aicmodel selectiontime seriesvector-autoregression
Suppose I have three stationary economic time series $y_i$ that are not cointegrated and I want to investigate the relationship between them. I happen to be unsure about the "endogenousness" of $y_3$ so I first fit a VAR model with all three variables, but I also fit a VARX model with $y_1$ and $y_2$ endogenous but $y_3$ exogenous.
Can I use an information criterion like the AIC to help me select the better model? Or if not, what are my other options?
I would not use the guidance in that video. It is extremely poor advice to use automatic model selection strategies. To help understand this point, it may help you to read my answer here: Algorithms for automatic model selection
That having been said, the answer to your specific question is that you are misunderstanding what is being shown in the video. What the R output displayed in the video means is that the AIC listed on the far right is what the model would have if you dropped the variable in question. Lower AIC values are still better, both in the Wikipedia article and in the video. In the middle of the video, the presenter walks through reading the output and shows that dropping C2004 would lead to a new model with AIC = 16.269. This is the lowest AIC possible, so it is the best model, so the variable you should drop is C2004. The presenter is not saying that you should drop that model, but that you should drop C2004 from the current model to get that model. The second model, under step can be seen on the same screen. You can see that model does not include the variable C2004 and has AIC=16.27. (Again, for the record, using the AIC in this way is invalid, I'm just explaining what the video is recommending.)
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
Best Answer
You cannot use AIC because the dependent variables of the two models are not the same: $(y_1,y_2,y_3)$ vs. $(y_1,y_2)$. However, you could test whether the lags of $(y_1,y_2)$ have zero coefficients in the equation for $y_3$ using an $F$-test. If you do not reject $H_0$ of the coefficients jointly being zero, you may consider $y_3$ exogenous in the system of $(y_1,y_2,y_3)$.