金融时序数据建模的模型设定问题分析

W．J．Granger与D．F．Hendry（2004）关于建模思路的对话引起了国际计量经济学界关于模型设定问题的争论，本文就这一问题分析讨论了在金融时序数据实证研究中得以广泛应用的ARCH／GARCH模型的设定问题，认为在金融时序数据的建模中，ARMA族模型不宜作为数据生成过程的模型设定，其统计性质也不能直接扩展到ARMA-GARCH族数据生成过程。虽然ARCH／GARCH族模型作为金融时序数据的生成过程有着良好的统计性质，但不宜单纯采用一般到特殊的建模思路，而应是一般到特殊和特殊到一般两种建模思路的结合。ARCH／GARCH族模型的设定应当包含事前检验、事后检验等设定检验步骤。     

Modelling optimal strategies for the allocation of wealth in multicurrency investments

This paper analyzes rates of return on financial assets denominated in five major currencies and provides a framework for the determination of optimal strategies for the allocation of wealth in multicurrency investments. Three models are estimated: a univariate autoregressive conditional heteroskedasticity (ARCH) model, an extended ARCH model using the random coefficient (RC) procedure, and a pure RC model. A comparison of the forecasts of these models with those generated by a random walk model demonstrates that forecasts based on the RC/extended ARCH procedure are superior to those based on the random walk model and those based on direct ARCH estimation. These results could be useful for both international investors for the allocation of their wealth among fixed-income investment securities and central banks for the management of their external reserve assets.     

Estimation of variance of housing prices using spatial conditional heteroskedasticity (SARCH) model with an application to Boston housing price data

In this paper we investigate housing price volatility within a spatial econometrics setting. We propose an extended spatial regression model of the real estate market that includes the effects of both conditional heteroskedasticity and spatial autocorrelation. Our suggested model has features similar to those of autoregressive conditional heteroskedasticity (ARCH) in the time-series context. We utilize the spatial ARCH (SARCH) model to analyze Boston housing price data used by Harrison and Rubinfeld (1978) and Gilley and Pace (1996). We show that measuring the variability of housing prices is an important issue and our SARCH model captures the conditional spatial variability of Boston housing prices. We argue that there is a different source of spatial variation, which is independent of traditional housing and neighborhood characteristics, and is captured by the SARCH model.     

ARCH MODELS: PROPERTIES, ESTIMATION AND TESTING

Abstract. The aim of this survey paper is to provide an account of some of the important developments in the autoregressive conditional heteroskedasticity (ARCH) model since its inception in a seminal paper by Engle (1982). This model takes account of many observed properties of asset prices, and therefore, various interpretations can be attributed to it. We start with the basic ARCH models and discuss their different interpretations. ARCH models have been generalized in different directions to accommodate more and more features of the real world. We provide a comprehensive treatment of many of the extensions of the original ARCH model. Next we discuss estimation and testing for ARCH models and note that these models lead to some interesting and unique problems. There have been numerous applications and we mention some of these as we present different models. The paper includes a glossary of the acronyms for the models we describe.     

Time Series Based Errors and Empirical Errors in Fertility Forecasts in the Nordic Countries*

We use ARCH time series models to derive model based prediction intervals for the Total Fertility Rate (TFR) in Norway, Sweden, Finland, and Denmark up to 2050. For the short term (5–10 yrs), expected TFR‐errors are compared with empirical forecast errors observed in historical population forecasts prepared by the statistical agencies in these countries since 1969. Medium‐term and long‐term (up to 50 years) errors are compared with error patterns based on so‐called naïve forecasts, i.e. forecasts that assume that recently observed TFR‐levels also apply for the future.     

The stochastic volatility in mean model: empirical evidence from international stock markets

In this paper we present an exact maximum likelihood treatment for the estimation of a Stochastic Volatility in Mean (SVM) model based on Monte Carlo simulation methods. The SVM model incorporates the unobserved volatility as an explanatory variable in the mean equation. The same extension is developed elsewhere for Autoregressive Conditional Heteroscedastic (ARCH) models, known as the ARCH in Mean (ARCH‐M) model. The estimation of ARCH models is relatively easy compared with that of the Stochastic Volatility (SV) model. However, efficient Monte Carlo simulation methods for SV models have been developed to overcome some of these problems. The details of modifications required for estimating the volatility‐in‐mean effect are presented in this paper together with a Monte Carlo study to investigate the finite sample properties of the SVM estimators. Taking these developments of estimation methods into account, we regard SV and SVM models as practical alternatives to their ARCH counterparts and therefore it is of interest to study and compare the two classes of volatility models. We present an empirical study of the intertemporal relationship between stock index returns and their volatility for the United Kingdom, the United States and Japan. This phenomenon has been discussed in the financial economic literature but has proved hard to find empirically. We provide evidence of a negative but weak relationship between returns and contemporaneous volatility which is indirect evidence of a positive relation between the expected components of the return and the volatility process. Copyright © 2002 John Wiley & Sons, Ltd.     

