Showing papers on "Heteroscedasticity published in 1993"
Book•
01 Jan 1993
TL;DR: In this paper, the authors propose a nonlinear regression model based on the Gauss-Newton Regression for least squares, and apply it to time-series data and show that the model can be used for regression models for time series data.
Abstract: 1. The Geometry of Least Squares 2. Nonlinear Regression Models and Nonlinear Least Squares 3. Inference in Nonlinear Regression Models 4. Introduction to Asymptotic Theory and Methods 5. Asymptotic Methods and Nonlinear Least Squares 6. The Gauss-Newton Regression 7. Instrumental Variables 8. The Method of Maximum Likelihood 9. Maximum Likelihood and Generalized Least Squares 10. Serial Correlation 11. Tests Based on the Gauss-Newton Regression 12. Interpreting Tests in Regression Directions 13. The Classical Hypothesis Tests 14. Transforming the Dependent Variable 15. Qualitative and Limited Dependent Variables 16. Heteroskedasticity and Related Topics 17. The Generalized Method of Moments 18. Simultaneous Equations Models 19. Regression Models for Time-Series Data 20. Unit Roots and Cointegration 21. Monte Carlo Experiments
4,912 citations
TL;DR: In this paper, the authors derived low frequency, say weekly, models implied by high frequency, such as ARMA models with symmetric GARCH errors, and they showed that low frequency models exhibit conditional heteroskedasticity of the GARCH form as well.
Abstract: The authors derive low frequency, say weekly, models implied by high frequency, say daily, ARMA models with symmetric GARCH errors. They show that low frequency models exhibit conditional heteroskedasticity of the GARCH form as well. The parameters in the conditional variance equation of the low frequency model depend upon mean, variance, and kurtosis parameters of the corresponding high frequency model. Moreover, strongly consistent estimators of the parameters in the high frequency model can be derived from low frequency data. The common assumption in applications that rescaled innovations are independent is disputable, since it depends upon the available data frequency. Copyright 1993 by The Econometric Society.
563 citations
TL;DR: In this paper, the authors introduce a class of statistical tests for the hypothesis that some feature that is present in each of several variables is common to them, which are data properties such as serial correlation, trends, seasonality, heteroscedasticity, auto-regression, and excess kurtosis.
Abstract: This article introduces a class of statistical tests for the hypothesis that some feature that is present in each of several variables is common to them. Features are data properties such as serial correlation, trends, seasonality, heteroscedasticity, autoregressive conditional hetero-scedasticity, and excess kurtosis. A feature is detected by a hypothesis test taking no feature as the null, and a common feature is detected by a test that finds linear combinations of variables with no feature. Often, an exact asymptotic critical value can be obtained that is simply a test of overidentifying restrictions in an instrumental variable regression. This article tests for a common international business cycle.
550 citations
TL;DR: In this article, the authors enlarge the class of threshold Heteroscedastic Models (TARCH) introduced by Zakoian (1991a) and show that it is possible to relax the positivity constraints on the parameters of the conditional variance.
Abstract: This paper attempts to enlarge the class of Threshold Heteroscedastic Models (TARCH) introduced by Zakoian (1991a). We show that it is possible to relax the positivity constraints on the parameters of the conditional variance. Unconstrained models provide a greater generality of the paths allowing for nonlinearities in the volatility. Cyclical behaviour is permitted as well as different relative impacts of positive and negative shocks on volatility, depending on their size. We give empirical evidence using French stock returns.
513 citations
TL;DR: In this paper, the authors take advantage of the time-varying structure of stock-returns variances to investigate whether two international stock markets share the same volatility process.
Abstract: In this article, we take advantage of the time-varying structure of stock-returns variances to investigate whether two international stock markets share the same volatility process. We use a test recently developed by Engle and Kozicki. This test is also used to assess the validity of a one-factor autoregressive conditional heteroscedasticity model. We find that some international stock markets have the same time-varying volatility.
321 citations
TL;DR: In this paper, conditions for co-persistence to occur in the multivariate linear GARCH model are presented, parallel the conditions for linear co-integration in the mean, as developed by Engle and Granger (1987).
