ARCH Errors Research Articles

Second-Order Least Squares Estimation in Nonlinear Time Series Models with ARCH Errors

Many financial and economic time series exhibit nonlinear patterns or relationships. However, most statistical methods for time series analysis are developed for mean-stationary processes that require transformation, such as differencing of the data. In this paper, we study a dynamic regression model with nonlinear, time-varying mean function, and autoregressive conditionally heteroscedastic errors. We propose an estimation approach based on the first two conditional moments of the response variable, which does not require specification of error distribution. Strong consistency and asymptotic normality of the proposed estimator is established under strong-mixing condition, so that the results apply to both stationary and mean-nonstationary processes. Moreover, the proposed approach is shown to be superior to the commonly used quasi-likelihood approach and the efficiency gain is significant when the (conditional) error distribution is asymmetric. We demonstrate through a real data example that the proposed method can identify a more accurate model than the quasi-likelihood method.

Oracally efficient estimation and testing for an ARCH model with trend

This article investigates an ARCH model with trend and proposes a two-step estimation method and develops an ARCH effect test. The trend is estimated by the linear B-spline method, and then the least squares estimation (LSE) and a Lagrange multiplier (LM) test based on the residuals are presented. Under mild conditions, it is shown that the proposed LSE and the test statistic enjoy the oracle properties; namely, they perform as well as if the true trend function were known and then removed to obtain the ARCH errors. Simulation studies assess the finite-sample performance of the LSE and the test statistic. An empirical example on the US gross private saving series is analyzed to illustrate the usefulness of our procedure.

The impact of patent citation information flow regarding economic innovation on common stock returns: Volume vs. patent citations

This study examines whether the number of forward patent citations (along with alternative patent data)—when used as a proxy for the mixing variable—could infer the aggregate amount of economic-innovation information arriving at the New York Stock Exchange (NYSE) in the United States. The results show that the number of forward patent citations, when used as a mixing variable, fails to eliminate total volatility persistence in the conditional variance equation of the exponential generalized autoregressive conditional heteroscedastic (EGARCH) model. However, the trading volume successfully eliminates total volatility persistence, thus confirming the validity of the framework used. When the volatility is modeled with an expectation of mean return, the persistence of conditional variance is deterministically increased, and the sum of the volatility coefficients exceeds unity. The inclusion of trading volume with a time trend in the variance equation rectifies the deterministic increase in the conditional volatility. These findings suggest that the form of heteroscedasticity (i.e., as per the autoregressive conditional heteroscedastic model, ARCH model) in NYSE portfolio returns is based on the type of shocks to volatility (e.g., deterministic vs. stochastic), which manifests as news arrivals (i.e., new information arrivals proxied by trading volume) at the stock market. The volume therefore reflects the time dependence in the innovations to the ARCH error generation process. The response of volatility to volume persists over time when the volatility estimates are derived from the EGARCH model with an expectation for the mean of return. Backward patent citations, patent applications, and patents issued have been found to interact somewhat with trading volume, suggesting that each of these variables could play the role of an absorptive capacity variable as the new information flow associated with economic innovation (i.e., flow of firms’ stock of new knowledge) could be picked up by the trading volume.

Partial Functional Linear Models with ARCH Errors

In this paper, the estimation of the parameters in partial functional linear models with ARCH(p) errors is discussed. With employing the functional principle component, a hybrid estimating method is suggested. The asymptotic normality of the proposed estimators for both the linear parameter in the mean model and the parameter in the ARCH error model is obtained, and the convergence rate of the slope function estimate is established. Besides, some simulations and a real data analysis are conducted for illustration, and it is shown that the proposed method performs well with a finite sample.

The spurious effect of ARCH errors on linearity tests: a theoretical note and an alternative maximum likelihood approach

AbstractLinearity tests against smooth transition nonlinearity are typically based on the standard least-squares (LS) covariance matrix estimator. We derive an expression for the bias of the LS estimator in the presence of ARCH errors. We show that the bias is downward, and increases dramatically with the persistence of the variance process. As a consequence, conventional tests spuriously indicate nonlinearity. Next, we examine an alternative maximum likelihood approach. Our findings suggest that this approach has substantially better size properties than tests based on least-squares and heteroskedasticity-consistent matrix estimators, and performs comparably to a bootstrap technique.

Inference for the tail index of a GARCH(1,1) model and an AR(1) model with ARCH(1) errors

ABSTRACTFor a GARCH(1,1) sequence or an AR(1) model with ARCH(1) errors, one can estimate the tail index by solving an estimating equation with unknown parameters replaced by the quasi maximum likelihood estimation, and a profile empirical likelihood method can be employed to effectively construct a confidence interval for the tail index. However, this requires that the errors of such a model have at least a finite fourth moment. In this article, we show that the finite fourth moment can be relaxed by employing a least absolute deviations estimate for the unknown parameters by noting that the estimating equation for determining the tail index is invariant to a scale transformation of the underlying model.

