Publication: Bootstrap forecasts of multivariate time series
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2014-06
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2014-07-18
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Abstract
En esta tesis se estudia el desempeño de procedimientos que tienen por objetivo la aproximación
de densidades de predicción y sus respectivos intervalos y regiones de confianza en series de
tiempos multivariantes. En concreto, desarrollamos procedimientos bootstrap para predecir los
modelos VAR y DCC, utilizados a menudo en la modelización y predicción de series temporales
macroeconómicas y financieras. La metodología bootstrap analizada en esta tesis es atractiva debido
a que no necesita supuestos distribucionales y es apropiada para incorporar la incertidumbre
de los parámetros y del modelo.
En el Capítulo 1 se describen los modelos VAR y DCC y el enfoque tradicional para construir
densidades de predicción con los mismos. Los problemas que surgen con este enfoque nos motiva
a considerar alternativas, algunas de ellas basadas en bootstrapping. Para entonces será un
momento propicio para presentar brevemente la metodología bootstrap en el marco de series de
tiempo así como su aplicación en problemas de predicción.
En el Capítulo 2 se establece la validez asintótica y se analiza el desempeño en muestras
pequeñas de un procedimiento bootstrap propuesto para construir densidades de predicción multivariante
en el contexto de modelos VAR no Gausianos. Este procedimiento bootstrap no necesita
de la representación backward usada por las alternativas existentes en la literatura y, por lo tanto,
se puede utilizar para obtener densidades de predicción multivariante en, por ejemplo, modelos
VARMAo VAR-GARCH. En el contexto de un VAR bivariado y estacionario, desarrollamos varios
experimentos de Monte Carlo con el objetivo de estudiar sus propiedades en muestras pequeñas,
obteniendo que las mismas son comparables a las de las alternativas basadas en la representación backward. Por tanto, nada se pierde cuando se abandonan los procedimientos bootstrap que hacen
uso de esta última. Este resultado también lo sugiere una de nuestras aplicaciones empíricas
en la que construimos densidades de predicción conjunta para la inflación, el desempleo y el crecimiento
del producto trimestrales de EEUU y sus correspondientes regiones de predicción con sus
coberturas empíricas, éstas últimas obtenidas en base a un esquema rolling window. Por último,
reproducimos un ejemplo de libro que utiliza la metodología de Gaussiana para predecir la inversión, el consumo y los ingresos en Alemania Occidental para después añadir los intervalos de
predicción bootstrap con fines de comparación.
El modelo, los parámetros y la distribución del error rara vez son conocidos con exactitud
por el usuario y, por tanto, la incertidumbre causada por la implementación del modelo estimado
debe tenerse en cuenta la hora de hacer predicciones. Distintos métodos bootstrap se han desarrollado
con éxito para hacer frente a las diferentes fuentes de incertidumbre en el contexto de
los modelos VAR. Por esta razón, en el Capítulo 3 se compara el desempeño de las regiones de
predicción construídas en base a la metodología tradicional Gaussiana y diversas variantes del
procedimiento bootstrap que incorporan sucesivamente la incertidumbre en la distribución del
error, del parámetro, la corrección por sesgo y la incertidumbre del modelo. Nuestro experimento
de Monte Carlo sugiere que la incertidumbre de los parámetros juega un papel preponderante
cuando se predicen modelos VAR altamente persistentes.
En cuanto a las predicciones de los modelos DCC, hay dos problemas que merecen atención.
En primer lugar, la no Gaussianidad de los rendimientos requiere de formas alternativas para
aproximar sus densidades de predicción. En segundo lugar, sólo se pueden obtener predicciones
puntuales de volatilidades, covarianzas y correlaciones en cada momento del tiempo. Por tanto,
en el Capítulo 4 se presenta un procedimiento bootstrap para predecir los rendimientos, volatilidades,
covarianzas y correlaciones en los modelos DCC. También llevamos a cabo simulaciones de
Monte Carlo para evaluar en muestras pequeñas las propiedades del procedimiento propuesto.
