Financial TIme Series Analysis: Part II

Transcription

1 Department of Mathematics and Statistics, University of Vaasa, Finland January 29 February 13, 2015 Feb 14, 2015

2 1 Univariate linear stochastic models: further topics Unobserved component model Signal extraction Impulse response function Variance ratio

3 1 Univariate linear stochastic models: further topics Unobserved component model Signal extraction Impulse response function Variance ratio

4 1 Univariate linear stochastic models: further topics Unobserved component model Signal extraction Impulse response function Variance ratio

5 Unobserved component model Decompose the times series y t as y t = z t + u t, (1) where u t WN(0, σ 2 u) and z t constitutes an unobserved component (UC) of the model. Modeling z t in different ways leads to a rich family of alternatives. An example is the deterministic trend model such that y t is trend stationary. z t = α + βt, (2)

6 Unobserved component model In finance applications, an interesting alternative is an unobserved (stochastic trend) component model with z t selected in (1) as z t = µ + z t 1 + v t, (3) where v t WN(0, σ 2 v ) and is independent of u t. Then y t = µ + v t + u t u t 1 (4) such that the first order autocorrelation becomes σ2 u ρ 1 = σu 2 + 2σv 2 (5) with 1 ρ 1 0, ρ k = 0 for all k > 1.

7 Unobserved component models If σ 2 u = 0, y t becomes random walk with ρ 1 = 0. Thus, we can write y t as an MA(1) process y t = µ + e t θe t 1. (6) Define κ = σ2 v σu 2 which is called signal-to-noise variance ratio. (7)

8 Unobserved component models Then we obtain θ = 1 2 ( κ + 2) ) κ 2 + 4κ, (8) κ 0, θ < 1, and κ = (1 θ)2, (9) θ σ 2 u = θσ 2 e, (10) σ 2 v = (1 θ) 2 σ 2 e (11) [Exercise: Derive equations (10) and (11).] Thus, κ = 0 corresponds θ = 1 and implies that y t is stationary, while κ =, or θ = 0 implies that y t I(1). Testing can be worked out by ADF.

9 Unobserved component models A more general formulation of the UC model is and z t = µ + γ(b)v t (12) u t = λ(b)u t, (13) where v t WN(0, σ 2 v ) and a t WN(0, σ 2 a) are independent, and γ(b) and λ(b) are polynomials. We do not elaborate this model further here, but consider how we can get an idea of z t.

10 1 Univariate linear stochastic models: further topics Unobserved component model Signal extraction Impulse response function Variance ratio

11 Signal extraction Estimating the unobserved component z t from the observed series y t in (1) is called signal extraction. There are several methods to estimate z t, including Kalman filter and different kinds of other smoothing algorithms. We introduce here and EWMA-based approach by Pierce (1979) (Annals of Statistics).

12 Signal extraction Given a finite time series y t, t = 1,..., T, the estimator of z t, denoted as ẑ t is selected such that the error z t ẑ t becomes minimized in some sense. Because y t is only observed, the estimation must be based on it, and is generally of the form ẑ t = ν(b)y t (14)

13 Signal extraction For the random walk model of z t in (3), Pierce (1979) shows that ν(b) = (1 θ) (θb) j, (15) where θ is the MA(1) parameter of the model in equation (6). This leads to EWMA (exponentially weighted moving average) (details are given in the classroom) j=0 ẑ t = θẑ t 1 + (1 θ)y t. (16) The parameter θ can be estimated from the MA(1) model in (6). Thus, given ˆθ and selecting ẑ 1 = y t, we obtain recursively the rest of ẑ t values.

14 Signal extraction Example 1 Let y t in (1) be the observed real rate of inflation and z t is the (unobserved) expected real rate, which under the rational expectations hypothesis can be modeled as a random walk z t = z t 1 + v t (17) with v t WN(0, σ 2 v ). By equation (6), we have y t = (1 θb)e t. For the UK Tbill quarterly real rates from 1942Q1 to 2005Q4 estimation of the model parameters yields (with R arima package) ŷ t = ( B)e t with s.e(ˆθ) = and ˆσ 2 e =

15 Signal extraction From (10) and from (11) By (16) ˆσ 2 u = ˆθˆσ 2 e = 12.61, ˆσ 2 v = (1 ˆθ) 2ˆσ 2 e = ẑ 1 = y 1 (18) ẑ t = 0.801ẑ y t (19) for t = 2,..., T.

16 Signal extraction Observed UK Real Interest Rate [1952q1 2005q4] % per annum Quarter Expected real UK Real Interest Rates [1952q1 2005q4] % per annum Quarter Unexpected real UK Real Interest Rates [1952q1 2005q4] % per annum Quarter

17 1 Univariate linear stochastic models: further topics Unobserved component model Signal extraction Impulse response function Variance ratio

18 1 Univariate linear stochastic models: further topics Unobserved component model Signal extraction Impulse response function Variance ratio

19 Impulse response function Consider the Wold decomposition x t = µ + ψ(b)u t = µ + where ψ 0 = 1 and u t WN(0, σ 2 u). ψ j u t j, (20) The impact of a shock u t in period t on the change x t+k in x in period t + k is ψ k. As a consequence, the impact on the level x t is 1 + ψ ψ k. The ultimate impact on the level is j=0 ψ(1) = 1 + ψ 1 + ψ 2 + = ψ j (21) j=0

