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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 Time Series Data Analysis Using R
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
3 Time Series Data Analysis Using R
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
•
4 Time Series Data Analysis Using R
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
Dynamic Linear Models
• Local Level Model
– State Space Model Representation
5 Time Series Data Analysis Using R
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
Dynamic Linear Models
• Local Trend Model
– State Space Model Representation
6 Time Series Data Analysis Using R
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
Dynamic Linear Models
• Time Varying Regression Parameters
– State Space Model Representation
7 Time Series Data Analysis Using R
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
Dynamic Linear Models
• Model Estimation
– Filtering (filtered estimate of )
– Smoothing (smoothed estimate of )
8 Time Series Data Analysis Using R
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
9 Time Series Data Analysis Using R
'
( | )
[( )( ) | )
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
10 Time Series Data Analysis Using R
| 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.
11 Time Series Data Analysis Using R
' 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.
12 Time Series Data Analysis Using R
*
| 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
13 Time Series Data Analysis Using R
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,…
15 Time Series Data Analysis Using R
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?
16 Time Series Data Analysis Using R
Example 2
• Chinese Yuan vs. U.S. Dollar
– Exchange Rate Time Series
• Trend
• Intervention
– Dynamic Linear Model
• Correlation with Stock Market?
• ARMA
17 Time Series Data Analysis Using R
