Introduction to Local Level Model and Kalman Filter

Introduction to
Local Level Model and Kalman Filter
S.J. Koopman
http://staff.feweb.vu.nl/koopman
January 2011

U.K. Yearly Inflation : signal extraction
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Classical Decomposition
Basic Model
A basic model for representing a time series is the additive model
yt = µt + γt + εt, t = 1, . . . , n,
also known as the Classical Decomposition.
yt = observation,
µt = slowly changing component (trend),
γt = periodic component (seasonal),
εt = irregular component (disturbance).
Unobserved Components Time Series Model
In a Structural Time Series Model (STSM) or Unobserved
Components Model (UCM), the various components are modelled
explicitly as stochastic processes.

Local Level Model
◮ Components can be deterministic functions of time (e.g.
polynomials), or stochastic processes;
◮ Deterministic example: yt = µ + εt with εt ∼ N ID(0, σ 2 ε).
◮ Stochastic example: the Random Walk plus Noise, or
Local Level model:
yt = µt + εt, εt ∼ N ID(0, σ 2 ε)
µt+1 = µt + ηt, ηt ∼ N ID(0, σ 2 η),
◮ The disturbances εt, ηs are independent for all s, t;
◮ The model is incomplete without a specification for µ1 (note
the non-stationarity):
µ1 ∼ N (a, P)

Local Level Model
General framework
yt = µt + εt, εt ∼ N ID(0, σ 2 ε)
µt+1 = µt + ηt, ηt ∼ N ID(0, σ 2 η),
µ1 ∼ N (a, P)
◮ The level µt and the irregular εt are unobservables;
◮ Parameters: σ 2 ε and σ 2 η;
◮ Trivial special cases:
◮ σ 2 η = 0 =⇒ yt ∼ N ID(µ1, σ 2 ε) (WN with constant level);
◮ σ 2 ε = 0 =⇒ yt+1 = yt + ηt (pure RW);
◮ Local Level is a model representation for EWMA forecasting.

Simulated LL Data
6
4
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0
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σ 2 =0.1 ση
2 =1 y μ
ε
0 10 20 30 40 50 60 70 80 90 100

Simulated LL Data
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σ 2 =1 ση
2 =1
ε
0 10 20 30 40 50 60 70 80 90 100

Simulated LL Data
6
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σ 2 =1 ση
2 =0.1
ε
0 10 20 30 40 50 60 70 80 90 100

Simulated LL Data
5 σε
2 =0.1 ση
2 =1 y μ
0
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0 10 20 30 40 50 60 70 80 90 100
5 σ 2
ε =1 ση
2 =1
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2 σε
2 =1 ση
2 =0.1
0
0 10 20 30 40 50 60 70 80 90 100

Local Level Model
Its properties
yt = µt + εt, εt ∼ N ID(0, σ 2 ε),
µt+1 = µt + ηt, ηt ∼ N ID(0, σ 2 η),
◮ First difference is stationary:
∆yt = ∆µt + ∆εt = ηt−1 + εt − εt−1.
◮ Dynamic properties of ∆yt:
E(∆yt) = 0,
γ0 = E(∆yt∆yt) = σ 2 η + 2σ 2 ε,
γ1 = E(∆yt∆yt−1) = −σ 2 ε,
γτ = E(∆yt∆yt−τ ) = 0 for τ ≥ 2.

Properties of the LL model
◮ The ACF of ∆yt is
ρ1 = −σ2 ε
σ 2 η + 2σ 2 ε
ρτ = 0, τ ≥ 2.
◮ q is called the signal-noise ratio;
= − 1
q + 2 , q = σ2 η/σ 2 ε,
◮ The model for ∆yt is MA(1) with restricted parameters such
that
−1/2 ≤ ρ1 ≤ 0
i.e., yt is ARIMA(0,1,1);
◮ Write ∆yt = ξt + θξt−1, ξt ∼ N ID(0, σ 2 ) to solve θ:
θ = 1
q2 + 4q − 2 − q .
2

Local Level Model
◮ The model parameters are estimated by Maximum Likelihood;
◮ Advantages of model based approach: assumptions can be
tested, parameters are estimated...;
◮ The model with estimated parameters is used for the signal
extraction of components;
◮ The estimated level µt is effectively a locally weighted average
of the data;
◮ The distribution of weights can be compared with Kernel
functions in nonparametric regressions;
◮ On basis of model, the methods yield minimum mean square
error (MMSE) forecasts and the associated confidence
intervals.

Nile Data: decomposition
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Nile−Irregular
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Nile Data: decomposition weights
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Nile−Level
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Nile Data: forecasts
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Kalman Filter
◮ The Kalman filter calculates the mean and variance of the
unobserved state, given the observations.
◮ The state is Gaussian: the complete distribution is
characterized by the mean and variance.
◮ The filter is a recursive algorithm; the current best estimate is
updated whenever a new observation is obtained.
◮ To start the recursion, we need a1 and P1, which we assumed
given.
◮ There are various ways to initialize when a1 and P1 are
unknown, which we will not discuss here. See discussion in
DK book, Chapter 2.

