1 Class Organization. 2 Introduction

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1 Time Series Analysis, Lecture 1, Class Organization Course Description Prerequisite Homework and Grading Readings and Lecture Notes Course Website: wechat group 2 Introduction 1. What are time series? Sequence of random variables { x1, x 2,...x T,... }, {x t } T t=1, {x t} + t=

2 Time Series Analysis, Lecture 1, Or a mapping, x(t, ω) : ([0, + ]; (Ω, F, P)) R, (Ω, F, P) is the proabability space. For fixed t, x t (ω) is a random variable, or ensemble series For a sequence of realization of {ω t }, x(t, ω t ) is a time series 2. Distribution, stationary and ergodicity Ideally, we want to estimate joint distribution of (x 1, x 2,...x T,...), but we don t have enough data Solution: (a) Nonparametric: use histograms (for high-frequency data) (b) Impose some structures, use parametric models for joint distribution, e.g. linear ARMA (c) In order to use historic data to predict future ensemble series, we need two assumptions: i. {x t } is stationary, weak (covariance) stationary or strong (distribution) stationary, - A process {x t } is said to be covariance stationary if both mean and variancecovariance matrix Σ t are constant in t Σ t+j,t depends on the time gap j but not on time t. - A process {x t } is said to be distribution stationary if the distributions of x t are constant in t

3 Time Series Analysis, Lecture 1, ii. Ergodicity: - A covariance-stationary process {x t } is said to be ergodic for the mean if 1 T T t=1 x t P E ω (x t ) - {x t } is ergodic for the second moment if 1 T j T t=j+1 (x t µ)(x t j µ) P γ j E ω [(x t µ)(x t j µ)] for all j = 1, 2,...T 1 - Suffi cient condition for ergodic for the mean: the autocovariances {γ j } j=0 satisfy j=0 γ j <.If {x t } is a stationary Gaussian process, j=0 γ j < is suffi cient to ensure ergodicity for all moments 3. Example: How should we interpret the distributions defined by the following? y t+1 = φ 1 y t + φ 2 y t 1 + φ 3 y t 2 + φ 4 y t 3 + σw t+1, w t+1 N(0, I), φ j < 1

4 Time Series Analysis, Lecture 1, which can be rewrite as a linear state-space model: x t+1 = Ax t + Cw t+1 y t = Gx t x 0 N(µ 0, Σ 0 ) x t = [ y t y t 1 y t 2 y t 3 ] G = [ ] A = φ 1 φ 2 φ 3 φ , C = σ Let µ t = E(x t ) unconditional mean of x t then µ t+1 = Aµ t E(y t ) = Gµ t

5 Time Series Analysis, Lecture 1, Figure 1: 20 Simulated Path of {y t } T t=1 and histogram of y T, Let Σ t = E[(x t µ t )(x t µ t ) ] unconditional variance-convariance matrix of x t then Σ t+1 = AΣ t A + CC with Σ 0 given var(y t ) = GΣ t G Intuitively, the probabilities in a distribution correspond to relative frequencies in a large population drawn from that distribution. Let s apply this idea to our setting, focusing on the distribution of y T for fixed T,using an independent set of shocks each time, 20 time series

6 Time Series Analysis, Lecture 1, Figure 2: 100 Simulated Path of {y t } T t=1 and histogram of y T, Ensemble means: the ensemble or cross-sectional average mean approximate the ensemble mean ȳ T = 1 N ΣN i=1 yi T E(y T ) = Gµ T

7 Time Series Analysis, Lecture 1, Figure 3: Historgram of 500,000 Simulated y T

8 Time Series Analysis, Lecture 1, Figure 4: Simulation comparing the ensemble averages and population means at time points t = 0,..., 50 (N = 20)

9 Time Series Analysis, Lecture 1, Figure 5: Stationary: Visualizing Stability, cross-sectional distributions for y at times T, T, T

10 Time Series Analysis, Lecture 1, Figure 6: Start (y 0, y 1, y 2, y 3 ) at the stationary distribution, now the differences in the observed distributions at T, T and T come entirely from random fluctuations due to the finite sample size

