Week 5 Quantitative Analysis of Financial Markets Characterizing Cycles

Transcription

1 Week 5 Quantitative Analysis of Financial Markets Characterizing Cycles Christopher Ting Christopher Ting : christopherting@smu.edu.sg : : LKCSB 5036 November 2, 2017 Christopher Ting QF 603 November 2, /28

2 Lesson Plan 1 Introduction 2 Covariance Stationary 3 White Noise 4 Lag Operator 5 Wold s Theorem 6 Estimation & Inference 7 Takeaways Christopher Ting QF 603 November 2, /28

3 Introduction A Cycles are defined as, in a general and all-encompassing way, any sort of dynamics not captured by trends or seasonals. A Cycles are based on some dynamics, some persistence, some way in which the present is linked to the past, and the future to the present. A It s crucial that we know how to model and forecast cycles, because their history conveys information regarding their future. A Trend and seasonal dynamics are simple, so we can capture them with simple models. Cyclical dynamics, however, are more complicated. Christopher Ting QF 603 November 2, /28

4 QA-12 Characterizing Cycles Chapter 7. Francis X. Diebold, Elements of Forecasting, 4th Edition (Mason, Ohio: Cengage Learning, 2006). A Define covariance stationary, autocovariance function, autocorrelation function, partial autocorrelation function and autoregression. A Describe the requirements for a series to be covariance stationary. A Explain the implications of working with models that are not covariance stationary. A Define white noise, independent white noise, and normal (Gaussian) white noise. Christopher Ting QF 603 November 2, /28

5 QA-12 Characterizing Cycles (cont d) A Explain the characteristics of the dynamic structure of white noise. A Explain how a lag operator works. A Describe Wold s theorem. A Define a general linear process. A Relate rational distributed lags to Wold s theorem. A Calculate the sample mean and sample autocorrelation, and describe the Box-Pierce Q-statistic and the Ljung-Box Q-statistic. A Describe sample partial autocorrelation. Christopher Ting QF 603 November 2, /28

6 Definitions B A realization of a time series is an ordered set, {..., y 2, y 1, y 0, y 1, y 2,...}. B In theory, a time series realization begins in the infinite past and continues into the infinite future. B In practice, of course, the data we observe is just a finite subset of a realization, {y 1,..., y T }, called a sample path. B A time series is said to be covariance stationary if its mean and covariance structure (that is, the covariances between current and past values) are stable over time. Christopher Ting QF 603 November 2, /28

7 Covariance Stationary B The mean of the series at time t is E ( y t ) = µt. B If the mean is stable over time, as required by covariance stationarity, then we can write E ( y t ) = µ. B The autocovariance at displacement τ is the covariance between y t and y t τ γ(t, τ) = C ( y t, y t τ ) = E [ (y t µ)(y t τ µ) ]. B If the covariance structure is stable over time, as required by covariance stationarity, then the autocovariances depend only on displacement, τ, not on time, t, and we write, for all t, γ(t, τ) = γ(τ). Christopher Ting QF 603 November 2, /28

8 Properties of Autocovariance Function of τ B Symmetric: γ(τ) = γ( τ) for all τ. B Implication of symmetry: It doesn t matter whether we go forward or backward. B Special case of γ(τ): γ(0) = C(y t, y t ) = V(y t ). B Many series that are clearly nonstationary in levels appear covariance stationary in growth rates. Christopher Ting QF 603 November 2, /28

9 Autocorrelation Function B The correlation for a covariance stationary time series is C ( ) y t, y t τ ρ(τ) := V (y t ) V(y t τ ) = γ(τ) = γ(τ) γ(0) γ(0) γ(0). B Hence, the autocorrelation function is obtained by dividing the autocovariance function by the variance, ρ(τ) := γ(τ), for τ = 0, 1, 2,.... γ(0) B Note that any series is perfectly correlated with itself, i.e., ρ(0) = γ(0) γ(0) = 1. Christopher Ting QF 603 November 2, /28

10 Partial Autocorrelation Function B The autocorrelations are just the simple or regular correlations between y t and y t τ. B The partial autocorrelations, on the other hand, measure the association between and after controlling for the effects of y t 1, y t 2,..., y t τ+1. That is, they measure the partial correlation between y t and y t τ. B All of the covariance stationary processes that we will study subsequently have autocorrelation and partial autocorrelation functions that approach zero, one way or another, as the displacement τ gets large. Christopher Ting QF 603 November 2, /28

