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## Dates for term tests

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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))**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**Autoregressive Time series of order p, 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.**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|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.**Mean value, variance, autocovariance function,**autocorrelation function of anARMA(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.**Assume that the ARMA(p,q) time series{xt|t T} is**stationary: Let m = E(xt). Then or**The Autocovariance function, s(h), of a stationary mixed**autoregressive-moving average time series {xt|t T} be determined by the equation: Thus**The autocovariance function s(h) satisfies:**For h = 0, 1. … , q: for h > q:**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.**Example:The autocovariance function, s(h), for an ARMA(1,1)**time series: For h = 0, 1: or for h > 1:**Substituting s(0) into the second equation we get:**or Substituting s(1) into the first equation we get:**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**A useful tool in time series analysis**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.**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: The partial auto correlation function at lag k**is defined to be: Using Cramer’s Rule**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**A General Recursive Formula for Autoregressive Parameters**and the Partial Autocorrelation function (PACF)**Let**denote the autoregressive parameters of order k satisfying the Yule Walker equations: