I'm working with some panel data, and I'm interested in estimating the following model:

$$\Delta y_{t+1} = \alpha + \delta t+\beta_1 \ln y_t + \beta_2 \ln x_t$$

Where $y_t \sim I(1)$ (and $\Delta y_t \sim I(0)$). Hence, we have a cointegration relationship:

$$\alpha + \delta t+\beta_1 \ln y_t + \beta_2 \ln x_t \sim I(0)$$

Similar work approaches this estimation problem by using error correction models using the above relationship as the co-integration relationship. Why should I do this, as opposed to simply estimating the model directly?

I'm working with some panel data, and I'm interested in estimating the parameters of the following process:

$$\Delta y_{t+1} = \alpha + \delta t+\beta_1 \ln y_t + \beta_2 \ln x_t+\epsilon_t$$

Where $y_t \sim I(1)$ (and $\Delta y_t \sim I(0)$). Hence, we have a cointegration relationship:

$$\alpha + \delta t+\beta_1 \ln y_t + \beta_2 \ln x_t \sim I(0)$$

Similar work approaches this estimation problem by using error correction models using the above relationship as the co-integration relationship. Why should I do this, as opposed to simply estimating the model directly?

My intuition says that the direct approach should be valid because both sides are $I(0)$ to begin with.