ECON217_HW_ARMA_Key

# ECON217_HW_ARMA_Key - ECON217_HW_ARMA – Suggested...

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Unformatted text preview: ECON217_HW_ARMA – Suggested Solutions 1. If a time series {X t } is covariance stationary, what do we know about E(X t ) and COV(X t , X t-k ) for t = 1, ..., T and k = 0, 1, 2, ..? As. E(X t ) denotes the mean of X t . If {X t } is covariance stationary, E(X t )is time invariant; i.e. E(X t ) equals to a constant, say μ , for all t. COV(X t , X t-k ) is the covariance between X t and X t-k . For k = 0, we have the variance of X t (i.e. COV(X t , X t )). If {X t } is covariance stationary, COV(X t , X t-k ) is time invariant, and it depends on the “distance” k between X t and X t-k but not the “time” t at which it is measured. γ (k) is usually used to represent COV(X t , X t-k ); a notation that highlights the dependence on k. 2. If {X t } is a white noise process, what do we know about E(X t ), and COV(X t , X t-k ) for for t = 1, ..., T and k = 0, 1, 2, ..? As. Usually, a white noise process is referred to a zero mean white noise process; that is, E(X t ) = 0 for all t. For k = 0, COV(X t , X t-k ) = σ 2 , a standard notation for the variance of a random variable. For k ≠ 0, COV(X t , X t-k ) = 0. This is, a white noise series is a sequence of zero mean, constant variance and uncorrelated random variables. 3. Define and compare the autocorrelation function and the partial autocorrelation function of a stationary time series. As. For a time series {X t }, the Autocorrelation Function, ρ (k), is defined as ρ (k) = γ (k)/ γ (0), where γ (k) is COV(X t , X t-k ). It can be shown that ρ (k) = ρ (-k) and ρ (k) ≤ 1, ρ (0)= 1. Partial Autocorrelation Function, φ kk , is defined by the following regression equation: X t = φ 1k X t-1 + ... + φ kk X t-k + ω t . Note that, depends on the true stochastic property of X t , ω t is not necessarily a white noise process. Both the autocorrelation function and partial autocorrelation function measure the association of the variables in a time series. In contrast to ρ (k), φ kk eliminates the effects of the intervening values X t-1 through X t-k+1. 4. Suppose Y t follows Y t = φ Y t-1 + ε t ; ε t ~ WN (0 , σ 2 ) . a. State the assumption(s) on φ that will make {Y t } stationary. b. Assuming {Y t } is stationary. Find the autocorrelation function and the partial autocorrelation function. As. Stationarity condition: | φ | < 1 and t is large. Note that Y t = φ Y t-1 + ε t ; Y o = 0. Y t = φ t Y o + φ t-1 ε 1 + φ t-2 ε 2 + ... + φ 1 ε t-1 + ε t . E(Y t ) = 0. V(Y t ) = ( φ 2(t-1) + φ 2(t-2) + ... + φ 2 + 1) σ 2 = σ 2 (1- φ 2t )/(1- φ 2 ). If Y t has an infinite history (i.e., we do not have the initial condition Y = 0), then V(Y t ) = σ 2 /(1- φ 2 ). Autovariance: γ (1) = E(Y t Y t-1 ) as E(Y t ) = 0 = E( φ Y 2 t-1 + Y t-1 ε t ) = φ E(Y 2 t-1 ) = φ γ (0)....
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ECON217_HW_ARMA_Key - ECON217_HW_ARMA – Suggested...

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