Smooth Transition Autoregressive Research Articles

STAR and ANN models: forecasting performance on the Spanish “Ibex-35” stock index

This paper studies whether it is possible to exploit the nonlinear behaviour of daily returns on the Spanish Ibex-35 stock index returns to improve forecasts over short and long horizons. In this sense, we examine the out-of-sample forecast performance of smooth transition autoregression (STAR) models and artificial neural networks (ANNs). We use one-step (obtained by using recursive and nonrecursive regressions) and multi-step-ahead forecasting methods. The forecasts are evaluated with statistical and economic criteria. In terms of statistical criteria, we compared the out-of-sample forecasts using goodness of forecast measures and various testing approaches. The results indicate that ANNs consistently surpass the random walk model and, although the evidence for this is weaker, provide better forecasts than the linear AR model and the STAR models for some forecast horizons and forecasting methods. In terms of the economic criteria, we assess the relative forecast performance in a simple trading strategy including the impact of transaction costs on trading strategy profits. The results indicate a better fit for ANN models, in terms of the mean net return and Sharpe risk-adjusted ratio, by using one-step-ahead forecasts. These results show there is a good chance of obtaining a more accurate fit and forecast of the daily stock index returns by using one-step-ahead predictors and nonlinear models, but that these are inherently complex and present a difficult economic interpretation.

Nonlinear Purchasing Power Parity under the Gold Standard

1. IntroductionRecent theoretical analysis of purchasing power deviations (see, e.g., Dumas 1992; Sercu, Uppal, and Van Hull 1995; and O'Connell and Wei 1997) demonstrates how transactions costs or the sunk costs of international arbitrage induce nonlinear adjustment of the real exchange rate to purchasing power parity (PPP). Globally mean-reverting this nonlinear process has the important property of exhibiting near unit root behavior for small deviations from PPP since small deviations from PPP are left uncorrected if they are not large enough to cover the transactions costs or the sunk costs of international arbitrage.A parametric nonlinear model, suggested by the theoretical literature, that captures the nonlinear adjustment process in aggregate data is the exponential smooth transition autoregression model (ESTAR) of Ozaki (1985). A smooth, rather than discrete, adjustment mechanism is motivated by the theoretical analysis of Dumas (1992). Also, as postulated by Terasvirta (1994) and demonstrated theoretically by Berka (2002), in aggregate data regime changes may be smooth, rather than discrete, given that heterogeneous agents do not act simultaneously even if they make dichotomous decisions.1 Recent empirical work (e.g., Michael, Nobay, and Peel 1997; Taylor, Peel, and Sarno 2001; Peel and Venetis 2002) has reported empirical results that suggest that the ESTAR model provides a parsimonious fit into a variety of data sets, particular for monthly data for the interwar and postwar floating period as well as for annual data spanning 200 years, as reported in Lothian and Taylor (1996). In addition, nonlinear impulse response functions derived from the ESTAR models show that although the speed of adjustment for small shocks around equilibrium will be highly persistent, larger shocks mean-revert much faster than the glacial rates previously reported for linear models (Rogoff 1996). In this respect, the ESTAR models provide some solution to the PPP puzzle outlined in Rogoff (1996).2The ESTAR model can also provide an explanation of why PPP deviations analyzed from a linear perspective appear to be described by either a nonstationary integrated I(1) process, or alternatively, described by fractional processes (see, e.g., Diebold, Husted, and Rush 1991). Taylor, Peel, and Sarno (2001), and Pippenger and Goering (1993) show that the Dickey-Fuller tests have low power against data simulated from an ESTAR model. Michael, Nobay, and Peel (1997) and Byers and Peel (2003) show that data that is generated from an ESTAR process can appear to exhibit the fractional property. That this would be the case was an early conjecture by Acosta and Granger (1995). Given that the ESTAR model has a theoretical rationale, whereas the fractional process is a relatively nonintuitive one, the fractional property might reasonably be interpreted as a misleading linear property of PPP deviations (Granger and Terasvirta 1999).Whereas the empirical work employing ESTAR models provides some explanation of the glacially slow adjustment speeds obtained in linear models, there is one aspect of the empirical work that is worthy of further attention. A second way of explaining the Rogoff puzzle, raised by Rogoff himself,3 is to relax the assumption that the equilibrium real exchange rate is a constant (see, e.g., Canzoneri, Cumby, and Diba 1996; and Chinn and Johnston 1996). Theoretical models, such as that of Balassa (1964) and Samuelson (1964), imply a nonconstant equilibrium in the real exchange rate if real productivity growth rates differ between countries.4 Nonlinear models that incorporate proxies for these effects are found to parsimoniously fit post-Bretton Woods data for the main real exchange rates (see Venetis, Paya, and Peel 2002; and Paya, Venetis, and Peel 2003). Naturally, models that ignore this effect may generate misleading speeds of PPP adjustment to shocks. In this regard, the empirical results of Hegwood and Papell (2002) for the Gold Standard period are particularly interesting. …

Unit roots or nonlinear stationarity in Turkish real exchange rates

The objective of this paper is to test if Turkish real exchange rates have a linear unit root or are generated by an Exponential Smooth Transition Autoregressive Model for the post-1980 period. Using two real exchange rates, one with the USA and the other with Germany, strong evidence of nonlinear stationarity was found for the US CPI-based series but no such evidence for the DM CPI-based series. When compared with earlier results in a previous paper where the alternative of the linear unit root test was stationarity with multiple shifts in the deterministic terms, it was found that similar results were obtained for the US CPI-based series but not for the DM CPI-based one, possibly implying that the multiple shifts approach may be more appropriate for the Turkish series.

Detecting Mean Reversion in Real Exchange Rates from a Multiple Regime STAR Model

Recent studies on general equilibrium models with transaction costs show that the dynamics of the real exchange rate are necessarily nonlinear. Our contribution to the literature on nonlinear price adjustment mechanisms is threefold. First, we model the real exchange rate by a Multi-Regime Logistic Smooth Transition AutoRegression (MR-LSTAR), allowing for both ESTAR-type and SETAR-type dynamics. This choice is motivated by the fact that even the theoretical models, which predict a smooth behavior for the real exchange rate, do not rule out the possibility of a discontinuous adjustment as a limit case. Second, we propose two classes of unit-root tests against this MR-LSTAR alternative, based respectively on the likelihood and on an auxiliary model. Their asymptotic distributions are derived analytically. Third, when applied to 28 bilateral real exchange rates, our tests reject the null hypothesis of a unit root for eleven series bringing evidence in favor of the purchasing power parity.

Linearity tests and stationarity

Summary This paper shows that the asymptotic distributions of LM-type linearity tests against Smooth Transition Autoregressive (STAR) models, in the presence of a unit root, are non-standard and using standard χ2 critical values may lead to incorrect inference as the tails of the distribution of tests will be thicker than the χ2. This finding also indicates that one needs to test for stationarity prior to applying linearity tests.

ON SINGAPORE DOLLAR–U.S. DOLLAR AND PURCHASING POWER PARITY

This study re-examines the validity of the relationship between the Singapore dollar–U.S. dollar exchange rate and relative prices using the latest econometric methodologies that account for non-linearity. Among others, this study finds Exponential Smooth Transition Autoregressive (ESTAR)-type non-linear mean-reverting adjustment process of the nominal Singapore dollar–U.S. dollar rate towards the consumer price index ratio. Unlike previous findings of a linear cointegration relationship between the nominal Singapore dollar–U.S. dollar exchange rate and consumer price index ratio, this study shows that the relationship is in fact non-linear in nature. The major economic implications of our findings are: (1) policy makers need to take non-linearity into consideration in their policy decisions; (2) the Monetary Authority of Singapore (MAS) is able to maintain the macroeconomic equilibrium despite the authority's strong dollar policy; and (3) one should keep track of Singapore's monetary policy and other innovations in aggregate demand in order to closely monitor the movement of the Singapore exchange rate.