Kalman Filter
The unobserved variable µt can be estimated from the
observations with the Kalman filter:
vt = yt − at,
Ft = Pt + σ 2 ε,
Kt = PtF −1
t ,
at+1 = at + Ktvt,
Pt+1 = Pt + σ 2 η − K 2 t Ft,
for t = 1, . . . , n and starting with given values for a1 and P1.
◮ Writing Yt = {y1, . . . , yt}, define
at+1 = E(µt+1|Yt), Pt+1 = var(µt+1|Yt).

Kalman Filter
Local level model: µt+1 = µt + ηt, yt = µt + εt.
◮ Writing Yt = {y1, . . . , yt}, define
at+1 = E(µt+1|Yt), Pt+1 = var(µt+1|Yt);
◮ The prediction error is
vt = yt − E(yt|Yt−1)
= yt − E(µt + εt|Yt−1)
= yt − E(µt|Yt−1)
= yt − at;
◮ It follows that vt = (µt − at) + εt and E(vt) = 0;
◮ The prediction error variance is Ft = var(vt) = Pt + σ 2 ε.

Regression theory
The proof of the Kalman filter uses lemmas from the multivariate
Normal regression theory.
Lemma 1
Suppose x, y are jointly Normally distributed vectors. Then
E(x|y) = E(x) + ΣxyΣ −1
y y,
var(x|y) = Σxx − ΣxyΣ −1
yy Σ ′ xy.
Lemma 2
Suppose x, y and z are jointly Normally distributed vectors with
E(z) = 0 and Σyz = 0. Then
E(x|y, z) = E(x|y) + ΣxzΣ −1
zz z,
var(x|y, z) = var(x|y) − ΣxzΣ −1
zz Σ ′ xz.

Derivation Kalman Filter
Local level model: µt+1 = µt + ηt, yt = µt + εt.
◮ We have Yt = {Yt−1, yt} = {Yt−1, vt} and E(vtyt−j) = 0 for
j = 1, . . . , t − 1;
◮ The lemma is E(x|y, z) = E(x|y) + ΣxzΣ −1
zz z.
In our case, take x = µt+1, y = Yt−1 and
z = vt = (µt − at) + εt;
◮ E(x|y) implies that
E(µt+1|Yt−1) = E(µt|Yt−1) + E(ηt|Yt−1) = at;
◮ Further, Σxz provides the expression
E(µt+1vt) = E(µtvt) + E(ηtvt) = E[(µt − at)(yt − at)] +
E(ηtvt) = E[(µt −at)(µt −at)]+E[(µt −at)εt]+E(ηtvt) = Pt;
◮ Since Σzz = Ft, we can apply lemma and obtain the state
update
at+1 = E(µt+1|Yt−1, yt)
= at + PtF −1
t vt
= at + Ktvt; with Kt = PtF −1
t .

Kalman Filter Derived
◮ Our best prediction of yt based on its past is at. When the
actual observation arrives, calculate the prediction error
vt = yt − at and its variance Ft = Pt + σ 2 ε.
◮ The best estimate of the state mean for the next period is
based on both the current estimate at and the new
information vt:
at+1 = at + Ktvt,
similarly for the variance:
◮ The Kalman gain
Pt+1 = Pt + σ 2 η − KtFtK ′ t.
Kt = PtF −1
t
is the optimal weighting matrix for the new evidence.
◮ You should be able to replicate the proof of the Kalman filter
for the Local Level Model (DK, Chapter 2).

Kalman filter for Nile Data: (i) at; (ii) Pt; (iii) vt and (iv) Ft.
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Steady State Kalman Filter
Kalman filter converges to a positive value, say Pt → ¯ P. We would
then have
Ft → ¯P + σ 2 ε, Kt → ¯P/(¯P + σ 2 ε).
The state prediction variance updating leads to
¯P = ¯P
1 −
which reduces to the quadratic
¯P
¯P + σ 2 ε
x 2 − xq − q = 0,
+ σ 2 η,
where x = ¯ P/σ 2 ε and q = σ 2 η/σ 2 ε, with solution
¯P = σ 2
ε q + q2 + 4q /2.

Smoothing
◮ The filter calculates the mean and variance conditional on Yt;
◮ The Kalman smoother calculates the mean and variance
conditional on the full set of observations Yn;
◮ After the filtered estimates are calculated, the smoothing
recursion starts at the last observations and runs until the
first.
ˆµt = E(µt|Yn), Vt = var(µt|Yn),
rt = weighted sum of future innovations, Nt = var(rt),
Lt = 1 − Kt.
Starting with rn = 0, Nn = 0, the smoothing recursions are given
by
rt−1 = F −1
t vt + Ltrt, Nt−1 = F −1
t + L 2 t Nt,
ˆµt = at + Ptrt−1, Vt = Pt − P 2 t Nt−1.

