Volatility of Stock Market Indices - An Analysis based on SEMIFAR Models

By applying SEMIFAR models (Beran, 1999), we examine 'long memory' in the volatility of worldwide stock market indices. Our analysis yields strong evidence of 'long memory' in stock market volatility, either in terms of stochastic long-range dependence or in form of deterministic trends. In some cases, both components are detected in the data. Thus, at least partially, there appears to be even stronger and more systematic 'long memory', than suggested by a stationary model with long-range dependence.

Temporal aggregation of stationary and nonstationary FARIMA (p, d, 0) models

We consider temporal aggregation of stationary and nonstationary time series with short memory, long memory and antipersistence, within the framework of fractional autoregressive processes. Asymptotically, long memory and antipersistence are preserved whereas short memory components vanish. In the case of integrated processes, the results extend Tiao's [15] to the fractional case.

SEMIFAR models

Recent results on so-called SEMIFAR models introduced by Beran (1997) are discussed. The nonparametric deterministic trend is estimated by a kernel method. The differencing- and fractional differencing parameters as well as the autoregressive coefficients are estimated by an approximate maximum likelihood approach. A data-driven algorithm for estimating the whole model is proposed based on the iterative plug-in idea for selecting bandwidth in nonparametric regression with long-memory. Prediction for SEMIFAR models is also discussed briefly. Two examples illustrate the potential usefulness of these models in practice

SEMIFAR Models, with Applications to Commodities, Exchange Rates and the Volatility of Stock Market Indices

The distinction between stationarity, difference stationarity, deterministic trends as well as between short- and long-range dependence has a major impact on statistical conclusions, such as confidence intervals for population quantities or point and interval forecasts. In this paper, recent results on so-called SEMIFAR models introduced by Beran(1999) are summarized and their potential usefulness for economic time series analysis is illustrated by analyzing several commodities, exchange rates, the volatility of stock market indices and some simulated series. SEMIFAR models provide a unified approach that allows for simultaneous modelling of and distinction between deterministic trends, difference stationarity and stationarity with short- and long-range dependence. An iterative data-driven algorithm combines MLE and kernel estimation. Predictions combine stochastic prediction of the random part with functional extrapolation of the deterministic part

SEMIFAR Models, with Applications to Commodities, Exchange Rates and the Volatility of Stock Market Indices

The distinction between stationarity, difference stationarity, deterministic trends as well as between short- and long-range dependence has a major impact on statistical conclusions, such as confidence intervals for population quantities or point and interval forecasts. In this paper, recent results on so-called SEMIFAR models introduced by Beran(1999) are summarized and their potential usefulness for economic time series analysis is illustrated by analyzing several commodities, exchange rates, the volatility of stock market indices and some simulated series. SEMIFAR models provide a unified approach that allows for simultaneous modelling of and distinction between deterministic trends, difference stationarity and stationarity with short- and long-range dependence. An iterative data-driven algorithm combines MLE and kernel estimation. Predictions combine stochastic prediction of the random part with functional extrapolation of the deterministic part.

Stationäre und nichtstationäre Farima-Modelle - Modellwahl, Vorhersage, Aggregation und Intervention

Die vorliegende Dissertation befasst sich mit der Modellwahl,Vorhersage, temporalen Aggregation und Interventionsanalysestationärer und nichtstationärer fraktioneller autoregressiverProzesse, sowie einer extensiven Anwendung auf weltweiteFinanzmarktdaten.Stationäre fraktionelle autoregressive Modelle wurden zuerst vonGranger und Joyeux (1980), sowieHosking (1981) eingeführt. Sie dienen vor allem zurstochastischen Modellierung stationärer Zeitreihen mitlangfristigen Abhängigkeiten (oder langem Gedächtnis, bzw.Persistenz), die aber nicht so stark sind, dass eine einfacheDifferenzenbildung im Rahmen traditioneller Box-Jenkins Modelleadäquat wäre.Unglücklicherweise ist die stochastische Theorie dieser Modelletypischerweise auf den stationären Bereich desDifferenzenparameters d beschränkt. Ineinem aktuellen Artikel zeigte Beran (1995) jedoch, dassjedes reellwertige d>-0,5 durch einen approximativenMaximum-Likelihood-Schätzer bestimmt werden kann.Insbesondere kann dadurch die mit der Schätzung desDifferenzenparameters d>-0,5 verbundene Unsicherheit in denKonfidenzintervallen der autoregressiven Parameterberücksichtigt werden. Beran (1995) zeigte dies aber nur für den Fall,dass die autoregressive Ordnung a prioribekannt sei. Eine entsprechende Verallgemeinerung findet sichjedoch in Beran, Bhansali und Ocker (1998).Wir entwickelten eine Version des Akaike-Informationskriteriums(AIC) zur Bestimmung der autoregressiven Ordnung, wenn sowohl dals auch die autoregressiven Parameter simultan geschätztwerden. Die Resultate in Beran und Ocker (1999)über die Vorhersage fraktioneller autoregressiver Prozesse rundeten schliesslichdiesen vereinheitlichten Ansatz zur simultanen Modellierung undPrädiktion stationärer und nichtstationärer Prozesse mitkurzfristigen und langfristigen Abhängigkeiten ab

SEMIFAR forecasts, with applications to foreign exchange rates

SEMIFAR Models introduced in Beran (1999) provide a semiparametric modelling framework that enables the data analyst to separate deterministic and stochastic trends as well as short- and long-memory components in an observed time series. A correct distinction between these components, and in particular, the decision which of the components may be present in the data have an important impact on forecasts. In this paper, forecasts are based on an extrapolation of the nonparametric trend function and optimal forecasts of the stochastic component. In the data analytical part of the paper, the proposed method is applied to foreign exchange rates from Europe and Asia

Pricing of cap-interest rates based on renewal processes

Pricing of cap insurance contracts is considered for political mortgage rates. A simple stochastic process for mortgage rates is proposed. The process is based on renewal processes for modelling the length of periods with downward and upward trend respectively. Prices are calculated by simulation of conditional future sample paths. Future conditional quantiles can be obtained to assess the risk of a contract. The method is illustrated by applying it to observed quarterly mortgage rates of the Swiss Union of Raiffeisenbanks for the years 1970 to 2001

Temporal aggregation of stationary and nonstationary FARIMA (p, d, 0) models

We consider temporal aggregation of stationary and nonstationary time series with short memory, long memory and antipersistence, within the framework of fractional autoregressive processes. Asymptotically, long memory and antipersistence are preserved whereas short memory components vanish. In the case of integrated processes, the results extend Tiao's [15] to the fractional case