## Tuesday, September 6, 2022

### NARDL in EViews 13: A Study of Bosnia's Tourism Sector

EViews 13 introduces several new features to extend the analysis of the well-known autoregressive distributed lag (ARDL) model (see our 3-part ARDL blog series: Part I, Part II, and Part III). In particular, estimation of ARDL models now accommodates asymmetric distributed lag (DL) regressors which extend traditional ARDL models to the increasingly popular nonlinear ARDL (NARDL) models. The latter allow for more complex dynamics which focus on modeling asymmetries both among the cointegrating (long-run) as well as the dynamic adjustment (short-run) relationships. To demonstrate these features, we will examine whether tourist arrivals and their length of stay (popular measures of tourism sector development) have asymmetric effects on the overall economic development (measured as gross domestic product (GDP)) in Bosnia and Herzegovina.

### Introduction

Tourism is a crucial source of revenue for many economies. In case of Bosnia and Herzegovina (BiH), despite having incredible historical and natural touristic appeal, tourism was an inconsequential contributor to GDP growth in the period preceding Bosnia's horrific period of aggression in the early 1990s. Bosnia's economy at that time was highly reliant on natural resource exploitation (particularly metal ores and forestry), hydroelectric production, and manufacturing. While Bosnia managed to resurrect some of these industries following the end of the aggression in 1996, it also struggled to reinvigorate several highly prospective ones, such as industrial production. In their stead, tourism has been evolving into an increasingly more significant contributor to Bosnia's GDP development.

Much of the expansion in foreign tourist interest in Bosnia stems from its active marketing campaigns and encouragement from the European Union (see the EU's 2007/145-210 Project). Bosnia's capital, Sarajevo, having hosted the 1984 winter olympics, is also home to the widely popular Sarajevo Film Festival, whereas the UNESCO protected city of Mostar in the south, hosts the Red Bull Cliff Diving World Series and is often on top lists of the most most beautiful cities to visit in Europe. Moreover, as a silver lining to its bloody history at the onset of the 90s, the country today boasts a diaspora of 2.2 million individuals (or 55% of its pre-aggression population), who live and work in major economies around the globe. These factors, in combination with the country's relatively low cost of living, makes Bosnia and Herzegovina a particularly appealing destination for tourists hailing from relatively larger economies.

On the other hand, Bosnia's domestic tourism sector, while lagging behind its foreign counterpart, witnessed a similar revival. As the country benefits from both summer and winter destinations, beach activities in costal cities along the Adriatic Sea and winter skiing activities in the mid-west, both present opportunities for locals to enjoy their homeland year round. Domestic tourism is further stimulated by the nearly 200 local hiking and alpine clubs which are also slowly using these platforms to offer foreign tourists multi-day guided tours of the region. Finally, it's worth noting that Bosnia had a significant middle-age domicile generation which remained in the country during the aggression period. Economic opportunities in the recovery years for this demographic segment were greatly stunted and today many of them cannot afford international travel; instead confining themselves to domestic excursions.

There are, in fact, numerous articles which demonstrate that tourism has a positive effect on economic growth. The general idea driving this body of research is that tourism is a positive externality which stimulates infrastructural development, foreign direct investments, and the fabric of modern internet and mobile connectivity. Our objective here is to study the magnitude and asymmetries in Bosnia's tourism sector may exert on its overall economic development.

Traditionally, these dynamics can be explored through vector error-correction (VEC) models. Nevertheless, this class of models generally assumes that all system variables are integrated of at least order 1, and does not preclude the possibility of multiple equally possible cointegrating relationships. Alternatively, classical ARDL models (see our blog series Part I, Part II, and Part III) allow for varying degrees of integration among the system variables (provided the maximum order of integration is less than 2), and assume a single unique cointegrating relationship among the variables of interest. Nevertheless, this framework assumes that the long-run (cointegrating) relationships is a symmetric linear combination of regressors.

While the classical ARDL framework is perfectly reasonable for many applications, it cannot, however, accommodate behavioral finance and economics research on nonlinearity and asymmetry, which are also often encountered in practice; see the seminal contributions by Kahneman and Tversky (1979) and Shiller (2005). To address this limitation, Shin, Yu, and Greenwood-Nimmo (2014) propose a nonlinear ARDL (NARDL) framework in which short-run and long-run non-linearities are modeled as positive and negative partial sum decompositions of the distributed lag variables. Recall that any given variable $z_t$ may be decomposed as $z_t = z_0 + z_t^{+} + z_t^{-}$ where $z_0$ is the initial value and $z_t^{+}$ and $z_t^{-}$ are the partial sum processes of positive and negative changes in $z_t$, respectively: \begin{align*} z_t^{+} &= \sum_{s = 1}^{t} \max \left( \Delta z_s, 0 \right)\\ z_t^{-} &= \sum_{s = 1}^{t} \min \left( \Delta z_s, 0 \right) \end{align*} Note that when $z_t$ is a distributed lag variable and $z_t^{+} \neq z_t^{-}$, the distributed lag variable exhibits asymmetric effects where positive changes have a different impact on the dependent variable than their negative counterparts. On the other hand, when $z_t^{+} = z_t^{-}$, the distributed lag variable exhibits symmetric effects on the dependent variable and reduces to the classical ARDL effect.

The NARDL framework also provides asymmetric dynamic multipliers. These constructs, which are similar to impulse-response curves in the VAR literature, trace asymmetric paths of adjustment of each nonlinear distributed lag regressor to its long-run (cointegrating) state.

