On adaptive estimation in nonstationary ARMA models with GARCH errors

This paper considers adaptive estimation in nonstationary autoregressive moving average models with the noise sequence satisfying a generalised autoregressive conditional heteroscedastic process. The locally asymptotic quadratic form of the log-likelihood ratio for the model is obtained. It is shown that the limit experiment is neither LAN nor LAMN, but is instead LABF. Adaptivity is discussed and it is found that the parameters in the model are generally not adaptively estimable if the density of the rescaled error is asymmetric. For the model with symmetric density of the rescaled error, a new efficiency criterion is established for a class of defined Mv-estimators. It is shown that such efficient estimators can be constructed when the density is known. Using the kernel estimator for the score function, adaptive estimators are constructed when the density of the rescaled error is symmetric, and it is shown that the adaptive procedure for the parameters in the conditional mean part uses the full sample without splitting. These estimators are demonstrated to be asymptotically efficient in the class of Mv-estimators. The paper includes the results that the stationary ARMA-GARCH model is LAN, and that the parameters in the model with symmetric density of the rescaled error are adaptively estimable after a reparameterisation of the GARCH process

Modelling the determinants of international tourism demand to Australia

Prior to the recent Asian currency and economic crises, tourism from Asia had rapidly become Australia's major tourism export industry. Tourists from Singapore, which is Australia's fifth major market, represented 6% of international tourist arrivals to Australia in 1996. The average annual growth rate of tourist arrivals from Singapore of around 20% over 1990-96 far exceeded the 10.5% average annual percentage growth rate of all tourist arrivals to Australia over the same period (Australian Bureau of Statistics, 1997). Despite the Asian currency and economic crises in 1997-98, tourist arrivals to Australia from Singapore has continued to grow slowly. It is imperative from the tourism marketing SWOT (strengths, weaknesses, opportunities and threats) analysis to consider the economic factors influencing international tourism demand for Australia by Singapore, and to undertake a primary competitor analysis of New Zealand. The purpose of the paper is to estimate the income, price and transportation cost elasticities of inbound tourism from Singapore to Australia using seasonally unadjusted quarterly data, to determine if Australia and New Zealand are substitute or complementary destinations for Singaporean tourists, and to examine issues such as nonstationarity, cointegration and spurious regressions

Estimating smooth transition autoregressive models with GARCH errors in the presence of extreme observations and outliers

This paper investigates several empirical issues regarding quasimaximum likelihood estimation of Smooth Transition Autoregressive (STAR) models with GARCH errors, specifically STAR-GARCH and STAR-STGARCH. Convergence, the choice of different algorithms for maximising the likelihood function, and the sensitivity of the estimates to outliers and extreme observations, are examined using daily data for S&P 500, Heng Seng and Nikkei 225 for the period January 1986 to April 2000

Comparing tests of autoregressive versus moving average errors in regression models using Bahadur's asymptotic relative efficiency

The purpose of this paper is to use Bahadur's asymptotic relative efficiency measure to compare the performance of various tests of autoregressive (AR) versus moving average (MA) error processes in regression models. Tests to be examined include non-nested procedures of the models against each other, and classical procedures based upon testing both the AR and MA error processes against the more general autoregressive-moving average model

Time series forecasts of international tourism demand for Australia

This paper examines stationary and nonstationary time series by formally testing for the presence of unit roots and seasonal unit roots prior to estimation, model selection and forecasting. Various Box-Jenkins Autoregressive Integrated Moving Average (ARIMA) models are estimated over the period 1975(1)-1989(4) for tourist arrivals to Australia from Hong Kong, Malaysia and Singapore. The mean absolute percentage error (MAPE) and root mean squared error (RMSE) are used as measures of forecast accuracy. As the best fitting ARIMA model is found to have the lowest RMSE, it is used to obtain post-sample forecasts. Tourist arrivals data for 1990(1) to 1996(4) are compared with the forecast performance of the ARIMA model for each origin market. The fitted ARIMA model forecasts tourist arrivals from Singapore between 1990(1)-1996(4) very well. Although the ARIMA model outperforms the seasonal ARIMA models for Hong Kong and Malaysia, the forecast of tourist arrivals is not as accurate as in the case of Singapore

Testing multiple non-nested factor demand systems

Empirical factor demand analysis typically involves making a choice from among several competing non-nested functional forms. Each of the commonly used factor demand systems, such as Translog, Generalized Leontief, Quadratic, and Generalized McFadden, can provide a valid and useful empirical description of the underlying production structure of the firm. As there is no theoretical guidance on selecting the model which is best able to capture the relevant features of the data, formal testing procedures can provide additional information. Paired and joint univariate nonnested tests of a null model against both single and multiple alternatives have been discussed at length in the literature, whereas virtually no attention has been paid to either paired or joint multivariate non-nested tests. This paper shows how some multivariate non-nested tests can be derived from their univariate counterparts, and examines how to use these tests empirically to compare alternative factor demand systems. The empirical application involves the classical Berndt-Khaled annual data set for the U.S. manufacturing sector over the period 1947-1971. A statistically adequate empirical specification is determined for each competing factor demand system. The empirical results are interpreted for each system, and the models are compared on the basis of multivariate paired and joint non-nested procedures

Asymptotic properties of the estimator of the long-run coefficient in a dynamic model with integrated regressors and serially correlated errors

In this paper we examine the asymptotic properties of the estimator of the long-run coefficient (LRC) in a dynamic regression model with integrated regressors and serially correlated errors. We show that the OLS estimators of the regression coefficients are inconsistent but the OLS-based estimator of the LRC is superconsistent. Furthermore, we propose an alternative consistent estimator of the LRC, compare the two estimators through a Monte Carlo experiment, and þnd that the proposed estimator is MSE-superior to the OLS-based estimator

Asymptotic theory for a vector ARMA-GARCH model

This paper investigates the asymptotic theory for a vector ARMA-GARCH model. The conditions for the strict stationarity, ergodicity, and the higherorder moments of the model are established. Consistency of the quasi-maximum likelihood estimator (QMLE) is proved under only the second-order moment condition. This consistency result is new, even for the univariate ARCH and GARCH models. Moreover, the asymptotic normality of the QMLE for the vector ARCH model is obtained under only the second-order moment of the unconditional errors, and the finite fourth-order moment of the conditional errors. Under additional moment conditions, the asymptotic normality of the QMLE is also obtained for the vector ARMA-ARCH and ARMA-GARCH models, as well as a consistent estimator of the asymptotic covariance

A survey of 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

Estimation and testing for unit root processes with GARCH(1,1) errors: Theory and Monte Carlo evidence

Least squares (LS) and maximum likelihood (ML) estimation are considered for unit root processes with GARCH (1, 1) errors. The asymptotic distributions of LS and ML estimators are derived under the condition alpha + beta < 1. The former has the usual unit root distribution and the latter is a functional of a bivariate Brownian motion, as in Ling and Li (1998). Several unit root tests based on LS estimators, ML estimators, and mixing LS and ML estimators, are constructed. Simulation results show that tests based on mixing LS and ML estimators perform better than Dickey-Fuller tests which are based on LS estimators, and that tests based on the ML estimators perform better han the mixed estimators