Class 1: Stationary Time Series Analysis

Transcription

1 Class 1: Stationary Time Series Analysis Macroeconometrics - Fall 2009 Jacek Suda, BdF and PSE February 28, 2011

2 Outline Outline: 1 Covariance-Stationary Processes 2 Wold Decomposition Theorem 3 ARMA Models 4 AR(1) Model 5 Auto-Correlation Function (ACF)

3 Stochastic Process Stochastic process: a collection of random variables {..., Y 1, Y 0, Y 1, Y 2,..., Y T,...} = {Y t }. Observed series {y 1, y 2,..., y T } realizations of a stochastic process. We want a model for {Y t } to explain observed realizations {y t } T 1.

4 Covariance-Stationary Definition {Y t } is covariance-stationary (weak stationary) if (i) E[Y t] = µ t (ii) Cov(Y t, Y t j) = E[(Y t µ)(y t j µ)] = γ j, t, j

5 Covariance-Stationary Definition {Y t } is covariance-stationary (weak stationary) if (i) E[Y t] = µ t (ii) Cov(Y t, Y t j) = E[(Y t µ)(y t j µ)] = γ j, t, j Note: mean is time-invariant

6 Covariance-Stationary Definition {Y t } is covariance-stationary (weak stationary) if (i) E[Y t] = µ t (ii) Cov(Y t, Y t j) = E[(Y t µ)(y t j µ)] = γ j, t, j Note: covariance doesn t depend on t

7 Covariance-Stationary Definition {Y t } is covariance-stationary (weak stationary) if (i) E[Y t] = µ t (ii) Cov(Y t, Y t j) = E[(Y t µ)(y t j µ)] = γ j, t, j Note: Var(Y t ) = γ 0 variance is also constant.

8 Covariance-Stationary Definition {Y t } is covariance-stationary (weak stationary) if (i) E[Y t] = µ t (ii) Cov(Y t, Y t j) = E[(Y t µ)(y t j µ)] = γ j, t, j It is weak stationarity because it only relates to the first two moments. Higher moments can be time-variant.

9 Covariance-Stationary Definition {Y t } is covariance-stationary (weak stationary) if (i) E[Y t] = µ t (ii) Cov(Y t, Y t j) = E[(Y t µ)(y t j µ)] = γ j, t, j It is weak stationarity because it only relates to the first two moments. Higher moments can be time-variant. Examples 1 Y t iid(0, σ 2 ) {Y t } white noise (WN). 2 Y t iidn(0, σ 2 ) Gaussian white noise.

10 Strict (Strong) Stationary Definition {Y t } is (strictly/strongly) stationary if for any values of j 1, j 2,..., j n the joint distribution of (Y t, Y t+j1, Y t+j2,..., Y t+jn ) depends only on the intervals separating the dates (j 1, j 2,..., j n ) and not on date itself (t).

11 Strict (Strong) Stationary Definition {Y t } is (strictly/strongly) stationary if for any values of j 1, j 2,..., j n the joint distribution of (Y t, Y t+j1, Y t+j2,..., Y t+jn ) depends only on the intervals separating the dates (j 1, j 2,..., j n ) and not on date itself (t). For all τ, t 1, t 2,..., t n : F Y (y t1, y t2,..., y tn ) = F Y (y t1+τ, y t2+τ,..., y tn+τ )

12 Strict (Strong) Stationary Definition {Y t } is (strictly/strongly) stationary if for any values of j 1, j 2,..., j n the joint distribution of (Y t, Y t+j1, Y t+j2,..., Y t+jn ) depends only on the intervals separating the dates (j 1, j 2,..., j n ) and not on date itself (t). For all τ, t 1, t 2,..., t n : F Y (y t1, y t2,..., y tn ) = F Y (y t1+τ, y t2+τ,..., y tn+τ ) If a process is strictly stationary with a finite second moment it is also covariance-stationary.

13 Strict (Strong) Stationary Definition {Y t } is (strictly/strongly) stationary if for any values of j 1, j 2,..., j n the joint distribution of (Y t, Y t+j1, Y t+j2,..., Y t+jn ) depends only on the intervals separating the dates (j 1, j 2,..., j n ) and not on date itself (t). For all τ, t 1, t 2,..., t n : F Y (y t1, y t2,..., y tn ) = F Y (y t1+τ, y t2+τ,..., y tn+τ ) If a process is strictly stationary with a finite second moment it is also covariance-stationary. Normality strong stationarity: whole distribution depends on the first two moments.

