Download - Linear Gaussian State Space Models - Portland State …web.pdx.edu/~crkl/WISE2016/TSAR-2.pdfDynamic Linear Models •Local Trend Model 2 –State Space Model Representation Time Series

Linear Gaussian State Space Models

• Structural Time Series Models

– Level and Trend Models

– Basic Structural Model (BSM)

• Dynamic Linear Models

– State Space Model Representation

• Level, Trend, and Seasonal Models

• Time Varying Regression Model

– Extensions

• Multivariate Time Series Analysis

• Bayesian Time Series Analysis

1 Time Series Data Analysis Using R

Structural Time Series Models

• Local level Model

• Local Trend Model


2

1

2

1

, ~ (0, )

, ~ (0, )

t t t t v

t t t t w

y v v N

w w N

2

1

2

1 1 , ,

2

1 , ,

, ~ (0, )

, ~ (0, )

, ~ (0, )

t t t t v

t t t t t

t t t t

y v v N

w w N

w w N

Structural Time Series Models

• Basic Structural Model (BSM)

• Forecasting


2

1 1, 1

2

1 1 , ,

2

1 , ,

12

1, , 1 , ,

2

, , 1

, ~ (0, )

, ~ (0, )

, ~ (0, )

, ~ (0, )

, 2,..., 1

t t t t t v

t t t t t

t t t t

p

t j t t t

j

j t j t

y v v N

w w N

w w N

w w N

j p

1| 1

|

ˆ , 1,2,...,

ˆ , 1,2,...

t t t t t p

n h n n n n h p

y a b s t n

y a hb s h p

Dynamic Linear Models

• Observation Equation

• State Equation

• Initial State Distribution

•


11 1 1 1

, ~ (0 , )t t t t t tmm m p p m m m m

y F v v iid N V

111 1 1 1

, ~ (0 , )t t t t t tpp p p p p p p p

G w w iid N W

0 0 0

' '

0 0

'

~ ( , )

( ) ( ) 0

( ) 0 ,

t t

t sm p

N m C

E v E w t

E v w t s


• Local Level Model



2

2

1

, ~ (0, )

, ~ (0, )

t t t t v

t t t t w

y v v N

w w N

1

2 2

( 1, 1)

, 1, 1, ,

t t t t

t t t t

t t t t t v t w

y F v

G w m p

F G V W


• Local Trend Model



2

2

1 1 , ,

2

1 , ,

, ~ (0, )

, ~ (0, )

, ~ (0, )

t t t t v

t t t t t

t t t t

y v v N

w w N

w w N

1

2

2

2

( 1, 2)

01 1, 1 0 , , ,

00 1

t t t t

t t t t

t

t t t t v t

t

y F v

G w m p

F G V W

,1

,1

1 1

0 1

tt t

tt t

w

w


• Time Varying Regression Parameters



2

2

1 , ,

2

1 , ,

, ~ (0, )

, ~ (0, )

, ~ (0, )

t t t t t t v

t t t t

t t t t

y x v v N

w w N

w w N

2

2, ,1

2, ,1

1 01 1 , ,

0 1

01 0, ,

00 1

t

t t t t t t t v

t

t tt t t

t t t

t tt t t

y x v F x G V

w ww W

w w


• Model Estimation

– Filtering (filtered estimate of )

– Smoothing (smoothed estimate of )


1

{ }t t t t

t

t t t t

y F vy

G w

1( | { ,..., })t t tE I y y

1( | { ,..., })t T TE I y y

Model Estimation

• The Kalman Filter is a set of recursion equations for determining the optimal estimates of t given It. The filter consists of two sets of equations:

– Prediction Equation

– Update Equation

• Using the following notations


'

( | )

[( )( ) | )

t t t t t

t t t t t t t

m E I optimal estimator of based on I

C E m m I MSE matrix of m

Model Estimation

• Prediction Equations – Given mt-1 and Ct-1 at t-1, the optimal predictor of t and its

MSE matrix are

– The corresponding optimal predictor of yt at t-1 is

– The predictive error and its MSE matrix are


| 1 1 1

' '

| 1 1 1 1 1

( | )

[( )( ) | )

t t t t t t

t t t t t t t t t t t

m E I G m

C E m m I G C G W

| 1 1 | 1[ | ]t t t t t t ty E y I Fm

| 1 | 1 | 1

' '

| 1

( )

( )

t t t t t t t t t t t t t

t t t t t t t t

e y y y F m F m v

E e e Q FC F V

Model Estimation

• Update Equations – When new observation yt become available, the optimal

predictor mt|t-1 and its MSE matrix are updated using

– Kalman Gain Matrix gives the weight on new information et in the update equation for mt.