Common risk factors in the U.S. and UK interest rate swap markets: Evidence from a nonlinear vector autoregression approach

AbstractThis paper produces evidence in support of the existence of common risk factors in the U.S. and UK interest rate swap markets. Using a multivariate smooth transition autoregression (STVAR) framework, we show that the dynamics of the U.S. and UK swap spreads are best described by a regime‐switching model. We identify the existence of two distinct regimes in U.S. and UK swap spreads; one is characterized by a “flat” term structure of U.S. interest rates and the other is characterized by an “upward” sloping U.S. term structure. In addition, we show that there exist significant asymmetries on the impact of the common risk factors on the U.S. and UK swap spreads. Shocks to UK oriented risk factors have a strong effect on the U.S. swap markets during the “flat” slope regime but a very limited effect otherwise. On the other hand, U.S. risk factors have a significant impact on the UK swap markets in both regimes. © 2004 Wiley Periodicals, Inc. Jrl Fut Mark 24:221–250, 2004

Arbitrage Bounds and the Time Series Properties of the Discount on UK Closed-End Mutual Funds

The persistence of discounts on UK (and US) closed-end mutual funds (investment trusts) is well known, both in the academic literature and in the financial sector. In a dataset of 135 UK ITC's, sampled at weekly intervals, it is shown that while prices and net asset values clearly have unit roots, the evidence suggests that discounts are for the most part not stationary, a finding which supports Cheng, Copeland and O'Hanlon (1994) and more recently Gemmill and Thomas (2002). On the other hand, it is shown here that discounts almost certainly do not have a unit root. A possible statistical reconciliation would be a long memory process, and results are presented to show that for most UK investment trusts the discount is well described by a fractionally-integrated process. This type of model has been widely applied to financial markets ever since it was introduced by Granger and Joyeux (1980). In most cases, the ITC discount process appears to have a root of around 0.7, which implies nonstationarity, but reversion to the mean. In other words, ITC discounts have a long run mean level (of around 13%) to which they eventually return, but disturbances are so persistent that reversion may be an extremely slow, protracted process. This conclusion is in some respects unsatisfactory, given that many of the ITC's in the sample are large, frequently traded, and quite liquid stocks. The long run mean discount estimated here is entirely consistent with the Gemmill and Thomas (2002) arbitrage-induced level. But if arbitrage is at all effective, it is difficult to reconcile with such slow adjustment in the aftermath of disturbances. A possible explanation is to be found in the broader arbitrage bounds generated by the threat of open-ending. If the discount process is actually a bounded random walk, then it is shown here that it will mimic a fractionally-integrated process, a conclusion which may well have relevance to a number of other financial markets which have been shown to display apparent long memory behaviour. The implication is that a more appropriate model of the typical ITC discount would take explicit account of the arbitrage forces at work. Specifically, it is argued that the process would be better characterised by a smooth-transition autoregression (STAR), implying that the speed of adjustment is an increasing function of the gap between the current discount and its long run level. Two specific formats are considered here: the exponential STAR (ESTAR) model, which postulates a symmetric adjustment process, and the logistic (LSTAR) model, which allows for asymmetry. The latter has to be a candidate in the present context, because of the possibility that the speed of adjustment to above-average and below-average discounts are not necessarily the same, since the arbitrage involved is different in the two cases and may involve different transaction costs. In the event, tests on 5 ITC's provide clear evidence of nonlinearity in the discount process, but they are inconclusive about whether the ESTAR or LSTAR model better describes the process.

Asian real interest rates, nonlinear dynamics, and international parity

This study tests for nonlinearities in the real interest differentials of four South East Asian economies with respect to Japan and the United States. The logistic and exponential smooth transition regression models are applied to monthly data over the sample period 1977M1–2000M3. There is evidence of nonlinearities in Asian real interest differentials where nonlinearities are often captured by the logistic smooth transition autoregressive (STAR) model. The extent of nonlinearities varies across the sample with the Singapore–Japan and Thailand–Japan differentials exhibiting the sharpest transition from one regime to another. Large shocks to real interest parity (RIP) are more likely to lead to the reestablishment of parity at a faster rate than small shocks. Modeling the nonlinear stochastic dynamics of RIP can thus be useful for policymaking purposes in recovering information on monetary and financial crises.

