In this section only first order models

In this section, only first-order models will be considered, while in Section 4 general nonlinear models will be analyzed. Consider the following set of assumptions.
Assumption 1 is standard. Note that we do not assume symmetry of the distribution, which is particularly useful when modelling financial time series. Assumption 2 forces the model to be of first-order. This will be crucial to the results in this section, but will be relaxed in Section 4. The restrictions (R.1a)–(R.1c) in Assumption 3 are important to guarantee that the model is globally identifiable. Restriction (R.2) is a sufficient condition for h>0 with probability one. Note that z=(y, h, η)′ is a Markov chain with homogenous transition probability, expressed aswhereand e=(ɛ, 0, η)′.
If the conditions of the above theorems are met, the processes y and h have the following causal expansions:
Parameter estimation and asymptotic theory
Set , where is the vector of parameters of the conditional mean, as defined in Section 2, and =(ω, α, β)′ is the vector of parameters of the conditional variance. As the distribution of η is unknown, the parameter vector is estimated by the quasi-maximum likelihood (QML) method. Consider the following assumption.
The quasi-log-likelihood function of the NAR-GARCH model is given by:Note that the processes y and h, t≤0, are unobserved, and hence are only arbitrary constants. Thus, is a quasi-log-likelihood function that is not conditional on the true , making it Romidepsin suitable for practical applications. However, to prove the asymptotic properties of the QMLE, it is more convenient to work with the unobserved process . The unobserved quasi-log-likelihood function conditional on isThe main difference between and is that the former is conditional on any initial values, whereas the latter is conditional on an infinite series of past observations. In practical situations, the use of (17) is not possible. Letand Define . In the following subsection, we discuss the existence of and the identifiability of the NAR-GARCH models. Then, in Section 4.2, we prove the consistency of and . We first prove the strong consistency of , and then show thatso that the consistency of follows. Asymptotic normality of both estimators is considered in Section 4.3. We prove the asymptotic normality of . The proof of is straightforward.
Monte Carlo simulations In this section we report the results of a simulation study designed to evaluate the finite sample properties of the QMLE. We consider three different model specifications as described below: The results are illustrated in Table 1. The table shows the average and the standard deviation of the estimates over 1000 replications. As we can see from the table, the estimates are rather precise and improve as the sample size increases.
Concluding remarks
Introduction Both academics and market participants have been interested for a long time in knowing whether or not thecodonts is possible to build sensible portfolios with only few assets in the so-called cardinality-constrained approach. The literature dates back to Evans and Archer (1968) and Jacob (1974), who were among the first to study the characteristics of small portfolios. This problem seems particularly relevant to investors with financial constraints as well as to financial institutions that customize portfolios for clienteles with specific needs. Blog et al. (1983), for instance, point out that the small investor that wishes to come up with an optimal risk-return portfolio will be constrained as the efficient portfolios in the mean-variance setting of Markowitz (1952) sometimes contain too many securities to be attractive to the small investor. Therefore, the cardinality-constrained approach is mostly suitable for the small investor wishing to optimize the risk-return tradeoff for a limited number of assets. Despite its practical appeal, obtaining a mean-variance portfolio with a constraint of the number of assets is a challenging optimization problem. The main difficulties are the non-differentiability and the discontinuity that arise due to the inclusion of a cardinality or counting function constraint. In this sense, a very large number of studies have been proposing alternative optimization methods to efficiently handle this NP-hard problem (see, for instance, Jacob, 1974; Faaland, 1974; Blog et al., 1983; Chang et al., 2000; Jobst et al., 2001; Jansen and Van Dijk, 2002; Maringer and Kellerer, 2003; Li et al., 2006; Maringer and Oyewumi, 2007; Cura, 2009; Brodie et al., 2009; Kopman et al., 2009; Bertsimas and Shioda, 2009; Canakgoz and Beasley, 2009; Wang et al., 2012; Chen and Kwon, 2012). These references suggest that a fairly large number of approaches ranging from integer programming methods to heuristic-based algorithms are currently available in over to overcome the difficulties in solving the cardinality-constrained problem.