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Festschrift in Honor of Peter Schmidt pp 281–314Cite as

Modelling Asymmetric Cointegration and Dynamic Multipliers in a Nonlinear ARDL Framework

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Abstract

We develop a cointegrating nonlinear autoregressive distributed lag (NARDL) model in which short- and long-run nonlinearities are introduced via positive and negative partial sum decompositions of the explanatory variables. We demonstrate that the model is estimable by OLS and that reliable long-run inference can be achieved by bounds-testing regardless of the integration orders of the variables. Furthermore, we derive asymmetric dynamic multipliers that graphically depict the traverse between the short- and the long-run. The salient features of the model are illustrated using the example of the nonlinear unemployment-output relationship in the US, Canada and Japan.

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Notes

  1. 1.

    The present version of the paper is a substantially revised version of Shin and Yu (2004), which has benefited greatly from a sequence of incremental improvements and additions arising from the constructive comments of conference and seminar participants and from editorial feedback. Earlier versions of the paper circulated under the titles “An ARDL Approach to an Analysis of Asymmetric Long-run Cointegrating Relationships” and “Modelling Asymmetric Cointegration and Dynamic Multipliers in an ARDL Framework”. By virtue of its wide circulation and prolonged availability as a working paper, our research has informed the development of a subsequent literature that we now discuss. In all cases, however, the development of the NARDL model is properly credited.

  2. 2.

    The presence of long-run asymmetry will induce a ratchet mechanism if the respective positive and negative regime probabilities are approximately equal and the shocks under each regime are of comparable magnitude. In the more general case in which these conditions are not satisfied, no such simple conclusion may be drawn.

  3. 3.

    Consider the threshold ECM as an example, in which case the choice of the transition variable is of importance both theoretically and empirically. In general, the asymptotic distribution of the test statistic for the null of linearity or symmetry is not only non-standard but also depends on these transition variables.

  4. 4.

    The concept of asymmetric cointegration is easily conceptualised by use of a simple example. Consider the output-unemployment relationship. In a standard cointegrating regression, one models y t and x t subject to a common stochastic trend. As this relationship is assumed to hold in the long-run, it represents the equilibrium to which the system returns after a perturbation (i.e. it acts as a global attractor). However, in our framework, the long-run relationship between y t and x t is modelled as piecewise linear subject to the decomposition of x t . Suppose that \({\vert \beta }^{+}\vert < {\vert \beta }^{-}\vert \) in (9.1). This suggests that the long-run effect of a unit negative change in output will increase unemployment by a greater amount than a unit positive change would reduce it. Thus, our model includes a regime-switching cointegrating relationship in which regime transitions are governed by the sign of \(\Delta x_{t}\). The economic implication of this line of reasoning is that equilibrium need not be unique in a globally linear sense. The link to the path dependency literature is apparent.

  5. 5.

    In the special case where v t is normally distributed with zero mean and constant variance \(\sigma _{v}^{2}\), it is well-established that the censored normal variates, \(v_{t}^{+} =\max \left [0,v_{t}\right ]\) and \(v_{t}^{-} =\min \left [0,v_{t}\right ]\), will have \(E\left (v_{t}^{+}\right ) = \frac{\sigma _{v}} {\sqrt{2\pi }}\), \(E\left (v_{t}^{-}\right ) = - \frac{\sigma _{v}} {\sqrt{2\pi }}\), and \(V ar\left (v_{t}^{+}\right ) = V ar\left (v_{t}^{-}\right ) = \frac{\sigma _{v}^{2}} {2} \frac{\pi -1} {\pi }\). We are grateful to Jinseo Cho for pointing this issue out and encouraging us to provide a more general result in Theorem 1.

  6. 6.

    Notice that the analysis of short-run dynamic asymmetries is not straightforward in the context of the static regression model employing the semiparametric approach.

  7. 7.

    In some cases, most notably where the growth rates of the series in \(\boldsymbol{x}_{t}\) are predominantly positive (negative), the use of a zero threshold may result in one regime containing an undesirably low number of effective observations. In such situations, an obvious candidate for an alternative threshold is the mean growth rate. We discuss such issues further in a separate paper (Greenwood-Nimmo et al. 2012).

  8. 8.

    For convenience we employ the same lag order, q. One may also allow for feedback effects from the lagged \(\Delta y\)’s on \(\Delta x_{t}\) in (9.8).

