Which model to use between VAR and VECM for the following problems (conditions)? I have three variables (monthly for 25 years) including wages (e.g. skilled and unskilled) and food price (P). I am interested to see if there any relationship exist between them, either short or long term and which one has the greater influence on other.
All the variables are non-stationary at levels but stationary at first differences. But when I check the cointegration rank in 'Stata' ('vec rank', for your information, I run this command with the variables in their levels ) then it suggests no cointegration. In that case,
 1. What should I do? Will I go for a VAR model of the first-differenced series?
 2. Why there is no cointegration among the variables?( while I found through 
    unit root rest that all the series are stationary after first difference)
3. When I check the cointegration of the first-differenced series then I find the cointegration rank equal to 1. But is it feasible to run a VECM model to first- differenced series?
 A: I would use a VAR in levels if you are only interested in the Impulse Response Functions (IRF) and not specifically in the cointegrating relationship, which seems plausible from your description.
As mentionend in the comments if your first differenced variables are stationary they cannot be cointegrated. Also, don't take every statistical test result at face-value.
You can have a look into the papers below for arguments to estimate a system of integrated (and possibly cointegrated) variables in levels. The classic reference would be the Sims, Stock and Watson paper. Definetly also look into Lütkepohl, he is an authority when it comes to SVARS.

Sims, C. A., Stock, J. H., & Watson, M. W. (1990). Inference in linear time series models with some unit roots. Econometrica: Journal of the Econometric Society, 113-144.
Ashley, R. A., & Verbrugge, R. J. (2009). To difference or not to difference: a Monte Carlo investigation of inference in vector autoregression models. International Journal of Data Analysis Techniques and Strategies, 1(3), 242-274.
Phillips, P. C., & Durlauf, S. N. (1986). Multiple time series regression with integrated processes. The Review of Economic Studies, 53(4), 473-495.
Lütkepohl, H. (2011). Vector autoregressive models. In International Encyclopedia of Statistical Science (pp. 1645-1647). Springer Berlin Heidelberg.
Christiano, L. J., Eichenbaum, M., & Evans, C. (1994). The effects of monetary policy shocks: some evidence from the flow of funds (No. w4699). National Bureau of Economic Research.
Doan, T. A. (1992). RATS: User's manual. Estima.ote

