modelling and forecasting the volatility of long-stay tourist arrivals
TRANSCRIPT
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Modelling and Forecasting the Volatility of Long-Stay Tourist Arrivals
by
Troy Lorde
Department of Economics
University of the West Indies, Cave Hill Campus
P.O. Box 64
Bridgetown, Barbados
Tel. (246) 417-4272; Fax: (246) 417-4270
Email: [email protected]
and
Winston Moore∗
Department of Economics
University of the West Indies, Cave Hill Campus
P.O. Box 64
Bridgetown, Barbados
Tel. (246) 417-4275; Fax: (246) 417-4270
Email: [email protected]
∗ Corresponding author.
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Modelling and Forecasting the Volatility of Long-Stay Tourist Arrivals
Abstract
Volatility is an important characteristic of most tourist economies. Fluctuations in tourist arrivals
can occur due to unanticipated events, such as natural disasters, crime, the threat of terrorism, and
business cycles in key source markets. This study exploits recent time series modelling
techniques to model and forecast the volatility in monthly international tourist arrivals to
Barbados. The results show that models which allow mean volatility to change over time tend to
have the best forecasting performance. However, relatively simple models, such as the
RiskMetrics approach can also produce predictions that are statistically at least as good as more
complex models.
Keywords: Volatility; Tourist arrivals; Forecasting; Caribbean
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1. Introduction
As a result of changes in economic fortunes abroad, natural disasters, ethnic conflicts, crime,
terrorist incidents, and other exogenous factors, there have been periods of considerable
fluctuation in international tourism demand to many small island developing states. These
fluctuations can and do have a significant impact on the solvency of hotels, employment in the
industry and overall economic activity. It is therefore imperative that tourism planners and
policymakers have models to explain and forecast the volatility of tourist arrivals.
Modelling the volatility of tourism demand is a relatively new sphere of leisure studies.
Nevertheless, there have been some recent attempts in this area, most notably Chan, Lim and
McAleer (2005), Chan, Hoti, Shareef and McAleer (2005) and Shareef and McAleer (2005).
Chan, Lim and McAleer (2005) examine the volatility of the logarithm of the monthly tourist
arrivals from the four leading source countries – Japan, New Zealand, UK and USA – to
Australia between 1975 and 2000. The authors estimate three multivariate volatility models: the
symmetric CCC-MGARCH approach of Bollerslev (1990), the symmetric vector ARMA-
GARCH framework of Ling and McAleer (2003) and the asymmetric vector ARMA-AGARCG
method of Chan, Hoti and McAleer (2002). Chan, Lim and McAleer (2005) find the presence of
interdependent effects in the conditional variances between the four leading source countries, and
asymmetric effects in arrivals from Japan and New Zealand. The authors report that their
estimates are robust to the alternative specifications of the multivariate conditional variance.
Chan et al. (2005) use several techniques to investigate the conditional volatility in monthly
international tourist arrivals to Barbados (1973-2002), Cyprus (1976-2002) and Fiji (1968-2002).
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The authors estimate a constant volatility linear regression model by OLS as a baseline for
comparison with three time-varying conditional volatility models – ARCH, GJR and EGARCH.
Overall, Chan et. al. report evidence of short run persistence, and occasionally long run
persistence, of shocks to international tourist arrivals. The authors also find evidence of
asymmetric effects of shocks for Barbados using the EGARCH specification.
Shareef and McAleer (2005) model both the volatility in monthly international tourist arrivals
and the growth rate of monthly tourist arrivals for six small island tourist economies: Barbados,
Cyprus, Dominica, Fiji, Maldives and Seychelles during the period 1980-2000 using
GARCH(1,1) and GJR(1,1). While estimates for the conditional mean and variance in monthly
international tourist arrivals for the countries were similar, the GARCH(1,1) and GJR(1,1)
estimates varied somewhat across countries. A similar result held when the growth rate of
monthly tourist arrivals was modelled. Using the log-moment and second moment conditions,
they obtain support for the statistical adequacy of the GARCH(1,1) and GJR(1,1) models.
This paper extends the literature surveyed above by estimating several models of tourism
volatility and evaluating the predictive ability of these models using monthly data for the period
1977-2005 for Barbados. Small island economies, due to their size and dependence on tourism
are more vulnerable to external shocks. These countries would therefore be interested in
understanding the characteristics of tourism volatility and have a system in place to adequate
forecast future volatility. The results obtained from the study could assist tourism planners in
these states to build their own models of tourism volatility and provides a guide as to which are
superior in terms of forecasting performance.
