stochastic process - supplementary

8/20/2019 Stochastic Process - SUPPLEMENTARY

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George Alogoskoufis, Dynamic Macroeconomic Theory, 2015

Mathematical Annex 4

Stochastic Processes

In this mathematical annex we introduce linear stochastic processes in discrete time. Stochastic

processes are extremely useful tools for studying dynamic stochastic economic problems, such as

aggregate fluctuations, which combine time and randomness. As Lucas (1977) had observed,

“Movements about trend in gross national product in any country can be well described by a

stochastically disturbed difference equation of very low order.”.

“Stochastically disturbed difference equations” are nothing more than difference equations that are

affected by random shocks, and are otherwise called stochastic difference equations.

To analyze them, we need an introduction to stochastic processes.

We also introduce the concept of impulse response functions (IRF) which refers to the response of a

dynamic system to exogenous shocks.

A4.1 Stochastic Processes

A stochastic process yt is defined as a collection of random variables that depend on the time index.

A random variable corresponds to each moment of time t ! T .

We shall assume that the probability law that corresponds to each stochastic process is characterized

by the set of mathematical expectations (means) of the stochastic process yt , and by the set of

covariances of y in different time periods.

The mean of the stochastic process y is given by,

E y = µ , t ! T (A4.1)

where E is the mathematical expectations operator.

The covariances are given by,

! [ (y - µ )( y - µ ) ] = " (A4.2)

A stochastic process is considered stationary, when the mean µt is independent of t, and the

covariance " t,s only depends on t-s.

t

t

t

t

t

s

s

t ,s



George Alogoskoufis, Dynamic Macroeconomic Theory, 2015 Mathematical Annex 4

The basic stochastic process, which is the building block of all the stochastic processes that we shall

analyze here, is the so called white noise process. In this process, the mean is equal to zero, the

variance is constant, and the covariance is equal to zero for t " s.

Thus, the white noise process satisfies the following properties.

! ( # ) = 0 # t

E ( # ) = " # t (A4.3)

! ( # # ) = 0 # s " t

As one can see, in the white noise process there is no time dependence, as the covariances are equal

to zero for different time periods.

A4.2 Autoregressive Stochastic Processes

Another useful category of stochastic processes is the category of autoregressive stochastic

processes (AR).

We shall first introduce the zero mean first order linear autoregressive process (AR(1)), which is

defined by,

y = $ y + # (A4.4)

where # is a “white noise” stochastic process.

As one can see from (A4.4), an AR(1) process is a first order difference equation, plus a “white

noise” process.

From (A4.4) it follows that,

y = # = (A4.5)

We can deduce from (A4.5) that this stochastic process has the white noise process # as its building

block. It follows that,

! ( y ) = E( # ) = 0, # t

E ( y ) = " # t (A4.6)

E (y y ) = " # t, s

t

t

2

!

2

t

s

t

t !1

t

t

t

1

1! " L

t

! i

i=0

"

# $ t % i

t

t

1

1! "

t

t

2

1

1! " 2

!

2

t

t !s

! |s|

1" ! 2

!

2

2




From (A4.6) the AR(1) process is stationary if | $| < 1.

If $ = 1, this AR(1) process is non stationary, and takes the form,

y = y + # (A4.7)

This non stationary stochastic process is called a random walk. The first difference of a random

walk is a white noise process, as from(A4.7) it follows that,

% y = y - y = # (A4.8)

% is the first difference operator. The random walk is a special case of homogeneous non stationary

stochastic processes, which become stationary when we transform them, by taking their first

differences one or more times.

The second order linear autoregressive process (AR(2)) has the form of a second order linear

difference equation plus a white noise stochastic process, and can be depicted as,

(A4.9)

where #t is a white noise process, and a,b,c are constant parameters.

(A4.9) can be rewritten using the lag operator, as,

(A4.10)

(A4.10) can be transformed into,

(A4.11)

where, and .

The conditions for (A4.11) to be stationary are analogous to the conditions for convergence in a

simple second order difference equation.

We shall distinguish among three possible cases.

Case 1: b2>-4c

Then, the roots are real and distinct , and take the form,

,

We shall have stationarity if | $1| < 1, and | $2| < 1.

t

t !1

t

t

t

t !1

t

yt = a + byt !1 + cyt !2 + " t

(1!bL

!cL2 ) yt = a + !

t

(1! ! 1 L)(1! !

2 L) yt = a + !

t

! 1 + !

2 = b !

1! 2 = "c

! 1 =

b + b2+ 4c

2! 2 =

b ! b2+ 4c

2

3




The solution will take the form,

(A4.12)

Case 2: b2

=-4c

In this case, we shall have two equal real roots of the form,

We shall have stationarity if | $| <1 which requires |b| < 2.

Case 3: b2<-4c

In this case we shall have two complex roots, which will take the form of a pair of complex

conjugates. We shall have stationarity if |c| < 1 which also implies in this case that |b| < 2.

A4.3 Moving Average Stochastic Processes

Another category of stochastic processes is the category of moving average processes ( MA). The

first order moving average stochastic process (MA(1)) is defined by,

y = # - & # = (1 - & L) # (A4.13)

From (A4.13) it follows that,

! ( y ) = ( 1-& ) E( # ) = 0, # t

E ( y ) = ( 1+& ) " # t (A4.14)

E (y y ) = -& " # t, | s| = 1

= 0 # t, | s| > 1

From (A4.14) it follows that this stochastic process is stationary.

A4.4 Autoregressive-Moving Average Stochastic Processes

Autoregressive and moving average processes can by combined. For example, the combined first

order autoregressive moving average process is called ARMA(1,1) and is defined by,

y = $ y + # - & # = # (A4.15)

yt =

a

(1! ! 1)(1! !

2)+

" t

(1! ! 1 L)(1! !

2 L)

! 1 = !

2 = ! =

b

2

t

t

t !1

t

t

t

t

2

2

!

2

t

t !s

!

2

t

t !1

t

t !1

1!" L

1! # L

t

4




This ARMA process is stationary if | $| < 1.

For a more general and detailed treatment of linear stochastic processes see Chapter 11, in Sargent

(1987).

References

Sargent T.J. (1987), Macroeconomic Theory, (2nd Edition), New York, Academic Press.

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