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ECON 5340 Class Notes
Chapter 2. The Classical Multiple Linear Regression Model
1 Introduction
The multiple linear regression model can be written as
yi = β 1 xi 1 + β 2 xi 2 + ... + β k xik + i i = 1, ..., n (1)
where yi is the dependent variable, xij is the j th explanatory variable and i is the error term.
- There are k explanatory variables.
- There are n observations.
- xi 1 is often set equal to one, i = 1, ..., n, so β 1 is an intercept.
- The βs are coefficients or parameters to be estimated.
Using matrices, model (1) can be written more compactly as
Y = Xβ + (2)
where Y is an n × 1 column vector of dependent variables, X is an n × k matrix of explanatory variables, β
is a k × 1 column vector of parameters and is an n × 1 column vector of errors.
For example:
Y =
y 1
y 2
yn
X =
x 11 x 12 · · · x 1 k
x 21 x 22 · · · x 2 k
xn 1 xn 2 · · · xnk
β =
β 1
β 2
. . .
β k
2 Data-Generating Assumptions
There are six data-generating assumptions associated with model (1) or (2).
2.1 Linearity
The model must take the form of (1) so that it is linear in the parameters (β) and the error term ().
- The model need not be linear in the Xs or Y s. However, it must be transformable into a form such
as (1).
- For example, after taking natural logs, yi = exp(β 1 xi + i) can be transformed into ln(yi) = β 1 xi + i,
which is in the form of (1) with an appropriate redefining of y.
2.2 Full Rank
The columns of X need to be linearly independent and there must be at least k observations.
- In other words, Rank(X) = k.
2.3 Mean-Zero Errors
Conditional on X, the error terms are mean zero.
- In other words, E[|X] = 0.
- This implies that E[Y |X] = Xβ, (i.e., the regression of Y on X is the conditional mean Xβ).
- Including a constant term will guarantee this assumption holds. Assume yi = β 0 + β 1 xi + i has the
property E[i] = μ 6 = 0. By redefining the constant term,
∗ i =^ i^ −^ μ, and intercept^ β
∗ 0 =^ β 0 +^ μ, the
model can be written with mean-zero errors, yi = β
∗ 0 +^ β 1 xi^ +^
∗ i.
2.4 Spherical Disturbances
The error terms should display homoscedasticity (i.e., error variances are constant across observations)
and no autocorrelation (i.e., errors are uncorrelated across observations).
- Homoscedasticity: V ar[i] = σ
2 for all i = 1, ..., n.
- No autocorrelation: Cov[i, j ] = 0 for all i 6 = j.
- Matrix representation: V ar[] = E[ 0 ] =
V ar[ 1 ] Cov[ 2 , 1 ] · · · Cov[n, 1 ]
Cov[ 1 , 2 ] V ar[ 2 ] · · · Cov[n, 2 ]
Cov[ 1 , n] Cov[ 2 , n] · · · V ar[n]
= σ 2 In.
