Effects of outliers on the identification and estimation of GARCH models

Abstract. This paper analyses how outliers affect the identification of conditional heteroscedasticity and the estimation of generalized autoregressive conditionally heteroscedastic (GARCH) models. First, we derive the asymptotic biases of the sample autocorrelations of squared observations generated by stationary processes and show that the properties of some conditional homoscedasticity tests can be distorted. Second, we obtain the asymptotic and finite sample biases of the ordinary least squares (OLS) estimator of ARCH(p) models. The finite sample results are extended to generalized least squares (GLS), maximum likelihood (ML) and quasi‐maximum likelihood (QML) estimators of ARCH(p) and GARCH(1,1) models. Finally, we show that the estimated asymptotic standard deviations are biased estimates of the sample standard deviations.     

Using least squares to generate forecasts in regressions with serial correlation

Abstract. The topic of serial correlation in regression models has attracted a great deal of research in the last 50 years. Most of these studies have assumed that the structure of the error covariance matrix Ω was known or could be consistently estimated from the data. In this article, we describe a new procedure for generating forecasts for regression models with serial correlation based on ordinary least squares and on an approximate representation of the form of the autocorrelation. We prove that the predictors from this specification are asymtotically efficient under some regularity conditions. In addition, we show that there is not much to be gained in trying to identify the correct form of the serial correlation since efficient forecasts can be generated using autoregressive approximations of the autocorrelation. A large simulation study is also used to compare the finite sample predictive efficiencies of this new estimator vis‐à‐vis estimators based on ordinary least squares and generalized least squares.     

Fixed Bandwidth Inference for Fractional Cointegration

In a fractional cointegration setting we derive the fixed bandwidth limiting theory of a class of estimators of the cointegrating parameter which are constructed as ratios of weighted periodogram averages. These estimators offer improved limiting properties over those of more standard approaches like ordinary least squares or narrow band least squares estimation. These advantages have been justified by means of traditional asymptotic theory and here we explore whether these improvements still hold when considering the alternative fixed bandwidth theory and, more importantly, whether this latter approach provides a more accurate approximation to the sampling distribution of the corresponding test statistics. This appears to be relevant, especially in view of the typical oversizing displayed by Wald statistics when confronted to the standard limiting theory. A Monte Carlo study of finite‐sample behaviour is included.     

Inference for pth‐order random coefficient integer‐valued autoregressive processes

Abstract. A pth‐order random coefficient integer‐valued autoregressive [RCINAR(p)] model is proposed for count data. Stationarity and ergodicity properties are established. Maximum likelihood, conditional least squares, modified quasi‐likelihood and generalized method of moments are used to estimate the model parameters. Asymptotic properties of the estimators are derived. Simulation results on the comparison of the estimators are reported. The models are applied to two real data sets.     

Explosive Random‐Coefficient AR(1) Processes and Related Asymptotics for Least‐Squares Estimation

Abstract. Large sample properties of the least‐squares and weighted least‐squares estimates of the autoregressive parameter of the explosive random‐coefficient AR(1) process are discussed. It is shown that, contrary to the standard AR(1) case, the least‐squares estimator is inconsistent whereas the weighted least‐squares estimator is consistent and asymptotically normal even when the error process is not necessarily Gaussian. Conditional asymptotics on the event that a certain limiting random variable is non‐zero is also discussed.     

Robust and powerful serial correlation tests with new robust estimates in ARX models

Abstract. We consider robust serial correlation tests in autoregressive models with exogenous variables (ARX). Since the least squares estimators are not robust when outliers are present, a new family of estimators is introduced, called residual autocovariances for ARX (RA‐ARX). They provide resistant estimators that are less sensible to abnormal observations in the output variable of the dynamic model. Such ‘bad’ observations could be due to unexpected phenomena such as economic crisis or equipment failure in engineering, among others. We show that the new robust estimators are consistent and we can consider robust and powerful tests of serial correlation in ARX models based on these estimators. The new one‐sided tests of serial correlation are obtained in extending Hong's (1996) approach in a framework resistant to outliers. They are based on a weighted sum of robust squared residual autocorrelations and on any robust and n^1/2‐consistent estimators. Our approach generalizes Li's (1988) test statistic, that can be interpreted as a test using the truncated uniform kernel. However, many kernels deliver a higher power. This is confirmed in a simulation study, where we investigate the finite sample properties of the new robust serial correlation tests in comparison to some commonly used robust and non‐robust tests.     