  9. 9.

    While the associated critical values can be tabulated easily using stochastic simulation, it is impractical to provide a meaningful set of critical values covering all possible combinations. It is generally straightforward, however, to compute the appropriate p-values by means of standard bootstrap techniques.

  10. 10.

    It is straightforward to extend similar reasoning to the more general case with multiple regressors decomposed into partial sum processes.

  11. 11.

    The level parameters are obtained as follows:

    $$\displaystyle{ \phi _{1} =\rho +1 +\varphi _{1};\ \phi _{i} =\varphi _{i} -\varphi _{i-1},\ i = 2,\ldots,p - 1;\ \phi _{p} = -\varphi _{p-1}; }$$
    $$\displaystyle{ \boldsymbol{\theta }_{0}^{\ell} =\boldsymbol{\pi }_{ 0}^{\ell};\ \boldsymbol{\theta }_{ 1}^{\ell} {=\boldsymbol{\theta } }^{\ell} -\boldsymbol{\pi }_{ 0}^{\ell} +\boldsymbol{\pi }_{ 1}^{\ell};\ \boldsymbol{\theta }_{ i}^{\ell} =\boldsymbol{\pi }_{ i}^{\ell} -\boldsymbol{\pi }_{ i-1}^{\ell},\ i = 2,\ldots,q - 1;\ \boldsymbol{\theta }_{ q}^{\ell} = -\boldsymbol{\pi }_{ q-1}^{\ell},\ \ell = +,-. }$$
  12. 12.

    The dynamic multipliers, \(\boldsymbol{\lambda }_{j}^{+}\) and \(\boldsymbol{\lambda }_{j}^{-}\) for j = 0, 1, …, can be evaluated using the following recursive relationships in which \(\boldsymbol{\lambda }_{0}^{\ell} =\boldsymbol{\theta }_{ 0}^{\ell}\), ϕ j  = 0 for j < 1 and \(\boldsymbol{\lambda }_{j}^{\ell} =\boldsymbol{ 0}\) for j < 0:

    $$\displaystyle{ \boldsymbol{\lambda }_{j}^{\ell} =\phi _{ 1}\boldsymbol{\lambda }_{j-1}^{\ell} +\phi _{ 2}\boldsymbol{\lambda }_{j-2}^{\ell} +\ldots +\phi _{ j-1}\boldsymbol{\lambda }_{1}^{\ell} +\phi _{ j}\boldsymbol{\lambda }_{0}^{\ell} +\boldsymbol{\theta }_{ j}^{\ell},\ \ell = +,-,\ j = 1, 2,\ldots, }$$
  13. 13.

    The final specification in Borenstein et al. (1997) differs slightly from (9.14) as the lagged \(\Delta y_{t}\)’s on the right hand side are also decomposed into positive and negative changes. However, their derivation is rather ad hoc.

  14. 14.

    Short-run symmetry restrictions (especially the pair-wise restrictions) may be excessively restrictive in many applications although they may be useful in providing more precise estimation results, particularly when estimating a long-run asymmetric relationship in small samples. The additive symmetry restrictions are somewhat weaker and have been discussed in the literature in terms of assessing the validity of the liquidity constraint where \(\sum _{i=0}^{q-1}\boldsymbol{\pi }_{i}^{+} <\sum _{ i=0}^{q-1}\boldsymbol{\pi }_{i}^{-}\) (e.g. Van Treeck 2008).

  15. 15.

    Webber (2000) utilises a similar approach in his analysis of the asymmetric pass-through from exchange rates, decomposed as the partial sum processes of appreciations and depreciations, to import prices.

  16. 16.

    Full results are available on request.

  17. 17.

    We employ a non-parametric bootstrapping routine and use 50,000 replications after rejecting those for which \(\rho > -1 \times 1{0}^{-4}\). Full details are available on request.

  18. 18.

    Earlier drafts of the paper include an additional illustration which has subsequently been removed to conserve space. Previously, the NARDL model was used to investigate to the so-called ‘rockets-and-feathers’ hypothesis associated with Bacon (1991), which describes how retail gasoline prices tend to react asymmetrically to changes in the price of crude oil (an exhaustive survey is provided by Grasso and Manera 2007). Working with Korean data spanning the period 1991q1–2007q2, our results confirm that gasoline prices respond more rapidly to increases in the price of crude oil than to decreases. Furthermore, our results suggest that the gasoline price is more sensitive to exchange rate depreciations than to appreciations and that gasoline price adjustments are approximately symmetric in the long-run. A complete discussion is available from the authors on request.

