# Vector Autoregressive Regression in Levels

Various textbooks suggest that it is essential to test the variables used for stationarity before a VAR anaylse. If the tests give an indication of I(1) variables, these variables should be transformed. This can be achieved by logarithms and differentiation. This can be checked again with stationarity tests.

But now I have seen several well published papers by an author who only uses the logarithm of the variables. As far as I can understand, it should NOT be enough for the variables to become I(0). It is not tested for cointegration. In the paper there is only a reference to Sims, 1990. The author replies by mail, that "If you estimate a VAR in (log) levels, you always have consistent estimates."

• The estimation would only be significantly influenced if first differences were estimated, but the data were co-integrated.

or

• A VECM is estimated whose long-term involvement is not properly captured. Furthermore, the strength of the tests would be controversial.

"It is therefore better to estimate VARs in (log) levels, unless you are really sure about stationarity and/or long-run relationships in the data."

I would agree about the test power, but everything else doesn't seem to be very consistent with my previous knowledge. Am I missing something here? Is there a special case when it is possible to estimate a VAR with I(1) variables?

You need to distinguish between estimation and inference. As Sims writes, we may consistently estimate the coefficients of the VAR when estimating in levels:

As maybe the simplest possible example of a VAR, take a univariate AR(1) $$y_t=\rho y_{t-1}+u_t$$ with $\rho=1$, so a random walk, such that $y_t$ is $I(1)$.

It is well-known that $T(\hat\rho-1)$ converges in distribution (to a Dickey-Fuller distribution), with $\hat\rho$ the OLS estimator of a regression of $y_{t}$ on $y_{t-1}$ - i.e., we estimated "in levels".

This implies that $$T(\hat\rho-1)=O_p(1),$$ i.e. is bounded in probability. Hence, $$\hat\rho-1=O_p(T^{-1})$$ so that $\hat\rho-1=o_p(1)$ or $$\hat\rho\to_p1,$$ i.e. OLS is consistent.

We do have issues for inference, however - e.g., if we want to test $H_0:\rho=1$, we need to use special unit root tests.

• So if I understand the explanation correctly, the standard errors can be distorted, which leads to tests (F-test, t-test, Portmanteau test etc) having problems. So can I appreciate consistent Impulse Response functions in levels? Can I trust the confidence intervals? They are related to the standard errors. – Martin Sep 10 '18 at 10:39
• Yes, CIs are certainly affected, too. IRs should be OK, but I have to give that another thought. – Christoph Hanck Sep 10 '18 at 10:48
• For the sake of completeness and for later readers: Benkwitz et al 2007 and Ashley 2000 – Martin Sep 11 '18 at 6:52