On a double-threshold autoregressive heteroscedastic time series model

Tong's threshold models have been found useful in modelling nonlinearities in the conditional mean of a time series. The threshold model is extended to the so-called double-threshold ARCH(DTARCH) model, which can handle the situation where both the conditional mean and the conditional variance specifications are piecewise linear given previous information. Potential applications of such models include financial data with different (asymmetric) behaviour in a rising versus a falling market and business cycle modelling. Model identification, estimation and diagnostic checking techniques are developed. Maximum likelihood estimation can be achieved via an easy-to-use iteratively weighted least squares algorithm. Portmanteau-type statistics are also derived for checking model adequacy. An illustrative example demonstrates that asymmetric behaviour in the mean and the variance could be present in financial series and that the DTARCH model is capable of capturing these phenomena.     

Combining competing forecasts of inflation using a bivariate arch model

This paper sets out the basic structure of the bivariate generalization of Engle's ARCH model. Conditions which guarantee that the conditional covariance matrix is well defined are summarized, as are estimation and hypothesis testing.The process is used to combine forecasts where the weights are allowed to vary over time. Forecast errors from competing models are treated as a bivariate ARCH process so that the conditional covariance matrix adapts over time. At each point in time these conditional estimates of the variances and covariances are used to construct the optimal weights for combining the forecasts. Consequently, when one model is fitting well, its variance will be reduced and its weight will be increased.Two models of US inflation are constructed; one is a stylized monetarist model while the other is a mark-up model. The forecast errors are modeled as a simple bivariate ARCH process. Diagnostic tests reveal that this has overly restricted the parameterization of the covariance matrix. An approximation to the theoretically anticipated factor structure model is then estimated. The results in both cases show the weights varying over the sample period in moderately interpretable fashion.     

Recent Theoretical Results for Time Series Models with GARCH Errors

This paper provides a review of some recent theoretical results for time series models with GARCH errors, and is directed towards practitioners. Starting with the simple ARCH model and proceeding to the GARCH model, some results for stationary and nonstationary ARMA–GARCH are summarized. Various new ARCH–type models, including double threshold ARCH and GARCH, ARFIMA–GARCH, CHARMA and vector ARMA–GARCH, are also reviewed.     

Inconsistency of the MLE and inference based on weighted LS for LARCH models

This paper considers a class of finite-order autoregressive linear ARCH models. The model captures the leverage effect, allows the volatility to be arbitrarily close to zero and to reach its minimum for non-zero innovations, and is appropriate for long memory modeling when infinite orders are allowed. However, the (quasi-)maximum likelihood estimator is, in general, inconsistent. A self-weighted least-squares estimator is proposed and is shown to be asymptotically normal. A score test for conditional homoscedasticity and diagnostic portmanteau tests are developed. Their performance is illustrated via simulation experiments. It is also investigated whether stock market returns exhibit some of the characteristic features of the linear ARCH model.     

A Note on the Power of Money-Output Causality Tests

This study suggests that some empirical findings against money-output causality can be the consequence of ignoring autoregressive conditional heteroskedastic (ARCH) errors. Monte Carlo results confirm that ARCH effects drastically reduce the power of the standard causality test. The maximum likelihood approach allowing for ARCH effects, on the other hand, provides a good power performance. Using different specifications and sample period, Friedman and Kuttner (1993) and Thomas (1994) report limited evidence of money causing output. We detect significant ARCH effects in the models considered by these studies. Once ARCH effects are explicitly accounted for, we find that the monetary effect is significant though its magnitude is quite small.     

Testing for ARCH in the presence of nonlinearity of unknown form in the conditional mean

Tests of ARCH are a routine diagnostic in empirical econometric and financial analysis. However, it is well known that misspecification of the conditional mean may lead to spurious rejection of the null hypothesis of no ARCH. Nonlinearity is a prime example of this phenomenon. There is little work on the extent of the effect of neglected nonlinearity on the properties of ARCH tests. We investigate this using new ARCH testing procedures that are robust to the presence of neglected nonlinearity. Monte Carlo evidence shows that the problem is serious and that the new methods alleviate this problem to a very large extent. We apply the new tests to exchange rate data and find substantial evidence of spurious rejection of the null hypothesis of no ARCH.     

