where −
= 0 as n → ∞
so that
) = (1 + + 2 + 3 + ⋯ ) = 1−
(
) =
Var(
1−
, ≥ 0 This model satisfies the conditions of weak stationarity as long as || < 1 so that all mean, variance and autocovariances are constant over time. Cov( , − ) = 𝜎 2 (1 + 2 + 4 + 6 + ⋯ ) = 𝜎 2 𝑠 1− 2
Therefore there three different situations for different values of β:
1. < 1 ⇒ → 0 𝑇 → ∞ So the shocks to the system gradually die away. 2. > 1 ⇒ = 1 ∀ 𝑇 So shocks persist in the system and never die away. We obtain: = 0 + ∑ ∞ =0 (2) So just an infinite sum of past shocks plus some starting value of 0 3. > 1 , now shocks become more influential as time goes on since if > 1 , 3 > 2 > etc. Now consider what happens to the properties of the model if β = 1 , the process started at t = 0 with fixed initial condition 0 = 0 . Then, = + −1 + , = 1,2, … , 𝑇 (3) Similarly, substitute for lagged ( −1 , −2 , −3 , ) = + ∑ ( 2 =1 ) + 0 (4) Therefore in this model E(Y t ) = tc so it is not constant over time. Thus this process is no longer stationary and is known as a random walk with drift. Note that the random walk process can be stationary by transforming the dependent variable by first differencing since ∆ = − −1 = + (5)
is a stationary process
A series that can be made stationary by differencing once is said to be integrated of order one, or to possess a unit root.
b. Dickey Fuller integration test
The early and pioneering work on testing for a unit root in time series was done by Dickey and Fuller. The reason why we focus on DF test is that it is simple
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