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The structure of the paper is as follows: After the introduction, Section 2 discusses the patterns of
tourist arrivals to Barbados. Section 3 describes the volatility models estimated in this study and
alternative loss functions are used to evaluate the forecasts from each model. Section 4 presents
the empirical results and Section 5 concludes with policy recommendations for tourism officials
in small island economies.
2. Volatility
The study uses monthly observations on the logarithm of deseasonalized total monthly tourist
arrivals for the period 1977-2005. The series is deseasonalized using Census X12 – the US
Census Bureau seasonal adjustment algorithm. The augmented Dickey-Fuller (ADF) test (1979)
and the Phillips-Perron (PP) tests (1988) of the unit root hypothesis, conducted using EViews 5.0,
both suggest the absence of a unit root in the log of the monthly deseasonalized series (see Table
1). These tests are robust to changes in lag length and auxiliary equation specification.
Figure 1 plots the log of the monthly deseasonalized arrival rate to Barbados between 1977 and
2005. The cyclicality of the series is apparent. The peaks in the cycle correspond to the boom in
the latter half of the 1970s and the recovery from the recession early in the 1990s, while the
troughs correspond to the slowdown caused by the second oil price shock in 1979 and the two
Persian Gulf conflicts.
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Following Chan, Lim and McAleer (2005), volatility is calculated as the square of the estimated
residuals 2t
ε from an autoregressive process. The correlogram of the log arrival rate suggested
that an AR(1) or an AR(2) would be suitable. Diagnostic checking confirmed that the AR(1)
with a deterministic time trend is a more suitable description of the process:
tt timeARTA εϕ ++= )1()log( (1)
2( )t tVol ε ε= (2)
where TA is the total monthly international tourist arrivals at time t and time = 1,…,T, where T =
324.
Figure 2, which plots the estimated volatility series, shows that tourist arrivals volatility is
characterised by volatility clustering: a shock is likely to be followed by other (smaller) shocks.
These volatility clusters correspond to the peaks and troughs of the cycles described previously.
The figure also shows that monthly tourist arrivals were more volatile in the first half of the
sample. This finding corresponds to the Butler (1980) lifecycle model, which predicts that in the
early stages of a destination’s lifecycle, growth is likely to be positive, but erratic.
3. Volatility Models
The authors specify a number of volatility models (see Appendix). The forecasts from more
complex models are all compared to those from the naïve model to test whether they add any
additional information. The naïve model assumes that the best n-step ahead forecast of the
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conditional variance is its past n-month average. This naïve model (NVOL) can be estimated by
regressing the realized volatility on a constant.
The RiskMetrics volatility model is a popular tool employed to measure risk. The framework has
two main advantages: (1) it is fairly simple, and; (2) it only requires a small number of
observations. The RiskMetrics volatility is calculated as follows:
21
22 )1( −+−= ttt brb σσ (3)
where 2tσ is the volatility at time t and 2
tr is the squared return at time (month-on-month change
in arrivals). Usually, the weighting parameter ( b ) is set at 0.97 for monthly data. To obtain n-
period ahead forecasts, one can use the following expression:
21
22 )1( −+++ +−= ntntnt brb σσ
The RiskMetrics approach is a special case of a generalized autoregressive conditional
heteroskedasticity (GARCH) model. GARCH models, introduced by Engle (1982) and
generalized by Bollerslev (1986) and Taylor (1986), are specifically designed to model and
forecast conditional variances. Volatility is modelled as a function of past values of volatility and
errors from the mean regression.
In general form the GARCH(p,q) model can be written as:
∑∑=
−=
− ++=q
j
jtj
p
j
jtjt
1
2
1
22 σβεαϖσ (4)
where Equation (4) states that the conditional variance of tourist arrivals depends on a constant
(ϖ ), the previous period’s squared random component of tourist arrivals (referred to as ARCH
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effects or the short-run persistence of shocks) and the previous period’s variance, where the
contribution of shocks to long-run persistence is given by βα + . Non-negativity of 2tσ requires
that ϖ , α and β are non-negative, while stationarity requires that 1<+ βα . (It is also
possible to consider so-called integrated GARCH models where 1=+ βα . However, in these
models volatility shocks have permanent effects (see Engle and Bollerslev, 1986), which is not
likely to be the case for tourist arrivals.) A value of βα + close to zero therefore implies that the
persistence in volatility is high. The GARCH model is suitable when large changes in returns are
likely to be followed by further large changes.
