ARMA models with time-varying coefficients. Periodic case.

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1 ARMA models with time-varying coefficients. Periodic case. Agnieszka Wy lomańska Hugo Steinhaus Center Wroc law University of Technology

2 ARMA models with time-varying coefficients. Periodic case. 1 Some time series models used in the analysis of nonstationary processes ARIMA models If d is a nonnegative integer, then {X t } is an ARIMA(p,d,q) process if Y t = (1 B) d X t is an ARMA(p,q) process, where BX t = X t 1. It means that {X t } satisfies a difference equation of the form φ (B)X t = φ(b)(1 B) d X t = θ(b)z t, {Z t } W N(0, σ 2 ). The models are useful for representing data with trend as like without trend. Estimation φ, θ and σ 2 is based on the observed differences (1 B) d X t.

3 ARMA models with time-varying coefficients. Periodic case. 2 Some time series models used in the analysis of nonstationary processes Example 1 Let consider an ARIMA(1,1,0) process for some φ ( 1, 1): (1 φb)(1 B)X t = Z t {Z t } W N(0, σ 2 ). A realization of {X 1, X 2,... X 500 } with X 0 = 0, φ = 0.3 and σ 2 = 1, the autocorrelation and partial autocorrelation functions are shown in the next figures.

4 ARMA models with time-varying coefficients. Periodic case. 3 2 ARIMA(1,1,0) model time

5 ARMA models with time-varying coefficients. Periodic case. 4

6 ARMA models with time-varying coefficients. Periodic case. 5 Some time series models used in the analysis of nonstationary processes SARIMA models If d and D are nonnegative integers, then {X t } is a seasonal ARIMA(p,d,q) (P,D,Q) process with period s if the differenced series Y t = (1 B) d (1 B s ) D X t is an ARMA(p,q) process defined by: φ(b)φ(b s )Y t = θ(b)θ(b s )Z t, {Z t } W N(0, σ 2 ). The models are often used to the analysis of the seasonal data. Seasonal ARIMA models allow for randomness in the seasonal pattern from one cycle to the next.

7 ARMA models with time-varying coefficients. Periodic case. 6 Some time series models used in the analysis of nonstationary processes Example 2 Let consider a SARIMA(1,1,0) (0,1,0) process with season s = 4 given by the equation: (1 φb)(1 B s )X t = Z t {Z t } W N(0, σ 2 ). A realization of {X 1, X 2,... X 500 } with X 0 = 0, φ = 0.8 and σ 2 = 1 and the autocorrelation function are shown in the next figures.

8 ARMA models with time-varying coefficients. Periodic case. 7

9 ARMA models with time-varying coefficients. Periodic case. 8

10 ARMA models with time-varying coefficients. Periodic case. 9 Why ARMA models with periodic coefficients are so important? As many authors observed, processes with periodic correlation should not be modeled with the widely used econometrics seasonal autoregressive moving-average class. This is because SARMA models, contrary to their name, are actually stationary models with large (in absolute value) autocovariance at lags that are multipliers of the period. A flexible class of models that have the desirable properties is the class of PARMA models. Analogous to ARMA models and short memory stationary series, PARMA models are fundamental periodic and periodically correlated time series models.

11 ARMA models with time-varying coefficients. Periodic case. 10 ARMA models fixed coefficients The sequence {X t } is an ARMA(p,q) process if {X t } is stationary and if for every t X t p φ j X t j = Z t + j=1 q θ i Z t i {Z t } W N(0, σ 2 ) i=1 and the polynomials (1 φ 1 z... φ p z p ) and (1 + θ 1 z +... θ q z q ) have no common factors. Theorem 1 (Brockwell, Davis, 1996) An unique stationary solution of ARMA(p,q) system exists iff for all z = 1. φ(z) = 1 φ 1 z... φ p z p 0

12 ARMA models with time-varying coefficients. Periodic case. 11 ARMA models fixed coefficients Example 3 Let consider an AR(1) model given by the equation: X t bx t 1 = Z t {Z t } W N(0, 1). If b < 1, then the the solution of such system is unique and stationary. Moreover it is causal and has the form: X t = b j Z t j j=0 The covariance function for b R and k 0 is given by R X (t, t + k) = b k j=0 b 2j = bk 1 b 2.