Below, we will apply the NARDL framework to identify the long-run (cointegrating) and short-run (adjusting) dynamics which relate Bosnia's tourism sector to its state of the economy.

### Data and Motivation

To conduct the analysis, we will collect data directly from the Agency for Statistics of Bosnia and Herzegovina (ASBH). In particular, we are interested in 5 different time series:
• GDP: gross domestic product
• FTA: foreign tourist arrivals.
• DTA: domestic tourist arrivals
• FTS: foreign tourist length of stay
• DTS: domestic tourist length of stay
Note that ASBH defines a foreign tourist as any "person with permanent residence outside of BiH who temporarily resides in BiH and who spends at least one night in a hotel or [similar] accommodation establishment." Similarly, it defines a domestic tourist as any "person with permanent residence inside BiH who spends at least one night in a hotel or [similar] accommodation establishment outside their place or residence." It also defines tourist arrivals as "the number of persons (tourists) who arrived and registered their stay in an accommodation establishment", and tourist length of stay as the number of "registered overnight stays of a person (tourist) in an accommodation establishment."

It's also important to mention a few caveats regarding our data. First, Bosnia's GDP, measured using the expenditure approach using previous year prices, is collected quarterly. Furthermore, while GDP data does exists from Q1 2000 to Q4 2021, ASBH has labeled data after 2021 as forecasted and not actual. As a precaution, we've decided to ignore this data altogether and shorten the GDP series to cover the last actual measurements made in Q4 of 2020.

In contrast, tourism sector variables are collected monthly and date back to January 2008. This clearly presents a challenge in terms of using all variables in the same framework simultaneously. While methods such as MIDAS do exist to address these issues, we've opted to handle the problem using manual frequency conversion using the official EViews frequency conversion tutorial. In particular, to benefit from the longer time series available among tourism variables, we've decided to convert GDP from its low quarterly frequency, into the higher monthly frequency, using the Denton method. As a convenience, we have made available a pre-processed version of this data as an EViews workfile, which may be downloaded from here.

Before engaging in any advanced analysis, it's also encouraged to understand the data we're dealing with. This will not only give us some idea about the state of Bosnia's tourism sector, but also help us identify meaningful patterns we may try to exploit later.

While Bosnia's foreign and domestic tourist arrivals have both seen considerable activity, over the reporting period between 2008 and 2022, foreign tourism seems to have undergone a nearly exponential transformation. The stark contrast is illustrated in Figure 1a below. In fact (see Figure 1b), in the period preceding the COVID-19 pandemic years, the average year-on-year percent change of annual tourist arrivals was 13% and 4.12% for foreign and domestic sectors, respectively. In 2015, the year-on-year growth in tourist arrivals hovered around 26.53% among foreigners, and 13.1% among locals.

 Figure 1a: Domestic vs. Foreign Annual Tourist Arrivals Figure 1b: Domestic vs. Foreign YoY Percent Change in Tourist Arrivals

Comparing Bosnia's foreign and domestic tourism sectors, the summaries above suggest that foreign tourist arrivals have significantly more clout than their domestic counterparts. We can gain further insight by looking at the monthly time series in Figure 2.

 Figure 2: Domestic vs. Foreign Tourist Arrivals Time Series

What stands out in Figure 2 is that both domestic and foreign tourist arrivals exhibit predictable seasonal effects.They both reach troughs early in the year, and reach crests at the start and end of the tourism peak season, respectively. More importantly the magnitude of arrivals in peak tourism months among foreign tourists simply towers over its domestic equivalent. For further context, Figure 3 and Table 1 illustrate the distribution of Bosnia's annual tourist arrivals from abroad.

 Figure 3: Annual Total Foreign Tourist Arrivals

 Table 1: Annual Total Foreign Tourist Arrivals

As tourism is often tightly tied to discretionary time, for much of the working population, this resource will remain steady across a good number of years. We therefore expect behavioural patterns driving length of visits to Bosnia to be little changed across our reporting period. In particular, in the period preceding the COVID-19 pandemic, the average (median) length of stay was 2.15 (2.15) and 2.13 (2.12) days for domestic and foreign tourists, respectively, whereas standard deviations hovered around 0.32 and 0.21, respectively. This is confirmed in Figure 4a below. In Figure 5b, we seek to understand the seasonality of domestic and foreign tourist stays. As is the case with tourist arrivals, these curves also exhibit seasonality. Nevertheless, there is an interesting pattern which indicates that while domestic tourists stay longer in the summer months as opposed to winter months, the opposite is true for foreign tourists. In fact, in the years before 2020, the mean (median) length of stay in quarter 1 was 1.98 (1.95) and 2.33 (2.35) days for domestic and foreign tourists, respectively. On the other hand, in the same period, the mean (median) length of stay in quarter 3 for domestic and foreign tourists was 2.56 (2.52) and 2.20 (2.19) days, respectively.

 Figure 4a: Domestic vs. Foreign Average Annual Length of Stay per Tourist Figure 4b: Domestic vs. Foreign Tourist Length of Stay Time Series

In case of foreign tourists exclusively, Figure 5 and Table 2 summarize the distribution of annual length of stays per tourist, across various countries.

