key: cord-0804171-l21a1hsg authors: Balli, Hatice Ozer; Tsui, Wai Hong Kan; Balli, Faruk title: Modelling the volatility of international visitor arrivals to New Zealand date: 2019-03-31 journal: Journal of Air Transport Management DOI: 10.1016/j.jairtraman.2018.10.002 sha: 5f9a87b9cf932142d715198dabdcb7ffc31cd858 doc_id: 804171 cord_uid: l21a1hsg Abstract International visitor arrivals are considered to be a major source of foreign exchange, tourism-related employment and other tourism-related activities. This study used SARIMAX/(E)GARCH volatility models to forecast visitor arrivals by air transport to New Zealand from its eight key tourist source markets (Australia, Canada, China, Japan, South Korea, Germany, the United Kingdom (UK) and the United States (US)) and control macroeconomic factors together with global and regional structural changes. The empirical models reveal that the macroeconomic factors contributed at various levels to different markets, and the models we provided made accurate and reliable forecasts for visitor arrivals by air transport from all studied markets. The results from the markets for Germany, Japan, South Korea and the UK showed that significantly negative tourism demand shocks increased the volatility of tourism arrivals, more than positive tourism demand shocks of equal magnitude. Accordingly, the findings of this study will allow policy-makers in the New Zealand tourism sector and other stakeholders (e.g. airline management) to better understand the impacts on the volatility of visitor arrivals to New Zealand. International tourism is one of the key drivers for boosting New Zealand's aviation industry as well as its economy via generation of goods and services, attracting foreign currency and stimulating tourismrelated employment and investment. According to Statistics New Zealand, New Zealand's tourism sector grew from 1.16 million visitors in 1993 to 2.86 million in 2014, equalling a growth rate of 146.97% during this period. Importantly, the unique location of New Zealand (an island country) makes air transport the most convenient way to visit New Zealand. An average of 98.97% of visitor arrivals using air transport visited New Zealand during this period. As of 2014, direct tourism value added 2.3% towards New Zealand's gross domestic product (GDP) and international tourism expenditure in New Zealand was approximately NZ$9567 million (Statistics New Zealand, 2015) . Therefore, in the New Zealand context, a substantial body of literature has discussed the international tourism demand (e.g. Balli and Tsui, 2015; Becken and Gnoth, 2004; Chan et al., 2005; Chang et al., 2011a Chang et al., , 2011b Turner and Witt, 2001; Yeoman et al., 2012) . This study's contribution to policy institutions, policy-making and the academic literature emerges from an accurate estimate of the volatility of monthly international tourist arrivals to New Zealand based on a long historical dataset of tourist arrivals. Recent studies (e.g. Chang et al., 2011a Chang et al., , 2011b Coshall, 2009; Divino and McAleer, 2010; Kim and Wong, 2006; McAleer, 2005, 2007) have shown that volatility modelling has been undertaken by researchers for different countries. From strategic management and planning perspective for the tourism sector, it is important to have accurate and reliable forecasts and estimations of the volatility of visitor arrivals from New Zealand's key tourist source markets. The foremost important reason is that the patterns of volatility have significant impact on reliable forecasts. Such forecasts can create the forecast bounds (upper and lower bounds) needed to evaluate the impacts of severe shocks and risks on New Zealand's international inbound tourism demand. Indeed, the volatility forecasts and the analysis of determinants is important for New Zealand's tourism and airline industries and other stakeholders. Volatility of tourist arrivals to New Zealand experienced by these tourismdependent industries has significant implications for capital investment, resource and yield management. The forecasts might be used by stakeholders to utilise these resources more efficiently. In addition, forecasting and estimation of volatility in international tourist arrivals assists the New Zealand government, airline management, airport managers, Tourism New Zealand, tourism operators and destination managers to design and implement effective policies and/or approaches to cater to the needs of international visitors travelling to https://doi.org/10.1016/j.jairtraman.2018.10.002 Received 5 August 2017; Received in revised form 23 October 2018; Accepted 24 October 2018 New Zealand. Reliable forecasts would eventually increase the quality of the service that international tourists to New Zealand will obtain. It should be noted that the tourism sector in New Zealand has become the largest export earner in terms of foreign exchange earnings in 2016 (Tourism New Zealand, 2016) . Therefore, it is necessary for the New Zealand government and tourism-related industries to realise whether there will be any fluctuations in tourism demand from its key tourist source markets in response to any specific shocks. Well-planned tourism policies and initiatives may enhance the competitiveness of New Zealand as one of the popular tourist destinations for international tourists (particularly the booming Chinese outbound market). These aims may be achieved through the provision of higher quality tourismrelated services and products as well as facilities, and thus the level of customer satisfaction can be improved for tourists visiting New Zealand. Importantly, these initiatives will benefit the future growth of New Zealand's