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

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Presentation on theme: "Dates for term tests 1.Friday, February 07 2.Friday, March 07 3.Friday, March 28."— Presentation transcript:

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

2 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))

3 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

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

5 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))

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

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

8 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

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

10 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.

11 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

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

13 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.

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

15 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.

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

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

18 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.

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

20 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.

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

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

23 Hence

24

25 We need to calculate:

26

27 h  ux (h)

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

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

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

32 for h > 1:

33 The Backshift Operator B

34 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 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 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.

37 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.

38 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

39 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

40 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

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

43 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.

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

47 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.

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

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

51 Then it can be shown that: and

52 Proof: The Yule Walker equations:

53 In matrix form:

54 The equations for

55 and The matrix A reverses order

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

57 Substituting this into the second equation or and

58 Hence and or

59 Some Examples

60 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.

61 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

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

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

64

65 Graph: Partial Autocorrelation function  kk

66 Exercise: Use the recursive method to calculate  kk and

67 Exercise: Use the recursive method to calculate  kk and

68 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.

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

70 hence

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

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

73 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.

74 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.

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

76 etc. where

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

78 The autocovariance, autocorrelation functions

79 Spectral Theory for a stationary time series


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