Abstract: Since the introduction of the autoregressive conditional heteroskedastic (ARCH) model in Engle (1982), numerous applications of this modeling strategy have already appeared. A common finding in many of these studies with high frequency financial or monetary data concerns the presence of an approximate unit root in the autoregressive polynomial in the univariate time series representation for the conditional second order moments of the process, as in the so-called integrated generalized ARCH (IGARCH) class of models proposed in Engle and Bollerslev (1986). In the IGARCH models shocks to the conditional variance are persistent, in the sense that they remain important for forecasts of all horizons. This idea is readily extended to a multivariate framework. Even though many time series may exhibit persistence in variance, it is likely that several different variables share the same common long-run component. In that situation, the variables are naturally defined to be co-persistent in variance, and the co-persistent linear combination is interpretable as a long-run relationship. Conditions for co-persistence to occur in the multivariate linear GARCH model are presented. These conditions parallel the conditions for linear co-integration in the mean, as developed by Engle and Granger (1987). The presence of co-persistence has important implications for asset pricing relationships and in optimal portfolio allocation decisions. An empirical example relating to the time series properties of nominal U.S. dollar exchange rates for the deutschemark and the British pound provides a simple illustration of the ideas.
301 citations
TL;DR: In this article, the effects of heteroscedasticity on the parameters of frontier regression models were investigated and a Monte Carlo experiment was employed to determine that heteroscaledasticity leads to overestimation of intercept and underestimation of slope coefficients.
279 citations
TL;DR: This article examined the relation between risk premiums and conditional variances or covariances of asset returns in numerical versions of a dynamic asset-pricing theory and showed that it can be increasing, decreasing, flat, or nonmonotonic.
Abstract: Many statistical models of time-varying risk premiums, including the autoregressive conditional heteroscedasticity (ARCH)-in-mean, attempt to exploit a relation between risk premiums and conditional variances or covariances of asset returns. We examine this relation in numerical versions of a dynamic asset-pricing theory and show that it can be increasing, decreasing, flat, or nonmonotonic. Its shape depends on both the preferences of the representative agent and the stochastic structure of the economy. Without additional structure, the theory does not provide either a general foundation for ARCH-in-mean specifications or a simple interpretation of their parameters.
233 citations
TL;DR: The authors found strong statistical evidence that higher levels of inflation are less predictable, although innovations in inflation are somewhat better predictors of future volatility than the actual level of inflation, and they argued that previous failures to find an inflation-uncertainty relationship are due to two factors.
Abstract: Milton Friedman proposed that there is a positive relationship between inflation and uncertainty about the future path of inflation. In contrast to previous studies of this hypothesis, we find strong statistical evidence that higher levels of inflation are less predictable, although innovations in inflation are somewhat better predictors of future volatility than the actual level of inflation. We argue that previous failures to find an inflation-uncertainty relationship are due to two factors. First, none of the previous work directly tested Friedman's hypothesis by including the level of inflation in the model of the conditional variance. Second, these studies also used symmetric models, which appears inconsistent with Friedman's hypothesis. Our results are robust to different sample periods and to assumptions about the presence of a unit root in inflation. To test the inflation-uncertainty hypothesis, we use state-dependent models (SDM's) of conditional moments to estimate the time-varying conditional v...
216 citations
TL;DR: In this paper, the weekly rates of the European Monetary System (EMS) vis-a-vis the Deutsche mark from April 1979 to March 1991 are modeled as a combined generalized autoregressive conditional heteroscedasticity (GARCH) and jump process.
Abstract: Weekly rates of the European Monetary System (EMS) vis-a-vis the Deutsche mark from April 1979 to March 1991 are modeled as a combined MA (1)–GARCH(1, 1)–jump process. The moving average (MA) part accounts for mean reversion required for the rates to stay inside the target zone. The generalized autoregressive conditional heteroscedasticity (GARCH) part accounts for changing volatility, whereas the jump process models parity changes and other erratic movements. Using an adjusted Pearson chi-squared goodness-of-fit test, we find similar results for the Bernoulli and the Poisson jump processes. In those cases in which the Bernoulli–normal distribution does not pass the goodness-of-fit test, a mixture of three normals does. Finally the MA(1)–GARCH(1, 1)–Bernoulli jump models are jointly estimated assuming a constant contemporaneous correlation matrix for the disturbances and a common jump probability for all the currencies.
