Time Series Analysis

Transcription

1 Time Series Analysis Christopher Ting Quantitative Finance Singapore Management University March 3, 2017 Christopher Ting Week 9 March 3, / 37

2 Table of Contents 1 Introduction 2 Stationary Processes 3 AR, MA, & ARMA 4 Sample ACF 5 Yule-Walker Equations 6 Partial ACF Christopher Ting Week 9 March 3, / 37

3 Introduction Learning Objectives Understand thoroughly the nature of white noise Describe AR, MA, and ARMA time series using white noise as the building block Compute the unconditional and conditional means, as well as variances of these stationary processes Describe the autocorrelation function and how the function is estimated and tested for statistical significance (Box and Pierce Q-statistic, Ljung and Box statistic) Define back-shift operator B n and describe the relationship between MA and AR( ) processes Thoroughly understand the Yule-Walker equations and their applications to obtain partial autocorrelation function Christopher Ting Week 9 March 3, / 37

4 Introduction Components of a Time Series In general, a time series Y t has the following components: (a) trend-cycle (TC) component: the combined long-term and growth cycle movement of the time series (b) seasonal (S) component: the systematic variations of the time series (c) irregular (I) component: the random fluctuations of short-term movements of the time series. Christopher Ting Week 9 March 3, / 37

5 Introduction Time Series Trend-Cycle Component Seasonal Component Irregular Component Seasonality Calendar Effects Moving-Holiday Effect Trading-Day Effect Christopher Ting Week 9 March 3, / 37

6 Introduction Example: Nike s EPS Christopher Ting Week 9 March 3, / 37

7 Stationary Processes Basic Building Block: White Noise ˇ ( Properties of white noise Independently identically distributed process {u t } Stationary E(u t ) = 0 V(u t ) = σ 2 u C(u t, u t+k ) = 0 for any k 0 Stronger definition: u t is independent of u t+k for all k 0 ˇ ( How doe white noise sound like? Generate white noise with Matlab: u = randn(8192*10, 1); Create an audioplayer object: pu = audioplayer(u, 8192); Listen to white noise: play(pu); Christopher Ting Week 9 March 3, / 37

8 Stationary Processes AR, MA, and ARMA With basic white noise process {u t }, t =,...,, the following processes are generated: Autoregressive order one: AR(1) process Y t = θ + λ Y t 1 + u t Moving Average order one: MA(1) process Y t = θ + u t + α u t 1 Autoregressive Moving Average order one: ARMA(1, 1) process Y t = θ + λ Y t 1 + u t + α u t 1 Christopher Ting Week 9 March 3, / 37

9 AR, MA, & ARMA Autoregressive Process AR(1) Repeated substitution for Y t in the AR(1) process leads to For each t, provided λ < 1, Y t = θ + λ(θ + λy t 2 + u t 1 ) + u t Y t = (1 + λ + λ 2 + )θ + (u t + λu t 1 + λ 2 u t 2 + ) E(Y t ) = (1 + λ + λ 2 + )θ = θ 1 λ V(Y t ) = V(u t + λu t 1 + λ 2 u t 2 + ) = σ 2 u(1 + λ 2 + λ 4 + ) = σ 2 u 1 λ 2 σ 2 u C(Y t, Y t 1 ) = C ( ) θ + λy t 1 + u t, Y t 1 = λ 1 λ 2 Christopher Ting Week 9 March 3, / 37

10 AR, MA, & ARMA Autocorrelation Coefficients of AR(1) Autocovariance at lag k is a function of k only k 1 k 1 C(Y t, Y t k ) = C θ λ j + λ k Y t k + λ j u t j, Y t k = λk V(Y t ) γ(k) j=0 j=0 Accordingly, the serial correlation is Corr(Y t, Y t k ) = C(Y t, Y t k ) V(Y t ) = λ k For a symmetrical time shift t t + k, Corr(Y t+k, Y t ) = C(Y t+k, Y t ) V(Y t+k ) = λ k Hence ρ(k) Corr ( Y t, Y t±k ) = λ k Christopher Ting Week 9 March 3, / 37

11 AR, MA, & ARMA Illustrative Example of an AR(1) Process Suppose Y t = Y t 1 + u t, where E(u t ) = 0, V(u t ) = 3 1 What is the mean of Y t? 2 What is the variance of Y t? 3 What is the first-order autocovariance of this AR(1) process? 4 What is the first-order autocorrelation of this AR(1) process? 5 What is the third-order autocorrelation of this AR(1) process? Christopher Ting Week 9 March 3, / 37

