TS Module 7 ø weights (filter representation)

(The attached PDF file has better formatting.)

Cryer and Chan use the term ø weights; other statisticians use the term filter representation. The term filter representation in any practice problems means ø weights.

We use ö parameters for autoregressive models and è parameters for moving average models.

           The ö parameters relate future time series values to past time series values.

           The è parameters relate future time series values to past residuals.

Moving average parameters have a finite memory, and autoregressive parameters have an infinite memory.

           For an MA(1) process, a random fluctuation in period T affects the time series value in period T+1 only.

           For an AR(1) process, a random fluctuation in period T affects the time series value in all future periods.

We can convert a ö parameter to an infinite series of è parameters.

Illustration: A φ₁ = 0.500 is equivalent to an infinite series of è parameters

è₁ =  –0.500, è₂ = – 0.250, è₃ = –0.125, … where è_j = –(0.500^j).

One might wonder: Why convert a single parameter to an infinite series?

Answer:  Each è parameter affects one future value. To estimate variances of forecasts, we convert autoregressive parameters into sets of moving average parameters. We call the new model a filter representation and represent the new parameters by ø weights.

Take heed: The ø parameters have the opposite sign of the è parameters: è = 0.450 is ø = –0.450. The model is the same, but the signs of the coefficients are reversed.

y_t = ä + ε_t – θ₁ ε_t-1 is the same as  y_t = ä + ε_t + ø₁ ε_t-1

The general form of a filter representation is y_t – ì = ψ₀ ε_t + ψ₁ ε_t-1 + ψ₂ ε_t-2 + …

           For a moving average model, ì = è₀.

           Cryer and Chan often use time series with a mean of zero.

           For the values of other time series, add the mean.

See Cryer and Chan, chapter 4, page 55, equation 4.1.1.

Both moving average and autoregressive processes have filter representations.

           If the time series has only moving average parameters, ø_j = –è_j.

           If the time series has autoregressive parameters, each ö_j is a series of ø_j’s.

The exercise below emphasizes the intuition. Once you master the intuition, the formulas are easy.

We examine the filter representation for autoregressive models and mixed models.

** Exercise 7.1: AR(1) ø weights (filter representation)

An AR(1) model with φ₁ = 0.6 is converted to a filter representation

y_t – ì = ψ₀ ε_t + ψ₁ ε_t-1 + ψ₂ ε_t-2 + …

A.      What is ø₀?

B.      What is ø₁?

C.      What is ø₂?

D.     What is ø_j?

Part A: ψ₀ is one for all ARIMA models. See Cryer and Chan, chapter 4, page 55.

Part B: If the current error term increases by 1 unit, the current value increases by one unit.  The one period ahead forecast changes by 1 × φ₁ = 1 × 0.6 = 0.6, so ψ₁ = ö₁.

Part C: If the one period ahead forecast changes by 1 × ö₁ = 1 × 0.6 = 0.6, the two periods ahead forecast changes by 0.6 × ö₁ = 0.6², so ψ₂ = ö₁².

Part D: The same reasoning shows that ψ_j = (ö₁)^j.

** Exercise 7.2: ARMA(1,1) ø weights (filter representation)

An ARMA(1,1) model with φ₁ = 0.6, θ₁ = 0.4 is converted to a filter representation y_t – ì = ψ₀ ε_t + ψ₁ ε_t-1 + ψ₂ ε_t-2 + …

Cryer and Chan drop the subscripts from parameters of AR(1), MA(1), and ARMA(1,1) processes, calling them ö and è instead of ö₁ and è₁.

A.      What is ø₀?

B.      What is ø₁?

C.      What is ø₂?

D.     What is ø_j?

Part A: ψ₀ is one for all ARIMA models.

Part B: Suppose the current error term increases by 1 unit.

           The moving average part of the ARMA process changes the forecast by 1 × –è₁ = 1 × –0.4 = –0.4.

           If the current error term increases by one unit, the current value increases by one unit.

           The autoregressive part of the ARMA process changes the forecast by 1 × ö₁ = 1 × 0.6 = 0.6.

The combined change in the forecast is –0.4 + 0.6 = 0.2. The change in the one period ahead forecast is φ₁ – θ₁.

Take heed: The negative sign reflects the convention that moving average parameters are the negative of the moving average coefficients.

Part C: The one period ahead forecast increases 0.2 units (the result in Part B), so the two periods ahead forecast increases 0.2 × φ₁ = 0.2 × 0.6 = 0.12 units.

Part D: Repeating the reasoning above gives ø_j = 0.6^j-1 × 0.2.

** Exercise 7.3: ARMA(2,1) ø weights (filter representation)

An ARMA(2,1) model with φ₁ = 0.6, φ₂ = –0.3, θ₁ = 0.4 is converted to a filter representation y_t – ì = ψ₀ ε_t + ψ₁ ε_t-1 + ψ₂ ε_t-2 + …

A.      What is ø₀?

B.      What is ø₁?

C.      What is ø₂?

D.     What is ø₃?

Part A: ψ₀ is one for all ARIMA models.

Part B: Suppose the current error term increases by 1 unit.

           The moving average part of the ARMA process changes the forecast by 1 × –è₁ = 1 × –0.4 = –0.4.

           If the current error term increases by one unit, the current value increases by one unit.

           The autoregressive part of the ARMA process changes the forecast by 1 × ö₁ = 1 × 0.6 = 0.6.

The combined change in the forecast is –0.4 + 0.6 = 0.2. The change in the one period ahead forecast is φ₁ – θ₁.

Part C: A 1 unit increase in the current error term increases the two periods ahead forecast two ways in this exercise:

           The one period ahead forecast increases 0.2 units (the result in Part A), so the two periods ahead forecast increases 0.2 × ö₁ = 0.2 × 0.6 = 0.12 units.

           The current value increases 1 unit, so the ö₂ parameter causes the two periods ahead forecast to increase –0.3 units.

The change in the two periods ahead forecast is 0.12 – 0.3 = –0.18 units, so ψ₂ = –0.18.

Take heed: The θ₁ parameter does not affect forecasts two or more periods ahead: an MA(1) process has a memory of one period.  In contrast, an AR(1) process has an infinite memory.  The φ₁ parameter affects all future forecasts.

Part D: If the number of periods ahead is greater than the maximum of p and q (2 and 1 in this exercise), the direct effects of the parameters is zero. We compute the combined effects: ψ₃ = φ₁ × ψ₂ + φ₂ × ψ₁ = 0.6 × –0.18 – 0.3 × 0.2 = –0.168.

** Exercise 7.4: AR(2) process ø weights (filter representation)

An AR(2) model y_t – ì = φ₁ (y_t-1 – ì) + φ₂ (y_t-2 – ì) + ε_t has φ₁ = 0.4 and φ₂ = –0.5. We convert this model to an infinite moving average model, or the filter representation

y_t – ì = ε_t + ψ₁ ε_t-1 + ψ₂ ε_t-2 + …

A.      What is ø₁?

B.      What is ø₂?

C.      What is ø₃?

Part A: Suppose the residual in Period T increases one unit. We examine the effect on the value in Period T+1.

           The current value increases 1 unit.

           The ö₁ coefficient causes next period’s value to increase 0.4 units.

Part B: Suppose the residual in Period T increases one unit. We examine the effect on the value in Period T+2.

           The current value increases 1 unit.

           The ö₂ coefficient causes the two periods ahead value to increase –0.5 units.

           The ö₁ coefficient has a two step effect. It causes next period’s value to increase 0.4 units and the value in the following period to increase 0.4 × 0.4 = 0.16 units.

The net change in the two periods ahead value is –0.5 + 0.16 = –0.34.

           The AR(2) formula is: ø₂ = ö₁² + ö₂ = 0.4² – 0.5 = -0.340. 

           The explanation above is the intuition for this formula.

Part C: We use all permutations: φ₁ × φ₁ × φ₁, φ₁ × φ₂, and φ₂ × φ₁ =

0.4³ + 2 × 0.4 × –0.5 = -0.336

For this part of the exercise, the subscript of ø is greater than the order of the ARMA process. Instead of working out all the permutations, we multiply each ö_j coefficient by the ø_k-j coefficient.  We multiply ö₁ by ø₂ and ö₂ by ø₁ = 0.4 × –0.34 + –0.5 × 0.4 = –0.336

Take heed: The formulas are simple permutations.

           Focus on the intuition, not on memorizing formulas.

           The final exam problems can all be solved with first principles.