Cointegration testing and seasonality

I run the multiple regression on the monthly data. To remove the linear trends I subtracted it using detrend function in R (differencing data was not an option because I wanted to check the relationship between the exact variables). I run the kpss test which didn't reject the null hypothesis that the variables are stationary. I understand that this test will not detect seasonality. And that's my problem, because there is a clear seasonal component of order 12 in one of the variables:

I decided to run a regression anyway: I have 4 explanatory variables, including one auto-regressive component. It is like this: $Y_t = aY_{t-1} + bX_t + cZ_t +d A_t$

All of the variables were significant, R squared isn't that big (0.7), picked explanatory variables are intuitively reasonable in explaining Y. Residuals are white noise and normally distributed. Durbin Watson statistic is higher than R squared. But I am still worried that it is spurious regression because of this seasonality. I wanted to check for cointegration, which would solve my problem, however the seasonality is of the order of 12, therefore the variable is not I(1), so I can't use Johansen or bound test. Is there any test for cointegration for the variables of higher orders of integration?

P.S 11 dummies made the majority of the variable insignificant.

• You may be interested in: 1) tests for seasonal stability, the Canova and Hansen test statistic is the seasonal counterpart of the KPSS test, 2) seasonal cointegration tests, 3) this post linked on the right side of this page. – javlacalle Jan 5 '18 at 18:52