Augmented Dickey-Fuller Test # of Lags as part of a time series analysis for my master thesis I want to test whether a time series is stationary with the Augmented Dickey-Fuller test in Python. I attached the result for one time series. While I do understand that this result means that the series is stationary, I do not get what the # of lags are. Does this mean that the series is only stationary if 13 lags are used? Do I need to transform the data? Help would be highly appreciated!

import statsmodels
from statsmodels.tsa.stattools import adfuller
import pandas as pd
import statsmodels.api as sm
from statsmodels.tsa.filters.hp_filter import hpfilter

class StationaryTests:
    def __init__(self, significance=.05):
        self.Significance
        self.pValue = None
        self.isStationary = None
    
    def ADF_Stationarity_Test(self, timeseries, printResults = True):
        #Dicky-Fuller tests:
        #adfTest = adfuller(timeseries, maxlag = 1)
        adfTest = adfuller(timeseries, autolag='AIC')
        
        self.pValue = adfTest[1]
        
        if (self.pValue<self.SignificanceLevel):
            self.isStationary = True
        else:
            self.isStationary = False
        
        if printResults:
            dfResults = pd.adfTest[0:4], index=['ADF Test Statistic', 'P-Value', '# Lags Used', '# Observations Used'])
            #Add Critical Values
            for key,value in adfTest[4].items():
                dfResults['Critical Value (%s)' %key] = value
            print('Augmented Dickey-Fuller Test Results:')
            print(dfResults)

 A: The lags are the reason for the word "Augmented" in the Augmented Dickey Fuller test. Without the lags, you'd be doing a Dickey Fuller test, like this one: $\Delta y_t=\alpha+\theta y_{t-1}+e_t$ testing whether $\theta=0$ where $\theta=\rho-1$ obtained by subtracting $y_{t-1}$ from both sides of the following equation $y_t=\alpha+\rho y_{t-1}+e_t$, in which we want to test for unit root (i.e. $\rho=1$). Under the null hypothesis $H_0:\theta=0$, the left side is $I(0)$ while the right side is $I(1)$, therefore this is an unbalanced regression so we already get the sense that $\theta$ may not have a t-distribution. The distribution of $\theta$ under the $H_0$ in a one-tailed test is called the $\tau$ distribution. Some values of this distribution have been tabulated under the assumption that the error terms are normally and identically distributed, but most crucially - under the assumption that the error terms are serially uncorrelated. So if in the Dickey Fuller equation the errors $e_t$ are normally and identically distributed and serially uncorrelated then you can compute the value of your sample's $\tau$ statistic and compare it to the $\tau$ critical values to decide if you reject the null of a unit root. But what if the errors $e_t$ are serially correlated? Well, then the distribution of $\theta$ will not be $\tau$. It will be some other distribution; perhaps an unnamed one. So, how do we fix it? We have to change the regression model  $\Delta y_t=\alpha+\theta y_{t-1}+e_t$ so that $e_t$ are serially uncorrelated. How do we do that? With the ADF test, under the assumption that $e_t$ is an invertible ARMA process, which recall means that the process has an AR representation, we can just add AR terms one at a time until $e_t$ are serially uncorrelated. Once we've added n AR terms where n is the order of the AR representation of the assumed ARMA process for $e_t$, we are back to the conditions under which the $\tau$ distribution has been tabulated so we can use the $\tau$ distribution once again. While the ADF test introduced AR terms to correct for serial correlation in $e_t$ other tests address it differently (for example the Phillips-Perron test makes a non-parametric correction to the t-test statistic). What if our assumption that $e_t$ follows an invertible ARMA process (in which the MA polynomial does not have a large negative root) is incorrect? Then the ADF test will overreject the null hypothesis of a unit root.
So, returning to your question. You've added the lags in the ADF test because you want the estimator of interest to follow a known (i.e. $\tau$ distribution). Your conclusion here pertains to whether your series has a unit root or not. Note that a series may not have a unit root but may still not be stationary due to, for example, non-constant variance. If you reject the null hypothesis of the ADF test, you conclude that your series does not have a unit root, or in other words is $I(0)$ or Integrated of order 0.
