Joint and Marginal Diagnostic Tests for Conditional Mean and Variance Specifications

Joint and marginal specification tests for conditional mean and variance models

Bootstrapping non-stationary stochastic volatility

Autoregressive Conditional Skewness

It is well known that time series of returns are characterized by volatility clustering and excess kurtosis. Therefore, when modelling the dynamic behavior of returns, inference and prediction methods, based on independent and/or Gaussian observations may be inadequate. As bootstrap methods are not, in general, based on any particular assumption on the distribution of the data, they are well suited for the analysis of returns. This paper reviews the application of bootstrap procedures for inference and prediction of financial time series. In relation to inference, bootstrap techniques have been applied to obtain the sample distribution of statistics for testing, for example, autoregressive dynamics in the conditional mean and variance, unit roots in the mean, fractional integration in volatility and the predictive ability of technical trading rules. On the other hand, bootstrap procedures have been used to estimate the distribution of returns which is of interest, for example, for Value at Risk (VaR) models or for prediction purposes. Although the application of bootstrap techniques to the empirical analysis of financial time series is very broad, there are few analytical results on the statistical properties of these techniques when applied to heteroscedastic time series. Furthermore, there are quite a few papers where the bootstrap procedures used are not adequate.

We present a pricing kernel that summarizes well the main features of the dynamics of interest rates and risk in postwar U.S. data and use it to uncover how the pricing kernel has moved with the short rate in this data. Our findings imply that standard monetary models miss an essential link between the central bank instrument and the economic activity that monetary policy is intended to affect, and thus we call for a new approach to monetary policy analysis. We sketch a new approach using an economic model based on our pricing kernel. The model incorporates the key relationships between policy and risk movements in an unconventional way: the central bank's policy changes are viewed as primarily intended to compensate for exogenous business cycle fluctuations in risk that threaten to push inflation off target. This model, while an improvement on standard models, is considered just a starting point for their revision. It leads to critical questions that researchers need to answer as they continue to revise their approach to monetary policy analysis. Copyright © 2009 by the National Bureau of Economic Research.

A general Markov‐Switching autoregressive conditional mean model, valued in the set of non‐negative numbers, is considered. The conditional distribution of this model is a finite mixture of non‐negative distributions whose conditional mean follows a GARCH‐like dynamics with parameters depending on the state of a Markov chain. Three different variants of the model are examined depending on how the lagged‐values of the mixing variable are integrated into the conditional mean equation. The model includes, in particular, Markov mixture versions of various well‐known non‐negative time series models such as the autoregressive conditional duration model, the integer‐valued GARCH (INGARCH) model, and the Beta observation driven model. For the three variants of the model, conditions are given for the existence of a stationary and ergodic solution. The proposed conditions match those already known for Markov‐switching GARCH models. We also give conditions for finite marginal moments. Applications to various mixture and Markov mixture count, duration and proportion models are provided.

To be utilized in the smart charging of plug-in electric vehicles, this paper proposes semiparametric conditional mean and variance models for the daily density forecasting of the electricity load. The mean is modeled by means of an autoregressive moving average model with exogenous inputs (ARMAX), whereas several options for the variance evolution are investigated, starting with modeling the variance as a power of the conditional mean, then as a piecewise constant function, and finally as a generalized autoregressive conditional heteroskedasticity (GARCH) model. Due to the possible non-Gaussianity of the distribution of the stochastic components, a quasi-maximum likelihood estimation (QMLE) with the Pearson type IV (P-IV) distribution is also considered, apart from the estimation with the Gaussianity assumption. Moreover, the daily density forecasts are generated in a non-parametric manner by propagating samples from the stochastic components iteratively through the available model. These strategies, involving different options for variance modeling and estimation, are compared in terms of their forecast performances on two representative phase currents from the low voltage cables of medium-to-low voltage transformers in the Netherlands. The results indicate that the QMLE with the Gaussianity assumption performs better due to the additional complexity of the P-IV distribution on estimation. Concerning the variance models, the piecewise constant and GARCH models are more preferable when processing phase currents exhibiting only daily seasonality, and the model as a power of the conditional mean outperforms the others if both daily and weekly seasonality and hence more complexity is present.

Neftci (1984) proposed a nonparametric test procedure for determining whether business cycles are asymmetric in the sense that contractions are steeper than expansions. He found evidence of this type of asymmetry in postwar quarterly U.S. unemployment variables.' This comment identifies a probable error in Neftci's empirical work that reverses the significance of his results for the unemployment rate, and it suggests that this test may have low power and be sensitive to measurement error.

The purpose is to document the significance of non-linearity in the conditional mean and variance of the long-run annual UK GDP growth rate. In particular, an examination is made of asymmetry in conditional variance with respect to past shocks which implies that the conditional variance is greater, and the conditional variance function steeper, following negative shocks to output growth. Simulations of the estimated model also suggest the potential for complex dynamics including substantial amplitude changes.

