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The Spectral Representation of Stationary Time Series

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Stationary time series satisfy the properties: 1.Constant mean (E(x t ) = ) 2.Constant variance (Var(x t ) = 2 ) 3.Correlation between two observations (x t, x t + h ) dependent only on the distance h. These properties ensure the periodic nature of a stationary time series

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and X 1, X 1, …, X k and Y 1, Y 2, …, Y k are independent independent random variables with where 1, 2, … k are k values in (0, ) Recall is a stationary Time series

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With this time series and We can give it a non-zero mean, , by adding to the equation

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We now try to extend this example to a wider class of time series which turns out to be the complete set of weakly stationary time series. In this case the collection of frequencies may even vary over a continuous range of frequencies [0, ].

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The Riemann integral The Riemann-Stiltjes integral If F is continuous with derivative f then: If F is is a step function with jumps p i at x i then:

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First, we are going to develop the concept of integration with respect to a stochastic process. Let {U( ): [0, ]} denote a stochastic process with mean 0 and independent increments; that is E{[U( 2 ) - U( 1 )][U( 4 ) - U( 3 )]} = 0 for 0 ≤ 1 < 2 ≤ 3 < 4 ≤ . and E[U( ) ] = 0 for 0 ≤ ≤ .

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In addition let G( ) =E[U 2 ( ) ] for 0 ≤ ≤ and assume G(0) = 0. It is easy to show that G( ) is monotonically non decreasing. i.e. G( 1 ) ≤ G( 2 ) for 1 < 2.

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Now let us define: analogous to the Riemann-Stieltjes integral

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Let 0 = 0 < 1 < 2 <... < n = be any partition of the interval. Let. Let i denote any value in the interval [ i-1, i ] Consider: Suppose that and there exists a random variable V such that *

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Then V is denoted by:

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Properties:

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The Spectral Representation of Stationary Time Series

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Let {X( ): [0, ]} and {Y( ): l [0, ]} denote a uncorrelated stochastic process with mean 0 and independent increments. Also let F( ) =E[X 2 ( ) ] =E[Y 2 ( ) ] for 0 ≤ ≤ and F(0) = 0. Now define the time series {x t : t T}as follows:

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Then

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Also

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Thus the time series {x t : t T} defined as follows: is a stationary time series with: F( ) is called the spectral distribution function: If f( ) = F ˊ ( ) is called then is called the spectral density function:

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Note The spectral distribution function, F( ), and spectral density function, f( ) describe how the variance of x t is distributed over the frequencies in the interval [0, ]

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The autocovariance function, (h), can be computed from the spectral density function, f( ), as follows: Also the spectral density function, f( ), can be computed from the autocovariance function, (h), as follows:

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Example: Let {u t : t T} be identically distributed and uncorrelated with mean zero (a white noise series). Thus and

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Graph:

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Example: Suppose X 1, X 1, …, X k and Y 1, Y 2, …, Y k are independent independent random variables with Let 1, 2, … k denote k values in (0, ) Then

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If we define {X( ): [0, ]} and {Y( ): [0, ]} Note: X( ) and Y( ) are “random” step functions and F( ) is a step function.

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Another important comment In the case when F( ) is continuous then

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in this case Sometimes the spectral density function, f( ), is extended to the interval [- , ] and is assumed symmetric about 0 (i.e. f s ( ) = f s (- ) = f ( )/2 ) It can be shown that

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Hence From now on we will use the symmetric spectral density function and let it be denoted by, f( ).

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Linear Filters

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Let {x t : t T} be any time series and suppose that the time series {y t : t T} is constructed as follows:: The time series {y t : t T} is said to be constructed from {x t : t T} by means of a Linear Filter. input x t output y t Linear Filter a s

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Let x (h) denote the autocovariance function of {x t : t T} and y (h) the autocovariance function of {y t : t T}. Assume also that E[x t ] = E[y t ] = 0. Then::

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Hence where of the linear filter

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Note: hence

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Spectral density function Moving Average Time series of order q, MA(q) Let 0 =1, 1, 2, … q denote q + 1 numbers. Let {u t |t T} denote a white noise time series with variance 2. Let {x t |t T} denote a MA(q) time series with = 0. Note: {x t |t T} is obtained from {u t |t T} by a linear filter.

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Now Hence

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Example: q = 1

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Example: q = 2

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Spectral density function for MA(1) Series

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Spectral density function Autoregressive Time series of order p, AR(p) Let 1, 2, … p denote p + 1 numbers. Let {u t |t T} denote a white noise time series with variance 2. Let {x t |t T} denote a AR(p) time series with = 0. Note: {u t |t T} is obtained from {x t |t T} by a linear filter.

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Now Hence

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Example: p = 1

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Example: p = 2

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Example : Sunspot Numbers ( )

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Autocorrelation function and partial autocorrelation function

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Spectral density Estimate

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Assuming an AR(2) model

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A linear discrete time series Moving Average time series of infinite order

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Let 0 =1, 1, 2, … denote an infinite sequence of numbers. 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 Linear discrete time series. Comment: A linear discrete time series is a Moving average time series of infinite order

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The AR(1) Time series Let {x t |t T} be defined by the equation. Then

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where and An alternative approach using the back shift operator, B. The equation: can be written

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Now since The equation: has the equivalent form:

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The time series {x t |t T} can be written as a linear discrete time series where and For the general AR(p) time series: [ (B)] -1 can be found by carrying out the multiplication

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can be written: where Thus the AR(p) time series: Hence This called the Random Shock form of the series

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can be written: where Thus the AR(p) time series: Hence This called the Random Shock form of the series

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An ARMA(p,q) time series {x t |t T} satisfies the equation: where and The Random Shock form of an ARMA(p,q) time series:

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Again the time series {x t |t T} can be written as a linear discrete time series namely where (B) =[ (B)] -1 [ (B)] can be found by carrying out the multiplication

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Thus an ARMA(p,q) time series can be written: where

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The inverted form of a stationary time series Autoregressive time series of infinite order

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An ARMA(p,q) time series {x t |t T} satisfies the equation: where and Suppose that This will be true if the roots of the polynomial all exceed 1 in absolute value. The time series {x t |t T} in this case is called invertible.

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Then where or

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Thus an ARMA(p,q) time series can be written: where This is called the inverted form of the time series. This expresses the time series an autoregressive time series of infinite order.

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