tourism sector and economy. Having said that, it is also important to determine the volatility of visitor arrivals to New Zealand but only a few prior tourism studies have explored the impacts of the volatility of visitor arrivals to New Zealand Chan et al., 2005; Chang et al., 2011a; 2011b) . To the best knowledge of authors, this study will be the first empirical study to utilise a combination of the Box-Jenkins seasonal (S), autoregressive (AR), integrated (I), moving average (MA), alongside the identified factors and/or interventions (x ) (SARIMAX) methodology and the (exponential) general autoregressive conditional heteroscedasticity ((E)GARCH) model to forecast visitor arrivals for a small country (tourism-oriented country) from its eight key tourist source markets and investigate how volatility affects visitor arrivals. From a theoretical perspective, there are two primary reasons that make the SARIMAX methodology and the (E)GARCH volatility model suitable for forecasting and modelling the volatility of New Zealand's international tourist arrivals in this study. First, the SARIMAX model is a times series model that has demonstrated its strengths in accurately forecasting the tourism demand of different countries in recent tourism literature (Baldigara and Mamula, 2015; Alsumair and Tsui, 2017) , and adding time series terms to a causal model is likely to improve its forecasting ability (Morley, 2009) . Second, the GARCH methodology has been applied in the tourism literature to model the volatility of tourism demand (e.g. Balli and Tsui, 2015; Chan et al., 2005; Chang et al., 2011a; 2011b; Shareef and McAleer, 2007; Song and Li, 2008) . The empirical models of this study have revealed that the macroeconomic factors contributed at various levels to different markets, and the models we provided made fairly accurate forecasts for the visitors originating from the selected markets. The volatility modelling results show that negative tourism demand shocks significantly increased the volatility of New Zealand's tourism demand, larger than positive tourism demand shocks of equal magnitude for some of the destination markets, namely Germany, Japan, South Korea and the UK. The study results pinpoint the factors that affect visitor arrivals to New Zealand and assist policy-makers (e.g. Tourism New Zealand and tourism-related operators) to realise the problems of tourism policy generation for New Zealand's tourist source markets while considering their inherent differences. The structure of this article is presented as follows. Section 2 provides a literature review of the determinants that affect New Zealand's inbound tourism demand and recent developments in tourism forecasting methodology. Section 3 presents the time series data for monthly visitor arrivals by air transport from New Zealand's eight key tourist source markets (Australia, Canada, China, Germany, Japan, South Korea, the UK and the US) and discusses the volatility of visitor arrivals by air transport, as well as presenting the data for analysis. Section 4 describes the empirical models used to forecast visitor arrivals by air transport from those eight countries. Section 5 presents the empirical findings of the study and Section 6 summarises the key findings. Two important aspects -New Zealand's inbound tourism and recent developments of tourism forecasting methodology -are examined in the following review sections, commencing with the key determinants affecting New Zealand's inbound tourism, followed by recent developments in tourism forecasting methodology. It is important to understand the determinants of New Zealand's international inbound tourism demand. These factors include the 9/11 terrorist attacks, the severe acute respiratory syndrome (SARS) outbreak (Yeoman et al., 2012) , higher aviation fuel prices (e.g. Becken, 2008; Becken et al., 2009; Becken and Lennox, 2012; Lennox, 2012; Schiff and Becken, 2011) , the Christchurch earthquakes of 2011/12 and the Rugby World Cup of 2011 (a major sport tournament) (Tsui et al., 2014; Yeoman et al., 2012) , seasonality and other specific calendarrelated holidays such as the New Year holiday period , regional and global economic crises (Yeoman et al., 2012) , air ticket prices (airfares), the availability of direct air services (Duval and Schiff, 2011; Schiff and Becken, 2011; Turner and Witt, 2001) and exchange rate fluctuations (e.g. Stavarek, 2007; Schiff and Becken, 2011; Turner and Witt, 2001; Yeoman et al., 2012) . Apart from the determinants that affect New Zealand's inbound tourism discussed above, this section reviews recent developments in tourism forecasting methodology that could be applied for forecasting New Zealand's visitor arrivals in this study. In relation to tourism demand modelling, numerous studies (e.g. Goh and Law, 2011; Lim, 1997; Peng et al., 2014; Song and Li, 2008; Witt and Witt, 1995) have offered structured reviews of the forecasting methodologies used to forecast tourism demand. In particular, Song and Li (2008) reviewed the most popular times series and econometric models (e.g. the simple naïve models, ARIMA-based methods, GARCH-based models and the vector error correction model -(VECM)) as well as a number of new quantitative and qualitative techniques (e.g. the almost ideal demand system (AIDS), artificial intelligence (AI) methods and artificial neural networks (ANN)) for tourism forecasting. They suggested that there is no single forecasting model that can consistently outperform other forecasting models in all situations. Similarly, Peng et al. (2014) also mentioned that no agreement has been reached regarding the superiority and advantages of different