1. Estimation of the spectral density function

2. The spectral density function, f( ) The spectral density function, f(x), is a symmetric function defined on the interval [- ,  ] satisfying and The spectral density function, f(x), can be calculated from the autocovariance function and vice versa.

3. Some complex number results: Use

5. Expectations of Linear and Quadratic forms of a weakly stationary Time Series

6. Expectations, Variances and Covariances of Linear forms

7. TheoremLet {x t :t  T} be a weakly stationary time series. Let Then and where andS r = {1,2,..., T-r}, if r ≥ 0, S r = {1- r, 2 - r,..., T} if r ≤ 0.

8. Proof

9. Also since Q.E.D.

10. TheoremLet {x t :t  T} be a weakly stationary time series. Let and

11. Then where and Also S r = {1,2,..., T-r}, if r ≥ 0, S r = {1- r, 2 - r,..., T} if r ≤ 0.

12. Expectations, Variances and Covariances of Quadratic forms

13. TheoremLet {x t :t  T} be a weakly stationary time series. Let Then

14. and

16. and S r = {1,2,..., T-r}, if r ≥ 0, S r = {1- r, 2 - r,..., T} if r ≤ 0,  (h,r,s) = the fourth order cumulant = E[(x t -  )(x t+h -  )(x t+r -  )(x t+s -  )] - [  (h)  (r-s)+  (r)  (h-s)+  (s)  (h-r)] Note  (h,r,s) = 0 if {x t :t  T}is Normal.

17. TheoremLet {x t :t  T} be a weakly stationary time series. Let Then

19. and where

20. Examples The sample mean

21. and Thus

22. Also

23. and where

24. Thus Compare with

25. Basic Property of the Fejer kernel: If g() is a continuous function then : Thus

29. The sample autocovariance function The sample autocovariance function is defined by:

30. where or if  is known

31. where or if  is known

32. TheoremAssume  is known and the time series is normal, then: E(C x (h))=  (h),

33. and

34. Proof Assume  is known and the the time series is normal, then: and

38. where

39. since

40. hence

41. Thus

42. and Finally

43. Where

44. Thus

45. Estimation of the spectral density function

46. The Discrete Fourier Transform

47. Let x 1,x 2,x 3,...x T denote T observations on a univariate one-dimensional time series with zero mean (If the series has non-zero mean one uses in place of x t ). Also assume that T = 2m +1 is odd. Then

48. where with k = 2  k/T and k = 0, 1, 2,..., m.

49. The Discrete Fourier transform: k = 0, 1,2,...,m.

50. Note:

51. Since

52. Thus

53. Summary: The Discrete Fourier transform k = 0, 1,2,...,m.

54. Theorem with k  k/T) E[X k ] = 0 with k  k/T) and h  h/T)

55. where

56. Proof Note Thus

58. where

59. Thus Also

60. with  =2  (k/T)+ with  =2  (h/T)+

61. Thus and

62. Defn: The Periodogram: k = 0,1,2,..., m with k = 2  k/T and k = 0, 1, 2,..., m.

63. Periodogram for the sunspot data

64. note:

66. Theorem

68. In addition: If k ≠ 0 If k ≠ h

69. Proof Note Let

72. Consistent Estimation of the Spectral Density function f( )

73. Smoothed Periodogram Estimators

74. Defn: The Periodogram: k = 0,1,2,..., m

75. Properties: If k ≠ 0 If k ≠ h

76. Spectral density Estimator

77. Properties: If k ≠ 0 The second properties states that: is not a consistent estimator of f( ):

78. Examples of using packages SPSS, Statistica

79. Example 1 – Sunspot data

80. Using SPSS Open the Data

81. Select Graphs- > Time Series - > Spectral

82. The following window appears Select the variable

83. Select the Window Choose the periodogram and/or spectral density Choose whether to plot by frequency or period

86. Smoothed Estimators of the spectral density

87. The Daniell Estimator

88. Properties 1. 2. 3.

89. Now let T  ∞, d  ∞ such that d/T  0. Then we obtain asymptotically unbiased and consistent estimators, that is

90. Choosing the Daniell option in SPSS

91. k = 5

93. k = 9

94. k = 5

95. Other smoothed estimators

96. More generally consider the Smoothed Periodogram and where

97. Theorem (Asymptotic behaviour of Smoothed periodogram Estimators ) and Let where {u t } are independent random variables with mean 0 and variance  2 with Let d T be an increasing sequence such that

98. and Then Proof (See Fuller Page 292)

99. Weighted Covariance Estimators Recall that where

100. The Weighted Covariance Estimator where {w m (h): h = 0, ±1,±2,...} are a sequence of weights such that: i) 0 ≤ w m (h) ≤ w m (0) = 1 ii) w m (-h) = w m (h) iii) w m (h) = 0 for |h| > m

101. The Spectral Window for this estimator is defined by: i) W m ( ) = W m (- ) ii) Properties :

102. also (Using a Reimann-Sum Approximation) = the Smoothed Periodogram Estimator Note:

103. 1. Asymptotic behaviour for large T 2. 3.

104. 1. Bartlett Examples w m (h) = w(h/m) Note:

105. 2. Parzen w(x) = 1 -2 a + 2a cos(  x) 3. Blackman-Tukey with a = 0.23 (Hamming), a = 0.25 (Hanning)

106. DaniellTukey Parzen Bartlett

107. 1. Approximate Distribution and Consistency 2. 3.

108. 1. Note: If W m ( ) is concentrated in a "peak" about = 0 and f( ) is nearly constant over its width, then 2. and

109. Confidence Limits in Spectral Density Estimation

110. 1. Satterthwaites Approximation: 2. where c and r are chosen so that

111. Thus = The equivalent df (EDF)

112. and Now Thus

113. Then a [1-  100 % confidence interval for f( ) is: Confidence Limits for The Spectral Density function f( ) : Let  and  denote the upper and lower critical values for the Chi-square distribution with r d.f. i.e.