In highly volatile market conditions it's always difficult to predict returns using heteroscedastic Garch models. This paper tries to investigate the impact of sample data inputs over forecast using nested conditional mean ARMAX(2,2,0) and conditional variance Garch(1,1), Gjr-garch(1,1) and Egarch(1,1) models. Research also tries to indentify relationship between outcome of formal hypothesis tests, the Ljung-Box-Pierce Q-test and Engle's ARCH test, sample input and forecast results. Study is conducted over two diversified stock markets index of America (NASDAQ Composite) and Asia (Nikkei 225). Returns are forecasted for the volatile month of aug-08 using sample inputs of last one month, two months up to seven months before aug-08 and a complete year before inputi.e from 1-aug-07 to 31-jul-08 of 246 trading days has also been taken. Graphical and correlative comparison of forecasted and observed returns is also done to identify any trend followed by Garch models in relation to size of sample inputs and forecast. Results show that forecast of mean of returns is far more accurate with Nikkei as compared to Nasdaq.

variable, given a scalar linear measurement with an additive Cauchy noise, has a conditional mean and a nite conditional variance, both being functions of the measurement. For scalar discrete linear systems with additive process and measurement noises that have Cauchy probability density functions, characteristic functions are used to generate a recursive process from which the conditional mean and variance are easily obtained. The conditional mean and variance are generated by a growing sum that at each measurement update increases by one term, constructed from four new parameters. In particular, the characteristic function of the un-normalized conditional probability density function is considered where the dynamics of the parameters are propagated by linear dynamics. These parameters are shown to decay, allowing an approximate nite dimensional recursion. Since radar measurement noise and target motion process noise are not Gaussian, but are better characterized by heavy tailed density functions, a simple homing missile problem using proportional navigation is used to demonstrate the Cauchy estimator.

Granular modelling of signals: A framework of Granular Computing

Testing Non-Linear Dependence in the Hedge Fund Industry

This study aims to determine whether daily television audience dynamics exhibit statistically significant conditional variance dependence that is systematically overlooked in conventional ARIMA-based broadcasting forecasts and to assess the incremental empirical value of integrating ARIMA with GARCH modeling. Using 1,096 consecutive daily observations (2022–2024) of viewers for a nationally broadcast program, we implement a diagnostic-first framework that jointly evaluates conditional mean and variance processes. Stationarity is confirmed through the Augmented Dickey–Fuller test (ADF = −3.4693, p = 0.0088), and an MA(1) specification is selected for the conditional mean (AIC = 1302.76). Residual diagnostics reveal pronounced ARCH effects (ARCH-LM = 78.4602, p < 0.001), justifying second-moment modeling. Among competing variance specifications, GARCH(2,2) yields the lowest information criterion (AIC = 1060.321) and indicates near-unit volatility persistence (Σα + Σβ = 0.9856), evidencing durable intertemporal uncertainty transmission. Out-of-sample forecast evaluation demonstrates low relative error (MAPE ≈ 1.0%), supporting empirical robustness. Unlike prior ARIMA-centered broadcasting studies that prioritize point accuracy under homoscedastic assumptions, this integration explicitly models volatility clustering as an object of inference, aligning media analytics with established volatility frameworks without overstating cross-domain novelty. The findings show that incorporating conditional variance dynamics provides measurable gains in risk-sensitive forecasting, offering a replicable approach for advertising allocation and scheduling decisions in competitive media environments.

Test statistics for checking the independence between the innovations of several time series are developed. The time series models considered allow for general specifications for the conditional mean and variance functions that could depend on common explanatory variables. In testing for independence between more than two time series, checking pairwise independence does not lead to consistent procedures. Thus a finite family of empirical processes relying on multivariate lagged residuals are constructed, and we derive their asymptotic distributions. In order to obtain simple asymptotic covariance structures, Möbius transformations of the empirical processes are studied, and simplifications occur. Under the null hypothesis of independence, we show that these transformed processes are asymptotically Gaussian, independent, and with tractable covariance functions not depending on the estimated parameters. Various procedures are discussed, including Cramér–von Mises test statistics and tests based on non‐parametric measures. The ranks of the residuals are considered in the new methods, giving test statistics which are asymptotically margin‐free. Generalized cross‐correlations are introduced, extending the concept of cross‐correlation to an arbitrary number of time series; portmanteau procedures based on them are discussed. In order to detect the dependence visually, graphical devices are proposed. Simulations are conducted to explore the finite sample properties of the methodology, which is found to be powerful against various types of alternatives when the independence is tested between two and three time series. An application is considered, using the daily log‐returns of Apple, Intel and Hewlett‐Packard traded on the Nasdaq financial market.The Canadian Journal of Statistics40: 447–479; 2012 © 2012 Statistical Society of Canada