 Figure 5: Average Annual Foreign Tourist Length of Stay

 Table 2: Average Annual Foreign Tourist Length of Stay

Next, let's glance at Bosnia's GDP. In Figure 6a below, we can identify trending growth with seasonal effects which closely mimic those of foreign tourist arrivals; namely, GDP ebbs in January and peaks in July. For further insight, Figure 6b plots the standardized monthly GDP and both domestic and foreign tourist arrivals. What this plot illustrates is that tourist arrivals fluctuate about their mean values significantly more than GDP. Furthermore, whereas domestic tourism seems to dominate these fluctuations at the start of the reported sample, deviations in foreign tourism seem to dominate the end of the reported sample. More importantly, we see a strong positive correlation between GDP and foreign tourist arrivals.

 Figure 6a: Monthly GDP Figure 6b: Standardized Monthly GDP and Tourist Arrivals

Before proceeding with estimation, it's also prudent to study the orders of integration of our series. As mentioned in the Introduction, ARDL estimation is not valid in the presence of I(2) variables, but does accommodate a mixture of I(0) and I(1) variables. To identify integration orders, it's easiest to create an EViews group with all the variables and perform a unit root test by clicking on View/Unit Root Tests/Cross-Sectionally Independent. The first test will perform the Im, Pesaran, and Shin (IPS) test with a constant and trend on series in levels. As the null hypothesis of this test is a unit root, tests which have $p$-values near zero will reject the null and identify the series as I(0). Otherwise tests with $p$-values significantly far away from zero will fail to reject the null and identify the series as I(1). By extension, I(2) series can be identified by repeating the tests on the first difference of the series. Figures 7a and 7b summarize these tests.

 Figure 7a: Unit Root Tests in Levels Figure 7b: Unit Root Tests in First Differences

The unit root tests in levels identify domestic tourist length of stays and the GDP as integrated of order 1, whereas the remaining variables are identified as integrated of order 0. On the other hand, every null hypothesis for the unit root tests in first differences is rejected, and therefore no variable is identified as I(2).

### Estimation / Asymmetry

Our objective in this section is to estimate a NARDL model linking the relationship of Bosnia's tourist variables DTA, DTS, FTA, FTS to its sate of the economy, measured as the variable GDP. Formally, the NARDL model - expressed below in its conditional error correction (CEC) form (see the EViews manual for details) - we are interested in studying is: \begin{align*} \class{bold} { \Delta \ln(GDP) } & \class{bold} { = } \class{bold col_red}{ \phi_{\scriptsize \text{GDP}} \ln(\text{GDP})_{t - 1} }\\ &\class{bold col_red}{ + \phi_{\scriptsize{\text{DTA}}}^{+} \ln(\text{DTA})_{t - 1}^{+} + \phi_{\scriptsize{\text{DTA}}}^{-} \ln(\text{DTA})_{t - 1}^{-} + \phi_{\scriptsize{\text{FTA}}}^{+} \ln(\text{FTA})_{t - 1}^{+} + \phi_{\scriptsize{\text{FTA}}}^{-} \ln(\text{FTA})_{t - 1}^{-} }\\ &\class{bold col_red}{ + \phi_{\scriptsize{\text{DTS}}}^{+} \text{DTS}_{t - 1}^{+} + \phi_{\scriptsize{\text{DTS}}}^{-} \text{DTS}_{t - 1}^{-} + \phi_{\scriptsize{\text{FTS}}}^{+} \text{FTS}_{t - 1}^{+} + \phi_{\scriptsize{\text{FTS}}}^{-} \text{FTS}_{t - 1}^{-} }\\ &\class{bold col_blue}{ + \sum_{j = 1}^{p - 1} \gamma_{\scriptsize{\text{GDP}} \normalsize{, \, j}} \Delta \ln(\text{GDP})_{t - j} }\\ &\class{bold col_blue}{ + \sum_{k_1 = 1}^{q_1 - 1} \left( \gamma_{\scriptsize{\text{DTA}} \normalsize{, \, k_1}}^{+} \Delta \ln(\text{DTA})_{t - k_1}^{+} + \gamma_{\scriptsize{\text{DTA}} \normalsize{, \, k_1}}^{-} \Delta \ln(\text{DTA})_{t - k_1}^{-} \right) + \sum_{k_2 = 1}^{q_2 - 1} \left( \gamma_{\scriptsize{\text{FTA}} \normalsize{, \, k_2}}^{+} \Delta \ln(\text{FTA})_{t - k_2}^{+} + \gamma_{\scriptsize{\text{FTA}} \normalsize{, \, k_2}}^{-} \Delta \ln(\text{FTA})_{t - k_2}^{-} \right) }\\ &\class{bold col_blue}{ + \sum_{k_3 = 1}^{q_3 - 1} \left( \gamma_{\scriptsize{\text{DTS}} \normalsize{, \, k_3}}^{+} \Delta \ln(\text{DTS})_{t - k_3}^{+} + \gamma_{\scriptsize{\text{DTS}} \normalsize{, \, k_3}}^{-} \Delta \ln(\text{DTS})_{t - k_3}^{-} \right) + \sum_{k_4 = 1}^{q_4 - 1} \left( \gamma_{\scriptsize{\text{FTS}} \normalsize{, \, k_4}}^{+} \Delta \ln(\text{FTS})_{t - k_4}^{+} + \gamma_{\scriptsize{\text{FTS}} \normalsize{, \, k_4}}^{-} \Delta \ln(\text{FTS})_{t - k_4}^{-} \right) }\\ &\class{bold col_green}{ + \alpha_0 + \alpha_1 t + \sum_{i = 1}^{11} \delta_{i} m_i + \epsilon_t } \end{align*} This describes a NARDL$(p, q_1, q_2, q_3, q_4)$ model where GDP enters as an autoregressive process of order $p$, and DTA, FTA, DTS, FTS enter as asymmetric distributed lag variables with orders $q_1, q_2, q_3, q_4$, respectively. For easier identification, we have also coloured some portions of the CEC relation as red, blue, and green. The latter characterize the cointegrating (levels or long-run), the adjusting (differences or short-run), and the deterministic (seasonality) dynamics, respectively. Furthermore, variables with + and - superscripts denote, respectively, the positive and negative partial sum decompositions of the underlying distributed lag variable. The positive and negative partial sums here explicitly model how asymmetries in Bosnia's tourism sector, both in the long- and short- run, reflect on its economic development.

