Proyecto Orquideas

Chapter 3
Further development and analysis of the classical linear regression model

‘Introductory Econometrics for Finance’ © Chris Brooks 2008

1

Generalising the Simple Model to Multiple Linear Regression
• Before, we have used the model yt    xt  ut t = 1,2,...,T • But what if our dependent (y) variable depends on more than one independent variable? For example the number ofcars 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-1regressors (independent variables).

‘Introductory Econometrics for Finance’ © Chris Brooks 2008

Multiple Regression and the Constant Term

• Now we write

yt  1   2 x2t  3 x3t  ...   k xkt  ut , 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 1 x1       1

1 is thecoefficient attached to the constant term (which we called  before).

‘Introductory Econometrics for Finance’ © Chris Brooks 2008

Different Ways of Expressing the Multiple Linear Regression Model
• We could write out a separate equation for every value of t:

y1  1   2 x21   3 x31  ...   k xk1  u1 y2  1   2 x22   3 x32  ...   k xk 2  u2    yT  1   2 x2T   3 x3T ...   k xkT  uT
• 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 2008

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:

 y1  1 x21   u1   y  1 x      1  u2  22   2              2         yT  1 x2T  uT 
T 1 T2 21 T1

• Notice that the matrices written in this way are conformable.

‘Introductory Econometrics for Finance’ © Chris Brooks 2008

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 notation, we have
ˆ  u1 u  ˆ ˆ u 2      ˆ u T 

• The RSS would be given by
ˆ ˆ ˆ ˆ u ' u  u1 u2 ˆ  u1  u  ˆ ˆ ˆ ˆ2 ˆ2 ˆ  uT  2   u12  u2  ...  uT   ut2     ˆ uT 

‘Introductory Econometrics for Finance’ © Chris Brooks 2008

The OLS Estimator for the Multiple Regression Model
• In order to obtain the parameter estimates, 1, 2,..., k, we would minimise the RSS with respect toall the s.

• It can be shown that
ˆ  1    ˆ ˆ    2   ( X X ) 1 X  y       ˆ  k   

‘Introductory Econometrics for Finance’ © Chris Brooks 2008

The OLS Estimator for the Multiple Regression Model
• Use the ceosal2 file and run the model: �������������� = �� + ��1 �������� + ��2 ������������ + ���� . Using matrices, and excel, get parameters: ��, ��1 and ��2 .
ˆ 1    ˆ ˆ    2   ( X X ) 1 X  y       ˆ  k   

‘Introductory Econometrics for Finance’ © Chris Brooks 2008

Calculating the Standard Errors for the Multiple Regression Model
 • Check the dimensions:  is k  1 as required.
• But how do we calculate the standard errors of the coefficient estimates? • Previously, to estimate the variance of the errors, 2, we used s2
ˆ u 
2 t

T 2

.

•

  2  u' u Now using the matrix notation, we use s

Tk

• where k = number of regressors. It can be proved that the OLS estimator of the variance of  is given by the diagonal elements of s2 ( X ' X )1 , so that the variance of 1 is the first element, the variance of 2 is the second  element, and …, and the variance of  k is the kth diagonal...