12 Simulation parameters: AR, MA, & ARMA Simulating an AR(1) Process number of samples 5,000 σ 2 u = 3 θ = 2.5, λ 1 = 0.5 n = 5000 su2 = 3 theta, lambda1 = 2.5, 0.5 np.random.seed( ) u = np.random.normal(0, 1, n) mu, su = np.mean(u), np.std(u, ddof=1) u -= mu # ensure that mean of u is really 0 u /= su; # ensure that variance of u is really 1 u *= np.sqrt(su2) Y = np.zeros(n, dtype=float) Y[0] = theta + u[0] for t in range(1, n): Y[t] = theta + lambda1*y[t-1] + u[t] Christopher Ting Week 9 March 3, / 37

13 AR, MA, & ARMA Sample Autocorrelation Function of AR(1) Christopher Ting Week 9 March 3, / 37

14 AR, MA, & ARMA Another Example of AR(1), λ = 0.5 Christopher Ting Week 9 March 3, / 37

15 AR, MA, & ARMA Moving Average Process MA(1) For MA(1) process Y t = θ + u t + α u t 1, E ( Y t ) = θ V ( Y t ) = (1 + α 2 )σ 2 u C ( ) ( ) Y t, Y t 1 = C θ + ut + αu t 1, θ + u t 1 + αu t 2 = α σ 2 u Corr ( ) α Y t, Y t 1 = 1 + α 2 Corr ( ) Y t, Y t k = 0 for k > 1 Hence, MA(1) process is covariance-stationary with constant mean θ, constant variance σ 2 u(1 + α 2 ) and autocorrelation function of k = 1 only nonzero. Christopher Ting Week 9 March 3, / 37

16 AR, MA, & ARMA Illustrative Example of an MA(1) Process Suppose Y t = u t + 0.5u t 1, where E(u t ) = 0, V(u t ) = 3 1 What is the mean of Y t? 2 What is the variance of Y t? 3 What is the first-order autocovariance of this MA(1) process? 4 What is the first-order autocorrelation of this MA(1) process? 5 What is the third-order autocorrelation of this MA(1) process? Christopher Ting Week 9 March 3, / 37

17 AR, MA, & ARMA Simulating an MA(1) Process Simulation parameters: number of samples 5,000 σ 2 u = 3 θ = 2.5, α = 0.5 n = 5000 su2 = 3 theta, alpha = 2.5, 0.5 np.random.seed( ) u = np.random.normal(0, 1, n) mu, su = np.mean(u), np.std(u, ddof=1) u -= mu # ensure that mean of u is really 0 u /= su; # ensure that variance of u is really 1 u *= np.sqrt(su2) Y = np.zeros(n, dtype=float) Y[0] = theta + u[0] for t in range(1, n): Y[t] = theta + alpha*u[t-1] + u[t] Christopher Ting Week 9 March 3, / 37

18 AR, MA, & ARMA Sample ACF of MA(1) Process Christopher Ting Week 9 March 3, / 37

19 AR, MA, & ARMA Autoregressive Moving Average Process ARMA(1,1) Repeated substitution of Y t = θ + λy t 1 + u t + α u t 1 leads to Y t = θ λ i + u t + (λ + α) λ i u t i i=0 For each t, provided λ < 1 E(Y t ) = V(Y t ) = σ 2 u θ 1 λ 1 + (λ + α)2 i=0 C(Y t, Y t 1 ) = λ V(Y t k ) + α σ 2 u C(Y t, Y t k ) = λ k V(Y t k ), for k > 1 i=0 λ 2i = σ2 u (1 + ) (λ + α)2 1 λ 2 Christopher Ting Week 9 March 3, / 37

20 AR, MA, & ARMA Autoregressive Moving Average Process ARMA(1,1) (cont d) Hence, ARMA(1,1) process is covariance-stationary ) with constant θ mean 1 λ, constant variance (λ + σ2 α)2 u (1 + 1 λ 2, and autocovariance a function of k only. Christopher Ting Week 9 March 3, / 37

21 AR, MA, & ARMA Sample ACF of ARMA(1,1) Process Christopher Ting Week 9 March 3, / 37

22 AR, MA, & ARMA Autocorrelation Function of ARMA(1,1) Question 1: Derive the autocorrelation function of ARMA(1,1) process Question 2: If λ = α, what is the ACF of ARMA(1,1) at lag 1? Christopher Ting Week 9 March 3, / 37