  19. 19.

    Further examples of the use of positive/negative decompositions in the modelling of asymmetry in the unemployment-output relationship include Lee (2000) and Virén (2001).

  20. 20.

    Seasonally-adjusted monthly data for unemployment and industrial production covering the range 1982m2–2003m11 were collected from the OECD’s Main Economic Indicators. Although not presented here, ADF testing lends overwhelming support to the hypothesis that all variates are I(1).

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Acknowledgements

This is a substantially revised version of an earlier working paper by Shin and Yu (2004). Earlier versions circulated under the titles “An ARDL Approach to an Analysis of Asymmetric Long-Run Cointegrating Relationships” and “Modelling Asymmetric Cointegration and Dynamic Multipliers in an ARDL Framework”. We are grateful to Badi Baltagi, Jinseo Cho, Ana-Maria Fuertes, Liang Hu, John Hunter, Minjoo Kim, Soyoung Kim, Gary Koop, Kevin Lee, Camilla Mastromarco, Emi Mise, Viet Nguyen, Neville Norman, Hashem Pesaran, Kevin Reilly, Laura Serlenga, Ron Smith, Till van Treeck and participants at the ESEM conference (Vienna 2006), the ICAETE conference (Hyderabad 2009), and research seminars at the IMK, the Bank of Korea, and the Universities of Bari, Lecce, Leeds, Leicester, Korea and Yonsei for their helpful comments. This paper has been widely circulated and the methodology adopted by a number of authors – we are pleased to acknowledge their valuable feedback, comments and discussion. Shin acknowledges partial financial support from the ESRC (Grant No. RES-000-22-3161). Yu is grateful for the hospitality of Leeds University Business School during his visit. The usual disclaimer applies.

Author information

Authors and Affiliations

  1. Department of Economics and Related Studies, University of York, Heslington, York, YO10 5DD, UK

    Yongcheol Shin

  2. Department of International Trade (BB-1408), School of Management, Dong-A University, Bumin-Dong, 2 Ga-1 Seo-gu, Busan, South Korea

    Byungchul Yu

  3. Department of Economics, University of Melbourne, 111 Barry Street, Carlton, VIC, 3053, Australia

    Matthew Greenwood-Nimmo

Authors
  1. Yongcheol Shin
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  2. Byungchul Yu
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  3. Matthew Greenwood-Nimmo
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Corresponding author

Correspondence to Yongcheol Shin .

Editor information

Editors and Affiliations

  1. Department of Economics, Rice University, Houston, Texas, USA

    Robin C. Sickles

  2. Department of Economics, Syracuse University, Syracuse, New York, USA

    William C. Horrace

Appendix

Appendix

9.1.1 Proof of Theorem 1

The OLS estimator, \(\hat{\boldsymbol{\beta }}:= {{(\hat{\beta }}^{+}{,\hat{\beta }}^{-})}^{{\prime}}\), in (9.1) is obtained by

$$\displaystyle{ \hat{\boldsymbol{\beta }}={ \left [\begin{array}{cc} \sum _{t=1}^{T}{\left (x_{t}^{+}\right )}^{2} & \sum _{t=1}^{T}x_{t}^{+}x_{t}^{-} \\ \sum _{t=1}^{T}x_{t}^{+}x_{t}^{-}& \sum _{t=1}^{T}{\left (x_{t}^{-}\right )}^{2} \end{array} \right ]}^{-1}\left [\begin{array}{c} \begin{array}{c} \sum _{t=1}^{T}x_{ t}^{+}y_{ t} \\ \sum _{t=1}^{T}x_{t}^{-}y_{t} \end{array} \end{array} \right ], }$$

so that

$$\displaystyle{ \hat{\boldsymbol{\beta }}-\boldsymbol{\beta } = \frac{1} {D_{T}}\left [\begin{array}{cc} \sum _{t=1}^{T}{\left (x_{t}^{-}\right )}^{2} & -\sum _{t=1}^{T}x_{t}^{+}x_{t}^{-} \\ -\sum _{t=1}^{T}x_{t}^{+}x_{t}^{-}& \sum _{t=1}^{T}{\left (x_{t}^{+}\right )}^{2} \end{array} \right ]\left [\begin{array}{c} \begin{array}{c} \sum _{t=1}^{T}x_{t}^{+}u_{t} \\ \sum _{t=1}^{T}x_{t}^{-}u_{t} \end{array} \end{array} \right ] = \frac{1} {D_{T}}\left [\begin{array}{c} A_{T} \\ B_{T} \end{array} \right ], }$$

