Regression forecasts when disturbances are autocorrelated

A Monte Carlo simulation is used to study the quality of forecasts obtained from regression models with various degrees of autocorrelation present in the disturbances. The methods used to estimate the model parameters include least squares, full maximum likelihood, Prais-Winsten, Cochrane-Orcutt and Bayesian estimation. Results indicate that the Cochrane-Orcutt method should be avoided. The full maximum likelihood, Prais-Winsten and Bayesian methods are relatively more efficient than least squares when the degree of autocorrelation is high (greater than or equal to 0.5) and show little efficiency loss when the degree is low. These results hold for both trended and untrended independent variables.     

Bicorrelations and cross‐bicorrelations as non‐linearity tests and tools for exchange rate forecasting

This paper proposes and implements a new methodology for forecasting time series, based on bicorrelations and cross‐bicorrelations. It is shown that the forecasting technique arises as a natural extension of, and as a complement to, existing univariate and multivariate non‐linearity tests. The formulations are essentially modified autoregressive or vector autoregressive models respectively, which can be estimated using ordinary least squares. The techniques are applied to a set of high‐frequency exchange rate returns, and their out‐of‐sample forecasting performance is compared to that of other time series models. Copyright © 2001 John Wiley & Sons, Ltd.     

Exponential smoothing: Behavior of the ex-post sum of squares near 0 and 1

When using simple exponential smoothing on a given time series the nature of the relationship between the optimal smoothing constant and the autocorrelation structure of the series remains an unresolved question. Although numerical search routines can easily be used to find optimal values of the smoothing constant, they offer little insight into the nature of the relationship between the estimated smoothing constant and the structure of the underlying time series. We suggest that renewed investigations of the ex-post sum of squares function may prove helpful in this pursuit. Results are presented that illustrate how the optimal smoothing constant depends upon the value used to initialize the smoothing and upon the sample autocorrelation coefficients of the observed series. These results are based on a new formula for the derivative of the ex-post sum of squares function. In particular, the derivative is examined near 0 and 1, where great simplifications occur in its form, thereby facilitating investigations near these points. A necessary and sufficient condition is stated for when the ex-post sum of squares must have a positive derivative at 0 and the autocorrelation coefficients of the differenced series are shown to affect the sign of the derivative near 1. Based on these results, a general algorithm is presented as an alternative to grid search routines for minimizing the ex-post sum of squares.     

A Robust Test for Threshold‐Type Nonlinearity in Multivariate Time Series Analysis

There is growing interest in exploring potential forecast gains from the nonlinear structure of multivariate threshold autoregressive (MTAR) models. A least squares‐based statistical test has been proposed in the literature. However, previous studies on univariate time series analysis show that classical nonlinearity tests are often not robust to additive outliers. The outlier problem is expected to pose similar difficulties for multivariate nonlinearity tests. In this paper, we propose a new and robust MTAR‐type nonlinearity test, and derive the asymptotic null distribution of the test statistic. A Monte Carlo experiment is carried out to compare the power of the proposed test with that of the least squares‐based test under the influence of additive time series outliers. The results indicate that the proposed method is preferable to the classical test when observations are contaminated by outliers. Finally, we provide illustrative examples by applying the statistical tests to two real datasets. Copyright © 2015 John Wiley & Sons, Ltd.     

Bounds for the least squares extrapolation in non‐linear AR(1) processes

It is proved that formula for least squares extrapolation in stationary non‐linear AR(1) process is valid also for non‐stationary non‐linear AR(1) processes. This formula depends on the distribution of the corresponding white noise. If the non‐linear function used in the model is non‐decreasing and concave, upper and lower bounds are derived for least squares extrapolation such that the bounds depend only on the expectation of the white noise. It is shown in an example that the derived bounds in some cases give a good approximation to the least squares extrapolation. Copyright © 2001 John Wiley & Sons, Ltd.     