Semiparametric Estimation in Time‐Series Regression with Long‐Range Dependence

Abstract. We consider semiparametric estimation in time‐series regression in the presence of long‐range dependence in both the errors and the stochastic regressors. A central limit theorem is established for a class of semiparametric frequency domain‐weighted least squares estimates, which includes both narrow‐band ordinary least squares and narrow‐band generalized least squares as special cases. The estimates are semiparametric in the sense that focus is on the neighbourhood of the origin, and only periodogram ordinates in a degenerating band around the origin are used. This setting differs from earlier studies on time‐series regression with long‐range dependence, where a fully parametric approach has been employed. The generalized least squares estimate is infeasible when the degree of long‐range dependence is unknown and must be estimated in an initial step. In that case, we show that a feasible estimate which has the same asymptotic properties as the infeasible estimate, exists. By Monte Carlo simulation, we evaluate the finite‐sample performance of the generalized least squares estimate and the feasible estimate.     

On Residual Variance Estimation in Autoregressive Models

In this paper we consider time series models belonging to the autoregressive (AR) family and deal with the estimation of the residual variance. This is important because estimates of the variance are involved in, for example, confidence sets for the parameters of the model, estimation of the spectrum, expressions for the estimated error of prediction and sample quantities used to make inferences about the order of the model. We consider the asymptotic biases for moment and least squares estimators of the residual variance, and compare them with known results when available and with those for maximum likelihood estimators under normality. Simulation results are presented for finite samples     

Statistical analysis of a spatio‐temporal model with location‐dependent parameters and a test for spatial stationarity

Abstract. In this article, we define a spatio‐temporal model with location‐dependent parameters to describe temporal variation and spatial nonstationarity. We consider the prediction of observations at unknown locations using known neighbouring observations. Further, we propose a local least squares‐based method to estimate the parameters at unobserved locations. The sampling properties of these estimators are investigated. We also develop a statistical test for spatial stationarity. To derive the asymptotic results, we show that the spatially nonstationary process can be locally approximated by a spatially stationary process. We illustrate the methods of estimation with some simulations.     

Evaluating Specification Tests for Markov‐Switching Time‐Series Models

Abstract. We evaluate the performance of several specification tests for Markov regime‐switching time‐series models. We consider the Lagrange multiplier (LM) and dynamic specification tests of Hamilton (1996) and Ljung–Box tests based on both the generalized residual and a standard‐normal residual constructed using the Rosenblatt transformation. The size and power of the tests are studied using Monte Carlo experiments. We find that the LM tests have the best size and power properties. The Ljung–Box tests exhibit slight size distortions, though tests based on the Rosenblatt transformation perform better than the generalized residual‐based tests. The tests exhibit impressive power to detect both autocorrelation and autoregressive conditional heteroscedasticity (ARCH). The tests are illustrated with a Markov‐switching generalized ARCH (GARCH) model fitted to the US dollar–British pound exchange rate, with the finding that both autocorrelation and GARCH effects are needed to adequately fit the data.     

