How To Interpret ARDL Results? This is a very frequently asked question on our social media groups for Econometrics, Statistics and Research. Some of the students ask about How To Interpret ARDL Results using Stata? and many others ask about How To Interpret ARDL Results using Eviews. We will use Eviews to estimate Eviews for this tutorial but the interpretation does not depend on softwares but the statistic/calculated estimates we have. So, the results from Stata can equally be interpreted relevantly.
How To Write ARDL Equations?
We can help you learn how to interpret ARDL Results in the following few steps. But first we should understand what is ARDL in few lines and how we can estimate ARDL in Eviews. The ARDL is a modified regression model to test and estimate the cointegrated relationships between time series variables. Using bounds test instead of Johenson Jusuleus Cointegration Test, the presence of long run relation between the time series variables can be predicted and that too using the F statistic. The ARDL equation is given in the following:
yt = β0 + β1yt-1 + .......+ βkyt-p + α0xt + α1xt-1 + α2xt-2 + ......... + αqxt-q + εt , (1)
The ECM equation from ARDL setup is:
Δyt = β0 + Σ βiΔyt-i + ΣγjΔx1t-j + ΣδkΔx2t-k + φzt-1 + et ; (2)
The conditional ECM (Pesaran et al. 2001) is written like:
Δyt = β0 + Σ βiΔyt-i + ΣγjΔx1t-j + ΣδkΔx2t-k + θ0yt-1 + θ1x1t-1 + θ2 x2t-1 + et ; (3)
The Cointegrated Equation can be written as:
yt = α0 + α1x1t + α2x2t + vt ; (4)
and the bounds test can be conducted on the coefficients (H0: θ0 = θ1 = θ2 = 0) from the equation 3. The critical values of this F Statistic should be looked upon from the Tables in (Pesaran et al. 2001) or Narayan (2005):
Now, we will interpret the estimated results from an actual Eviews output. To estimate ARDL using Eviews (Learn Time Series Analysis Theory Here), one needs to open the WORKFILE or data in Eviews. Then click on Quick, then click on Estimate Equation and a new small window will appear. We select ARDL from the Method section of this small window which is near the bottom and below the list of variables textbox. Once ARDL is selected the options on this Window changes. We enter our list of variables in order of DV and IVs and intercept. The rest of the options can be selected from same Window like whether to include intercept, trend or both, lags for both the dependent variables and regressors and
Eviews allows us to specify the equation in form of regression models with general list of coefficients and estimated values in form the regression equation like this:
DEBT = C(1)*DEBT(-1) + C(2)*DEBT(-2) + C(3)*DEBT(-3) + C(4)*GDP + C(5)*GDP(-1) + C(6)*GFC + C(7)*GFC(-1) + C(8)*GFC(-2) + C(9)*GFC(-3) + C(10)*TRADE + C(11)
DEBT = 0.9172*DEBT(-1) - 0.4375*DEBT(-2) + 0.3484*DEBT(-3) - 0.0614*GDP - 0.0955*GDP(-1) + 0.3101*GFC + 0.1751*GFC(-1) + 0.7765*GFC(-2) + 0.3646*GFC(-3) - 2864*TRADE + 1375446667.23
The Cointegrated equation from above model becomes where we truncted the coefficients at 4 decimal points:
D(DEBT) = 1375446667.2237 -0.1718*DEBT(-1) -0.1569*GDP(-1) + 1.6265*GFC(-1) -28649248.9413*TRADE** + 0.0890*D(DEBT(-1)) -0.3484*D(DEBT(-2)) -0.0614*D(GDP) + 0.3101*D(GFC) -1.1411*(DEBT - (-0.9135*GDP(-1) + 9.4649*GFC(-1) -166714193.3055*TRADE(-1) + 8003926456.4992 ) -0.3646*D(GFC(-2)) )
Then we can get the first estimation table which looks like this:
In this results, the first part is summary of the information, Eviews has worked on. We can see that Dependent variable and the method has been reported in the first two lines respectively. Then the time and information about the sample time period and number of observations is given. Also, we can see that Eviews estimated a few regressions models to come up to the selected model based on information criteria. AIC selected the ARDL with Model with 3 lags for the dependent (DEBT) and the independent variables (GDP GFC and DEBT)were included with their level and selected lags of 1, 3 and 0 respectively. 0 lags of an independent variable means it will be added to list of regressor in level only. The second part of the main equation gives values of coefficients, standard errors and t statistics with p-values. We can interpret this an AR and DL equations like we do in ADL models. We can either interpret the coefficients as simple regression coefficients keeping mind the nature of X and Y variables like in log or percentage etc or we can conduct an F test to jointly determine if the X with its lags has any effect on the Y which works like causality test (not what Granger Causality is). We can generalize this step to conduct Granger Causality test on given set of coefficients as well upon confirmation that the hypothesis test matches the one which Granger causality is based upon (hint for those who wish to conduct Granger Causality after ARDL).
The main results of the ARDL Regression model is given in the central table in Eviews results output window. We can see this portion contain few columns each on list of variables in the model with their lags, coefficients values, standard errors, t or Z statistics and corresponding p-values. We can interpret these as conventional regression models coefficients are interpreted like if the variables are in logs or simple measurement. One has to keep in mind the sign and size of coefficients for interpretation if the objective is inferential study of predictive study respectively. A positive coefficient on the level variable means that the current change in X affects current level of Y positively and negative sign of the level variable means the current change in X affects current level of Y negatively. The coefficient of a lag 1 of X means the effect of changes in past years values of X results in a change in current values of Y or the current changes in X affects values of Y in next time periods. This can also be positive or negative. The coefficient of second lag means the change in X today will cause changes in Y two years from today or a change in X two years ago will affect Y today. It can be positive and negative as well as we can see it in level changes or first lag changes.
The Standard Error of the coefficients are given the sampling distribution of the coefficients. We can see these value the are the standard deviation of the coefficients from different samples of the data if the same coefficients are estimated from each sample and taken as a sample itself. This gives us a margin of error or limits within which the value of coefficients can vary within a limit on average. We will need this determine the t-Statistic and Z statistic to define the hypothesis testing for the given coefficients.
The next column is t Statistics. We use this column to test the null hypothesis that
Interpret ARDL Results: Some Basic Results:
Interpret ARDL Results for Short Run Relationship
Interpret ARDL Results for Long Run Relationship
Pesaran, M. H. and Y. Shin, 1999. An autoregressive distributed lag modelling approach to cointegration analysis. Chapter 11 in S. Strom (ed.), Econometrics and Economic Theory in the 20th Century: The Ragnar Frisch Centennial Symposium
. Cambridge University Press, Cambridge. (Discussion Paper version
Pesaran, M. H. and R. P. Smith, 1998. Structural analysis of cointegrating VARs. Journal of Economic Surveys, 12, 471-505.
Toda, H. Y and T. Yamamoto (1995). Statistical inferences in vector autoregressions with possibly integrated processes. Journal of Econometrics, 66, 225-250.