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### Conditions for stationarity

### Mean value, variance, autocovariance function, autocorrelation function of anARMA(p,q) series

### The Backshift Operator autocorrelation function of anB

### The autocorrelation function of anpartial autocorrelation function

### A General Recursive Formula for Autoregressive Parameters and the Partial Autocorrelation function (PACF)

### Some Examples and the Partial Autocorrelation function (PACF)

Dates for term tests

- Friday, February 5
- Friday, March 5
- Friday, March 26

Let {xt|t T} be defined by the equation.

The Moving Average Time series of order q, MA(q)

where {ut|t T} denote a white noise time series with variance s2.

Then {xt|t T} is called a Moving Average time series of order q. (denoted by MA(q))

The mean value for an MA(q) time series

The autocovariance function for an MA(q) time series

The autocorrelation function for an MA(q) time series

Let {xt|t T} be defined by the equation.

The Autoregressive Time series of order p, AR(p)

where {ut|t T} is a white noise time series with variance s2.

Then {xt|t T} is called a Autoregressive time series of order p. (denoted by AR(p))

The mean value of a stationary AR(p) series

The Autocovariance function s(h) of a stationary AR(p) series

Satisfies the equations:

The Autocorrelation function r(h) of a stationary AR(p) series

Satisfies the equations:

with

for h > p

and

where r1, r2, … , rp are the roots of the polynomial

and c1, c2, … , cp are determined by using the starting values of the sequence r(h).

Autoregressive Time series of order p, AR(p)

If b1 = 1 and d = 0.

The value of xt increases in magnitude and ut eventually becomes negligible.

The time series {xt|t T} satisfies the equation:

The time series {xt|t T} exhibits deterministic behaviour.

For a AR(p) time series, consider the polynomial

with roots r1, r2 , … , rp

then {xt|t T} is stationary if |ri| > 1 for all i.

If |ri| < 1 for at least one i then {xt|t T} exhibits deterministic behaviour.

If |ri| ≥ 1 and |ri| = 1 for at least one i then {xt|t T} exhibits non-stationary random behaviour.

and |r1 |>1, |r2 |>1, … , | rp |> 1 for a stationary AR(p) series then

i.e. the autocorrelation function, r(h), of a stationary AR(p) series “tails off” to zero.

Let {xt|t T} be defined by the equation.

with root r1= 1/b1

- {xt|t T} is stationary if |r1| > 1 or |b1| < 1 .

- If |ri| < 1 or |b1| > 1 then {xt|t T} exhibits deterministic behaviour.

- If |ri| = 1 or |b1| = 1 then {xt|t T} exhibits non-stationary random behaviour.

Let {xt|t T} be defined by the equation.

where r1 and r2 are the roots of b(x)

- {xt|t T} is stationary if |r1| > 1 and |r2| > 1 .

This is true if b1+b2 < 1 , b2 –b1 < 1 and b2 > -1.

These inequalities define a triangular region for b1 and b2.

- If |ri| < 1 or |b1| > 1 then {xt|t T} exhibits deterministic behaviour.

- If |ri| ≥ 1 for i = 1,2 and |ri| = 1 for at least on i then {xt|t T} exhibits non-stationary random behaviour.

Patterns of the ACF and PACF of AR(2) Time Series

In the shaded region the roots of the AR operator are complex

b2

Let b1, b2, … bp , a1, a2, … ap , d denote p + q +1 numbers (parameters).

The MixedAutoregressive Moving Average Time Series of order p, ARMA(p,q)

- Let {ut|tT} denote a white noise time series with variance s2.
- independent
- mean 0, variance s2.

Let {xt|t T} be defined by the equation.

Then {xt|t T} is called a Mixed Autoregressive- Moving Average time series - ARMA(p,q) series.

Similar to an autocorrelation function of anAR(p) time series, for certain values of the parameters b1, …, bp an ARMA(p,q) time series may not be stationary.

An ARMA(p,q) time series is stationary if the roots (r1, r2, … , rp ) of the polynomial

b(x) = 1 – b1x – b2x2 - … - bpxp

satisfy | ri| > 1 for all i.