The GARCH model assumes that negative shocks have the same impact on future volatility
(symmetry) as a large positive shock of the same magnitude, i.e. a terrorist attack on the tourist
destination would have the same impact on volatility as hosting a major sporting event. To allow
for asymmetry, i.e. negative shocks have a larger impact on future volatility than positive shocks,
one can use Nelson’s (1990) exponential GARCH model (EGARCH) which allows positive
shocks to generate differential effects on volatility relative to negative shocks. Asymmetry can
also be modelled by using the threshold GARCH (TRGARCH) model, introduced independently
by Zakoïan (1994) and Glosten, Jaganathan and Runkle (1993), which adjusts the forecasts of
tourism volatility to account for the past negative or positive shocks.
Ding et. al. (1993) also introduced the Power ARCH (APARCH) specification to deal with
asymmetry. In the APARCH model the power parameter is estimated rather than imposed, and
optional parameters are added to capture asymmetry. The APARCH model, under certain
assumptions can also encompass the TS-GARCH model of Taylor (1986) and Schwert (1990),
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the GJR-GARCH of Glosten, Jagannathan and Runkle (1993) and the N-GARCH of Higgens and
Bera (1992). To abstract from asymmetry in tourist arrivals, one can instead consider the
AGARCH model that uses the absolute value of the threshold term, while the log-GARCH model
uses the logarithm of the conditional standard deviation rather than the variance.
Rather than assume that the conditional variance shows mean reversion to some average level of
volatility, one can instead estimate a model that allows mean reversion to a varying level of
volatility. This approach assumption is can incorporated into the modelling of tourism volatility
by estimating a component GARCH model (CGARCH). The CGARCH model would be
appropriate if policies implemented by tourism officials can result in reduced volatility in the
industry. The CGARCH model can also be estimated with a threshold, denoted by CGARCHT.
None of the models outlined above provide the ‘best’ forecast of the conditional variance in every
time period. One model may perform well in one period, while another will have superior
performance during another period. In this situation, a combination of forecasts may perform
better over time than those generated using a single model (see Clemen, 1989 and Mahmond,
1984 for surveys of this literature).
The authors use the linear regression weighting approach suggested by Granger and Ramanathan
(1984), which gives:
∑=
++ +=m
i
ntiint
1,σθτσ (5)
where m is the number of forecasting models considered. The weights are based on the 1-month
horizon in-sample results. Bivariate combination forecasts are considered for each of the
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GARCH-type models presented in this Section and a model where all the predictions are included
in the regression and only those models with significant weights are retained.
4. Forecasting Tourism Volatility
4.1 Comparison Criteria
A number of criteria exist for evaluating the forecasting performance of volatility models (see
Lopez, 2001, for a survey of these approaches.). Each method has its own drawbacks, so rather
than choose between the approaches, the authors choose six loss functions:
( )∑=
− −=n
t
tt hnMSE1
211 σ (6)
( )∑=
− −=n
t
tt hnMSE1
22212 σ (7)
( )∑=
− +=n
t
ttt hhnQLIKE1
2221 /)log( σ (8)
( )2
1
2212 )/log(∑=
−=n
t
tt hnLOGR σ (9)
∑=
− −=n
t
tt hnMAD1
11 σ (10)
∑=
− −=n
t
tt hnMAD1
2212 σ (11)
11
Equations (9) and (10) are the usual mean squared error criteria. The metric QLIKE is the loss
implied by a Gaussian likelihood, while the R2LOG loss function penalises volatility forecasts
asymmetrically in low volatility and high volatility periods. The final two criteria (Equations 13
and 14), are the mean absolute deviation, which tend to be more robust in the presence of outliers
than the MSE criteria.
The authors also use the superior predictive ability (SPA) test of Hansen and Lunde (2005),
which tests the hypothesis of whether some benchmark model is as good as any other model in
terms of expected loss. It provides a useful framework within which to test relatively simple
models (for example RiskMetrics and GARCH specifications) against more complex alternatives.