' 1

| 1 | 1 | 1

' 1

| 1 | 1

' 1

| 1 | 1 | 1

' 1

| 1

( )

:

t t t t t t t t t t t

t t t t t t t

t t t t t t t t t t

t t t t t

m m C F Q y F m

m C F Q e

C C C F Q FC

Note K C F Q Kalman Gain Matrix

Model Estimation

• Kalman Smoother – Once all data IT is observed, the optimal estimators E(t|IT)

can be computed using the backwards Kalman smoothing recursions

– The algorithm starts by setting mT|T = mT and CT|T = CT and then proceed backwards for t = T-1, …,1.


*

| 1| 1

' * *'

| | | 1| 1|

* ' 1

1 1|

( | ) ( )

[( )( ) | ) ( )

t T t T t t t T t t

t t T t t T T t T t t T t t t

t t t t t

E I m m C m G m

E m m I C C C C C

C C G C

Maximum Likelihood Estimation

• For a linear Gaussian state space model, let y denote the parameters of the model (embedded in the system matrices Ft, Gt, Vt, and Wt). The prediction error decomposition of the Gaussian log-likelihood function is


1

1

' 1

1 1

ln ( | ) ln ( | ; )

1 1ln(2 ) ln | ( ) | ( ) ( ) ( )

2 2 2

ˆ arg max ln ( | )

T

t t

t

N N

t t t t

t t

MLE

L y f y I

NTQ e Q e

L yy

y y

y y y y

y y

| 1 | 1

| 1 | 1

1/2 ' 1

| 1

~ ( ( ) ( ), ( ))

( ) ( ) ( ) ~ (0, ( ))

1( ; ) (2 ( )) exp ( ) ( ) ( )

2

t t t t t t

t t t t t t t t t

t t t t t t

y N F m Q

e y y y F m N Q

f y Q e Q e

y y y

y y y y

y y y y y

Forecasting

• The Kalman filter prediction equations produces in-sample 1-step ahead forecasts and MSE matrices.

• Out-of-sample h-step ahead predictions and MSE matrices can be computed from the prediction equations by extending the data set y1, …, yT with a set of h missing values.

• When yt is missing the Kalman filter reduces to the prediction step so a sequence of h missing values at the end of the sample will produce a set of h-step ahead forecasts

Time Series Data Analysis Using R

Forecasting

• One-Step Ahead Forecast at t = p, p+1,…


1| 1

1|0 0 0 1

2|1 1 1 2

| 1 1 1 0

ˆ

ˆ

ˆ

...

ˆ

t t t t t p

p

p

p p p p

y a b s

y a b s

y a b s

y a b s

1| 1 1|

1|

1| 1

2

1|

2

1|

2

1| 1

ˆ ˆ

ˆ(| |)

ˆ100 (| / |)

ˆ( )

ˆ( )

ˆ100 [( / ) ]

t t t t t

t t

t t t

t t

t t

t t t

y y

MAE mean

MAPE mean y

MSE mean

RMSE mean

RMSPE mean y

Example 1 (Continued)

• China Shanghai Common Stock

– High Frequency Daily Index

– Monthly Index Time Series

• Trend, Seasonality

– Dynamic Linear Model

• Correlation with Exchange Rate?


Example 2

• Chinese Yuan vs. U.S. Dollar

– Exchange Rate Time Series

• Trend

• Intervention

– Dynamic Linear Model

• Correlation with Stock Market?

• ARMA


Download - Linear Gaussian State Space Models - Portland State …web.pdx.edu/~crkl/WISE2016/TSAR-2.pdfDynamic Linear Models •Local Trend Model 2 –State Space Model Representation Time Series

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