If \(Y_1\),…,\(Y_n\) is a time series, denote \(p(y_y |y^{1,t-1})=\Pr\{Y_t<y_t |Y^{1,t-1}=y^{1,t-1}\}\) (here \(y^{s,t}\) means \((y_s,\dots,y_t)\)). Then it is well known that \(u_t=p(Y_t |Y^{1,t-1})\) are i.i.d. uniform on [0,1] and \(v_t=\Phi^{-1}(u_t)\) are i.i.d. \(N(0,1)\). The authors propose to use this transform for model diagnostics in the case when \(Y^{1,t}\) depends on a vector of latent variables \(\vartheta^{1,t}\). A special algorithm is constructed to evaluate \(p(y_t |y^{1,t-1})\) if the conditional probabilities \(p(y^{1,t} |\vartheta_i)\) and the prior distribution of \(\vartheta^{1,t}\) are known. The algorithm is based on a combination of the Markov chain Monte-Carlo method and importance sampling.
Two-sided CUSUM, Shapiro-Wilks, Ljung-Box and Studentized range statistics are used for testing i.i.d. \(N(0,1)\) distributions of the model “residuals” \(v_t\). Results of simulation studies are considered for different kinds of autoregression models with shifts.
The authors’ conclusion is: “For any given departure from the model care needs to be taken with the choice of test statistics to ensure that they have considerable power”.