The test compares a sequence of forecasts from the model under consideration to the conditional
variance using a loss function. Let the first model be the benchmark model that is compared to
the other models. Each model provides a sequence of losses and relative performance variables.
The null hypothesis of the test is that the expected loss from the benchmark model is no more
than the expected loss from any other model. The test has an asymptotic normal distribution.
4.2 Out-of-Sample Forecast Results
Estimation of the various volatility models and statistical tests are conducted using the computing
package EViews 5.0 for the period 1977Q2 to 2003Q12. The threshold GARCH model is
obtained by assuming that the errors have a generalized error distribution while all the remaining
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models presume that the conditional distribution of the errors is normal. The coefficient
estimates for all models are obtained by maximum likelihood.
All the empirical models are then used to provide out-of-sample forecasts for the period 2004Q1
to 2005Q12. The predictions from these models are then compared over 1, 3, 6, 12 and 24 month
horizons using the six comparison criteria outlined earlier (MSE1, MSE2, R2LOG, QLIKE,
MAE1, MAE2). Relative forecast evaluation statistics – the forecast evaluation statistic of the
model under consideration as a ratio of that for the naïve model – are averaged across the six
comparison criteria and the results are presented in Table 2. A value less than one suggest that
the model outperforms the naïve model. Given that each evaluation criteria may differ in its
choice of the ‘best’ model, the authors also compute the average rank for each model based on
their forecasting accuracy. Therefore the best model receives a score of 1, the second best a score
of 2, and so on.
Over short horizons (1 to 3 months), Table 2 shows that most models outperform the naïve
model, the only exceptions being the APARCH, TAYLOR and TRGARCH models. The
RiskMetrics approach and models that allowed for mean reversion to different levels of volatility,
i.e. CGARCH and CGARCHT, usually outperformed all the other techniques. For longer
horizons, however, the predictive ability of the RiskMetrics approach was quite poor: over a 6 to
24 months horizon, the model was usually one of the four worst forecasting techniques. For
longer horizons, the CGARCH, CGARCHT also provided relatively accurate forecasts along
with the ARCH and EGARCH models. These four approaches consistently outperformed both
the naïve and other volatility techniques over longer horizons.
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Table 2 also provides an average ranking for each approach based on the forecast evaluation
statistics. The CGARCHT (mean reverting threshold component GARCH) model had the best
overall average ranking, indicating that the model was consistently one of the top four models
over the five forecast horizons considered. The other models in the top 5 were the CGARCH,
EGARCH, ARCH, GARCH, GJR-GARCH, PARCH and Riskmetrics.
Since each of the models have their shortcomings, one can combine the out-of-sample forecast
from the ‘best’ models to identify whether they generate more accurate forecasts than any
individual model. The authors consider six combination models: (1) CGARCH and CGARCHT;
(2) CGARCHT and ARCH; (3) CGARCHT and EGARCH; (4) CGARCHT and GARCH; (5)
CGARCHT and GJRGARCH, and; (6) CGARCH, GARCH, EGARCH, APARCH, AGARCH,
LOGGARCH, the naïve model (NAÏVE) and the RiskMetrics approach (RSK). The forecast
evaluation statistics as well as the rankings are provided for comparison purposes in Table 3.
Models 2, 4 and 5 are the top performing models over both short and long horizons, with Model 5
performing relatively better over long horizons, while Model 2 provides superior forecasts over
relatively short horizons.
The results so far indicate that the CGARCH, CGARCHT, EGARCH and ARCH models and
model combinations tend to generally provide superior predictions. However, these approaches
are complex and might be beyond the resources of tourism officials in small countries.
Therefore, the authors evaluate whether simple models, such as the RiskMetrics approach and a
GARCH(1,1) can provide forecasts that are at least as good as those from more sophisticated
techniques. To do this, one can use the superior predictive ability test of Hansen and Lunde
(2005) and various loss functions. The results are provided in Table 4.
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Table 4 contains the results of the model comparisons in the form of p-values, where the null
hypothesis is that the benchmark model is at least as good as the other models. The naïve p-value
compares the best performing model to the benchmark model without controlling for the full set
of models. The SPA tests control for the full set of models, where SPAc are asymptotic p-values,
while SPAu and SPAl are upper and lower bounds, respectively.
