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This paper applies the Hafner and Herwartz (2006) (hereafter HH) approach to the analysis of multivariate GARCH models using volatility impulse response analysis. The data set features ten years of daily returns series for the New York Stock Exchange Index and the FTSE 100 index from the London Stock Exchange, from 3 January 2005 to 31 January 2015. This period captures both the Global Financial Crisis (GFC) and the subsequent European Sovereign Debt Crisis (ESDC). The attraction of the HH approach is that it involves a novel application of the concept of impulse response functions, tracing the effects of independent shocks on volatility through time, while avoiding typical orthogonalization and ordering problems. Volatility impulse response functions (VIRF) provide information about the impact of independent shocks on volatility. HH’s VIRF extends a framework provided by Koop et al. (1996) for the analysis of impulse responses. This approach is novel because it explores the effects of shocks to the conditional variance, as opposed to the conditional mean. HH use the fact that GARCH models can be viewed as being linear in the squares, and that multivariate GARCH models are known to have a VARMA representation with non-Gaussian errors. They use this particular structure to calculate conditional expectations of volatility analytically in their VIRF analysis. A Jordan decomposition of Σt is used to obtain independent and identically distributed innovations. A general issue in the approach is the choice of baseline volatilities. VIRF is defined as the expectation of volatility conditional on an initial shock and on history, minus the baseline expectation that conditions on history. This makes the process endogenous, but the choice of the baseline shock within the data set makes a difference. We explore the impact of three different shocks, the first marking the onset of the GFC, which we date as 9 August 2007 (GFC1). This began with the seizure in the banking system precipitated by BNP Paribas announcing that it was ceasing activity in three hedge funds that specialised in US mortgage debt. It took a year for the financial crisis to come to a head, but it did so on 15 September 2008, when the US government allowed the investment bank Lehman Brothers to go bankrupt (GFC2). The third shock is 9 May 2010, which marked the point at which the focus of concern switched from the private sector to the public sector. A further contribution of this paper is the inclusion of leverage, or asymmetric effects. Our modelling is undertaken in the context of a multivariate GARCH model featuring pre-whitened return series, which are then analysed using both BEKK and diagonal BEKK models with the t-distribution. A key result is that the impact of negative shocks is larger, in terms of the effects on variances and covariances, but shorter in duration, in this case a difference between three and six months, in the context of the return series.

__Abstract__ Energy and agricultural commodities and markets have been examined extensively, albeit separately, for a number of years. In the energy literature, the returns, volatility and volatility spillovers (namely, the delayed effect of a returns shock in one asset on the subsequent volatility or covolatility in another asset), among alternative energy commodities, such as oil, gasoline and ethanol across different markets, have been analysed using a variety of univariate and multivariate models, estimation techniques, data sets, and time frequencies. A similar comment applies to the separate theoretical and empirical analysis of a wide range of agricultural commodities and markets. Given the recent interest and emphasis in bio-fuels and green energy, especially bio-ethanol, which is derived from a range of agricultural products, it is not surprising that there is a topical and developing literature on the spillovers between energy and agricultural markets. Modelling and testing spillovers between the energy and agricultural markets has typically been based on estimating multivariate conditional volatility models, specifically the BEKK and DCC models. A serious technical deficiency is that the Quasi-Maximum Likelihood Estimates (QMLE) of a full BEKK matrix, which is typically estimated in examining volatility spillover effects, has no asymptotic properties, except by assumption, so that no statistical test of volatility spillovers is possible. Some papers in the literature have used the DCC model to test for volatility spillovers. However, it is well known in the financial econometrics literature that the DCC model has no regularity conditions, and that the QMLE of the parameters of DCC has no asymptotic properties, so that there is no valid statistical testing of volatility spillovers. The purpose of the paper is to evaluate the theory and practice in testing for volatility spillovers between energy and agricultural markets using the multivariate BEKK and DCC models, and to make recommendations as to how such spillovers might be tested using valid statistical techniques. Three new definitions of volatility and covolatility spillovers are given, and the different models used in empirical applications are evaluated in terms of the new definitions and statistical criteria.

