Consider the regression model \(Y_{t}=\beta_{0}+\beta_{1} X_{t}+u_{t}\), where \(u_{t}\) follows the stationary \(\operatorname{AR}(1)\) model \(u_{t}=\phi_{1} u_{t-1}+\widetilde{u}_{t}\) with \(\widetilde{u}_{t}\) i.i.d. with mean 0 and variance \(\sigma_{\tilde{u}}^{2}\) and \(\left|\phi_{1}\right|<1\).

a. Suppose that \(X_{t}\) is independent of \(\widetilde{u}_{j}\) for all \(t\) and \(j\). Is \(X_{t}\) exogenous (past and present)? Is \(X_{t}\) strictly exogenous (past, present, and future)?

b. Suppose that \(X_{t}=\widetilde{u}_{t+1}\). Is \(X_{t}\) exogenous? Is \(X_{t}\) strictly exogenous?