192 citations
TL;DR: In this article, the authors extend the standard unobserved component time series model to include Hamilton's Markov-switching heteroscedasticity and apply a generalized version of the model to investigate the link between inflation and its uncertainty (U.S. data, gross national product deflator, 1958:1-1990:4).
Abstract: In this article, I first extend the standard unobserved-component time series model to include Hamilton's Markov-switching heteroscedasticity. This will provide an alternative to the unobserved-component model with autoregressive conditional heteroscedasticity, as developed by Harvey, Ruiz, and Sentana and by Evans and Wachtel. I then apply a generalized version of the model to investigate the link between inflation and its uncertainty (U.S. data, gross national product deflator, 1958:1–1990:4). I assume that inflation consists of a stochastic trend (random-walk) component and a stationary autoregressive component, following Ball and Cecchetti, and a four-state model of U.S. inflation rate is specified. By incorporating regime shifts in both mean and variance structures, I analyze the interaction of mean and variance over long and short horizons. The empirical results show that inflation is costly because higher inflation is associated with higher long-run uncertainty.
TL;DR: In this paper, the authors consider the finite-sample accuracy of the Dickey-Fuller root tests when the errors are conditionally heteroskedastic and consider the specific case that the error variance follows a GARCH (1, 1) model.
TL;DR: A review of the use of bootstrapping in econometrics can be found in this paper, where the authors present a review of several applications of bootstrap in economics.
Abstract: Publisher Summary This chapter presents a review of the several applications of bootstrap in econometrics. Almost every type of model used in econometric work has been bootstrapped: regression models with heteroskedastic and autocorrelated errors, seemingly unrelated regression models, models with lagged dependent variables, state–space models and the Kalman filter, panel data models, simultaneous equation models, logit, probit, tobit, and other limited dependent variable models, generalized autoregressive conditional heteroskedasticity (GARCH) models, robust estimators (LAD estimators), data mining, pretesting, James–Stein estimation, semi-parametric estimators, and so on. Estimation of standard errors of parameters, confidence intervals for parameters and generating forecast intervals (for multiperiod forecasts), have been considered in the chapter. In some models, the asymptotic theory of the estimator is intractable (Manski's maximum score estimator). In such cases, bootstrap provides a tractable method of deriving confidence intervals and so on. The computational advances, like the use of balanced sampling, importance sampling, antithetic variates, and so, do not seem to have been implemented in econometric work. These methods, if properly used, would substantially increase the efficiency of bootstrap computations, and it would be possible to use more bootstrap samples with no extra computational burden. An important point to remember is that bootstrapping defective models is of no value. Bootstrap does not rescue bad models.
TL;DR: In this article, the authors extend and generalize the latent variables methodology of Gibbons and Ferson (1985) to assume that expected returns vary over time as functions of a small number of risk premiums, which are common across assets.
Abstract: The methods of Gibbons and Ferson (1985) are extended, relaxing the assumption that expected returns are linear functions of predetermined instruments. A model of conditional mean-variance spanning generalizes Huberman and Kandel (1987). The empirical results indicate that more than a single risk premium is needed to model expected stock and bond returns, but the number of common factors in the expected returns is small. However, when size-based common stock portfolios proxy for the risk factors, we reject the hypothesis that four of them describe the conditional expected returns of the other assets. THIS PAPER MAKES TWO contributions to the burgeoning literature on tests of asset-pricing models with changing expected returns. First, we extend and generalize the latent variables methodology of Gibbons and Ferson (1985). Latent variable models assume that expected returns vary over time as functions of a small number of risk premiums, which are common across assets. The expected risk premiums are treated as unobserved latent variables. Numerous studies have applied such models to study the expected returns of stocks, bonds, foreign exchange, and other assets.' Our general model allows conditional heteroskedasticity and does not assume a functional form for the conditional expected returns. This is attractive because in models that assume a functional form, misspecification of the functional form can contaminate inferences about the number of common risk premiums. While our model is more general, it should also have improved power compared with other models. We illustrate how the approach can be further extended by examining models that allow limited (seasonal) fluctuations in conditional betas. Using individual common stocks or using portfolios based on size rankings and industry affiliation, we reject models with fixed betas and a single risk
Journal Article•
TL;DR: The robustness and power of four commonly used MANOVA statistics (the Pillai-Bartlett trace (V), Wilks' Lambda (W), Hotelling's trace (1), Roy's greatest root (R)) are reviewed and their behaviours demonstrated by Monte Carlo simulations using a one-way fixed effects design as mentioned in this paper.