14 Nonstationary Processes Examples: 1 Y t = β t + ε t, ε t WN

15 Nonstationary Processes Examples: 1 Y t = β t + ε t, ε t WN t - time dummy

16 Nonstationary Processes Examples: 1 Y t = β t + ε t, ε t WN deterministic part

17 Nonstationary Processes Examples: 1 Y t = β t + ε t, ε t WN stochastic component

18 Nonstationary Processes Examples: 1 Y t = β t + ε t, ε t WN E[Y t] = β t depends on t But, X t = Y t βt is covariance stationary.

19 Nonstationary Processes Examples: 1 Y t = β t + ε t, ε t WN E[Y t] = β t depends on t But, X t = Y t βt is covariance stationary. 2 Y t = Y t 1 + ε t, ε t WN, Y 0 constant Random walk

20 Nonstationary Processes Examples: 1 Y t = β t + ε t, ε t WN E[Y t] = β t depends on t But, X t = Y t βt is covariance stationary. 2 Y t = Y t 1 + ε t, ε t WN, Y 0 constant Solving recursively : Y t = t ε j + Y 0. j=1 E[Y t] = Y 0, time-invariant mean. But Var(Y t) = t σ 2 depends on t. X t = Y t Y t 1 is covariance stationary.

21 Wold s Decomposition Theorem Any covariance stationary {Y t } has infinite order, moving-average representation: Y t = ψ j ε t j + κ t, j=0 ψ 0 = 1, ε t WN.

22 Wold s Decomposition Theorem Any covariance stationary {Y t } has infinite order, moving-average representation: Y t = ψ j ε t j + κ t, j=0 ψ 0 = 1, ε t WN. Linear combination of ε s (innovations over time) Weights does not depend on time t, they only depend on j, i.e. how long ago the shock ε occurred.

23 Wold s Decomposition Theorem Any covariance stationary {Y t } has infinite order, moving-average representation: Y t = ψ j ε t j + κ t, j=0 ψ 0 = 1, ε t WN. j=0 ψ2 j <, ε t WN(O, σ 2 ), κ t deterministic term (perfectly forecastable). Example: κ t = µ, constant mean.

24 Wold s Decomposition Theorem - Illustration Let X t = Y t κ t. Then, E[X t ] = E[X 2 t ] = ψ j E[ε t j ] = 0, j=0 ψj 2 E[ε 2 t j] = σ 2 j=0 j=0 ψ 2 j <, as ε t are independent; we have constant finite variance. E[X t X t j ] = E[(ε t + ψ 1 ε t 1 + ψ 2 ε t )(ε t j + ψ 1 ε t j 1 + ψ 2 ε t j 2 = σ 2 (ψ j + ψ j+1 ψ 1 + ψ j+2 ψ ) = σ 2 ψ k ψ k+j, depends on j not t. k=0 So we have a covariance stationary process in mean and variance.

25 ARMA Models Approximate Wold form with finite number of parameters. Wold Form: Y t µ = ψ j ε t j, ε t WN, j=0 ARMA(p,q): Y t µ = φ 1 (Y t 1 µ)+...+φ p (Y t p µ)+ε t +θ 1 ε t θ q ε t q.

26 Lag Operator Define the operator L as LX t X t 1, L 2 X t = L LX t = X t 2. In general, If c is a constant, Also, L k X t = X t k. Lc = c. L 1 X t X t+1, X t = (1 L)X t = X t X t 1.

27 Lag Operator It satisfies and, when φ < 1, L(αX t + βy t ) = αx t 1 + βy t 1 (a L + b L 2 ) X t = = ax t 1 + bx t 2, lim (1 + φl + j φ2 L φ j L j ) = (1 φl) 1

28 Lag Operator It satisfies and, when φ < 1, L(αX t + βy t ) = αx t 1 + βy t 1 (a L + b L 2 ) X t = = ax t 1 + bx t 2, lim (1 + φl + j φ2 L φ j L j ) = (1 φl) 1

29 ARMA Models in Lag Notaion ARMA(p, q): Y t µ = φ 1 (Y t 1 µ)+...+φ p (Y t p µ)+ε t +θ 1 ε t θ q ε t q. With lag operator: φ(l)(y t µ) = θ(l)ε t, where φ(l) = 1 φ 1 L φ 2 L 2... φ p L p, θ(l) = 1 + θ 1 L + θ 2 L θ q L q.

