Dates for term tests. Friday, February 07 Friday, March 07 Friday, March 28. 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 • Friday, February 07 • Friday, March 07 • Friday, March 28

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))

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))

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.

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.

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.

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 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.

Consider the time series {xt : 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: • 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 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: • 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 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 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 xt= b1xt-1 +b2xt-2 +... +bpxt-p+ d +ut + a1ut-1 +a2ut-2 +... +aqut-q can be written b(B)xt= a(B)ut + d where a(B)= a0I + a1B +a2B2 +... +aqBq and b(B)= I - b1B - b2B2 -... - bpBp

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 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: 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.

Comment: The partial auto correlation function, Fkk is determined from the auto correlation function, r(h) The partial auto correlation function at lag k, Fkk is the last auto-regressive parameter, . if the series was assumed to be an AR(k) series. If the series is an AR(p) series then An AR(p) series is also an AR(k) series with k > p with the auto regressive parameters zero after p.

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