Abstract: The robustness and power of four commonly used MANOVA statistics (the Pillai-Bartlett trace (V), Wilks' Lambda (W), Hotelling's trace (1), Roy's greatest root (R)) are reviewed and their behaviours demonstrated by Monte Carlo simulations using a one-way fixed effects design in which assumptions of the model are violated in a systematic way under different conditions of sample size (n), number of dependent variables (P), number of groups (k), and balance in the data. The behaviour of Box's M statistic, which tests for covariance heterogeneity, is also examined. The behaviours suggest several recommendations for multivariate design and for application of MANOVA in marine biology and ecology, viz. (1) Sample sizes should be equal.
(2) p, and to a lesser extent k, should be kept to a minimum insofar as the hypothesis permits. (3) Box's M statistic is rejected as a test of homogeneity of covariance matrices. A suitable alternative is Hawkins' (1981) statistic that tests for heteroscedasticity and non-normality simultaneously. (4) To improve agreement with assumptions, and thus reliability of tests, reduction of p (e.g. by PCA or MDS methods) and/or transforming data to stabilise variances should be attempted. (5) The V statistic is recommended for general use but the others are more appropriate in particular circumstances. For Type I errors, the violation of the assumption of homoscedasticity is more serious than is nonnormality and the V statistic is clearly the most robust to variance heterogeneity in terms of controlling level. Kurtosis reduces the power of all statistics considerably. Loss of power is dramatic if assumptions of normality and homoscedasticity are violated simultaneously. (6) The preferred approach to multiple comparison procedures after MANOVA is to use Bonferroni-type methods in which the total number of comparisons is limited to the fewest possible. If all possible comparisons are required an alternative is to use the V statistic in the overall test and the R statistic in a follow-up simultaneous test procedure. We recommend following a significant MANOVA result with a canonical discriminant analysis. (7) Classical parametric MANOVA should not be used with data in which high levels of variance heterogeneity cannot be rectified or in which sample sizes are unequal and assumptions are not satisfied. We discuss briefly alternatives to parametric MANOVA.
TL;DR: In this paper, the authors consider the twin problems of testing for autoregressive conditional heteroscedasticity (ARCH) and generalized ARCH disturbances in the linear regression model and construct a test that exploits this one-sided aspect.
Abstract: This article considers the twin problems of testing for autoregressive conditional heteroscedasticity (ARCH) and generalized ARCH disturbances in the linear regression model. A feature of these testing problems, ignored by the standard Lagrange multiplier test, is that they are onesided in nature. A test that exploits this one-sided aspect is constructed based on the sum of the scores. The small-sample-size and power properties of two versions of this test under both normal and leptokurtic disturbances are investigated via a Monte Carlo experiment. The results indicate that both versions of the new test typically have superior power to two versions of the Lagrange multiplier test and possibly also more accurate asymptotic critical values.
TL;DR: The authors examined both time variation and truncation of futures price changes and concluded that previous rejections of the unbiasedness hypothesis in the foreign exchange futures market are not substantively altered by inclusion of price limits but may be attributed to potentially biased testing procedures.
Abstract: Daily price limits, an institutional feature of futures markets, truncate the distribution of price changes and dampen the variance. Previous tests of the unbiasedness hypothesis using daily foreign exchange futures prices have accounted for the observed conditional heteroscedasticity in the data but have neglected to adequately incorporate the additional effects of daily price limits. This article examines both time variation and truncation of futures price changes. Empirical results suggest that previous rejections of the unbiasedness hypothesis in the foreign exchange futures market are not substantively altered by inclusion of price limits but may be attributed to potentially biased testing procedures. Copyright 1993 by University of Chicago Press.
TL;DR: In this paper, the authors review Efron's method called the bootstrap and briefly mention its relation to the jackknife, with a particular emphasis on econometric applications.
