Yes

Yes, because they are assumed nonstationary in levels

Hi all,

Thanks for your replies.

1) Can I conduct a cointegration using by checking unit root on a residual of  Y=ax1 + bx2 + time trend + c ?

2) If I conduct a root test as "level" and "trend and intercept", I found that the predictand is not stationary and predictors are stationary (many predictors are not stationary). SO, I found a situation where all variables are stationary in the 1st difference and not all variables are stationary when a root test of as "level" and "trend and intercept" is conducted. What does this mean ?

Bhavesh's comments are excellent. 

@Chul and Bhavesh,

Really appreciate your replies. The issue  here is many papers use "1st difference" and other papers use "intercept and trend". In other words, relevant papers relate to my research use different approach. So, please answer my question as follows:

If I conduct a root test as "level" and "trend and intercept", I found that the predictand is not stationary and predictors are stationary (many predictors are not stationary). So, I found a situation where all variables are stationary in the 1st difference and not all variables are stationary when a root test of as "level" and "trend and intercept" is conducted. What does this mean ?

Let me explain this way; Let us assume that you are studying the long-run relationship between Y F G and N. 

You can start by testing them using c and time so for Y you create DY and run DY c time Y(-1) and lags of DY the critical value is -3.44 (ADF). so when you look at the t-stat associated with Y(-1) and its absolute value is below  3.44 , you can conclude that the Y variable is the first-difference stationary, however, it is much clearer to the reader if you said that Y has a unit root. 

Then, you do the same for F G and N. Assume now that all are I(1) then the next thing to do is to see whether a linear combination of the variables is I(0) (i.e. stationary. This is where you run

Y c F G and N and get the residual (e)

then, create De and then run De (no constant) e(-1) and lagged of De as I explained before.

Suppose that Y and F are (!) but G and N are I(0), you have a different case and you have 2 choices (a) ARDL and (b) ECM to follow

Hi Muhamed: Did we satisfy you? Just asking 

Yes and she are cointegrated, you can made a cointegration regresion: FMOLS; DOLS o CCR

Manue1 Jaen is correct. Yes because you have confirmed that the times are stationary in first differences and you can do FMOLS, CCR or DOLS. Bear in mind, that their estimates may vary a little. You may play around with AR(1) type FMOLS and you can do also test the stability of our cointegration equation by using Hansen Instability test (Lc) click the residual after using FMOLS.  

@anyone,

My observed data is time series and I construct 2 statistical models. The first model is models with first differenced models (independent and dependent variables are first differenced variables) and the second model is models with variables (in their level) with time variables. Note variables/time series are not stationary in their levels but stationary after first differencing.

I checked both models pass statistical assumptions/tests i.e. (no) autocorrelation, (no) heteroscdasticity, etc. As my data is time series, shall I conduct a stationary test (although my models pass those statistical assumptions) ?

Your first model (first differenced model) relates to the short run and your second model in the levels of the variables relates to the long-run impacts. Yes, you need to conduct stationarity test by conducting cointegration on the level variables. Economic theory deals most with the long-run. Therefore, my suggestion is to combine the two models:

Therefore, run the  following model  DY c  Dx DZ DR Dy(-1) y(-1) X(-1) Z(-1) R(-1)  

you may need to include the lags of the first differenced terms if you have autocorrelation. The most important thing here is to look at the t-ratio for y(-1) it should be negative and high if you have cointegration. This will make you smile because you can make statements about the long-run relationship.

An alternative is:

Run y c x z R and save residual (ecm)

Run another regression Dy c ecm(-1) Dy(-1) DX DZ DR (see what I said about autocorrelation. Now you can look at the t-ratio for ecm (-1). It should be negative and large. I can explain more but I do not want it to be too long. The significance of the coefficient of Ecm(-1) means support for cointegration. 

@Chuck, so if residuals of models with time variables are stationary, this means the model with time variables is stationary ? is that correct ?

Yes to your question but to be sure we understand each other. Look at my alternative above. I mean you Run Y constant X Z R and you get the residual (ecm)

You can test this residual to see if it stationary by doing another regression:

Decm ecm(-1) Decm(-1)    (you can add lags of Decm(-1) but remember no constant but even if you include a constant it would be significant.  Note that Decm = Ecm-ecm(-1) 

Yes, I agree with Chuck and Manuel. You should go for cointegration. For your second question, if all variables are stationary at first difference, the model with those variables will be stationary.

@anyone, can you tell me how to read a White test result as follows:

F-statistics                  2.9                          Prob. F(17,18) ... 0.0138

Obs*R-squared         26.5                        Prob. Chi-Square(17) .... 0.0654

Scaled explained      18.7                       Prob. Chi-Square(17) .... 0.3445

Your white test says you have no significant heteroskedastic errors at the 5 percent level because 0.05 is not greater than 0.0654. Although the F-statistics of 2.9 is significant, it may be because of small sample size. Since the scaled explained is not significant, you shouldn't worry. Focus only on scaled result. The important thing is that available disease in your model is not significant. 

@ Chuck,

could you please tell me the difference(s) between statistics and econometrics?

Thanks in advance

@Chuck I am much satistfied

@Muhamed Thank you for being a decent and hard working person! Blessings follow those who earnestly are goal oriented. I wish you the very best!

@Sabyasachi,

so, if independent variables are the first difference variables (stationary), should the models with the first difference predictors and the first difference predictand are stationary too ? if yes, please give me papers to cite

From your unit root, all variables are stationary in the 1st difference I(1), do the cointegration test but use the Johanson cointegration, if cointegrated, use the ECM,