where \(D_{T}:=\sum _{ t=1}^{T}{\left (x_{t}^{+}\right )}^{2}\sum _{t=1}^{T}{\left (x_{t}^{-}\right )}^{2} -{\left (\sum _{t=1}^{T}x_{t}^{+}x_{t}^{-}\right )}^{2}\), \(A_{T}:=\sum _{ t=1}^{T}{\left (x_{t}^{-}\right )}^{2}\) \(\sum _{t=1}^{T}x_{t}^{+}u_{t} -\sum _{t=1}^{T}x_{t}^{+}x_{t}^{-}\sum _{t=1}^{T}x_{t}^{-}u_{t}\), and \(B_{T}:= -\sum _{t=1}^{T}x_{t}^{+}x_{t}^{-}\sum _{t=1}^{T}x_{t}^{+}u_{t} +\sum _{ t=1}^{T}{\left (x_{t}^{+}\right )}^{2}\sum _{t=1}^{T}x_{t}^{-}u_{t}\). We now let

$$\displaystyle{ w_{t}^{+}:=\max [0,v_{ t}] {-\mu }^{+},\;\;\;\;w_{ t}^{-}:=\min [0,v_{ t}] {-\mu }^{-}, }$$

where \({\mu }^{+}:= E\left [\max [0,v_{t}]\right ]\) and \({\mu }^{-}:= E\left [\min [0,v_{t}]\right ]\), so that

$$\displaystyle{ x_{t}^{+} \equiv {t\mu }^{+} +\sum _{ j=1}^{t}w_{ j}^{+},\;\;\;\;x_{ t}^{-}\equiv {t\mu }^{-} +\sum _{ j=1}^{t}w_{ j}^{-} }$$

Hence, we obtain:

$$\displaystyle\begin{array}{rcl} D_{T}& =& \left \{\sum _{t=1}^{T}{t}^{2}\right \}\left \{\sum _{ t=1}^{T}\left [{\mu }^{+2}{\left (\sum _{ j=1}^{t}w_{ j}^{-}\right )}^{2} {+\mu }^{-2}{\left (\sum _{ j=1}^{t}w_{ j}^{+}\right )}^{2}\right.\right. {}\\ & & -\,{2\mu }^{+}\left.\left.{\mu }^{-}\left (\sum _{ j=1}^{t}w_{ j}^{-}\right )\left (\sum _{ j=1}^{t}w_{ j}^{+}\right )\right ]\right \} {}\\ & & -\left \{{\mu }^{+2}{\left (\sum _{ t=1}^{T}t\sum _{ j=1}^{t}w_{ j}^{-}\right )}^{2}\right. {}\\ & & +\,\left.{\mu }^{-2}{\left (\sum _{ t=1}^{T}t\sum _{ j=1}^{t}w_{ j}^{+}\right )}^{2} - {2\mu {}^{+}\mu }^{-}\left (\sum _{ t=1}^{T}t\sum _{ j=1}^{t}w_{ j}^{-}\right )\left (\sum _{ t=1}^{T}t\sum _{ j=1}^{t}w_{ j}^{+}\right )\right \} {}\\ & & +\,o_{P}({T}^{5}). {}\\ \end{array}$$

Here, o P (T 6) terms are canceled off, and the remaining next-order terms are stated as above. We now note that

$$\displaystyle{ \frac{1} {{T}^{3}}\sum\limits_{t=1}^{T}{t}^{2} = \frac{1} {3} + o(1), }$$
$$\displaystyle{{ \mu }^{+2}{\left (\sum _{ j=1}^{t}w_{ j}^{-}\right )}^{2} {+\mu }^{-2}{\left (\sum _{ j=1}^{t}w_{ j}^{+}\right )}^{2} - {2\mu {}^{+}\mu }^{-}\left (\sum _{ j=1}^{t}w_{ j}^{-}\right )\left (\sum _{ j=1}^{t}w_{ j}^{+}\right ) ={ \left (\sum _{ j=1}^{t}s_{ j}\right )}^{2} }$$