11 Time Series Analysis, Lecture 1, Linear ARMA Models 3.1 White Noise and Martingale Difference Sequences 1. White Noise: building blocks of any process, no predictability, no autocorrelation, homoscedasticity E(ε t ) = E(ε t ε t 1, ε t 2,...) = E(ε t F t 1 ) = 0 (1) E(ε t ε t j ) = cov(ε t, ε t j ) = 0, for all j (2) var(ε t ) = var(ε t ε t 1, ε t 2,...) = var(ε t F t 1 ) = σ 2 (3) If in addition, ε t N(0, σ 2 ε) then we have Gaussian white noise process. 2. Martingale Difference Sequence: no predictability A sequence of random scalars {ε t } t=1 is a martingale difference with respect to {F t } if E(ε t F t 1 ) = 0 and E(ε t ) = 0 for all t, where F t denote information available at date t, which includes current and lagged values of ε. For example: where ν t is a second random variable. F t = {ε t, ε t 1,..., ε 1, ν t,...ν 1 },

12 Time Series Analysis, Lecture 1, Where no information set is specified, F t is presumed to consist solely of current and lagged values ε : F t = {ε t, ε t 1,..., ε 1 } then we will say simply that {ε t } is a martingale difference sequence (with respect to itself) A sequence of random vectors {Y t } t=1 satisfying E(Y t F t 1 ) = 0 and E(Y t ) = 0 for all t is said to form a vector martingale difference sequence MDSs while serially uncorrelated are not necessarily independent. CLT holds for mds What do we gain by relaxing the independence assumption in favor of the no serial correlation? It will allow our models to have autoregressive, conditionally heteroskedastic (ARCH) disturbances. Digression: Why is this useful? Many time series, especially high-frequency financial time series, seem to be serially uncorrelated in their levels but serially correlated in their squared levels E [x t x t 1, x t 2,...] = µ E [ x 2 t x t 1, x t 2,... ] = σ 2 f(x t 1, x t 2,...)

13 Time Series Analysis, Lecture 1, The past doesn t help us predict whether today s x will be above or below normal but it may help us predict whether today s x will be unusually large or small in an absolute sense. Example: P t = P t 1 + v t, where E [v t P t 1, P t 2,...] = 0 so v t is a m.d.s, serially uncorrelated. However, it is easy to show that v t = ε t α 0 + α 1 v 2 t 1, α 0 > 0 and 0 α 1 < 1 E [v t v t 1, v t 2,...] = 0 E [ v 2 t v t 1, v t 2,... ] = α 0 + α 1 v 2 t 1 hence v t is an m.d.s, but v 2 t is serially correlated. In fact, we show that v t is stationary and ergodic mds with E(v t ) = 0 and E(v 2 t ) = α o/(1 α 1 )

14 Time Series Analysis, Lecture 1, Basic (Stationary) ARMA Process 1. AR(p): x t = φ 1 x t 1 + φ 2 x t φ p x t p + ε t In particular: AR(1) x t = φ 1 x t 1 + ε t 2. MA(q): x t = ε t + θ 1 ε t 1 + θ 2 ε t θ q ε t q In particular: MA(1) x t = ε t + θ 1 ε t 1 3. ARMA(p,q): x t = φ 1 x t 1 + φ 2 x t φ p x t p + ε t + θ 1 ε t 1 + θ 2 ε t θ q ε t q 4. Demean and deterministic trends 5. Estimation: Is linear regression valid?

15 Time Series Analysis, Lecture 1, Lag Operators and Polynomials 4.1 Definition Lx t = x t 1 L j x t = x t j L j x t = x t+j Lag polynomials: a(l) = a 0 + a 1 L a j L j +... Express ARMA using lag polynomials: 1. AR(p): a(l)x t = ε t, a(l) = 1 φ 1 L... φ p L p 2. MA(q): x t = b(l)ε t, b(l) = 1 + θ 1 L + θ 2 L θ q L q 3. ARMA(p,q): a(l)x t = b(l)ε t

16 Time Series Analysis, Lecture 1, Conversion of AR(p) and MA( ) 1. AR(1): a(l)x t = ε t, a(l) = 1 φl, when φ < 1 = φl < 1, a(l)x t is invertible, and x t = a(l) 1 ε t = (1 + φl + φ 2 L )ε t which is a MA( ). 2. invertible of AR(p): factorize the polynomial 3. ARMA(p,q): a(l)x t = b(l)ε t x t = d(l)ε t where d(l) = a(l) 1 b(l) or b(l) = a(l)d(l), hence we can solve d(l) recursively by matching the coeffi cients 4.3 Multivariate ARMA models X t = [ x1t x 2t ], ε t N(0, Σ)

17 Time Series Analysis, Lecture 1, ARMA(p,q): A(L)X t = B(L)ε t X t = A(L) 1 B(L)ε t In particular, VAR(1): (I ΦL)X t = ε t = X t = (I ΦL) 1 ε t = j=0 Φ j ε t j Question: Why MA? Why AR? 5 Homework 1 Read Linear State Space Models of do Exercises 1-4