Threshold cointegration in R with package tsDyn Matthieu Stigler - PowerPoint PPT Presentation

Threshold cointegration in R with package tsDyn Matthieu Stigler Matthieu.Stigler at gmail.com 8 July 2009 National Institute for Public Finance and Policy, India Agroscope, Federal Office for Agriculture, Switzerland Matthieu Stigler Matthieu.Stigler at gmail.com () Threshold cointegration in R with package tsDyn 8 July 2009 1 / 26

Outline Cointegration (linear) 1 Threshold cointegration 2 Areas of application 3 Implementation in R 4 Matthieu Stigler Matthieu.Stigler at gmail.com () Threshold cointegration in R with package tsDyn 8 July 2009 2 / 26

Outline Cointegration (linear) 1 Threshold cointegration 2 Areas of application 3 Implementation in R 4 Matthieu Stigler Matthieu.Stigler at gmail.com () Threshold cointegration in R with package tsDyn 8 July 2009 3 / 26

Background Non-stationnary variables with unit root: I(1) Spurious regression when I(1) regressed on I(1): ◮ R 2 → 1 ◮ Statistical dependance among independant variables ◮ Wrong conclusions! Matthieu Stigler Matthieu.Stigler at gmail.com () Threshold cointegration in R with package tsDyn 8 July 2009 4 / 26

Background Non-stationnary variables with unit root: I(1) Spurious regression when I(1) regressed on I(1): ◮ R 2 → 1 ◮ Statistical dependance among independant variables ◮ Wrong conclusions! Matthieu Stigler Matthieu.Stigler at gmail.com () Threshold cointegration in R with package tsDyn 8 July 2009 4 / 26

Cointegration Definition (Cointegration (Engle, Granger 1982)) If two (or more) variables are non-stationary , but there exist a linear combination of them which is stationary , there are said to be cointegrated Example X and Y as I(1), Take X t − aY t = ε t X and Y cointegrated ⇔ ε is I(0) Matthieu Stigler Matthieu.Stigler at gmail.com () Threshold cointegration in R with package tsDyn 8 July 2009 5 / 26

Cointegration Definition (Cointegration (Engle, Granger 1982)) If two (or more) variables are non-stationary , but there exist a linear combination of them which is stationary , there are said to be cointegrated Example X and Y as I(1), Take X t − aY t = ε t X and Y cointegrated ⇔ ε is I(0) Matthieu Stigler Matthieu.Stigler at gmail.com () Threshold cointegration in R with package tsDyn 8 July 2009 5 / 26

Interest of linear cointegration Stable long-run relationship between random walk variables. Error-correction mechanisms pushing deviations back towards the long-run equilibirum. Example (VECM model with cointegrated variables) � 0 . 02 � � � � � − 0 . 01 ∆ X t − 1 ∆ X t � � ECT t − 1 + ( 0 . 04 0 . 02 = + 0 . 31 0 . 07 ) − 0 . 01 ∆ Y t 0 . 08 ∆ Y t − 1 Where ECT (error-correction term) represents deviations from the long-run relationship ECT t − 1 = Y t − bX t Matthieu Stigler Matthieu.Stigler at gmail.com () Threshold cointegration in R with package tsDyn 8 July 2009 6 / 26

Interest of linear cointegration Stable long-run relationship between random walk variables. Error-correction mechanisms pushing deviations back towards the long-run equilibirum. Example (VECM model with cointegrated variables) � 0 . 02 � � � � � − 0 . 01 ∆ X t − 1 ∆ X t � � ECT t − 1 + ( 0 . 04 0 . 02 = + 0 . 31 0 . 07 ) − 0 . 01 ∆ Y t 0 . 08 ∆ Y t − 1 Where ECT (error-correction term) represents deviations from the long-run relationship ECT t − 1 = Y t − bX t Matthieu Stigler Matthieu.Stigler at gmail.com () Threshold cointegration in R with package tsDyn 8 July 2009 6 / 26

Interest of linear cointegration Stable long-run relationship between random walk variables. Error-correction mechanisms pushing deviations back towards the long-run equilibirum. Example (VECM model with cointegrated variables) � 0 . 02 � � � � � − 0 . 01 ∆ X t − 1 ∆ X t � � ECT t − 1 + ( 0 . 04 0 . 02 = + 0 . 31 0 . 07 ) − 0 . 01 ∆ Y t 0 . 08 ∆ Y t − 1 Where ECT (error-correction term) represents deviations from the long-run relationship ECT t − 1 = Y t − bX t Matthieu Stigler Matthieu.Stigler at gmail.com () Threshold cointegration in R with package tsDyn 8 July 2009 6 / 26

Interest of linear cointegration Stable long-run relationship between random walk variables. Error-correction mechanisms pushing deviations back towards the long-run equilibirum. Example (VECM model with cointegrated variables) � 0 . 02 � � � � ∆ X t − 1 ∆ X t � − 0 . 01 ECT t − 1 + ( 0 . 04 0 . 02 � � = + 0 . 31 0 . 07 ) ∆ Y t − 0 . 01 ∆ Y t − 1 0 . 08 Whith ECT t − 1 = Y t − bX t Matthieu Stigler Matthieu.Stigler at gmail.com () Threshold cointegration in R with package tsDyn 8 July 2009 7 / 26