13 ARMA models with time-varying coefficients. Periodic case Autcovariance function for AR(1) lag

14 ARMA models with time-varying coefficients. Periodic case. 13 ARMA models time-varying coefficients The model ARMA(p,q) with time-varying coefficients is defined by the following difference equation X t p b j (t)x t j = j=1 q 1 i=0 a i (t)z t i {Z t } W N(0, σ 2 (t)). (1) The processes are useful especially when the sequences (b j (t)), j = 1, 2,... p, (a i (t)), i = 0, 1,..., q 1 and (σ 2 (t)) are periodic in t with the same period T. The periodic ARMA models have found wide applications in many fields such as climatology, hydrology, economics, electrical engineering.

15 ARMA models with time-varying coefficients. Periodic case. 14 ARMA models time-varying coefficients Example 4 Let consider a PAR(1) model with period T = 2: X t b(t)x t 1 = Z t {Z t } W N(0, 1), where b(1) = 0.3, b(2) = 0.6. A realization of {X 1, X 2,... X 500 } and the autocovariance function are shown in the figures.

16 ARMA models with time-varying coefficients. Periodic case PAR(1) model time

17 ARMA models with time-varying coefficients. Periodic case. 16 Periodically correlated processes Conventional time series analysis is heavily dependent on the assumption of stationarity. This assumption is unsatisfactory for many physical processes of interest. Periodically correlated (PC) processes offer an alternative. They are non-stationary but possess many of the properties of stationary processes. Hence, the numerous attempts to apply PC processes in various areas of science and technology.

18 ARMA models with time-varying coefficients. Periodic case. 17 Periodically correlated processes PC processes exhibit a periodic rhytm that is generally much more complicated than periodicity in the mean (which is a manifestation of the classical notion of periodocity). This is due to the fact that for a stochastic sequence to be periodically correlated its autocovariance function R X (m, m + k) = E[(X m µ X (m))(x m+k µ X (m + k))] has to be periodic in m with period T and for all integers m and k. µ X (m) = E(X m ) = µ X (m + T )

19 ARMA models with time-varying coefficients. Periodic case. 18 Simple models for PC sequences. Connection between PC and stationary processes. If {X t } L 2 (Ω, F, P) is a T -periodic sequence ( X t X t+t L2 = 0), then {X t } is PC with period T. If {X t }, t N is wide sense stationary with µ X (t) 0 and f(t) is a scalar periodic sequence (f(t) = f(t + T )), then Y t = f(t) + X t is PC with period T.

20 ARMA models with time-varying coefficients. Periodic case. 19 Simple models for PC sequences. Connection between PC and stationary processes. If {X t }, t N is a PC sequence with period T and Θ is a random variable independent of {X t } uniformly distributed over the interval [0, T 1], then Y t = X t+θ is wide sense stationary. If {X t }, t N is wide sense stationary with µ X (t) 0 and f(t) is a scalar periodic sequence (f(t) = f(t + T )), then Y t = f(t)x t is PC with period T.

21 ARMA models with time-varying coefficients. Periodic case. 20 The connection between PC processes and ARMA models with time-varying coefficients. Theorem 2 (Makagon, Weron, Wy lomańska, 2004) The sequence {X t }, t N is an unique bounded PC solutin of system ARMA(1,q) with time-varying coefficients defined by X t b(t)x t 1 = q 1 j=0 a j (t)z t j, {Z t } W N(0, 1) iff the sequences {b(t)} and {a j (t)}, j = 0, 2,..., q 1 are periodic with the same period T and P = b(1)b(2)... b(t ) 1. Moreover if P < 1, then the solution is causal and if P > 1, then it is invertible with respect to the innovations.