Abstract: Publisher Summary This chapter reviews Efron's method called the bootstrap, and briefly mentions its relation to the jackknife, with a particular emphasis on econometric applications. Bootstrap literature has made tremendous progress in solving an old statistical problem: making reliable confidence statements in complicated small sample, multi-step, dependent, and non-normal cases. Resampling methods provide radically new solutions to several modeling problems involving interdependence, simultaneity, nonlinearity, nonstationarity, instability, nonnormality, heteroscedasticity, small or missing data, Hawthorne effect, and more solutions. The bootstrap handles these problems nonparametrically and intuitively, avoiding complicated power functions, Cramer–Rao lower bounds, bias corrections for Wald or Lagrange multiplier tests, and such. Many early applications of the bootstrap in econometrics attempts to provide an alternative to asymptotic standard error estimates. The jackknife is also used to find improved estimates of the standard errors. The bootstrap offers a potentially valuable insight into the sampling distributions, beyond simpler and improved estimation of standard errors. When two or more statistical tests are used, their power is difficult to determine analytically. The bootstrap sampling distribution can eliminate the need for tedious computations of the power in some cases. The chapter also discusses the post hoc technique for cleverly manipulating the bootstrap replications, computational aspects of bootstrap methods, and simultaneous equation and dynamic econometric models which require a special setup different from the traditional bootstrap.
TL;DR: In this paper, diagnostic tests and plots are proposed for detecting heteroscedasticity in nonparametric regression, and the large and small sample power properties are studied for a class of test statistics for the hypothesis of homogeneous variances.
Abstract: SUMMARY Diagnostic tests and plots are proposed for detecting heteroscedasticity in nonparametric regression The large and small sample power properties are studied for a class of test statistics for the hypothesis of homogeneous variances New diagnostic plots are also developed and illustrated
TL;DR: In this article, a class of general quadratic forms in the dependent variable for estimating the variance function in a nonparametric heteroscedastic fixed design regression model is proposed.
TL;DR: In this article, a delete-group jackknife method is used to produce consistent variance estimators irrespective of within-group correlations, unlike the delete-one jackknife variance estimator or those based on the customary 8-method assuming within group independence.
Abstract: Inference on the regression parameters in a heteroscedastic linear regression model with replication is considered, using either the ordinary least-squares (OLS) or the weighted least-squares (WLS) estimator. A delete-group jackknife method is shown to produce consistent variance estimators irrespective of within-group correlations, unlike the delete-one jackknife variance estimators or those based on the customary 8-method assuming within-group independence. Finite-sample properties of the delete-group variance estimators and associated confidence intervals are also studied through simulation.
TL;DR: A Bayesian extension of the estimation procedure of the dispersion parameters is presented which consists of determining the mode of their marginal posterior distribution using log inverted chi-square or Gaussian distributions as priors.
Abstract: Summary - A statistical method for identifying meaningful sources of heterogeneity of residual and genetic variances in mixed linear Gaussian models is presented. The method is based on a structural linear model for log variances. Inference about dispersion parameters is based on the marginal likelihood after integrating out location parameters. A likelihood ratio test using the marginal likelihood is also proposed to test for hypotheses about sources of variation involved. A Bayesian extension of the estimation procedure of the dispersion parameters is presented which consists of determining the mode of their marginal posterior distribution using log inverted chi-square or Gaussian distributions as priors. Procedures presented in the paper are illustrated with the analysis of muscle development scores at weaning of 8575 progeny of 142 sires in the Maine-Anjou breed. In this analysis, heteroskedasticity is found, both for the sire and residual components of variance.
TL;DR: In this article, the adaptive use of a conceptual model for real-time flow forecasting is investigated, and the performance of maximum likelihood techniques for autocorrelated (AMLE) and heteroscedastic (HMLE) errors is analyzed jointly with that provided by the commonly used ordinary least squares estimation (OLSE) technique.
Abstract: The adaptive use of a conceptual model for real-time flow forecasting is investigated. Maximum likelihood and ordinary least squares estimation criteria are considered, and the performance of maximum likelihood techniques for autocorrelated (AMLE) and heteroscedastic (HMLE) errors is analyzed jointly with that provided by the commonly used ordinary least squares estimation (OLSE) technique. Streamflow forecasts are compared for three rivers in central Italy, obtained by AMLE, HMLE, and OLSE adaptive calibration of a simple conceptual model describing the rainfall-runoff transformation by accounting for Hortonian infiltration and linear basin response to rainfall excess. Although model residuals display both autocorrelation and heteroscedasticity, OLSE is found to provide a rather satisfactory performance. Because the OLSE technique also requires less computational effort compared to that for AMLE and HMLE, one could consider OLSE as a suitable option for real-time model operation.
