Modelling Asymmetric Cointegration and Dynamic Multipliers in a Nonlinear ARDL Framework

Yongcheol Shin; Byungchul Yu; Matthew Greenwood-Nimmo

doi:10.1007/978-1-4899-8008-3_9

Festschrift in Honor of Peter Schmidt pp 281-314 | Cite as

Modelling Asymmetric Cointegration and Dynamic Multipliers in a Nonlinear ARDL Framework

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Yongcheol Shin
Byungchul Yu
Matthew Greenwood-Nimmo

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First Online: 05 February 2014

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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.

JEL Classification:

C12 C13 J64

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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.

Appendix

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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Yongcheol Shin
- 1
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Byungchul Yu
- 2
Matthew Greenwood-Nimmo
- 3

1.Department of Economics and Related StudiesUniversity of YorkHeslingtonUK
2.Department of International Trade (BB-1408), School of ManagementDong-A UniversityBusanSouth Korea
3.Department of EconomicsUniversity of MelbourneCarltonAustralia

Modelling Asymmetric Cointegration and Dynamic Multipliers in a Nonlinear ARDL Framework

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