How to interpret lags in cointegration test for constructing the mean reverting series 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? 
 A: Good detailed coverage of cointegration and cointegration testing is available in Chapter 12 of Zivot & Wang's "Modeling Financial Time Series with S-PLUS" (2006). I will base my answer on it, but I will skip most of the details that are not directly relevant with regards to the original question.
Engle & Granger's cointegration testing procedure consists of two stages:


*

*Obtaining the cointegration residual $\varepsilon_t=y_t-\beta x_t$

*Testing $\varepsilon_t$ for a unit root.


The stage relevant for your question is the second one. It is common to use the augmented Dickey-Fuller (ADF) test in this stage. The word "augmented" indicates that the test regression may include lags of the dependent variable (with $\delta$ coefficients in front):
$$ \Delta \varepsilon_t = \alpha + \beta t + \gamma \varepsilon_{t-1} + \delta_1 \Delta \varepsilon_{t-1} + \dotsc + \delta_{p-1} \Delta \varepsilon_{t-p+1} + u_t $$
($\alpha$ and $\beta t$ may or may not be included depending on the first stage of the Engle & Granger procedure; see the reference above for details.) The motivation for including the lags of the dependent variable can be found in Chapter 4 of the same textbook. Essentially, they are there to make the test regression better represent the underlying process being tested for a unit root.
In my understanding, the phrase

a lag is used in the ADF test, say of length $n$

means that there are $n$ lags of the dependent variable in the ADF test equation, i.e. $p-1=n$ or $p=n+1$. That is, you have an equation of the form
$$ \Delta \varepsilon_t = \alpha + \beta t + \gamma \varepsilon_{t-1} + \delta_1 \Delta \varepsilon_{t-1} + \dotsc + \delta_{n} \Delta \varepsilon_{n} + u_t $$
(with or without $\alpha$ and $\beta t$.)
Meanwhile, the first stage of the Engle & Granger procedure is not affected by the quoted statement and remains as indicated above.