forecasting models. However, the ARMA-based models are still widely used to forecast tourism demand. For example, Chu (2009) used three different ARMAbased models to forecast worldwide visitors to nine countries (Australia, Hong Kong, Japan, Korea, New Zealand, Taiwan, Singapore, Thailand and the Philippines). Baldigara and Mamula (2015) also used the Box-Jenkins SARIMAX model to forecast German tourism demand in Croatia and identified the determinants affecting the number of tourist arrivals. In addition, Tsui and Balli (2015) used the Box-Jenkins SARIMAX model to forecast and predict air passenger throughput for Hong Kong International Airport. Recently, Alsumair and Tsui (2017) also used the Box-Jenkins SARIMAX models to model and forecast the international inbound tourism of Saudi Arabia. It should be noted that the Box-Jenkins SARIMAX models used in these studies are significantly accurate for forecasting tourism demands. Furthermore, the volatility models in the finance literature have been adapted and applied to estimate tourist numbers for several countries in tourism studies (e.g. Chan et al., 2005 , Chang et al., 2011a , 2011b Coshall, 2009; Hoti et al., 2007; Kim and Wong, 2006; Divino and McAleer, 2010) . Note that the GARCH methodology was developed for investigating and modelling time-varying volatility in financial time series data and stock returns (e.g. Balli et al., 2013; Bollerslev, 1986; Engle et al., 1990; Poon and Granger, 2003) . However, Chang et al. (2011b) mentioned that the analysis of tourism volatility is still a relatively new approach within tourism research and literature. Importantly, the GARCH methodology has been applied for modelling volatility because tourism demands, just like macroeconomic indicators, exhibit varying degrees of volatility and heteroskedastic behaviour Shareef and McAleer, 2007; Song and Li, 2008) . New Zealand's international inbound tourism is largely dependent on long-distance inbound markets (except for Australia) (Schiff and Becken, 2011) . Eight major tourist source markets (i.e. Australia, Canada, China, Japan, South Korea, Germany, the UK and the US) for New Zealand's tourism sector took a share of 77.08% of the total inbound tourists by air transport in 2013 (Statistics New Zealand, 2015) . The key variable of interest in this study is the monthly visitor arrivals by air transport from New Zealand's eight major tourist source markets (see Table 1 ). Fig. 1 shows the deseasonalised monthly visitor arrivals by air transport from the eight major tourist source markets between January 1993 and December 2013. The time series of the monthly visitor arrivals by air transport for these eight tourist source markets exhibit different patterns, alongside the possibility of seasonal patterns. Regarding the volume of visitor arrivals by air transport to New Zealand, Australia (1,207,952), China (228,592), the UK (189,072), the US (187,504), Japan (74,160) and Germany (68,992) were the largest six tourist source markets for New Zealand's tourism industry during 2013. Visitor arrivals by air transport from the smallest markets (South Korea and Canada) were 50,832 and 45,776 during the same period. On average, the largest share of visitor arrivals by air transport originated from Australia (37.35%), followed by the UK (10.40%), the US (8.94%), Japan (6.48%), South Korea (3.92%), China (3.83%), Germany (2.77%) and Canada (1.91%) (see Table 2 ). Considering the average monthly growth rate of the monthly visitor arrivals by air transport, China was New Zealand's most important market with approximately 1.75% of the total arrivals, followed by Australia (0.68%). The significant increase in Chinese visitor arrivals by air transport to New Zealand suggests China's increasingly significant role in aiding New Zealand's tourism development over recent years . During the analysis period, Australia was New Zealand's largest international tourism market, accounting for almost half of international visitors (approximately 37.5% of New Zealand's total international inbound visitors) (Tourism New Zealand, 2017). The positive growth rate of Australian tourists is largely caused by the close proximity of New Zealand and Australia (i.e. it is a short-haul inbound market), 1 and their close social/historical and economic ties (e.g. the Closer Economic Relationships Programme between New Zealand and Australia) (Chan et al., 2005; Schiff and Becken, 2011) . The growth rates of five other countries (Canada, Germany, South Korea, the UK and the US) presented modest increases of 0.26─0.39% throughout the study period. However, Japan showed a negative growth rate (−0.13%) in the monthly visitor arrivals by air transport to New Zealand during the analysis period (see Table 2 ). The key reasons that contributed to the declining number of visits by Japanese tourists to New Zealand were Japan's continued economic recession, the rise in strength in the New Zealand dollar against the Japanese Yen, and the aging Japanese population, alongside the preference of Japanese outbound tourists to visit short-haul (closer) and/or cheaper destinations in Asia over recent years Mak et al., 2005) . Note that only the monthly average growth rates of China (1.74%) and Australia's visitor arrivals (0.68%) by air transport were above the monthly average growth rate of New Zealand's total visitor arrivals by air transport (0.49%) 2 during the analysis period; the remaining key tourist source markets had figures below New Zealand's average growth rate (see Table 2 ). Looking at the standard deviation, the variability of the monthly visitor arrivals by air transport is dispersed across the countries; the standard