B) Local trend approximation 3 страница

so that

x_t = D (x_{t− 1}) − g (x_{t− 1}) (^yt). ^{r x}t− 1

We may observe that D (x_t) becomes linear in the state variables when f (x_t)and g (x_t)are linear in x_t and w (x_t) = r (x_t). Further, when the vector g (x_t)/ r (x_t)does not depend on the state variables, the transition equationsgiven in (4.3) reduce to

x_t = Dx_{t− 1} − gy_t.

It then follows that the model is stable in the sense of Sect. 3.3.1. These con-ditions may seem restrictive, but they correspond to an important class of heteroscedastic models, as we shall see below.

The conditional expectation, which is also the one-step-ahead point forecast y ˆ _{t|t− 1}, is given by:

E(yt |yt− 1,..., y 1, ^x 0) = E(yt |^xt− 1) = y ^ˆ t|t− 1 = w (^xt− 1),

so that the recursive relationships may be summarized as:

^{y ˆ} t|t− 1= ^w (^xt− 1)^,

ε t = (yt − y ^ˆ t|t− 1)/ r (^xt− 1),

x_t = f (x_{t− 1}) + g (x_{t− 1}) ε _t.

Once the model has been fully specified, updating proceeds directly using these equations. This approach is in stark contrast to models based on the Kalman filter using multiple independent sources of error. In that case, no direct updating is feasible in general, and we must make use of var-ious approximations such as the extended Kalman filter (West and Harrison 1997, pp. 496–497). The present form is too general for any meaningful dis-cussion of particular model properties, and so we will consider these on a case-by-case basis.

56 4 Nonlinear and Heteroscedastic Innovations State Space Models

As with the linear version in Sect. 3.1, the probability density function for y may be written in a relatively simple form as:

n

p (y | x ₀) =∏ p (y_t | x_{t− 1})

t =1

n

= ∏ p (ε _t)/ |r (x_{t− 1}) |.

t =1

If we assume that the distribution is Gaussian, this expression becomes:

p (y) = (2π σ ^2) ^−^n ^/2

_

n

_

− 1

n

ε ^2

/ σ ^2.

∏

r (x

t

)

exp

∑

t =1

−

−

t =1

_

_

_

_

_

_

4.2 Basic Special Cases

We now explore some of the special cases that can be used to model time series, recognizing as always that such models are at best an approxima-tion to the data generating process. We observed in Chap. 2 that models in the ETS(M, ∗, ∗) class give rise to the same point forecasts as those in the ETS(A, ∗, ∗) class, because we are deploying the same recursive relation-ships given in Table 2.1. The stochastic elements of the process determine whether to use the additive or multiplicative version. If the error process is homoscedastic, the constant variance assumptions in Chap. 3 are appro-priate, and the predictions intervals for h -steps ahead have constant widths regardless of the current level of the process. On the other hand, if the pro-cess is heteroscedastic, and, in particular, the error variance is proportional to the current level of the process, the nonlinear schemes introduced in Sect. 4.1 are appropriate. Clearly other assumptions concerning the nature of the vari-ance are possible, but we restrict our attention to the two options, additive or multiplicative, for the present discussion. Some extensions are considered briefly in Sect. 4.4.5.

We may justify the local models as the leading terms of a Taylor series expansion, and the only difference we would see relative to Fig. 3.1 is that the superimposed prediction intervals would widen when the observed value was high and narrow when it was low. Why does this matter? Think for the moment in terms of forecasting sales. During periods of high sales volume, the inappropriate use of a constant variance model would lead to underesti-mation of the level of uncertainty, and hence to a safety stock level that was too small. Similarly, in periods of low demand, the analysis would lead to carrying excess inventory. In either case, a loss of net revenue results. This effect is not always easy to recognize. For example, an empirical investiga-tion of the coverage provided by prediction intervals might simply count the number of times the actual value fell within the prediction interval. Because

4.2 Basic Special Cases

the interval is too wide when the level is low, and too narrow at high lev-els, the overall count might well come out close to the nominal level for the interval. We need to track the coverage at a given level to be sure that the prediction intervals are constructed appropriately.