A Bayesian nonlinear support vector machine error correction model

The use of linear error correction models based on stationarity and cointegration analysis, typically estimated with least squares regression, is a common technique for financial time series prediction. In this paper, the same formulation is extended to a nonlinear error correction model using the idea of a kernel‐based implicit nonlinear mapping to a high‐dimensional feature space in which linear model formulations are specified. Practical expressions for the nonlinear regression are obtained in terms of the positive definite kernel function by solving a linear system. The nonlinear least squares support vector machine model is designed within the Bayesian evidence framework that allows us to find appropriate trade‐offs between model complexity and in‐sample model accuracy. From straightforward primal–dual reasoning, the Bayesian framework allows us to derive error bars on the prediction in a similar way as for linear models and to perform hyperparameter and input selection. Starting from the results of the linear modelling analysis, the Bayesian kernel‐based prediction is successfully applied to out‐of‐sample prediction of an aggregated equity price index for the European chemical sector. Copyright © 2006 John Wiley & Sons, Ltd.     

Using wavelets to obtain a consistent ordinary least squares estimator of the long‐memory parameter

We develop an ordinary least squares estimator of the long‐memory parameter from a fractionally integrated process that is an alternative to the Geweke and Porter‐Hudak (1983) estimator. Using the wavelet transform from a fractionally integrated process, we establish a log‐linear relationship between the wavelet coefficients' variance and the scaling parameter equal to the log‐memory parameter. This log‐linear relationship yields a consistent ordinary least squares estimator of the long‐memory parameter when the wavelet coefficients' population variance is replaced by their sample variance. We derive the small sample bias and variance of the ordinary least squares estimator and test it against the GPH estimator and the McCoy–Walden maximum likelihood wavelet estimator by conducting a number of Monte Carlo experiments. Based upon the criterion of choosing the estimator which minimizes the mean squared error, the wavelet OLS approach was superior to the GPH estimator, but inferior to the McCoy–Walden wavelet estimator for the processes simulated. However, given the simplicity of programming and running the wavelet OLS estimator and its statistical inference of the long‐memory parameter we feel the general practitioner will be attracted to the wavelet OLS estimator. Copyright © 1999 John Wiley & Sons, Ltd.     

Forecasting local currency bond risk premia of emerging markets: The role of cross-country macrofinancial linkages

In this paper, we forecast local currency debt of five major emerging market countries (Brazil, Indonesia, Mexico, South Africa, and Turkey) over the period January 2010 to January 2019 (with an in-sample period: March 2005 to December 2009). We exploit information from a large set of economic and financial time series to assess the importance not only of “own-country” factors (derived from principal component and partial least squares approaches), but also create “global” predictors by combining the country-specific variables across the five emerging economies. We find that, while information on own-country factors can outperform the historical average model, global factors tend to produce not only greater statistical and economic gains, but also enhance market timing ability of investors, especially when we use the target variable (bond premium) approach under the partial least squares method to extract our factors. Our results have important implications not only for fund managers but also for policymakers.     

Factor‐Augmented Bridge Models (FABM) and Soft Indicators to Forecast Italian Industrial Production

This paper presents a new forecasting approach straddling the conventional methods applied to the Italian industrial production index. Specifically, the proposed method treats factor models and bridge models as complementary ingredients feeding a unique model specification. We document that the proposed approach improves upon traditional bridge models by making efficient use of the information conveyed by a large amount of survey data on manufacturing activity. Different factor algorithms are compared and, under the provision that a large estimation window is used, partial least squares outperform principal component‐based alternatives. Copyright © 2016 John Wiley & Sons, Ltd.     