Practical improvements to autocovariance least‐squares

Identifying disturbance covariances from data is a critical step in estimator design and controller performance monitoring. Here, the autocovariance least‐squares (ALS) method for this identification is examined. For large industrial models with poorly observable states, the process noise covariance is high dimensional and the optimization problem is poorly conditioned. Also, weighting the least‐squares problem with the identity matrix does not provide minimum variance estimates. Here, ALS method to resolve these two challenges is modified. Poorly observable states using the singular value decomposition (SVD) of the observability matrix is identified and removed, thus decreasing the computational time. Using a new feasible‐generalized least‐squares estimator that approximates the optimal weighting from data, the variance of the estimates is significantly reduced. The new approach on industrial data sets provided by Praxair is successfully demonstrated. The disturbance model identified by the ALS method produces an estimator that performs optimally over a year‐long period. © 2015 American Institute of Chemical Engineers AIChE J, 61: 1840–1855, 2015     

SEMI‐PARAMETRIC ESTIMATION OF LINEAR COINTEGRATING MODELS WITH NONLINEAR CONTEMPORANEOUS ENDOGENEITY

This article considers linear cointegrating models with unknown nonlinear short‐run contemporaneous endogeneity. Two estimators are proposed to estimate the linear cointegrating parameter after the nonlinear endogenous component is estimated by local linear regression approach. Both the proposed estimators are shown to have the same mixed normal limiting distribution with zero mean and smaller asymptotic variance than the fully modified ordinary least squares and instrumental variables estimators. Monte Carlo simulations are used to evaluate the finite sample performance of our proposed estimators, and an empirical application is also included.     

Ensemble locally weighted partial least squares as a just‐in‐time modeling method

The predictive ability of soft sensors, which estimate values of an objective variable y online, decreases due to process changes in chemical plants. To reduce the decrease of predictive ability, adaptive soft sensors have been developed. We focused on just‐in‐time soft sensors, especially locally weighted partial least squares (LWPLS) regression. Since a set of hyperparameters in an LWPLS model has to be set beforehand and there is only onedataset, a traditional LWPLS model is difficult to accurately predict y‐values in multiple process states. In this study, we propose to combine LWPLS and ensemble learning, and predict y‐values with multiple LWPLS models, whose datasets and sets of hyperparameters are different. The weights of LWPLS models are determined based on Bayes’ theorem, considering their predictive ability. We confirmed that the proposed model has higher predictive accuracy than traditional models through numerical simulation data and two industrial data analyses. © 2015 American Institute of Chemical Engineers AIChE J, 62: 717–725, 2016     

Multivariate Portmanteau Test For Autoregressive Models with Uncorrelated but Nonindependent Errors

Abstract. We study the asymptotic behaviour of the least squares estimator, of the residual autocorrelations and of the Ljung–Box (or Box–Pierce) portmanteau test statistic for multiple autoregressive time series models with nonindependent innovations. Under mild assumptions, it is shown that the asymptotic distribution of the portmanteau tests is that of a weighted sum of independent chi‐squared random variables. When the innovations exhibit conditional heteroscedasticity or other forms of dependence, this asymptotic distribution can be quite different from that of models with independent and identically distributed innovations. Consequently, the usual chi‐squared distribution does not provide an adequate approximation to the distribution of the Box–Pierce goodness‐of‐fit portmanteau test in the presence of nonindependent innovations. Hence we propose a method to adjust the critical values of the portmanteau tests. Monte carlo experiments illustrate the finite sample performance of the modified portmanteau test.     

Fuzzy multi‐model based adaptive predictive control and its application to thermoplastic injection molding

Many chemical processes are inherently nonlinear. A single linear model is ineffective for these processes. Several local linear models may be developed for different operating conditions. A combination of these local models, through a fuzzy logic representation, results in an overall model for a wider operation range. In this paper, on‐line improvements and a fuzzy multi‐model have been proposed for predictive control implementation. Firstly, assuming that the premises of the fuzzy rules keep their original structures, the linear parameters in the rule consequents are on‐line updated by a weighted recursive least squares algorithm at each sample interval. Secondly, a batch learning algorithm is proposed to tune the fuzzy rule premises using a competitive learning algorithm. The effectiveness of the proposed improvements is demonstrated with experimental applications to the filling velocity control of thermoplastic injection molding     