11 Definition of White Noise s Suppose that y t = ɛ t and ɛ t (0, σ 2 ), where the shock, ɛ t is uncorrelated over time. s We say that ɛ t, and hence y t, is serially uncorrelated. s Such a process, with zero mean, constant variance, and no serial correlation, is called zero-mean white noise, or simply white noise. ɛ t WN ( 0, σ 2). Christopher Ting QF 603 November 2, /28

12 Definition of I.I.D. White Noise s In addition to being serially uncorrelated, y t is serially independent, then we say that y t is independent white noise. y t iid ( 0, σ 2 ). s y t is independently and identically distributed with zero mean and constant variance. s If y t is serially uncorrelated and normally distributed, then it follows that y t is also serially independent, and we say that y t is normal white noise, or Gaussian white noise. y t iid N ( 0, σ 2 ). Recall that zero correlation implies independence only in the normal case. Christopher Ting QF 603 November 2, /28

13 Dynamic Stochastic Structure of White Noise s The unconditional mean and unconditional variance of y t are, respectively, E ( y t ) = 0, V(y t ) = σ 2. Note that the unconditional mean and variance are constant. s The autocovariance γ(τ), autocorrelation ρ(τ), and partial autocorrelation p(τ) functions for a white noise process are, respectively, γ(τ) = { σ 2, τ = 0; 0, τ 1., ρ(τ) = { 1, τ = 0; 0, τ 1., p(τ) = { 1, τ = 0; 0, τ 1. Christopher Ting QF 603 November 2, /28

14 Importance of White Noise s Processes with much richer dynamics are built up by taking simple transformations of white noise. s 1-step-ahead forecast errors from good models should be white noise. s If such forecast errors aren t white noise, then they re serially correlated, which means that they re, in principle, forecastable, and if forecast errors are forecastable then the forecast can t be very good. Christopher Ting QF 603 November 2, /28

15 Conditional Mean and Variance s Information set Ω t 1 = {y t 1, t t 2,...} s For the independent white noise process, the conditional mean is E ( y t Ω t 1 ) = 0. s The conditional variance is V ( ) [ (yt y t Ω t 1 = E E ( )) 2 ] y t Ω t 1 Ω t 1 = σ 2. s Conditional and unconditional means and variances are identical for an independent white noise series; there are no dynamics in the process, and hence no dynamics in the conditional moments to exploit for forecasting. Christopher Ting QF 603 November 2, /28

16 Definition of Lag Operator y Lag operator, L, operates on a time series by lagging it. y Similarly, Ly t := y t 1. L 2 y t = L ( Ly t ) = Lyt 1 = y t 2, and so on. y A lag operator polynomial of degree m is just a linear function of powers of L, up through the m-th power, B(L) := b 0 + b 1 L + b 2 L b m L m. Christopher Ting QF 603 November 2, /28

17 Examples y A well-known operator, the first-difference operator, is actually a first-order polynomial in the lag operator, y t = (1 L)y t = y t y t 1. y Second-order lag operator polynomial ( L + 0.6L 2 ) ( L + 0.6L 2 ) = y t + 0.9y t y t 2, which is a weighted sum, or distributed lag, of current and past values. y All forecasting models, one way or another, must contain such distributed lags, because they ve got to quantify how the past evolves into the present and future. Christopher Ting QF 603 November 2, /28

18 Infinite-Order Lag Operator Polynomial y Nothing to stop us from extending to infinity. B(L) = b 0 + b 1 L + b 2 L 2 + = b i L i. i=0 y To denote an infinite distributed lag of current and past shocks, B(L)ɛ t = b 0 ɛ t + b 1 ɛ t 1 + b 2 ɛ t 2 + = b i ɛ t i. i=0 y It turns out that models involving infinite distributed lags are central to time series modeling and forecasting. Christopher Ting QF 603 November 2, /28

19 Wold s Theorem h Let {y t } be any zero-mean covariance-stationary process. Then we can write it as, when b 0 = 1 and b 2 i <, i=0 y t = B(L)ɛ t = b i ɛ t i, i=0 where ɛ t WN(0, σ 2 ). h In short, the correct model for any covariance stationary series is some infinite distributed lag of white noise, called the Wold representation. Christopher Ting QF 603 November 2, /28

