I was wondering if someone could help me clarify something from my lecture notes. It concerns the last step. I was wondering why we test if
$\frac{\hat{\beta}}{\textrm{SE}(\hat{\beta}}<0$
and what the rejection rule is in this case. I am quite confused by this.
My own thoughts were to test $H_0:\beta=0$ against $H_1:\beta \neq 0$ using the $t$-statistic $\frac{\hat{\beta}}{\textrm{SE}(\hat{\beta}}$ (we compare this to the critical values of a $t$ distribution with $N-K$ degrees of freedom where $N$ is the number of observations and $K$ the number of parameters; here $K=3$ since we estimate $\beta,\tilde{\alpha}$ and $\tilde{\gamma}$). Then, if we do not reject $H_0$, this thus means that the model is the same as the pure linear trend model, which corresponds to the special case of $\rho=0$. Hence, if we do not reject $H_0:\beta=0$ then we do reject $H_0:\rho=1$.
(1) If $\beta=0$ then the specification you were testing simplifies to
$$x_t=x_{t-1}+\tilde\alpha+\tilde\gamma t+\epsilon_t,$$
which is the equation at the top for $\rho=1$, correct?
How is that a pure linear trend model?
(2) One tests against $\rho<1$ (and hence $\beta<0$) because roots greater than unity are typically ruled out.
(3) One beef I have with the lecture notes is that one tests coefficient values, not values of estimates.
(4) Correcting for degrees of freedom and using t-distributions only makes sense if you're willing to assume that your data are drawn from a normal distribution, the $\epsilon_t$'s are iid, and your sample size is small.