Recent Developments in Recursive Estimation for Time Series Models

Recently there has been a growing interest in joint estimation of the location and scale parameters using combined estimation functions. Combined estimating functions had been studied in Liang et al. (2011) for models with finite variance errors and in Thavaneswaran et al. (2013) for models with infinite variance<br />stable errors. In this paper, first a theorem on recursive estimation based on estimating functions is extended to multi-parameter setup and it is shown that the unified approach can be used to estimate the location parameter recursively for models with finite variance/infinite variance errors. The method is applied for the joint estimation of the location and scale parameters for regression models with ARCH errors and RCA models with GARCH errors.

Stationary Bootstrapping for the Nonparametric AR-ARCH Model

We consider a nonparametric AR(1) model with nonparametric ARCH(1) errors. In order to estimate the unknown function of the ARCH part, we apply the stationary bootstrap procedure, which is characterized by geometrically distributed random length of bootstrap blocks and has the advantage of capturing the dependence structure of the original data. The proposed method is composed of four steps: the first step estimates the AR part by a typical kernel smoothing to calculate AR residuals, the second step estimates the ARCH part via the Nadaraya-Watson kernel from the AR residuals to compute ARCH residuals, the third step applies the stationary bootstrap procedure to the ARCH residuals, and the fourth step defines the stationary bootstrapped Nadaraya-Watson estimator for the ARCH function with the stationary bootstrapped residuals. We prove the asymptotic validity of the stationary bootstrap estimator for the unknown ARCH function by showing the same limiting distribution as the Nadaraya-Watson estimator in the second step.

On the stationary tail index of iterated random Lipschitz functions

Let Ψ,Ψ1,Ψ2,… be a sequence of i.i.d. random Lipschitz maps from a complete separable metric space (X,d) with unbounded metric d to itself and let Xn=Ψn∘⋯∘Ψ1(X0) for n=1,2,… be the associated Markov chain of forward iterations with initial value X0 which is independent of the Ψn. Provided that (Xn)n≥0 has a stationary law π and picking an arbitrary reference point x0∈X, we will study the tail behavior of d(x0,X0) under Pπ, viz. the behavior of Pπ(d(x0,X0)>t) as t→∞, in cases when there exist (relatively simple) nondecreasing continuous random functions F,G:R≥→R≥ such that F(d(x0,x))≤d(x0,Ψ(x))≤G(d(x0,x)) for all x∈X and n≥1. In a nutshell, our main result states that, if the iterations of i.i.d. copies of F and G constitute contractive iterated function systems with unique stationary laws πF and πG having power tails of order ϑF and ϑG at infinity, respectively, then lower and upper tail index of ν=Pπ(d(x0,X0)∈⋅) (to be defined in Section 2) are falling in [ϑG,ϑF]. If ϑF=ϑG, which is the most interesting case, this leads to the exact tail index of ν. We illustrate our method, which may be viewed as a supplement of Goldie’s implicit renewal theory, by a number of popular examples including the AR(1)-model with ARCH errors and random logistic transforms.

Asymptotic inference in multiple-threshold double autoregressive models

This paper investigates a class of multiple-threshold models, called Multiple Threshold Double AR (MTDAR) models. A sufficient condition is obtained for the existence and uniqueness of a strictly stationary and ergodic solution to the first-order MTDAR model. We study the Quasi-Maximum Likelihood Estimator (QMLE) of the MTDAR model. The estimated thresholds are shown to be n-consistent, asymptotically independent, and to converge weakly to the smallest minimizer of a two-sided compound Poisson process. The remaining parameters are n-consistent and asymptotically multivariate normal. In particular, these results apply to the multiple threshold ARCH model, with or without AR part, and to the multiple threshold AR models with ARCH errors. A score-based test is also presented to determine the number of thresholds in MTDAR models. The limiting distribution is shown to be distribution-free and is easy to implement in practice. Simulation studies are conducted to assess the performance of the QMLE and our score-based test in finite samples. The results are illustrated with an application to the quarterly US real GNP data over the period 1947–2013.

Resolving spurious regressions and serially correlated errors

This study investigates the spurious regression phenomenon for two independent stationary and non-stationary processes and illustrates, using a Monte Carlo analysis, that estimation of the spurious regression in first differences or with a lagged dependent variable eliminates the spurious regression problem. Moreover, the results also apply in eliminating the problem of serially correlated errors as well as the problem of ARCH(1) errors.

Empirical likelihood inference for partial linear models with ARCH(1) errors

We consider the application of the empirical likelihood method to a partial linear model with ARCH(1) errors. It is shown that the empirical log-likelihood ratio at the true parameter converges to the standard chi-square distribution and that the confidence intervals can be constructed accordingly. Some simulations are conducted to illustrate the proposed method.