__Abstract__ The Basel Committee on Banking Supervision (BCBS) (2013) recently proposed shifting the quantitative risk metrics system from Value-at-Risk (VaR) to Expected Shortfall (ES). The BCBS (2013) noted that “a number of weaknesses have been identified with using VaR for determining regulatory capital requirements, including its inability to capture tail risk” (p. 3). For this reason, the Basel Committee is considering the use of ES, which is a coherent risk measure and has already become common in the insurance industry, though not yet in the banking industry. While ES is mathematically superior to VaR in that it does not show “tail risk” and is a coherent risk measure in being subadditive, its practical implementation and large calculation requirements may pose operational challenges to financial firms. Moreover, previous empirical findings based only on means and standard deviations suggested that VaR and ES were very similar in most practical cases, while ES could be less precise because of its larger variance. In this paper we find that ES is computationally feasible using personal computers and, contrary to previous research, it is shown that there is a stochastic difference between the 97.5% ES and 99% VaR. In the Gaussian case, they are similar but not equal, while in other cases they can differ substantially: in fat-tailed conditional distributions, on the one hand, 97.5%-ES would imply higher risk forecasts, while on the other, it provides a smaller down-side risk than using the 99%-VaR. It is found that the empirical results in the paper generally support the proposals of the Basel Committee.

__Abstract__ Two of the fastest growing frontiers in econometrics and quantitative finance are time series and financial econometrics. Significant theoretical contributions to financial econometrics have been made by experts in statistics, econometrics, mathematics, and time series analysis. The purpose of this special issue of the journal on “Frontiers in Time Series and Financial Econometrics” is to highlight several areas of research by leading academics in which novel methods have contributed significantly to time series and financial econometrics, including forecasting co-volatilities via factor models with asymmetry and long memory in realized covariance, prediction of Lévy-driven CARMA processes, functional index coefficient models with variable selection, LASSO estimation of threshold autoregressive models, high dimensional stochastic regression with latent factors, endogeneity and nonlinearity, sign-based portmanteau test for ARCH-type models with heavy-tailed innovations, toward optimal model averaging in regression models with time series errors, high dimensional dynamic stochastic copula models, a misspecification test for multiplicative error models of non-negative time series processes, sample quantile analysis for long-memory stochastic volatility models, testing for independence between functional time series, statistical inference for panel dynamic simultaneous equations models, specification tests of calibrated option pricing models, asymptotic inference in multiple-threshold double autoregressive models, a new hyperbolic GARCH model, intraday value-at-risk: an asymmetric autoregressive conditional duration approach, refinements in maximum likelihood inference on spatial autocorrelation in panel data, statistical inference of conditional quantiles in nonlinear time series models, quasi-likelihood estimation of a threshold diffusion process, threshold models in time series analysis - some reflections, and generalized ARMA models with martingale difference errors.

__Abstract__ Of the two most widely estimated univariate asymmetric conditional volatility models, the exponential GARCH (or EGARCH) specification can capture asymmetry, which refers to the different effects on conditional volatility of positive and negative effects of equal magnitude, and leverage, which refers to the negative correlation between the returns shocks and subsequent shocks to volatility. However, the statistical properties of the (quasi-) maximum likelihood estimator (QMLE) of the EGARCH parameters are not available under general conditions, but only for special cases under highly restrictive and unverifiable conditions, such as EGARCH(1,0) or EGARCH(1,1), and possibly only under simulation. A limitation in the development of asymptotic properties of the QMLE for the EGARCH(p,q) model is the lack of an invertibility condition for the returns shocks underlying the model. It is shown in this paper that the EGARCH(p,q) model can be derived from a stochastic process, for which the invertibility conditions can be stated simply and explicitly. This will be useful in re-interpreting the existing properties of the QMLE of the EGARCH(p,q) parameters.