The inadequacy of linear autoregressive model for real exchange rates: empirical evidence from Asian economies

Utilizing the formal linearity test of Luukkonen, Saikkonen and Teräsvirta (Biometrika, 75, 491-499, 1998) as diagnostic tool, the empirical finding suggests that the linear autoregressive (AR) model is inadequate in describing the real exchange rates behaviour of 11 Asian economies. It is noted that the conventional battery of diagnostic tests is capable of identifying the inadequacy of the linear model in only three of these series. Moreover, the linearity nature of this behaviour has been formally rejected in favour of the non-linear smooth transition autoregressive (STAR) model. The finding of non-linearity in the data generating process of these real exchange rates warrants that the use of linear framework in empirical modelling and statistical testing procedures in the field of exchange rates may lead to an inappropriate policy conclusions.

Diagnostic Checking in a Flexible Nonlinear Time Series Model

This paper considers a sequence of misspecification tests for a flexible nonlinear time series model. The model is a generalization of both the smooth transition autoregressive (STAR) and the autoregressive artificial neural network (AR‐ANN) models. The tests are Lagrange multiplier (LM) type tests of parameter constancy against the alternative of smoothly changing ones, of serial independence, and of constant variance of the error term against the hypothesis that the variance changes smoothly between regimes. The small sample behaviour of the proposed tests is evaluated by a Monte‐Carlo study and the results show that the tests have size close to the nominal one and a good power.

Discriminating between competing STAR models

We propose a pseudo-true score encompassing test to distinguish between competing smooth transition autoregressive (STAR) models. The proposed test is easy to implement and applicable to all STAR models. Our Monte Carlo simulations show that the proposed test significantly outperforms the model selection rules of Teräsvirta and Anderson [Journal of Applied Econometrics, 7 (1992) S119] and Teräsvirta [Journal of the American Statistical Association, 89 (1994) 208].

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

The paper investigates several empirical issues regarding quasi-maximum likelihood estimation of smooth transition autoregressive (STAR) models with GARCH errors (STAR-GARCH) and STAR models with smooth transition GARCH errors (STAR-STGARCH). Empirical evidence is provided to show that different algorithms produce substantially different estimates for the same model. Consequently, the interpretation of the model can differ according to the choice of algorithm. Convergence, the choice of different algorithms for maximizing the likelihood function, and the sensitivity of the estimates to outliers and extreme observations, are examined using daily data for S&P 500, Hang Seng and Nikkei 225 for the period January 1986 to April 2000.

The net barter terms of trade: A smooth transition approach

AbstractThis paper analyses the net barter terms of trade measured by the primary commodity price index relative to the indexes of unit values of export of manufactures from industrial countries. The starting point is that the series is stationary but possibly non‐linear. Statistical tests indicate that the logarithmed series is non‐linear and we estimate a Smooth Transition Autoregressive model to describe the process. The dynamics of the model are illustrated by use of parametrically estimated local spectra and generalized impulse response functions. Our model encompasses models from several previous studies, and our conclusion is that the starting point very much decides the outcome. Copyright © 2003 John Wiley & Sons, Ltd.

Nonlinear Time Series Modelling: An Introduction

Recent developments in nonlinear time series modelling are reviewed. Three main types of nonlinear models are discussed: Markov Switching, Threshold Autoregression and Smooth Transition Autoregression. Classical and Bayesian estimation techniques are described for each model. Parametric tests for nonlinearity are reviewed with examples from the three types of models. Finally, forecasting and impulse response analysis is developed.

Maximum likelihood estimation of STAR and STAR‐GARCH models: theory and Monte Carlo evidence

AbstractTheoretical and practical interest in non‐linear time series models, particularly regime switching models, have increased substantially in recent years. Given the abundant research activity in analysing time‐varying volatility through Generalized Autoregressive Conditional Heteroscedasticity (GARCH) processes (see Engle, 1982; Bollerslev, 1986), it is important to analyse regime switching models with GARCH errors. A popular specification in this class is the (stationary) Smooth Transition Autoregressive–GARCH (STAR‐GARCH) model. Little is presently known about the structure of the model, or the consistency, asymptotic normality and finite sample properties of the estimators. The paper develops the structural and statistical properties of the STAR‐GARCH model, and investigates the finite sample properties of maximum likelihood estimation (MLE) of STAR and STAR‐GARCH models through numerical simulation. The effects of fixing the threshold value and/or the transition rate for the STAR model, misspecification of the conditional mean and the transition function of the STAR‐GARCH model, and the finite sample properties of the MLE for the STAR‐GARCH model, are also examined. These numerical results are used as a guide in empirical research, with an application to Standard and Poor's Composite 500 Index returns for alternative STAR‐GARCH models. Copyright © 2002 John Wiley & Sons, Ltd.

