ECONOMICS 240C

6-06-2006 Lecture 17 XXX

I. Harvey Structural Time Series Models

Structural time series models use one or more white noise processes as inputs or building blocks. Consequently, they can include ARIMA models as well as be more complex than ARIMA models. The components of a structural time series can include "level", "slope", "irregular", and "periodic". The latter can de used for modeling a seasonal and/or cyclical component. These various components can be excluded(none) or included, and can be fixed(deterministic) or stochastic. The various possibilities are indicated schematically in the following table.

Structural Time Series Components: Inclusion Options
Component / None / Fixed / Stochastic
Level
Irregular / N. A.
Slope
Seasonal
Cycle

Examples of some of the possibilities follow. To begin with, we set irregular, seasonal and cyclical components to zero and concentrate on the nine combinations available from level and slope.

A. Level and Slope Component Combinations

The level and slope component can each take one of three forms for a total of nine possible models, as illustrated in the following table.

Level and Slope Component Combinations and Models: Examples

SLOPE
None / Fixed / Stochastic
None / 1, zero / 3, trend / 7, integrated random walk
LEVEL / Fixed / 2, constant / 4, constant +
trend / 8, constant + integrated random walk
Stochastic / 5, random walk / 6, random walk + trend

B. Harvey Structural Models

The general structural model framework can be summarized with three equations, the specification of three potential stochastic components, and the specification of two initial conditions.

1. three equations

i. time series(t) = level(t) + irregular(t)

ii. level(t) = level(t-1) + slope(t) +stochastic level(t)

iii. slope(t) = slope(t-1) + stochachistic slope(t)

2. three potential stochastic components

i. irregular(t) = WN1(t) or 0.

ii. stochastic level(t) = WN2(t) or 0.

iii. stochastic slope(t) = WN3(t) or 0.

3. two initial conditions

i. level(0) = constant, which could be zero.

ii. slope(0) = constant, which could be zero.

C. Various Possibile Specifications

#1. No components: Time series, y(t), is zero.

y(t) = level(t)

level(t) = level(t-1) = 0 = constant

y(t) = 0, all t.

For example, applying the specifications above:

1. three equations that define the relevant components

i. time series(t) = level(t)

ii. level(t) = level(t-1) + slope(t)

iii. slope(t) = slope(t-1)

2. three potential stochastic components

i. irregular(t) = 0.

ii. stochastic level(t) = 0.

iii. stochastic slope(t) = 0.

3. two initial conditions

i. level(0) =0.

ii. slope(0) =0.

#2. Deterministic constant level only, no other components, y(t) is constant.

y(t) = level(t).

level(t) = level(t-1) = level(0) = constant

time series is constant, y(t) = level(0)

Specifying the conditions in detail:

1. three equations

i. time series(t) = level(t)

ii. level(t) = level(t-1) + slope(t)

iii. slope(t) = slope(t-1)

2. three potential stochastic components

i. irregular(t) = 0.

ii. stochastic level(t) = 0.

iii. stochastic slope(t) = 0.

3. two initial conditions

i. level(0) = constant.

ii. slope(0) =0.

#3. Deterministic trending level, constant slope, no other components.

y(t) = level(t)

level(t) = level(t-1) + slope(t),

where slope(0) = constant, and level(0) = 0,

thus level(1) = level(0) + slope(0) = slope(0),

level(2) = level(1) + slope(0) = 2*slope(0),

level(t) = t*slope(0)

y(t) = t*slope(0), i.e. the time series is trend.

or specifying the conditions in detail:

1. three equations

i. time series(t) = level(t)

ii. level(t) = level(t-1) + slope(t)

iii. slope(t) = slope(t-1)

2. three potential stochastic components

i. irregular(t) = 0.

ii. stochastic level(t) = 0.

iii. stochastic slope(t) = 0.

3. two initial conditions

i. level(0) = 0.

ii. slope(0) = constant..

#4. Same as example 3, except the time series is trend plus a constant.

y(t) = level(t)

level(t) = level(t-1) + slope(t),

where slope(0) = constant, and level(0) = constant,

thus level(1) = level(0) + slope(0),

level(2) = level(1) + slope(0) = level(0) + 2*slope(0),

level(t) = level(0) + t*slope(0)

y(t) = level(0) + t*slope(0), i.e. the time series is trend + constant.

#5. No slope; trend has a white noise input, leading to a stochastic y(t).

y(t) = level(t)

level(t) = level(t-1) + slope(t) + stochastic level(t),

where slope(0) = 0, and level(0) = 0, and stochastic level(t) is white noise, WN2(t)

thus level(1) = level(0) + slope(0) + WN2(1) = WN2(1),

level(2) = level(1) + WN2(2) = WN2(1) + WN2(2), i.e. level is

the sum of white noise or a random walk. Note that in this example,

level(t) = level(t-1) + WN2(t), i.e.

level(t) = WN2(t)/[1-Z] = RW2(t), and y(t) = RW2(t) .