__Abstract__ Of the two most widely estimated univariate asymmetric conditional volatility models, the exponential GARCH (or EGARCH) specification can capture asymmetry, which refers to the different effects on conditional volatility of positive and negative effects of equal magnitude, and leverage, which refers to the negative correlation between the returns shocks and subsequent shocks to volatility. However, the statistical properties of the (quasi-) maximum likelihood estimator (QMLE) of the EGARCH parameters are not available under general conditions, but only for special cases under highly restrictive and unverifiable conditions. A limitation in the development of asymptotic properties of the QMLE for EGARCH is the lack of an invertibility condition for the returns shocks underlying the model. It is shown in this paper that the EGARCH model can be derived from a stochastic process, for which the invertibility conditions can be stated simply and explicitly. This will be useful in re-interpreting the existing properties of the QMLE of the EGARCH parameters.

__Abstract__ One of the most popular univariate asymmetric conditional volatility models is the exponential GARCH (or EGARCH) specification. In addition to asymmetry, which captures the different effects on conditional volatility of positive and negative effects of equal magnitude, EGARCH can also accommodate leverage, which is the negative correlation between returns shocks and subsequent shocks to volatility. However, there are as yet no statistical properties available for the (quasi-) maximum likelihood estimator of the EGARCH parameters. It is often argued heuristically that the reason for the lack of statistical properties arises from the presence in the model of an absolute value of a function of the parameters, which does not permit analytical derivatives or the derivation of statistical properties. It is shown in this paper that: (i) the EGARCH model can be derived from a random coefficient complex nonlinear moving average (RCCNMA) process; and (ii) the reason for the lack of statistical properties of the estimators of EGARCH is that the stationarity and invertibility conditions for the RCCNMA process are not known.

This paper investigates the stock returns and volatility size effects for firm performance in the Taiwan tourism industry, especially the impacts arising from the tourism policy reform that allowed mainland Chinese tourists to travel to Taiwan. Four conditional univariate GARCH models are used to estimate the volatility in the stock indexes for large and small firms in Taiwan. Daily data from 30 November 2001 to 27 February 2013 are used, which covers the period of Cross-Straits tension between China and Taiwan. The full sample period is divided into two subsamples, namely prior to and after the policy reform that encouraged Chinese tourists to Taiwan. The empirical findings confirm that there have been important changes in the volatility size effects for firm performance, regardless of firm size and estimation period. Furthermore, the risk premium reveals insignificant estimates in both time periods, while asymmetric effects are found to exist only for large firms after the policy reform. The empirical findings should be useful for financial managers and policy analysts as it provides insight into the magnitude of the volatility size effects for firm performance, how it can vary with firm size, the impacts arising from the industry policy reform, and how firm size is related to financial risk management strategy.

This paper estimates the long memory volatility model for 16 agricultural commodity futures returns from different futures markets, namely corn, oats, soybeans, soybean meal, soybean oil, wheat, live cattle, cattle feeder, pork, cocoa, coffee, cotton, orange juice, Kansas City wheat, rubber, and palm oil. The class of fractional GARCH models, namely the FIGARCH model of Baillie et al. (1996), FIEGACH model of Bollerslev and Mikkelsen (1996), and FIAPARCH model of Tse (1998), are modelled and compared with the GARCH model of Bollerslev (1986), EGARCH model of Nelson (1991), and APARCH model of Ding et al. (1993). The estimated d parameters, indicating long-term dependence, suggest that fractional integration is found in most of agricultural commodity futures returns series. In addition, the FIGARCH (1,d,1) and FIEGARCH(1,d,1) models are found to outperform their GARCH(1,1) and EGARCH(1,1) counterparts.

