I assume your question is to know how to measure the relationship between variables that do not vary over time. If this is the case, when you have time-invariant (or stationary) variables in your time-series data, you should use OLS  regression to find the underlying relationship. This is because OLS regression better captures the impact of time-invariant (stationary) variables. I hope this helps

what is wish to know is that if Y variable is I(1) and X is I(2) and Z is I(1), how do I fit in regression between them?

In this case, regression may not help you much because you have three variables and you did not specify which is the dependent variable and independent variables. Perhaps, correlation analysis is useful if your intention is to observe whether any kind of relationship exists between the three variable. But note that correlation analysis will not tell whether the relationship is linear or non-linear, also it will not tell you whether a change in X cause a change in the values of Y  or a change in Z cause a change in the values of Y.. To investigate a causal relationship, you'll need to divide the three variables into dependent and independent variables, that the multiple regression works!

To fit the regression, let's assume that 'Y' is your dependent variable and X and Z are your dependent variables. You may fit it this way:

Y = f(X,Z)

Y = a + X + Z + e

where 'a' is the error term and 'e' is the error term

The fact that the dependent variable 'Y' and independent variable 'Z' have the same values should not be a problem. OLS regression can deal with that.

If two or more independent variables have the same values in the time-series, then, the problem of  multicollinearity will arise. 

so if y is dependent variable and other two are independent variables but X is I(2) and Z is I(1) like Y, in such a situation can we have a meaningful regression or it may be spurious?

If two variables are integrated of different orders (I(1) or I(2)). The I (1) variables will be diferenced once while the I(2) variables will be differenced twice, after an estimation is made using an OLS. Most of economic variables are I(1) and we can have few instances of I(2) variables. Differencing the variables before estimation will help eliminate spurious correlation. I hope this addresses your needs/question, thanks and Good Luck

To investigate the relationship between the three variables, you should use the stationary variables, e g. The first differences for the variables integrated of order one and the second differences for those integrsted of order two.

If Y is I(1), X is I(2) and Z is I(1) I would start by looking for cointegration between Y, the first difference of X and Z probably using a Vecm  analysis (If this makes economic sense). If there was no cointegration I would use a VAR (or other appropriate analysis involving the first differences of Y and Z and the second difference if X. I would also consider if the original assumptions about the time series properties of X, Y and Z were in accordance with economic theory.  

If Y is I(1), X is I(2) and Z is I(1), there will be two possible regressional solutions:

1. apply the second differences for all variables or,

2. run the regression using the first differences for Y and Z and, the second differences for X.

Solution 1 is more realistic and helpful in terms of consistancy in  result interpretation. But if you apply the second solutions, you have to consider two different methods of interpretation: one for the relationship between Z & Y and the second between X and Y.

Good luck.

Fayq

If data series are of different order of integration, then by time series stationarity properties, linear combination of the series in a regression analysis should be at the highest order of integration. For example, if you have X, Y, Z series where X is I(0), Y is I(1), and Z is I(2), then all the series (i.e. X,Y, Z) must be at I(2) when you are combining them in a regression analysis. Meanwhile interpreting series that are integrated of the second order I(2) might be difficult as there is hardly economic theory that support that. Therefore, I would suggest you perform a log-transform of the series before checking for stationarity. However, if you are considering estimating the series (at levels) without transforming them to stationarity, you should go for ARDL Bound Testing approach as put forward by Pesaran, Shin, and Schmidt (2001) to check for cointegration and actual estimation. Hope you find the explanations useful. Good luck

In this case, you can use ARDL model to know the long run relationship among variables.

One of the fundamental precondition for ARDL method is none of the variables are I(2). I am facing same problem. but I'm not sure whether your data is I(2) after considering structural breaks or not as you didnt mentioned that in your question. For this you may go for Zivot Andrews test or IO test. If your test results found that there is a structural break or multiple break, use dummy variable for that. After considering break if variable contain unit root proceed with ARDL model and will be absolutely fine. However if your variable still I(2) after considering break than you have to address that first.

I also want to ask if OLS can be applied to variables of different order of stationarity? Also when does one go for ARDL?

@ Joseph Sello Kau, the OLS estimation procedure is only applicable to series integrated of the same order; either all be level or first difference stationary (dependent variable inclusive). However, the ARDL approach is applicable only when the dependent variable is I(1) integrated, while the regressors can be either I(1) or I(0), or a combination of both; but definitely not I(2). Hope this helps!!!

In a situation like yours, you may want to try other methods of analysis such as VAR and VECM. These methods enable you to dynamically measure variables with a combination of orders.

If All variables are stationarity at the level or differences(first, second.....), called the integration of different order up to order P, the appropriate model is VAR or OLS but if non- of the variables is stationary after conducting a non-stationary test or unit root test, then Johansen cointegration or non-cointegration applied and if the variables are stationary at mixed order of integration, the ARDL is applicable.

Mine is a question not answer. If after one has run the unit tests for variables and the results indicated mixtures of cointegration order eg I(2), (I1) and I(0). Is it appropriate to use Johansen cointegration technique and ECM?

Ogunlana Olarewaju Fatai It is possible to use a variation of the Johansen Cointegration Technique when your variables are integrated of order 0, 1, and 2. The procedure is covered in Juselius (2006), The Cointegrated VAR model, OUP.

I know of two implementations of the procedures. Otherwise, you will have to do a lot of programming.

You will need to put a lot of work into an analysis such as this. You must have or acquire a good knowledge of stationary processes, non-stationary processes. order of integration and cointegration at the level of the Juselius book.

You should also speak to your lecturer/supervisor to check what he is expecting.

Mine is question, How do I fit the appropriate model after performing autocorrelation, stationarity test of order 0,1,2 and cointegration analysis.

if a variable is not becoming stationary even on second difference than what should we do?

Walid Hussain - Consider some asset price series. Economic theory would imply that this is non-stationary. Any (measurable) function of a non-stationary variable is also non-stationary. Therefore log(real GDP) is also non-stationary. Economic (or Finance) theory and statistical tests imply that returns on assets are stationary. The return on an asset can be estimated as the first difference of log(asset price). Thus log(asset price) is found to be I(1). However, there is no reason why any level of differencing will make the asset price itself stationary. and it is likely not to be integrated of any order.

Is your process multiplicative? Does a shock to these it by a constant amount (an additive process) or is it likely to increase it by some percentage (multiplicative process)? If it is a multiplicative effect then you should consider using the log of the variable in your analysis.

hi

my order of variables integrations are 0 , 1 , 3 (I(0) , I(1) , I(3))

y -> I(3) x1 -> I(1) x2 -> I(0)

I made normal OLS and it's residual was I(0) (order of integration was 0)

so I understood that cointegration is ok so long relation is ok

now I need to make ECM to check the short relation

what should I do?

I did two methods. (ecm is residual of cointegration regression)

1. d(y,3) d(y(-1),3) d(x1,3) d(x1(-1),3) d(x2,3) d(x2(-1),3) ecm(-1) c

2. d(y,3) d(y(-1),3) d(x1,1) d(x1(-1),1) x2 x2(-1) ecm(-1) c

but i don't know which one is acceptable

I need some reference about them

please help me

thanks a lot

Yes it is time series dat

If I understand you have three variables y, x1, and x2. y is I(3), x1 is I(1), and x2 is I(0). You estimate the regression

y = b0 + b1 x1 +b2 x2 +u

and find that u is I(0). This does not make sense.

First, you must understand what combination of variables are cointegrated

It is more complicated if you have I(3) variables and or a greater number of variables. An I(3) variable can not be cointegrated with an I(1) and or anI(0) variable.

Johansen, Engle-Granger, DOLS, and ARDL are methods of estimating an econometric model involving non-stationary variables defined by economic theory and common sense. A model found in that way is likely to be stationary It should not be used as a method to find a model that fits your data. Economic theory/common sense should indicate the degree of integration of your variables. For example, the log of real GDP is regarded as being I(1), and growth rates which are the first difference of the log of real GDP as I(0). These results can be confirmed by unit root tests. I have never seen an argument that an economic variable is I(3). While the log of real GDP is I(1) real GDP is not integrated of any order. It is possible that if you difference it enough it will appear stationary.

To summarize I would suggest that you return to your economics and consider your model. Does theory imply that there is an equilibrium relationship? Does theory imply anything about the integration status of the variables in your model?