Using a VAR over a VECM (in spite of of existing cointegration)

Question

Is there ever a reason to use a first differenced VAR over a VECM when all your variables are I(1) and co integration exists?

The reason why I ask is because I see in the most recent Bank of Canada Analytic note on the minimum wage increase (page 15) it seems that opt for a reduced form VAR over a VECM. $$\Delta \text{CPI}_t^p=\Sigma_{i=0}^4\eta_i \Delta\text{MW}_{t-i}+\Delta \alpha_1\text{CPI}_{t-1}^{p}+\alpha_2\Delta\text{UR}_t^p+\mu^m+\mu^y+\mu^p+\mathcal{E}_t$$

Is my understanding of the model correct? Is this theoretically sound?

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The answer I got from the bank of Canada with regard to this question is the following:

Equation A1 (the equation in the question) is estimated with ordinary least squares (OLS). Data are pooled.

The equation above is neither a VAR or a VECM. It seems a little odd, but that is the model which is employed.

Answer 1

There is at least one reason -- the bias-variance trade-off. You might prefer a wrong model as long as it gives you better forecasts.

Suppose VECM is the true model. Then VAR in first differences is wrong because it misses a variable, namely, the error correction term*. Suppose also that the loading (the coefficient) on the error correction term cannot be estimated with any decent precision given the dataset at hand. Then it might make sense to exclude that variable if the purpose of modelling is forecasting, because the variable might be estimated so imprecisely that, when included in the model, it would reduce forecast accuracy. Excluding the variable means introducing a bias, but it is motivated by a reduction in estimation variance; hence, the bias-variance trade-off.

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*There could be one or more error correction terms depending on the rank of cointegration, but that does not change the essence.