Time series factorial models
with uncertainty measures:
applications to ARMA processes
and financial data
V. Terraza, C. Toque
(speaker: [email protected])
On the topic of time series identification
• the (justified) domination of ARMA methods
(autocorrelations, corner method, eigenvalue … )
√ these techniques are fundamentally numerical
• we consider alternative techniques based on a much
more qualitative perspective
√ to identify structural changes or discontinuities in time
series
A novel approach (1)
• using entropies or uncertainty measures
√ to identify time series factorial models
• applied to ARMA process identification
√ for calibration purposes
• applied to financial time series
√ for decision rules
A novel approach (2)
• based on an alphabet of states of a chronological
time series
• integrates at the same time
√ the technical analysis of oscillators (Wilder, 1978)
√ the examination of turning points and Shannon information
theory (Shannon, 1948)
• along with factorial techniques
of visualization like MCA
Data
• 36 simulated time series : stationary AR(1)
and MA(1)
• a set of 1500 equity funds with both
Morningstar and Europerformace agencies
The simulated AR(1) and MA(1)
• 18 AR(1) and 18 MA(1) processes
√ with coefficients between -0.9 and +0.9
and with a step of 0.1
√ for each model: length=500
• the temporal matrix is of dimensions
(36x500)
The financial data
• 1500 equity funds domiciled
√ in Belgium, Britain, France, Ireland, Italy,
Luxembourg, Spain and Switzerland
• the period of observation
√ from 1 January 2005 to 31 december 2007
• each fund is described by
√ one series of daily returns
√ two series of monthly ratings from 1 to 5
(Morningstar) or 6 (Europerformance) stars
The structural analysis (1)
• The series of symbols: 1st step
√ the ‘direct’ symbols on the monthly rating series:
m=6 with 1 for 1*, …, 5 for 5*, and 6 for 5h
√ the ‘computed’ symbols by first differences on simulated
AR(1) and MA(1) or on the daily returns :
m=2 with 0 for a decline and 1 for an advance
• The probabilities of k-symbols: 2nd step
√ the sequences of k-symbols, in number mk
√ the probabilities
Pk   pk ,1 , pk ,2 ..., pk , mk 
The structural analysis (2)
• How to qualify this information ? 3rd step
√ the entropy measures of order k, also denoted Shk
mk
- min. incertitude if all the probabilities are
concentrated in one modality
i 1
- max. incertitude if all the probabilities are
equal (1/m or equiprobability)
H k ( Pk )   pk ,i log 2 pk ,i
√ the conditional entropies of order k, also denoted Condk
h1  H1 ( P1 )
hk  H k ( Pk / Pk 1 )  H k ( Pk )  H k 1 ( Pk 1 )
√ the residual entropies of order k, also denoted Resk
or reductions of incertitude
d k  hk  hk 1
The structural analysis (3) –
AR(1) and MA(1)
• Vectors of entropies and MCA: 4rth
step
√ for each series : a vector of shannon, conditonal and residual
entropies of dimension 3k-1
√ for the simulated processes: a matrix of dimension (36x 3k-1
√ on this matrix of entropies, a MCA
on three modalities is built
The first graphic reference model:
scores graphic for AR(1) and MA(1) processes
Results (1)
• the model of series and varaibles is of better
quality in (F1,F2)
• we distinguish 3 classes of processes:
√ the class of negative AR(1) and positive MA(1)
√ the class of positive AR(1) and negative MA(1)
√ a class of ‘weak’ processes with coefficients <0.3
characterised by a residual entropy of order1
(mod 1, 2 or 3 for the weak processes)
The structural analysis (3) –
financial data
• Vectors of entropies and MCA: 4rth
step
√ for each series ( daily returns, 2 ratings): a vector of conditonal
entropies (Cond1, …, Cond(k)) of dimension k
√ for each fund: a vector of entropies of dimension 2k or 3k
√ for the panel with all the funds: a matrix of dimension (1500x2k )
or (1500x3k)
√ on each matrix of entropies, a MCA
on three modalities is built
MCA scores for 1500 European funds with uncertainties
measured on Morningstar and Europerformance ratings: 2005 - 2007
Results (2)
• the F1 axis mainly expresses the uncertainty measured on
Europerformance ratings
• the F2 axis mainly expresses the uncertainty measured on
Morningstar ratings
• we find groups of funds with a minimum uncertainty on the
Morningstar ratings but a maximum uncertainty on
Europerformance …
and conversely.
• the Europerformance ratings are not compatible
with Morningstar ratings
MCA scores for 1500 European funds …
with in more the returns: 2005 - 2007
Results (3)
• the minimum uncertainty measured on on the
returns of year 2007 is associated to the minimum
uncertainties measured on the Morningstar ratings
for the period 2005 to 2007 (but not for
Europerformance)
• we identify a group for which the Morningstar
ratings reflect the future returns.