4.2.1 Local Level Model: ETS(M,N,N)

This model is described by (4.2a, b).Upon inspection of these equations, we see that the conditional variance of y_t given __{t− 1} is σ ² _ ² _{t− 1}, reflecting the com-ments made earlier. The state equation reveals that the quantity α__{t− 1} ε _t has a persistent effect, feeding into the expected level for the next time period. When α = 0, the mean level does not change, so that the additive and multi-plicative models are then identical apart from the way the parameters were specified. When α = 1, the model reverts to a form of random walk with the reduced form y_t = y_{t− 1}(1 + ε _t); the complete effect of the random error is passed on to the next period. In general, the one-step-ahead predictions are, as for the additive scheme in (3.12), given by:

t− 1

y ˆ _{t +1 |t} = (1 − α) ^t _ ₀+ α ∑(1 − α) ^j y_t−j. j =0

The stability condition is satisfied provided 0 < α < 2.

A natural question to ask is what difference does it make if we select the multiplicative rather than the additive form of the local level model? Indeed, plots of simulated series look very similar to those given in Fig. 3.2, so mean-ingful comparisons must be sought in other ways. One approach is to look at the conditional variances given the initial conditions. Using the subscripts A and M for the additive and multiplicative schemes, we arrive at:

V_A (y_t |x ₀) = σ_A ²[1+ (t − 1) α ²],

V_M (y_t |x ₀) = x ₀²[(1+ σ_M ²)(1+ α ² σ_M ²) ^{t− 1} − 1].

In order to compare the two, we set the one-step-ahead variances equal by putting σ_A = x ₀ σ_M. We may then compute the ratio V_M / V_A for different values of t and α:

^σM	0.03	0.03	0.03	0.12	0.12	0.12
α	0.1	0.5	1.5	0.1	0.5	1.5
t =5	1.000	1.001	1.004	1.001	1.010	1.058
t =10	1.000	1.001	1.009	1.001	1.020	1.149
t =20	1.000	1.002	1.019	1.006	1.040	1.364

Perusal of the table indicates that there are only substantial differences in the variances for longer horizons with high α and relatively high values of σ_M.

58 4 Nonlinear and Heteroscedastic Innovations State Space Models

When σ_M = 0.30, the multiplicative error has a mean that is about three times its standard deviation, and the differences become noticeable quite quickly, as shown in this table for t = 10:

σM	0.03	0.12	0.30
α =0.1	1.000	1.001	1.008
α =0.5	1.001	1.020	1.134
α =1.0	1.004	1.067	1.519
α =1.5	1.009	1.149	2.473

The examination of stock price volatility represents an area where the distinction could be very important. Based on the efficient market hypothesis we would expect that α = 1. The process might be observed at hourly or even minute intervals, yet the purpose behind the modeling would be to evaluate the volatility (essentially as measured by the variance) over much longer periods. In such circumstances, the use of an additive model when a multiplicative model was appropriate could lead to considerable under-estimation of the risks involved. Interestingly, if we start out with this form of the random walk model and consider the reduced form of (4.2), we can rewrite the measurement equation as:

^y t ^{− y}t− 1_{= ε t .}

^yt− 1

That is, the one-period return on an investment follows a white noise process. As we observed in Sect. 4.1, the Gaussian distribution is not a valid assumption for strictly positive processes such as these multiplicative mod-els. In applications, the prediction interval appears to be satisfactory pro-vided the h -step-ahead forecast root mean squared error is less than about one-quarter of the mean. In other cases, or for simulations, a more careful

specification of the error distribution may be needed (see Chap. 15).

