Article Selecting appropriate methodological framework for time seri...

I very rare to see a data integrated of order 3 let alone order 4 level of integration. Try to use the appropriate unit root test and check for structiral breaks and seasonality if you are using monthly data..

You can try to reduce the degree of integration (4) before you start building cointegration models

Hello everyone thank you so much for your contributions.

In fact I don't know why but when I was working with annual data I found I(4) and I(2) like I mentionned previously ,but now I changed my data and I dicided to work with quarltely data so I am dealing now with I(1) and I(2),

so my first question is: why you recommend me to build cointegration ,because from what I know cointegration is just a test for relationship between variables?and since know my dependant variable Y is I(1) and my indepedant variable X is I(2) what model should I use in order to modelise the following equation

Yt=a*Yt-1+b*Xt+c?

Olga M. Pisareva Hamid Muili Thomas Dimpfl

Thank u Mr Ertimi Basem I am still reading that article

Based on your Quaterly data. Kindly use Toda-Yamamoto (1995) Approach. All the best

9I would comment on your first sentence.

• A stationary series is said to be I(0)
• An I(1) series is not stationary. Its first difference is stationary

If you wish to use the ARDL methodology -

You can not estimate the equation

Yt=a*Yt-1+b*Xt+c?

where y is I(1) and X is I(2). There is no economic theory that can relate an I(1) variable to an I(2).

Before you start estimation you should use your theory and common sense to set out a balanced model. In a balanced model you have all I(0) variables or I(1) or very rarely I(2). In a cointegration analysis variables are all first differences of I(1) variables and therefore stationary or error correction terms which are also stationary. The form of these must accord with your economic theory. You can achieve balance by considering first differencing your I(2) variable or integrating your I(1) variable to make it I(2). I have never seen an economic theory that supported or suggested an I(4). Even I(2) variables are rare. A mixture of I(0), I(1), and I(2) variables can be used in a "generalization" of the Johansen methodology. The is intermediate to advanced analysis and requires a good knowledge of the methodology.

Perhaps if you reconsider your economics and describe your problem, economic theory and data-set you might get a better answer

Thank you Mr John C Frain for your answer.

Well to put you in the context what I am trying to model is the Phillips curve where Yt is inflation rate and a*Yt-1 is the anticipation of inflation and Xt is unemployment rate.

so from what I understand since I have I(2) and I(1) a Jhoansen model is the proper approach

And here are my results Mr John C Frain

And by the way my data is going from 2010Q1 to 2020Q1

I have deleted yesterday's brief answer and am replacing it.

As regards your econometrics I would have two comments

I have not worked on the Phillips curve for many years. At that time there were many criticisms of the Phillips curve. It is possible that in the interim some of these problems have been solved. I would think that, in your first analysis, you might do better drawing the kind of graphs of inflation versus unemployment that appears in many textbooks, and joining consecutive points to see what semblance of a Phillips curve is present in the data. You might then describe, in words, how an expectations augmented or New Keynesian Phillips Curve might explain the features of the data. You may also need to estimate a Natural Rate of Unemployment. There are lots of things that can be done without advanced econometrics.

Thank you so much Mr John C Frain for your detailed answer and I'll take your valuable advices into consideration.

When beginning to analyse economic relations, (co)integration of variables is a rather unimportant problem. The first may be to choose of find a theory and try to find a convenient model. This step needs no data and no statistics, but of course one should try to test one's model. It may be necessary to transform it to a specification which can be estimated. In this respect, I support John's proposal to simply analyse the data and their relations with simple "methods" (diagrams, correlations) first before applying complicated methods. May be OLS estimates are sufficient.

By the way, I think and hope, that you do not use the price index itself in your specification, because it is only its percentage change which is relevant. For low inflation, %P is practically equal to dlnP, but I have never heard an economic discussion or negotiation arguing in logs. Most people do not even know, what a log is or how to calculate with them. I am not sure, if the lagged inflation (Yt-1= %P of the quarter before) multiplied by some coefficient (a) can be seen as expected inflation. I assume that the unemployment rate (X) is a seasonally adjusted variable, otherwise you would have to find an adjustment or relations which take that into account.