and there is no uniformly better alternative. The basic objective of this test is to test the level of integration of the time series using autoregressive model: 5 = −1 + (6) The majority if economic and financial series contain a single unit root, although some are stationary and consumer prices have been argued to have two unit roots. Here to simplify the situation we have the hypothesises: 0 : 1 :
We usually use the regression in the form:
∆
= −1
+
(7)
So that a test of β=1 is equivalent to a test of δ=0 (since δ-1=β)
The null and alternative hypothesises are:
0 1
: :
= −1
+
= −1 , < 1 This is a test for a random walk against a stationary autoregressive process of order one(AR(1)) +
i. Computing the DF test statistic
Fuller (1976) and Dickey and Fuller (1979, 1981) proposed a test based on the t-ratio t(α) in the Ordinary Least Squares (OLS) regression.
The test statistics are defined as:
𝛿̂ (𝛿̂ )̂
test statistic =
Where SE means standard deviation. The test statistic does not follow the usual t-distribution under the null since the null is one of non-stationarity, but rather follows a non-standard distribution. Critical values are derived from Monte Carlo experiments, the critical values are shown in the table 4.1.
c. Durbin Watson test for Cointegration
i. The Meaning of Cointegration
If I(1) variables are cointegrated, this means that although they are individually non-stationary, they are moving together so that there is some long run
5 Armstrong, J. S. (ed.) (2001) Principles of Forecasting: A Handbook for Researchers and Practitioners . 1st edn. Boston, MA: Kluwer Academic Publishers.
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