The deterministic dynamics deserve a brief comment as well. In particular, $\alpha_0$ and $\alpha_1$ respectively capture the effect of the constant and linear trend. To also capture monthly seasonality, the coefficients $\delta_{i}$ are associated with seasonal dummy variables $m_i$ for month $i$.

We will start the analysis by estimating the model above, treating all tourism variables as asymmetric in both the adjusting and cointegrating dynamics. To identify the autoregressive and distributed lag orders $p, q_1, q_2, q_3, q_4$, we will perform automatic lag selection, allowing at most 3 lags for the dependent variable and each of the regressors (the default options). This effectively states that all variables depend on values at most 3 periods (months) in the past; in other words, a single quarter.

The variables we need are located in the workfile page tourism_monthly. To avoid the complications of the pandemic years, we will only use data in the years prior to 2020. We can do so in the Command window by typing
smpl if @year<2020
Next, bring up the NARDL dialog (see Figure 8a) and enter the specifications as follows:
1. From the main EViews menu, click on Quick/Estimate Equation...
2. Change the Method dropdown to ARDL - Auto-regressive Distributed Lag Models (including NARDL)
3. Under Linear dynamic specification specify @log(gdp)
4. Under Long and short-run asymmetry specify @log(dta) @log(fta) dts fts
5. Under Fixed regressors specify @expand(@month, @droplast)
6. Set the Trend specification to Constant
7. Set both of the Max. lags dropdowns to 3
8. Click on OK
 Figure 8a: Full Asymmetry NARDL Dialog Figure 8b: Full Asymmetry NARDL(2,1,3,1,0) Output

Figure 8b summarizes the estimation output. The table header lists a number of important estimation parameters, the most important of which is the set of optimally selected lag order;namely $2, 1, 3, 1, 0$. In other words, $\class{wfvar} { \text{@log(gdp)} }$ enters the optimal model with lag 2; $\class{wfvar} { \text{@log(dta)}^{+} }$ and $\class{wfvar} { \text{@log(dta)}^{-} }$ each enter with lag 1; $\class{wfvar} { \text{@log(fta)}^{+} }$ and $\class{wfvar} { \text{@log(fta)}^{-} }$ each enter with lag 3; $\class{wfvar} { \text{dts}^{+} }$ and $\class{wfvar} { \text{dts}^{-} }$ each enter with lag 1; and $\class{wfvar} { \text{fts}^{+} }$ and $\class{wfvar} { \text{fts}^{-} }$ each enter with lag 0. Recall that the optimal lag order is selected by identifying the model (among the $768 = 3 \times (3 + 1)^4$ estimated) which achieves the optimal information criterion - in this case the minimal value of the Akaike Information Criterion (AIC). We can also visualize (see Figure 9) the lag selection criteria by clicking on View/Model Selection Summary/Criteria Table.

 Figure 9: Full Asymmetry NARDL(2,1,3,1,0) Lag Selection Criteria

Returning to the main estimation table, just below the header, the first 9 coefficients characterize the cointegrating (long-run) dynamics (the coefficients in red); the next 11 coefficients characterize the adjusting (short-run) dynamics (the coefficients in blue); the remaining coefficients characterize the linear trend and seasonal dynamics (the coefficients in green). The table footer rounds off the output with a number of estimation summary statistics.

Following estimation, the aim of the first inferential exercise is to formally validate assumptions on asymmetry. Although we have written the model above to reflect that all distributed lag variables are asymmetric both among adjusting and cointegrating dynamics, the NARDL model is flexible enough to accommodate partially asymmetry. This manifests when variables enter asymmetrically either the adjusting or cointegrating dynamics, but not both. For instance, consider an arbitrary variable $z_t$ with asymmetric decompositions $z_t^{-}$ and $z_t^{+}$ and associated asymmetric level coefficients $\phi^{-}$ and $\phi^{+}$, and associated asymmetric difference coefficients $\gamma_k^{-}$ and $\gamma_k^{+}$, for $k = 1, \ldots, q$. Partial asymmetry in this framework manifests by imposing the restrictions below: \begin{align*} \text{Partial Short-run asymmetry (Long-run Symmetry):}& \quad \phi = \phi^{-} = \phi^{+} \\ \text{Partial Long-run asymmetry (Short-run Symmetry):}& \quad \gamma_k = \gamma_k^{-} = \gamma_k^{+} \\ \end{align*} As NARDL models are typically estimated using least-squares, (partial) asymmetry can be formally tested. These tests reduce to the usual Wald-like hypotheses on the equivalence of positive and negative asymmetry coefficients. Formally, \begin{align*} \text{Long-run symmetry only } H_0 &:\quad \phi^{-} = \phi^{+} \\ \text{Short-run symmetry only } H_0 &:\quad \begin{cases} \gamma_k^{-} = \gamma_k^{+} \text{ for each } k \\ \\ \text{or}\\ \\ \sum_{k = 1}^q \gamma_k^{-} = \sum_{k = 1}^q \gamma_k^{+} \end{cases} \\ \text{Joint Short- and Long- run symmetry } H_0 &:\quad \begin{cases} \gamma_k^{-} = \gamma_k^{+} \text{ for each } k \text{ and } \phi^{-} = \phi^{+} \\ \\ \text{or}\\ \\ \sum_{k = 1}^q \gamma_k^{-} = \sum_{k = 1}^q \gamma_k^{+} \text{ and } \phi^{-} = \phi^{+} \end{cases} \end{align*} In EViews, these tests are performed after estimating a NARDL model by clicking on View/ARDL Diagnostics/Symmetry Test. Figure 10 below summarizes the output for the regression above.

