A test of conditional heteroscedasticity in time series (Q1283077)

Let \((X_{t})_{t\in Z}\) be an autoregressive time series with finite second moments and conditional heteroscedasticity, denoted as AR-ARCH. That is \[ X_{t}=\phi_{1}X_{t-1}+...+\phi_{p}X_{t-p}+\varepsilon_{t},\;E\{\varepsilon _{t}|X_{t-1},...,X_{t-p}\}=0\;\text{a.s.}, \] \[ E\{\varepsilon_{t}^{2} |X_{t-1} ,...,X_{t-p}\}=a_{0}+a_{1}\varepsilon_{t-1}^{2}+...+a_{p}\varepsilon_{t-p} ^{2} \text{a.s.}, \] where \(1-\phi_{1}z-...-\phi_{p}z^{p}\neq 0,\) for all \(\left|z\right|\leq 1,\) \(a_{0}>0,\) \(a_{i}\geq 0,\) \(i=1,...,p\). This work deals with a test of conditional heteroscedasticity whether the conditional variance of the one-step forecast error is constant or not. More precisely, the null hypothesis is \(H_{0}:a_{i}=0,\) \(i=1,...,p\), vs. \(H_{1}:\) there exists \(i=1,...,p\) such that \(a_{i}>0\). The authors propose a new test based on a Cramér - von Mises type statistic and a goodness of fit statistic. The asymptotic consistency of the new test statistic is shown. The work is mathematically demanding, and hard to read for professionals not possessing good working knowledge in measure theory, stochastic processes, etc. Future work providing computational implementations of the proposed test will allow its assessment with real data and/or with Monte Carlo experiences.