deviation is between 1718 visitors (Canada) and 27,649 visitors (Australia) (see Table 2 ). All the time series are skewed to the right, with half of the distribution showing a positive excess kurtosis, denoting heavy-tailed distribution, and the other half showing negative excess kurtosis, denoting light-tailed distribution. Overall for all countries, the skewness and kurtosis are at moderate levels. Fig. 2 contains the volatility in New Zealand's tourism demand from its eight major tourist source markets. This study proxies the volatility by calculating the first differences of the monthly visitor arrivals by air transport to New Zealand from the corresponding markets. The patterns of the studied countries in Fig. 2 clearly show some seasonality patterns in the volatility. In particular, the figures for Australia, Canada, The logarithm of visitor arrivals by air transport from a tourist source market to New Zealand in month t. The logarithm of the GDP per capita of the tourist source country at month t (in US dollars). OECD National Accounts Statistics* ln(Bilateral trade volumes) t The logarithm of bilateral trade volumes between New Zealand and a tourist source market in month t. The jet fuel price per gallon in month t (in US dollars per barrel). Datastream Exchange rate t The change in the nominal exchange rate between New Zealand and the currency of a tourist source market in month t. A dummy variable that takes 1 for the period of November 2002-July 2003 and 0 otherwise. A dummy variable that takes 1 for the period of the global financial crisis from January 2008 to December 2008 and 0 otherwise. Germany, the UK and the US have similar seasonality patterns, but there are some fluctuations in the data that cannot be explained by seasonality. Fig. 2 also indicates that for the Asian tourist source markets of New Zealand (China, Japan and South Korea), it is more obvious that seasonality is not the main reason for these fluctuations in the data. Accordingly, it would be more appropriate to explore the tourism demand volatility patterns of New Zealand's eight key tourist source markets. In addition to this analysis above, the Jarque and Bera (1980) test in Table 2 also confirmed the non-normal patterns for all the time series. In addition, the significant Q(1) and Q(12) statistics for all the time series data indicated that the time series are serially correlated up to the 12th lag. Furthermore, the significant Q 2 (1) and Q 2 (12) statistics also showed the sign of strong second-moment dependencies (conditional heteroscedasticity) in the distribution of time series data. In this study, the Box-Jenkins SARIMAX methodology and the (E)GARCH volatility model were used to model and forecast the monthly visitor arrivals by air transport to New Zealand from its key tourist source markets (Australia, Canada, China, Japan, South Korea, Germany, the UK and the US). Two main reasons prompted this study to use this method: (1) a quick glance at the information in both Fig. 2 and Table 2 indicates that there are some strong non-seasonal patterns of the volatility in New Zealand's visitor arrivals by air from its major tourist source markets; (2) Columns 7-10 in Table 2 contain information about the autocorrelation of order one (Q(1)) and autocorrelation of order 12 (Q(12)) of the residual of the series from the AR(1) model along with the same order of autocorrelation of the squared residuals (Q 2 (1) & Q 2 (12)). For all the time series, the Q(1) and Q(12) statistics in Table 2 are highly significant, presenting strong evidence for the existence of autocorrelation in the logarithmic distribution of tourism demand. The Q 2 (1)) and Q 2 (12) statistics are also highly significant for tourism demand, suggesting the presence of strong second-moment dependencies (conditional heteroscedasticity) in the time series data. Again, all these significant values indicate strong evidence for autoregressive and conditional heteroscedasticity patterns within the time series data, and therefore our study uses the ARCH/GARCH models to model the volatility of visitor arrivals from New Zealand's eight key tourist source markets. However, starting with the naïve volatility model (GARCH), because of stability and convergence problems within the volatility of some tourism time series data, this study has also used a more flexible version, namely the EGARCH volatility model. The ventions (x ) t (e.g. Box and Jenkins, 1976; Gujarati and Porter, 2009; Tsui et al., 2014) . 3 In this study, the effects of interventions x ( ) t on visitor arrivals by air transport to New Zealand are incorporated. Following the approach of Tsui et al. (2014) , the SARIMAX model in this study is written as shown in Equation (1): (1) where Y t denotes the dependent variable, denotes the constant, t denotes the error tem, p ( ) represents the non-seasonal AR process of order p, P ( ) represents the seasonal AR process of order P, represents the non-seasonal MA process of order q, Q ( ) represents the seasonal MA process of order Q, and d s D represent the differencing level for both the non-seasonal and seasonal processes, respectively, and x t represents the permanent effect or the temporary effect of interventions and/or exogenous shocks incorporated in the forecasting process. For the second component of the forecasting models in this study, the GARCH volatility models were added into the forecasting process because the number of visitor arrivals by air transport to New Zealand from the eight tourist source markets is volatile and susceptible to different macroeconomic factors and tourism demand shocks (positive and negative shocks) such as the source country's