Question

     Let A and B be two independent white-noise processes. What can you say about the time series properties of the process Y = A+B? How will your answer change if the processes A and B are correlated at lag one?

 

Answer 1

A stochastic process is a set of random variables with time 1 2 +1 = = as the index. In most applications, the time index is a regularly spaced index that reflects calendar time (e.g., days, months, years, etc.). When modeling time-series data, the ordering enforced by the time index is critical because we often wish to capture any temporal connections between the random variables in the stochastic process. The sequence in which the random variables that make up the sample appear is irrelevant when sampling from a population since they are independent.
A stochastic process employing observations is realised by the sequence of observed data 1 = 1 2 = 2 = = ==1.
1
We'll commonly refer to the stochastic process as a simple = to save on notation.
CONCEPTS OF TIME SERIES IN CHAPTER 1
The goal of time series modeling is to provide a clear picture of the probabilistic behavior of the underlying stochastic process that is considered to have produced the observed data. Furthermore, we want to be able to use the observed sample to estimate critical aspects of a time series model, such as measures of time dependence. To do so, we must make a number of assumptions regarding the combined behavior of the stochastic process' random variables, such that the stochastic process may be compared to a random sample from a given population.