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Dates for term tests 1.Friday, February 07 2.Friday, March 07 3.Friday, March 28

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The Moving Average Time series of order q, MA(q) where {u t |t T} denote a white noise time series with variance 2. Let {x t |t T} be defined by the equation. Then {x t |t T} is called a Moving Average time series of order q. (denoted by MA(q))

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The autocorrelation function for an MA(q) time series The autocovariance function for an MA(q) time series The mean value for an MA(q) time series

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The autocorrelation function for an MA(q) time series Comment “cuts off” to zero after lag q. q

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The Autoregressive Time series of order p, AR(p) where {u t |t T} is a white noise time series with variance 2. Let {x t |t T} be defined by the equation. Then {x t |t T} is called a Autoregressive time series of order p. (denoted by AR(p))

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The mean value of a stationary AR(p) series The Autocovariance function (h) of a stationary AR(p) series Satisfies the equations:

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with for h > p The Autocorrelation function (h) of a stationary AR(p) series Satisfies the equations: and

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or: and c 1, c 2, …, c p are determined by using the starting values of the sequence (h). where r 1, r 2, …, r p are the roots of the polynomial

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Conditions for stationarity Autoregressive Time series of order p, AR(p)

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For a AR(p) time series, consider the polynomial with roots r 1, r 2, …, r p then {x t |t T} is stationary if |r i | > 1 for all i. If |r i | < 1 for at least one i then {x t |t T} exhibits deterministic behaviour. If |r i | ≥ 1 and |r i | = 1 for at least one i then {x t |t T} exhibits non-stationary random behaviour.

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since: i.e. the autocorrelation function, (h), of a stationary AR(p) series “tails off” to zero. and |r 1 |>1, |r 2 |>1, …, | r p | > 1 for a stationary AR(p) series then

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Special Cases: The AR(1) time Let {x t |t T} be defined by the equation.

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Consider the polynomial with root r 1 = 1/ 1 1.{x t |t T} is stationary if |r 1 | > 1 or | 1 | < 1. 2.If |r i | 1 then {x t |t T} exhibits deterministic behaviour. 3.If |r i | = 1 or | 1 | = 1 then {x t |t T} exhibits non- stationary random behaviour.

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Special Cases: The AR(2) time Let {x t |t T} be defined by the equation.

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Consider the polynomial where r 1 and r 2 are the roots of (x) 1.{x t |t T} is stationary if |r 1 | > 1 and |r 2 | > 1. 2.If |r i | 1 then {x t |t T} exhibits deterministic behaviour. 3.If |r i | ≤ 1 for i = 1,2 and |r i | = 1 for at least on i then {x t |t T} exhibits non-stationary random behaviour. This is true if 1 + These inequalities define a triangular region for 1 and 2.

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Patterns of the ACF and PACF of AR(2) Time Series In the shaded region the roots of the AR operator are complex 22

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The Mixed Autoregressive Moving Average Time Series of order p,q The ARMA(p,q) series

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The Mixed Autoregressive Moving Average Time Series of order p, ARMA(p,q) Let 1, 2, … p, 1, 2, … p, denote p + q +1 numbers (parameters). Let {u t |t T} denote a white noise time series with variance 2. –independent –mean 0, variance 2. Let {x t |t T} be defined by the equation. Then {x t |t T} is called a Mixed Autoregressive- Moving Average time series - ARMA(p,q) series.

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

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Similar to an AR(p) time series, for certain values of the parameters 1, …, p an ARMA(p,q) time series may not be stationary. An ARMA(p,q) time series is stationary if the roots (r 1, r 2, …, r p ) of the polynomial (x) = 1 – 1 x – 2 x 2 - … - p x p satisfy | r i | > 1 for all i.

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Assume that the ARMA(p,q) time series {x t |t T} is stationary: Let = E(x t ). Then or

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

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Hence

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We need to calculate:

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h ux (h)

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The autocovariance function (h) satisfies: For h = 0, 1. …, q: for h > q:

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We then use the first (p + 1) equations to determine: (0), (1), (2), …, (p) We use the subsequent equations to determine: (h) for h > p.

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Example:The autocovariance function, (h), for an ARMA(1,1) time series: For h = 0, 1: for h > 1: or

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Substituting (0) into the second equation we get: or Substituting (1) into the first equation we get:

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for h > 1:

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The Backshift Operator B

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Consider the time series {x t : t T} and Let M denote the linear space spanned by the set of random variables {x t : t T} (i.e. all linear combinations of elements of {x t : t T} and their limits in mean square). M is a vector space Let B be an operator on M defined by: Bx t = x t-1. B is called the backshift operator.

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Note: 1. 2.We can also define the operator B k with B k x t = B(B(...Bx t )) = x t-k. 3.The polynomial operator p(B) = c 0 I + c 1 B + c 2 B c k B k can also be defined by the equation. p(B)x t = (c 0 I + c 1 B + c 2 B c k B k )x t. = c 0 Ix t + c 1 Bx t + c 2 B 2 x t c k B k x t = c 0 x t + c 1 x t-1 + c 2 x t c k x t-k

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4.The power series operator p(B) = c 0 I + c 1 B + c 2 B can also be defined by the equation. p(B)x t = (c 0 I + c 1 B + c 2 B )x t = c 0 Ix t + c 1 Bx t + c 2 B 2 x t +... = c 0 x t + c 1 x t-1 + c 2 x t If p(B) = c 0 I + c 1 B + c 2 B and q(B) = b 0 I + b 1 B + b 2 B are such that p(B)q(B) = I i.e. p(B)q(B)x t = Ix t = x t than q(B) is denoted by [p(B)] -1.

