Week 5.2: Two types of stationarity-2

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From the course by National Research University Higher School of Economics

Stochastic processes

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National Research University Higher School of Economics

Stochastic processes

76 ratings

The purpose of this course is to equip students with theoretical knowledge and practical skills, which are necessary for the analysis of stochastic dynamical systems in economics, engineering and other fields. More precisely, the objectives are 1. study of the basic concepts of the theory of stochastic processes; 2. introduction of the most important types of stochastic processes; 3. study of various properties and characteristics of processes; 4. study of the methods for describing and analyzing complex stochastic models. Practical skills, acquired during the study process: 1. understanding the most important types of stochastic processes (Poisson, Markov, Gaussian, Wiener processes and others) and ability of finding the most appropriate process for modelling in particular situations arising in economics, engineering and other fields; 2. understanding the notions of ergodicity, stationarity, stochastic integration; application of these terms in context of financial mathematics; It is assumed that the students are familiar with the basics of probability theory. Knowledge of the basics of mathematical statistics is not required, but it simplifies the understanding of this course. The course provides a necessary theoretical basis for studying other courses in stochastics, such as financial mathematics, quantitative finance, stochastic modeling and the theory of jump - type processes.

From the lesson

Week 5: Stationarity and Linear filters

Upon completing this week, the learner will be able to determine whether a given stochastic process is stationary and ergodic; determine whether a given stochastic process has a continuous modification; calculate the spectral density of a given wide-sense stationary process and apply spectral functions to the analysis of linear filters.

Week 5.1: Two types of stationarity-119:50

Week 5.2: Two types of stationarity-28:59

Week 5.3: Spectral density of a wide-sense stationary process-17:49

Week 5.4: Spectral density of a wide-sense stationary process-24:58

Week 5.5: Stochastic integration of the simplest type10:11

Week 5.6: Moving-average filters-15:59

Week 5.7: Moving-average filters-212:03

Week 5.8: Moving-average filters-38:30

Meet the Instructors

Vladimir Panov
Assistant Professor
Faculty of economic sciences, HSE

Our next example is a so-called moving average process.

This process is very popular in the context of a econometric.

This process actually means that Yt up is a linear combination of the variance of

the whiteners process Xt from previous time points,

that is Yt is equal to Xt plus a1 X t minus one plus so on,

plus aq X t minus q.

Here a1 and so on aq are real numbers.

And for simplicity we'll assume there is also a coefficient here,

a zero which is equal to one.

So this is a very popular model and in the context of our course,

it will be very natural to ask whether this process is tertiary honored.

First of all, let me mention that mathematical expectation

of Yt is actually equal to zero.

This is a very simple corollary of the fact

that the mathematical expectation of Xt is equal to zero.

As for the covariance function,

it is equal to covariance between the sum j from 0

to q aj X t minus j.

And this sum k from zero to q ak X s minus k. Now,

we can guess linearity of the covariance function again is equal to the sum,

j from zero to q and sum k from

zero to q of the aj ak,

and here I should write covariance between x t minus j and x s minus

k. You know that xt is

a white noise and we have already calculated the covariance function of this was process.

Therefore, I just used that result is equal to sigma squared

multiplied by the indicator that t minus s is equal

to j minus k. This formula means that

the function kts can be represent as the function gamma at time point T minus s,

because only the difference between the arguments play some role here.

In particular, if we consider a partial case of this process which is

mainly denoted by m a of q so there's q parameters.

So the case when q is equal to one,

we get this auto covariance function is not equal to zero, only three points.

So it's some number multiplied by the indicator.

It's an absolute value of X is equal to one plus some value

multiplied by the indicators that X is equal to zero.

So finally will conclude that the moving average process is

a weakly stationary process and

the function gamma auto covariance function can be written in closed form.

Our last example will be the auto regressive model.

It's denoted as AR of p.

This model assumes there's the process Yt is defined as follows.

So this is B1 Y t minus one and so on,

Bp Y t minus p plus some white

noise epsilon t. Normally,

it is also assumed that the covariance between epsilon t and Ys is qual

to zero for all t largrer than s. So this is also

a very popular model in econometrics and this

also would be very natural to ask whether this process is stationary or not.

Mathematical expectation of Yt and

other objects are not very clear from this representation,

and actually it turns out that this equation has various solutions.

And let me consider more precisely the case when p is equal to one.

That this same model is Yt minus some B multiplied by

Y t minus one is equal to epsilon t. Here b can be any real number.

It turns out that even this equation has

various solutions and one of the solutions is the

following: so Yt is equal to the sum j from zero to infinity.

B is a power J multiply it by

epsilon t minus j.

So this is a close form for one of the solutions.

It's very simple to show that these Y t is exactly solution of

this equation and it turns out that

this solution is stationary under some very simple assumption.

To show this assumption,

let's mentions that methodical expectation of Yt is

definitely equal to zero and this was a covariance function.

We can use exactly the same ideas as in our previous example when we considered

the moving average process and what we have here is double sum j k

from 0 to infinity B to the power J plus K multiplied by sigma squared and

multiplied by the indicator that t minus s is equal

to g minus K and here,

if you employ the same ideas as in the previous example,

if you will take t minus s equal to some specific number.

For instance, if we take t minus s equal to zero.

That is, t is equal to s,

then j should be equal to K,

and this sum would be

the sum of b to k. K from 0 to infinity.

You know there this sum converges if and only if absolute value of b is smaller than one.

And if you will think about the situation you will

immediately realize that only in this case,

the corresponding infinite sums converge.

This is exactly is the answer.

The absolute value of b is smaller than one,

then definitely this process Yt is weakly stationary.

In more general case when you have arbitrary p,

you should consider some equation related to

this coefficients and you should consider all complex roots of this equation,

and if all of them lie inside the unit circle,

then this process is stationary.

I think the details can be found in any course

on econometrics and I would like to skip them now.

This was an introduction to the concept of stationarity.

I hope very much that my examples helped you to understand

this topic and it was found that even if a process Xt is weakly stationary,

this property helps a lot to understand some other more complicated properties of Xt.

I will discuss many further properties of stochastic processes later,

but it will be very helpful for you to understand what does

the notion of stochastic integration mean and in the next subsection,

I will give you an intuition about this notion at least in simple situations.

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