It is common practice to evaluate fixed-event forecast revisions in macroeconomics by regressing current revisions on one-period lagged revisions. Under weak-form efficiency, the correlation between the current and one-period lagged revisions should be zero. The empirical findings in the literature suggest that the null hypothesis of zero correlation between the current and one-period lagged revisions is rejected quite frequently, where the correlation can be either positive or negative. In this paper we propose a methodology to be able to interpret such non-zero correlations in a straightforward manner. Our approach is based on the assumption that forecasts can be decomposed into both an econometric model and expert intuition. The interpretation of the sign of the correlation between the current and one-period lagged revisions depends on the process governing intuition, and the correlation between intuition and news.

This paper examines risk transmission and migration among six US measures of credit and market risk during the full period 2004-2011 period and the 2009-2011 recovery subperiod, with a focus on four sectors related to the highly volatile oil price. There are more long-run equilibrium risk relationships and short-run causal relationships among the four oil-related Credit Default Swaps (CDS) indexes, the (expected equity volatility) VIX index and the (swaption expected volatility) SMOVE index for the full period than for the recovery subperiod. The auto sector CDS spread is the most error-correcting in the long run and also leads in the risk discovery process in the short run. On the other hand, the CDS spread of the highly regulated, natural monopoly utility sector does not error correct. The four oil-related CDS spread indexes are responsive to VIX in the short- and long-run, while no index is sensitive to SMOVE which, in turn, unilaterally assembles risk migration from VIX. The 2007-2008 Great Recession seems to have led to “localization” and less migration of credit and market risk in the oil-related sectors.

Macroeconomic forecasts are often based on the interaction between econometric models and experts. A forecast that is based only on an econometric model is replicable and may be unbiased, whereas a forecast that is not based only on an econometric model, but also incorporates an expert’s touch, is non-replicable and is typically biased. In this paper we propose a methodology to analyze the qualities of combined non-replicable forecasts. One part of the methodology seeks to retrieve a replicable component from the non-replicable forecasts, and compares this component against the actual data. A second part modifies the estimation routine due to the assumption that the difference between a replicable and a non-replicable forecast involves a measurement error. An empirical example to forecast economic fundamentals for Taiwan shows the relevance of the methodological approach.

A risk management strategy is proposed as being robust to the Global Financial Crisis (GFC) by selecting a Value-at-Risk (VaR) forecast that combines the forecasts of different VaR models. The robust forecast is based on the median of the point VaR forecasts of a set of conditional volatility models. This risk management strategy is GFC-robust in the sense that maintaining the same risk management strategies before, during and after a financial crisis would lead to comparatively low daily capital charges and violation penalties. The new method is illustrated by using the S&P500 index before, during and after the 2008-09 global financial crisis. We investigate the performance of a variety of single and combined VaR forecasts in terms of daily capital requirements and violation penalties under the Basel II Accord, as well as other criteria. The median VaR risk management strategy is GFC-robust as it provides stable results across different periods relative to other VaR forecasting models. The new strategy based on combined forecasts of single models is straightforward to incorporate into existing computer software packages that are used by banks and other financial institutions.

This paper estimates the effects of short and long haul volatility (or risk) in monthly Japanese tourist arrivals to Taiwan and New Zealand, respectively. In order to model appropriately the volatilities of international tourist arrivals, we use symmetric and asymmetric conditional volatility models that are commonly used in financial econometrics, namely the GARCH (1,1), GJR (1,1) and EGARCH (1,1) models. The data series are for the period January 1997 to December 2007. The volatility estimates for the monthly growth in Japanese tourists to New Zealand and Taiwan are different, and indicate that the former has an asymmetric effect on risk from positive and negative shocks of equal magnitude, while the latter has no asymmetric effect. Moreover, there is a leverage effect in the monthly growth rate of Japanese tourists to New Zealand, whereby negative shocks increase volatility but positive shocks of similar magnitude decrease volatility. These empirical results seem to be similar to a wide range of financial stock market prices, so that the models used in financial economics, and hence the issues related to risk and leverage effects, are also applicable to international tourism flows.