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Other operators closely related to B: 1. F = B -1,the forward shift operator, defined by Fx t = B -1 x t = x t+1 and = I - B,the first difference operator, defined by x t = (I - B)x t = x t - x t-1.

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The Equation for a MA(q) time series x t = 0 u t + 1 u t-1 + 2 u t q u t-q + can be written x t = (B) u t + where (B) = 0 I + 1 B + 2 B q B q

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The Equation for a AR(p) time series x t = 1 x t-1 + 2 x t p x t-p + + u t can be written (B) x t = + u t where (B) = I - 1 B - 2 B p B p

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The Equation for a ARMA(p,q) time series x t = 1 x t-1 + 2 x t p x t-p + + u t + 1 u t-1 + 2 u t q u t-q can be written (B) x t = (B) u t + where (B) = 0 I + 1 B + 2 B q B q and (B) = I - 1 B - 2 B p B p

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Some comments about the Backshift operator B 1.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; 2.It is also useful for making certain computations related to the time series described above;

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The partial autocorrelation function A useful tool in time series analysis

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The partial autocorrelation function Recall that the autocorrelation function of an AR(p) process satisfies the equation: x (h) = 1 x (h-1) + 2 x (h-2) p x (h-p) For 1 ≤ h ≤ p these equations (Yule-Walker) become: x (1) = 1 + 2 x (1) p x (p-1) x (2) = 1 x (1) + p x (p-2)... x (p) = 1 x (p-1)+ 2 x (p-2) p.

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In matrix notation: These equations can be used to find 1, 2, …, p, if the time series is known to be AR(p) and the autocorrelation x (h)function is known.

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In this case If the time series is not autoregressive the equations can still be used to solve for 1, 2, …, p, for any value of p ≥ 1. are the values that minimizes the mean square error:

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Definition: The partial auto correlation function at lag k is defined to be: Using Cramer’s Rule

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Comment: The partial auto correlation function, kk is determined from the auto correlation function, (h) The partial auto correlation function at lag k, kk 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.

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Some more comments: 1.The partial autocorrelation function at lag k, kk, can be interpreted as a corrected autocorrelation between x t and x t-k conditioning on the intervening variables x t-1, x t-2,...,x t-k+1. 2.If the time series is an AR(p) time series than kk = 0 for k > p 3.If the time series is an MA(q) time series than x (h) = 0 for h > q

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

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

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Then it can be shown that: and

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Proof: The Yule Walker equations:

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In matrix form:

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The equations for

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and The matrix A reverses order

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The equations may be written Multiplying the first equations by or

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Substituting this into the second equation or and

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Hence and or

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Some Examples

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Example 1: MA(1) time series Suppose that {x t |t T} satisfies the following equation: x t = u t u t – 1 where {u t |t T} is white noise with = 1.1. Find: 1.The mean of the series, 2.The variance of the series, 3.The autocorrelation function. 4.The partial autocorrelation function.

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Solution Now {x t |t T} satisfies the following equation: x t = u t u t – 1 Thus: 1.The mean of the series, = 12.0 The autocovariance function for an MA(1) is

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Thus: 2.The variance of the series, (0) = and 3.The autocorrelation function is:

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4.The partial auto correlation function at lag k is defined to be: Thus

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Graph: Partial Autocorrelation function kk

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Exercise: Use the recursive method to calculate kk and

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Exercise: Use the recursive method to calculate kk and

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Example 2: AR(2) time series Suppose that {x t |t T} satisfies the following equation: x t = 0.4 x t – x t – u t where {u t |t T} is white noise with = 2.1. Is the time series stationary? Find: 1.The mean of the series, 2.The variance of the series, 3.The autocorrelation function. 4.The partial autocorrelation function.

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1.The mean of the series 3.The autocorrelation function. Satisfies the Yule Walker equations

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hence

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2.the variance of the series 4.The partial autocorrelation function.

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The partial autocorrelation function of an AR(p) time series “cuts off” after p.

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Example 3: ARMA(1, 2) time series Suppose that {x t |t T} satisfies the following equation: x t = 0.4 x t – u t u t – u t – 2 where {u t |t T} is white noise with = 1.6. Is the time series stationary? Find: 1.The mean of the series, 2.The variance of the series, 3.The autocorrelation function. 4.The partial autocorrelation function.

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x t = 0.4 x t – u t u t – u t – 1 white noise std. dev,. = 1.6. Is the time series stationary? (x) = 1 – 1 x = 1 – 0.4x has root r 1 =1/0.4 =2.5 Since |r 1 | > 1, the time series is stationary Find: 1.The mean of the series.

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The autocovariance function (h) satisfies: For h = 0, 1, 2 for h > q: i.e. For h = 0, 1, 2 for h > q:

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etc. where

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We use the first two equations to find 0 and 1 Then we use the third equation to find 2

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The autocovariance, autocorrelation functions

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Spectral Theory for a stationary time series

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