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Stationary Time Series AMS 586 1

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The Moving Average Time series of order q, MA(q) where {Z 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)) 2

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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 3

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The autocorrelation function for an MA(q) time series Comment cuts off to zero after lag q. q 4

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The Autoregressive Time series of order p, AR(p) where {Z t |t T} is a white noise time series with variance 2. Let {X t |t T} be defined by the equation. Then {Z t |t T} is called a Autoregressive time series of order p. (denoted by AR(p)) 5

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The mean of a stationary AR(p) Assuming {X t |t T} is stationary, and take expectations of the equation, we obtain the mean μ: 6 Now we can center (remove the mean of) the time series as follows :

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Computing the autocovariance of a stationary AR(p) Now assuming {X t |t T} is stationary with mean zero: 7 Multiplying by X t-h, h 0, and take expectations of the equation, we obtain the Yule-Walker equations for the autocovariance. Note, for a zero mean sequence:

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Note: For h > 0, we have: The Autocovariance function (h) of a stationary AR(p) series Satisfies the equations: 8 For h = 0, we have:

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with for h > p The Autocorrelation function (h) of a stationary AR(p) series Satisfies the equations: and 9

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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 10

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Conditions for stationarity Autoregressive Time series of order p, AR(p) 11

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The value of X t increases in magnitude and Z t eventually becomes negligible. If 1 = 1 and = 0. The time series {X t |t T} satisfies the equation: The time series {X t |t T} exhibits deterministic behavior. 12

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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 behavior. If |r i | 1 and |r i | = 1 for at least one i then {X t |t T} exhibits non-stationary random behavior. 13

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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 14

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Special Cases: The AR(1) time Let {X t |t T} be defined by the equation. 15

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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 behavior. 3.If |r i | = 1 or | 1 | = 1 then {X t |t T} exhibits non- stationary random behavior. 16

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Special Cases: The AR(2) time Let {X t |t T} be defined by the equation. 17

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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 behavior. 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 behavior. This is true if These inequalities define a triangular region for 1 and 2. 18

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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 2 19

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The Mixed Autoregressive Moving Average Time Series of order p,q The ARMA(p,q) series 20

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The Mixed Autoregressive Moving Average Time Series of order p and q, ARMA(p,q) Let 1, 2, … p, 1, 2, … p, denote p + q +1 numbers (parameters). Let {Z t |t T} denote a white noise time series with variance 2. –uncorrelated –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. 21

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Mean value, variance, autocovariance function, autocorrelation function of an ARMA(p,q) series 22

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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. 23

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Assume that the ARMA(p,q) time series {X t |t T} is stationary: Let = E(X t ). Then or 24

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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 25

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Hence 26

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We need to calculate: etc 28

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The autocovariance function (h) satisfies: For h = 0, 1. …, q: for h > q: 29

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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. 30

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Example:The autocovariance function, (h), for an ARMA(1,1) time series: For h = 0, 1: for h > 1: or 31

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Substituting (0) into the second equation we get: or Substituting (1) into the first equation we get: 32

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for h > 1: 33

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The Backshift Operator B 34

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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. 35

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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 36

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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)]

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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 2. = I - B,the first difference operator, defined by X t = (I - B)X t = X t - X t-1. 38

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The Equation for a MA(q) time series X t = 0 Z t + 1 Z t Z t q Z t-q + can be written X t = (B) Z t + where (B) = 0 I + 1 B + 2 B q B q 39

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The Equation for a AR(p) time series X t = 1 X t X t p X t-p + + Z t can be written (B) X t = + Z t where (B) = I - 1 B - 2 B p B p 40

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The Equation for a ARMA(p,q) time series X t = 1 X t X t p X t-p + + Z t + 1 Z t Z t q Z t-q can be written (B) X t = (B) Z t + where (B) = 0 I + 1 B + 2 B q B q and (B) = I - 1 B - 2 B p B p 41

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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; 42

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The partial autocorrelation function A useful tool in time series analysis 43

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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) = 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. 44

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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. 45

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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: 46

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Definition: The partial auto correlation function at lag k is defined to be: 47

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Comment: The partial auto correlation function, kk is determined from the auto correlation function, (h) 48

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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 49

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A General Recursive Formula for Autoregressive Parameters and the Partial Autocorrelation function (PACF) 50

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Let denote the autoregressive parameters of order k satisfying the Yule Walker equations: 51

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Then it can be shown that: and 52

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Proof: The Yule Walker equations: 53

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In matrix form: 54

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The equations for 55

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and The matrix A reverses order 56

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The equations may be written Multiplying the first equations by or 57

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Substituting this into the second equation or and 58

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Hence and or 59

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Some Examples 60

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Example 1: MA(1) time series Suppose that {X t |t T} satisfies the following equation: X t = Z t Z t – 1 where {Z 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. 61

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Solution Now {X t |t T} satisfies the following equation: X t = Z t Z t – 1 Thus: 1.The mean of the series, = 12.0 The autocovariance function for an MA(1) is 62

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Thus: 2.The variance of the series, (0) = and 3.The autocorrelation function is: 63

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4.The partial auto correlation function at lag k is defined to be: Thus 64

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Graph: Partial Autocorrelation function kk 66

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Exercise: Use the recursive method to calculate kk and 67

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Exercise: Use the recursive method to calculate kk and 68

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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 – Z t where {Z 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. 69

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1.The mean of the series 3.The autocorrelation function. Satisfies the Yule Walker equations 70

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hence 71

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2.the variance of the series 4.The partial autocorrelation function. 72

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The partial autocorrelation function of an AR(p) time series cuts off after p. 73

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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 – Z t Z t – Z t – 1 where {Z 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. 74

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Theoretical Patterns of ACF and PACF 75 Type of Model Typical Pattern of ACF Typical Pattern of PACF AR (p)Decays exponentially or with damped sine wave pattern or both Cut-off after lags p MA (q)Cut-off after lags q Declines exponentially ARMA (p,q)Exponential decay

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Reference GEP Box, GM Jenkins, GC Reinsel (1994) Time series analysis: Forecasting and control, Prentice- Hall. Brockwell, Peter J. and Davis, Richard A. (1991). Time Series: Theory and Methods. Springer-Verlag. We also thank colleagues who posted their notes as on-line open resources for time series analysis. 76

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