 Figure 10: Full Asymmetry NARDL(2,1,3,1,0) NARDL Symmetry Test

The output header summarizes the null hypothesis and degrees of freedom, followed by simple tests for long- and short- run symmetry, respectively; followed by the joint test for full symmetry. As short-run and full symmetry can be tested in one of two ways, EViews reports the second test in both cases. Note also that if a variable enters the model with zero-lag, the short-run symmetry test, and by extension the joint test, are not applicable.

Turning to specific insights, we reject the null hypothesis of long-run symmetry for @LOG(FTA) at all reasonable significance levels and for FTS at the 5% significance level. While we also reject the joint symmetry test for @LOG(FTA) at all reasonable significance levels, we cannot evaluate the joint test for FTS as it enters the current model with zero lags. We also fail to reject the null hypothesis for the remaining coefficients. This suggests that the model we ought to consider next assumes the form:

\begin{align*} \class{bold} { \Delta \ln(GDP) } & \class{bold} { = } \class{bold col_red}{ \phi_{\scriptsize \text{GDP}} \ln(\text{GDP})_{t - 1} + \phi_{\scriptsize \text{DTA}} \ln(\text{DTA})_{t - 1} + \phi_{\scriptsize \text{DTS}} \ln(\text{DTS})_{t - 1} }\\ &\class{bold col_red}{ + \phi_{\scriptsize{\text{FTA}}}^{+} \ln(\text{FTA})_{t - 1}^{+} + \phi_{\scriptsize{\text{FTA}}}^{-} \ln(\text{FTA})_{t - 1}^{-} }\\ &\class{bold col_red}{ + \phi_{\scriptsize{\text{FTS}}}^{+} \text{FTS}_{t - 1}^{+} + \phi_{\scriptsize{\text{FTS}}}^{-} \text{FTS}_{t - 1}^{-} }\\ &\class{bold col_blue}{ + \sum_{j = 1}^{p - 1} \gamma_{\scriptsize{\text{GDP}} \normalsize{, \, j}} \Delta \ln(\text{GDP})_{t - j} }\\ &\class{bold col_blue}{ + \sum_{k_1 = 1}^{q_1 - 1} \gamma_{\scriptsize{\text{DTA}} \normalsize{, \, k_1}} \Delta \ln(\text{DTA})_{t - k_1} + \sum_{k_2 = 1}^{q_2 - 1} \gamma_{\scriptsize{\text{FTA}} \normalsize{, \, k_2}} \Delta \ln(\text{FTA})_{t - k_2} }\\ &\class{bold col_blue}{ + \sum_{k_3 = 1}^{q_3 - 1} \gamma_{\scriptsize{\text{DTS}} \normalsize{, \, k_3}} \Delta \ln(\text{DTS})_{t - k_3} + \sum_{k_4 = 1}^{q_4 - 1} \gamma_{\scriptsize{\text{FTS}} \normalsize{, \, k_4}} \Delta \ln(\text{FTS})_{t - k_4} }\\ &\class{bold col_green}{ + \alpha_0 + \alpha_1 t + \sum_{i = 1}^{11} \delta_{i} m_i + \epsilon_t } \end{align*} We estimate this model next; see Figures 11a and 11b.

 Figure 11a: Partial Asymmetry NARDL Dialog Figure 11b: Partial Asymmetry NARDL(2,1,2,3,0) Output

As in the first regression, our first inferential exercise here is to confirm the asymmetry assumptions made earlier. The results, summarized in Figure 12, reinforce our understanding that both @LOG(FTA) and FTS are partially asymmetric in the long-run, although the conclusion for FTS holds at all significance levels roughly greater than 7.2%.

 Figure 12: Partial Asymmetry NARDL(2,1,2,3,0) Symmetry Test

Returning to estimation results, a comparison of summary statistics from the first full asymmetry regression with the partial asymmetry regression above, indicates that the R-squared and adjusted R-squared statistics in the latter model are slightly worse, whereas information criteria in the latter model are slightly bigger. While these summary statistics suggest a slight downgrade in model preference, the difference is so slight it can be safely ignored. We therefore continue the remaining inreference exercises using the partial asymmetry NARDL(2,1,2,3,0) model.

### Bounds Test / Cointegration

Now that we have settled on a model, we'll test for cointegration among the system variables using the famous bounds test. We proceed by clicking on View/ARDL Diagnostics/Bounds Test; the results are summarized in Figure 13.

