Page 122 - Textos de Matemática Vol. 47

112 M. E. SILVA
a spectral density function or the autocorrelation function. Here, the autocor- relation properties are of interest. Thus M artificial data sets, with the same length of the original data set, are generated from the fitted model. Based on these, M sample autocorrelation functions (ACF) are obtained. For each fixed lag of the ACF, the (1 ↵/2) and ↵/2 quantiles of the empirical distribution, denominated acceptance bounds, are computed. Then, a probability interval is obtained for each lag and an envelope is obtained for the ACF. If the fitted model is adequate at each lag, the sample ACF of the original data should be largely within the envelope. Plotting the envelope and the sample ACF of the data jointly, gives rise to a graphical display that can be used to assess the overall goodness of fit of the fitted model with respect to the serial correlation properties. A model is considered to be adequately reproducing the correlation structure of the data, if the sample ACF of the observed data lies within these acceptance bounds. Note that since the sample ACF at di↵erent lags are corre- lated, the acceptance envelope considered is not a joint 100(1 ↵)% confidence interval of the sample ACF.
3.2.2. Residual based methods. The dynamic structure in the mean and dis- persion properties may be checked using tools based on the Pearson residuals defined by
rt = Xt E(Xt|Xt 1), (3.3) Var(Xt |Xt 1 )1/2
where the population quantities are replaced by their estimated counterparts. If the model is correctly specified, these residuals should exhibit mean zero and variance one and no (significant) serial correlation.
However, the structure of the INAR(1) model suggests additional residu- als checks. In fact, the INAR(1) model maybe seen as a structural model in the sense that it considers the data to be composed of a set of unobserved components each of which captures a feature of the data: the first component ↵⇧Xt 1 specifies the random number departures or its complement the random number of survivors from the past while ✏t represents the new arrivals at the system at time t. This interpretation leads to a residual decomposition that allows to check the adequacy of each component. For details on the residual decomposition and subsequent testing procedures see [16].
3.2.3. Methods based on the predictive distributions. A useful tool to check the adequacy of the distributional assumptions of the models is a suitably speci- fied and modified version of the probability integral transform, PIT, originally proposed by [40]. This device has been used in the assessment of predictive dis- tributions of a continuous type by [40], [14] and recently by [18] and [19]. The PIT has a uniform distribution when the underlying model is continuous. For