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TL;DR: In this paper, a stochastic variance model is estimated by quasi-maximum likelihood procedure by transforming to a linear state space form and the properties of observations corrected for heteroscedasticity can be derived.
Abstract: A stochastic variance model may be estimated by quasi-maximum likelihood procedure by transforming to a linear state space form. The properties of observations corrected for heteroscedasticity can be derived. A model with explanatory variables can be handled by correcting the observations for heteroscedasticity after estimating a stochastic variance model from the OLS residuals and then constructing a feasible GLS estimator. A model with stochastic variance, or standard deviation, as an explanatory variable can also be formulated. The paper explores the properties of these procedures and shows how they may be used as part of a model specification strategy/ It is argued that the approach is relatively robust since distributions need not be specified for the disturbances.
TL;DR: In this article, the White information matrix (IM) test is applied to the linear regression model with autocorrelated errors and a special case of one component of the test is found to be identical to the Engle Lagrange multiplier (LM) test for autoregressive conditional heteroskedasticity.
Abstract: We apply the White information matrix (IM) test to the linear regression model with autocorrelated errors. A special case of one component of the test is found to be identical to the Engle Lagrange multiplier (LM) test for autoregressive conditional heteroskedasticity (ARCH). Given Chesher's interpretation of the IM test as a test for parameter heterogeneity, this establishes a connection among the IM test, ARCH and parameter variation. This also enables us to specify conditional heteroskedasticity in a more general and convenient way. Other interesting by-products of our analysis are tests for the variation in conditional and static skewness which we call tests for "heterocliticity".
01 May 1993
TL;DR: In this paper, the authors introduce ARFIMA-ARCH models which simultaneously incorporate fractional differencing and conditional heteroskedasticity and develop the likelihood function and a numerical estimation procedure for this model class.
Abstract: We introduce ARFIMA-ARCH models which simultaneously incorporate fractional differencing and conditional heteroskedasticity We develop the likelihood function and a numerical estimation procedure for this model class Two ARCH models - Engle- and Weiss-type - are explicitly treated and stationarity conditions are derived Finite-sample properties of the estimation procedure are explored by Monte Carlo simulation An application to the Standard & Poor 500 Index indicates existence of intermediate memory (d<0) for the 1980's and no fractional differencing (d=0) but strong conditional heteroskedastic effects for the 1960's For the latter time period, contrary to the suggestion of long memory by Mandelbrot, we only found evidence for a positive first-order autoregressive parameter;
TL;DR: In this paper, conditional moment tests based on covariances between pairs of orthonormal polynomials are derived for a wide variety of discrete and continuous bivariate and multivariate regression equations.
Abstract: Tests of independence between variables in a wide variety of discrete and continuous bivariate and multivariate regression equations are derived using results from the theory of series expansions of joint distributions in terms of marginal distributions and their related orthonormal polynomials. The tests are conditional moment tests based on covariances between pairs of orthonormal polynomials. Examples include tests of serial independence against bilinear and/or autoregressive conditional heteroscedasticity alternatives, tests of dependence in multivariate normal regression models, and dependence in count-data models. Monte Carlo simulation based on bivariate count models is used to evaluate the. size and power properties of the proposed tests. A multivariate count-data model for Australian health-care-utilization data is used to empirically illustrate the tests.
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01 Jan 1993
TL;DR: In this paper, a stochastic variance model is estimated by quasi-maximum likelihood procedure by transforming to a linear state space form and the properties of observations corrected for heteroscedasticity can be derived.
Abstract: A stochastic variance model may be estimated by quasi-maximum likelihood procedure by transforming to a linear state space form. The properties of observations corrected for heteroscedasticity can be derived. A model with explanatory variables can be handled by correcting the observations for heteroscedasticity after estimating a stochastic variance model from the OLS residuals and then constructing a feasible GLS estimator. A model with stochastic variance, or standard deviation, as an explanatory variable can also be formulated. The paper explores the properties of these procedures and shows how they may be used as part of a model specification strategy/ It is argued that the approach is relatively robust since distributions need not be specified for the disturbances.