 Figure 13: Partial Asymmetry NARDL(2,1,2,3,0) Bounds Test

The output is a spool object with the first table summarizing test statistics and the second summarizing the critical values. For a practical review of bounds testing, please refer to Part III of our ARDL blog. In the current framework, the F-bounds test statistic is 11.60, well beyond the I(1) critical value bound, and a clear rejection of the null hypothesis of no cointegration, when all variables are I(1). Recall that the rejection of the bounds test null hypothesis leads to 3 possible alternative hypotheses, only one of which confirms the existence of a useful cointegrating relationship. To ascertain which of the three alternatives emerge, an additional t-bounds test on parameter significance of the lagged dependent variable, namely $\phi_{\scriptsize \text{GDP}}$ must be performed. In this case, the t-bounds statistic is -8.40, also well below the I(1) critical value bound, and a clear rejection of the null hypothesis that no cointegrating relationship exists when all variables are I(1). As outlined in Pesaran et al. (2001), the rejection of the t-bounds test in the secondary stage confirms the existence of a cointegrating relationship, but does not preclude that that it is degenerate. To rule out degenerate cointegration, a joint test of parameter significance on all coefficients associated with distributed lag variables in levels, ought to be inspected. This can be done with a simple Wald-test (see Figures 14a and 14b) by clicking on View/Coefficient Diagnostics/Wald Test - Coefficient Restrictions... and entering
C(2)=0, C(3)=0, C(4)=0, C(5)=0, C(6)=0, C(7)=0
 Figure 14a: Wald Test Dialog Figure 14b: Wald Test Output

With a p-value of 0.00, this test rejects the null hypothesis that all tested coefficients are jointly zero, and by extension, confirms that the cointegrating relationship which emerges, is in fact sensible and not degenerate.

Given the existence of a non-degenerate cointegrating relationship, we can identify the normalized, long-run coefficients in the cointegrating space which are associated with each of the distributed lag variables. Recall that if $\phi_{\scriptsize \text{DEP}}$ and $\phi_{\scriptsize \text{k}}$ are the coefficients associated with the dependent variable $y_t$ and the k$^{\text{th}}$ distributed-lag variable $x_{k, t}$ in levels in the (N)ARDL CEC form, respectively, the normalized, long-run distributed lag coefficient in the cointegration space is defined as \begin{align*} \beta_{k} \equiv - \frac{\phi_k}{\phi_{DEP}} \end{align*} In other words, for a NARDL model with $K$ distributed lag variables, the cointegrating relationship is formalized as: \begin{align*} \class{bold}{ \text{CE} = \ln(\text{GDP})_{t - 1} - \sum_{r = 1}^{K} \beta_{r} \, x_{r, t - 1} } \end{align*} We can estimate these values (see Figure 15) for our concrete model above by clicking on View/ARDL Diagnostics/Cointegrating Relation.

 Figure 15: Partial Asymmetry NARDL(2,1,2,3,0) Cointegrating Relation

The output is a spool object with two tables and a graph. The first table provides the cointegrating specification; the second table provides derived long-run (cointegrating) coefficients for each distributed-lag regressors; the graph plots the cointegrating specification as a series. We can also run the error-correction regression in which the long-run variables in the model are replaced by the cointegrating relation series defined in the first table of Figure 15. To do this, click on View/ARDL Diagnostics/Error Correction Results and look at the second table of the spool object which is produced. The latter is reproduced in Figure 16 below.

 Figure 16: Partial Asymmetry NARDL(2,1,2,3,0) Error Correction Regression

### Dynamic Multipliers

An important exercise in classical regression analysis is estimating the causal effect or multiplier of a regressor on the dependent variable, ceteris paribus; recall also that this is just the partial derivative of the dependent variable with respect to (wrt) a regressor. While (N)ARDL models can be cast into a classical regression framework, they are dynamic in that lagged values of both the dependent and distributed-lag regressors affect the current state of the dependent variable. Accordingly, (N)ARDL models lend themselves to the derivation of dynamic causal effects or dynamic multipliers - causal effects which can be traced over time.

In practice, dynamic causal effects can be thought of as analogous to impulse response curves in classical VAR / VEC models. They can be derived as response curves to unitary positive shocks in distributed-lag variables. In particular, for any distributed-lag regressor $x_{t}$, holding all other regressors unchanged, a single positive unitary shock is introduced at $T −h$, where $T$ is the length of the estimation sample, and the evolution of the dependent variable, $y_t$, is measured through the period $[T − h, T]$ where $h \geq 0$ is some horizon length. This can also be derived as the difference $\tilde{y}_t - \hat{y}_t$, where $\tilde{y}_t$ is the in-sample dynamic forecasts of $y_t$ when $x_t$ is perturbed at $t = T - h$ to equal $x_{T - h} + 1$, and $\hat{y}_t$ is the in-sample dynamic forecasts of $y_t$ when $x_t$ is left unchanged.

A natural extension of the dynamic multiplier is the cumulative dynamic multiplier (CDM)- a cumulative sum of dynamic multipliers at each point in time on the interval $[T - h, T]$. In fact, as $h \rightarrow \infty$, the cumulative dynamic multiplier converges to the long-run (cointegrating) coefficients discussed in the previous section. In other words, we can trace out the adjustment patterns (the short-run dynamics) as they evolve to converge to their cointegrating (long-run) equilibrium state. See Shin, Yu, and Greenwood-Nimmo (2014) for details.

