When fitting a Holt-Winters model, I usually take the approach of retrospectively "predicting" some known historical values for the series, and optimising the coefficients for the parameters by changing these until I reach values which minimise the SSE for the retrospective forecast period.
I usually try to ensure that the coefficients chosen cause the parameters to have commonsensical values - e.g. a level which is broadly close to the actual level over time, etc. However, I have found that even if you input coefficient values which cause the parameters to go haywire and return nonsensical values, the model still actually has good predictive power, provided the parameters and coefficients chosen come close to minimising SSE.
Intuitively, this does make sense, since there is no reason why a crazy coefficient value and crazy parameter value can't collectively cause the prediction equation to generate sensible results - results which are close to optimal.
I guess my question is, apart from being obviously "wrong", what is actually wrong with this in a statistical sense? Is there anything to stop someone from using any old coefficient values, provided they come close to minimising SSE and generate accurate forward predictions?