For bivariate time series cases, the Engle-Granger two step cointegration test is essentially testing the linear combination for a unit root. The format of the error term is thus: $$ y_t - \gamma x_t = \epsilon_t $$
When a lag is used in the ADF test, say of length $n$, is the mean reverting series: $$ y_t - \gamma x_{t-n} = \epsilon_t $$
Or is it something else? I am trying to understand how the lag changes the mean reverting series. Also, does this generalize to other cointegration tests and models?
Edit
Example: Let $y$ = housing prices in San Francisco and $x$ = housing prices in New York City. The approximation process is given by the linear combination $y_t-\gamma x_t = \epsilon_t$. Testing $\epsilon_t$ for a unit root via ADF results in a high probability of cointegration with a lag of $n$.
Does this mean the mean reversion process is $y_t - \gamma x_{t-n} = \epsilon_t$, meaning there is a lag in the relationship such that the housing prices in San Francisco are cointegrated with the housing prices in New York $n$ time steps ago? If not, how does the lag express itself in the relationship of the housing prices?