Los resultados muestran un buen desempeño con diferentes tamaños de muestras y distribuciones
del error. Finalmente, empleamos el algoritmo bootstrap en dos sistemas de rendimientos financieros. En primer lugar, construímos predicciones fuera de la muestra de rendimientos,
volatilidades, covarianzas y correlaciones para un sistema de rendimientos de tipos de cambio
diarios, Euro, Yen japonés y Dólar australiano respecto el Dólar de EE.UU. Asimismo, obtuvimos
intervalos de predicción bootstrap, dentro de la muestra, de la correlación condicional de los
rendimientos obtenidos con los índices de mercados S&P500 y NASDAQ. Ambas aplicaciones a
datos reales sugieren que nuestro procedimiento brinda información adicional que enriquece la
predicción de modelos DCC.
Por último, en el Capítulo 5 se concluye y presentan las líneas de investigación que quedan
abiertas.
In this thesis we study the performance of bootstrap procedures to approximate forecast densities and their intervals and regions for multivariate time series data. In particular, we develop bootstrap procedures for VAR and DCC models which are often implemented when modeling and forecasting macroeconomic and financial time series. The bootstrap methodology considered in this thesis is attractive since it is free of distributional assumptions and well suited to incorporate the parameter uncertainty and can be even designed to deal with the model uncertainty. Chapter 1 introduces the VAR and DCC models and the standard approach to construct forecast densities within them. The problems that arise when forecasting these models push us to consider alternatives, some of them based on bootstrapping. By then, it will be a propitious time to briefly introduce the bootstrap methodology for time dependent data and review its implementation to forecast problems. Chapter 2 establishes the asymptotic validity and analyses the finite sample performance of a simple bootstrap procedure to construct multi-step multivariate forecast densities in the context of non-Gaussian unrestricted VAR models. This bootstrap procedure avoids the backward representation used by existing alternatives and, consequently, can be implemented to obtain multivariate forecast densities in, for example, VARMA or VAR-GARCH models. In the context of bivariate stationary VAR models, we carry out several Monte Carlo experiments to study its finite sample properties, finding that these are comparable to those of alternatives based on the backward representation. Hence we remark that nothing is lost when we abandon the more complicated backward representation. This result is also suggested by one of the empirical applications, in which we construct joint forecast densities of US quarterly inflation, unemployment and GDP growth and the corresponding forecast regions with their empirical coverages obtained using a rolling window scheme. Finally, we reproduce a textbook example that applies the standard Gaussian methodology to forecast West German investment, consumption and income and then, for comparison purposes, we include the bootstrap forecast intervals. The model, the parameters and the error distribution are rarely known without uncertainty by the forecaster and, thus, the sampling variability caused by the use of the estimated model needs to be taken into account. Bootstrap methods are successfully designed to deal with different sources of uncertainties in the context of forecasting VAR models. For this reason, Chapter 3 compares the forecast performance of the regions constructed using the traditional Gaussian methodology and several variants of the bootstrap procedure that successively incorporate error distribution, parameter uncertainty, bias correction and lag order uncertainty. Our Monte Carlo study suggests that the parameter uncertainty plays a prominent role when forecasting highly persistent VAR models. Regarding DCC forecast, there are two problem that deserve attention. First, the non-Gaussianity of returns demands alternative ways of approximating its forecast density. Second, only point forecasts of volatilities, covariances and correlations can be obtained at each moment of time. These issues encouraged us to developed bootstrap procedure to forecast returns, volatilities, covariances and correlations in corrected DCC models, which is described in Chapter 4. We conduct Monte Carlo simulations in order to evaluate its finite sample properties, which show a rather good performance of the bootstrap procedure under different sample sizes and error distributions. We apply the proposed bootstrap algorithm to two systems of returns. First, we obtain out-of-sample forecast of returns, volatilities, covariances and correlations in the context of a system of daily exchange rates returns of the Euro, Japanese Yen and Australian Dollar against the US Dollar. Second, we construct within sample forecast intervals of the conditional correlation of S&P500 and NASDAQ returns. Both empirical applications point out that our bootstrap algorithm can provide additional information that enriches the DCC forecast approach. Finally, Chapter 5 concludes and presents the research lines that are still open.