Assume that the ARMA(p,q) time series autocorrelation function of an{xt|t T} is stationary:

Let m = E(xt). Then

or

The autocorrelation function of anAutocovariance function, s(h), of a stationary mixed autoregressive-moving average time series{xt|t T} be determined by the equation:

Thus

Hence autocorrelation function of an

We need to calculate: autocorrelation function of an

etc

The autocovariance function autocorrelation function of ans(h) satisfies:

For h = 0, 1. … , q:

for h > q:

We then use the first ( autocorrelation function of anp + 1) equations to determine:

s(0), s(1), s(2), … , s(p)

We use the subsequent equations to determine:

s(h) for h > p.

Example: autocorrelation function of anThe autocovariance function, s(h), for an ARMA(1,1) time series:

For h = 0, 1:

or

for h > 1:

Substituting autocorrelation function of ans(0) into the second equation we get:

or

Substituting s(1) into the first equation we get:

for autocorrelation function of anh > 1:

Consider the time series { autocorrelation function of anxt : tT} and Let Mdenote the linear space spanned by the set of random variables {xt : tT}

(i.e. all linear combinations of elements of {xt : tT} and their limits in mean square).

Mis a vector space

Let B be an operator on M defined by:

Bxt = xt-1.

B is called the backshift operator.

Note: autocorrelation function of an

- We can also define the operator Bk with
Bkxt = B(B(...Bxt)) = xt-k.

- The polynomial operator
p(B) = c0I + c1B + c2B2 + ... + ckBk

can also be defined by the equation.

p(B)xt = (c0I + c1B + c2B2 + ... + ckBk)xt .

= c0Ixt + c1Bxt + c2B2xt + ... + ckBkxt

= c0xt + c1xt-1 + c2xt-2 + ... + ckxt-k

- The power series operator autocorrelation function of an
p(B) = c0I + c1B + c2B2 + ...

can also be defined by the equation.

p(B)xt= (c0I + c1B + c2B2 + ... )xt

= c0Ixt + c1Bxt + c2B2xt + ...

= c0xt + c1xt-1 + c2xt-2 + ...

- If p(B) = c0I + c1B + c2B2 + ... and q(B) = b0I + b1B + b2B2 + ... are such that
p(B)q(B) = I

i.e. p(B)q(B)xt = Ixt = xt

than q(B) is denoted by [p(B)]-1.

Other operators closely related to B: autocorrelation function of an

- F = B-1 ,the forward shift operator, defined by Fxt = B-1xt = xt+1and
- D = I - B ,the first difference operator, defined by Dxt = (I - B)xt = xt - xt-1 .

The Equation for a MA(q) time series autocorrelation function of an

xt= a0ut + a1ut-1 +a2ut-2 +... +aqut-q+ m

can be written

xt= a(B)ut + m

where

a(B)= a0I + a1B +a2B2 +... +aqBq

The Equation for a AR(p) time series autocorrelation function of an

xt= b1xt-1 +b2xt-2 +... +bpxt-p+ d +ut

can be written

b(B)xt= d + ut

where

b(B)= I - b1B - b2B2 -... - bpBp

The Equation for a ARMA(p,q) time series autocorrelation function of an

xt= b1xt-1 +b2xt-2 +... +bpxt-p+ d +ut + a1ut-1 +a2ut-2 +... +aqut-q

can be written

b(B)xt= a(B)ut + m

where

a(B)= a0I + a1B +a2B2 +... +aqBq

and

b(B)= I - b1B - b2B2 -... - bpBp

Some comments about the Backshift operator autocorrelation function of anB

- It is a useful notational device, allowing us to write the equations for MA(q), AR(p) and ARMA(p, q) in a very compact form;
- It is also useful for making certain computations related to the time series described above;

A useful tool in time series analysis

The partial autocorrelation function autocorrelation function of an

Recall that the autocorrelation function of an AR(p) process satisfies the equation:

rx(h) = b1rx(h-1) + b2rx(h-2) + ... +bprx(h-p)

For 1 ≤ h ≤ p these equations (Yule-Walker) become: rx(1) = b1 + b2rx(1) + ... +bprx(p-1)

rx(2) = b1rx(1) + b2 + ... +bprx(p-2)

...

rx(p) = b1rx(p-1)+ b2rx(p-2) + ... +bp.