Realized Volatility Forecast of Stock Index Under Structural Breaks

We investigate the realized volatility forecast of stock indices under the structural breaks. We utilize a pure multiple mean break model to identify the possibility of structural breaks in the daily realized volatility series by employing the intraday high‐frequency data of the Shanghai Stock Exchange Composite Index and the five sectoral stock indices in Chinese stock markets for the period 4 January 2000 to 30 December 2011. We then conduct both in‐sample tests and out‐of‐sample forecasts to examine the effects of structural breaks on the performance of ARFIMAX‐FIGARCH models for the realized volatility forecast by utilizing a variety of estimation window sizes designed to accommodate potential structural breaks. The results of the in‐sample tests show that there are multiple breaks in all realized volatility series. The results of the out‐of‐sample point forecasts indicate that the combination forecasts with time‐varying weights across individual forecast models estimated with different estimation windows perform well. In particular, nonlinear combination forecasts with the weights chosen based on a non‐parametric kernel regression and linear combination forecasts with the weights chosen based on the non‐negative restricted least squares and Schwarz information criterion appear to be the most accurate methods in point forecasting for realized volatility under structural breaks. We also conduct an interval forecast of the realized volatility for the combination approaches, and find that the interval forecast for nonlinear combination approaches with the weights chosen according to a non‐parametric kernel regression performs best among the competing models. Copyright © 2014 John Wiley & Sons, Ltd.     

Forecasting Simultaneously High‐Dimensional Time Series: A Robust Model‐Based Clustering Approach

This paper considers the problem of forecasting high‐dimensional time series. It employs a robust clustering approach to perform classification of the component series. Each series within a cluster is assumed to follow the same model and the data are then pooled for estimation. The classification is model‐based and robust to outlier contamination. The robustness is achieved by using the intrinsic mode functions of the Hilbert–Huang transform at lower frequencies. These functions are found to be robust to outlier contamination. The paper also compares out‐of‐sample forecast performance of the proposed method with several methods available in the literature. The other forecasting methods considered include vector autoregressive models with ∕ without LASSO, group LASSO, principal component regression, and partial least squares. The proposed method is found to perform well in out‐of‐sample forecasting of the monthly unemployment rates of 50 US states. Copyright © 2013 John Wiley & Sons, Ltd.     

On the existence of generalized inverse estimators in a singular system of equations

This paper deals with estimation problems in the context of singular systems of equations. It provides the necessary and sufficient conditions for the existence of restricted estimators as a routine extension of the standard theory of restricted least squares estimation. The paper also provides the means for carrying out tests of hypotheses on subsets of restrictions imposed on the system by explicitly providing an expression for the (appropriate) Lagrange multipliers.     

Local to unity,long‐horizon forecasting thresholds for model selection in the AR(1)

This article introduces a novel framework for analysing long‐horizon forecasting of the near non‐stationary AR(1) model. Using the local to unity specification of the autoregressive parameter, I derive the asymptotic distributions of long‐horizon forecast errors both for the unrestricted AR(1), estimated using an ordinary least squares (OLS) regression, and for the random walk (RW). I then identify functions, relating local to unity ‘drift’ to forecast horizon, such that OLS and RW forecasts share the same expected square error. OLS forecasts are preferred on one side of these ‘forecasting thresholds’, while RW forecasts are preferred on the other. In addition to explaining the relative performance of forecasts from these two models, these thresholds prove useful in developing model selection criteria that help a forecaster reduce error. Copyright © 2004 John Wiley & Sons, Ltd.     

A comparison of forecasts from least absolute value and least squares regression

A Monte Carlo simulation is used to compare forecasts from least absolute value and least squares estimated regression equations. When outliers are present, the least absolute value forecasts are shown to be superior to least squares forecasts. The results emphasize the importance of exercising caution when using forecasts from least squares estimated regressions. Use of least absolute value regression (or some other robust regression method) instead of, or as an adjunct to, least squares is recommended. The comparison of forecasts from the two methods provides one way of assessing whether the least squares forecasts have been adversely affected by outliers. If outliers are present, the least absolute value regression forecasts can be used.     

Finite‐sample properties of tests for equal forecast accuracy

This study examines the small‐sample properties of some commonly used tests of equal forecast accuracy. The paper considers the size and power of different tests and the performance of different heteroscedasticity and autocorrelation‐consistent (HAC) variance estimators. Monte Carlo experiments show that the tests all suffer some size distortions in small samples, with the distortions varying across tests. The experiments also show that, adjusted for size distortions, the tests have broadly similar power, although some small differences exist. Finally, the experiments indicate that the size and power performances of HAC estimators vary with the features of the data. Copyright © 1999 John Wiley & Sons, Ltd.     

