attached to the constant term (which we called before). ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 Different Ways of Expressing the Multiple Linear Regression Model We could write out a separate equation for every value of t: We can write this in matrix form y = X +u where y is T 1 X is T k is k 1 u is T 1 ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 Inside the Matrices of the Multiple Linear Regression Model e.g. if k is 2, we have 2 regressors, one of which is a column of ones: T 1 T 2 2 1 T 1 Notice that the matrices written in this way are conformable. ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 How Do We Calculate the Parameters (the ) in this Generalised Case? Previously, we took the residual sum of squares, and minimised it w.r.t. and . In the matrix