Common Risk Factors in the US and UK Interest Rate Swap Markets: Evidence From a Non-linear Vector Autoregression Approach

This paper produces evidence in support of the existence of common risk factors in the US and UK interest rate swap markets. Using a multivariate smooth transition autoregression (STVAR) framework, we show that the dynamics of the US and UK swap spreads are best described by a regime-switching model. We identify the existence of two distinct regimes in US and UK swap spreads; one characterized by a flat term structure of US interest rates and the other characterized by an upward slopping US term structure. In addition, we show that there exist significant asymmetries on the impact of the common risk factors on the US and UK swap spreads. Shocks to UK oriented risk factors have a strong effect on the US swap markets during the flat slope regime but a very limited effect otherwise. On the other hand, US risk factors have a significant impact on the UK swap markets in both regimes. Despite their added flexibility, the STVAR models do not consistently produce superior forecasts compared to less sophisticated autoregressive (AR) and vector autoregressive (VAR) models

Robust Tests Against Smooth Transition Autoregressive Models

In this paper, we consider testing for linearity against a well-known class of regime switching models known as the smooth transition autoregressive (STAR) models. Apart from the model selection issues, one reason for interest in testing for linearity in time-series models is that non-linear models such as the STAR are considerably more difficult to use. This testing problem is non-standard because a nuisance parameter becomes unidentified under the null hypothesis. In this paper, we further explore the class of tests proposed by Luukkonen, Saikonnen and Terasvirta (1988). Luukkonen et al . (1988) proposed LM tests for linearity against STAR models. A potential difficulty here is that the linear approximation introduces high leverage points, and hence outliers are likely to be quite influential. To overcome this difficulty, we use the same approximating linear model of Luukkonen et al . (1988), but we apply Wald and F -tests based on l 1 - and bounded influence estimates. The efficiency gains of this procedure cannot be easily deduced from the existing theoretical results because the test is based on a misspecified model under H 1 . Therefore, we carried out a simulation study, in which we observed that the robust tests have desirable properties compared to the test of Luukkonen et al . (1988) for a range of error distributions in the STAR model, in particular the robust tests have power advantages over the LM test.

Estimating Asymmetric Output Cost of Lowering Inflation for Australia

The purpose of this paper is to estimate the output cost associated with lowering inflation for Australia. The paper is particularly motivated by a strand of theoretical and empirical evidence in the literature suggesting nonlinearity in the output‐inflation relationship, namely, a nonlinear Phillips curve. To accommodate this potentially important departure from linearity, a vector autoregression (VAR) model of output, inflation, and the terms of trade is augmented with logistic smooth transition autoregression specifications. My empirical results indicate that the model captures the nonlinear features present in the data well. Based on this nonlinear approximation, the output costs for reducing inflation are found to vary, depending critically on the state of the economy, the size of intended inflation change, and whether policymakers seek to disinflate or prevent inflation from rising. This implies that inferences based on the conventional linear Phillips curve may provide misleading signals about the cost of lowering inflation and thus the appropriate policy stance.

Estimating Asymmetric Output Cost of Lowering Inflation for Australia

The purpose of this paper is to estimate the output cost associated with lowering inflation for Australia. The paper is particularly motivated by a strand of theoretical and empirical evidence in the literature suggesting nonlinearity in the output-inflation relationship, namely, a nonlinear Phillips curve. To accommodate this potentially important departure from linearity, a vector autoregression (VAR) model of output, inflation, and the terms of trade is augmented with logistic smooth transition autoregression specifications. My empirical results indicate that the model captures the nonlinear features present in the data well. Based on this nonlinear approximation, the output costs for reducing inflation are found to vary, depending critically on the state of the economy, the size of intended inflation change, and whether policymakers seek to disinflate or prevent inflation from rising. This implies that inferences based on the conventional linear Phillips curve may provide misleading signals about the cost of lowering inflation and thus the appropriate policy stance.