Time Series – Estimating Second Order Differences Stationary Series via VECM Model Without Economic Theory

Question

Because some series contain negative raw data, and data is normalized by MinMax.
Can VECM be used after second-order difference where the series are stationary? If can, whether the number of lagged difference is the p-1 or p-2 through VAR(p)?

From Comments:

integrated of order two (Δ2𝑥𝑡) for all series. My data is not only integrated of order one, but must two to keep all series stationary. According to Lutkepohl "New Introduction to Multiple Time Series Analysis", it only gives proof of integrated of order one, but no proof more than one. When I get through the chapter, it seems that Δd Xt works. Because so far I am not writing paper but just the aim to conduct a robust model without residual autocorrelation and not-bad generalized model. To some degree, the model need not be so strict with theory or mathematic proof.

Answer 1

It looks like you have $x_1$ and $x_3$ being I(1) and $x_2$ being I(2). This is what I would do.

1. Take the first difference of $x_2$ to make it I(1) and call that $y$: $y:=\Delta x_2$.
2. Inspect whether $(x_1,y,x_3)$ are cointegrated.
3. If they are cointegrated, use a VEC model for them. Select the lag order as you would if $(x_1,y,x_3)$ were raw data.
   If they are not cointegrated, inspect whether any pair is cointegrated.
   ___ If no pairs are cointegrated, use a VAR on $(\Delta x_1,\Delta y,\Delta x_3)$.
   ___ If one pair is cointegrated, without loss of generality say it is $(x_1,x_3)$. Use a restricted VEC for $(x_1,y,x_3)$ where $y$ is absent from the cointegrating relationship. For details, see "VAR or VECM for a mix of stationary and nonstationary variables?".
   ___ If two pairs are cointegrated, then the triple must be cointegrated, which contradicts the previous step.

Instead of 2. and 3. you could also just use a VAR on $(x_1,y,x_3)$, as this turns out to be a rather good option regardless of cointegration, according to Ashley & Verbrugge "To difference or not to difference: a Monte Carlo investigation of inference in vector autoregression models" (2009).

What I would not do is difference each series twice and try to fit a VECM. First, two of the series will be overdifferenced and as such will contain unit-root moving average components. Second, VECM will not be appropriate as there can be no cointegration between variables that, after the differencing, are at most I(0).