so we cannot validly undertake hypothesis tests about the regression parameters. 4
What are the conditions for a stationary series:
) = 𝜇 ( )
E(
) = ( )
Var(
) = () ≠ ()
Covarience(
, −
ii. Mathematical Modelling
This is the model which has been frequently used to characterise non- stationarity: the random walk with drift: = + −1 + ~ (0, 𝜎 2 )
(1)
where −1 < < 1
is independent identically distributed.
Consider the above autoregressive model, the error process usual assumptions of the classical model, which are:
retains all the
) = 0
Zero mean: E(
Normality: the error terms follow a normal distribution Homoscedasticity: the variance of the error term is always the same Var( ) = 𝜎 2 Independence: the error terms , − associated with two different settings of the predictor variables are statistically independent Covarience( , − ) = () ≠ () We can write: −1 = + −2 + −1 −2 = + −3 + −2 Substitute −1 into (∗) yields: = + ( + −2 + −1 ) + = (1 + ) + 2 −2 + −1 + Then substitute −2 into (∗) yields: = (1 + ) + 2 ( + −3 + −2 ) + −1 + = (1 + + 2 ) + 3 −3 + 2 −2 + −1 +
Successive substitutions of this type lead to:
= (1 + + 2 + 3 + ⋯ ) +
+ 2 −2
+ 3 −3
+ −1
+ ⋯
4 Brooks, C. (2008) Introductory Econometrics for Finance . 2nd edn. New York: Cambridge University Press.
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