ARMA (and ARIMA) models are often expressed in backshift notation.

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1 Backshift Notation ARMA (and ARIMA) models are often expressed in backshift notation. B is the backshift operator (also called the lag operator ). It operates on time series, and means back up by one time unit. Examples: Bz t = z t 1 Bz t 2 = z t 3 Ba t = a t 1 Ba t 1 = a t 2 B k means backshift k times. Examples B k z t = z t k B k a t 2 = a t 2 k

2 Backshift Algebra As an operation on time series, the backshift operator B obeys certain algebraic rules. Let {X t } and {Y t } be time series, and b, c, d be constants. B (X t + Y t ) = BX t + BY t (= X t 1 + Y t 1 ) B c = c ( ) B cx t = cbx t (= cx t 1 ) B j B k X t = B j+k X t (= X t j k ) To understand rule ( ), think of the constant c as standing for the constant time series..., c, c, c,.... When you backshift this series, nothing changes. All these rules follow from the definition: BX t = X t 1.

3 These rules can be used in combination. Here are some examples: B(bX t + cy t + d) = bx t 1 + cy t 1 + d B(C + φ 1 z t 1 + φ 2 z t 2 + a t θ 1 a t 1 ) = C + φ 1 z t 2 + φ 2 z t 3 + a t 1 θ 1 a t 2

4 Backshift Polynomials A backshift polynomial is a polynomial involving powers of B. Examples: 1 + B 2 + 3B + 4B 2 b + cb + db 2 bb + cb 3 + db 5 A backshift polynomial represents an operation performed on time series. Examples: (1 + B)Z t = Z t + BZ t = Z t + Z t 1 (2 + 3B + 4B 2 )Z t = 2Z t + 3BZ t + 4B 2 Z t = 2Z t + 3Z t 1 + 4Z t 2 (b + cb + db 2 )Z t = bz t + cz t 1 + dz t 2 (bb + cb 3 + db 5 )Z t = bz t 1 + cz t 3 + dz t 5

5 Manipulation of Backshift Polynomials Backshift polynomials may be treated as both polynomials and as operations on time series. That is, B may be viewed as a number or as the backshift operation (Bz t = z t 1 ). Backshift polynomials obey the usual rules of algebra. You may add them and multiply them, etc.

6 ARMA Processes Expressed in Backshift Notation Example: We can write an ARMA(2,3) in backshift form as follows. so that z t = C + φ 1 z t 1 + φ 2 z t 2 + a t θ 1 a t 1 θ 2 a t 2 θ 3 a t 3 z t φ 1 z t 1 φ 2 z t 2 = C + a t θ 1 a t 1 θ 2 a t 2 θ 3 a t 3 z t φ 1 Bz t φ 2 B 2 z t = C + a t θ 1 Ba t θ 2 B 2 a t θ 3 B 3 a t (1 φ 1 B φ 2 B 2 )z t = C + (1 θ 1 B θ 2 B 2 θ 3 B 3 )a t which we abbreviate as φ(b)z t = C + θ(b)a t where we define φ(b) = 1 φ 1 B φ 2 B 2 and θ(b) = 1 θ 1 B θ 2 B 2 θ 3 B 3

7 Summary of Backshift Notation and its Uses Definition: B k z t = z t k B k a t = a t k ARMA(p, q) process in backshift notation: (1 φ 1 B φ 2 B 2 φ p B p )z t = C +(1 θ 1 B θ 2 B 2 θ q B q )a t or in mean-centered form (which eliminates C): (1 φ 1 B φ 2 B 2 φ p B p ) z t = (1 θ 1 B θ 2 B 2 θ q B q )a t If we define polynomials: φ(b) =1 φ 1 B φ 2 B 2 φ p B p θ(b) =1 θ 1 B θ 2 B 2 θ q B q we may write this more briefly as φ(b)z t = θ(b)a t.

8 Stationarity An ARMA(p, q) process is stationary if and only if all the solutions of φ(b) = 0 lie strictly outside the unit circle in the complex plane. Notes: The solutions of φ(b) = 0 are called the zeros or roots of φ(b). The unit circle is the circle of radius 1 about (0, 0). The modulus of a complex number z = x + yi is z = x 2 + y 2, the distance of (x, y) from the origin. A point z = x + yi lies strictly outside the unit circle if z > 1. For real numbers, modulus is the same as absolute value.

9 Checking Stationarity AR(1) Processes: In this case, the AR polynomial has a single zero: φ(b) = 1 φ 1 B = 0 iff B = 1 φ 1 This lies strictly outside the unit circle when 1 = 1 φ 1 > 1 or φ 1 < 1 φ 1 which is the stationarity condition stated earlier.

10 AR(2) Processes: The AR(2) polynomial has two roots which are found by solving the quadratic equation: φ(b) = 1 φ 1 B φ 2 B 2 = 0. From the known expressions for the roots, it is possible (but not so easy) to verify that the roots are outside the unit circle iff φ 2 < 1, φ 2 + φ 1 < 1, φ 2 φ 1 < 1. AR(p) Processes: For p 3, the roots of φ(b) = 0 can be found numerically. Note: If φ 1 + φ φ p 1, the process is always non-stationary. (Why?)

11 Stationary ARMA Processes Written in MA( ) Form A stationary ARMA(p, q) process may be written in MA( ) form: z t = a t + ψ 1 a t 1 + ψ 2 a t 2 + ψ 3 a t 3 + = k=0 ψ ka t k (with ψ 0 = 1) where ψ k 0 as k and k ψ k <. The values ψ j (eventually) decay to zero in an exponential fashion, possibly with sinusoidal oscillation. The coefficients {ψ k } are called the ψ-weights. In backshift notation, we may write ( ) z t = ψ k a t k = ψ k B k a t = ψ(b)a t k=0 k=0 where we define ψ(b) = k=0 ψ kb k.

12 Derivation : φ(b) z t = θ(b)a t z t = θ(b) φ(b) a t = ψ(b)a t where we define ψ(b) = θ(b) φ(b). The ψ-weights {ψ k } may be found be expanding ψ(b) = θ(b)/φ(b). This may be done in various ways (e.g., Taylor series or recursions). We can always expand ψ(b) = θ(b)/φ(b) as an infinite series so that even non-stationary ARMA processes can be written in MA( ) form k ψ ka t k. However, for non-stationary processes the coefficients ψ k no longer converge to zero as k, and it is not clear if the infinite sum has any mathematical meaning. But the ψ-weights {ψ k } obtained by expanding ψ(b) = θ(b)/φ(b) are still useful in computing forecast variances.

13 Useful Fact: A stationary ARMA(p,q) process with mean zero can be expressed as θ(b) φ(b) a t