To test if the time series is covariance stationary, we use Dickey Fuller (DF). In DF, transform the AR model to run a simple regression by subtracting x(t-1) from both sides. Rather than testing whether the original coefficient is different from 1, we test whether the new transformed coefficient is different from 0 using a modified t-test. If the new coefficient is not significantly different from 0 (i.e. is not significant), it must be equal to 1 and the series has a unit root. If coefficient is significant, reject the null, and conclude that it is covariance stationary and has no unit root. (The null is that there is a unit root and is non-stationary). Can someone please explain the part in bold? So we are testing if the coefficient is = 0. If it is not significant (so = 0), then we conclude it = 1?
i guess yes …
Because you take the first difference, you modify the equation. For the new equation, the implication is that the coefficient is rho minus 1 (just algebra if you want to write the equations out). You now test if this coefficient is different from zero. If it is not, then rho must be 1 (statistically speaking, to satisfy the equality in the null hypothesis below), indicating a unit root.
Consider the null hypothesis below: c1 is the coeffiecient in the new differenced equation
Ho: c1=0 where c1= (rho -1) ----> if (rho -1) =0 and we fail to reject this null hypothesis, then we can rearrange (+1 to both sides) to show that rho=1
Hope this helps!