Nonlinear ARDL (NARDL) Model With Eviews

Before you start reading, If you are not familiar with NARDL see A Quick Explanation of Recent NARDL where i formulated a food import model (for India for the illustration purpose). In this post, i estimated the NARDL model with Eviews 8/9.  The  raw data used to demonstrate with Eviews can be downloaded from google drive. Download Workfile. Or send a hi to my Email: hssanhanif@gmail.com or meet me at dept. of Economics and Banking, IIUC.  This file contains the yearly data of inflation rate and ''percentage of food import to total import'' of India . The food model has the following form: 

   Food_t=C+ INF_t+U_t

 Where, Food_t indicates % of Food Import of total import. (dependent variable) and  INF_t indicates inflation rate (independent variable).

Since NARDL is a recently developed model, Eviews 8, Eviews 9 or other current versions do not have any built-in option yet to estimate NARDL. So, 

Follow these steps:
F

Step 1. Perform unit root tests to make sure 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.

Step1: run unit root test to make sure that data are not I(2).

Like the ARDL error correction model by pesaran et al (2001), the NARDL does not allow I(2) variables. To perform unit root test in Eviews:  ⟾click on the series unit root test ⟾select level and ⟾select intercept and trend now see the p-value. If it is less than 0.05 the series is said to be ststionary or I(0) at 5% significance level. and if not repeat the steps and this time select the choice first difference. If p-value is less than 0.05 then series is said to be stationary at first difference or I(1). And if it is more than 0.05 the series is not I(1). And should not be entered in NARDL model. I have checked each series and found that Foodt is  integrated of order 1 or I(1) and inflation is found to be the integrated of order zero I(0).  So we can run NARDL. You can verify the order of integration with the Eviews workfile i provided in first paragraph. 

Step 2 Calculate partial sum of positive and negative change:

To do this, copy each line of the commands given in the box paste it into the command area of Eviews then press ENTER in keyboard. For example after putting  genr dinf = inf-inf(-1) press ENTER than put the 2nd line genr dfood=food-food(-1) then press ENTER and so on. To understand these commands, what  these commands to is that they create first difference variable of inf and food: dinf and dfood, then create a series for the values of dinf which are not negative and another  series for negatives. Then they calculate the cumulative sum of each after multiplying positive and negative series by the first difference. The resulting cumulative sums are partial sum of positive and negative changes in inflation, which denoted by inf_p and inf_n, respectively. 

Eviews Commands to decompose independent variable in NARDL.  Note: throughout this Eviews illustration,  INFt+ is denoted by inf_p and INFt– denoted by inf_n

genr dfood=food-food(-1) genr dinf = inf-inf(-1) genr pos = dinf >=0 genr dinf_p = pos*dinf genr dinf_n = (1-pos)*dinf genr inf_p = @cumsum(dinf_p) genr inf_n = @cumsum(dinf_n)

Eviews commands for decomposing a series into  partial sum of positive and negative change 

Step 3:  Run the Non linear ECM under NARDL 

 Step 3.1  Estimating NECM under NARDL framework

Now ⟾ click on quick estimate equation ⟾ choose two steps least square from drop down list and specify the equations in both of the boxes as shown the picture below. 

d(food) c food(-1) inf_p(-1) inf_n(-1) dfood(-1 to -4) dinf_p(-0 to -4) dinf_n(-0 to -4)

NARDL specification 

Explanation of the specification: The specifications in first box contains the dependent variable d(food), constant C, followed by 3 independent variables—the  first lagged term of food: food(-1),  the first lagged term of partial sum of positive change in inf which is denoted by inf_p(-1), and the first lagged term of partial sum of negative change in inf which is denoted by inf_n(-1). The second box contains the set of differenced variables upto 4 lags. ( i chose maximum lag 4 since the data are yearly). By specifying (-1 to -4), all the lagged terms of dfood from first lag to fourth lag are included. So as for the rest. 0 lag implies no lag. The 0 lag of Xt, for example,  is the Xt itself. Note here that the 0 lag is not included in case of dfood because dfood is already acting as dependet variable in the model. 

But, before you click OK, click ‘options’ and set the settings as shown in the picture below

This setting will drop the insignificant stationary variables from NARDL  model as suggested by Shin et al (2014)

Choose unidirectional, backwards, p-values, set p-value 0.10 (or 0.05 or even 0.01.) I chose 0.10 here. Also choose uni directional  and backwards.  The interpretation of this setting is that it will remove the variables that are not significat even at 10% level while it will retain the variables of upper panel of the box even if they are not significant this was chosen in shin' study. However, this setting is not hard and fast rule. The goal is to select the appropriate model specification with appropriate lags for the differenced regressors. You can choose other criteria to choose appropriate lags. 

