(6.34) and (6.35) is located at the bottom, thank you

o show that the Jacobian of the transformations 16.39) is equal to 1 liig show that 16.35) holds, TIME SERIES MODEL EXAMPLE (CHARN MODEL) A stochastic process {X} = (X1,1,..., Xm,1)':t Z} is said to follow a conditional heteroscedastic autoregressive nonlinear model (CHARN) if it satisfies X+ = F(X:-1, ..., Xt-p) + H2(X:-1,...,X-)U(6.26) where U, = (Uite..., Um, '~.i.d.O,V), Fe : RMP R" and He : RM R" XR" are measurable functions, and o=(0,...,0)' ECR' is an unknown parameter (Hardle, Tsybakov and Yang (1998)). This model is very general, and can be applied to analysis of brain and muscular waves as well as financial time series analysis (Kato, Taniguchi and Honda (2006)). M. TANIGUCHI TIME SERIES ANALYSIS 2/12 TIME SERIES MODEL When we analyze the nonlinear time series models mentioned, their stationarity is a fundamental and important condition. In what follows, we state some sufficient conditions for typical nonlinear models to be stationary. THEOREM (CHEN AND AN (1998)) If the GARCH model (6.19) satisfies 4 (6.27) i=1 La;+b; 0 a.e., u ER". There exist ai; > 0, bi > 0,1 sismasi sp, such that 12 m P |F(x) 0 such that {the minimum eigenvalue of H.(x)} > 1 for all x ERP. max La+EU bi 0 a.e., u ER". There exist ai; > 0, bi > 0,1 sismasi sp, such that 12 m P |F(x) 0 such that {the minimum eigenvalue of H.(x)} > 1 for all x ERP. max La+EU bi