23 AR, MA, & ARMA Changing Conditional Means AR(1) ℵ At t + 1, information about Y t is already known, so for AR(1) process, conditional mean is a random variable E ( Y t+1 Yt ) = θ + λyt + E ( u t+1 Yt ) = θ + λyt ℵ Conditional variance is constant MA(1) V ( Y t+1 Yt ) = V ( ut+1 Yt ) = σ 2 u < σ2 u 1 λ 2 θ 1 λ ℵ Conditional mean is stochastic but conditional variance is constant: E ( Y t+1 Yt ) = θ + αut V ( Y t+1 Yt ) = σ 2 u < σ 2 u(1 + α 2 ) Christopher Ting Week 9 March 3, / 37

24 Sample ACF Sample Autocorrelation Function Sample autocovariance lag k, for k = 0, 1, 2,..., p, with p < T/4. c(k) = 1 T k ( Yt Y )( Y t+k Y ) T The estimator c(k) is consistent, as lim T c(k) = C ( Y k, Y t k ). t=1 Sample autocorrelation lag k is then r(k) = c(k) c(0) = T k t=1 which is also consistent, as lim T r(k) = ρ(k). ( Yt Y )( Y t+k Y ) Tt=1 ( Yt Y ) 2 Variance of r(k) for AR(1) process V ( r(k) ) 1 ( (1 + λ 2 )(1 λ 2k ) ) 2kλ 2k T 1 λ 2 Christopher Ting Week 9 March 3, / 37

25 Sample ACF Test for ρ = 0, AR(1) Test hypothesis H 0 : ρ(j) = 0 for all j > 0 or H 0 = λ = 0. Then V ( r(k) ) 1 T To test that the j th autocorrelation is zero, the test statistic is z j = r(j) 0 1/T d N(0, 1) Reject H 0 : ρ(j) = 0 at 5% significance level if z > 1.96 Christopher Ting Week 9 March 3, / 37

26 Sample ACF Test for ρ = 0, MA(q) For MA(q) processes, the variance of r(k) is V ( r(k) ) 1 q T ρ(i) 2 Thus for MA(1) process, V ( r(k) ) for k > 1 is estimated by i=1 1 ( 1 + 2r(1) 2 ) > 1 T T and rejection of the null hypothesis allows the identification of MA(1). Christopher Ting Week 9 March 3, / 37

27 Sample ACF Joint Test Statistic To test if all autocorrelations are jointly zero, the null hypothesis is H 0 : ρ(1) = ρ(2) = = ρ(m) = 0 The Box and Pierce Q-statistic for asymptotic test is m m Q m = T r(k) 2 ( ) = T r(k) 2 = m z 2 k k=1 k=1 k=1 d χ 2 m The Ljung and Box test statistic provides an approximate correction for finite sample: Q m = T(T + 2) m k=1 r(k) 2 T k d χ 2 m Christopher Ting Week 9 March 3, / 37

28 Sample ACF ARMA (p, q) Processes AR(p) process is ARMA (p, 0); MA(q) process is ARMA(0, q). Backward shift operator B: BY t Y t 1 Applying the backward shift operator k times, AR(p) process B k Y t = Y t k Y t = θ + λ 1 Y t 1 + λ 2 Y t λ p Y t p + u t = θ + φ ( B ) p Y t 1 λ i B i Y t = θ + u t i=1 p λ i B i Y t + u t The equation φ(b) = 0 is called the characteristic equation of the AR(p) process. For AR(p) to be stationary, the roots of the characteristic equation must lie outside of the unit circle. Christopher Ting Week 9 March 3, / 37 i=1

29 Sample ACF Representation of MA(1) MA(1) process is representable as AR( ) process Y t = θ + u t + α u t 1 = θ + (1 + α B)u t (1 + αb) 1 Y t = (1 + αb) 1 θ + u t ( 1 αb + α 2 B 2 α 3 B 3 + )Y t = c + u t where c (1 + αb) 1 θ. Hence Y t = c + αy t 1 α 2 Y t 2 + α 3 Y t 3 α 4 Y t u t Stationarity requires (1 + αb) = 0 lie outside the unit circle, which is satisfied if α < 1. A stationary MA(q) process is said to be invertible if it can be represented as a stationary AR( ) process. What s the point of inverting an MA process? Christopher Ting Week 9 March 3, / 37

