18.S096 Problem Set 4 Fall 2013 Time Series Due Date: 10/15/2013

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1 18.S096 Problem Set 4 Fall 2013 Time Series Due Date: 10/15/ Covariance Stationary AR(2) Processes Suppose the discrete-time stochastic process {X t } follows a secondorder auto-regressive process AR(2) : X t = φ 0 + φ 1 X t 1 + φ 2 X t 2 + η t, where {η t } is W N(0, σ 2 ), with σ 2 > 0, and φ 0, φ 1, φ 2, are the parameters of the autoregression. (a) If {X t } is covariance stationary with finite expectation µ = E[X t ] show that φ0 µ = 1 φ 1 φ 2 (b) For the autocovariance function γ(k) = Cov[X t, X t k ], show that γ(k) = φ 1 γ(k 1) + φ 2 γ(k 2), for k = 1, 2,... (c) For the autocorrelation function ρ k = corr[x t, X t k ], show that ρ k = φ 1 ρ k 1 + φ 2 ρ k 2, for k = 1, 2,... (d) Yule-Walker Equations Define the two linear equations for φ 1, φ 2 in terms of ρ 1, ρ 2 given by k = 1, 2 in (c): ρ 1 = φ 1 ρ 0 + φ 2 ρ 1 ρ 2 = φ 1 ρ 1 + φ 2 ρ 0 Using the facts that ρ 0 = 1, and ρ k = ρ k, this gives ρ 1 = φ 1 + φ 2 ρ 1 ρ 2 = φ 1 ρ 1 + φ 2 which is equivalent to: [ ] [ ] [ ] ρ 1 1 ρ 1 φ = 1 ρ 2 ρ 1 1 φ 2 Solve for φ1 and φ2 in terms of ρ 1, and ρ 2. 1

2 (e) Solve for ρ 1, and ρ 2 in terms of φ 1 and φ 2. (f) Derive complete formulas for ρ k, k > 2. Hint: Solve this part by referring to the answer to problem 4(b) below. 2. Difference equations associated with an AR(p) process. An AR(p) process, X t = φ 0 + φ 1 X t 1 + φ 2 X + t 2 + φ p X t p + η t where {η t } is W N(0, σ 2 ), with σ 2 > 0, and auto-regression parameters φ 0,..., φ p, can be written using the polynomial-lag operator as follows: φ(l) = (I φ 1 L φ 2 L 2 φ p L p ) φ(l)x t = η t. Consider the homogeneous difference equations: φ(l)g(t) = 0. for a discrete-time function g(t). Such equations arise in AR(p) models when analyzing auto-covariance, auto-correlation, and prediction functions. (a) If µ = E[X t ], show that φ0 µ = 1 p i=1 φ i So, µ = 0, if and only if φ 0 = 0. It is common practice to demean a time series which has the result of eliminating the autoregression parameter associated with the mean, i.e., φ 0. (b) For the auto-covariance function, γ(t) = Cov(X t, X 0 ), t = 0, 1,..., prove that φ(l)γ(t) = 0, for all t; (c) For the auto-correlation function, ρ(t) = Corr(X t, X 0 ), t = 0, 1,..., prove that φ(l)ρ(t) = 0, for all t. (d) Suppose that the process is observed up to time t, so the values (x 0, x 1,..., x t ) are known. Let g t (h), for h = 1, 2,..., be forecasts of the process: 2

3 g t (h) = E[X t +h (X t,..., X 1, X 0 ) = (x t,..., x 1, x 0 )] With the notation xˆt (h) = g t (h), for h > 0 and defining xˆt (h) = x t +h, for h 0, show that xˆt (1) = φ 0 + p i=1 φ ixˆt (1 i) p xˆt (2) = φ 0 + i=1 φ i xˆt (2 i) xˆt (h). = p φ 0 + i=1 φ i xˆt (h i), for any h > 0. (e) For the de-meaned process in (c) (i.e., φ 0 = 0), show that the forecast function g t (t) satisfies φ(l)g t (t) = 0, for t = 1, 2, Solutions to Difference Equations Associated with AR(p) Processes In the previous problem, the autocovariance, autocorrelation and forecast functions satisfy the difference equations: φ(l)g(t) = 0, for t = 0, 1, 2,... (3.1) (a) As a possible solution, consider the function: g(t) = Ce λt, for some constants C and λ. Show that such a function g() is a solution if z = e λ is a root of the characteristic equation: φ(z) = 1 p i=1 φ iz p = 0. (b) More generally, consider the function: g(t) = CG t, for some constants C and G (this is the generalization of (a) where G = e λ is constrained to be real and positive). Show that such a function g() is a solution if G 1 is a root of the characteristic equation. (c) For an AR(p) process, suppose the p roots of the characteristic equation z i = (G i 1 ) are distinct. Show that for constants C 1, C 2,..., C p, the function g(t) = C 1 G t 1 + C 2G t C pg t p satisfies φ(l)g(t) = 0. (d) For an AR(p) process, if all roots z i are outside the complex unit circle, prove that any solution g() given in (b) is bounded. 3