In matrix notation: autocorrelation function of an

These equations can be used to find b1, b2, … , bp, if the time series is known to be AR(p) and the autocorrelation rx(h)function is known.

If the time series is not autoregressive the equations can still be used to solve for b1, b2, … , bp, for any value of p 1.

In this case

are the values that minimizes the mean square error:

Definition: still be used to solve for The partial auto correlation function at lag k is defined to be:

Comment: still be used to solve for

The partial auto correlation function, Fkk is determined from the auto correlation function, r(h)

- Some more comments: still be used to solve for
- The partial autocorrelation function at lag k, Fkk, can be interpreted as a corrected autocorrelation between xt and xt-k conditioning on the intervening variables xt-1, xt-2, ... ,xt-k+1 .
- If the time series is an AR(p) time series than
- Fkk = 0 for k > p
- If the time series is an MA(q) time series than
- rx(h) = 0 for h > q

Let and the Partial Autocorrelation function (PACF)

denote the autoregressive parameters of order k satisfying the Yule Walker equations:

Then it can be shown that: and the Partial Autocorrelation function (PACF)

and

The Yule Walker equations: and the Partial Autocorrelation function (PACF)

Proof:

In matrix form: and the Partial Autocorrelation function (PACF)

The equations for and the Partial Autocorrelation function (PACF)

and and the Partial Autocorrelation function (PACF)

The matrix A reverses order

The equations may be written and the Partial Autocorrelation function (PACF)

Multiplying the first equations by

or

Example 1: and the Partial Autocorrelation function (PACF)MA(1) time series

Suppose that {xt|t T} satisfies the following equation:

xt = 12.0 + ut + 0.5 ut – 1

where {ut|t T} is white noise with s = 1.1.

Find:

- The mean of the series,
- The variance of the series,
- The autocorrelation function.
- The partial autocorrelation function.

Solution and the Partial Autocorrelation function (PACF)

Now {xt|t T} satisfies the following equation:

xt = 12.0 + ut + 0.5 ut – 1

Thus:

- The mean of the series,
m= 12.0

The autocovariance function for an MA(1) is

Thus: and the Partial Autocorrelation function (PACF)

- The variance of the series,
s(0)= 1.5125

and

- The autocorrelation function is:

- The and the Partial Autocorrelation function (PACF)partial auto correlation function at lag k is defined to be:

Thus

Graph: Partial Autocorrelation function and the Partial Autocorrelation function (PACF)Fkk

Exercise: and the Partial Autocorrelation function (PACF)Use the recursive method to calculate Fkk

and

Exercise: and the Partial Autocorrelation function (PACF)Use the recursive method to calculate Fkk

and

Example 2: and the Partial Autocorrelation function (PACF)AR(2) time series

Suppose that {xt|t T} satisfies the following equation:

xt = 0.4 xt – 1+ 0.1 xt – 2+ 1.2 + ut

where {ut|t T} is white noise with s = 2.1.

Is the time series stationary?

Find:

- The mean of the series,
- The variance of the series,
- The autocorrelation function.
- The partial autocorrelation function.

- The mean of the series and the Partial Autocorrelation function (PACF)

- The autocorrelation function.
- Satisfies the Yule Walker equations

hence and the Partial Autocorrelation function (PACF)

- the variance of the series and the Partial Autocorrelation function (PACF)

- The partial autocorrelation function.

The partial autocorrelation function of an AR( and the Partial Autocorrelation function (PACF)p) time series “cuts off” after p.

Example 3: and the Partial Autocorrelation function (PACF)ARMA(1, 2) time series

Suppose that {xt|t T} satisfies the following equation:

xt = 0.4 xt – 1+ 3.2 + ut + 0.3 ut – 1 + 0.2 ut – 1

where {ut|t T} is white noise with s = 1.6.

Is the time series stationary?

Find:

- The mean of the series,
- The variance of the series,
- The autocorrelation function.
- The partial autocorrelation function.

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