Backward‐in‐Time Selection of the Order of Dynamic Regression Prediction Model

We investigate the optimal structure of dynamic regression models used in multivariate time series prediction and propose a scheme to form the lagged variable structure called Backward‐in‐Time Selection (BTS), which takes into account feedback and multicollinearity, often present in multivariate time series. We compare BTS to other known methods, also in conjunction with regularization techniques used for the estimation of model parameters, namely principal components, partial least squares and ridge regression estimation. The predictive efficiency of the different models is assessed by means of Monte Carlo simulations for different settings of feedback and multicollinearity. The results show that BTS has consistently good prediction performance, while other popular methods have varying and often inferior performance. The prediction performance of BTS was also found the best when tested on human electroencephalograms of an epileptic seizure, and for the prediction of returns of indices of world financial markets.Copyright © 2013 John Wiley & Sons, Ltd.     

Validating multiple‐period density‐forecasting models

This paper examines the problem of how to validate multiple‐period density forecasting models. Such models are more difficult to validate than their single‐period equivalents, because consecutive observations are subject to common shocks that undermine i.i.d. The paper examines various solutions to this problem, and proposes a new solution based on the application of standard tests to a resample that is constructed to be i.i.d. It suggests that this solution is superior to alternatives, and presents results indicating that tests based on the i.i.d. resample approach have good power. Copyright © 2007 John Wiley & Sons, Ltd.     

Projection of population structure in China using least squares support vector machine in conjunction with a Leslie matrix model

China is a populous country that is facing serious aging problems due to the single‐child birth policy. Debate is ongoing whether the liberalization of the single‐child policy to a two‐child policy can mitigate China's aging problems without unacceptably increasing the population. The purpose of this paper is to apply machine learning theory to the demographic field and project China's population structure under different fertility policies. The population data employed derive from the fifth and sixth national census records obtained in 2000 and 2010 in addition to the annals published by the China National Bureau of Statistics. Firstly, the sex ratio at birth is estimated according to the total fertility rate based on least squares regression of time series data. Secondly, the age‐specific fertility rates and age‐specific male/female mortality rates are projected by a least squares support vector machine (LS‐SVM) model, which then serve as the input to a Leslie matrix model. Finally, the male/female age‐specific population data projected by the Leslie matrix in a given year serve as the input parameters of the Leslie matrix for the following year, and the process is iterated in this manner until reaching the target year. The experimental results reveal that the proposed LS‐SVM‐Leslie model improves the projection accuracy relative to the conventional Leslie matrix model in terms of the percentage error and mean algebraic percentage error. The results indicate that the total fertility ratio should be controlled to around 2.0 to balance concerns associated with a large population with concerns associated with an aging population. Therefore, the two‐child birth policy should be fully instituted in China. However, the fertility desire of women tends to be low due to the high cost of living and the pressure associated with employment, particularly in the metropolitan areas. Thus additional policies should be implemented to encourage fertility.     

A comparison of adaptive structural forecasting methods for electricity sales

This paper presents the results of a study to determine whether new forecasting technologies might be of use to electric utilities for sales forecasting up to 3 years into the future. The methods considered included ordinary least squares on dynamic structural models, autocorrelated error models, adaptive variance and adaptive parameter models. Overall, the more adaptive models performed best, but most of the methods proved vastly superior to simple least squares models which do not take dynamics into account.     

Improvement of the Liu‐type Shiller estimator for distributed lag models

The problem of multicollinearity produces undesirable effects on ordinary least squares (OLS), Almon and Shiller estimators for distributed lag models. Therefore, we introduce a Liu‐type Shiller estimator to deal with multicollinearity for distributed lag models. Moreover, we theoretically compare the predictive performance of the Liu‐type Shiller estimator with OLS and the Shiller estimators by the prediction mean square error criterion under the target function. Furthermore, an extensive Monte Carlo simulation study is carried out to evaluate the predictive performance of the Liu‐type Shiller estimator.