Nonlinear spectral analysis: A local Gaussian approach

The spectral distribution $f(\omega)$ of a stationary time series $\{Y_t\}_{t\in\mathbb{Z}}$ can be used to investigate whether or not periodic structures are present in $\{Y_t\}_{t\in\mathbb{Z}}$, but $f(\omega)$ has some limitations due to its dependence on the autocovariances $\gamma(h)$. For example, $f(\omega)$ can not distinguish white i.i.d. noise from GARCH-type models (whose terms are dependent, but uncorrelated), which implies that $f(\omega)$ can be an inadequate tool when $\{Y_t\}_{t\in\mathbb{Z}}$ contains asymmetries and nonlinear dependencies. Asymmetries between the upper and lower tails of a time series can be investigated by means of the local Gaussian autocorrelations introduced in Tj{\o}stheim and Hufthammer (2013), and these local measures of dependence can be used to construct the local Gaussian spectral density presented in this paper. A key feature of the new local spectral density is that it coincides with $f(\omega)$ for Gaussian time series, which implies that it can be used to detect non-Gaussian traits in the time series under investigation. In particular, if $f(\omega)$ is flat, then peaks and troughs of the new local spectral density can indicate nonlinear traits, which potentially might discover local periodic phenomena that remain undetected in an ordinary spectral analysis.Comment: Version 4: Major revision from version 3, with new theory/figures. 135 pages (main part 32 + appendices 103), 11 + 16 figure

Local Lead–Lag Relationships and Nonlinear Granger Causality: An Empirical Analysis

The Granger causality test is essential for detecting lead–lag relationships between time series. Traditionally, one uses a linear version of the test, essentially based on a linear time series regression, itself being based on autocorrelations and cross-correlations of the series. In the present paper, we employ a local Gaussian approach in an empirical investigation of lead–lag and causality relations. The study is carried out for monthly recorded financial indices for ten countries in Europe, North America, Asia and Australia. The local Gaussian approach makes it possible to examine lead–lag relations locally and separately in the tails and in the center of the return distributions of the series. It is shown that this results in a new and much more detailed picture of these relationships. Typically, the dependence is much stronger in the tails than in the center of the return distributions. It is shown that the ensuing nonlinear Granger causality tests may detect causality where traditional linear tests fail.publishedVersio

Testing for asymmetric dependency structures in financial markets: regime-switching and local Gaussian correlation

This paper examines asymmetric and time-varying dependency structures between financial returns, using a novel approach consisting of a combination of regime-switching models and the local Gaussian correlation (LGC). We propose an LGC-based bootstrap test for whether the dependence structure in financial returns across different regimes is equal. We examine this test in a Monte Carlo study, where it shows good level and power properties. We argue that this approach is more intuitive than competing approaches, typically combining regime-switching models with copula theory. Furthermore, the LGC is a semi-parametric approach, hence avoids any parametric specification of the dependence structure. We illustrate our approach using returns from the US-UK stock markets and the US stock and government bond markets. Using a two-regime model for the US-UK stock returns, the test rejects equality of the dependence structure in the two regimes. Furthermore, we find evidence of lower tail dependence in the regime associated with financial downturns in the LGC structure. For a three-regime model fitted to US stock and bond returns, the test rejects equality of the dependence structures between all regime pairs. Furthermore, we find that the LGC has a primarily positive relationship in the time period 1980-2000, mostly a negative relationship from 2000 and onwards. In addition, the regime associated with bear markets indicates less, but asymmetric dependence, clearly documenting the loss of diversification benefits in times of crisis

Pairwise local Fisher and naive Bayes: Improving two standard discriminants

Under embargo until: 2022-02-01The Fisher discriminant is probably the best known likelihood discriminant for continuous data. Another benchmark discriminant is the naive Bayes, which is based on marginals only. In this paper we extend both discriminants by modeling dependence between pairs of variables. In the continuous case this is done by local Gaussian versions of the Fisher discriminant. In the discrete case the naive Bayes is extended by taking geometric averages of pairwise joint probabilities. We also indicate how the two approaches can be combined for mixed continuous and discrete data. The new discriminants show promising results in a number of simulation experiments and real data illustrations.acceptedVersio

Multivariate and conditional density estimation using local Gaussian approximations

Paper 1 ”Bias and bandwidth for local likelihood density estimation”: A local likelihood density estimator is shown to have asymptotic bias depending on the dimension of the local parameterization. Comparing with kernel estimation it is demonstrated using a variety of bandwidths that we may obtain as good, and potentially even better estimates using local likelihood. Boundary effects are also examined. Paper 2 ”The locally Gaussian density estimator for multivariate data”: It is well known that the Curse of Dimensionality causes the standard Kernel Density Estimator to break down quickly as the number of variables increases. In non-parametric regression, this effect is relieved in various ways, for example by assuming additivity or some other simplifying structure on the interaction between variables. This paper presents the Locally Gaussian Density Estimator (LGDE), which introduces a similar idea to the problem of density estimation. The LGDE is a new method for the non-parametric estimation of multivariate probability density functions. It is based on preliminary transformations of the marginal observation vectors towards standard normality, and a simplified local likelihood fit of the resulting distribution with standard normal marginals. The LGDE is introduced, and asymptotic theory is derived. In particular, it is shown that the LGDE converges at a speed that does not depend on the dimension. Examples using real and simulated data confirm that the new estimator performs very well on finite sample sizes. Paper 3 ”Non-parametric estimation of conditional density functions: A new method”: Let X = (X1, . . . , Xp) be a stochastic vector having joint density function fX(x) with partitions X1 = (X1, . . . , Xk) and X2 = (Xk+1, . . . , Xp). A new method for estimating the conditional density function of X1 given X2 is presented. It is based on locally Gaussian approximations, but simplified in order to tackle the curse of dimensionality in multivariate applications, where both response and explanatory variables can be vectors. We compare our method to some available competitors, and the error of approximation is shown to be small in a series of examples using real and simulated data, and the estimator is shown to be particularly robust against noise caused by independent variables. We also present examples of practical applications of our conditional density estmator in the analysis of time series. Typical values for k in our examples are 1 and 2, and we include simulation experiments with values of p up to 6. Large sample theory is established under a strong mixing condition