# Working Papers del DiSIA

### Consistent non-Gaussian pseudo maximum likelihood estimators

### Gabriele Fiorentini, Enrique Sentana

We characterise the mean and variance parameters that distributionally misspecified maximum likelihood estimators can consistently estimate in multivariate conditionally heteroskedastic dynamic regression models. We also provide simple closed-form consistent estimators for the rest. The inclusion of means and the explicit coverage of multivariate models make our procedures useful not only for GARCH models but also in many empirically relevant macro and finance applications involving VARs and multivariate regressions. We study the statistical properties of our proposed consistent estimators, as well as their efficiency relative to Gaussian pseudo maximum likelihood procedures. Finally, we provide finite sample results through Monte Carlo simulations.

Working Papers del Dipartimento

### First-Birth Gains and Losses from the First Job in Italy: The Role of Employment Uncertainty

### Daniele Vignoli, Valentina Tocchioni, Alessandra Mattei

Previous research on the link between employment uncertainty and fertility has focused on the existence of a statistically significant effect of contractual arrangements on fertility, largely disregarding the magnitude of the effect in question. In this article, we address this oversight for Italy using retrospective data from the nationally representative 2009 Family and Social Subjects survey. We adopt the potential outcome approach to causal inference to quantify the net effect of first jobs (temporary or permanent) on the propensity to have a first child within the first five years of employment. We find that 7% of first-birth losses among women and 5% of first-birth losses among men are attributable to employment uncertainty: they would have had a first child if they had had a permanent job. For women, first-birth losses are especially elevated among those with higher education (reaching 16%), while for men first-birth losses are pronounced among those with low and medium educational attainment.

Working Papers del Dipartimento

### Algorithms for exact and approximate linear abstractions of polynomial continuous systems

### Michele Boreale

A polynomial continuous system S=(F, X_{0}) is specified by a polynomial vector field F and a set of initial conditions X_{0}. We study polynomial changes of bases that transform S into a linear system, called linear abstractions. We first give a complete algorithm to find all such abstractions that fit a user-specified template. This requires taking into account the algebraic structure of the set X_{0}, which we do by working modulo an appropriate invariant ideal. Next, we give necessary and sufficient syntactic conditions under which a full linear abstraction exists, that is one capable of representing the behaviour of the individual variables in the original system. We then propose an approximate linearization and dimension-reduction technique, that is amenable to be implemented "on the fly". We finally illustrate the encouraging results of a preliminary experimentation with the linear abstraction algorithm, conducted on challenging systems drawn from the literature.

Working Papers del Dipartimento

### Is the Impact of Employment Uncertainty on Fertility Intentions Channeled by Subjective Well-Being?

### Daniele Vignoli, Letizia Mencarini, Giammarco Alderotti

This article combines two apparently distinct strands of contemporary research on fertility: the literature on employment uncertainty and fertility and the literature on subjective well-being and fertility. We advance the hypothesis that the impact of term-limited work contracts and precarious jobs on fertility intentions is channeled by an individual’s level of subjective well-being. To test this hypothesis, we adopt a formal framework for causal inference and apply techniques of mediation analysis to data from two rounds of the European Social Survey (ESS 2004 and 2010). Our analysis clearly suggested that the impact of employment uncertainty on fertility intentions depended on the level of subjective well-being: the negative effect was found only when subjective well-being was relatively low (i.e. life satisfaction levels equal or below 6). Detailed results show that parents and younger individuals reduced their fertility intentions more than the childless and older individuals when experiencing employment uncertainty and facing low subjective well-being. We also found that in 2010 – while the economic crisis was underway – it was especially the deterioration in men’s position in the labor market that inhibited fertility planning.

Working Papers del Dipartimento

### Specification tests for non-Gaussian maximum likelihood estimators

### Gabriele Fiorentini, Enrique Sentana

We propose generalised DWH specification tests which simultaneously compare three or more likelihood-based estimators of conditional mean and variance parameters in multivariate conditionally heteroskedastic dynamic regression models. Our tests are useful for GARCH models and in many empirically relevant macro and finance applications involving Vars and multivariate regressions. To design powerful and reliable tests, we determine the rank deficiencies of the differences between the estimators' asymptotic covariance matrices under the null of correct specification, and take into account that some parameters remain consistently estimated under the alternative of distributional misspecification. Finally, we provide finite sample results through Monte Carlo simulations.

Working Papers del Dipartimento

### Crawling, (pretty) printing and graphing the OEIS

### Massimo Nocentini, Donatella Merlini

In this paper we present a suite of software tools that allows us to interact with the Online Encyclopedia of Integer Sequences; in particular, (i) a crawler fetches sequences recursively with respect to their relations, (ii) a pretty printer represents the same data stored in the archive using two different formats, namely the old console and modern Jupyter notebooks, (iii) a grapher shows connections among sequences by using graph structures.

Working Papers del Dipartimento

### Dynamic specification tests for dynamic factor models

### Gabriele Fiorentini, Enrique Sentana

We derive computationally simple expressions for score tests of misspecification in parametric dynamic factor models using frequency domain techniques. We interpret those diagnostics as time domain moment tests which assess whether certain autocovariances of the smoothed latent variables match their theoretical values under the null of correct model specification. We also reinterpret reduced-form residual tests as checking specific restrictions on structural parameters. Our Gaussian tests are robust to nonnormal, independent innovations. Monte Carlo exercises confirm the finite sample reliability and power of our proposals. Finally, we illustrate their empirical usefulness in an application that constructs a US coincident indicator.

Working Papers del Dipartimento

Ultimo aggiornamento 15 gennaio 2019.