Inference for 2-D GARCH models

Using a generalised method of moment approach and 2D‐generalised autoregressive conditional heteroscedasticity modelling for denoising ultrasound images

“…As we need fourth-order stationarity of the GARCH model. So in the following theorem we recall a sufficient condition for 2mthorder stationarity obtained in [18]. First, let…”

“…Proof: See [18]. □ However, in practice the most realistic situation is when the GARCH process is not directly observed, but it constitutes the innovation of an observed linear process.…”

“…□ The second step in the estimation procedure concerns the GARCH parameter φ. The author in [18] gave the GMM estimator of φ based on a sample {ε t , t ∈ S [1, N]} having the form…”

“…Next, authors in [13] have extended the standard GARCH model to two-dimensions one in order to take into account the space dependence of variance. An extensive review of 2D GARCH model has been presented in [17,18]. In the literature, most of studies dealing with a 2D GARCH modelling have used a maximum-likelihood (ML) method to estimate the model parameters.…”