Change point test for tail index for dependent data

To test for the constancy of tail index, Quintos et al. (Rev Econ Stud 68:633–663, 2001) proposed three types of change point tests for independent and ARCH type sequences. In this paper, we demonstrate that their tests can be successfully extended to a large class of dependent stationary sequences. Further, we designate a time-reverse version of those tests since the original tests produce very low powers in case the tail of distribution gets thinner. A simulation study is implemented for illustration.     

The exchange rate and macroeconomic determinants: Time-varying transitional dynamics

In this paper, I consider modeling the effects of the macroeconomic determinants on the nominal exchange rate to be channeled through the transition probabilities in a Markovian process. The model posits that the deviation of the exchange rate from its fundamental value alters the market's belief in the probability of the process staying in certain regime next period. This paper further takes into account the ARCH effects of the volatility of the exchange rate. Empirical results generally confirm that fundamentals can affect the evolution of the dynamics of the exchange rate in a nonlinear way through the transition probabilities. In addition, I find that the volatility of the exchange rate is associated with significant ARCH effects which are subject to regime changes.     

An empirical analysis of alternative parametric ARCH models

This paper presents empirical evidence on the effectiveness of eight different parametric ARCH models in describing daily stock returns. Twenty‐seven years of UK daily data on a broad‐based value weighted stock index are investigated for the period 1971–97. Several interesting results are documented. Overall, the results strongly demonstrate the utility of parametric ARCH models in describing time‐varying volatility in this market. The parameters proxying for asymmetry in models that recognize the asymmetric behaviour of volatility are highly significant in each and every case. However, the ‘performance’ of the various parameterizations is often fairly similar with the exception of the multiplicative GARCH model that performs qualitatively differently on several dimensions of performance. The outperformance of any model(s) is not consistent across different sub‐periods of the sample, suggesting that the optimal choice of a model is period‐specific. The outperformance is also not consistent as we change from in‐sample inferences to out‐of‐sample inferences within the same period. Copyright © 2000 John Wiley & Sons, Ltd.     

SIGN- AND VOLATILITY-SWITCHING ARCH MODELS: THEORY AND APPLICATIONS TO INTERNATIONAL STOCK MARKETS

This paper develops two conditionally heteroscedastic models which allow an asymmetric reaction of the conditional volatility to the arrival of news. Such a reaction is induced by both the sign of past shocks and the size of past unexpected volatility. The proposed models are shown to converge in distribution to absolutely continuous Itô diffusion processes, as happens for other heteroscedastic formulations. One of the schemes developed in the paper—the Volatility-switching ARCH—differs from the existing asymmetric models insofar as it is able to capture a particular aspect of the behaviour of the volatilities, i.e. the reversion of their asymmetric reaction to news. Empirical evidence from stock market returns in six countries shows that such a model outperforms traditional asymmetric ARCH equations. © 1997 by John Wiley & Sons, Ltd.     

证券投资基金业绩度量模型探析

证券投资基金是中国资本市场上一支重要的力量,对其业绩进行客观评价是一项极其复杂的系统工程。本文试图运用风险调整收益的思想,分别选取风险价值VAR与尾条件期望CVAR的加权平均、以及ARCH模型计算出来的条件异方差作为风险测度指标,建立基金业绩评价模型。     

Value‐at‐risk for long and short trading positions

In this paper we model Value‐at‐Risk (VaR) for daily asset returns using a collection of parametric univariate and multivariate models of the ARCH class based on the skewed Student distribution. We show that models that rely on a symmetric density distribution for the error term underperform with respect to skewed density models when the left and right tails of the distribution of returns must be modelled. Thus, VaR for traders having both long and short positions is not adequately modelled using usual normal or Student distributions. We suggest using an APARCH model based on the skewed Student distribution (combined with a time‐varying correlation in the multivariate case) to fully take into account the fat left and right tails of the returns distribution. This allows for an adequate modelling of large returns defined on long and short trading positions. The performances of the univariate models are assessed on daily data for three international stock indexes and three US stocks of the Dow Jones index. In a second application, we consider a portfolio of three US stocks and model its long and short VaR using a multivariate skewed Student density. Copyright © 2003 John Wiley & Sons, Ltd.     

Multivariate GARCH models: a survey

This paper surveys the most important developments in multivariate ARCH‐type modelling. It reviews the model specifications and inference methods, and identifies likely directions of future research. Copyright © 2006 John Wiley & Sons, Ltd.