GDP per capita, bilateral trade volumes, exchange rates, fuel prices, the SARS outbreak of 2003, the global financial crisis of 2008 and the Christchurch earthquakes of 2011/12 (see the description of the variables in Table 1 and the monthly average of the variables in Table 3 ). We aim to identify and examine the effects of these factors on visitor arrivals by air transport to New Zealand from the sampled countries. Note that the relevance of these factors for modelling and explaining New Zealand's international inbound tourism has been demonstrated in prior literature (see the Literature Review). Specification of the GARCH volatility model requires two equations: a 'mean equation' and a 'variance equation' (Bollerslev, 1986; Engle, 1982) . Basic GARCH modelling includes the 'mean equation', followed by modelling the conditional variance in the 'variance equation'. Following the approach of Bollerslev (1986) , the p q GARCH ( , ) volatility model is written as shown in Equation (2) where t 2 represents the time-varying conditional variance, 0 The table above reports the summary statistics for the monthly visitor arrivals by air transport from New Zealand's eight main tourist source markets. The table also presents autocorrelations of order 1 (Q(1)) and order 12 (Q(12)), and autocorrelations of the squared residuals of an AR(1) model of order 1 (Q 2 (1)) and order 4 (Q 2 (12)). The Ljung and Box, 1978 test statistics are significant at * p < 0.10, **p < 0.05 and ***p < 0.01. represents the mean of the volatility equation, i and j are the coefficients of the ARCH and GARCH effects, e t is the error term and t is the time. Equation (2) also allows t 2 to be based on past information. Prior studies (Bollerslev, 1986; Coshall, 2009 ) claimed that previous shortrun shocks are represented by the lag of the squared residuals (e t 2 ) obtained from the 'mean equation', and also by the previous long-term conditional variance t 2 , which can be obtained from the 'variance equation' in the GARCH volatility model. The generic GARCH (1,1) volatility model is often considered to be suitable for capturing the volatility in tourism time series data (Chen and Lian, 2005) . However, if the two stationary limitations cannot be satisfied (i.e. all of the ARCH and GARCH parameters are larger than zero, and the total of the ARCH and GARCH parameters should be less than or equal to 1) (Divino and McAleer, 2010; Kim and Wong, 2006) , the EGARCH volatility model can become another option for modelling the volatility of visitor arrivals by air transport to New Zealand from the eight tourist source markets, as it has no restrictions on the parameters of t 2 > 0, and it focuses on the positive and negative shocks asymmetrically (e.g. Bowden and Payne, 2008; Coshall, 2009; Divino and McAleer, 2010; Longmore and Robinson, 2004; McAleer et al., 2007; Nelson, 1991) . Following the approach of Coshall (2009), the p q EGARCH ( , ) volatility model is written as shown in Equation (3) i is the sign effect, indicating the presence of asymmetric effects in response to a shock (positive or negative tourism demand shocks) when i ≠ 0; if < 0 i , this indicates the presence of a leverage effect. The parameter for log t j 2 , , j indicates the degree of volatility persistence (Bowden and Payne, 2008; Nelson, 1991) . The EGARCH (1,1) volatility model is considered to fit well for the tourism time series dataset in this study. This section provides the empirical results of forecasting the monthly visitor arrivals by air transport for New Zealand's eight key tourist source markets. As mentioned in Section 4, the SARIMAX/(E)GARCH model has two components: the Box-Jenkins SARIMAX model and the (E)GARCH volatility model. In using the SARIMAX/(E)GARCH volatility model for modelling the monthly visitor arrivals by air transport from New Zealand's eight key tourist source markets, all of the time series for visitor arrivals by air transport need to be stationary. Fig. 2 shows that visitor arrivals by air transport from all tourist source markets are relatively volatile. Table 4 also shows the estimation results of the Augmented Dickey-Fuller (ADF) unit root tests and indicates that all the time series of visitor arrivals by air transport are non-stationary, except for South Korea. After we applied first-order differencing, all of the time series became stationary at above the 0.05 significance level. Furthermore, the estimation results of the Hylleberg, Engle, Granger and Yoo (HEGY) seasonal unit root tests are shown in Table 5 , which also verified that all of the time series are seasonally stationary. The SARIMAX models for New Zealand's key tourist source markets were determined by the lowest Akaike information criterion (AIC) and the lowest Schwarz information criterion (SIC) values. In addition, the autocorrelation function (ACF) and partial autocorrelation function (PACF) diagnostic correlograms as well as the Ljung-Box Q-statistics proved that the residual series of all of the time series in the SARIMAX model have 'white noise' characteristics. Table 6 shows the estimation results of the SARIMAX/(E)GARCH volatility models for New Zealand's eight key tourist source markets. All the volatility models provided accurate and reliable forecasting performance with lower mean absolute percentage error (MAPE), mean absolute error (MAE) and root mean square error (RMSE) values. In particular, the MAPE values for all countries are below 3.61%, and thus the forecasting performance accuracy of the SARIMAX/(E)GARCH volatility models is considered to be highly accurate according to the criteria set by findings of Lewis (1982) . A primary interest of this study is to