Introduction to Local Level Model and Kalman Filter · Kalman Filter I The Kalman lter calculates...
Transcript of Introduction to Local Level Model and Kalman Filter · Kalman Filter I The Kalman lter calculates...
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Introduction toLocal Level Model and Kalman Filter
S.J. Koopman
http://staff.feweb.vu.nl/koopman
January 2011
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U.K. Yearly Inflation : signal extraction
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Dow Jones
Dow Jones (Industrial Avg, Monthly Closures)
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LogReturn
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LogSqr LogReturns Dow Jones : signal extraction
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Classical Decomposition
Basic ModelA 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 UnobservedComponents Model (UCM), the various components are modelledexplicitly as stochastic processes.
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Local Level Model
I Components can be deterministic functions of time (e.g.polynomials), or stochastic processes;
I Deterministic example: yt = µ+ εt with εt ∼ NID(0, σ2ε).
I Stochastic example: the Random Walk plus Noise, orLocal Level model:
yt = µt + εt , εt ∼ NID(0, σ2ε)
µt+1 = µt + ηt , ηt ∼ NID(0, σ2η),
I The disturbances εt , ηs are independent for all s, t;
I The model is incomplete without a specification for µ1 (notethe non-stationarity):
µ1 ∼ N (a,P)
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Local Level Model
yt = µt + εt , εt ∼ NID(0, σ2ε)
µt+1 = µt + ηt , ηt ∼ NID(0, σ2η),
µ1 ∼ N (a,P)
General framework
I The level µt and the irregular εt are unobservables;
I Parameters: σ2ε and σ2η;I Trivial special cases:
I σ2η = 0 =⇒ yt ∼ NID(µ1, σ
2ε) (WN with constant level);
I σ2ε = 0 =⇒ yt+1 = yt + ηt (pure RW);
I Local Level is a model representation for EWMA forecasting.
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Simulated LL Data
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Simulated LL Data
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Simulated LL Data
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Simulated LL Data
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Local Level Model
yt = µt + εt , εt ∼ NID(0, σ2ε),
µt+1 = µt + ηt , ηt ∼ NID(0, σ2η),
Its properties
I First difference is stationary:
∆yt = ∆µt + ∆εt = ηt−1 + εt − εt−1.
I Dynamic properties of ∆yt :
E(∆yt) = 0,
γ0 = E(∆yt∆yt) = σ2η + 2σ2ε ,
γ1 = E(∆yt∆yt−1) = −σ2ε ,γτ = E(∆yt∆yt−τ ) = 0 for τ ≥ 2.
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Properties of the LL model
I The ACF of ∆yt is
ρ1 =−σ2ε
σ2η + 2σ2ε= − 1
q + 2, q = σ2η/σ
2ε ,
ρτ = 0, τ ≥ 2.
I q is called the signal-noise ratio;
I The model for ∆yt is MA(1) with restricted parameters suchthat
−1/2 ≤ ρ1 ≤ 0
i.e., yt is ARIMA(0,1,1);
I Write ∆yt = ξt + θξt−1, ξt ∼ NID(0, σ2) to solve θ:
θ =1
2
(√q2 + 4q − 2− q
).
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Local Level Model
I The model parameters are estimated by Maximum Likelihood;
I Advantages of model based approach: assumptions can betested, parameters are estimated...;
I The model with estimated parameters is used for the signalextraction of components;
I The estimated level µt is effectively a locally weighted averageof the data;
I The distribution of weights can be compared with Kernelfunctions in nonparametric regressions;
I On basis of model, the methods yield minimum mean squareerror (MMSE) forecasts and the associated confidenceintervals.
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Nile Data: decomposition
Nile Level +/− 2SE
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Nile Data: decomposition weights
Nile−Level
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Nile Data: forecasts
Nile Forecast−Nile +/− SE
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Kalman Filter
I The Kalman filter calculates the mean and variance of theunobserved state, given the observations.
I The state is Gaussian: the complete distribution ischaracterized by the mean and variance.
I The filter is a recursive algorithm; the current best estimate isupdated whenever a new observation is obtained.
I To start the recursion, we need a1 and P1, which we assumedgiven.
I There are various ways to initialize when a1 and P1 areunknown, which we will not discuss here. See discussion inDK book, Chapter 2.
