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# T2 - We want to see if is independently distributed as...

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STA4005B Time Series (2007-2008) Tutorial 2 (22/1, 23/1) (Time & Venue: T9, BMS LT; W5, MMW 704) Example 1 ( AR(1) model) Consider the stationary process , and ) , 0 ( ~ , 2 1 a t t t t WN a a Z Z σ φ µ + + = 1 | | < . Find and for ) ( t Z E ) , ( k t t Z Z Cov + K , 2 , 1 , 0 = k 1

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Estimation of a constant mean: Consider the model: , t t X Z + = µ ) 0 ( = t X E for all time t. where { is } t X stationary with zero mean, variance 0 γ , autocorrelation function (ACF) k ρ . = = = n t t Z n Z 1 1 ˆ is unbiased estimator of .( i.e. = ) ˆ ( E ) Then i) ii) + = = 1 1 0 ) 1 ( 2 1 ) ˆ ( n k k n k n Var iii) The standard error of ˆ is ) ˆ ( Var . Example 2 Suppose ) 1 , 0 ( ~ , 5 . 0 8 . 0 2 1 N a a a a Z t t t t t + = . Find . ) ( 10 1 = t t Z Var 2
Example 3 (MA(q) model) Suppose that are independent and identically distributed random variables with mean zero and finite variance and is a constant for If t a i b K , 2 , 1 , 0 ± ± = i , 1 1 0 q t q t t t a b a b a b X + + + = L then compute the autocorrelation function k ρ of { } t X for K , 2 , 1 , 0 = k 3

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Residual Analysis After model is fitted, we need to assess the model by a residual analysis using a graphical approach. Let be the residuals
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Unformatted text preview: . We want to see if is independently distributed as normal with constant variance. t t t Z Z e ˆ − = n t , , 2 , 1 K = t e (a) Whether the residuals look like independent. (Time series plot of the residuals) (b) Whether the residuals look normal. (Histogram or Q-Q plot of the residuals) (c) Whether the residuals look like to have constant variance. (Plot of residuals against the fitted values) The sample autocorrelation function For the stationary process 2 )] )( [( Z k t t k k Z Z E σ µ γ ρ − − = = + The sample autocorrelation function is defined as c c r k k = where , ) )( ( 1 1 ∑ − = + − − = k N t k t t k Z Z Z Z N c K k , , 1 , K = with K not larger than N/4 . is the most satisfactory estimate of k r k . 4...
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T2 - We want to see if is independently distributed as...

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