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A long-run model in econometrics estimates the relationship between variables when they reach a stable, long-term equilibrium. It is often used to analyze how dependent variables (e.g., output, income) respond to changes in independent variables (e.g., investment, technology) over time.
The long-run model is defined as follow:
\[\begin{equation*} \begin{split} {lexpger}_t = \alpha_0 + \alpha _1 {lpitr}_{t } + \alpha _2 {lrer}_{t } + \epsilon _t \end{split} \end{equation*}\]
The ECM is a type of econometric model used to estimate how quickly variables return to equilibrium after a short-term shock. It incorporates both short-term adjustments and long-term equilibrium relationships, allowing for more accurate modeling of time series data when variables are cointegrated.
The error correction model can be defined as:
\[\begin{equation*} \begin{split} \Delta {lexpger}_t = \beta_0 + \sum_{j=1}^{p} { \beta_{1j} \Delta {lexpger}_{t-j} }+ \sum_{j=0}^{q} { \beta_{2j} \Delta {lpitr}_{t-j} }+ \sum_{j=0}^{m} { \beta_{3j} \Delta {lrer}_{t-j} }+ \gamma_1 {trend}_{t} + \varphi t + \theta \epsilon _{t-1} + e_t \end{split} \end{equation*}\]
By embedding the long-run equilibrium relationship from the long-run model into the ECM, the model can account for both immediate changes and the gradual return to equilibrium, capturing both short-term fluctuations and the overarching long-run trend.
By using the long-run model into the ECM model we can have:
\[\begin{equation*} \begin{split} \Delta {lexpger}_t = \psi + \eta _0 {lexpger}_{t-1} + \eta _1 {lpitr}_{t-1} + \eta _2 {lrer}_{t-1} +\sum_{j=1}^{p} { \beta_{1j} \Delta {lexpger}_{t-j} }+ \sum_{j=0}^{q} { \beta_{2j} \Delta {lpitr}_{t-j} }+ \sum_{j=0}^{m} { \beta_{3j} \Delta {lrer}_{t-j} }+ \gamma_1 {trend}_{t} + \varphi t + e_t \end{split} \end{equation*}\]
The ARDL (Auto-Regressive Distributed Lag) model is a regression model that includes lags of both the dependent and independent variables. It is particularly useful in analyzing relationships when variables have different orders of integration, as it can handle both I(0) (stationary) and I(1) (non-stationary) series. It provides a framework to explore both long-term relationships and short-term dynamics.
We have ARDL model with following definiation:
\[\begin{equation*} \begin{split} ARDL(p,q,m) \end{split} \end{equation*}\]
By embedding the residuals from the long-run model into the ECM, new parameters can be introduced.
We used following modifications to obtain the ARDL model:
\[\begin{equation*} \begin{split} \psi = \beta_0 - \theta \alpha_0 \:,\: \eta _0 =\theta \:,\: \eta _1 = -{\theta \alpha _1 } \:,\: \eta _2 = -{\theta \alpha _2 } \end{split} \end{equation*}\]
To obtain estimated values for the long-run model from the ARDL model estimation, additional calculations are required.
Then, for reobtaining the long-run coefficients…:
\[\begin{equation*} \begin{split} \: \theta = \eta _0 \:,\: { \alpha _1 =- \frac { \eta _1 } {\theta }} \:,\: { \alpha _2 =- \frac { \eta _2 } {\theta }} \end{split} \end{equation*}\]
Asymmetric models capture different responses to positive and negative changes in independent variables. For example, a variable like oil prices might affect economic output differently during increases versus decreases. Asymmetric models allow for this kind of differential impact, which standard models may not capture.
