Modern methods The classical approach: MethodProsCons Time series regression Easy to implement...
-
date post
21-Dec-2015 -
Category
Documents
-
view
218 -
download
0
Transcript of Modern methods The classical approach: MethodProsCons Time series regression Easy to implement...
Modern methods
The classical approach:
Method Pros Cons
Time series regression • Easy to implement
• Fairly easy to interpret
• Covariates may be added (normalization)
• Inference is possible (though sometimes questionable)
• Static
• Normal-based inference not generally reliable
• Cyclic component hard to estimate
Decomposition • Easy to interpret
• Possible to have dynamic seasonal effects
• Cyclic components can be estimated
• Descriptive (no inference per def)
• Static in trend
Explanation to the static behaviour:
The classical approach assumes all components except the irregular ones (i.e. t and IRt ) to be deterministic, i.e. fixed functions or constants
To overcome this problem, all components should be allowed to be stochastic, i.e. be random variates.
A time series yt should from a statistical point of view be treated as a stochastic process.
We will interchangeably use the terms time series and process depending on the situation.
Stationary and non-stationary time series
20
10
0
100908070605040302010
Stationary
Index
3000
2000
1000
0
300200100
Non-stationary
Index
Characteristics for a stationary time series:
• Constant mean
• Constant variance
A time series with trend is non-stationary!
Auto Regressive,
Integrated,
Moving Average
Box-Jenkins models
A stationary times series can be modelled on basis of the serial correlations in it.
A non-stationary time series can be transformed into a stationary time series, modelled and back-transformed to original scale (e.g. for purposes of forecasting)
ARIMA – models
These parts can be modelled on a stationary series
This part has to do with the transformation
Different types of transformation
1. From a series with linear trend to a series with no trend:
First-order differences zt = yt – yt – 1
MTB > diff c1 c2
Note that the differences series varies around zero.
20
15
10
5
0
linear trendno trend
Variable
2. From a series with quadratic trend to a series with no trend:
Second-order differences
wt = zt – zt – 1 = (yt – yt – 1) – (yt – 1 – yt – 2) = yt – 2yt – 1 + yt – 2
MTB > diff 2 c3 c4
20
15
10
5
0
quadratic trendno trend 2
Variable
3. From a series with non-constant variance (heteroscedastic) to a series with constant variance (homoscedastic):
Box-Cox transformations (per def 1964)
Practically is chosen so that yt + is always > 0
Simpler form: If we know that yt is always > 0 (as is the usual case for measurements)
0 and 0for ln
0 and 0for 1
tt
tt
t
yy
yy
yg
asticity heterosced extreme if1
asticity heteroscedheavy if1
asticity heterosced pronounced ifln
- " -
asticity heteroscedmodest if4
t
t
t
t
t
t
y
y
y
y
y
yg
The log transform (ln yt ) usually also makes the data ”more” normally distributed
Example: Application of root (yt ) and log (ln yt ) transforms
25
20
15
10
5
0
originalrootlog
Variable
AR-models (for stationary time series)
Consider the model
yt = δ + ·yt –1 + at
with {at } i.i.d with zero mean and constant variance = σ2
and where δ (delta) and (phi) are (unknown) parameters
Set δ = 0 by sake of simplicity E(yt ) = 0
Let R(k) = Cov(yt,yt-k ) = Cov(yt,yt+k ) = E(yt ·yt-k ) = E(yt ·yt+k )
R(0) = Var(yt) assumed to be constant
Now:
R(0) = E(yt ·yt ) = E(yt ·( ·yt-1 + at ) = · E(yt ·yt-1 ) + E(yt ·at ) =
= ·R(1) + E(( ·yt-1 + at ) ·at ) = ·R(1) + · E(yt-1 ·at ) + E(at ·at )=
= ·R(1) + 0 + σ2 (for at is independent of yt-1 )
R(1) = E(yt ·yt+1 ) = E(yt ·( ·yt + at+1 ) = · E(yt ·yt ) + E(yt ·at+1 ) =
= ·R(0) + 0 (for at+1 is independent of yt )
R(2) = E(yt ·yt+2 ) = E(yt ·( ·yt+1 + at+2 ) = · E(yt ·yt+1 ) +
+ E(yt ·at+2 ) = ·R(1) + 0 (for at+1 is independent of yt )
R(0) = ·R(1) + σ2
R(1) = ·R(0) Yule-Walker equations
R(2) = ·R(1)
…
R(k ) = ·R(k – 1) =…= k·R(0)
R(0) = 2 ·R(0) + σ2
2
2
1)0(
R
Note that for R(0) to become positive and finite (which we require from a variance) the following must hold:
112
This in effect the condition for an AR(1)-process to be weakly stationary
Note now that
)0()(
)0()0(
)(
)()(
),(),(
RkR
RR
kR
yVaryVar
yyCovyyCorr
ktt
kttkktt
kk
k R
R
)0(
)0(
ρk is called the Autocorrelation function (ACF) of yt
”Auto” because it gives correlations within the same time series.
