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Information Engineering Assignment Help(( )) = µ, independent of (1, 2) is only a function of

Question No 1.

Give the definition of stationarity in strong and weak sense. Are the following processes stationary? Why? White noise (give the definition), Random walk (give the definition), MA(n) process (give the definition).

Answer:

Stationarity refers to time invariance of some, or all the statistics of a random process. The process could be mean, auto correlation, nth order distribution etc. A random process ( ) is said to be SSS (stationary random process), if all its finite order distributions are time invariant.

It simple means that the joint cfd of (1), (2), … … , ( ) is the same as for (1+∝), (2+∝), … . ( +∝), for all k, all t1, t2,…, tk and all time shifts .

So for a SSS process, the first order distribution is independent of t, and the second order distribution i.e. the distribution of any two samples

(1) (2) is the same as the joint distribution of (1 + ( −1)) (2 + ( −2)) = (1 + (2 1)).

Strong Sense Stationarity:

In mathematics and statistics, a stationary process or strictly stationary or strongly stationary is a process whose joint probability distribution does not change when shifted in time. It can be said that a stationary time series is one whose statistical properties such as mean, variance, auto correlation, etc. are all constant over time. Stationarity is used as a tool in time series analysis, where the raw data is often transformed to become stationary; for example, economic data are often seasonal and/or dependent on a non-stationary price level. An important type of non-stationary process that does not include a trend-likebehavior is the cyclostationary process.

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Mathematically, it can be stated as:

Formally, let

be a stochastic process and let

represent

the cumulative distribution function of the joint distribution of

at

times

. Then,

is said to be stationary if, for all

, for all , and for

all

,

Since does not affect , is not a function of time.

Wide Sense Stationarity:

A process is said to be Wide sense stationary, if its mean and autocorrelation functions are time invariant i.e.

( 2 1)

Since ( 1, 2) = ( 2, 1), if ( ) is WSS, ( 1, 2) is only a function of | 2 1| Clearly SSS , the converse however is not necessarily true.

For GRP, WSS , since the process is completely specified by its mean and autocorrelation functions.

Random walk is not WSS, since ( 2, 1) = ( 1, 2) is not time invariant – in fact no independent increment process can be WSS.

Is White a Stationary Series?

The time series is said to be a white noise with mean zero and variance 2, written as

~ (0, 2)

if and only if has zero mean and covariance function as

(ℎ) = { 2 ℎ = 0 0 ℎ ≠ 0

It is clear that a white noise process is stationary. Note that white noise assumption is weaker than identically independent distributed assumption.

To tell if a process is covariance stationary, we compute the unconditional first two moments, therefore, processes with conditional heteroskedasticity may still be stationary.

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Is Random Walk a Stationary Series?

Let be a random walk = ∑=0 with 0 = 0 and Xt is independent and identically distributed with mean zero and variance 2. Then for ℎ > 0,

( , + ℎ) = ( , +ℎ)

+ℎ

= (∑ ,

∑ )

=1

=1

= (∑ ) ( , ) = 0 ≠

=1

= 2

In this case, the autocovariance function depends on time t, therefore the random walk process St is not stationary.

Moving Average Process:

An MA process models [ | −1] with lagged error terms. An ( ) model involves lags. We keep the white noise assumption for .

Example: A linear MA(q) model:

= + − 1 −1 2 −2 − ⋯ − = − ∑ +

=1

= − ∑ + = + ( ) ,

=1

( ) = 1 − 1 2 2 − ⋯ −

stationary? Check the moments. WLOG, assume µ=0.

( ) = 0

( ) = (1

+ 2

+ 2

+ ⋯ + 2) 2

1

2

( − ) = [ , ] = [(−1 −1 −2 −2 − − ⋯ − − − + )]

= 2 [∑

]

| | ≤ ; ℎ ( − ) = 0

+| |

=1

It is easy to verify that the sums are finite => ( ) is stationary. Note that an ( )process can generate an AR process.

= + ( )

=> ( )−1

= ∗ +

Question No. 2

How to test stationarity of an empirically given time series? (you can use textbooks and lectures with references).

Answer:

Question No. 3

Give the definition of the autocorrelation function of a stationary time series.

Answer:

Autocorelation Function of Stationary Time Series:

The time series { , ∈ } (where Z is the integer set) is said to be stationary if

(I) (2) < ∞ ∀ ∈ .