The decomposition formula is as follow:
\[\begin{equation*} \begin{split} {lpitr}^{\text{+ } }_t = \sum_{i=1}^{t} { \Delta {lpitr}^{\text{+ } }_i } = \sum_{i=1}^{t} {max(\Delta {lpitr}_i ,0) } \:;\:{lpitr}^{\text{- } }_t = \sum_{i=1}^{t} { \Delta {lpitr}^{\text{- } }_i }= \sum_{i=1}^{t} {min(\Delta {lpitr}_i ,0) } \\ {lrer}^{\text{+ } }_t = \sum_{i=1}^{t} { \Delta {lrer}^{\text{+ } }_i } = \sum_{i=1}^{t} {max(\Delta {lrer}_i ,0) } \:;\:{lrer}^{\text{- } }_t = \sum_{i=1}^{t} { \Delta {lrer}^{\text{- } }_i }= \sum_{i=1}^{t} {min(\Delta {lrer}_i ,0) } \end{split} \end{equation*}\]
The asymmetric long-run model extends the concept of asymmetry to the long-term relationship, estimating how positive and negative changes in independent variables differently impact the dependent variable over the long run.
The long-run model containing asymmetric variables is defined as follow:
\[\begin{equation*} \begin{split} {lexpger}_t = \alpha_0 + \alpha^{\text{+ } } _1 {lpitr}^{\text{+ } }_{t } + \alpha^{\text{- } } _1 {lpitr}^{\text{- } }_{t } + \alpha^{\text{+ } } _2 {lrer}^{\text{+ } }_{t } + \alpha^{\text{- } } _2 {lrer}^{\text{- } }_{t } + \epsilon _t \end{split} \end{equation*}\]
In this context, an asymmetric model explicitly models how variables respond differently depending on the direction of the change, applicable in both short and long-run analyses.
Non-linear ARDL model is as follow:
\[\begin{equation*} \begin{split} \Delta {lexpger}_t = \psi + \eta _0 {lexpger}_{t-1} + \eta^{\text{+ } } _1 {lpitr}^{\text{+ } }_{t-1} + \eta^{\text{- } } _1 {lpitr}^{\text{- } }_{t-1} + \eta^{\text{+ } } _2 {lrer}^{\text{+ } }_{t-1} + \eta^{\text{- } } _2 {lrer}^{\text{- } }_{t-1} +\sum_{j=1}^{p} { \beta_{1j} \Delta {lexpger}_{t-j} }+ \sum_{j=0}^{q}{\beta^{\text{+ } }_{2j} \Delta {lpitr}^{\text{+ } }_{t-j}}+\sum_{j=0}^{q} {\beta^{\text{- } }_{2j} \Delta {lpitr}^{\text{- } }_{t-j}}+ \sum_{j=0}^{m}{\beta^{\text{+ } }_{3j} \Delta {lrer}^{\text{+ } }_{t-j}}+\sum_{j=0}^{m} {\beta^{\text{- } }_{3j} \Delta {lrer}^{\text{- } }_{t-j}}+ \gamma_1 {trend}_{t} + \varphi t + e_t \end{split} \end{equation*}\]
By embedding the residuals from the nonlinear long-run model into the ECM, which includes nonlinear independent variables, new parameters can be introduced.
We used following modifications to obtain the NARDL model:
\[\begin{equation*} \begin{split} \psi = \beta_0 - \theta \alpha_0 \:,\: \eta _0 =\theta \:,\: \eta^{\text{+ } } _1 = -{\theta \alpha^{\text{+ } } _1 } \:,\: \eta^{\text{- } } _1 = -{\theta \alpha^{\text{- } } _1 } \:,\: \eta^{\text{+ } } _2 = -{\theta \alpha^{\text{+ } } _2 } \:,\: \eta^{\text{- } } _2 = -{\theta \alpha^{\text{- } } _2 } \end{split} \end{equation*}\]
To get estiamted values for the long-run NARDL model’s values some calculations should be performed.
Then, for reobtaining the NARDL long-run coefficients…:
\[\begin{equation*} \begin{split} \: \theta = \eta _0 \:,\: \alpha^{\text{+ } } _1 =- { \frac { \eta^{\text{+ } } _1 } {\theta }} \:,\: \alpha^{\text{- } } _1 = -{ \frac{ \eta^{\text{- } } _1 } {\theta }} \:,\: \alpha^{\text{+ } } _2 =- { \frac { \eta^{\text{+ } } _2 } {\theta }} \:,\: \alpha^{\text{- } } _2 = -{ \frac{ \eta^{\text{- } } _2 } {\theta }} \end{split} \end{equation*}\]
The asymmetric short-run model analyzes the differing effects of positive and negative changes in independent variables in the short term. While nonlinearity exists in the short run, linearity is maintained in the long run.
