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. Expectations, Variances and Covariances of Linear forms Summary

12. 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.

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

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

15. Expectations, Variances and Covariances of Quadratic forms

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

17. and

19. 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.

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

22. and where

23. Examples The sample mean

24. and Thus

25. Also

26. and where

27. Thus Compare with

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

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

33. where or if  is known

34. where or if  is known

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

36. and

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

41. where

42. since

43. hence

44. Thus

45. and Finally

46. Where

47. Thus

48. Expectations, Variances and Covariances of Linear forms Summary

49. 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.

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

51. Expectations, Variances and Covariances of Quadratic forms

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

53. and

55. 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.

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

57. Estimation of the spectral density function

58. The Discrete Fourier Transform

59. 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

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

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

62. Note:

63. Since

64. Thus

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

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

67. where

68. Proof Note Thus

70. where

71. Thus Also

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

73. Thus and

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

75. Periodogram for the sunspot data

76. note:

78. Theorem

80. In addition: If k ≠ 0 If k ≠ h

81. Proof Note Let

84. Recall Basic Property of the Fejer kernel: If g() is a continuous function then : The remainder of the proof is similar

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

86. Smoothed Periodogram Estimators

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

88. Properties: If k ≠ 0 If k ≠ h

89. Spectral density Estimator

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

91. Periodogram Spectral density Estimator Properties: If k ≠ 0 The second property states that: is not a consistent estimator of f( ): Asymptotically unbiased

92. Examples of using packages SPSS, Statistica

93. Example 1 – Sunspot data

94. Using SPSS Open the Data

95. Select Graphs- > Time Series - > Spectral

96. The following window appears Select the variable

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

100. Periodogram Spectral density Estimator Properties: If k ≠ 0 The second property states that: is not a consistent estimator of f( ): Asymptotically unbiased

101. Smoothed Estimators of the spectral density

102. The Daniell Estimator

103. Properties 1. 2. 3.

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

105. Choosing the Daniell option in SPSS

106. k = 5

108. k = 9

109. k = 5

110. Other smoothed estimators

111. More generally consider the Smoothed Periodogram and where

112. 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

113. and Then Proof (See Fuller Page 292)

114. Weighted Covariance Estimators Recall that where

115. 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

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

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

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

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

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

121. DaniellTukey Parzen Bartlett

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

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

124. Confidence Limits in Spectral Density Estimation

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

126. Thus = The equivalent df (EDF)

127. and Now Thus

128. 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.

129. Estimation of the spectral density function Summary

130. The spectral density function, f( ) The spectral density function, f(x), is a symmetric function defined on the interval [- ,  ] satisfying and

131. Periodogram Spectral density Estimator Properties: If k ≠ 0 The second property states that: is not a consistent estimator of f( ): Asymptotically unbiased

133. Smoothed Estimators of the spectral density

134. Smoothed Periodogram Estimators and where The Daniell Estimator

135. 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

136. 1. Bartlett Choices for w m (h) = w(h/m) 2. Parzen w(x) = 1 -2 a + 2a cos(  x) 3. Blackman-Tukey with a = 0.23 (Hamming), a = 0.25 (Hanning)

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

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

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

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

141. 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.

142. and Now Thus

143. 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.

144. and