20 Impulse response function For a difference stationary series (i.e., I (1) series) ψ(1) 0, for a trend stationary series ψ(1) = 0, and a random walk ψ(1) = 1, since ψ 0 = 1 and ψ j = 0 for all j 1. The coefficients ψ k are generally called as a impulse response function (IRF) that trace a unit shock u t on future changes x t+k of x t. The sum k j=0 ψ j is called usually as the cumulative IRF. For and ARMA(p, q) process such that ψ(b) = θ(b)/φ(b) and x t = µ + θ(b) φ(b) u t (22) ψ(1) = 1 + θ θ q 1 φ 1 φ p. (23)

21 Impulse response function Example 2 Suppose i.e., x t AR(1). Then such that x t = φ 0 + φ 1 x t 1 + u t, (24) x t = φ φ 1 1 φ 1 B u t (25) 1 ψ(b) = 1 θ 1 B = (φ 1 B) j = 1 + φ 1 B + φ 2 1B 2 + (26) j=0

22 Impulse response function For example if φ 1 = 1/2, then ψ 1 = 1/2, ψ 2 = φ 2 1 = (1/2)2 = 1/4, ψ 3 = = 1/8,... µ = (φ 0 /(1 1/2) = 2φ 0, and the total effect of a unit shock on the level of x t is 1 ψ j = 2 j = 2. (27) j=0 j=0

23 1 Univariate linear stochastic models: further topics Unobserved component model Signal extraction Impulse response function Variance ratio

24 Variance ratio Cochrane (1988) 2 proposes a non-parametric measure of persistence defined in terms of the variance ratio V k = σ2 k σ1 2, (28) where σ 2 k = 1 k var[x t x t k ] = 1 k var[ kx t ]. (29) Note: k = 1 B k. 2 Cochran, John, H. (1988), How big is the random walk in GNP, Journal of Political Economy 96,

25 Variance ratio The variance ratio is based on the idea that if x t is a pure random walk with drift, such that then writing k = 1 B k x t = θ + u t, u t WN(0, σ 2 u), (30) = (1 B) + (B B 2 ) + (B 2 B 3 ) + + (B k 1 B k ) and we can write k x t = (x t x t 1 ) + (x t 1 x t (x t k+1 x t k ). (31)

26 Variance ratio Thus, utilizing (30), (31) becomes k x t = u t + u t u t k+1. Because u t are WN(0, σu) 2 for all t, i.e. are i.i.d random variables var[ k x t ] = var[u t ] + var[u t 1 ] var[u t k+1 ] = kσu. 2 (32) On the other hand is x t is trend stationary such that x t = β 0 + β 1 t + u t then such that k x t = β t k + u t u t k var[ k x t ] = var[u t ] + var[u t k ] = 2σ 2 u.

27 Variance ratio Cochrane: Plot (estimated) σ 2 k of equation (29) against k: If x t is random walk, the σk 2 plot is constant at σ2 u. If x t is trend stationary the σk 2 plot declines towards zero. If fluctuation in x t is partly permanent and partly permanent, the fluctuation of the σk 2 plot should settle down towards the variance of the innovation of the random walk component.

28 Variance ratio To account for finite sample biases in estimating σ 2 k, Cohcrane proposes estimator ˆσ 2 k = T k(t k)(t k + 1) T j=k Cochrane derives the standard error of ˆσ 2 k as ( x j x j k k ) 2 T (x T x 0 ) (33) asy.se(ˆσ 2 k ) = 4k 3T ˆσ2 k. (34) Finally, (33) the variance ratio in (28) can be estimated as ˆV k = ˆσ2 k ˆσ 1. (35)

29 Variance ratio Further, Cochrane shows that V k can be written also in terms of autocorrelations, ρ j, as k 1 V k = j=1 so that the limiting variance ratio can be written as V = lim k V k = k j ρ k, (36) k ρ j. (37) j=1

30 Variance ratio Furthermore, it can be shown that lim k σ2 k = ( ψj ) 2 ( ) 2 σ 2 ψ 2 1 = ψj σ 2 u = ψ(1) 2 σu, 2 (38) j ( ) where we have utilized the relation σ1 2 = ψ 2 j σ0 2. Thus, V in equation (37) can be written as V = σ2 u σ1 2 ψ(1) 2. (39) Recalling that in the impulse response approach the persistence measure was based on ψ(1), such that (39) defines a relation between the two persistence measures.

31 Variance ratio Writing we get R 2 = 1 σ2 u σ1 2, (40) V ψ(1) = 1 R 2, (41) which implies that ψ(1) V, such that the more predictable x t the grater the difference between the two measures.

32 Variance ratio Finally, because V k = 1 if x t is a random walk, i.e, x t = θ + x t 1 + u t, with u t WN(0, σu), 2 Lo and MacKinlay 3 show that under the null hypothesis of random walk, z 1 (k) = ( ˆV 3Tk k 1) N(0, 1) (42) 2(2k 1)(k 1) asymptotically. 3 Lo, A.W. and A.C. MacKinlay 1988, Stock prices do not follow random walks: Evidence from a simple specification test, Review of Financial Studies 1, Lo, A.W. and A.C. MacKinlay 1989, The size and power of variance ratio test in finite samples: A Monte Carlo investigation, Journal of Econometrics 40,