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Model-based clustering
Published in Elizabeth Ann Maharaj, Pierpaolo D'Urso, Jorge Caiado, Time Series Clustering and Classification, 2019
Elizabeth Ann Maharaj, Pierpaolo D'Urso, Jorge Caiado
Dealing with financial heteroskedastic time series, the comparison of the dynamics of the variances is pivotal. If the conditional variance follows a stochastic process, heteroskedastic time series can be represented by GARCH models. In this regards, in the literature different time series clustering models based on GARCH models have been proposed; see, for instance, Otranto (2008, 2010), Caiado and Crato (2010), D'Urso et al. (2013b, 2016). Heteroskedastic time series
Nonlinear Time Series Analysis
Published in Tucker S. McElroy, Dimitris N. Politis, Time Series, 2019
Tucker S. McElroy, Dimitris N. Politis
The conditional varianceσt2is often called the volatility.
Theoretical and practical foundations of liquidity-adjusted value-at-risk (LVaR)
Published in Noura Metawa, Mohamed Elhoseny, Aboul Ella Hassanien, M. Kabir Hassan, Expert Systems in Finance, 2019
To examine the relationship between asset expected returns and volatility, we can implement a conditional volatility approach to determine the risk parameters that are needed for the LVaR’s engine and thereafter for the estimation of daily asset market liquidity risk exposure and investable portfolio requirements. Indeed, the time-varying pattern of asset volatility has been widely recognized and modeled as a conditional variance within the GARCH framework, as originally developed by Engle (1982, 1995). Engle (1982) introduced a likelihood ratio test to ARCH effects and a maximum likelihood method to estimate the parameters in the ARCH model. This approach was generalized by Bollerslev (1986) and Engle and Kroner (1995). The following generalized autoregressive conditional heteroskedasticity in mean, GARCH-M (1,1) model, is used for the estimation of expected return and conditional volatility for each of the time-series variables:10Rit=ai+biσit+εit,σit2=ci+βi1σit−12+βi2εit−12, where Rit is the continuous compounding return of time series i, σit is the conditional standard deviation as a measure of volatility, and εit is the error term return for time series i. The denotations ai, bi, ci, βi1 and βi2 represent parameters to be estimated. The parameters representing variance are assumed to undertake a positive value.
Volatility forecasting for the shipping market indexes: an AR-SVR-GARCH approach
Published in Maritime Policy & Management, 2022
Jiaguo Liu, Zhouzhi Li, Hao Sun, Lean Yu, Wenlian Gao
where is the residual, is the conditional variance and represents the normal distribution. In practical application, the first-order GARCH model, that is, GARCH(1,1) specification, is sufficient to model the conditional variance (Chen, Härdle, and Jeong 2010). Then, Glosten, Jagannathan, and Runkle (1993) introduced the GJR model, the AR()-GJR(1,1) model can be written as follows:
A Conjugate Model for Dimensional Analysis
Published in Technometrics, 2018
Without distributional assumptions on Y, we resort to a regression approach instead of a likelihood approach. The main benefits include more freedom in the model and wider applications. These are especially important because the variation and the distribution of the response Y are often unknown. Here, we use least absolute deviations as the criterion in the projection pursuit regression. The least absolute deviations criterion is believed to be more robust to outliers, than the least-square criterion. Due to the multiplicative nature of the relationship between Y and Xi’s, the propagation of errors from Xi’s to Y is likely to generate residuals that are inhomogenous and associated with the conditional means. For example, if the observational errors are multiplicative, as . Even when p = 1 and E(ϵj) = 1, we have . The conditional variance is proportional to the squared conditional mean. If the errors εi are additive and small compared to values of Xi, then as , just like the multiplicative case. When p = 1 and E(ϵ′i) = 0, . The least absolute deviations protect against possible outliers (resulting from the inhomogenous errors) and are more robust in the projection pursuit regression. Note that the least absolute deviations estimates correspond to the maximum likelihood estimates under additive Laplace distributed errors. The complete procedure is as follows. Suppose the sample size is N, Y = (y1, …, yN)T, and Xi = (xi, 1, …, xi, N)T. Initialize current residuals and term counter Search for the next term in the model and the estimate , by minimizing the least absolute deviations criterion If the model is applied with DA, the estimate are achieved under dimensional constrains (1), that is, .Termination. If the criterion (8) is smaller than a threshold value, stop. Otherwise, update the current residuals and the term counter and go to Step 2.