establish accurate forecasting models for modelling and forecasting the monthly visitor arrivals by air transport from New Zealand's eight key tourist source markets. The adjusted-R 2 (Adj-R 2 ) values are high for most tourist source markets, indicating a good fit; however, the lower Adj-R 2 value for China might suggest that other factors are driving and affecting Chinese visitor arrivals by air transport to New Zealand. Section 4 presents the mean equation that includes the selected bestfit Box-Jenkins SARIMA models and other identified explanatory variables (i.e. the economic variables and exogenous shocks affecting New Zealand's international inbound tourism demand) (see Equation (1)). All the variables in the models were transformed to logarithmic form and therefore the coefficients of the explanatory variables can be interpreted as elasticities. The impacts of the economic variables and exogenous shocks on the monthly growth of visitor arrivals by air transport to New Zealand from its eight key tourist source markets are shown in the mean equation (Table 6) . By modelling the monthly visitor arrivals for each tourist source market, we either captured the current level of GDP growth or the previous months' GDP growth level (the lagged level). With the motivation of accessing the best forecasting model, we have chosen the best fitting one that produced the least error. For Canada and Germany only, we have included the current level of GDP growth because the forecasting models produced much better results than otherwise. For the rest of the countries, we included the previous months' GDP growth level (the lagged level) with the motivation that increased GDP might create more opportunity to spend on travelling in the upcoming period. The findings for the growth in 'GDP per capita' variable presented mixed results: only Balli et al. Journal of Air Transport Management 75 (2019) 204-214 the growth of GDP per capita 1 month before for South Korea was found to have a positive significant impact on increasing South Korean visitor arrivals to New Zealand, but this was not found for other key tourist source markets. This empirical finding indicate that the growth of South Korean GDP per capita increased the volatility of the monthly visitor arrivals by air transport to New Zealand. This finding is consistent with the Hakim and Merkert (2016) who stated that there is a directional causality from the GDP to air passengers and air freight volumes. Indeed, the GDP growth in a domestic economy would enhance the international trade relationships so do the volume of airfreight and business travels increase. This directional relationship developed by Hakim and Merkert (2016) is observed only for the tourism flows from South Korea to New Zealand. Indeed, the stable economic growth of South Korea with its worldwide brands including Kia, Samsung, LG, Hyundai, might have brought significant level of international business partnerships to the rest of the world. Accordingly, business travels from South Korea to the rest of the world (including New Zealand) has increased as well. However, Australia's insignificant positive coefficient of the growth in 'GDP per capita' (i.e. one of the key economic variables affecting tourism demand) may stem from other factors beyond the macroeconomic determinants affecting Australia's monthly visitor arrivals to New Zealand such as the strong cultural connection, closer proximity and flying distances across the Tasman Sea, and the provision of budget travel by low-cost carriers (e.g. Virgin and Jetstar). Australia always contributes the largest share of New Zealand's visitor arrivals because of their strong shared ties (e.g. economy, culture and family), which stimulates the growth in visitor arrivals from Australia (Ryan and Birks, 2006; Tsui and Balli, 2015) . The significant positive coefficients of the increasing level of 'bilateral trade volumes' or the bilateral business activity of 1 month before between five countries (Australia, Canada, Germany, Japan and the UK) and New Zealand indicated that visitor arrivals by air transport from these countries are sensitive to changes in bilateral trade volumes, or that bilateral trade volumes is an important factor stimulating and increasing visitor arrivals from these countries to New Zealand. In particular, this empirical finding is in line with the notion of causality between business travel and bilateral trade volumes that was suggested by Tsui and Fung (2016) and in a more broad manner by Baker et al. (2015) . 4 Note that the expected significant impact of bilateral trade volumes on visitor arrivals to New Zealand was not found for the other countries in this study. In addition, the expected negative impact of exchange rate on visitor arrivals by air transport to New Zealand (i.e. New Zealand's tourism demand) was not found to be significant in this study, and the empirical results of the change in the 'exchange rate' variable were not consistent with general acceptance and prior literature (e.g. Duval and Schiff, 2011; Dwyer et al., 2002; Schiff and Becken, 2011; Yeoman et al., 2012) . Only three countries (Australia, South Korea and the US) presented negative parameters for the change in the 'exchange rate' variable. The possible interpretation of this insignificant exchange rate factor would be that these countries are listed as the top tourist source markets of New Zealand and are relatively richer. The higher income factors and the inelastic tourism demand of New Zealand would overweigh some other factors that might be important for explaining New Zealand's inbound tourism demand. The impact of the change in 'fuel prices' variable (a