The following code comes from this book, Statistics and Data Analysis For Financial Engineering, which describes how to generate simulation data of ARCH(1) model.
library(TSA)
library(tseries)
n = 10200
set.seed("7484")
e = rnorm(n)
a = e
y = e
sig2 = e^2
omega = 1
alpha = 0.55
phi = 0.8
mu = 0.1
omega/(1-alpha) ; sqrt(omega/(1-alpha))
for (t in 2:n){
a[t] = sqrt(sig2[t])*e[t]
y[t] = mu + phi*(y[t-1]-mu) + a[t]
sig2[t+1] = omega + alpha * a[t]^2
}
plot(e[10001:n],type="l",xlab="t",ylab=expression(epsilon),main="(a) white noise")
My question is that why we need to discard the first 10000 simulation?
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Truncation is needed to deal with sampling bias introduced by the simulation model's initialization when the simulation output is a time series.
Not all simulations require truncation of initial data. If a simulation produces independent observations, then no truncation is needed. The problem arises when the simulation output is a time series. Time series differ from independent data because their observations are serially correlated (also known as autocorrelated). For positive correlations, the result is similar to having inertia—observations which are near neighbors tend to be similar to each other. This characteristic interacts with the reality that computer simulations are programs, and all state variables need to be initialized to something. The initialization is usually to a convenient state, such as "empty and idle" for a queueing service model where nobody is in line and the server is available to immediately help the first customer. As a result, that first customer experiences zero wait time with probability 1, which is certainly not the case for the wait time of some customer k where k > 1. Here's where serial correlation kicks us in the pants. If the first customer always has a zero wait time, that affects some unknown quantity of subsequent customer's experiences. On average they tend to be below the long term average wait time, but gravitate more towards that long term average as k, the customer number, increases. How long this "initialization bias" lingers depends on both how atypical the initialization is relative to the long term behavior, and the magnitude and duration of the serial correlation structure of the time series.
The average of a set of values yields an unbiased estimate of the population mean only if they belong to the same population, i.e., if E[Xi] = μ, a constant, for all i. In the previous paragraph, we argued that this is not the case for time series with serial correlation that are generated starting from a convenient but atypical state. The solution is to remove some (unknown) quantity of observations from the beginning of the data so that the remaining data all have the same expected value. This issue was first identified by Richard Conway in a RAND Corporation memo in 1961, and published in refereed journals in 1963 - [R.W. Conway, "Some tactical problems on digital simulation", Manag. Sci. 10(1963)47–61]. How to determine an optimal truncation amount has been and remains an active area of research in the field of simulation. My personal preference is for a technique called MSER, developed by Prof. Pres White (University of Virginia). It treats the end of the data set as the most reliable in terms of unbiasedness, and works its way towards the front using a fairly simple measure to detect when adding observations closer to the front produces a significant deviation. You can find more details in this 2011 Winter Simulation Conference paper if you're interested. Note that the 10,000 you used may be overkill, or it may be insufficient, depending on the magnitude and duration of serial correlation effects for your particular model.
It turns out that serial correlation causes other problems in addition to the issue of initialization bias. It also has a significant effect on the standard error of estimates, as pointed out at the bottom of page 489 of the WSC2011 paper, so people who calculate the i.i.d. estimator s2/n can be off by orders of magnitude on the estimated width of confidence intervals for their simulation output.