The assumption of linearity Implicit assumption: every small/big deviation from equilibirum leads to instantaneous correction . But economic theory suggests: Transaction costs (no adjustment when: deviations < transaction costs) Stickiness of the price Asymetries: + / − deviations don’t lead to same effect Matthieu Stigler Matthieu.Stigler at gmail.com () Threshold cointegration in R with package tsDyn 8 July 2009 8 / 26

The assumption of linearity Implicit assumption: every small/big deviation from equilibirum leads to instantaneous correction . But economic theory suggests: Transaction costs (no adjustment when: deviations < transaction costs) Stickiness of the price Asymetries: + / − deviations don’t lead to same effect Matthieu Stigler Matthieu.Stigler at gmail.com () Threshold cointegration in R with package tsDyn 8 July 2009 8 / 26

Outline Cointegration (linear) 1 Threshold cointegration 2 Areas of application 3 Implementation in R 4 Matthieu Stigler Matthieu.Stigler at gmail.com () Threshold cointegration in R with package tsDyn 8 July 2009 9 / 26

The threshold autoregressive (TAR) model Linear model: AR : ε t = ρε t − 1 + u t Regime-specific dynamics in the Threshold Autoregressive (TAR) model: � ρ L ε t − 1 + u t if ε t − 1 ≤ 0 TAR(2) : ε t = ρ H ε t − 1 + u t if 0 ≤ ε t − 1 ρ L ε t − 1 + u t ε t − 1 ≤ θ L  if  θ L ≤ ε t − 1 ≤ θ H ρ M ε t − 1 + u t TAR(3) : ε t = if θ H ≤ ε t − 1 ρ H ε t − 1 + u t if  Stationarity condition: | ρ L | < 1 , | ρ H | < 1 | ρ M | < ∞ (non-stationarity of middle regime doesn’t affect stationarity of whole proces) Matthieu Stigler Matthieu.Stigler at gmail.com () Threshold cointegration in R with package tsDyn 8 July 2009 10 / 26

The threshold autoregressive (TAR) model Linear model: AR : ε t = ρε t − 1 + u t Regime-specific dynamics in the Threshold Autoregressive (TAR) model: � ρ L ε t − 1 + u t if ε t − 1 ≤ 0 TAR(2) : ε t = ρ H ε t − 1 + u t if 0 ≤ ε t − 1 ρ L ε t − 1 + u t ε t − 1 ≤ θ L  if  θ L ≤ ε t − 1 ≤ θ H ρ M ε t − 1 + u t TAR(3) : ε t = if θ H ≤ ε t − 1 ρ H ε t − 1 + u t if  Stationarity condition: | ρ L | < 1 , | ρ H | < 1 | ρ M | < ∞ (non-stationarity of middle regime doesn’t affect stationarity of whole proces) Matthieu Stigler Matthieu.Stigler at gmail.com () Threshold cointegration in R with package tsDyn 8 July 2009 10 / 26

The threshold autoregressive (TAR) model Linear model: AR : ε t = ρε t − 1 + u t Regime-specific dynamics in the Threshold Autoregressive (TAR) model: � ρ L ε t − 1 + u t if ε t − 1 ≤ 0 TAR(2) : ε t = ρ H ε t − 1 + u t if 0 ≤ ε t − 1 ρ L ε t − 1 + u t ε t − 1 ≤ θ L  if  θ L ≤ ε t − 1 ≤ θ H ρ M ε t − 1 + u t TAR(3) : ε t = if θ H ≤ ε t − 1 ρ H ε t − 1 + u t if  Stationarity condition: | ρ L | < 1 , | ρ H | < 1 | ρ M | < ∞ (non-stationarity of middle regime doesn’t affect stationarity of whole proces) Matthieu Stigler Matthieu.Stigler at gmail.com () Threshold cointegration in R with package tsDyn 8 July 2009 10 / 26

The threshold autoregressive (TAR) model Linear model: AR : ε t = ρε t − 1 + u t Regime-specific dynamics in the Threshold Autoregressive (TAR) model: � ρ L ε t − 1 + u t if ε t − 1 ≤ 0 TAR(2) : ε t = ρ H ε t − 1 + u t if 0 ≤ ε t − 1 ρ L ε t − 1 + u t ε t − 1 ≤ θ L  if  θ L ≤ ε t − 1 ≤ θ H ρ M ε t − 1 + u t TAR(3) : ε t = if θ H ≤ ε t − 1 ρ H ε t − 1 + u t if  Stationarity condition: | ρ L | < 1 , | ρ H | < 1 | ρ M | < ∞ (non-stationarity of middle regime doesn’t affect stationarity of whole proces) Matthieu Stigler Matthieu.Stigler at gmail.com () Threshold cointegration in R with package tsDyn 8 July 2009 10 / 26

TAR with three regimes 1.0 0.5 Mean reversion zone: ρ = 0.4 0.0 No mean reversion (random walk): ρ = 1 −0.5 Mean reversion zone: ρ = 0.3 −1.0 0 50 100 150 200 Time Matthieu Stigler Matthieu.Stigler at gmail.com () Threshold cointegration in R with package tsDyn 8 July 2009 11 / 26

Threshold cointegration Definition (Threshold cointegration) If two (or more) variables are I(1), but there exist a linear combination of them which is ” threshold stationary ” , there are said to be ” threshold cointegrated ” Two main features: Allows no-adjustment band Allows asymetries: different +/- adjustment speeds ( ρ H � = ρ L ) Threshold effects in: Long-run (LR) relationship VECM Matthieu Stigler Matthieu.Stigler at gmail.com () Threshold cointegration in R with package tsDyn 8 July 2009 12 / 26