Does the level matter in non-stationary models?

We have been led to believe that the level of the data is important in non-stationary models. We have been working with models that include a random walk plus drift specification for technological progress, A:

A/(A(-1)==dA = exp(gamma + error)

It seems reasonable to suppose that the level will matter in a world of co-integrating macroeconomic time series where the relationships between the levels contain important information; think of the ratio consumption/output=c/y, for example if we estimate with two series.

Although our experience with simulated data suggests that the level does matter in these models, it is not clear to us how this can be the case given how the data is entered into Dynare.

For example, we estimate using observed output==Y_obs, where the output divided by the trending technological progress is modeled in Dynare as

Ouput/A==y = (k(-1)/dA)^chi * lab^(1 - chi),

where k is the capital stock, lab is the labor supply, and chi is the Cobb-Douglas production function parameter. To enter the observation into Dynare we include the observer equation

Y_obs/Y_obs(-1) = dA * y/y(-1)

We only see ratios here. It looks like the level information is being thrown away. If Dynare is somehow using the level information, what is it doing?

We are convinced that Dynare does use the level information when estimating the model. We have done simulation studies where we divide all our consumption observations C_obs by 2. When produce estimates based on this wrong C_obs, and on Y_obs, and lab data (the last two having the correct levels), this dramatically affects some of the parameter estimates - so it would seem that the level is used somewhere.

Are we correct that the level matters? Just out of curiosity, can you give us a reference explaining how this is done, or tell us where to look in the code?

Thank you.