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Dates for term tests. Friday, February 5 Friday, March 5 Friday, March 26. Let { x t | t  T } be defined by the equation. The Moving Average Time series of order q, MA(q). where { u t | t  T } denote a white noise time series with variance s 2.

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dates for term tests
Dates for term tests
  • Friday, February 5
  • Friday, March 5
  • Friday, March 26
slide2

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))

slide3

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

slide4

Comment

The autocorrelation function for an MA(q) time series

“cuts off” to zero after lag q.

q

slide5

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))

slide6

The mean value of a stationary AR(p) series

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

Satisfies the equations:

slide7

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

Satisfies the equations:

with

for h > p

and

slide8

or:

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).

conditions for stationarity

Conditions for stationarity

Autoregressive Time series of order p, AR(p)

slide10

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.

slide11

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.

slide12

since:

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.

slide13

Special Cases: The AR(1) time

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

slide14

Consider the polynomial

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.
slide15

Special Cases: The AR(2) time

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

slide16

Consider the polynomial

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.
slide17

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

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

b2

slide18
The MixedAutoregressive Moving Average

Time Series of order p,q

The ARMA(p,q) series

slide19

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|tT} 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.

mean value variance autocovariance function autocorrelation function of an arma p q series

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

slide21
Similar to an AR(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.

slide23

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

Thus

slide27

The autocovariance function s(h) satisfies:

For h = 0, 1. … , q:

for h > q:

slide28

We then use the first (p + 1) equations to determine:

s(0), s(1), s(2), … , s(p)

We use the subsequent equations to determine:

s(h) for h > p.

slide30

Substituting s(0) into the second equation we get:

or

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

slide33
Consider the time series {xt : tT} and Let Mdenote the linear space spanned by the set of random variables {xt : tT}

(i.e. all linear combinations of elements of {xt : tT} 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.

slide34
Note:
  • 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

slide35
The power series operator

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
Other operators closely related to B:
  • 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
The Equation for a MA(q) time series

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
The Equation for a AR(p) time series

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
The Equation for a ARMA(p,q) time series

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 b
Some comments about the Backshift operator B
  • 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;
the partial autocorrelation function

The partial autocorrelation function

A useful tool in time series analysis

the partial autocorrelation function1
The partial autocorrelation function

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.

slide43
In matrix notation:

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.

slide44

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:

slide46

Comment:

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

slide47

Some more comments:

  • 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
slide48

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

slide49
Let

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

slide54

and

The matrix A reverses order

slide55

The equations may be written

Multiplying the first equations by

or

slide57
Hence

and

or

example 1 ma 1 time series
Example 1: 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
Solution

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

slide61
Thus:
  • The variance of the series,

s(0)= 1.5125

and

  • The autocorrelation function is:
example 2 ar 2 time series
Example 2: 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.
slide68
The mean of the series
  • The autocorrelation function.
  • Satisfies the Yule Walker equations
slide70
the variance of the series
  • The partial autocorrelation function.
example 3 arma 1 2 time series
Example 3: 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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