where \(s_{j} {\equiv \mu }^{+}w_{j}^{-}{-\mu }^{-}w_{j}^{-}\) by the definitions of \(w_{j}^{-}\) and \(w_{j}^{+}\). Hence, by Donsker’s FCLT

$$\displaystyle{ {T}^{-1/2}\sum\limits_{ j=1}^{T(\cdot )}s_{ t}/\sigma _{s} \Rightarrow W_{\tilde{s}}(\cdot ), }$$

where \(\sigma _{s}^{2}:= V ar\left (s_{t}\right )\), ⇒ indicates weak convergence, and \(W_{\tilde{s}}(r)\) is the standard Brownian motions defined on \(r \in \left [0,1\right ]\). Therefore,

$$\displaystyle{ {T}^{-2}\sum\limits_{ t=1}^{T}{\left (\sum\limits_{ j=1}^{t}s_{ j}\right )}^{2} \Rightarrow \sigma _{ s}^{2}\int _{ 0}^{1}W_{\tilde{ s}}{(r)}^{2}dr }$$

by the CMT (e.g. Eq. (17.3.22) of Hamilton (1994), p. 486). Also notice that

$$\displaystyle\begin{array}{rcl} & & {\mu }^{+2}{\left (\sum _{ t=1}^{T}t\sum _{ j=1}^{t}w_{ j}^{-}\right )}^{2} {+\mu }^{-2}{\left (\sum _{ t=1}^{T}t\sum _{ j=1}^{t}w_{ j}^{+}\right )}^{2} {}\\ & & \quad - {2\mu {}^{+}\mu }^{-}\left (\sum _{ t=1}^{T}t\sum _{ j=1}^{t}w_{ j}^{-}\right )\left (\sum _{ t=1}^{T}t\sum _{ j=1}^{t}w_{ j}^{+}\right ) {}\\ & =&{ \left (\sum _{t=1}^{T}t\sum _{ j=1}^{t}\left ({\mu }^{+}w_{ j}^{-}{-\mu }^{-}w_{ j}^{+}\right )\right )}^{2} ={ \left (\sum _{ t=1}^{T}t\sum _{ j=1}^{t}s_{ j}\right )}^{2}, {}\\ \end{array}$$

then it follows that

$$\displaystyle{ {T}^{-\frac{5} {2} }\sum\limits_{t=1}^{T}t\sum\limits_{j=1}^{t}s_{j} \Rightarrow \sigma _{s}\int _{0}^{1}rW_{\tilde{s}}(r)dr }$$

by the CMT. Collecting all these results we obtain:

$$\displaystyle{ {T}^{-5}D_{ T} \Rightarrow \sigma _{s}^{2}\left [\frac{1} {3}\int _{0}^{1}W_{\tilde{ s}}{(r)}^{2}dr -{\left (\int _{ 0}^{1}rW_{\tilde{ s}}(r)dr\right )}^{2}\right ]. }$$
(9.25)

Next, we consider the asymptotic weak limit of the numerator of \({\hat{\beta }}^{+} {-\beta }^{+}\). For this, we note that the O P (T 9∕2) terms cancel off and that the remaining next-order terms are O p (T 4) so that

$$\displaystyle\begin{array}{rcl} A_{T}&:=& \sum _{t=1}^{T}{\left (x_{ t}^{-}\right )}^{2}\sum _{ t=1}^{T}x_{ t}^{+}u_{ t} -\sum _{t=1}^{T}x_{ t}^{+}x_{ t}^{-}\sum _{ t=1}^{T}x_{ t}^{-}u_{ t} \\ & =& \left \{{\mu }^{-2}\sum _{ t=1}^{T}{t}^{2}\sum _{ t=1}^{T}u_{ t}\sum _{j=1}^{t}w_{ j}^{+} + {2\mu {}^{-}\mu }^{+}\sum _{ t=1}^{T}tu_{ t}\sum _{t=1}^{T}\sum _{ j=1}^{t}w_{ j}^{-}\right \} \\ & &-\left \{{\mu {}^{+}\mu }^{-}\sum _{ t=1}^{T}{t}^{2}\sum _{ t=1}^{T}u_{ t}\sum _{j=1}^{t}w_{ j}^{+} +\sum _{ t=1}^{T}t\sum _{ j=1}^{t}{{(\mu }^{+}w_{ j}^{-} {+\mu }^{-}w_{ j}^{+})\mu }^{-}\sum _{ t=1}^{T}tu_{ t}\right \} \\ & & \quad + o_{P}({T}^{4}) \\ & =& {\mu }^{-}\left \{-\left (\sum _{ t=1}^{T}{t}^{2}\right )\left (\sum _{ t=1}^{T}u_{ t}\sum _{j=1}^{t}s_{ j}\right ) + \left (\sum _{t=1}^{T}t\sum _{ j=1}^{t}s_{ j}\right )\left (\sum _{t=1}^{T}tu_{ t}\right )\right \} + o_{P}({T}^{4}){}\end{array}$$
(9.26)