The NARDL model in the case of linearity in the long-run is as follow:
\[\begin{equation*} \begin{split} \Delta {lexpger}_t = \psi + \eta _0 {lexpger}_{t-1} + \eta _1 {lpitr}_{t-1} + \eta _2 {lrer}_{t-1} +\sum_{j=1}^{p} { \beta_{1j} \Delta {lexpger}_{t-j} }+ \sum_{j=0}^{q}{\beta^{\text{+ } }_{2j} \Delta {lpitr}^{\text{+ } }_{t-j}}+\sum_{j=0}^{q} {\beta^{\text{- } }_{2j} \Delta {lpitr}^{\text{- } }_{t-j}}+ \sum_{j=0}^{m}{\beta^{\text{+ } }_{3j} \Delta {lrer}^{\text{+ } }_{t-j}}+\sum_{j=0}^{m} {\beta^{\text{- } }_{3j} \Delta {lrer}^{\text{- } }_{t-j}}+ \gamma_1 {trend}_{t} + \varphi t + e_t \end{split} \end{equation*}\]
This variant of the asymmetric model considers long-term effects, focusing on how variables might have different impacts on the dependent variable in the long run, based on the direction of changes.
The NARDL model in the case of linearity in short-run
\[\begin{equation*} \begin{split} \Delta {lexpger}_t = \psi + \eta _0 {lexpger}_{t-1} + \eta^{\text{+ } } _1 {lpitr}^{\text{+ } }_{t-1} + \eta^{\text{- } } _1 {lpitr}^{\text{- } }_{t-1} + \eta^{\text{+ } } _2 {lrer}^{\text{+ } }_{t-1} + \eta^{\text{- } } _2 {lrer}^{\text{- } }_{t-1} +\sum_{j=1}^{p} { \beta_{1j} \Delta {lexpger}_{t-j} }+ \sum_{j=0}^{q} { \beta_{2j} \Delta {lpitr}_{t-j} }+ \sum_{j=0}^{m} { \beta_{3j} \Delta {lrer}_{t-j} }+ \gamma_1 {trend}_{t} + \varphi t + e_t \end{split} \end{equation*}\]
Asymmetric dynamics describe the process by which variables adjust over time, capturing differing speeds or magnitudes of response to positive versus negative shocks. These dynamics are critical in accurately modeling the evolution of economic or financial systems under asymmetric influences.
The dynamic multipliers formulas are as follow:
\[\begin{equation*} \begin{split} m^{\text{+ } } _h = \sum_{i=0}^{h}{{\frac{ \partial {lexpger} _{t+i} } {\partial {lpitr}^{\text{+ } } _t} }} \:;\: m^{\text{- } } _h = \sum_{i=0}^{h} { \frac{ \partial {lexpger} _{t+i} } {\partial {lpitr}^{\text{- } } _t} } \\ \lim_ { h \to \infty } m^{\text{+ } } _h = \alpha^{\text{+ } } _1 \:,\:\lim_ { h \to \infty } m^{\text{- } } _h = \alpha^{\text{- } } _1 \\ m^{\text{+ } } _h = \sum_{i=0}^{h}{{\frac{ \partial {lexpger} _{t+i} } {\partial {lrer}^{\text{+ } } _t} }} \:;\: m^{\text{- } } _h = \sum_{i=0}^{h} { \frac{ \partial {lexpger} _{t+i} } {\partial {lrer}^{\text{- } } _t} } \\ \lim_ { h \to \infty } m^{\text{+ } } _h = \alpha^{\text{+ } } _1 \:,\:\lim_ { h \to \infty } m^{\text{- } } _h = \alpha^{\text{- } } _1 \end{split} \end{equation*}\]