For pairs of different time series one can define the Cross correlation function which gives correlations at different lags between series.
By studying the ACF it might be possible to identify the approximate magnitude of
Examples: ACF for AR(1), phi=0.1
0
0.2
0.4
0.6
0.8
1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
k
ACF for AR(1), phi=0.3
0
0.2
0.4
0.6
0.8
1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
k
ACF for AR(1), phi=0.5
0
0.2
0.4
0.6
0.8
1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
ACF for AR(1), phi=0.8
0
0.2
0.4
0.6
0.8
1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
ACF for AR(1), phi=0.99
0
0.2
0.4
0.6
0.8
1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
ACF for AR(1), phi=-0.1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
ACF for AR(1), phi=-0.5
-1-0.8-0.6-0.4-0.20
0.20.40.60.81
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
ACF for AR(1), phi=-0.8
-1-0.8-0.6-0.4-0.20
0.20.40.60.81
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
The look of an ACF can be similar for different kinds of time series, e.g. the ACF for an AR(1) with = 0.3 could be approximately the same as the ACF for an Auto-regressive time series of higher order than 1 (we will discuss higher order AR-models later)
To do a less ambiguous identification we need another statistic:
The Partial Autocorrelation function (PACF):
υk = Corr (yt ,yt-k | yt-k+1, yt-k+2 ,…, yt-1 )
i.e. the conditional correlation between yt and yt-k given all observations in-between.
Note that –1 υk 1
A concept sometimes hard to interpret, but it can be shown that
for AR(1)-models with positive the look of the PACF is
and for AR(1)-models with negative the look of the PACF is
0.00
1.00
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
k
-1
0
1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
k
Assume now that we have a sample y1, y2,…, yn from a time series assumed to follow an AR(1)-model.
Example:
Monthly exchange rates DKK/USD 1991-1998
0
2
4
6
8
10
The ACF and the PACF can be estimated from data by their sample counterparts:
Sample Autocorrelation function (SAC):
if n large, otherwise a scaling
might be needed
Sample Partial Autocorrelation function (SPAC)
Complicated structure, so not shown here
n
tt
kt
kn
tt
k
yy
yyyyr
1
2
1
)(
))((
The variance function of these two estimators can also be estimated
Opportunity to test
H0: k = 0 vs. Ha: k 0
or
H0: k = 0 vs. Ha: k 0
for a particular value of k.
Estimated sample functions are usually plotted together with critical limits based on estimated variances.
Example (cont) DKK/USD exchange:
SAC:
SPAC: Critical limits
Ignoring all bars within the red limits, we would identify the series as being an AR(1) with positive .
The value of is approximately 0.9 (ordinate of first bar in SAC plot and in SPAC plot)
Higher-order AR-models
AR(2): or
yt-2 must be present
AR(3):
or other combinations with 3 yt-3
AR(p):
i.e. different combinations with p yt-p
tttt ayyy 2211
ttt ayy 22
ttttt ayyyy 332211
tptptt ayyy ...11
Stationarity conditions:
For p > 2, difficult to express on closed form.
For p = 2:
The values of 1 and 2 must lie within the blue triangle in the figure below:
tttt ayyy 2211
Typical patterns of ACF and PACF functions for higher order stationary AR-models (AR( p )):
ACF: Similar pattern as for AR(1), i.e. (exponentially) decreasing
bars, (most often) positive for 1 positive and alternating for 1 negative.
PACF: The first p values of k are non-zero with decreasing
magnitude. The rest are all zero (cut-off point at p )
(Most often) all positive if 1 positive and alternating if 1 negative
Examples:
AR(2), 1 positive:
AR(5), 1 negative:
PACF
0
1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
ACF
0
1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
PACF
-1
0
1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
ACF
-1
0
1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15