where we also employ the definition of \(s_{j}:{=\mu }^{+}w_{j}^{-}{-\mu }^{-}w_{j}^{+}\). Then, by the CMT (e.g. Eqs. (f) on p. 548 and (17.3.19) on p. 486 of Hamilton (1994), respectively) we have:

$$\displaystyle{ {T}^{-1}\sum\limits_{ t=1}^{T}u_{ t}\sum\limits_{j=1}^{t}s_{ j} \Rightarrow \sigma _{s}\sigma _{u}\int _{0}^{1}W_{\tilde{ s}}(r)dW_{\tilde{u}}(r) }$$
(9.27)
$$\displaystyle{ {T}^{-\frac{3} {2} }\sum _{t=1}^{T}tu_{t} \Rightarrow \sigma _{u}\left (W_{\tilde{u}}(1) -\int _{0}^{1}W_{\tilde{u}}(r)dr\right ) }$$
(9.28)

where \(W_{\tilde{u}}(\cdot )\) is a standard Brownian motion independent of \(W_{\tilde{s}}(\cdot )\). Collecting all these results and (9.28) and plugging them into A T , we obtain by the CMT:

$$\displaystyle\begin{array}{rcl}{ T}^{-4}A_{ T}& \Rightarrow & {\mu }^{-}\sigma _{ s}\sigma _{u} \\ & & \times \left \{-\frac{1} {3}\int _{0}^{1}W_{\tilde{ s}}(r)dW_{\tilde{u}}(r) +\int _{ 0}^{1}rW_{\tilde{ s}}(r)dr\left (W_{\tilde{u}}(1) -\int _{0}^{1}W_{\tilde{ u}}(r)dr\right )\right \} \\ & & {}\end{array}$$
(9.29)

We now examine the numerator of \({(\hat{\beta }}^{-}{-\beta }^{-})\) in a similar manner. That is,

$$\displaystyle{ B_{T}:{=\mu }^{+}\sigma _{ s}\sigma _{u}\left \{\left (\sum\limits_{t=1}^{T}{t}^{2}\right )\left (\sum\limits_{ t=1}^{T}u_{ t}\sum\limits_{j=1}^{t}s_{ j}\right ) -\left (\sum\limits_{t=1}^{T}t\sum\limits_{ j=1}^{t}s_{ j}\right )\left (\sum\limits_{t=1}^{T}tu_{ t}\right )\right \} + o_{P}({T}^{4}),\qquad }$$
(9.30)

and

$$\displaystyle\begin{array}{rcl}{ T}^{-4}B_{ T}{\Rightarrow \mu }^{+}\sigma _{ s}\sigma _{u}\left \{\frac{1} {3}\int _{0}^{1}W_{\tilde{ s}}(r)dW_{\tilde{u}}(r)-\int _{0}^{1}rW_{\tilde{ s}}(r)dr\left (W_{\tilde{u}}(1)-\int _{0}^{1}W_{\tilde{ u}}(r)dr\right )\right \}& & \\ & &{}\end{array}$$
(9.31)

Combining (9.29) and (9.31) respectively with (9.25) we obtain the main results.

Next, from (9.26) and (9.30), it is easily seen that

$$\displaystyle{{ \mu }^{+}A_{ T} {+\mu }^{-}B_{ T} = o_{P}({T}^{4}), }$$

which proves the final result in Theorem 1.

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Shin, Y., Yu, B., Greenwood-Nimmo, M. (2014). Modelling Asymmetric Cointegration and Dynamic Multipliers in a Nonlinear ARDL Framework. In: Sickles, R., Horrace, W. (eds) Festschrift in Honor of Peter Schmidt. Springer, New York, NY. https://doi.org/10.1007/978-1-4899-8008-3_9

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