Kalman smoothing for Nile Data: (i) ˆµt; (ii) Vt; (iii) rt and (iv) Nt.
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Missing Observations
Missing observations are very easy to handle in Kalman filtering:
◮ suppose yj is missing
◮ put vj = 0, Kj = 0 and Fj = ∞ in the algorithm
◮ proceed further calculations as normal
The filter algorithm extrapolates according to the state equation
until a new observation arrives. The smoother interpolates
between observations.

Nile Data with missing observations : (i) at, (ii) Pt, (iii) ˆµt and
(iv) Vt.
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Forecasting
Forecasting requires no extra theory: just treat future observations
as missing:
◮ put vj = 0, Kj = 0 and Fj = ∞ for j = n + 1, . . . , n + k
◮ proceed further calculations as normal
◮ forecast for yj is aj

Nile Data: forecasting
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Parameters in Local Level Model
We recall the Local Level Model as
General framework
yt = µt + εt, εt ∼ N ID(0, σ 2 ε)
µt+1 = µt + ηt, ηt ∼ N ID(0, σ 2 η),
µ1 ∼ N (a, P)
◮ The unknown µt’s can be estimated by prediction, filtering
and smoothing;
◮ The other parameters are given by the variances σ 2 ε and σ 2 η;
◮ We estimate these parameters by Maximum Likelihood;
◮ Parameters can be transformed : σ 2 ε = exp(ψε) and
σ 2 η = exp(ψη);
◮ Parameter vector ψ = (ψε , ψη) ′ .

Parameter Estimation by ML
The parameters in any state space model can be collected in some
vector ψ. When model is linear and Gaussian; we can estimate ψ
by Maximum Likelihood.
The loglikelihood af a time series is
log L =
n
log p(yt|Yt−1).
t=1
In the state space model, p(yt|Yt−1) is a Gaussian density with
mean at and variance Ft:
log L = − n 1
log 2π −
2 2
n
t=1
log Ft + F −1
t v 2 t
with vt and Ft from the Kalman filter. This is called the prediction
error decomposition of the likelihood. Estimation proceeds by
numerically maximising log L.
,

Diagnostics
◮ Null hypothesis: standardised residuals
vt/ √ F t ∼ N ID(0, 1)
◮ Apply standard test for Normality, heteroskedasticity, serial
correlation;
◮ A recursive algorithm is available to calculate smoothed
disturbances (auxilliary residuals), which can be used to detect
breaks and outliers;
◮ Model comparison and parameter restrictions: use likelihood
based procedures (LR test, AIC, BIC).

Nile Data: diagnostics
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0 5 10

Three exercises
1. Consider LL model (see slides, see DK chapter 2).
◮ Reduced form is ARIMA(0,1,1) process. Derive the
relationship between signal-to-noise ratio q of LL model and
the θ coefficient of the ARIMA model;
◮ Derive the reduced form in the case ηt = √ qεt and notice the
difference in the general case.
◮ Give the elements of the mean vector and variance matrix of
y = (y1, . . . , yn) ′ when yt is generated by a LL model.
◮ Show that the forecasts of the Kalman filter (in a steady state)
are the same as those generated by the exponentially weighted
moving average (EWMA) method of forecasting:
ˆyt+1 = ˆyt + λ(yt − ˆyt) for t = 1, . . . , n. Derive the relationship
between λ and the signal-to-noise ratio q ?

Three exercises (cont.)
2. Derive a Kalman filter for the local level model
yt = µt + εt, εt ∼ N(0, σ 2 ε), ∆µt+1 = ηt ∼ N(0, σ 2 η),
with E(εtηt) = σεη = 0 and E(εtηs) = 0 for all t, s and t = s.
Also discuss the problem of missing obervations in this case.
3. Write Ox program(s) that produce all Figures in Ch 2 of DK
except Fig. 2.4. Data:
http://www.ssfpack.com/dkbook.html

Selected references
◮ A.C.Harvey (1989). Forecasting, Structural Time Series
Models and the Kalman Filter. Cambridge University Press
◮ G.Kitagawa & W.Gersch (1996). Smoothness Priors Analysis
of Time Series. Springer-Verlag
◮ J.Harrison & M.West (1997). Bayesian Forecasting and
Dynamic Models. Springer-Verlag
◮ R.H. Shumway and D.S. Stoffer (2000). Time Series Analysis
and its Applications. Springer-Verlag.
◮ J.Durbin & S.J.Koopman (2011). Time Series Analysis by
State Space Methods. Second Edition, Oxford University
Press
◮ J.Commandeur & S.J.Koopman (2007). An Introduction to
State Space Time Series Analysis. Oxford University Press