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If you forecast dlog(y) there are no dynamics in the equation, so static=dynamic.

If you forecast y, then forecasts of y depend upon previous values of y (through the dlog), so dynamic and static will differ.

Thanks for the response.

I think what I am grappling with is understanding the difference between a forecast of y and forecasting dlog(y) then calculating what forecasts of y are. How exactly does the dynamics get into the equation when forecasting y? Does Eviews estimate an AR() term in the background?

If you know what the value of dlog(y) is, you can work out what the value of y is, as long as you know the initial value. You just undo the differences.

The difference between static and dynamic is simply what value of the lag of Y you use.

That is exactly what I have done, and that is what I think is confusing about this process. As my exercise described above is all in-sample, so in both cases dynamic and static the value of y is known. I guess my question is, under the dynamic case, how is the forecast value of y determined and why is it different to the dynamic case when dlog(y) is forecast and then working out what y is?

Doesn't matter whether Y is known or not, dynamic forecasting will use the estimated values of lagged Y (that's the definition of dynamic forecasting). Static will always use actual values of lagged Y (and if Y isn't known, static forecasting can't be performed).