ARDL long-run coefficient

[Mahana Noorma]

Dear all,

I have a question regarding the calculation of long-run coefficient from an ARDL model.

I would like to estimated the following ARDL model and then test whether y_t and x_1t are cointegrated by means of bounds testing approach :

Δy_t = β₀ + Σ β_iΔy_t-i + Σγ_jΔx_1t-j + θ₀y_t-1 + θ₁x_1t-1 + e_t

regress d.y  c  l.d.y  l(0/1).d.x1  l.y  l.x1

From this equation, I know that the long-run coefficient for x₁ is -(θ₁/ θ₀).
(the bounds test say that the two variables of interest are cointegrated)


I would like to know whether this long-run coefficient is equivalent to the long-run coefficient in the corresponding long-run relationship between y_t and x_1t (the levels model):

y_t = β₀ + a₁x_1t + v_t (levels model)

regress y c  x1

When I perfomed the estimates, the two different calculations of the long-run coefficient do not give the same results. I am confused and wonder where I could have made a mistake. Or must a₁ not necessarily be equal to -(θ₁/ θ₀)?

Thank you so much for your help

Here are the Stata results:

Source | SS df MS Number of obs = 36 -------------+------------------------------ F( 5, 30) = 16.94 Model | 63.2903203 5 12.6580641 Prob > F = 0.0000 Residual | 22.4119664 30 .747065547 R-squared = 0.7385 -------------+------------------------------ Adj R-squared = 0.6949 Total | 85.7022867 35 2.44863676 Root MSE = .86433 ------------------------------------------------------------------------------ D. | y | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- y | LD. | .2882683 .1202836 2.40 0.023 .0426164 .5339202 x1 | D1. | -.4332363 .065969 -6.57 0.000 -.567963 -.2985096 LD. | .1881575 .0844541 2.23 0.034 .0156793 .3606358 y | L1. | -.9508915 .1854258 -5.13 0.000 -1.329582 -.5722014 x1 | L1. | -.0168967 .0079469 -2.13 0.042 -.0331265 -.0006669 _cons | 2.646874 .6757397 3.92 0.000 1.266829 4.026918

Source | SS df MS Number of obs = 38 -------------+------------------------------ F( 1, 36) = 8.80 Model | 15.5955962 1 15.5955962 Prob > F = 0.0053 Residual | 63.8147884 36 1.77263301 R-squared = 0.1964 -------------+------------------------------ Adj R-squared = 0.1741 Total | 79.4103846 37 2.14622661 Root MSE = 1.3314 ------------------------------------------------------------------------------ y | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- x1 | -.029125 .0098192 -2.97 0.005 -.0490393 -.0092108 _cons | 2.919511 .5483467 5.32 0.000 1.807412 4.03161 ------------------------------------------------------------------------------

We can see that -(-.0168967/-.9508915 ) is not equal to -.029125

[Sebastian Kripfganz]

The short answer is that the results differ because the regressions are different. I advise against estimating the levels model because it suffers from the omitted short-run dynamics. You can find a discussion about ARDL estimation and the bounds test in the Statalist topic concerning the ardl command:
ARDL in Stata

[Emad Shehata]

From this equation, I know that the long-run coefficient for x₁ is -(θ₁/ θ₀).

Dear Mahana

your long-run coefficient formula is not correct

long-run coefficient =

θ1/(1-Σβi)

[Sebastian Kripfganz]

Sorry Emad,
but Mahana's formulation of the long-run coefficient is correct because her dependent variable is Δy_t and not the level y_t. Her model is already in error-correction form.

[Mahana Noorma]

Dear Sebastian,

Thank you so much for your reply. I was very confused...now I can go foward with my analyses. If I have further questions regarding ARDL model, I will post them directly in the ARDL discussion you mentioned.

Thanks again and kind regards

[Emad Shehata]

Dear Sebastian

I think there is some confusions in answers

Your answer is correct with respect to first difference or ECM form

And my answer is also correct with respect to level equation form

But Mahan's question is can she apply the formula as in first difference in level form

As she wrote:

I would like to know whether this long-run coefficient is equivalent to the long-run coefficient in the corresponding long-run relationship between yt and x1t (the levels model):

yt = β0 + a1x1t + vt (levels model)

So, my answer to here, is not correct to use the same formula in both models.

for summary:

(1) first difference equation:

Δyt = β0 + Σ βiΔyt-i + ΣγjΔx1t-j + θ0yt-1 + θ1x1t-1 + et

long-run coefficient = -(θ1/ θ0).

(2) level equation:

yt = β0 + Σ βiyt-i + θ1x1t-1 + et
long-run coefficient = θ1/(1-Σβi)