An ARCH model without intercept

- An ARCH process without constant either converges to zero a.s., diverges to infinity a.s., or behaves like a random walk.
- In all three cases, the MLE is consistent and asymptotically normal under mild conditions.
- The paper proposes a test of the random walk hypothesis and investigates its finite sample properties.
- Applications to low-frequency financial returns are shown to be promising.

While theory of autoregressive conditional heteroskedasticity (ARCH) models is well understood for strictly stationary processes, some recent interest has focused on the nonstationary case. In the classical model including a positive intercept parameter, the volatility process diverges to infinity at least in probability, and it has been shown that no consistent estimator of the full parameter vector, including intercept, exists. This paper considers a nonstationary ARCH model which arises by setting the intercept term to zero. Unlike nonstationary ARCH models with positive intercept, this model includes the interesting case of log volatility following a random walk, which is called the stability case. For the ARCH(1) model without intercept, the paper derives asymptotic theory of the maximum likelihood estimator and proposes a test of the stability hypothesis. Numerical evidence illustrates the finite sample properties of the maximum likelihood estimator and the stability test.