A Quick Explanation of Recent NARDL

The News of a positive change  in the character of a villain (did something very good) of your area may surprise you, but his negative change may not.  These two sides of changes do not have an uniform impact on you. Again, think of a very moral person you like. His  negative change may shock you but his positive change won’t. These two changes in opposite directions (positive and negative) do not have the same power to move you, in both examples. And the effects are not really symmetric or equivalent; they are asymmetric or non-equivalent. Asymmetry is one type of non-linearity. The asymmetries and other other forms of non-linearity are also frequent in economic variables. For example, an increase (positive change) in oil price is said to have stronger effect on particular macroeconomic variables than decrease (negative change). In fact, ‘’nonlinearity is endemic within the social sciences and that asymmetry is fundamental to the human condition’’ Shin et al. (2014)

A conventional time series regression model contains constant parameters and assumes that a change in explanatory variable has the same effect over time which may not be appropriate in all cases as shown in the oil example earlier.  Again, the popular cointegration tecniques such as EG-ECM, VECM, Bound testing etc. imply a constant speed of adjustment ( i.e a constant ECT) to long-run equilibrium after a shock (change). But this dos not hold true always when there is market frictions. (see G Dufrénot, V Mignon 2012). Estimating a relationship which possibly has asymmetry with symmetric techniques seems unfair and may leads one to some serious inappropriate  policy conclusions (Enders 2014)). Since the conventiona cointeration test doesn’t allow one to capture the asymmetries in macroeconomic variables. Various techniques have been introduced so far to account this asymmetry, Threshold ECM, Smooth transition regression ECM. Markov-switching ECM etc. But the recent NARDL or Non-linear Autoregressive Model proposed by shin et al (2014) incorporate asymmetries both in the long run and in short run relationships, and at the same time, it captures the asymmetries in the dynamic adjustment.  Moreover,  it allows the regressors of mixed order of I(0) and I(1).

An illustration: If inflation rate rises in a country you may expect that the domestic foods becomes expensive and there will be a tendency to import foods from foreign countries. The relationship is positive. Again if inflation rate falls the consumers find domestic foods cheaper and people reduce buying foreign foods, the food imports decline (positive relationship). Although in both cases the food imports react positively to the inflation rate, are the magnitudes of reactions same in both cases ? maybe not;  maybe food import response more to positive change or otherwise. A time series regression specification with a constant parameter will tell us that the reaction is same in both direction. Here comes the NARDL. NARDL (also other asymmetric regression techniques) explicitly distinguishes the reactions of both directions. 
I denote food import by food_t and inflation rate by INF_t, intercept by C and residuals by U_t. For simplicity, ignore the other regressors that may influence the food import. The simple OLS two-variable model takes the following form:

Foodt=C+βINFt + Ut.

To capture the possible asymmetric effects of inflation on food import NARDL technique decomposes the inflation rate series into two parts 1)partial sum of positive change in inflation rate denoted by INF_t⁺   and 2) partial sum of positive change in inflation rate denoted by INF_t^-   and include both of them as separate regressors in the model and the model becomes:

Foodt=C+β1 INFt+ + β2 INFt - + Ut.

Clearly, this is now a three-variable OLS model. 
If we now represent this equation in (linear) ARDL model proposed by pesaran et. Al (2001) the final model takes the form as show in picture below. The model shown in the picture is the general form of NARDL. (Non linear Autoregressive Distributed Lag Model. See the explanation of each term in the picture below. 

🔍The long run coefficients: We can calculate the long run coefficient of INF⁺_t by dividing the the negative of the coefficient of INF⁺_{t ,} θ⁺    by the coefficient of Food_{t-1 ,}  ρ, and   the the long run coefficient of INF^- _t  by dividing the negative of the coefficient of INF^-_{t ,} θ^-    by the coefficient of Food_{t-1 ,}  ρ

   (-θ⁺  / ρ) and   (-θ^-  / ρ)  are the long run coefficients of of INF_t⁺   and INF_t^-  , respectively.

The summation notation  Σ  implies that NARDL consider inclusion of differenced variables into model upto some lags. For example, in case of  ∆Food_t-1, NARDL considers the incusion of  its first lagged term upto maximum lag you choose, if appropriate. And in case of  ∆INF^-_t it consider the the inclusion of its zero lag (∆INF^-_t itself)  upto the maximum lag you choose, if appropriate.  

🔍Asymmetric Cointegration test:  A long run relationship or cointegration is present if  the joint null hypothesis,

             ρ =θ⁺  =  θ^- =zero is rejected. The critical value are the same critical values for ARDL.  

🔍Testing Symmetry: Clearly, if the long-run coefficients  (-θ⁺  / ρ) and   (-θ^-  / ρ)  are not same then there is asymmetry in the long run. So we test the null hypothesis of  (-θ⁺  / ρ)  =  (-θ^-  / ρ). If the null is rejected then there is an evidence of long-run asymmetry in the model. 


To estimate NARDL, follow these steps: 

Steps:
F

Step 1. Perform unit root tests to justify that non of the variables are I(2).

Step 2. Generate INFt+ and INFt–  from  INFt

Step 3. Run the Non linear ECM under NARDL

Step 4. Test the ‘non linear cointegration’ test with F-test

Step 5. check the asymmetries.

See my another post on estimating Nonlinear ARDL (NARDL) with Eviews.

NARDL With Eviews

Shin et al. (2104): 

Researchgate link
https://www.researchgate.net/publication/228275564_Modelling_Asymmetric_Cointegration_and_Dynamic_Multipliers_in_a_Nonlinear_ARDL_Framework


Some papers which applied NARDL:

1. Abdlaziz, Rizgar Abdlkarim, Khalid Abdul Rahim, and Peter Adamu. "Oil and Food Prices Co-integration Nexus for Indonesia: A Nonlinear ARDL Analysis." International Journal of Energy Economics and Policy 6.1 (2016).
2. Ndoricimpa, Arcade. "Analysis of asymmetries in the nexus among energy use, pollution emissions and real output in South Africa." Energy (2017).
3. Zhang, Zan, Su-Ling Tsai, and Tsangyao Chang. "New Evidence of Interest Rate Pass-through in Taiwan: A Nonlinear Autoregressive Distributed Lag Model." Global Economic Review (2017): 1-14.











References: 

1. Dufrénot, Gilles, and Valérie Mignon. Recent developments in nonlinear cointegration with applications to macroeconomics and finance. Springer Science & Business Media, 2012.  link

1.    Enders, Walter. Applied Econometric Time Series. Hoboken: Wiley, 2015. Print. 

1. Pesaran, M. Hashem, Yongcheol Shin, and Richard J. Smith. "Bounds testing approaches to the analysis of level relationships." Journal of applied econometrics 16.3 (2001): 289-326. link

1. Shin, Y., Yu, B., Greenwood-Nimmo, M.J., 2014. Modelling Asymmetric Cointegration and Dynamic Multipliers in a Nonlinear ARDL Framework. In William C. Horrace and Robin C. Sickles (Eds.), Festschrift in Honor of Peter Schmidt: Econometric Methods and Application, pp. 281-314. New York (NY): Springer Science & Business Media.  link

Eviews official website:

http://www.eviews.com/home.html