I have been implementing dickey fuller test after gaining an understanding of its procedure.. The relevant equations are
I have obtained the value for t-stat as +ve and hence when comparing with the table for alpha= 0.05 the result shows it is not stationary. I am confident that my data is stationary because on plotting the data it's stationary.
What I dont understand is how can the t-stat value ever be negative when numtr and denmtr are positive. The theta value or p_hat is positive(by OLS estimation) and SE as it has squares in the equation will also yield a +ve value.
My assumption:
Can it be because, rooting in SE gives -ve value(as square of a number, both +ve and -ve is the same). Or my theta value has to be -ve but can that be?
Any help on this great thanx.
Edit made: Image After seasonality removed for m = 4
Edit done(2) Before seasonality
The Dickey Fuller test tests between two cases: unit root (integrated time series), which is the null hypothesis and a stationary time series which is the alternative hypothesis. In writing the model for the test (by differencing the AR model), the coefficient for the $y_{t-1}$ term is $p=\rho-1$. $$\Delta y_t=p y_{t-1} + \epsilon_t$$
$p$ being positive means $\rho > 1$ which is explosive growth, which you said from a plot of the series is obviously not the case. So the test is being used to only distinguish between $p<0$ and the null, $p=0$.
It's not true that the numerator in the test statistic is positive. It's negative and the denominator is always positive.
As to why you got it positive in your OLS, Did you apply differencing? Enough number of times?
Btw, are you using the right one of the three versions of the test? https://en.m.wikipedia.org/wiki/Dickey–Fuller_test