Cumulative dynamic multipliers are particularly interesting for asymmetric distributed lag-variables, such as those characterizing NARDL models. They allow researchers to study the evolution of adjustment patterns following negative and positive shocks to asymmetric regressors and quantify the path of asymmetry as CDMs evolve towards their respective (cointegrating) equilibrium states. Furthermore, confidence intervals for the evolution of asymmetry can also be derived via non-parametric bootstrapping.

To derive CDMs for the model we estimated earlier with a 95% confidence interval derived over 999 bootstrap replications, we can proceed as follows:
1. From the estimated equation object, click on View/ARDL Diagnostics/Dynamic Multiplier Graph...
2. Change the Horizon to 50
3. Set the evolution type to Shock or Dynamic multiplier
4. Leave the rest at their default values and click on OK
There are several things to unpack before analyzing the output. First, note that options for confidence intervals is only available for NARDL models with asymmetric regressors as the asymmetry path for symmetric variables will by construction always be zero.

Next, note that there are two options for the evolution type: 1) Shock and 2) Dynamic multiplier. As noted in the EViews manual, this distinction is only relevant for NARDL models with asymmetric regressors, and only affects asymmetric negative response curves. In particular, both shock evolution and dynamic multiplier evolution plot the response to a one unit positive change in the symmetric and positive asymmetric cumulated differences. However, unlike the shock evolution framework which plots the response to a one unit negative change in cumulated differences in the negative asymmetric case, the dynamic multiplier evolution framework plots an “improvement” producing a one unit positive increase (reduction of one unit of negative change) in the negative cumulative differences. In fact, the shock evolution plot can be derived from the dynamic multiplier evolution plot by reflecting the negative response curve in the dynamic multiplier evolution plot along the x-axis.

While the dynamic multiplier evolution framework is reasonable from a technical perspective, it is not ideal if we wish to study the properties of the models under parallel unit increases in the absolute amount of positive and negative asymmetry, as in when determining whether an increase in positive asymmetry has the same effect as an increase in negative asymmetry. Furthermore, as the dynamic multiplier framework is better aligned to a technical analysis, in contrast to the shock evolution framework, it will also plot the long-run coefficient values to which the CDMs converge as the horizon lengths approaches infinity.

We will start with the Dynamic multiplier evolution and display the output in Figures 17a through 17d.

 Figure 17a: CDM - @LOG(DTA) Figure 17b: CDM - DTS Figure 17c: CDM - @LOG(FTA) Figure 17d: CDM - FTS

First, notice that every plot, in addition to the response curves, also displays the long-run values to which the evolution converges in equilibrium. These dotted lines correspond to the long-run coefficient values outlined in the second table of Figure 15. Furthermore, notice that Figures 17a and 17b display the CDM for symmetric regressors. As such, they have a single response without confidence interval computations. On the other hand, Figures 17c and 17d display the response curves for the positive and negative changes, a response curve for the asymmetry between the two, as well as a confidence interval band around the asymmetry response. In particular, as the zero line is not located between the lower and upper bands in either Figures 17c or 17d, the asymmetric effects of those variables is significant at the 5% level.

Let's also plot the same information in terms of a shock evolution framework.

 Figure 18a: CDM - @LOG(DTA) Figure 18b: CDM - DTS Figure 18c: CDM - @LOG(FTA) Figure 18d: CDM - FTS

The curves in Figures 18 present similar information. In fact, Figures 17 and Figures 18 are effectively identical apart from the negative response curve which is reflected along the x-axis in Figures 18.

### Interpretation and Policy Implications

We devote this section to a brief interpretation of the results above and suggestions for policy implementations. First, recall that the results of the bounds test summarized in Figure 13 and 14b confirm the existence of a cointegrating relationship between Bosnia's tourist variables and its gross domestic product. In fact, this cointegrating relationship is itself significant as the COINTEQ coefficient in the error-correction regression in Figure 16 is highly significant. On the other hand, only certain components of the cointegrating relationship are significant themselves. Note that the second table in Figure 15 indicates that while domestic tourist arrivals and foreign tourist stays are insignificant in the long-run, domestic tourist length of stay and foreign tourist arrivals are indeed significant in the equilibrium.

We can gain further insight into how each tourist variables contributes the evolution of Bosnia's GDP by looking at their response curves plotted in Figures 17 and 18. In particular, Figures 17a and 18a illustrate that a 1% positive shock to domestic tourist arrivals increases GDP by 0.04% in roughly two years (25 months), with a considerable short-run boost in the first 6 months.

Similarly, as shown in Figures 17b and 18b, a single unit positive shock to domestic tourist length of stay (in other words, a single day prolonged stay) lifts GDP by 0.10% in roughly 2 years, with another significant short-run boost in the first 5 months.

On the other hand, Figures 17c and 18c indicate that a 1% increase to foreign tourist arrivals produces a 0.17% increase in GDP in roughly 25 months. This is in contrast to a 1% decrease (see Figure 18c instead of 17c) in foreign tourist arrivals which will decrease GDP by 0.12% in again, 25 months. As shown by the asymmetry curve and its associated 95% confidence interval, this asymmetry is significant.

Lastly, Figures 17d and 18d show that a 1 unit positive shock to foreign tourist length of stay produces a 0.02% increase in GDP in approximately two years. Interestingly, as shown in Figure 18d, a 1 unit negative shock to foreign tourist length of stay will also increase GDP by 0.01% in roughly 25 months. This asymmetry is also significant.

