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I have constructed a VEC model to study real housing price dynamics in relation to demographic demand, real GDP and costs of mortgages. However, I am stuck with the choice of deterministic terms.

The different options available regarding the choice of deterministic terms (as first(?) documented by Johansen in his book "Likelihood-based Inference in Cointegrated Vector Autoregressive Models") is summed up nicely in e.g. Matlab's manual on VECM. A previous study (by Elias Oikarinen) on housing price dynamics says (on pages 68-69):

"The trend variable may be needed in the long-run relation if the growth rates of the distinct series are not equal. For example, in a pairwise Johansen test between housing prices and stock prices a trend term might be needed in the long-run model if stock prices grow faster than housing prices (or the other way round). Nevertheless, in previous literature the possibility of a need for a trend in the long-run relation has usually been neglected when studying cointegration between different asset prices"

The author is hereby referring to the deterministic term spec referred to as H* in Matlab's manual and Johansen's book, i.e. $$ \Delta y_t =A(B'y_{t-1}+c_0+d_0t)+c_1+... $$ The model has thus an unrestricted drift in the short-run as well as a trend in the long-run. While I understand the reasoning, should $ log$ variables be treated differently?

Normally, if say, the cointegrating vector between housing prices and GDP per capita was $ (1, -4) $, and housing prices would trend upwards at twice the speed of GDP, I understand the need for a trend term, as the cointegrating vector (plus a constant term) could not account for the differences in growth rates, and the residuals of the cointegrating relation would probably end up being non-stationary. However, if the same variables were in $log$ and the vector would in this case end up being $(1, -2)$ (plus the constant term), this would already account for housing prices growing at a double rate, compared to GDP.

So my question is, what would be the appropriate deterministic term specification to use with $log$ variables in the situation described above? Would the modification "equivalent" to normally using the H* (trend in the long-run, unrestricted drift in the short-run) be H1 (just the unrestricted drift) in the case of $log$ variables?

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