(II)= µ ∀ t ∈ Z.

(III) ( , ) = ( + ℎ, + ℎ) ∀ , , ℎ ∈ .

In other words, a stationary time series { } must have three features: finite variation, constant first moment, and that the second moment ( , ) only depends on ( − ) and not depends on s or t. In light of the last point, we can rewrite the autocovariance function of a stationary process as

(ℎ) = ( , +ℎ) , ℎ ∈ .

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Also, when is stationary, we must have

(ℎ) = (−ℎ).

When ℎ = 0, (0) = ( , ) is the variance of , so the autocorrelation function for a stationary time series { } is defined to be

(ℎ) =

(ℎ)

(0)

Question No. 4

Derive a formula for the autocorrelation function of a MA(n) process (you can use textbooks and lectures with references).

Answer:

Start with { } being white noise or purely random, mean zero, s.d. . is a moving average process of order q (written ( )) if for some constants 0, 1, … we have

= 0 + 1 −1 + ⋯

Usually 0 = 0

The mean and variance are given by:

( ) = 0, ( ) = 2 2

�=0

The process is weakly stationary because the mean is constant, the covariance does not depend on t, and the autocovariance depend only on the time lag.

Note the autocorrelation cuts off at lag q. For the MA(1) process with 0 = 1

1 = 0= {1/(1 + 12) = ±1

0 ℎ

If the s are normal then so is the process, and it is then strictly stationary. The autocorrelation is:

1 = 0

=0

+

= 1, … ,

=

2

=0

0 >

{< 0

Note the autocorrelation cuts off at lag q. For the (1) process with 0 = 1

1 = 0= {1/(1 + 12) = ±1

0 ℎ

Question No. 5

Derive autocorrelation function of the following MA(2) process:

X(t)=Z(t)+0.5Z(t-1)+0.25Z(t-2)?

Solution:

The coefficients are θ1= 0.5 and θ2= 0.25.

Because this is an MA(2), the theoretical ACF will have nonzero values only at lags 1 and 2. Autocorrelation function (ACF) is given by:

+

=

+

+

=

+

+

Values of the two nonzero autocorrelation functions are:

. + ( . ∗ . )

.

=

=

= .

+ . + .

.

.

.

=

=

= .

+ . + .

.

Question No 6

Under which condition a general AR(n) process is stationary?

Answer:

Assume ( ) is white noise with mean zero and standard deviation

Then the autoregressive process of order or ( ) process is :

X(t) = 1X(t − 1) + 2X(t − 2) … … . + X(t − n) + Z(t)

ℎ ( ) ∶

(1 − 1 2 2−. . . ) ( ) = ( )

Then the necessary and sufficient condition for stationarity is that the roots of the equation

∅( ) = ( − − −.. . ) =

Must lie outside the unit circle.

Question No 7

For which a the following processes are stationary: X(t)=aX(t-1)+2aX(t-2)+Z(t);X(t)=2aX(t-1)+aX(t- 2)+Z(t)?

Answer:

. (i)

X(t) = X(t − 1) + 2 X(t − 2) + Z(t)

Applying condition for stationarity as stated in question 6

=

=

So the equation becomes :

1 − − 22 = 0

Now for roots of the equation applying Sridhar Acharya formula :

=

− ±√

2−4

2

ℎ = , = −2 , = − , = 1

putting values and solving equation:

=

±√(− )2−4∗(−2 )∗1

2∗(−2 )

=

±

2 + 8

−4

So now for stationarity :

| | > 1

±√

|

2+8

| > 1

−4

±√

2+8

> 1 &

±√

2+8

< −1

−4

−4

Cross multiplying and finding range of values of a:

2 + 8 < 252 & 2 + 8 > 92

8 (3 − 1) > 0 & 8 ( − 1) < 0

∈ (−∞, 0) ∪ (

1

, ∞) & ∈ (0,1)

3

Combining both we get

∈ (−∞, 0) ∪ (0,∞)

(ii)

X(t) = 2 X(t − 1) + X(t − 2) + Z(t)

Applying condition for stationarity as stated in question 6

=

=

So the equation becomes :

1 − 2 − 2 = 0

Now for roots of the equation applying Sridhar Acharya formula :

=

− ±√

2−4