20 The General Linear Process h Wold s theorem tells us that when formulating forecasting models for covariance stationary time series we need only consider models of the form y t = B(L)ɛ t = b i ɛ t i, ɛ t WN(0, σ 2 ). i=0 h We call this the general linear process, general because any covariance stationary series can be written that way, and linear because the Wold representation expresses the series as a linear function of its innovations. Christopher Ting QF 603 November 2, /28

21 Mean and Variance of a General Linear Process (1) h Unconditional mean E ( ( ) ) y t = E b i ɛ t i = i=0 b i E(ɛ t i ) = 0. i=0 h Unconditional variance V ( ( ) ) y t = V b i ɛ t i = i=0 b 2 i V(ɛ t i ) = i=0 b 2 i σ 2 = σ 2 b 2 i. i=0 i=0 Christopher Ting QF 603 November 2, /28

22 Mean and Variance of a General Linear Process (2) h Conditional mean E ( ) y t Ω t 1 = b i E ( ) ɛ t i Ω t 1 = b i ɛ t i. i=0 Only b 0 E ( ɛ t Ω t 1 ) = 0. The rest of ɛt 1, ɛ t 2,... are in the information set Ω t 1. h Conditional variance V ( ) [ (yt y t Ω t 1 = E E ( )) y t 2 ] Ω t 1 Ω t 1 i=1 = E [ ɛ 2 ] t Ω t 1 = E ( ɛ 2 ) t = σ 2. The key insight is that the conditional mean moves over time in response to the evolving information set. Christopher Ting QF 603 November 2, /28

23 Rational Distributed Lags h The infinite polynomial B(L) may be a ratio of finite-order polynomials. B(L) = Θ(L) Φ(L), where the numerator polynomial is of degree q, Θ(L) = and the denominator polynomial is of degree p, Φ(L) = h There are only p + q parameters! q θ i L i, i=0 p φ i L i. i=0 Christopher Ting QF 603 November 2, /28

24 Sample Mean and Sample Autocorrelations p Our estimator for the population mean, given a sample of size T, is the sample mean, y = 1 T y t. T p Sample autocorrelation function (aka correlogram) t=1 ρ(τ) = 1 T T t=τ+1 1 T [ (yt y)(y t τ y) ] = T (y t y) 2 t=1 T [ (yt y)(y t τ y) ] t=τ+1. T (y t y) 2 t=1 Christopher Ting QF 603 November 2, /28

25 Sample Autocorrelation s Distribution p If a series is white noise, then the distribution of the sample autocorrelations in large samples is ( ρ(τ) N 0, 1 ). T p Under normality, taking plus or minus two standard errors yields an approximate 95% confidence interval. Thus, if the series is white noise, approximately 95% of the sample autocorrelations should fall in the interval of ± 2 T. Christopher Ting QF 603 November 2, /28

26 Chi-Square Tests of Sample Autocorrelation p Since T ρ(τ) N(0, 1), it follows that T ρ 2 (τ) χ 2 1. p The Box-Pierce Q-statistic is m Q BP = T ρ 2 (τ) χ 2 m. τ=1 The null hypothesis is that y is white noise. p Small sample adjusted = the Ljung-Box Q-statistic m ( ) 1 Q LB = T (T + 2) ρ 2 (τ) χ 2 T τ m. τ=1 p For the joint test of m displacements (lags) in practice, as a guide, m in range of T is reasonable. Christopher Ting QF 603 November 2, /28

27 Sample Partial Autocorrelations p If the fitted regression is ŷ t = ĉ + β 1 y t β τ y t τ, then the sample partial autocorrelation at displacement τ is p(τ) = β τ. p If the series is white noise, approximately 95% of the sample partial autocorrelations should fall in the interval of ± 2 T as well. Christopher Ting QF 603 November 2, /28

28 Takeaways V Time series of returns is often assumed to be covariance stationary. V A defining characteristic of a covariance stationary time series is that the autocorrelation function is a function of the displacement τ and not on time t. V Autocorrelation function versus partial autocorrelation function V White noise is the fundamental building block of linear time series. V Polynomial function of lag operators and Wold s theorem V Box-Pierce and Ljung-Box Q-statistics Christopher Ting QF 603 November 2, /28