Reliability and Risk Analysis
Dynamic Models for Risk Prediction
Introduction to time series analysis
Linear dynamic models
Introduction to time series analysis
Time series – a finite sequence of real values of a monitored indicator measured at
certain time intervals
Example: dollar exchange rate, production volume per month, etc.
Jiří Neubauer
Dynamic Models for Risk Prediction
Introduction to time series analysis
Linear dynamic models
Introduction to time series analysis
Jiří Neubauer
Dynamic Models for Risk Prediction
Introduction to time series analysis
Linear dynamic models
Introduction to time series analysis
Jiří Neubauer
Dynamic Models for Risk Prediction
Introduction to time series analysis
Linear dynamic models
Introduction to time series analysis
Jiří Neubauer
Dynamic Models for Risk Prediction
Introduction to time series analysis
Linear dynamic models
Introduction to time series analysis
Objective of time series analysis: understand the mechanism that determines the values
of the monitored variables and predict its evolution.
We will use several time series models to understand evolution of analyzed variables –
the model is usually described by stochastic equations.
Jiří Neubauer
Dynamic Models for Risk Prediction
Introduction to time series analysis
Linear dynamic models
Introduction to time series analysis
Methods of time series analysis:
expert methods
graphical analysis
time series decomposition
econometric models
Box–Jenkins methodology
spectral analysis
Jiří Neubauer
Dynamic Models for Risk Prediction
Introduction to time series analysis
Linear dynamic models
Linear dynamic models
Example of the linear dynamic model
Ct = α + βCt−1 + γXt + δPt + t ,
where public expenditure on the purchase of consumer goods Ct in year t are explained
using past values of Ct−1 and also using disposable population cash income Xt and the
consumer goods price index Pt (α, β, γ and δ are the parameters t denotes white noise)
Jiří Neubauer
Dynamic Models for Risk Prediction
Introduction to time series analysis
Linear dynamic models
Linear dynamic models
Consider a single-equation linear models expressed in the form of a single equation
Yt = β1 Xt1 + β2 Xt2 + · · · + βk X tk + t , where t = 1, 2, . . . , n.
Yt is response variable Y in time t, Xt1 , . . . , Xtk are explanatory variables X1 , . . . , Xk in
time t, β1 , . . . , βk are unknown parameters (see OLS), t is a error term.
Jiří Neubauer
Dynamic Models for Risk Prediction
Introduction to time series analysis
Linear dynamic models
Linear dynamic models
Example. Logging can be influenced by the following factors: reforestation, forest
fertilization vegetation, forest fires and damage by animals. Based on the annual time
series (1995–2004) assess the real contribution of these factors and construct an
econometric model.
Logging (1000 m3 )
Reforestation (ha)
forest fertilization (1000 ha)
forest fires (des. ha)
damage by animals (mil. CZK)
1995
47.52
46
7.86
22.7
41.8
1996
51.64
89
5.13
51.9
53.8
1997 1998 1999 2000 2001
79.65 101.42 105.35 83.45 80.54
66
100
101
46
47
3.49 4.48
3.79 23.67 17.23
19.5 34.2
18.9 21.5 6.8
61.1
8.2
25.8 36.4 34.5
2002
72.79
61
14.31
6.6
65.3
2003
60.45
84
5.25
35.1
27.4
2004
67.62
50
7.11
17.7
33.0
We get the model
Yt = β1 + β2 Xt2 + β3 Xt3 + β4 Xt4 + β5 Xt5 + t , where t = 1, 2, . . . , 10,
Y denotes logging , X2 reforestation, X3 forest fertilization, X4 forest fires and X5
damages by animals.
Jiří Neubauer
Dynamic Models for Risk Prediction
Introduction to time series analysis
Linear dynamic models
Linear dynamic models
Jiří Neubauer
Dynamic Models for Risk Prediction
Introduction to time series analysis
Linear dynamic models
Lineární dynamické modely
We can rewrite the model in the form
Y = Xβ + ,
where



Y=

y1
y2
..
.
yn






, = 


1
2
..
.
n



,


1
 .
X =  ..
1
Jiří Neubauer
x12
..
.
xn2
···
..
.
···


x1k


.. 
. , β = 

xnk
Dynamic Models for Risk Prediction
β1
β2
..
.
βk



.

Introduction to time series analysis
Linear dynamic models
Linear dynamic models
OLS parameter estimates are
b = X0 X −1 X0 Y.
β
intercept
reforestation
fertilization
fires
damages
Estimate
46.7804
0.8147
1.0182
−1.0373
−0.3353
St. error
29.4463
0.2876
0.8280
0.3749
0.2605
t-test
1.59
2.83
1.23
−2.77
−1.29
p-value
0.1730
0.0365
0.2735
0.0395
0.2544
intercept
reforestation
fires
Estimate
50.5821
0.7372
−1.1242
St. error
14.8608
0.2523
0.4149
t-test
3.40
2.92
−2.71
p-value
0.0114
0.0223
0.0302
bt = 50.5821 + 0.7372X2t − 1.1242X4t
Y
Jiří Neubauer
Dynamic Models for Risk Prediction
Introduction to time series analysis
Linear dynamic models
Model verification
histogram,
QQ plot,
Shapiro-Wilk test (shapiro.test), Lilliefors test (lillie.test), Jarque-Bera test
(jarque.bera.test) and so on.
Jiří Neubauer
Dynamic Models for Risk Prediction
Introduction to time series analysis
Linear dynamic models
Residual analysis
Jiří Neubauer
Dynamic Models for Risk Prediction
Introduction to time series analysis
Linear dynamic models
Residual analysis
Residuals et = yt − ybt should be uncorrelated. This assumption can be verified:
Durbin-Watson’s test (dwtest)
Pn
DW =
(et − et−1 )2
t=2P
.
n
2
t=1 et
autocorrelation and partial autocorrelation function, portmanteau test
Jiří Neubauer
Dynamic Models for Risk Prediction