22 ARMA models with time-varying coefficients. Periodic case. 21 The connection between PC processes and ARMA models with time-varying coefficients. Example 5 Let consider the system PARMA(1,1) with period T = 2 given by the equation X t b(t)x t 1 = a(t)z t j, {Z t } W N(0, 1), If P < 1, then the solution has the form X t = P s/2 b(t) smod(2) a(t s)z t s. s=0 The covariance functon for k > 0 and b(t), a(t) R is given by the following equation R X (t, t + k) = P 2 s/2 + k/2 b(t) smod(2) b(t s) (k+s)mod(2) a 2 (t s). s=0

23 ARMA models with time-varying coefficients. Periodic case. 22 The estimation of PARMA coefficients. There are two methods uded in the estimation of PARMA coefficients: Yule-Walker estimation method Least squares estimation method. The sequence modelled by PARMA(p,q) system can be described by PAR(p) sequence with higher p coefficients, therefore we concentrate on periodic autoregressive processes. It is connected with the simpler formulas for estimators of such sequences.

24 ARMA models with time-varying coefficients. Periodic case. 23 The estimation of PARMA coefficients Yule-Walker method. The estimator of periodic mean (T -the period, NT -number of the data): ˆ µ(v) = 1 N N 1 n=0 X nt +v v = 1, 2,..., T. Eliminate the period mean Y nt +v = X nt +v The Γ v (v = 1, 2,... T ) p p matrices ˆ µ(v) n = 0, 1,... N 1, v = 1, 2,... T. (Γ v ) i,j = R Y (v i, v j).

25 ARMA models with time-varying coefficients. Periodic case. 24 The R v (v = 1, 2,..., T ) vectors R v = [R Y (nt + v, nt + v 1),..., R Y (nt + v, nt + v p)]. The estimators of R Y (v, v l) ˆ R Y (v, v l) = 1 N N 1 n=0 Y nt +v Y nt +v l. The estimators of B v = [b 1 (v), b 2 (v),..., b p (v)] for v = 1, 2,... T Γ v B v = R v. The estimators for σ 2 (v) (v = 1, 2,... T ) σ 2 (v) = R Y (v, v) B vr v.

26 ARMA models with time-varying coefficients. Periodic case. 25 The estimation of PARMA coefficients Least squares method. a = [b 1 (1),...b p (1),...b 1 (T ),...b p (T )] The estimator of a minimalizes S( a) S( a) = N 1 n=0 T σ 2 (v)(znt +v( a)) 2. v=1 The ZnT +v ( a) obatined from Z nt +v( a) = Y nt +v p b i (v)y nt +v i. i=1

27 ARMA models with time-varying coefficients. Periodic case. 26 a is a solution of the pt -dimensional equation N 1 n=0 v=1 T σ 2 (v)z nt +v( a) δz nt +v ( a) δ a = 0 The form of δz nt +v ( a) δ a is known. The estimator of σ 2 (v) (v = 1, 2,... T ) σ 2 (v) = 1 N N 1 n=0 Z nt +v(ˆ a). N 1 n=0 v=1 T Z nt +v( a) δz nt +v ( a) δ a = 0. To find the p parameter we use the FPE, AIC and BIC selection criterions.

28 ARMA models with time-varying coefficients. Periodic case. 27 The estimation of PARMA coefficients. Example 6 Let consider PAR(2) model with period T = 2 with the coefficients b 1 (1) = 0.6, b 1 (2) = 0.2, b 2 (1) = 0.4, b 2 (2) = 0.3, σ 2 (1) = 0.16, σ 2 (2) = For the Yule-Walker and least squares estimation methods we obtained the results:

29 ARMA models with time-varying coefficients. Periodic case. 28 p b i (1) b i (2) σ 2 (1) σ 2 (2) BIC AIC FPE Yule-Walker estimaton method.

30 ARMA models with time-varying coefficients. Periodic case. 29 p b i (1) b i (2) σ 2 (1) σ 2 (2) BIC AIC FPE Least squares estimation method.

31 ARMA models with time-varying coefficients. Periodic case. 30 The estimation of PARMA coefficients. Example 7 We analize the real volume power trading data (in MWh). The hour data are from to We choose the data from one day in a week, because of two kind of seasonality. We assume T = 24.

32 ARMA models with time-varying coefficients. Periodic case x 104 The real data and the trend The real data The trend volume time Detrended data volume time

33 ARMA models with time-varying coefficients. Periodic case. 32 p BIC AIC FPE

34 ARMA models with time-varying coefficients. Periodic case x 104 Prediction of volume trading hour data 1.3 Prediction The real data One step prediction error

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