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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


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


Comment

The autocorrelation function for an MA(q) time series

“cuts off” to zero after lag q.

q


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


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)


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.


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.


Special Cases: The AR(1) time

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


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.


Special Cases: The AR(2) time

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


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.


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

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

b2


The MixedAutoregressive Moving Average

Time Series of order p,q

The ARMA(p,q) series


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


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:


The backshift operator b

The Backshift Operator autocorrelation function of anB


Consider the time series { autocorrelation function of anxt : 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.


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
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
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
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
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 b
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;


The partial autocorrelation function

The autocorrelation function of anpartial autocorrelation function

A useful tool in time series analysis


The partial autocorrelation function1
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


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


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


Substituting this into the second equation and the Partial Autocorrelation function (PACF)

or

and


Hence and the Partial Autocorrelation function (PACF)

and

or


Some examples

Some Examples and the Partial Autocorrelation function (PACF)


Example 1 ma 1 time series
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
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 ar 2 time series
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 autocorrelation function.

  • Satisfies the Yule Walker equations


hence 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 arma 1 2 time series
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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