In this thesis we study the performance of bootstrap procedures to approximate forecast densities and their intervals and regions for multivariate time series data. In particular, we develop bootstrap procedures for VAR and DCC models which are often implemented when modeling and forecasting macroeconomic and financial time series. The bootstrap methodology considered in this thesis is attractive since it is free of distributional assumptions and well suited to incorporate the parameter uncertainty and can be even designed to deal with the model uncertainty. Chapter 1 introduces the VAR and DCC models and the standard approach to construct forecast densities within them. The problems that arise when forecasting these models push us to consider alternatives, some of them based on bootstrapping. By then, it will be a propitious time to briefly introduce the bootstrap methodology for time dependent data and review its implementation to forecast problems. Chapter 2 establishes the asymptotic validity and analyses the finite sample performance of a simple bootstrap procedure to construct multi-step multivariate forecast densities in the context of non-Gaussian unrestricted VAR models. This bootstrap procedure avoids the backward representation used by existing alternatives and, consequently, can be implemented to obtain multivariate forecast densities in, for example, VARMA or VAR-GARCH models. In the context of bivariate stationary VAR models, we carry out several Monte Carlo experiments to study its finite sample properties, finding that these are comparable to those of alternatives based on the backward representation. Hence we remark that nothing is lost when we abandon the more complicated backward representation. This result is also suggested by one of the empirical applications, in which we construct joint forecast densities of US quarterly inflation, unemployment and GDP growth and the corresponding forecast regions with their empirical coverages obtained using a rolling window scheme. Finally, we reproduce a textbook example that applies the standard Gaussian methodology to forecast West German investment, consumption and income and then, for comparison purposes, we include the bootstrap forecast intervals. The model, the parameters and the error distribution are rarely known without uncertainty by the forecaster and, thus, the sampling variability caused by the use of the estimated model needs to be taken into account. Bootstrap methods are successfully designed to deal with different sources of uncertainties in the context of forecasting VAR models. For this reason, Chapter 3 compares the forecast performance of the regions constructed using the traditional Gaussian methodology and several variants of the bootstrap procedure that successively incorporate error distribution, parameter uncertainty, bias correction and lag order uncertainty. Our Monte Carlo study suggests that the parameter uncertainty plays a prominent role when forecasting highly persistent VAR models. Regarding DCC forecast, there are two problem that deserve attention. First, the non-Gaussianity of returns demands alternative ways of approximating its forecast density. Second, only point forecasts of volatilities, covariances and correlations can be obtained at each moment of time. These issues encouraged us to developed bootstrap procedure to forecast returns, volatilities, covariances and correlations in corrected DCC models, which is described in Chapter 4. We conduct Monte Carlo simulations in order to evaluate its finite sample properties, which show a rather good performance of the bootstrap procedure under different sample sizes and error distributions. We apply the proposed bootstrap algorithm to two systems of returns. First, we obtain out-of-sample forecast of returns, volatilities, covariances and correlations in the context of a system of daily exchange rates returns of the Euro, Japanese Yen and Australian Dollar against the US Dollar. Second, we construct within sample forecast intervals of the conditional correlation of S&P500 and NASDAQ returns. Both empirical applications point out that our bootstrap algorithm can provide additional information that enriches the DCC forecast approach. Finally, Chapter 5 concludes and presents the research lines that are still open.
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Keywords
Bootstrap, Análisis de series temporales, Análisis multivariante, Previsión