Asymptotic Properties of Weighted Least Squares Estimation in Weak PARMA Models

The aim of this work is to investigate the asymptotic properties of weighted least squares (WLS) estimation for causal and invertible periodic autoregressive moving average (PARMA) models with uncorrelated but dependent errors. Under mild assumptions, it is shown that the WLS estimators of PARMA models are strongly consistent and asymptotically normal. It extends Thm 3.1 of Basawa and Lund (2001) on least squares estimation of PARMA models with independent errors. It is seen that the asymptotic covariance matrix of the WLS estimators obtained under dependent errors is generally different from that obtained with independent errors. The impact can be dramatic on the standard inference methods based on independent errors when the latter are dependent. Examples and simulation results illustrate the practical relevance of our findings. An application to financial data is also presented.     

Least tail‐trimmed squares for infinite variance autoregressions

We develop a robust least squares estimator for autoregressions with possibly heavy tailed errors. Robustness to heavy tails is ensured by negligibly trimming the squared error according to extreme values of the error and regressors. Tail‐trimming ensures asymptotic normality and super‐‐convergence with a rate comparable to the highest achieved amongst M‐estimators for stationary data. Moreover, tail‐trimming ensures robustness to heavy tails in both small and large samples. By comparison, existing robust estimators are not as robust in small samples, have a slower rate of convergence when the variance is infinite, or are not asymptotically normal. We present a consistent estimator of the covariance matrix and treat classic inference without knowledge of the rate of convergence. A simulation study demonstrates the sharpness and approximate normality of the estimator, and we apply the estimator to financial returns data. Finally, tail‐trimming can be easily extended beyond least squares estimation for a linear stationary AR model. We discuss extensions to quasi‐maximum likelihood for GARCH, weighted least squares for a possibly non‐stationary random coefficient autoregression, and empirical likelihood for robust confidence region estimation, in each case for models with possibly heavy tailed errors.     

Large sample properties of parameter least squares estimates for time-varying arma models

Abstract.  This paper considers estimation of ARMA models with time-varying coefficients. The ARMA parameters belong to d different regimes. The changes in regime occur at irregular time intervals. Consistency and asymptotic normality of least squares and quasi-generalized least squares estimators are shown.     

K‐M theory of fabric knitted by three‐channel rotor spun wool yarn

Two‐component and three‐component color blended yarns were spun by red, yellow, and blue wool slivers using a three‐channel rotor spun machine, and the corresponding plain fabrics were knitted. The color‐matching models of K‐M theory were built with the relative method and the least squares method, respectively. Colors and blending ratios of the fabrics were predicted by the model. The results showed that the average color differences of the samples predicted by the two methods are both about 1.0 and the mean value of the proportional error is below 3%. The least squares method has a better color‐matching effect for the three‐component sample, and the relative value method has better color‐matching results for the two‐component sample. When the tolerance range is 2.0, the pass rates of the samples predicted by either the relative value method or the least squares method reach 100%.     

Polynomial Trend Regression With Long‐memory Errors

Abstract. For a time series generated by polynomial trend with stationary long‐memory errors, the ordinary least squares estimator (OLSE) of the trend coefficients is asymptotically normal, provided the error process is linear. The asymptotic distribution may no longer be normal, if the error is in the form of a long‐memory linear process passing through certain nonlinear transformations. However, one hardly has sufficient information about the transformation to determine which type of limiting distribution the OLSE converges to and to apply the correct distribution so as to construct valid confidence intervals for the coefficients based on the OLSE. The present paper proposes a modified least squares estimator to bypass this drawback. It is shown that the asymptotic normality can be assured for the modified estimator with mild trade‐off of efficiency even when the error is nonlinear and the original limit for the OLSE is non‐normal. The estimator performs fairly well when applied to various simulated series and two temperature data sets concerning global warming.