Wald tests for detecting multiple structural changes in persistence (Q2847584)

This paper focuses on the detection of multiple structural changes in the persistence of a univariate time series. The authors consider sup-Wald tests for the null hypothesis of the unit root against the alternative that process alternates between stationary and unit root regimes. In the brief literature review the authors give evidence about the novelty of the problem they investigate. They notice that most tests for changes in persistence are based on either partial sums of the data or on unit root statistics applied to various data samples. Also these tests often consider only one change point problem. The proposed tests are based on the difference between the sum of squared residuals from the unit root model and those from a model that allows shifts in persistence between stationary and nonstationary regimes.NEWLINENEWLINE Two models are considered in this paper. The first one is a first order autoregressive model with constants. NEWLINE\[NEWLINEy_t = c_i + \alpha_i y_{t-1} + u_{it}, \quad t \in [T_{i-1}+1,T_i], \quad i=1, \ldots , m+1,\;T_0=0,\;T_{m+1} = T. NEWLINE\]NEWLINE The vector of break functions is \(\lambda = (\lambda_1, \ldots, \lambda_m)\), with \(\lambda_i = T_i/T\). The errors \(\{u_t\}\) are generated by the stationary linear process. In this model two cases are analysed depending on whether the initial regime is \(I(1)\) or not, i.e.,NEWLINENEWLINE model 1a: \(c_i=0,\alpha_i=1\) in odd regimes and \(|\alpha_i| < 1\) in even regimes; model 1b is defined similarly, except that the first regime is stationary. Then Wald tests \(F_{1a}(\lambda,k)\) and \(F_{1b}(\lambda,k)\) are constructed in the case, when the alternative involves a fixed value \(k\) of changes. And a sup-Wald test is defined as NEWLINE\[NEWLINE\begin{aligned} F_{1a}(k) &= \sup_{\lambda \in \Lambda_{\epsilon}^k} F_{1a}(\lambda,k) \\ F_{1b}(k) &= \sup_{\lambda \in \Lambda_{\epsilon}^k} F_{1b}(\lambda,k), \end{aligned}NEWLINE\]NEWLINE where \(\Lambda_{\epsilon}^k = \{\lambda: |\lambda_{i+1}-\lambda_{i}| \geq \epsilon, \lambda_1 \geq \epsilon, \lambda_k \leq 1-\epsilon\}\).NEWLINENEWLINE Further a second type of tests is constructed. These tests are based on that one has no a priori information regarding whether the first regime contains unit roots or not. The tests are given by \(W_{1}(k)=\max[F_{1a}(k),F_{1b}(k)]\). Finally, if there is an unknown number of breaks (but not more than \(A\) breaks), the authors suggest the \(W\max_1 = \max_{1 \leq m \leq A} W_{1}(m)\) test statistic.NEWLINENEWLINE The second model is a first order autoregressive model with constants and trend: NEWLINE\[NEWLINEy_{t} = c_{i} + b_{i} t +\alpha_i y_{t-1} + u_{it}.NEWLINE\]NEWLINE The corresponding models are: model 2a: \(\alpha_i = 0, b_i = 0\) in odd regimes and \(|\alpha_i| < 1\) in even regimes; model 2b: \(\alpha_i = 0, b_i = 0\) in even regimes and \(|\alpha_i| < 1\) in odd regimes. Similarly as in the first model, Wald tests \(F_{2a}(\lambda,k)\) and \(F_{2b}(\lambda,k)\) are defined. Finally the statistics \(F_{2a}(k)\), \(F_{2b}(k)\), \(W_{2}(k)\) and \(W\max_2\) are defined in the same way as for the first model.NEWLINENEWLINE For both models the limit distribution under null hypothesis depends on functionals of a Wiener process. The limit distributions are different depending on the model and the initial regime specified.NEWLINENEWLINE To obtain the asymptotic critical values of the test statistics, the authors perform Monte Carlo simulations (they use Perron and Qu's dynamic programming algorithm). The critical values for model 1a and 2a are larger than those for model 1b and 2b, respectively. Also, it is observed that the critical values are not monotonically decreasing as \(k\) increases.NEWLINENEWLINE The consistency of the tests are investigated. Further some notes on hybrid testing procedures are given. The performed simulation experiments demonstrate that these tests have adequate finite sample properties.