3.B    NARDL output, long-run coefficients and interpretation: 

NARDL output in Eviews 

By a quick  glance to the output we can learn that Eviews removed some lags and that  0.308099 and 0.390456 are the coefficients of inf_p and inf_n, respectively. But they are not the long run coefficient. To calculate the long run coefficient divide the negative of coefficient of each inf_p and inf_n by the coiefficient of food(-1). So the long run coefficient of inf_p is -0.0308099 /-0.457567=0.673342  and long run coefficient of inf_n is -0.390456/-0.457567=0.853331 . clearly, both long-run coefficient are positive. hence So the long run equation or the cointegrating equation is :

            long run equation:        🔴   Food= 0.673342inf_p  + 0.853331inf_n +u

interpretation of NARDL output: 1 percent point increase in inflation rate leads to 0.67 percent point increase in food import India (positive realtion), and 1 percent point decrease in inflation rate leads to 0.85 percent point decrease in food import (also positive relation). (I said  percent point because the variables are rates) clearly, the food import response more to negative change because the coefficient is larger. 

Step 4: Asymmetric Cointegration test: 

Before drawing any conclusion regarding the estimated coefficients one needs to check if variables are co-inetgrated. The coefficients would be sporious if variables are not cointegrated.  For testing cointegration under NARDL, Shin at al recommended to use joint null hypothesis of level (non-diffrenced) variables and to compare the critical values of bound testing in pesran et al (2001). (You can download the paper from internet, and find the table in Page 300). If the calculated F statistics is found to be the greater than the upper critical value then there is evidence of co-integration. And if not, then evidence of cointegration is not found.

 To start cointegration test in Eviews click view on output window⟾coefficient diagonistics⟾wald test-coefficient restriction. ⟾Specify the restriction c(2)=c(3)=c(4)=0. Then click OK’. Clearly, this wald test perform the joint null hypothesis of the coefficient of food(-1) which is second coefficient in the list (hence the notation c(2)), and the coefficient of lnf_p which is c(3) and and the coefficient of inf_n which is c(4). Basically, this joint null hypothesis acts as the null hypothesis of no cointegration. The procedure is illustrated in the picture below. 

Asymmetric cointegration test under NARDL

After clicking OK you will see Wald test results as shown in the picture below.

The calculated F-statistics for this (asymmetric) cointegration test

The calculated F-statistics is 13.34022. now lets compare it with critical values of 

pesaran et al (2001).  Here is the screenshot. Of case III.  

Pesaran et al. (2001) critical values Case III. 

Case III was chosen because in the NARDL representation on my food model there is a constant.(intercept was not restricted). In the table, k denotes the number of long run regressors.  Note here that there are two independent variables in long run equation of my food model :inf_p and inf_n.  but I chose k=1. The reason is that, the variables actually came from from 1 variable, hence, the k lies between 1 and 2. Also notice in table that the critical value for small k is larger. If the null hypothesis of no cointegration is rejected by critical value of its smaller k, then according to shin et al(2014) there is a strong evidence of co-integration. In my food model i chose  k=1, but for large number of variables set k equal to number of regressors before decomposition. 

Decision: since The calculated F statisics is larger than  The crticical value 7.84 at 1% significance level, there is strong evidence of cointegration at 1% significance level. 

Step 5. Testing the presence of  asymmetry: 

both of the positive change and the negative change have the long-run positive effect on food import. But are they really (statically) different ?  an asymmetry test basically test if the coefficient are equal or not. If they are equal then threre is no asymmetry and if they are not then threre is evidence of asymmetry. Recall from step 3 .2 that we calculated long run coefiicient for inf_p and inf_n by –c(3)/c(2) and –c(4)/c(2), respectively.  So, To test the long run  asymmetry in Eviews click on wald test and write this command below in the restriction box of wald-test.

–c(3)/c(2)=–c(4)/c(2)

Eviews will show the wald-test results as in the picture below. 

Wald test results for asymmetry test. 

Decision:  Clearly, the null of hypothesis of equality is rejected as p-value is less than 0.05. Wald test indicates that there is asymmetry in the long run impact of inflation on food import in India.

The Last step: 

Check the robustness by various diagnostic tests. I will not explain theme here but I saved the results of what I have shown above including the results of diagnostic tests. If you are stuck please do not hesitate to ask me. 

Important Note: This is not a full fledged research, rather an illustration to show how to estimate NARDL.  This model may suffer from omitted variable bias or any type of model misspecification, I'm not claiming that these results are valid for India. 




Acknowledgement: 

I'm thankful to M.J. Greenwood-Nimmo, one of the author of Shin et al (2014),  and to Rizgar Abdlkarim, co-author "Oil and Food Prices Co-integration Nexus for Indonesia: A Nonlinear ARDL Analysis." International Journal of Energy Economics and Policy 6.1 (2016).

Eviews official website:
http://www.eviews.com/home.html