Macro-economic forecasts typically involve both a model component, which is replicable, as well as intuition, which is non-replicable. Intuition is expert knowledge possessed by a forecaster. If forecast updates are progressive, forecast updates should become more accurate, on average, as the actual value is approached. Otherwise, forecast updates would be neutral. The paper proposes a methodology to test whether forecast updates are progressive and whether econometric models are useful in updating forecasts. The data set for the empirical analysis are for Taiwan, where we have three decades of quarterly data available of forecasts and updates of the inflation rate and real GDP growth rate. The actual series for both the inflation rate and the real GDP growth rate are always released by the government one quarter after the release of the revised forecast, and the actual values are not revised after they have been released. Our empirical results suggest that the forecast updates for Taiwan are progressive, and can be explained predominantly by intuition. Additionally, the one-, two- and three-quarter forecast errors are predictable using publicly available information for both the inflation rate and real GDP growth rate, which suggests that the forecasts can be improved.

Macroeconomic forecasts are frequently produced, published, discussed and used. The formal evaluation of such forecasts has a long research history. Recently, a new angle to the evaluation of forecasts has been addressed, and in this review we analyse some recent developments from that perspective. The literature on forecast evaluation predominantly assumes that macroeconomic forecasts are generated from econometric models. In practice, however, most macroeconomic forecasts, such as those from the IMF, World Bank, OECD, Federal Reserve Board, Federal Open Market Committee (FOMC) and the ECB, are based on econometric model forecasts as well as on human intuition. This seemingly inevitable combination renders most of these forecasts biased and, as such, their evaluation becomes non-standard. In this review, we consider the evaluation of two forecasts in which: (i) the two forecasts are generated from two distinct econometric models; (ii) one forecast is generated from an econometric model and the other is obtained as a combination of a model, the other forecast, and intuition; and (iii) the two forecasts are generated from two distinct combinations of different models and intuition. It is shown that alternative tools are needed to compare and evaluate the forecasts in each of these three situations. These alternative techniques are illustrated by comparing the forecasts from the Federal Reserve Board and the FOMC on inflation, unemployment and real GDP growth

Tourism is a major source of service receipts for many countries, including Taiwan. The two leading tourism countries for Taiwan, comprising a high proportion of world tourist arrivals to Taiwan, are Japan and USA, which are sources of short and long haul tourism, respectively. As it is well known that a strong domestic currency can have adverse effects on international tourist arrivals, daily data from 1 January 1990 to 31 December 2008 are used to model the world price and US$ / New Taiwan $ and Yen/ New Taiwan $ exchange rates, and tourist arrivals from the world, USA and Japan to Taiwan, as well as their associated volatility. The sample period includes the Asian economic and financial crises in 1997, and part of the global financial crisis of 2008-09. Inclusion of the exchange rate allows approximate daily price effects on world, US and Japanese tourist arrivals to Taiwan to be captured. The Heterogeneous Autoregressive (HAR) model does not reproduce the theoretical hyperbolic decay rates associated with fractionally integrated (or long memory) time series models, but it can nevertheless approximate quite accurately and parsimoniously the slowly decaying correlations associated with such models. The HAR model is used to approximate long memory properties in daily exchange rates and international tourist arrivals, to test whether alternative short and long run estimates of conditional volatility are sensitive to the approximate long memory in the conditional mean, to examine asymmetry and leverage in volatility, and to examine the effects of temporal and spatial aggregation. The empirical results show that the conditional volatility estimates are not sensitive to the approximate long memory nature of the conditional mean specifications. The QMLE for the GARCH(1,1), GJR(1,1) and EGARCH(1,1) models for world, US and Japanese tourist arrivals to Taiwan, and the world price and US$ / New Taiwan $ and Yen/ New Taiwan $ exchange rates, are statistically adequate and have sensible interpretations. Asymmetry (though not leverage) is found for several alternative HAR models for the world, US and Japanese tourist arrivals to Taiwan. For policy purposes, these empirical results suggest that an arbitrary choice of data frequency or spatial aggregation will not lead to robust findings as they are generally not independent of the level of aggregation used.