Specifying the conditions in detail:

1. three equations

i. time series(t) = level(t)

ii. level(t) = level(t-1) + slope(t) +stochastic level(t)

iii. slope(t) = slope(t-1)

2. three potential stochastic components

i. irregular(t) = 0.

ii. stochastic level(t) = WN2(t).

iii. stochastic slope(t) = 0.

3. two initial conditions

i. level(0) = 0.

ii. slope(0) = 0.

Example 5 can be illustrated schematically:

#6. Fixed slope; trend has a white noise input.

1. three equations:

i. y(t) = level(t)

ii. level(t) = level(t-1) + slope(t) + stochastic level(t)

iii. slope(t) = slope(t-1) + stochastic slope(t)

2. three potential stochastic components:

i. irregular(t) = 0.

ii. stochastic level(t) = WN2(t).

iii. stochastic slope(t) = 0.

3. two initial conditions

i. level(0) = 0.

ii. slope(0) = constant.

slope(1) = slope(0), so slope(t) =slope(0).

level(t) = level(t-1) + slope(0) + WN2(t), so

level(t) = {slope(0)/[1-Z]} + WN2(t)/[1-Z] = trend + random walk

y(t) = {slope(0)/[1-Z]} + WN2(t)/[1-Z] .

#7. No level; stochastic slope

1. three equations:

i. y(t) = level(t)

ii. level(t) = level(t-1) + slope(t) + stochastic level(t)

iii. slope(t) = slope(t-1) + stochastic slope(t)

2. three potential stochastic components:

i. irregular = 0 ii. stochastic level(t) = 0

iii. stochastic slope(t) = WN3(t)

3. two initial conditions

i. level(0) = 0

ii. slope(0) = 0.

slope(t) = slope(t-1) + WN3(t), so slope(t) = WN3(t)/[1-Z] = RW3(t).

level(t) = level(t-1) + slope(t) = level(t-1) + WN3(t)/[1-Z],

level(t) = WN3(t)/[1-Z]2

y(t) = WN3(t)/[1-Z]2 .

#8. Fixed level; stochastic slope

1. three equations

i. time series(t) = level(t) + irregular(t)

ii. level(t) = level(t-1) + slope(t) +stochastic level(t)

iii. slope(t) = slope(t-1) + stochachistic slope(t)

2. three potential stochastic components

i. irregular(t) = 0.

ii. stochastic level(t) = 0.

iii. stochastic slope(t) = WN3(t).

3. two initial conditions

i. level(0) = constant.

ii. slope(0) = 0.

y(t) = level(t)

level(t) = level(t-1) + slope(t)

slope(t) = slope(t-1) + stochastic slope(t) = slope(t-1) + WN3(t)

so slope(t) = WN3(t)/[1-Z] = RW3(t).

level(1) = level(0) + slope(1) = level(0) + RW3(1)

level(2) = level(1) + slope(2) = level(0) + RW3(1) + RW3(2)

level(t) = level(0) + RW3(t)/[1-Z] = level(0) + WN3(t)/[1-Z]2

y(t) = level(0) + WN3(t)/[1-Z]2 .

#9. Stochastic slope; stochastic level

1. three equations

i. time series(t) = level(t) + irregular(t)

ii. level(t) = level(t-1) + slope(t) +stochastic level(t)

iii. slope(t) = slope(t-1) + stochachistic slope(t)

2. three potential stochastic components

i. irregular(t) = 0.

ii. stochastic level(t) = WN2(t).

iii. stochastic slope(t) = WN3(t).

3. two initial conditions

i. level(0) = 0.

ii. slope(0) = 0.

y(t) = level(t)

level(t) = level(t-1) + slope(t) +WN2(t)

slope(t) = slope(t-1) + WN3(t)

so slope(t) = WN3(t)/[1-Z] = RW3(t).

[1-Z]level(t) = {WN3(t)/[1-Z]} + WN2(t)

level(t) = {WN3(t)/[1-Z]2} + WN2(t)/[1-Z]

level(t) is an integrated random walk plus a random walk

y(t) = WN3(t)/[1-Z]2 + WN2(t)/[1-Z] .

D. Summary

In summary, the three formula for: (1) the time series in terms of the level and the irregular, (2) the level in terms of the slope, and (3) the slope, are as follows:

y(t) = level(t) + irregular(t)

level(t) = level(t-1) + slope(t) + stochastic level(t)

slope(t) = slope(t-1) + stochastic slope(t)

where an option is to model the irregular as white noise instead of zero,

irregular(t) = WN1(t),

so all of the nine examples above can be complicated by the addition of a white noise irregular component. The stochastic level and the stochastic slope, as we saw above, can be modeled as WN2(t) and WN3(t), respectively, or in eithe case as zero. Lastly the initial values of the level and the slope are constant, so one possible value is zero.

II. Estimation of Harvey Structural Time Series Models

Estimation as well as modeling is discussed in Andrew Harvey’s book, Forecasting, Structural Time Series Models and the Kalman Filter(1989). Estimation involves the Kalman filter and an updating procedure. As indicated from the models discussed above, this approach can deal with evolutuonary as well as stationary time series. A program, Stamp, for the DOS-Windows environment is available from Chapman & Hall for structural time series analysis.