commonly used proxy for airfares) on the growth of visitor arrivals by air transport from New Zealand's eight key tourist source markets is largely consistent with prior literature (e.g. Becken, 2008; Becken et al., 2009; Becken and Lennox, 2012; Lennox, 2012) , showing negative coefficients for five countries (Australia, China, Japan, South Korea and the US); more importantly, the largest tourist source market (Australia) and the fastest growing market (China) were sensitive to changes in fuel prices; more specifically, visitor arrivals by air transport from Australia and China were negatively affected by the changes in fuel prices. Furthermore, only China was reported to have a negative parameter for the 'SARS outbreak 2003' variable out of all of the tourist source markets. This empirical finding is in line with the fact that the SARS outbreak in 2003 originated from China but its significant negative impact on China's international outbound tourism to New Zealand was not found in the data. Interestingly, a significant positive 'SARS outbreak 2003' variable was also reported for Australia; logically, the number of Australian tourists visiting New Zealand should not be seriously and adversely affected by this epidemic happening in China and Hong Kong because of Australia and New Zealand's strong ties and proximity (see Fig. 1 ). The 'global financial crisis 2008' variable had the expected significant adverse impact on the growth of visitor arrivals by air transport from four key trading partners (China, Japan, the UK and the US) that have strong economic relationships and conduct the largest business 28.321*** 11.085*** 29.372*** 10.573*** 28.784*** 179.358*** 22.669*** 11.553*** Remarks: All of the time series above are stated in natural logarithm. The regression contains the constant, trend and seasonal dummies for the HEGY seasonal unit root tests. *, ** and *** indicate that the explanatory variable is significant at the 0.10, 0.05 and 0.01 significance level, respectively. i is the coefficient of the monthly seasonal time series. Remarks: The dependent variable is ln(visitor arrivals by air transport). The explanatory variables transformed into logarithmic form should be interpreted as growth in the variable. *, ** and *** indicate that the explanatory variable is significant at the 0.10, 0.05 and 0.01 significance level, respectively. t-statistics are printed in parentheses. The AR and MA terms included in the SARIMAX model aim to capture the autoregressive and moving average relationships in the tourism time series data. † indicates that the coefficients of the trend and fuel variables were multiplied by 100 for ease of presentation. Constant was included in regression analysis but has not been reported for space limitations. transactions and activities with New Zealand, but Australia, Canada and Germany still reported statistically insignificant and negative coefficients. Finally, the 'Christchurch earthquakes 2011/12' variable was found to have a significant negative impact, reducing the growth of visitor arrivals by air transport from two countries (Canada and Germany), but also with statistically insignificant coefficients with Australia, Japan, the UK and the US. Section 4 presents two different variance equations: Equation (2) (GARCH volatility model) and Equation (3) (EGARCH volatility model). In terms of the variance equation (see Table 6 ), the residuals of all of the (E)/GARCH volatility models are free of the problem of serial correlation and the ARCH effects, and are normally distributed, as confirmed by the Ljung-Box Q-statistics, the ARCH Lagrange multiplier test and the Jarque-Bera (normality) test. Only visitor arrivals by air transport from the US were modelled with the GARCH (1,1) volatility model; the rest of the tourist source markets in this study were modelled with the EGARCH volatility models. According to Kim and Wong (2006) , the summation of two coefficients, the ARCH effect ( ) i and the GARCH effect ( ), j for the GARCH (1,1) volatility model for the US should be less than 1, indicating the stability condition has been satisfied. This empirical finding for the US shows that the volatility of tourism demand from the US has statistically significant parameters of i and i but the sum of these two coefficients is not close to 1, thus indicating the non-persistent volatility of US tourist arrivals. For the other seven countries (Australia, Canada, China, Germany, Japan, South Korea and the UK), the EGARCH(1,1) volatility models were fitted for modelling visitor arrivals by air transport to New Zealand because using the GARCH(1,1) volatility model did not produce stationary results. Four countries (China, Japan, South Korea and the UK) were reported to have statistically significant and positive size effect parameters ( ) i , indicating the significant positive impact of volatility on their respective visitor arrivals by air transport to New Zealand. The parameter values of i are between 0.308 (the UK) and 0.838 (China). In addition, a negative and statistically significant sign effect ( i ) was reported for four countries (Germany, Japan, South Korea and the UK), but the other three countries (Australia, Canada and China) were still reported to have insignificant negative coefficients for the sign effect. These empirical findings show that visitor arrivals by air transport from Germany, Japan, South Korea and the UK to New Zealand were affected by an asymmetric effect alongside the leverage effect, as the coefficients met the conditions of i ≠ 0 and i < 0, which suggest that negative tourism demand shocks increased volatility compared with positive tourism demand shocks of equal