The results of the RiskMetrics model are presented first. The p-values in the SPAc column show
that forecasts from the RiskMetrics approach is not statistically outperformed by any of the
models at normal levels of testing, with the exception of the QLIKE loss function. This is in
contrast to the GARCH(1,1) model which is statistically outperformed by several of the models
considered. The final benchmark model is the CGARCHT, which as expected based on previous
results, is statistically at least as good as all the other models considered. These results indicate
that despite the simplicity of the RiskMetrics approach, it provides fairly accurate volatility
forecasts.
5. Conclusions
Many small island economies suffer from the effects of large sudden changes in tourist arrivals.
In this study, the authors evaluated the predictive ability of a number of volatility models applied
to monthly tourist arrivals to Barbados. The models are estimated over the period 1977Q2 to
2003Q12 and then used to provide forecasts for the period 2004Q1 to 2005Q12. The forecasts
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from these models are compared using six forecast evaluation statistics: MSE1,MSE2, R2LOG,
QLIKE, MAE1, MAE2. The results suggests that the CGARCH, CGARCHT, EGARCH and
ARCH specifications outperform most of the other models over both short and long horizons.
The authors also evaluated whether simple models, such as the RiskMetrics approach and a
GARCH(1,1) model are statistically at least as good as the more sophisticated models using the
Hansen and Lunde (2005) superior predictive ability test. The results suggest that the
RiskMetrics approach provides forecasts that are at least as good as those from more
sophisticated models, while the GARCH model is inferior to other models considered.
Tourist officials in small countries, therefore, who may not have the resources to implement more
complex models in their day-to-day forecasting, can still obtain fairly accurate predictions of
tourism volatility using the relatively simple RiskMetrics technique. The approach can easily be
implemented using a spreadsheet programme and only requires a historical series for tourist
arrivals. The findings in this study could also be used by managers of tourist attractions that are
interested in projecting the volatility of monthly sales.
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Table 1: Descriptive Statistics and Unit Root Tests of Log of Monthly Deseasonalized
Tourist Arrivals
Statistic Value
Mean 10.457 Maximum 10.471 Minimum 9.879 St. Dev 0.197 Skewness -0.366 Kurtosis 2.546 Jarque-Bera 10.777
[0.004] Observations 348 ARCH test (F-statistic) 5.827
[0.016] ADF test -4.542
[0.002] PP test -6.740
[0.000]
Note: P-value given in square parenthesis.
20
Table 2: Out-of-Sample Forecasting Accuracy Forecast Comparison Criteria Rank
Forecasting Horizon Forecasting Horizon
