Suppose the quarterly model has significant seasonality (seasonal errors have serial correlation), we might include a seasonal lag in the autoregressive model and estimate. But does it cause Time-Series Misspecification (including lagged dependent variables as independent variables in regressions with serially correlated errors)?
The other way around. Adding the 4th lag (4Q of the variable) increases correct specification. Remember that you are not including the dependent variable itself as an independent variable, but a lagged version of it. So it is like a “new different” variable.
The misspecification resides when you omit the seasonal lag, so the error term of that model is indeed correlated with the dependent variable and independent variables (because is a time-series model). Remember that the error term must not be correlated with the dependent variable nor the independent variables.