4.2.2 Local Trend Model: ETS(M,A,N)

We may augment the local level model by adding an evolving growth rate b_t to give the new model:

yt = (_t− 1	+ bt− 1)(1	+ ε t),	(4.4a)
_t = (_t− 1	+ bt− 1)(1	+ αε t),	(4.4b)
bt = bt− 1+ β (_t− 1+ bt− 1) ε t.	(4.4c)

This model has a state space structure with:

xt	= [ lt, bt ] ^_, w (xt− 1) = r (xt− 1) = _t− 1 + bt− 1,
f (xt− 1) = [ _t− 1	+ bt− 1, bt− 1] ^_, and g = [ α (_t− 1 + bt− 1), β (_t− 1 + bt− 1)] ^_.

4.2 Basic Special Cases

Given the multiplicative nature of the model, we take the process and the underlying level to be strictly positive; the slope may be positive, zero or negative. There are now two smoothing parameters α and β, and they are scaled by the current level of the process. Because the slope is typically quite small relative to the current level of the process, the value of β will often be small. The following special cases are worth noting:

• β =0: a global trend

• β =0, α =1: random walk with a constant trend element, often known asthe random walk with drift

• β =0, α =0: fixed level and trend, thereby reducing to a classical or globallinear trend model

We may represent the model in innovations form as:

^{y ˆ} t|t− 1= (^_t− 1+ ^bt− 1)^,

ε t = (yt − y ^ˆ t|t− 1)/(_t− 1+ bt− 1),

__t = (__{t− 1} + b_{t− 1})(1 + αε _t),

bt = bt− 1+ β (_t− 1+ bt− 1) ε t.

An equivalent form that is more convenient for updating is:

^{y ˆ} t|t− 1= (^_t− 1+ ^bt− 1)^,

ε t = (yt − y ^ˆ t|t− 1)/(_t− 1+ bt− 1),

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

where β^∗ = β / α. The state updates are of exactly the same algebraic form as those for the additive model in (3.14). Manipulation of the state equations provides the recursive relationships:

__t = αy_t + (1 − α)(__{t− 1} + b_{t− 1}), b_t = β (y_t − __{t− 1}) + (1 − β) b_{t− 1}.

Following Sect. 3.3, we may write the general form of the stability condition as constraints on

D (x_t) = f (x_t) − g (x_t) w (x_t)/ r (x_t)= f (x_t) − g (x_t)

when w (x_{t− 1}) = r (x_{t− 1}). Given the particular form of (4.4), this expression simplifies to

D (x_{t− 1}) = [(__{t− 1}+ b_{t− 1})(1 − α), b_{t− 1} − β (__{t− 1}+ b_{t− 1})]

= ^_1 − α 1 − α ^_

_{−β 1 − β} ^xt− 1^.

60 4 Nonlinear and Heteroscedastic Innovations State Space Models

This is the same general form as for the additive models discussed in Chap. 3. So, for the local trend model, the stability conditions remain: α > 0, β > 0 and 2 α + β < 4.

4.2.3 Local Multiplicative Trend, Additive Error Model: ETS(A,M,N)

Some of the boxes in Tables 2.2 and 2.3 correspond to models that might not occur to the model builder as prime candidates for consideration. Nev-ertheless, we consider a few of these forms for several reasons. First of all, if we select a model based on some kind of automated search, it is useful to have a rich set of potential models, corresponding to the many nuances the time series might display. Second, by exploring such models we gain a better understanding of the attributes of various nonlinear configurations. Finally, if our study of their properties reveals some undesirable characteristics, we are forewarned about such possibilities before jumping to inappropriate conclusions about usable models.

If we allow an additive error to be associated with a multiplicative trend (exponential growth or decay), we have the following ETS(A,M,N) model:

yt = _t− 1 bt− 1+ ε t, _t = _t− 1 bt− 1+ αε t,

bt = bt− 1+ βε t / _t− 1.

If we substitute for the error term, we arrive at the recursive relationships

(β = αβ^∗):

__t = (1 − α) __{t− 1} b_{t− 1} + αy_t,

b_t = (1 − β) b_{t− 1}+ βy_t / __{t− 1}= (1 − β^∗) b_{t− 1}+ β^∗__t / __{t− 1}.

It is no longer possible to derive simple expressions to ensure that the stability conditions are satisfied.