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Kalman Filter
The unobserved variable µt can be estimated from theobservations with the Kalman filter:
vt = yt − at ,
Ft = Pt + σ2ε ,
Kt = PtF−1t ,
at+1 = at + Ktvt ,
Pt+1 = Pt + σ2η − K 2t Ft ,
for t = 1, . . . , n and starting with given values for a1 and P1.
I Writing Yt = {y1, . . . , yt}, define
at+1 = E(µt+1|Yt), Pt+1 = var(µt+1|Yt).
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Kalman Filter
Local level model: µt+1 = µt + ηt , yt = µt + εt .
I Writing Yt = {y1, . . . , yt}, define
at+1 = E(µt+1|Yt), Pt+1 = var(µt+1|Yt);
I The prediction error is
vt = yt − E(yt |Yt−1)
= yt − E(µt + εt |Yt−1)
= yt − E(µt |Yt−1)
= yt − at ;
I It follows that vt = (µt − at) + εt and E(vt) = 0;
I The prediction error variance is Ft = var(vt) = Pt + σ2ε .
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Regression theoryThe proof of the Kalman filter uses lemmas from the multivariateNormal regression theory.
Lemma 1Suppose x , y are jointly Normally distributed vectors. Then
E(x |y) = E(x) + ΣxyΣ−1y y ,
var(x |y) = Σxx − ΣxyΣ−1yy Σ′xy .
Lemma 2Suppose x , y and z are jointly Normally distributed vectors withE(z) = 0 and Σyz = 0. Then
E(x |y , z) = E(x |y) + ΣxzΣ−1zz z ,
var(x |y , z) = var(x |y)− ΣxzΣ−1zz Σ′xz .
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Derivation Kalman FilterLocal level model: µt+1 = µt + ηt , yt = µt + εt .
I We have Yt = {Yt−1, yt} = {Yt−1, vt} and E(vtyt−j) = 0 forj = 1, . . . , t − 1;
I The lemma is E(x |y , z) = E(x |y) + ΣxzΣ−1zz z .In our case, take x = µt+1, y = Yt−1 andz = vt = (µt − at) + εt ;
I E(x |y) implies thatE(µt+1|Yt−1) = E(µt |Yt−1) + E(ηt |Yt−1) = at ;
I Further, Σxz provides the expressionE(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 ;
I Since Σzz = Ft , we can apply lemma and obtain the stateupdate
at+1 = E(µt+1|Yt−1, yt)
= at + PtF−1t vt
= at + Ktvt ; with Kt = PtF−1t .
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Kalman Filter Derived
I Our best prediction of yt based on its past is at . When theactual observation arrives, calculate the prediction errorvt = yt − at and its variance Ft = Pt + σ2ε .
I The best estimate of the state mean for the next period isbased on both the current estimate at and the newinformation vt :
at+1 = at + Ktvt ,
similarly for the variance:
Pt+1 = Pt + σ2η − KtFtK′t .
I The Kalman gainKt = PtF
−1t
is the optimal weighting matrix for the new evidence.
I You should be able to replicate the proof of the Kalman filterfor the Local Level Model (DK, Chapter 2).
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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 wouldthen have
Ft → P + σ2ε , Kt → P/(P + σ2ε).
The state prediction variance updating leads to
P = P
(1− P
P + σ2ε
)+ σ2η,
which reduces to the quadratic
x2 − xq − q = 0,
where x = P/σ2ε and q = σ2η/σ2ε , with solution
P = σ2ε(q +
√q2 + 4q
)/2.
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Smoothing
I The filter calculates the mean and variance conditional on Yt ;
I The Kalman smoother calculates the mean and varianceconditional on the full set of observations Yn;
I After the filtered estimates are calculated, the smoothingrecursion starts at the last observations and runs until thefirst.
µt = 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 givenby
rt−1 = F−1t vt + Ltrt , Nt−1 = F−1t + L2t Nt ,
µt = at + Ptrt−1, Vt = Pt − P2t Nt−1.
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Kalman smoothing for Nile Data: (i) µt ; (ii) Vt ; (iii) rt and (iv) Nt .