Crude oil price volatility has been analyzed extensively for organized spot, forward and futures markets for well over a decade, and is crucial for forecasting volatility and Value-at-Risk (VaR). There are four major benchmarks in the international oil market, namely West Texas Intermediate (USA), Brent (North Sea), Dubai/Oman (Middle East), and Tapis (Asia-Pacific), which are likely to be highly correlated. This paper analyses the volatility spillover and asymmetric effects across and within the four markets, using three multivariate GARCH models, namely the constant conditional correlation (CCC), vector ARMA-GARCH (VARMA-GARCH) and vector ARMA-asymmetric GARCH (VARMA-AGARCH) models. A rolling window approach is used to forecast the 1-day ahead conditional correlations. The paper presents evidence of volatility spillovers and asymmetric effects on the conditional variances for most pairs of series. In addition, the forecast conditional correlations between pairs of crude oil returns have both positive and negative trends. Moreover, the optimal hedge ratios and optimal portfolio weights of crude oil across different assets and market portfolios are evaluated in order to provide important policy implications for risk management in crude oil markets.

Both domestic and international tourism are a major source of service export receipts for many countries worldwide, and is also increasingly important in Taiwan. One of the three leading tourism source countries for Taiwan is the Republic of Korea, which is a source of short haul tourism. Daily data from 1 January 1990 to 31 December 2008 are used to model the Korean Won / New Taiwan $ exchange rate and tourist arrivals from Korea to Taiwan, as well as their associated volatility. The sample period includes the Asian economic and financial crises in 1997, and a significant part of the global financial crisis of 2008-09. Inclusion of the exchange rate allows approximate daily price effects on Korean tourism arrivals to Taiwan to be captured. The Heterogeneous Autoregressive (HAR) model is used to capture long memory properties in exchange rates and Korean tourist arrivals, to test whether alternative estimates of conditional volatility are sensitive to the long memory in the conditional mean, and to examine asymmetry and leverage in volatility. The empirical results show that the conditional volatility estimates are not sensitive to the long memory nature of the conditional mean specifications. The QMLE for the GARCH(1,1), GJR(1,1) and EGARCH(1,1) models for Korean tourist arrivals to Taiwan and the Korean Won / New Taiwan $ exchange rate are statistically adequate and have sensible interpretations. Asymmetry (though not leverage) is found for several alternative HAR models.

In this paper we consider a nonlinear model based on neural networks as well as linear models to forecast the daily volatility of the S&P 500 and FTSE 100 indexes. As a proxy for daily volatility, we consider a consistent and unbiased estimator of the integrated volatility that is computed from high frequency intra-day returns. We also consider a simple algorithm based on bagging (bootstrap aggregation) in order to specify the models analyzed in the paper.

Asia is presently the most important market for the production and consumption of natural rubber. World prices of rubber are not only subject to changes in demand, but also to speculation regarding future markets. Japan and Singapore are the major futures markets for rubber, while Thailand is one of the world’s largest producers of rubber. As rubber prices are influenced by external markets, it is important to analyse the relationship between the relevant markets in Thailand, Japan and Singapore. The analysis is conducted using several alternative multivariate GARCH models. The empirical results indicate that the constant conditional correlations arising from the CCC model of Bollerslev (1990) lie in the low to medium range. The results from the VARMA-GARCH model of Ling and McAleer (2003) and the VARMA-AGARCH model of McAleer et al. (2009) suggest the presence of volatility spillovers and asymmetric effects of positive and negative return shocks on conditional volatility. Finally, the DCC model of Engle (2002) suggests that the conditional correlations can vary dramatically over time. In general, the dynamic conditional correlations in rubber spot and futures returns shocks can be independent or interdependent.

