Multiple-Threshold Nonlinear ARDL (MT-NARDL)

Introduction

The Multiple Threshold Nonlinear ARDL method can be found in Verheyen (2013) as an extension of the ARDL to the nonlinearity ARDL (NARDL) model. Basically, the NARDL model decomposes the series into two around zero, implying NARDL is focused on the median value of the series as the threshold point. Rather than being focused on one threshold point (whether around the median or at the median value), the Multiple Threshold Nonlinear ARDL model is focused on more than one threshold point. The graph below gives an idea of what MTARDL does. Each vertical line is a threshold point in the distribution of the series. Arbitrary partioning can be done but is not justifiable. It is better to focus on certain partioning that is sensible. Partioning into quantiles is more appealing and that is what is done in this post.

                                                Fig1: Partioning of the series into sections


Let x be the series of interest to be portioned into subunits. Let the Q(τ|Δx) be the τ quantile of Δx. The portion of x below the quantile is given by

\[x_t ^{Q(\tau_b|\Delta x)}=\sum_{j=1}^t\{\Delta x_j<Q(\tau_b|\Delta x)\}\Delta x_j\]

Above a given quantile, the generated series is given by

\[x_t ^{Q(\tau_a|\Delta x)}=\sum_{j=1}^t\{\Delta x_j>Q(\tau_a|\Delta x)\}\Delta x_j\]

For the regime that lies in between, the generated series is given by

\[x_t ^{Q_{a|b}(\tau|\Delta x)}=\sum_{j=1}^t\{Q(\tau_a|\Delta x)<\Delta x_j<Q(\tau_b|\Delta x)\}\Delta x_j\]

Long-run Relationship

Suppose we are interested in the relationship between y and x. Specifically, suppose we want to study the quantile-based effect of x on y. The long-run relationship of interest is then given by

\[y_t =\alpha +\beta x_t^{Q(t_a)}+\chi x_t^{Q(t_b)}+\gamma x_t^{Q(t_{a|b})}+\epsilon_t\]

Adopting an ARDL framework for the model, we have an ARDL(k,l,m,n) as specified:

\[y_t =\lambda +\sum_{j=1}^k \varphi_j y_{t-j}+\sum_{j=0}^l \phi_j x_{t-j}^{Q(t_a)}+\sum_{j=0}^m \psi_j x_{t-j}^{Q(t_b)}+\sum_{j=0}^n \eta_j  x_{t-j}^{Q(t_{a|b})}+\epsilon_t\] 

A reparameterized version of the above model is given by:

\[\begin{multline*}\Delta y_t =\theta_0 + \theta_1 y_{t-1}+ \theta_2 x_{t-1}^{Q(t_a)}+ \theta_3 x_{t-1}^{Q(t_b)}+ \theta_4 x_{t-1}^{Q(t_{a|b})} +\cdots \\\sum_{j=1}^{k-1} \delta_{1,j} \Delta y_{t-j}+\sum_{j=0}^{l-1}  \delta_{2,j} \Delta  x_{t-j}^{Q(t_a)}+\sum_{j=0}^{m-1} \delta_{3,j} \Delta x_{t-j}^{Q(t_b)}+\sum_{j=0}^{n-1} \delta_{4,j} \Delta   x_{t-j}^{Q(t_{a|b})}+\mu_t\end{multline*}\] 

The FPSS seeks to test the hypothesis that

\[\theta_1 = \theta_2 = \theta_3 = \theta_4 =0\]

Rule out the degenerate cases by t-ratio tests of each of the level variables in the model. An inbuilt Eviews code needed most for the implementation of Multiple Threshold Nonlinear ARDL is:

                                                Q(τ|x)=@quantile(x,τ)

Although the model makes use of the quantile concept to deal with the problem at hand, this is not what has been termed Quantile ARDL (QARDL) in the literature. We will discuss the idea subsequently. But note that whereas the MTARDL uses the quantile on the regressors, the QARDL applies the same on the dependent variable.

Steps in implementing MTARDL in Eviews

1. Decide on the threshold variable to be decomposed (say x)
2. Compute the difference of the variable (i.e., Δx)
3. Compute the thresholds using the formula

                                series Q1 = @quantile(Δx,τ1)

                                series Q2 = @quantile(Δx,τ2)

            Note that the quantile is computed from Δx and NOT x

        4. Given the computed threshold value for Δx above, generate

            - series for the lower-tail regime:

                              series x_lw=@cumsum((Δx<Q1)*Δx)

            - series for the upper-tail regime:

                              series x_up=@cumsum((Δx>Q2)*Δx)

            - series for the inner-corridor regime:

                             series x_in=@cumsum(((Δx>Q1) and (Δx<Q2))*Δx)

            There can be more than one inner-corridor regime if there are more than two threshold points.                For example, if there is a third threshold Q3, the two inner-corridor regimes will be given by

                            series x_in1=@cumsum(((Δx>Q1) and (Δx<Q2))*Δx)

                            series x_in2=@cumsum(((Δx>Q2) and (Δx<Q3))*Δx)

      5. After computing the series for all the regimes, the model in (5) can then be estimated within the              ARDL equation environment. To do that, type

                                            y x_lw x_up x_in

         for two-threshold/three-regime model, and

                                            y x_lw x_up x_in1 x_in2

         for three-threshold/four-regime model

All the analyses that can be performed within Eviews environment for ARDL can also be carried out for MTARDL.

Eviews Implementation

Suppose we have three variables and we are seeking the LR relationship between stock price exchange rate and oil price. We'll illustrate the steps as follows:

STEP I: We decide on oil price

We are interested in the multiple threshold effect of oil price on stock market activity. We, therefore, need to find the thresholds (at 25% and 75%) for oil price.

STEPS II and III: Difference and thresholds

The screenshot below shows how we achieve this. We click on the genr button in the workfile and the Generate Series by Equation dialog box shows up. We input the expression for the first threshold, which is

                                oilthre25=@quantile(d(oilpr),0.25)

for 25 percentile and OK box. We do the same for the 75 percentile typing

                                oilthre75=@quantile(d(oilpr),0.75)

instead.

Note that we have implicitly computed the difference for oilpr in the expressions above. In other words, the first two steps are already taken care of.

STEP IV: Generate the series for the regimes

To generate series for three regimes, we need the difference and the thresholds as inputs. We achieve this through the Generate Series by Equation dialog box by typing the following:

For Regime One:

oilreg1=@cumsum((d(oilpr)<oilthre25)*d(oilpr))

For Regime Two:

oilreg2=@cumsum(((d(oilpr)>oilthre25) and (d(oilpr)<oilthre75))*d(oilpr))

For Regime Three:

oilreg3=@cumsum((d(oilpr)>oilp75)*d(oilpr))

We have the three regimes plotted in the figure below.

STEP V: Multiple Threshold ARDL

We are now in a position to do MTARDL. Go to Quick and select Estimate Equation…. From the Equation Estimation dialog box, change the method to ARDL – Auto-regressive Distributed Lag Models in the Estimation settings group. Having done that, input your variables as shown in the screenshot below:

The result is as shown below, yielding MTARDL(1,3,1,0,1) model

The ECM representation of the model is given below. Though no long-run exists for this example, the process of going about MTARDL is as demonstrated.

Other functionalities that come with Eviews for ARDL can be used.