A stochastic process is a mathematical model used to describe the evolution of random variables over
time or space. It represents a sequence of random variables indexed in some way, often time. Stochastic
processes are widely used in various fields, including finance, engineering, physics, biology, and
telecommunications.
Properties of a stochastic process include:
1. **Randomness**: The values of the process at any given time are not deterministic but are subject to
randomness or uncertainty.
2. **Indexing**: The process is indexed by a parameter representing time, space, or some other
dimension.
3. **Stationarity**: The statistical properties of the process (such as mean, variance, and covariance)
remain constant over time.
4. **Independence or Dependence**: The random variables in the sequence may be independent of
each other or exhibit some form of dependence.
5. **Markov Property**: The future behavior of the process depends only on its present state and not
on the sequence of events that preceded it.
WHITE NOISE PROCCESS:
a white noise process can be defined as a sequence of random variables {��}{εt} satisfying the
properties of independence, constant mean, and constant variance. These properties imply that the
white noise process exhibits random fluctuations with no autocorrelation or systematic patterns across
time.
It is important for two main reasons:
1. Predictability: If your time series is white noise, then, by definition, it is random. You
cannot reasonably model it and make predictions.
2. Model Diagnostics: The series of errors from a time series forecast model should ideally
be white noise
Gaussian process (GP)
a Gaussian process (GP) is a stochastic process where any finite collection of random
variables follows a joint Gaussian distribution. Gaussian processes are used to model the
uncertainty and correlation structure of time series data.
Properties:
1. Joint Gaussian Distribution: In a GP, any finite collection of observations forms a joint
Gaussian distribution. This property simplifies computations involving conditional
distributions and predictions, making Gaussian processes tractable for inference.
2. Mean Function: A GP is characterized by its mean function, which represents the
expected value of the process at each point in time. The mean function can capture
trends, seasonal patterns, or any systematic behavior in the time series.
3. Covariance Function (Kernel): The covariance function, also known as the kernel
function, describes the degree of similarity or correlation between observations at
different time points. It quantifies the dependence structure of the process over time,
indicating how the values of the process at different time points relate to each other.
4. Flexibility: Gaussian processes offer flexibility in modeling various time series patterns,
including trends, seasonality, and irregular fluctuations. By selecting appropriate mean
and covariance functions, Gaussian processes can capture complex temporal dynamics,
making them suitable for modeling diverse types of time series data.
5. Stationarity: Gaussian processes can exhibit different levels of stationarity. Weak
stationarity implies that the mean and covariance functions are time-invariant, indicating
that the statistical properties of the process do not change over time on average. Strong
stationarity requires that the entire distribution of the process is invariant under time
shifts, which imposes stronger constraints on the process's behavior.
A unit root process, also known as a unit root series or a non-stationary series, is a stochastic process
commonly encountered in time series analysis. It refers to a process where the root of the characteristic
equation (unit root) is equal to 1, indicating that the process exhibits a high degree of persistence and
lacks a stable mean over time.
The standard equation for a unit root process can be written as:
\[Y_t = Y_{t-1} + \epsilon_t\]
Where:
- \(Y_t\) represents the value of the time series at time \(t\).
- \(Y_{t-1}\) represents the lagged value of the time series at time \(t-1\).
- \(\epsilon_t\) represents the error term or the random shock at time \(t\).
Properties of a unit root process include:
1. **Non-Stationarity**: Unit root processes are non-stationary because their statistical properties such
as mean and variance are not constant over time. This non-stationarity arises due to the presence of the
unit root, which implies that the process has a trend or exhibits random walk behavior.
2. **High Persistence**: Unit root processes display high persistence, meaning that shocks or deviations
from the mean tend to persist over time. This persistence is evident in the fact that the current value of
the series is highly correlated with its past values.
3. **Trend or Random Walk**: Unit root processes often exhibit a trend or random walk behavior,
where the series evolves over time without a clear pattern or tendency to revert to a stable mean.
4. **Spurious Regression**: Due to the non-stationarity of unit root processes, regression analyses
involving such series can produce spurious results. This means that apparent relationships may arise
between variables solely due to their non-stationary nature, even if no true relationship exists.
5. **Differencing**: To render a unit root process stationary, it often requires differencing the series
(e.g., first-order differencing) to eliminate the unit root. Once differenced, the series becomes suitable
for further analysis using stationary time series methods like ARIMA modeling.
Understanding and addressing unit root processes are essential in time series analysis, particularly in
forecasting and econometrics, as failing to account for non-stationarity can lead to biased estimates and
erroneous conclusions.