30 Stochastic Difference Equation (SDE) Representation Let X t = Y t µ and w t = θ(l)ε t. Then φ(l)x t = w t, or X t = φ 1 X t φ p X t p + w t, is a pth-order stochastic difference equation.

31 SDE Representation - AR(1) Example Example: First-order SDE (AR(1)): X t = φx t 1 + ε t, ε t WN Solve for Wold Form (recursive substitution) X t = φ t+1 X 1 + φ t ε 0 + φ t 1 ε φε t 1 + ε t = φ t+1 X 1 + ψ iε t i. i=0 where X 1 is an initial condition and ψ i = φ i. We approximated Wold form with 1 parameter form for AR(1).

32 Dynamic Multiplier The dynamic multiplier measurers the effect of ε t on subsequent values of X τ : X t+j ε t For the X t being AR(1) process = X j ε 0 = ψ j. (1) X t+j ε t = ψ j = φ j. (2) The dynamic multiplier for any linear difference equations depends only on the length of time j, not on time t.

33 Impulse Response Function The impulse-response function is a sequence of dynamic multipliers as a function of time from the one time impulse on ε t

34 Cumulative impact Permanent increase in ε at time t, i.e. ε t = 1, ε t+1 = 1, ε t+2 = 1,... X t+j ε t + X t+j ε t+1 + X t+j ε t X t+j ε t+j = ψ j + ψ j ψ + 1 In the limit, as j lim j [ Xt+j ε t + X t+j X ] t+j = ε t+1 ε t+j ψ j = ψ(1), j=0 where ψ(1) = ψ(l = 1) = 1 + ψ 1 + ψ

35 AR(1) Model Recall X t = φx t 1 + ε t, ε t WN, = φ j ε t j = ψ j ε t j. Wold coefficients j=0 j=0 ψ j = φpsi j 1, ψ j = Y t+j ε t If φ < 1 X t is stationary solution to first-order SDE. If φ = 1 then ψ j = 1 j and X t = X 1 + t j=0 εj is neither stationary nor stable solution, and ψ(1) is infinite.

36 AR(1) lag notation AR(1): X t = φx t 1 + ε t, ε t WN, (1 φl)x t = ε t Multiply both sides by (1 φl) 1 : X t = (1 φl) 1 ε t = (1 + φl + φ 2 L 2 + φ 3 + L )ε t = φ j ε t j = ψ j ε t j, j=0 j=0 ψ(l) = (1 φl) 1.

37 AR(1): Long-run effects For AR(1), if φ < 1 the permanent increase in ε t equals X t+j ε t and as j + Xt+j ε t Xt+j ε t+j = 1 + φ + φ 2 + φ φ j. ψ(1) = 1 + φ + φ = 1/(1 φ) the cumulative consequences for X of a one-time change in ε, j=0 X t+j ε t = 1/(1 φ)

38 AR(1) Mean Intercept representation for Y t X t + µ Y t = c + φy t 1 + ε t, where c = µ(1 φ). Mean E[Y t] = c + φe[y t 1] + E[ε t], Since we have covariance stationary process, E[Y t] = E[Y t 1] and E[Y t] = c 1 φ µ.

39 AR(1) Variance Var(Y t ) = E[(Y t µ) 2 ] = E[(φ(Y t 1 µ) + ε t ) 2 ] = φ 2 E[(Y t 1 µ) 2 ] + 2φE[(Y t 1 µ)ε t ] + E[ε 2 t ]. Since Y t is covariance stationary and ε t is independently distributed, Var(Y t ) = Var(Y t 1 ) and, so Var(Y t ) = E[Y t 1 ε t ] = 0] σ2 1 φ = γ 0.

40 AR(1): Covariance Cov(Y t, Y t j ) = E[(Y t µ)(y t j µ)] = φe[(y t 1 µ)(y t j µ)] + E[ε t (Y t j µ)], Cov(Y t, Y t j ) γ j = φγ j 1.

41 AR(1): Auto-correlation Function (ACF) Define For AR(1), ρ j = φρ j 1. ρ j γ j γ 0 j th autocorrelation corr(y t, Y t j ) Figure For AR(1) ACF and IRF are the same. In general it not true. ACF < 1, 1 >. If φ < 0: Figure