30 Yule-Walker Equations Yule-Walker Equations (1) Multiply both sides of AR(p) process by Y t k, Y t k Y t = Y t k θ + λ 1 Y t k Y t 1 + λ 2 Y t k Y t λ p Y t k Y t p + Y t k u t Taking unconditional expectations on both sides and noting that E ( ) Y t k Y t = γ(k) + µ 2 for any k, then γ(k) + µ 2 ( = µθ + λ 1 γ(k 1) + µ 2 ) ( + λ 2 γ(k 2) + µ 2 ) ( + + λ p γ(k p) + µ 2 ) The unconditional means of an AR(p) process is µ = θ + λ 1 µ + + λ p µ. So µ 2 = µθ + λ 1 µ λ p µ 2. Hence for k > 1, γ(k) = λ 1 γ(k 1) + λ 2 γ(k 2) + + λ p γ(k p) Dividing both sides by γ(0) to obtain correlations: ρ(k) = λ 1 ρ(k 1) + λ 2 ρ(k 2) + + λ p ρ(k p) Christopher Ting Week 9 March 3, / 37

31 Yule-Walker Equations Yule-Walker Equations (2) Note that ρ(0) = 1 and ρ(j) = ρ( j) For each k, there is a corresponding equation with p parameters of λ 1, λ 2,..., λ p, resulting in the Yule-Walker equations: ρ(1) = λ 1 + λ 2 ρ(1) + + λ p ρ(p 1) ρ(2) = λ 1 ρ(1) + λ λ p ρ(p 2) ρ(3) = λ 1 ρ(2) + λ 2 ρ(1) + + λ p ρ(p 3). = ρ(p) = λ 1 ρ(p 1) + λ 2 ρ(p 2) + + λ p Christopher Ting Week 9 March 3, / 37

32 Yule-Walker Equations Yule-Walker Equations in Matrix Form (1) Replace the ACF ρ(p) by sample ACF r(k), the p linear equations can be solved as 1 r(1) r(2) r(p 1) r(1) r(1) 1 r(1) r(p 2) r(2) R. r(p), Φ r(2) r(1) 1 r(p 3) r(p 1) r(p 2) r(p 3) 1 Λ λ 1 λ 2. λ p Christopher Ting Week 9 March 3, / 37

33 Yule-Walker Equations Yule-Walker Equations in Matrix Form (2) Write compactly as R = ΦΛ, so the parameters can be estimated as along with Λ = Φ 1 R, µ = 1 T T t=1 Y t Christopher Ting Week 9 March 3, / 37

34 Partial ACF Partial Autocorrelation Function ˇ * The sample ACF allows the identification respectively of either an AR or an MA process depending on whether the sample r(k) decays slowly or are clearly zero after some lag k. ˇ * However, even if an AR(p) is identified, it is still difficult to identify the order of the lag, p, since all AR(p) processes show similar decay patterns of ACF. ˇ * Is it possible to identify the order of an AR process? ˇ * Yes, use the Yule-Walker equations to generate a sequence of λ kk, k = 1, 2,..., p which is the PACF. Christopher Ting Week 9 March 3, / 37

35 Partial ACF Sample PACF Calculation (1) Suppose the true process is AR(p). Take k = 1 and based on AR(1) Y t = θ + λ 11 Y t 1 + u t The Yule-Walker equation for k = 1 gives λ 11 = r(1) Next take k = 2, i.e., assume that it is AR(2). The Yule-Walker system is ) ( ) 1 ( ) ( λ 21 1 r(1) r(1) = λ 22 r(1) 1 r(2) Solution for λ 22 is λ 22 = r(2) r(1)2 1 r(1) 2 Christopher Ting Week 9 March 3, / 37

36 Partial ACF Sample PACF Calculation (2) In general, to calculate the p-th value, take k = p and the AR(p) is Y t = θ + λ p1 Y t 1 + λ p2 Y t λ kp Y t p + u t Solve the Yule-Walker equations Λ = Φ 1 R, If the true order is p, theoretically, λ 11 0, λ 22 0,..., λ pp 0 λ p+1p+1 = λ p+2p+2 = = 0 Christopher Ting Week 9 March 3, / 37

37 Partial ACF Duality between AR and MA, ACF and PACF While an AR(p) process has a decaying ACF infinite in extent, the PACF cuts off after lag p. Recall that an MA(1) process is invertible into AR( ). In general, this property holds for MA(q) processes. So while the ACF of an MA(q) process cuts off after lag q, the PACF is infinite in extent. Christopher Ting Week 9 March 3, / 37