magnitudethe negative leverage effect (Kim and Wong, 2006) . More importantly, these empirical findings show evidence of asymmetry and are consistent with the study of Coshall (2009) , which indicated that the asymmetrical impact on visitor arrivals to a country is apparently caused by the susceptibility of tourism demand to exogenous shocks (i.e. tourism demand shocks). For the degree of volatility persistence ( j ), Australia and Canada were reported to have significant negative parameters and four counties (China, Germany, South Korea and the UK) were reported to have significant positive parameters. The parameter values are: −0.988 (Australia), −0.688 (Canada), 0.293 (China), 0.484 (Germany), 0.736 (South Korea) and 0.922 (the UK). This study used the SARIMAX/(E)GARCH volatility models to forecast New Zealand's visitor arrivals by air transport from its eight key tourist source markets between January 1993 and December 2013. The empirical results suggest that all the forecasting models provide accurate and reliable forecasting results with lower MAPE, MAE and RMSE values. The seven tourism-related variables (the economic variables and exogenous shocks affecting New Zealand's international inbound tourism demand) incorporated in the SARIMAX/(E)GARCH volatility models presented mixed results for explaining visitor arrivals by air transport from the studied tourist source markets but involved the expected parameter signs. In particular, the growth of South Korean's GDP per capita 1 month before had a significant impact on South Korean visitors by air transport to New Zealand. The growth in bilateral trade volumes between five countries (Australia, Canada, Germany, Japan and the UK) and New Zealand had a significant positive impact on their respective visitor arrivals to New Zealand. However, the expected negative effect of exchange rate on visitor arrivals by air transport to New Zealand (i.e. New Zealand's tourism demand) was not found in this study. The negative impact of fuel prices on New Zealand's biggest tourist source market (Australia) and its fastest growing market (China) was found in this study. Three exogenous shocks (the SARS outbreak of 2003, the global financial crisis of 2008 and the Christchurch earthquakes of 2011/12) had different negative effects on visitor arrivals by air transport among the studied markets, but they were largely consistent with generally accepted paradigms and had significant negative parameters. In addition, the volatility models presented evidence that visitor arrivals by air transport to New Zealand from its eight key tourist source markets were affected by positive and negative tourism demand shocks. A significant negative leverage effect was found for Germany, Japan, South Korea and the UK, suggesting the effects of negative tourism demand shocks on their respective visitor arrivals by air transport to New Zealand are more volatile than the impacts of positive tourism demand shocks of equal degree. The key empirical findings of this study have several significant implications for strategic decision-making and policies by policy-makers (the New Zealand government, airline management, airport managers, Tourism New Zealand and tourism-related operators) such as the development of the best strategies and approaches based on accurate and reliable forecasting and estimation of the volatility of visitor arrivals to New Zealand from its key tourist source markets. In addition, the empirical findings can provide useful insights to policy-makers in New Zealand to understand the volatility impacts (the positive tourism demand shocks and/or the negative tourism demand shocks) on visitor arrivals to New Zealand and their important implications for tourism policy and tourism revenues. Particularly, the export revenues and foreign exchange flows are important items in the balance of payment balance sheets of the central banks (same as the Reserve Bank of New Zealand -RBNZ). Accordingly, the RBNZ would investigate the reasons for the volatility of tourist arrivals to New Zealand (same as for tourism revenues) to make reliable balance of payment forecasts. Due to the influence of the volatility of visitor arrivals to New Zealand's tourism-dependent industries, this signals the need for further research regarding the forecasting accuracy of the volatility of tourist arrivals. In fact, this critical issue in tourism demand has previously been voiced by the tourism literature (Coshall, 2009) , and it should be noted that prior studies in the field of finance literature (Bollerslev and Wright, 2001; Gospodinov et al., 2006; Yu, 2002) have suggested techniques for forecasting the volatility of a particular market. Therefore, from a strategic planning and decision-making perspective, further research into New Zealand's tourism demand should include thorough forecasting of the volatility of tourist source markets; as a result, this will assist New Zealand's policy-makers to better target the growing tourist markets with more cost-effective tourism marketing strategies. Another direction of further research would be modelling the volatility of tourism revenues and observe if there will be big deviations with the tourism demand volatilities in New Zealand. Considering that many stakeholders consider the tourism demand would be the main driving factor for the volatility of tourism revenues, and it would be important and interesting to observe and identify the global/countryspecific factors that would affect the tourism revenue volatilities and analyse the deviations between these two indicators in New Zealand and other parts of the world. 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