try the variable *dummy (1 for increasing and  0 for declining ) . this will be a new explanatory variable. like interaction variable

What I see is the function rises at an increasing rate, reaches an inflection point, and then rises at a decreasing rate. It reaches a maximum and then falls at an increasing rate, reaches an inflection point, and finally falls at a decreasing rate (looks like there might be an asymptotic level). I do not see a discontinuity in the data or a "kink." The first function I would try would be an exponential-quadratic function. That would look like Y=a exp(b+cX+dX^2). Taking logs you would get the following: lnY=lna+b+cX+dX^2. Note that the intercept is lna+b, you can't separate these things from each other. So you can just leave b out of the math and be just fine.

 ARDL is one of the family of AR (autoregressive) model. The basis of AR family of models is the long-run equilibrium of time dependent series. At a certain point, the series is faced with an exogeneous shock, the analyst then tries to measure the effect of that shock on the series, i.e. how is the new mean compared to the long-run mean. if the effect is permanent, it is said that the data series is integrated, i.e. the effect of the shock is integrated into the series. If so, then the old mean is not useful, we have to re-establish the new equilibrium point. Some times the effect of the shock is not permanent, the series is said to be mean reverting, i.e. go back to its original long-run mean. This is a case of non-integrated series, i.e. the effect of the exogeneous shock is not integrated into the series. by testing the effect of the shock at various lag length, on could could come to conclusion whether there is integration. using VECM, for instance, may be one of the tool coupled with Augmented Dickey-Fuller test for integration.

In the present case, the graph shows that the time series received several shocks: A, B, and C and after C, the trend line is downward. ARDL may not be able to capture the entire period: 1980 - 2010. One can break the data into two periods [see attached figure]: From A-B-C, ARDL or any member of AR family may work. Each shock, A, B and C, one can test for integration and work VEC into the right-hand side of the equation. However, from C-D and beyond, the direction of the slope changed. If the change of direction (from positive to negative slope) is permanent, ARDL may not be an appropriate modeling tool for the entire period: 1980 - 2010---unless we run ARDL-1 and ARDL-2 in two segments and analyze them separately as 1980-1995 (period 1) and 1995 - 2010 (period 2). You might consider a quadratic equation, i.e. second order polynomial as a possible modeling tool.

Prof. Dave Giles maintains active blog on econometrics. His ex.planation is quite detailed and accessible. The link to that blog is: http://davegiles.blogspot.com/2013/03/ardl-models-part-i.html

Some useful articles and links on ARDL. However, as stated above, seeing the plot of the data, ARDL might not be the best tool for modeling.

http://classes.ses.wsu.edu/EconS512/Yoder/lectures/pdfsectionsold/dynamics.pdf

http://people.umass.edu/arazmi/Unpublished_Appendix.pdf

cont.

of the variable I was asking the question. This ,however, means that I will mix  the right hand side variables with a linear,  a  quadratic and couple of logarithmic variables.

The first suggestion of professor Selliah for the inclusion of a dummy tmes the variable I tried and it does not work.

What I thougt it was interesting is that the coefficient of the variable in question changed sign whe it was used in another ARDL model that had a different  specification and different form of the dependent variable.

Again thank you for your help and I will follow carefully your suggestions and read the references provided by professor Louangrath.

Kind regards,

George K Zestos

please see my entire response below

Hello everybody for your responses to my question. It is obvious that the first part of my

response did not go through. I wrote there that the suggestion of professor Louagrath to

break the sample into two sabsamples and esimate the ARDL  model for the two

subperiods would had been ideal, but I cannot do this because my sample is too small.

Replacing the variable with a quadratic term i.e, lnY=lna+b+cX+dX^2.term  as

professors Colwell and Louangrath suggested it may work. This ,however, means that I

will mix  the right hand side variables with a linear,  a  quadratic and couple of

logarithmic variables, which may work.

The first suggestion of professor Selliah for the inclusion of a dummy tmes the variable,

I tried and it does not work.

What I thougt it was interesting is that the coefficient of the variable in question changed

sign when it was used in another ARDL model that had a different  specification and a

slightly different form of the dependent variable.

Again thank you for your help and I will follow carefully your suggestions and read the

references provided by professor Louangrath.

Kind regards,

George K Zestos

How can you handle a right hand side variable of an ARDL model which has a kink (peak) i.e., a distinct break? - ResearchGate. Available from: https://www.researchgate.net/post/How_can_you_handle_a_right_hand_side_variable_of_an_ARDL_model_which_has_a_kink_peak_ie_a_distinct_break#57831db448954ce6255c86d6 [accessed Jul 11, 2016].