timeseries presentation
TRANSCRIPT
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An Introduction to Time Series
Ginger Davis
VIGRE Computational Finance Seminar Rice University
November 26, 2003
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What is a Time Series?
• Time Series– Collection of observations
indexed by the date of each observation
• Lag Operator– Represented by the symbol L
• Mean of Yt = μt
Tyyy ,,, 21
1 tt xLx
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White Noise Process
• Basic building block for time series processes
0
022
t
t
t
tt
EE
E
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White Noise Processes, cont.
• Independent White Noise Process– Slightly stronger condition that and are
independent• Gaussian White Noise Process
2,0~ Nt
t
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Autocovariance
• Covariance of Yt with its own lagged value
• Example: Calculate autocovariances for:
jtjtttjt YYE
jttjttjt
tt
EYYEY
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Stationarity
• Covariance-stationary or weakly stationary process– Neither the mean nor the autocovariances depend on
the date t
jjtt
t
YYEYE
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Stationarity, cont.
• 2 processes– 1 covariance stationary, 1 not covariance
stationary
tt
tt
tYY
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Stationarity, cont.
• Covariance stationary processes– Covariance between Yt and Yt-j depends only on
j (length of time separating the observations) and not on t (date of the observation)
jj
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Stationarity, cont.
• Strict stationarity– For any values of j1, j2, …, jn, the joint
distribution of (Yt, Yt+j1, Yt+j2
, ..., Yt+jn) depends
only on the intervals separating the dates and not on the date itself
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Gaussian Processes
• Gaussian process {Yt}– Joint density
is Gaussian for any • What can be said about a covariance stationary
Gaussian process?
nnjtjt jtjttYYY yyyf
,,,111 ,,,
njjj ,,, 21
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Ergodicity
• A covariance-stationary process is said to be ergodic for the mean if
converges in probability to E(Yt) as
T
tty
Ty
1
1
T
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Describing the dynamics of a Time Series
• Moving Average (MA) processes• Autoregressive (AR) processes• Autoregressive / Moving Average (ARMA)
processes• Autoregressive conditional heteroscedastic
(ARCH) processes
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Moving Average Processes
• MA(1): First Order MA process
• “moving average”– Yt is constructed from a weighted sum of the two
most recent values of .
1 tttY
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Properties of MA(1)
0
1
2
2
212
22
11
2111
22
21
21
2
21
2
jtt
ttttttt
tttttt
tttt
ttt
t
YYE
E
EYYE
E
EYE
YE
for j>1
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MA(1)
• Covariance stationary– Mean and autocovariances are not functions of time
• Autocorrelation of a covariance-stationary process
• MA(1)0
j
j
222
2
1 11
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Autocorrelation Function for White Noise:
0.0
0.2
0.4
0.6
0.8
1.0
0 5 10 15 20
Lag
Autocorrelation
ttY
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Autocorrelation Function for MA(1): 18.0 tttY
0.0
0.2
0.4
0.6
0.8
1.0
0 5 10 15 20
Lag
Autocorrelation
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Moving Average Processesof higher order
• MA(q): qth order moving average process
• Properties of MA(q)
qtqttttY 2211
qj
qj
j
jqqjjjj
q
,0
,,2,1,
12
2211
2222
210
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Autoregressive Processes
• AR(1): First order autoregression
• Stationarity: We will assume• Can represent as an MA
ttt YcY 1
1
22
1
22
1
1 ttt
tttt
c
cccY
:)(
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Properties of AR(1)
2
2
242
22
21
20
1
1
1
ttt
t
E
YE
c
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Properties of AR(1), cont.
jj
j
j
j
jjj
jtjtjtjtj
ttt
jttj
E
YYE
0
22
242
242
22
122
1
1
1
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Autocorrelation Function for AR(1): ttt YY 18.0
0.0
0.2
0.4
0.6
0.8
1.0
0 5 10 15 20
Lag
Autocorrelation
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Autocorrelation Function for AR(1): ttt YY 18.0
-0.5
0.0
0.5
1.0
0 5 10 15 20
Lag
Autocorrelation
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Gaussian White Noise
0 20 40 60 80 100
-2-1
01
2
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AR(1),
0 20 40 60 80 100
-3-2
-10
12
5.0
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AR(1),
0 20 40 60 80 100
-20
24
9.0
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AR(1),
0 20 40 60 80 100
-4-2
02
49.0
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Autoregressive Processes of higher order
• pth order autoregression: AR(p)
• Stationarity: We will assume that the roots of the following all lie outside the unit circle.
tptpttt YYYcY 2211
01 221 p
p zzz
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Properties of AR(p)
• Can solve for autocovariances / autocorrelations using Yule-Walker equations
p
c
211
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Mixed Autoregressive Moving Average Processes
• ARMA(p,q) includes both autoregressive and moving average terms
qtqtt
tptpttt YYYcY
2211
2211
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Time Series Models for Financial Data
• A Motivating Example– Federal Funds rate– We are interested in forecasting not only the
level of the series, but also its variance.– Variance is not constant over time
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U. S. Federal Funds Rate
Time
1955 1960 1965 1970 1975
24
68
1012
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Modeling the Variance
• AR(p):• ARCH(m)
– Autoregressive conditional heteroscedastic process of order m
– Square of ut follows an AR(m) process
– wt is a new white noise process
tptpttt uyyycy 2211
tmtmttt wuuuu 22
222
112
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References
• Investopia.com• Economagic.com• Hamilton, J. D. (1994), Time Series
Analysis, Princeton, New Jersey: Princeton University Press.