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Missing Observations
Missing observations are very easy to handle in Kalman filtering:
I suppose yj is missing
I put vj = 0,Kj = 0 and Fj =∞ in the algorithm
I proceed further calculations as normal
The filter algorithm extrapolates according to the state equationuntil a new observation arrives. The smoother interpolatesbetween observations.
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Nile Data with missing observations : (i) at , (ii) Pt , (iii) µt and(iv) Vt .
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Forecasting
Forecasting requires no extra theory: just treat future observationsas missing:
I put vj = 0,Kj = 0 and Fj =∞ for j = n + 1, . . . , n + k
I proceed further calculations as normal
I forecast for yj is aj
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Nile Data: forecasting
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Parameters in Local Level Model
We recall the Local Level Model as
yt = µt + εt , εt ∼ NID(0, σ2ε)
µt+1 = µt + ηt , ηt ∼ NID(0, σ2η),
µ1 ∼ N (a,P)
General framework
I The unknown µt ’s can be estimated by prediction, filteringand smoothing;
I The other parameters are given by the variances σ2ε and σ2η;
I We estimate these parameters by Maximum Likelihood;
I Parameters can be transformed : σ2ε = exp(ψε) andσ2η = exp(ψη);
I Parameter vector ψ = (ψε , ψη)′.
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Parameter Estimation by ML
The parameters in any state space model can be collected in somevector ψ. When model is linear and Gaussian; we can estimate ψby Maximum Likelihood.
The loglikelihood af a time series is
log L =n∑
t=1
log p(yt |Yt−1).
In the state space model, p(yt |Yt−1) is a Gaussian density withmean at and variance Ft :
log L = −n
2log 2π − 1
2
n∑t=1
(log Ft + F−1t v2
t
),
with vt and Ft from the Kalman filter. This is called the predictionerror decomposition of the likelihood. Estimation proceeds bynumerically maximising log L.
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Diagnostics
I Null hypothesis: standardised residuals
vt/√
F t ∼ NID(0, 1)
I Apply standard test for Normality, heteroskedasticity, serialcorrelation;
I A recursive algorithm is available to calculate smootheddisturbances (auxilliary residuals), which can be used to detectbreaks and outliers;
I Model comparison and parameter restrictions: use likelihoodbased procedures (LR test, AIC, BIC).
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Nile Data: diagnostics
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Three exercises
1. Consider LL model (see slides, see DK chapter 2).I Reduced form is ARIMA(0,1,1) process. Derive the
relationship between signal-to-noise ratio q of LL model andthe θ coefficient of the ARIMA model;
I Derive the reduced form in the case ηt =√
qεt and notice thedifference in the general case.
I Give the elements of the mean vector and variance matrix ofy = (y1, . . . , yn)′ when yt is generated by a LL model.
I Show that the forecasts of the Kalman filter (in a steady state)are the same as those generated by the exponentially weightedmoving average (EWMA) method of forecasting:yt+1 = yt + λ(yt − yt) for t = 1, . . . , n. Derive the relationshipbetween λ and the signal-to-noise ratio q ?
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Three exercises (cont.)
2. Derive a Kalman filter for the local level model
yt = µt + εt , εt ∼ N(0, σ2ε), ∆µt+1 = ηt ∼ N(0, σ2η),
with E (εtηt) = σεη 6= 0 and E (εtηs) = 0 for all t, s and t 6= s.Also discuss the problem of missing obervations in this case.
3. Write Ox program(s) that produce all Figures in Ch 2 of DKexcept Fig. 2.4. Data:http://www.ssfpack.com/dkbook.html
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Selected references
I A.C.Harvey (1989). Forecasting, Structural Time SeriesModels and the Kalman Filter. Cambridge University Press
I G.Kitagawa & W.Gersch (1996). Smoothness Priors Analysisof Time Series. Springer-Verlag
I J.Harrison & M.West (1997). Bayesian Forecasting andDynamic Models. Springer-Verlag
I R.H. Shumway and D.S. Stoffer (2000). Time Series Analysisand its Applications. Springer-Verlag.
I J.Durbin & S.J.Koopman (2011). Time Series Analysis byState Space Methods. Second Edition, Oxford UniversityPress
I J.Commandeur & S.J.Koopman (2007). An Introduction toState Space Time Series Analysis. Oxford University Press