stochastic processes - nkd groupstochastic processes edited: february 2011 page 1 professor: nina...

Stochastic Processes Edited: February 2011 Professor: Nina Kajiji

Stochastic Processes

Stochastic Process – Non Formal Definition:

Non‐formal: A stochastic process (random process) is the opposite of a deterministic

process such as one defined by a differential equation. A stochastic process deals with

more than one possible reality of how a process might evolve. This means that even if

the starting point (initial condition) is known there are many paths that a process may

follow – some are more probable than others.

For processes in time, a stochastic process is simply a process that develops in time

according to probabilistic rules.

Stochastic Process – Formal Definition

Stochastic Process (X) is a family of random variables, dependent upon a parameter

which usually denotes time (T) and defined on some sample space (Ω).

Mathematically,

{Xt, t є T} = { Xt(ω), t є T, ω є Ω}

Of course the parameter does not have to always denote time. It could be a vector

representing location in space. In such a case the process will represent a random

variable that varies across two‐dimensional space. In our case we will not delve into

this level of details.


Stochastic Process – Discrete Time

When T is a set of integers, representing specific time points we have a stochastic

process in discrete time. In this case we generally define the random variable as Xn.

The random variable Xn will depend on earlier values of the process,

That is: Xn‐1, Xn‐2, …


Stochastic Process – Continuous Time

When T is the real line (or some interval of the real line) we have a stochastic process in

continuous time. We will focus on this definition for our study. The random variable

X(t) will depend on values of X(u) for u < t.


Examples of Stochastic Processes

Random Walk Models – such as exchange rate data. In a random walk model

changes in the “rate” are independent normal random variables with zero mean

and standard deviation of the actual data. In essence, the upward and

downward movement in the “rate” is equally likely and there is no scope for

profiteering by speculation except by luck.

Aside:

J.P. Morgan’s famous stock market prediction was that, “Prices will fluctuate.”

Bachelier’s Theory of Speculation in 1900 postulated that prices fluctuate

randomly.

These models make sense in a world where:

1. Most price changes result from temporary imbalances between buyers and

sellers.

2. Stronger price shocks are unpredictable.

3. Under efficient capital market hypothesis the current price of a stock reflects

all information about it.


These models do not make sense if:

1. One believes in technical analysis

2. Random walk models assume that returns are normally or log‐normally

distributed. Thus, frequency of extreme events is underestimated.

NOTE: These models will be the focus of our study.

Poisson Processes – such as photon emissions. A photon is a minute particle of

light measured by a special machine. The problem – the machine has a dead

time period. That is, after it counts the photon it has to recharge before it can

count the next photon. Therefore, the counts are underestimated. A stochastic

model would need to be formulated that not only deals with the emissions but

also the dead time based on some probability of occurrence. A homogenous

poisson process satisfies:

o Starts at zero

o It is stationary, and has independent increments

o For every t > 0, X(t) has a poisson distribution

NOTE: Because poisson processes are counting processes they inherently have a

jump.

Epidemics – such as SARS. The disease is detected, spread, and eventually

controlled to eradicate it. The question arises, how soon will it spread. How

widespread will it be. How many should you vaccinate – because some will die

anyway so the ultimate goal of eradication is reached. The graphs generally have

a peak when the epidemic is at its peak.


Point Processes – such as Earthquakes. Empirically we say that on average

there is one earthquake per year. However, in actuality there are years when

there have been no recorded earthquakes. Thus, there is a clustering effect in

the processes. The question would be, “is it related to magnitude?” “is it strictly

by chance?”, etc.

Reproduction Processes – Yeast cells. Mother yeast cells produces daughter

yeast cells after a random time. Each daughter cell has to evolve to a mother

stage before it can reproduce. But that evolution also takes a random time.

Each cell can only reproduce a fixed number of times before it dies. Question,

how many cells will there be give some fixed time.

Evolutionary Processes – Interest to evolutionary biologists. When they study

gene behavior of plants. These processes all exhibit the Markov property.


Definition of Terms Used in Stochastic Processes

Recall:

Financial Time Series for this course is a collection of financial measurements over

time. Example the log returns over time can be stated as:

{ rt} = {r1, r2, … , rT} for T observations

TimeInvariant

A system in which all quantities that make up the systems behavior remain constant

with time. That is, the system’s response to a given input does not depend on the time

it is applied. For example:

System A: · – not time in‐variant

System B: 10 · – time‐invariant

White Noise

White Noise is a simple type of stochastic process (random signal) whose terms are iid

with zero mean. However, the iid requirement is too restrictive in practice. A Gaussian

white noise is a stochastic process having the following characteristics:

Mean = zero.

Variance is finite.

Zero autocorrelations.

FiniteDimensional Disturbances (fidis)

A fidis of the stochastic process X are the distributions of the finite‐dimensional vectors

, … , , t1, … , tn є T for all possible choices of times ti and every n ≥ 1. A

collection of fidis is the distribution of the stochastic process.


Stationarity

A strict‐stationary process is a stochastic process whose joint probability distribution

does not change when shifted in time or space. Thus, Stationarity explores the time‐

invariant behavior of a time series. Determining the stationarity condition of the time

series allows for proper identification and development of forecasting models. Two

types of Stationarities:

Strict Stationarity – distributions are time‐invariant

Weak Stationarity – only the first 2 moments are time‐invariant. That is, the

data values fluctuate with constant variation around a constant level.

How used in Time Series Analysis

Raw data are often transformed to remove the trend effect (de‐trended).

If the time‐plot of { rt} varies around a fixed‐level with a finite range the series is

said to be weakly stationary.

The first 2 moments of future rt are the same as those of the data we infer the

series is weakly stationary.