4 Since the autocovariance function is a solution, this condition is necessary and sufficient for the autocovariance to be bounded (covariance stationary). 4. Consider an AR(2) process. (a) Solve for the 2 roots of the characteristic equation, z 1 and z 2. Detail the conditions under which z 1 and z 2 are Distinct and real. Coincident and real. Complex and conjugates of each other (i.e., z 1 = a + ib and z 2 = a ib, for two real constants a and b, and the imaginary i = 1. (b) Prove that the autocorrelation function satisfies: ρ k = C 1 G k 1 + C 2G k 2 where G 1 = z 1 1, and G 2 = z2 1 Using equations for two values of k, solve for C 1 and C 2 when z 1 and z 2 are distinct, and show that G1(1 G 2 2 ) G k 1 G k = 2(1 G 2 1 ρ )Gk 2 (G 1 G 2 )(1+G 1 G 2 ) (c) When distinct and real, show that the autocovariance function decreases geometrically/exponentially in magnitude. (d) In (c), under what conditions on φ 1 does the autocovariance/correlation function remain positive; and under what conditions does it alternate in sign. (e) Optional: When the two roots are complex conjugates, define d and f 0 so that: G 1 = de i2πf 0, and G 2 = de i2πf 0. Using part (b), it can be shown that sin(2πf 0 k+f ) ρ k =[sgn(φ 1 )] k d k sin(f ) where The damping factor d is d = φ 2 The frequency f 0 is such ( that ) φ1 2πf 0 = arccos 2 φ 2 4

5 The phase F is such that tanf =( 1+d2 1 d 2 )tan(2πf 0 ) 5. Moving Average Process MA(1) Suppose the discrete stochastic process {X t } follows a MA(1) model: X t = η t + θ 1 η t 1 = (1 + θ 1 L)η t, t = 1, 2,... (η 0 = 0) where {η t } is W N(0, σ 2 ). (a) Derive the auto-correlation function (ACF) of X t (b) For the following two model cases, solve for the first-order autocorrelation: θ 1 = 0.5 θ 1 = 2.0 (c) Using the formula for ρ 1 = corr(x t, X t 1 ), in (a), solve for the MA process parameter θ 1 in terms of ρ 1. Is the solution unique? (d) An MA(1) process is invertible if the process equation can be inverted, i.e., the process {X t } satisfies: (1 θ 1 L) 1 X t = η t For each model case in (b), determine whether the process is invertible, and if so, provide an explicit expression for the model process as an (infinite-order) autoregression. 6. Autoregressive Moving Average Process: ARM A(1, 1) Suppose the discrete stochastic process {X t } follows a covariance stationary ARMA(1,1) model: X t φ 1 X t 1 = φ 0 + η t + θ 1 η t 1 (1 φ 1 L)X t = φ 0 + (1 + θ 1 L)η t, t = 1, 2,... (η 0 = 0) where {η 2 t} is W N(0, σ ). (a) Prove that φ0 µ = E[X t ] = 1 φ 1 (b) Prove that σ 2 σ [1+θ = V ar(x ) = γ = 1+2φ 1 θ 1 ] [ = σ 2 ( θ X t 1 φ φ 1 ) φ 2 1 ] 5

6 (c) Prove that the auto-correlation function (ACF) of the covariance stationary ARM A(1, 1) process is given by the following recursions: ρ 1 = θ φ σ 2 γ 0 ρ k = φ 1 ρ k 1, k > 1 (d) Compare the ACF of the ARMA(1, 1) process to that of the AR(1) process with the same parameters φ 0, φ 1. Note that Both decline geometrically/exponentially in magnitude by the factor φ 1 from the second time-step on. If φ 1 > 0, both processes have an ACF that is always positive. If φ 1 < 0, the ACFs alternate in sign from the second timestep on. What pattern in the ACF function of an ARMA(1, 1) model is not possible with an AR(1) model? Suppose an economic index time series follows such an ARM A(1, 1) process. What behavior would it exhibit? 6

7 MIT OpenCourseWare 18.S096 Mathematical Applications in Financial Industry Fall 2013 For information about citing these materials or our Terms of Use, visit: