B) Local trend approximation 4 страница

We may use (4.5) to develop the recursive relationships for this scheme and we arrive at:

_{xt = f ( xt− 1) − g ( xt− 1)} ^w (^xt− 1) ₊ ^g (^xt− 1) _yt r (x_{t− 1}) r (x_{t− 1})

= [ F x_{t− 1} − g (x_{t− 1})] + ^{g ( xt− 1)} y_t. r (x_{t− 1})

If we substitute the specific functions derived above, we arrive at:

__t

b_t

s_t

^xt = s_{t− 1}

_...

			(1 − α)(_t− 1 + bt− 1)		α / st−m
bt− 1 − β (_t− 1+ bt− 1)	β / st−m
	=		(1 −s	γ) st−m		+		γ /(_t− 1+ bt− 1)	yt. (4.6)
				t− 1
				.
				.
				.

^st−m +1

Further, we may use the first equation in (4.6) to substitute for y_t in the expression for b_t. Thus, the final recursive relationships, in agreement with Table 2.1, are:

t−m) ε t,

t−m) ε t,

4.3 Nonlinear Seasonal Models

__t = (1 − α)(__{t− 1} + b_{t− 1}) + αy_t / s_t−m, b_t = β^∗ (__t − __{t− 1}) + (1 − β^∗) b_{t− 1},

s_t = (1 − γ) s_t−m + γy_t /(__{t− 1}+ b_{t− 1}).

The h -step-ahead forecasting function is:

^{y ˆ} t + h− 1 |t− 1= (^_t− 1+ ^hbt− 1) ^st−m− 1+ h^{∗ ,}

where h^∗ = h mod m. This set of forecasting relationships is known as the multiplicative Holt-Winters’ system for seasonal exponential smoothing; see the discussion in Sect. 1.3. Users of this method usually recommend the parameter ranges 0 < α, β, γ < 1, but the exact specification of an accept-able parameter space is extremely difficult. We defer further discussion until Chap. 8.

4.3.2 A Multiplicative Seasonal Model with Additive Errors: ETS(A,A,M)

The key feature of the seasonal model that we have just developed is the multiplicative nature of the interaction between the trend and seasonal components. Although we used a multiplicative error structure and would suggest that this assumption is more likely to be satisfied in practice, we could also develop the multiplicative seasonal structure with additive errors. The underlying model becomes:

yt = (_t− 1+ bt− 1) st−m + ε t,

_t = _t− 1+ bt− 1+ (α / s

b_t = b_{t− 1}+ (β / s

s_t = s_t−m + (γ /(__{t− 1}+ b_{t− 1})) ε _t.

Following the same reasoning as before, we arrive back at the recursive rela-tionships given in (4.6). There is an element of reverse engineering in this model; that is, we worked back from the recursive relationships to determine the form of the model. Making adjustments to the smoothing parameters in the manner indicated does not seem very plausible, yet logic dictates this is an implicit assumption if the ETS(A,A,M) scheme is to be used. Perhaps a better way to look at this conclusion is to recognize that one of the ben-efits of formulating an underlying model is that the process forces us to make our assumptions explicit. In this case, intuition guides us towards the ETS(M,A,M) model. As a practical matter, we recommend consideration of both schemes, so that resulting prediction intervals are consistent with the historical patterns in the series.

Similar adjustments enable us to consider the ETS(A,M,M) and ETS(M,M,M) schemes, and to arrive at the recursive relationships given in Table 2.1. These developments are left as end-of-chapter exercises.

64 4 Nonlinear and Heteroscedastic Innovations State Space Models

4.4 Variations on the Common Models

As might be expected, the number of usable models can be expanded con-siderably. In particular, we may incorporate damping factors or set specific parameters to special values. We first consider models without a seasonal component and then examine special cases of the model associated with the Holt-Winters method.

4.4.1 Local Level Model with Drift

If the growth rate is steady over time, we may simplify the local trend model by setting β = 0. This modification is often effective when the forecasting horizon is fairly short and growth is positive:

y_t = (__{t− 1}+ b)(1+ ε _t), __t = (__{t− 1} + b)(1 + αε _t).

The principal difference between this model and the additive error version is that the prediction intervals of this model gradually widen as the mean level increases.

4.4.2 Damped Trend Model: ETS(M,A_d,N)

The damped trend model has the reduced growth rate φb_{t− 1} at time t, where 0 ≤ φ < 1. The level tends to flatten out as time increases, and this feature can be useful for series whose trends are unlikely to be sustained over time. In particular, when the variable of interest is non-negative, a negative trend clearly has to flatten out sooner or later.

y_t = (__{t− 1}+ φb_{t− 1})(1+ ε _t), __t = (__{t− 1} + φb_{t− 1})(1 + αε _t),

bt = φbt− 1+ β (_t− 1+ φbt− 1) ε t.

The damped model may be combined with the previous scheme to produce a smoother trajectory towards a limiting expected value.

4.4.3 Local Multiplicative Trend with Damping: ETS(A,M_d,N)

If we try to introduce similar damping coefficients into multiplicative mod-els, the models are not well-behaved. In particular, such damping forces the expected value towards a limiting value of zero. To avoid this difficulty, we follow Taylor (2003a) and raise the growth rate to a fractional power, 0 ≤ φ < 1. The relevant model, as given in Table 2.2, is:

yt = _t− 1 b_t^φ_{− 1}+ ε t,

4.4 Variations on the Common Models

_t = _t− 1 b_t^φ_{− 1}+ αε t, bt = b_t^φ_{− 1}+ βε t / _t− 1.

The state variable b_t is now a growth index with a base of 1.0. On making the appropriate substitutions, the recursive relationships in Table 2.1 follow, and are left as an exercise.

4.4.4 Various Seasonal Models

A variety of special seasonal models may be obtained as special cases of Holt-Winters’ multiplicative scheme, and we present a few of these without going into great detail.

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