1 3 6 12 24 1 3 6 12 24 Average VOLF_AGARCH 2.120 0.951 0.942 0.761 0.694 12 10 9 8 7 9 VOLF_APARCH 1.456 1.110 0.963 0.822 0.754 11 11 10 11 11 11 VOLF_ARCH 0.805 0.744 0.754 0.689 0.661 8 6 5 3 4 5 VOLF_CGARCH 0.337 0.716 0.747 0.701 0.683 3 4 4 5 6 4 VOLF_CGARCHT 0.276 0.662 0.748 0.696 0.674 2 3 4 4 4 3 VOLF_EGARCH 0.775 0.682 0.746 0.689 0.672 7 3 4 3 5 4 VOLF_GARCH 0.527 0.765 0.781 0.739 0.711 4 7 7 7 8 7 VOLF_GJRGARCH 0.546 0.748 0.750 0.764 0.751 5 6 5 9 11 7 VOLF_LOGGARCH 0.568 0.787 0.787 0.767 0.740 6 9 7 9 10 8 VOLF_NAIVE 1.000 1.000 1.000 1.000 1.000 9 10 11 13 14 11 VOLF_PARCH 1.099 0.774 0.865 0.722 0.656 10 8 8 5 2 7 VOLF_RSK 0.216 0.763 0.982 0.797 0.814 1 7 9 10 10 7 VOLF_TAYLOR 3.356 1.225 1.118 0.820 0.722 14 12 12 10 9 11 VOLF_TRGARCH 2.516 0.980 1.138 0.813 0.686 13 10 12 10 6 10
21
Table 3: Out-of-Sample Forecasting Accuracy of Combination Models Comparison Criteria Rank
Forecasting Horizon Forecasting Horizon
1 3 6 12 24 1 3 6 12 24 Average
VOLF_COMB1 0.453 0.716 0.861 0.743 0.698
4
3
4
4
4
4
VOLF_COMB2 0.352 0.708 0.834 0.732 0.697
1
2
2
2
3
2
VOLF_COMB3 2.333 0.850 1.011 0.746 0.688
7
5
5
4
2
5
VOLF_COMB4 0.392 0.713 0.852 0.745 0.704
2
3
3
4
4
3
VOLF_COMB5 0.576 0.726 0.818 0.718 0.689
5
4
2
2
2
3
VOLF_COMB6 0.401 1.087 1.017 1.023 0.993
3
6
6
7
6
6
VOLF_NAIVE 1.000 1.000 1.000 1.000 1.000
6
5
6
6
7
6
22
Table 4: Superior Predictive Ability Tests Naïve SPAl SPAc SPAu Benchmark: RiskMetrics MSE1 0.149 0.222 0.222 0.222 MSE2 0.049 0.068 0.068 0.068 QLIKE 0.010 0.010 0.010 0.010 R2LOG 0.576 0.569 0.648 0.668 MAE1 0.164 0.243 0.243 0.254 MAE2 0.151 0.213 0.213 0.222 Benchmark: GARCH(1,1) MSE1 0.003 0.007 0.008 0.009 MSE2 0.005 0.010 0.010 0.012 QLIKE 0.004 0.008 0.009 0.011 R2LOG 0.232 0.011 0.011 0.012 MAE1 0.047 0.016 0.019 0.022 MAE2 0.065 0.018 0.018 0.026 Benchmark: CGARCHT(1,1) MSE1 0.031 0.085 0.089 0.158 MSE2 0.037 0.084 0.090 0.147 QLIKE 0.026 0.065 0.070 0.092 R2LOG 0.357 0.142 0.142 0.257 MAE1 0.200 0.150 0.150 0.263 MAE2 0.228 0.208 0.212 0.314
23
Figure 1: Log of Depersonalized Arrivals to Barbados
9.8
10.0
10.2
10.4
10.6
10.8
11.0
1978 1981 1984 1987 1990 1993 1996 1999 2002 2005
24
Figure 2: Volatility of Arrivals to Barbados
.00
.01
.02
.03
.04
.05
.06
1978 1981 1984 1987 1990 1993 1996 1999 2002 2005
25
Appendix: List of Volatility Models
Naïve ϖσ =2t
RiskMetrics 2
122 )1( −+−= ttt brb σσ
ARCH ∑
=− ++=
p
j
jtjt
1
22 εαϖσ
GARCH ∑∑
=−
=− ++=
q
j
jtj
p
j
jtjt
1
2
1
22 σβεαϖσ
Taylor/Schwert ∑∑
=−
=− ++=
q
j
jtj
p
j
jtjt
11
2 || σβεαϖσ
A-GARCH [ ] ∑∑
=−
=−− +++=
q
j
jtj
p
j
jtijtjt
1
2
1
22 σβεγεαϖσ
Thr.-GARCH ∑∑∑
=−−
=−
=− Γ+++=
r
j
jtjtj
q
j
jtj
p
j
jtjt
1
2
1
2
1
22 εγσβεαϖσ
GJR-GARCH { }[ ] ∑∑
=−
=−> +++=
−
q
j
jtj
p
j
jtijt tI
1
2
1
20
2
1σβεγαϖσ ε
Log-GARCH ∑∑
=−
=− ++=
q
j
jtj
p
j
jtjt
11
)log(||)log( σβεαϖσ
EGARCH ∑ ∑∑
= = −
−
−
−
=
− +++=p
j
r
j jt
jt
j
jt
jt
j
q
j
jtjt
1 11
22 )log()log(σ
εγ
σ
εασβϖσ
NGARCH ∑∑
=−
=− ++=
q
j
jtj
p
j
jtjt
11
|| δδδ σβεαϖσ
A-PARCH ∑∑
=
−
=
−− +−+=q
j
jtj
p
j
jtjjtjt
11
)|(| δδδ σβεγεαϖσ
CGARCH(1,1)a )()(
)()(2
12
11
21
21
2
−−−
−−
−+−+=
−+−+=−
tttt
tttt
mm
m
σεφωρω
ϖσβϖεαϖσ
a Model also estimated with threshold.