How is the proper number of lags for ACF or PACF displaying? How many lags should be used for ACF or PACF displaying if we have $S$ seasonality?
For example,
for 500 observations I have 25 lags
for 200 observations I have 22 lags
It is independent from frequency of seasonality (for $S = 7, 14, 50, 60, ...$ number of lags on the picture is the same). Is it enough for model estimation based on ACF/PACF for high frequencies?
We do not need to show lags for few periods; for example, for $S=50$ at least 170 lags (it contains 3 periods)? Sorry, but available materials show different approaches.
 A: There's no fixed rule. It is a function of the noise in the time-series. I would show at least until no data-point crosses a confidence interval or $\frac{n}{10}$, whichever comes first.
Shown below are realizations of seasonal trend at lag of 7, with white noise $\sim\mathbb N(0,0.1)$ and $\sim\mathbb N(0,0.3)$. The confidence intervals were calculated by $\frac{\text{PPF}(0.99)}{\sqrt{n}}$ assuming the noise is distributed normally.


EDIT:
Here's another plot of lag, with more of the ACF and PACF shown:

Here's another plot with a higher period.

The code used for the figures:
import numpy as np
from scipy import stats
from matplotlib import pyplot as plt
from statsmodels import api as sm


def main():
    prototype = np.random.random(60)
    for _ in xrange(560 / 60):
        prototype = np.concatenate((prototype, np.random.normal(0, 0.1, 60) + prototype[:60]))
    prototype = prototype[60:]
    n = prototype.shape[0]
    pa = sm.tsa.pacf(prototype, 100)
    acf = sm.tsa.acf(prototype, nlags=100)
    plt.figure()
    plt.subplot(4, 1, 1)
    plt.plot(prototype)
    plt.title("Time Series with a Lag of 60, White Noise of .1")
    plt.subplot(4, 1, 2)
    plt.plot(prototype.reshape(-1, 60).T)
    plt.title("Overlapping Windows")
    plt.subplot(4, 1, 3)
    plt.plot(acf)
    z = stats.norm.ppf(0.99)
    c = '#660099'
    plt.axhline(y=z / np.sqrt(n), linestyle='--', color=c)
    plt.axhline(y=-z / np.sqrt(n), linestyle='--', color=c)
    plt.title("ACF")
    plt.subplot(4, 1, 4)
    plt.plot(pa)
    plt.axhline(y=z / np.sqrt(n), linestyle='--', color=c)
    plt.axhline(y=-z / np.sqrt(n), linestyle='--', color=c)
    plt.title("PACF")
    plt.show()


if __name__ == '__main__':
    main()

