Page 121 - Textos de Matemática Vol. 47

MODELLING TIME SERIES OF COUNTS: AN INAR APPROACH 111
where fXt|Xt 1(xt|xt 1) is given by (2.2) and P(X1 = x1) represents the sta- tionary marginal distribution. For asymptotic analysis the standard approach is to consider the conditional log-likelihood function given by
Xn
`n(✓)= log fXt|Xt 1(xt|xt 1) . (3.2)
t=2
Based on the results of [5] for Markov processes, [25, 26] prove the following
result
Theorem 3.1. The maximum likelihood estimators, MLE, of the parameters ✓ in INAR(1) models are consistent, asymptotically normal and asymptotically e cient.
The proof requires that some regularity conditions hold ([26], pp. 318).
1/2 ˆ 1 The limit distribution of n (✓ ✓) is N(0, ⌃M LE ), ⌃M LE = ⌃
(✓) and
⌃(✓) = ( ij (✓)) is a non-singular s ⇥ s matrix with elements (✓) = E ⇣`˙ (✓;x ,x )`˙ (✓;x ,x )⌘.
ij ✓i12j12
In general, for numerical maximum likelihood estimation, a quasi-Newton method can be used with an input of the negative log-likelihood function and output of the MLE and inverse Hessian matrix at the MLE. The initial es- timates required by the optimization algorithm are based on the method of
ˆ
moments. The inverse Hessian evaluated at the maximum, ✓, can be used as
ˆ the estimated variance-covariance matrix of the ML estimator ✓.
3.2. Diagnostic tools. A crucial step in any statistical investigation is the assessment of the adequacy of the models proposed and fitted to the data under analysis. Various methods for model validation and diagnostics in discrete- valued time series have been proposed in the literature. These methods can be broadly classified as: parametric resampling methods; residual based methods; methods based on the predictive distributions; model comparisons using scores and information criteria.
3.2.1. Parametric resampling methods. [49] proposes a procedure based on para- metric bootstrap and special functionals designed to show the specific features of interest. Specifically, the fitted model is used to generate many samples, all with the same number of observations as the original data set. The samples generated are then used to construct an empirical distribution of the functional of interest. If the fitted model is adequate in describing the feature of inter- est, the functional quantity of the original data should be a reasonable point with respect to the empirical distribution. The functional of interest may be