Structure of a double autoregressive process driven by a hidden Markov chain

This paper considers a new so-called autoregressive process with ARCH(1) errors driven by a hidden Markov chain, Xt+1=α(Δt+1)Xt+ηt+1β(Δt+1)+λ(Δt+1)Xt2,t∈N, where (ηt) is a sequence of independent and identically distributed standard normal random variables, and (Δt) is a Markov chain with finite state space. Some structural properties of this new autoregressive process are considered. A sufficient condition for the existence of the strictly stationary and geometrically ergodic solution of the process is presented. The condition for this is only E[ln|α(Δt)+ηtλ(Δt)|]<0. Moreover, some simple conditions for the existence of the moments of the process are also derived.

Asymptotic Inference for the Weak Stationary Double AR(1) Model

An AR(1) model with ARCH(1) error structure is known as the first-order double autoregressive (DAR(1)) model. In this paper, a conditional likelihood based method is proposed to obtain inference for the two scalar parameters of interest of the DAR(1) model. Theoretically, the proposed method has rate of convergence O(n-3/2). Applying the proposed method to a real-life data set shows that the results obtained by the proposed method can be quite different from the results obtained by the existing methods. Results from Monte Carlo simulation studies illustrate the supreme accuracy of the proposed method even when the sample size is small.

Joint Estimation Using Quadratic Estimating Function

A class of martingale estimating functions is convenient and plays an important role for inference for nonlinear time series models. However, when the information about the first four conditional moments of the observed process becomes available, the quadratic estimating functions are more informative. In this paper, a general framework for joint estimation of conditional mean and variance parameters in time series models using quadratic estimating functions is developed. Superiority of the approach is demonstrated by comparing the information associated with the optimal quadratic estimating function with the information associated with other estimating functions. The method is used to study the optimal quadratic estimating functions of the parameters of autoregressive conditional duration (ACD) models, random coefficient autoregressive (RCA) models, doubly stochastic models and regression models with ARCH errors. Closed‐form expressions for the information gain are also discussed in some detail.

Ownership and competition in European transit: assessing efficiency

Ownership and competition and their effect on public transit system efficiency is a topic of considerable interest. In this article, we use a stochastic frontier error component model specification that incorporates a dynamic ARCH error structure to assess the effects of ownership and competition on European public transport systems using data from 15 European systems for a 10-year time period (1990–2000). Results indicate: (i) a clear rejection of the standard frontier model and existence of ARCH-related heteroskedastic technical inefficiency; (ii) transport operators in cities where competition has been introduced to demonstrate increased efficiency and (iii) efficiency findings vary between different error formulations.

A note on the geometric ergodicity of a nonlinear AR-ARCH model

This note studies the geometric ergodicity of nonlinear autoregressive models with conditionally heteroskedastic errors. A nonlinear autoregression of order p (AR( p )) with the conditional variance specified as the conventional linear autoregressive conditional heteroskedasticity model of order q (ARCH( q )) is considered. Conditions under which the Markov chain representation of this nonlinear AR-ARCH model is geometrically ergodic and has moments of known order are provided. The obtained results complement those of Liebscher [Liebscher, E., 2005. Towards a unified approach for proving geometric ergodicity and mixing properties of nonlinear autoregressive processes, Journal of Time Series Analysis, 26, 669–689] by showing how his approach based on the concept of the joint spectral radius of a set of matrices can be extended to establish geometric ergodicity in nonlinear autoregressions with conventional ARCH( q ) errors.

Periodic stationarity of random coefficient periodic autoregressions

This paper studies periodic stationarity of a random coefficient periodic autoregression (RCPAR) which generalizes the standard random coefficient autoregressive (RCAR) model to the case where the deterministic parameters and the disturbance variances are periodically time-varying. Sufficient conditions for the existence of a (strictly and second-order) periodically stationary solution to the RCPAR equation are proposed. As an application, we study periodic stationarity of a class of periodic bilinear models and a periodic autoregression with periodic ARCH errors.

Local Estimation in AR Models with Nonparametric ARCH Errors

In this article, we consider the problem of estimating the volatility function of a parametric AR(1) model with nonparametric ARCH(1) errors. Consistency and asymptotic normality of local constant and local log-linear estimators are established. Our simulation study and an application to finance lead to superior performance of the local log-linear estimator compared with the conventional treatments in the literature. A possible extension of the estimation procedure is described.

Weighted least absolute deviations estimation for an AR(1) process with ARCH(1) errors

The weighted least absolute deviations estimator is studied for an AR(1) process with ARCH(1) errors e-sub-t. Unlike for the quasi maximum likelihood estimator, the estimator's, limiting distribution is shown to be normal even when E(e-sub-t-super-4) = ∞. Furthermore, the estimator can be applied to examine the symmetry of the density of e-sub-t and to estimate the quantity E(log |α + λ-super-½ e-sub-t|), which are of crucial importance for conducting asymptotic inference for quasi maximum likelihood estimators and weighted least absolute deviations estimators. Copyright 2005, Oxford University Press.