levels versus first difference

[equestion]

I am estimating a three variable VAR. Two variables are I(1) and one variable is I(0). Isn't it the case that all variables in the VAR should be included in stationary form to avoid spurious results?

[TomDoan]

I am estimating a three variable VAR. Two variables are I(1) and one variable is I(0). Isn't it the case that all variables in the VAR should be included in stationary form to avoid spurious results?

Actually, no. In fact, if the two I(1) variables are cointegrated, the VAR done in differences is misspecified.

The "spurious regression" comes from estimating something like y=bx+u where y and x both have unit roots. If b is, in fact, zero, then u also has to have a unit root, and it's the combination of (uncorrelated) unit roots in both x and u that cause the spurious regression result. When you run a VAR, you should have enough lags that the residuals are close to being serially uncorrelated and at minimum you don't have a residual unit root.

[iloverats]

if the variables are all I(1) but not cointegrated, is the VAR done in levels right?

[TomDoan]

If the variables aren't cointegrated, then you can do the VAR either way.

[iloverats]

If the variables aren't cointegrated, then you can do the VAR either way.

but if the variales are unstationary , can level regresion do the right statistical Inference?
can the impusles response function make sense?

[TomDoan]

If the variables aren't cointegrated, then you can do the VAR either way.

but if the variales are unstationary , can level regresion do the right statistical Inference?
can the impusles response function make sense?

That depends upon what type of "statistical inference" you are trying to do. The results in Sims, Stock and Watson, Econometrica 1990 show that any tests of any hypothesis which doesn't restrict the unit root behavior of the process are asymptotically equivalent done with or without differencing. And any hypothesis not covered by that isn't being "tested" by differencing, it's being imposed.

The impulse responses make perfect sense.

[iloverats]

you said that the irf make sense

because their paper said "the OLS estimator is consistent"

am i right

[TomDoan]

That the coefficient estimates are consistent long pre-dates the SSW paper. The open question that they settled was which linear combinations of coefficients had standard vs non-standard asymptotic distributions.

[Henrique Andrade]

I am estimating a three variable VAR. Two variables are I(1) and one variable is I(0). Isn't it the case that all variables in the VAR should be included in stationary form to avoid spurious results?

Actually, no. In fact, if the two I(1) variables are cointegrated, the VAR done in differences is misspecified.

The "spurious regression" comes from estimating something like y=bx+u where y and x both have unit roots. If b is, in fact, zero, then u also has to have a unit root, and it's the combination of (uncorrelated) unit roots in both x and u that cause the spurious regression result. When you run a VAR, you should have enough lags that the residuals are close to being serially uncorrelated and at minimum you don't have a residual unit root.

Dear Tom, if I understood correctly, in that case it would be better estimate the VAR without differencing the series, i.e., put all variables in their level regardless of their order of integration. Or it would be better to difference only the two variables that are I(1)?

Best regards,

[TomDoan]

Dear Tom, if I understood correctly, in that case it would be better estimate the VAR without differencing the series, i.e., put all variables in their level regardless of their order of integration. Or it would be better to difference only the two variables that are I(1)?

Best regards,

If you're not interested in studying the unit root behavior of the series, then there is no real advantage to differencing the variables. A model with differences might be misspecified (if there are I(0) or cointegrated variables) and if the model in differences isn't misspecfied, the two sets of estimates are asymptotically equivalent.

[Henrique Andrade]

Dear Tom,

Thanks a lot for your valuable help

I think I have one last question:

I'm estimating a trivariate VAR using the following variables (using Brazilian monthly data for the period between 1999 and 2010):

(1) Output gap (stationary, I(0));
(2) Nominal interest rate (stationary, I(0));
(3) Price level (non stationary, I(1)).

The question is: Does it make any sense to do this estimation without differencing the price level (i.e. changing it into the inflation rate)? I´m asking this because the price level exhibits a trend, thus if I proceed the estimation without differencing I will have a trended series explaining detrended series. Am I correct?

Best regards,

[TomDoan]

The most common way to handle that combination of variables would be to use the inflation rate rather than the price level, which gives results that are easier to interpret. Since no variable in the model could possibly be cointegrated with the price level, you don't have to worry about the problem of misspecification by ignoring cointegration.

The theory is that there is no harm in modeling this using the price level because the lag polynomial in the price level in the equations for the stationary variables can always recover the difference as long as you have at least two lags. However, here there is also no harm in using the difference because of the collection of variables involved. Add log(m) and the model with differences of log(p) and log(m) might be misspecified by ignoring possible cointegration between p and m.

[nazif]

Dear Tom,

I have also similar problem. I am analyzing contractionary effects of oil prices on output based on a VAR model including oil and wholesale prices, nominal exchange rate and also interest rate. All variables in my VAR model are I(1) and they are cointegrated. The IRF results are in line with expectations, confirms the negative impact of oil prices on output when the model is estimated in levels. But the results are completely different when the model estimated with the first differenced variables. What is your suggestion in this case?

Could you please give me any reference dealing with this problem?

Thanks for your help.

Nazif

[moderator]

The suggestion is that you not estimate the VAR in first differences. If the variables are cointegrated, the VAR in first differences is misspecified. That goes back to the original Engle and Granger paper.