4.2.4 Local Multiplicative Trend, Multiplicative Error Model: ETS(M,M,N)

In a similar vein, we can make both components multiplicative:

yt = _t− 1 bt− 1(1+ ε t), _t = _t− 1 bt− 1(1+ αε t),

b_t = b_{t− 1}(1+ βε _t).

If we substitute for the error term, we arrive at the same updating equations:

__t = (1 − α) __{t− 1} b_{t− 1} + αy_t,

b_t = (1 − β) b_{t− 1}+ βy_t / __{t− 1}= (1 − β^∗) b_{t− 1}+ β^∗__t / __{t− 1}.

4.3 Nonlinear Seasonal Models

This model is of interest in that we can guarantee strictly positive values¹ for the series using a set of conditions such as:

0 < α < 1, 0 < β < 1 and 1 + ε _t > 0.

When we set β = 0, we have a constant trend term corresponding to a fixed growth rate, b. If we had b < 1 this would correspond to a form of damping, whereas b > 1 allows perpetual growth and could not sat-isfy the stability condition. Such a model might fit the past history of a time series, but we should be cautious about building such a relationship into the predictive recursion.

4.3 Nonlinear Seasonal Models

Much of the discussion for additive seasonal models applies equally to multi-plicative models. In addition to any extant trends, we must allow for seasonal variations in a series. As in Sect. 3.4.3, we represent the seasonal factors by s_t. A common feature of such seasonal patterns is that the variability is pro-portional to the general level of the series. For example, discussions about the volume of air passenger traffic or about retail sales usually speak of percentage changes rather than absolute shifts in the series. Such changes also apply when the seasonal pattern corresponds to days of the week (e.g., commuter traffic) or hours of the day (e.g., electricity usage). Consideration of these examples also indicates that the seasonal pattern may change over time. Perhaps the most dramatic examples are the differences in commuter traffic on public holidays or electricity usage between weekdays and week-ends. In such cases, multiple seasonal cycles may be needed to provide an effective picture (see Chap. 14). However, even in less volatile situations the need for evolving seasonal patterns is clearly evident.

4.3.1 A Multiplicative Seasonal and Error Model: ETS(M,A,M)

The seasonal variations are usually more dramatic within a short time frame than the longer-term changes in trend, so that the focus is primarily on the correct specification of the seasonal structure. We consider a model with mul-tiplicative effects for both the seasonal and error components: ETS(M,A,M). This model may be written as:

^yt = (_t− 1	+ bt− 1) st−m (1 + ε t),	(4.5a)
_t = (_t− 1	+ bt− 1)(1 + αε t),	(4.5b)
bt = bt− 1+ β (_t− 1+ bt− 1) ε t,	(4.5c)
st = st−m (1+ γε t).	(4.5d)

¹ See Chap. 15 for a detailed discussion of models for positive data.

^st−m +1

.

.

.

62 4 Nonlinear and Heteroscedastic Innovations State Space Models

The model has a state space structure with:

												_t
												bt
												st
											^st− ..
							xt	=		,
													.
										s	t		m +1
											−
f (xt− 1) = F xt− 1,	w (^xt− 1) = r (^xt− 1) = (_t− 1+ bt− 1) st−m, and
						α (_t 1+ bt	1)									1 1 0 0... 0 0
													0 1 0 0... 0 0
					β (_t^− 1+ bt^− 1)
					−	−									0 0 0 0... 0 1	.
g (x	t			) =		γst−m				,		where F =	0 0 1 0... 0 0
	−												0 0 0 1... 0 0
					.
						.										.....	..
						.										....	. ...
																......
																0 0 0 0... 1 0

Just as the level and seasonal terms in the additive model are only deter-mined up to an arbitrary additive constant, so the level, trend and seasonal terms in the present scheme are only determined up to an arbitrary multi-plicative constant. To resolve the indeterminacy, we usually set ∑ ^m_{i =} ⁻ ₀¹ s _{0 −i} = m. As before, we will find it preferable to apply this normalization through-out the series; the details are discussed in Chap. 8.

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