When dealing with market risk under the Basel II Accord, variation pays in the form of lower capital requirements and higher profits. Typically, GARCH type models are chosen to forecast Value-at-Risk (VaR) using a single risk model. In this paper we illustrate two useful variations to the standard mechanism for choosing forecasts, namely: (i) combining different forecast models for each period, such as a daily model that forecasts the supremum or infinum value for the VaR; (ii) alternatively, select a single model to forecast VaR, and then modify the daily forecast, depending on the recent history of violations under the Basel II Accord. We illustrate these points using the Standard and Poor’s 500 Composite Index. In many cases we find significant decreases in the capital requirements, while incurring a number of violations that stays within the Basel II Accord limits.

The El Niños Southern Oscillations (ENSO) is a periodical phenomenon of climatic interannual variability which could be measured through either the Southern Oscillation Index (SOI) or the Sea Surface Temperature (SST) Index. The main purpose of this paper is to analyze these two indexes in order to capture ENSO volatility. The empirical results show that both the ARMA(1,1)-GARCH(1,1) and ARMA(3,2)-GJR(1,1) models are suitable for modelling ENSO volatility. Moreover, 1998 is a turning point for the volatility of SOI, and the ENSO volatility has became stronger since 1998 which indicates that the ENSO strength has increased.

A government’s ability to forecast key economic fundamentals accurately can affect business confidence, consumer sentiment, and foreign direct investment, among others. A government forecast based on an econometric model is replicable, whereas one that is not fully based on an econometric model is non-replicable. Governments typically provide non-replicable forecasts (or, expert forecasts) of economic fundamentals, such as the inflation rate and real GDP growth rate. In this paper, we develop a methodology to evaluate non-replicable forecasts. We argue that in order to do so, one needs to retrieve from the non-replicable forecast its replicable component, and that it is the difference in accuracy between these two that matters. An empirical example to forecast economic fundamentals for Taiwan shows the relevance of the proposed methodological approach. Our main finding is that it is the undocumented knowledge of the Taiwanese government that reduces forecast errors substantially.

This paper estimates univariate and multivariate conditional volatility and conditional correlation models of spot, forward and futures returns from three major benchmarks of international crude oil markets, namely Brent, WTI and Dubai, to aid in risk diversification. Conditional correlations are estimated using the CCC model of Bollerslev (1990), VARMA-GARCH model of Ling and McAleer (2003), VARMA-AGARCH model of McAleer et al. (2009), and DCC model of Engle (2002). The paper also presents the ARCH and GARCH effects for returns and shows the presence of significant interdependences in the conditional volatilities across returns for each market. The estimates of volatility spillovers and asymmetric effects for negative and positive shocks on conditional variance suggest that VARMA-GARCH is superior to the VARMA-AGARCH model. In addition, the DCC model gives statistically significant estimates for the returns in each market, which shows that constant conditional correlations do not hold in practice.

The Amazon rainforest is one of the world’s greatest natural wonders and holds great importance and significance for the world’s environmental balance. Around 60% of the Amazon rainforest is located in the Brazilian territory. The two biggest states of the Amazon region are Amazonas (the upper Amazon) and Pará (the lower Amazon), which together account for around 73% of the Brazilian Legal Amazon, and are the only states that are serviced by international airports in Brazil’s North region. The purpose of this paper is to model and forecast sustainable international tourism demand for the states of Amazonas, Pará, and the aggregate of the two states. Economic progress of the region has been achieved at a cost of destroying large areas of the Amazon rain forest. In this scenario, the tourism industry would seem to have the potential to contribute to sustainable economic development in the North region of Brazil. The paper presents unit root tests for monthly and annual data, estimates alternative time series models and conditional volatility models of the shocks to international tourist arrivals, and provides forecasts for 2006 and 2007.