Most financial time series exhibit weak form stationarity.

Markov Property

Given the present (Xk‐1) the future (Xk) is independent of the past (Xk‐1, Xk‐3, … , X1).

In other words, the “lack of memory” property of a time series.

Counting Process

It is a process X(t) in discrete or continuous time for which the possible values of X(t)

are the natural numbers (0, 1, 2, …) with the property that X(t) is a non‐decreasing

function of t.


A Sample Path

A sample path of a stochastic process is a particular realization of the process. That

is, a particular set of values X(t) for all t, which is generated according to the

(stochastic) rules of the process.

Increments

An increments of a stochastic process are the changes X(t)‐X(s) between time points s

and t (s < t). Processes in which the increments for non‐overlapping time intervals are

independent and stationary are of important. Examples: Random Walks. Levy

processes such as Poisson processes; and Brownian Motion.


Statistical Measures for Linear Time Series

Gaussian Processes

A stochastic process is called Gaussian if all its fidis are multivariate Gaussian. The

distribution of a Gaussian stochastic process is determined only by the collection of the

expectations and covariance matrices of the fidis.

Mean or Expectation

Variance

Mean and Variance of Returns

1

11

To Test H0 : µ = 0 v/s H0 : µ 0 compute,

/ ~ 0,1

Reject H0 of zero mean if |t| > Zα/2

Covariance and Correlation

The covariance of random variables X and Y is defined: Cov(X,Y) = E[(X ‐ µx)(Y‐µy)]. If X

and Y are “in sync” the covariances will be high; If they are independent, the positive

and negative terms should cancel out to give a score around zero.


The lag‐l auto‐covariance Cov(rt, rt‐1) = γl has two interesting properties:

γ0 = Var(rt);

γl = γl‐1

The correlation coefficient of random variables X and Y, is defined:

, ,

,

It measures the strength of linear dependence between X and Y, and lies between ± 1.

Correlation and Causation

NOTE: Correlation does not imply causation.

The meaningfulness of the correlation can be evaluated by considering:

the number of pairs tested

the number of points in each time series

the “sniff” test

statistical tests

Lagk auto covariance

,

Serial (or auto) correlations

,

Existence of serial correlation implies that the return is predictable indicating market

inefficiency.

Aside: Market inefficiency is a condition that occurs when the current prices don’t

reflect the available information regarding securities.


Sample Autocorrelation Function (ACF)

∑

∑

Where: is the sample mean and T is the sample size

Test zero serial correlations (market efficiency)

Single Test

: 0 : 0

1⁄

√ ~ 0,1

Reject H0 if |t| > Zα/2

Joint Test – Ljung Box statistics

: 0 : 0

2 ~

Reject H0 if Q(m) >


What are we looking for when studying autocorrelations?

If stock returns are truly random, we expect all lags to show a correlation of

around zero.

Today’s volatility is a good predictor for tomorrow, so we expect high

autocorrelations for short lags.

Macro variables such as today’s sales are a good predictor for yesterday’s sales,

so we expect high autocorrelations for short lags.

If the analysis period is changed you may be able to study day‐of‐week effects,

or day‐of‐year effects. These will show up as lags of 7, and 365 respectively.


Univariate Time Series

Purpose

1. A model for

2. Understanding models for : properties, forecasting, etc.

Linear Time Series

is linear if

the predictable part is a linear function of Ft‐1, and

are independent and have the same distribution (iid).

That is,

Where:

: is a constant

1 – associated with impulse responses

is an iid sequence with mean zero and a well‐defined distribution. Generally these are shock (or innovations)

Univariate Linear Time Series Models

1. Autoregressive (AR) Models

2. Moving‐Average (MA) Models

3. Mixed ARMA Models

4. Seasonal Models

5. Regression Models with Time Series Errors

6. Fractionally differenced Models (long memory models)


Important Properties of a Model

1. Stationarity Conditions

2. Basic properties: mean, variance, and serial dependence

3. Empirical model building: specification, estimation, and checking

4. Forecasting

Consideration for Empirical Model Building

1. Ethical and Financial considerations

2. Art as much as a Science

3. How much detail – over / under specification

4. NOTE: “All models are wrong” ‐‐‐ that is they only approximate reality they are

not the reality

5. Constant “tweaking” is essential before one that stands the test of time is found


AR Models

Simple AR Models:

Similar to a Simple Linear Regression model with lagged variables.

AR(1) Model

An AR(1) model can be stated as follows:

Where: , : are real numbers, which are referred to as “parameters”. : is assumed to be a white noise series with mean = 0 and a finite variance.

AR(p) Model

An AR(p) model (similar to a multiple regression model with lagged variables) can be

stated as follows:

…

Both models suggest that the past period values jointly determine the conditional

expectation of today’s value.


Properties of AR(1) Model

Stationarity

Assume: weak stationarity – a necessary and sufficient condition is | 1| < 1

Recall: Under stationarity E(rt) = E(rt‐1) = µ. Therefore, taking the expectation of

We have,

And the Mean = μ

Recall: Under stationarity Var(rt) = Var(rt‐1). Therefore, taking the square and

expectation of:

μ μ

We have,

Or, the Variance =


Autocorrelations: ρ0 1, ρ1 1, ρ2 , etc. In general, ρk , and ACF ρk

decays exponentially as k increases. Note: if 1 0, the decay rate 1. If if 1 0

the decay rate


AR(2) Model

An AR(2) model can be stated as follows:

Where: , , : are real numbers, which are referred to as “parameters”. : is assumed to be a white noise series with mean = 0 and a finite variance.

Stationarity of AR(2)

stochastic processes - nkd groupstochastic processes edited: february 2011 page 1 professor: nina...

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