The analysis above paints a general picture of Bosnia's tourism sector and how it impacts its GDP. This picture suggests that Bosnia's domestic tourism, while lagging behind its foreign counterpart, has a symmetric effect on Bosnia's economy. This symmetry suggests that policies which equally bolster or hinder domestic tourism will have similar, but opposite effects on the GDP. Furthermore, considering that domestic tourist length of stay seems to have a larger impact on GDP than domestic tourist arrivals, policies should focus on encouraging longer domestic tourists stays, perhaps by encouraging infrastructural changes that will reduce travel times (think better intra-national highways or fast train services) or by offering services / deals (think hotel discounts for longer stays or hotel loyalty programs) which incentivize longer visits.

In contrast, Bosnia's foreign tourist industry has a statistically significant asymmetric impact on Bosnia's GDP. In particular, foreign tourist arrivals can benefit from policies which focus more on increasing tourist arrivals. Possible strategies here include international advertisement campaigns, easier tourist visa issuances and ideally a digitized process which can be completed upon arrival, and improved international airline connections. On the other hand, negative shocks to foreign tourist arrivals, while decreasing GDP, have a smaller impact on Bosnia's economy than positive shocks. This suggests that Bosnia can insulate from downtrends in foreign arrivals by attempting to recoup losses with policies which bolster arrivals.

In the end, it's Bosnia's foreign tourist length of stay which is the most interesting variable. This is a variable which also has a statistically significant asymmetric effect on Bosnia's long-run economy, but as the long-run coefficient on the negative partial sums of changes in FTS is negative, GDP actually benefits from a decrease in foreign tourist length of stay. This suggests that Bosnia benefits from higher foreign tourist turnover, perhaps due to revenues gained on tourist visas. Another possible explanation is overcrowding which can possibly lead to inefficiencies in providing services, as well as crowding out of domestic tourists which refuse to participate in the tourist market at times when it is overwhelmed by foreigners. This is certainly suggested in Figure 4b which shows that domestic and foreign tourist length of stays do not peak at the same time during the year.

### References

1. Kahneman, D. and Tversky, A., (1979). Prospect theory: an analysis of decision under risk. Econometrica, 47: 263–291.
2. Pesaran, M. H., Shin, Y., and Smith, R. J, (2001). Bounds testing approaches to the analysis of level relationships. Journal of Applied Econometrics, 16(3): 289–326.
3. Shiller, R. J., (2005). Irrational exuberance. Princeton University Press, Princeton, 2nd edition.
4. Shin Y., Yu B., Greenwood-Nimmo M., (2014). Modelling Asymmetric Cointegration and Dynamic Multipliers in a Nonlinear ARDL Framework Festschrift in honor of Peter Schmidt, 281–314.

1. Great to see all of these features now in EViews! This post is absolutely superb - thanks!

2. Amazing post!!!!!!!!!

3. Top! Thanks

4. Hi
I just got my license to do a NARDL analysis. I realized that whenever I want to perform a NARDL analysis, I get the response "invalid or duplicate specficiation". I use daily log returns of S&P 500, Apple, and VIX. I also tried to use the dataset here (Toursim in Bosnia) I get the same error. I exactly do the same thing that you do here. What should I do?
ARDL works but NARDL I get this error. I am on EViews 13.

Thanks

1. Hello there! I have just replicated every example in this post using the dataset in this post and had no such error come up. Please email support@eviews.com and attach the workfile you're using along with a screenshot of the NARDL dialog setup with the regressors you're using which causes this error. Thank you.

1- What if we only see short-run Asymmetry and joint asymmetry (But not a long-run asymmetry)?
2- In dealing with daily data, clearly there is heteroscedasticity (and even 12 lags might not help the issue), so is it fine if I only use White for the covariance matrix?

1. 1) This probably indicates that the test for short-run asymmetry is highly significant, but the test on long-run asymmetry is borderline insignificant. To be conservative, I'd conclude short-run asymmetry only.
2) That seems sensible.

2. Thanks again.
There are a few papers which use the idea of deleting the insignificant + and - lags based on "Krolzig, H. M., & Hendry, D. F. (2001). Journal of Economic Dynamics and Control", is there any way to separate the positive in EViews?

I know it is possible to choose lags in EViews, but do not know how to single out + vs - ones, since even I cannot save those lags or differenced variables to use them in the equation.

3. What you're asking cannot be done automatically. Nevertheless, since (N)ARDL estimation is estimated using least squares (LS), what you want can be done manually.

After estimating a NARDL equation, proceed to View/Representations. From here, copy the string after "Estimation Equation:". Use this string to remove the variables you don't want and use the remaining string to estimate a basic least squares regression.

6. hi-- If the coefficients for partial sum of positive and negative effects are NOT significant, but the asymmetry test shows that there is an asymmetric effect exists for that variable, what is the inference in this case?
tnx

1. The asymmetry test is a Wald test on the joint significance of all coefficients for a given variable in either the short-, the long-, or simultaneously short- and long- runs. These Wald tests are based on the coefficients of the CEC (default) regression output. In other words, there can only ever be a single situation in which you have the scenario you're talking about and that is that the individual coefficients are insignificant (probably very slightly only), but jointly, they are significant. In this case, you will take the results of the joint test to conclude that asymmetry is indeed valid.

7. Hi EViews team.
What should one do when there is autocorrelation based on LM test? Increasing the lag or differentiating the variable up to 3 three times does not resolve the issue! Interesting enough the unit root test says there is no unit root in level (daily frequency).