Download Integer Valued AR Processes with Explanatory Variables

Integer Valued AR Processes with
Explanatory Variables
V. Enciso-Mora, P. Neal & T. Subba Rao
First version: 12 December 2008
Research Report No. 20, 2008, Probability and Statistics Group
School of Mathematics, The University of Manchester
Integer valued AR processes with explanatory variables
Vı́ctor Enciso-Mora∗(University of Manchester), Peter Neal†(University of Manchester)
and T. Subba Rao (University of Manchester)
November 7, 2008
Address: School of Mathematics, University of Manchester, Alan Turing Building, Oxford Rd, Manchester, M13 9PL, United Kingdom.
E-mail: [email protected]
Abstract
Integer valued AR (INAR) processes are perfectly suited for modelling count data. We consider
the inclusion of explanatory variables into the INAR model to extend the applicability of INAR models
and give an alternative to Poisson regression models. An efficient MCMC algorithm is constructed to
analyze the model and incorporates both explanatory variable and order selection. The methodology
is illustrated by analyzing monthly polio incidences in the USA 1970-1983 and claims from the logging
industry to the British Columbia Workers’ Compensation Board 1985-1994.
Keywords: Integer valued time-series, Reversible jump MCMC, count data, explanatory variables.
AMS Primary: 62M10; AMS Secondary: 65C40.
1
Introduction
There has been considerable recent interest in developing models and statistical inference for integer
valued time series. The modeling has primarily been divided into two approaches, Poisson regression
models, see for example Davis et al. (2003) and Davis et al. (2005) and integer valued ARMA models,
see for example McCabe and Martin (2005) and Jung and Tremayne (2006).
∗ The
author would like to thank CONACYT (Consejo Nacional de Ciencia y Tecnologı́a, México) for funding this
research.
† corresponding author
1
Poisson regression models assume that an integer valued time series {Xt } given explanatory variables
wt is generated by Xt ∼ P o(exp(wtT β)), where β is a vector of parameters. (Throughout this paper all
vectors are taken to be column vectors.) This has been extended in Davis et al. (2000) to incorporate
a more explicit dependence between the {Xt } with Xt |Wt ∼ P o(exp(wtT β + Wt )), where {Wt } is a real
valued stationary process with E[exp(Wt )] = 1. This model is very flexible and statistical inference is
usually relatively straightforward. In contrast, integer valued AR (INAR) models are discrete analogues
of the (standard, real valued) AR model. The INAR(p) model is given by
Xt =
p
X
j=1
αj ◦ Xt−j + Zt ,
(1.1)
where the {Zt } are independent and identically distributed Poisson random variables and ’◦’ denotes a
D
binomial operator. That is, for any 0 ≤ γ ≤ 1, and nonnegative integer random variable Y , γ ◦ Y =
Bin(Y, γ). Statistical inference for INAR models is generally more complicated than for the Poisson
regression models and most attention has been devoted to the INAR(1) model. However, recently in
Neal and Subba Rao (2007) an efficient MCMC algorithm has been developed for INARMA processes
of known AR and MA orders with extensions to unknown AR and MA orders given in Enciso-Mora et
al. (2008). The main benefit of the INARMA models over the Poisson regression models is their similarity
to standard ARMA models. In fact, the stationary and invertibility conditions of ARMA models hold
for INARMA models, see Latour (1997) and Latour (1998).
The aim of this paper is to extend the analysis of INAR models to incorporate explanatory variables.
That is, we look to increase the flexibility of INAR models whilst maintaining the AR structure of the
model. Thus seeking an alternative to Poisson regression models for integer valued time series. Moreover,
the two approaches are complimentary and are appropriate to different types of data with the Poisson
regression models being multiplicative in structure through eWt , whilst the INAR model has additive
structure through the binomial operator. As with our earlier work, Neal and Subba Rao (2007) and
Enciso-Mora et al. (2008), an MCMC algorithm will be introduced to perform statistical inference for
the model. The only previous work, that we are aware of, which incorporates explanatory variables into
an INAR model is Brännäs (1995).
The paper is structured as follows. The model is introduced in Section 2. In Section 3, the MCMC
algorithm is constructed. The algorithm includes explanatory variable and AR order selection which
enables us to determine the most parsimonious model. Finally, in Section 4 the MCMC algorithm is
applied to two real-life data sets, the total number of monthly cases of polio in the USA from January
2
1970 to December 1983 and the total number of monthly claims from the logging industry to the British
Columbia Workers’ Compensation Board from January 1985 to December 1994.
2
Model
Suppose that for each time point there are r explanatory variables. For t ∈ Z and i = 1, 2, . . . , r, let
wt,i denote the value of the ith explanatory variable at time t and let wt = (wt,0 , wt,1 , . . . , wt,r ), where
wt,0 = 1 for all t ∈ Z. Let p denote the maximum AR order of the model and for j = 1, 2, . . . p let
δ j = (δj,0 , δj,1 , . . . , δj,r ). Let γ = (γ0 , γ1 , . . . , γr ). Then for t ∈ Z, the INAR(p) model with explanatory
variables is given by
p
X
αt,j ◦ Xt−j + Zt ,
(2.1)
ª−1
©
αt,j = 1 + exp(wtT δ j )
(2.2)
λt = exp(wtT γ).
(2.3)
Xt =
j=1
where Zt ∼ P o(λt ),
and
The special case where p = 1 was considered in Brännäs (1995). In Brännäs (1995), separate explanatory
variables were used for αt,1 and λt and is covered by the current set up by fixing some of the components
of δ j and γ equal to 0. The binomial operator could be replaced by some generalised Steutel and van
Harn operator (see, Steutel and van Harn (1979) and Latour (1997)). Also alternative forms for αt,j and
λt can be used so long as for all t ∈ Z and j = 1, 2, . . . , p, 0 ≤ αt,j ≤ 1 and λt ≥ 0. However, for ease of
exposition we restrict attention to the model defined by (2.1), (2.2) and (2.3). The model defined by (2.1)
is the full model. It will often be the case that a simpler model which does not include all the explanatory
variables or all the AR terms will suffice. Therefore we assume that there exists R ⊆ {1, 2, . . . , r} and
A ⊆ {1, 2, . . . , p} such that for i 6∈ A, αt,i = 0 and for j 6∈ R, δi,j = 0 and γj = 0, where A and R are
unknown and are parameters in the model to be estimated.
Finally, it is important to note that the explanatory variables can be used to model a linear trend or
periodicity. Unlike standard AR processes for INAR processes trends and periodicity can not easily
be removed by transforming the original time series since any transformation would need to preserve
3
the integer nature of the data. Therefore if there are trends and periodicity in the data these can be
incorporated through explanatory variables.
Suppose that αt,j ≡ 0 for all t ∈ Z and j = 1, 2, . . . , p. Then Xt ∼ P o(exp(wtT γ)) and there is no
dependence between successive time points. This is equivalent to the Poisson regression model with
eWt ≡ 1 for all t ∈ Z. Thus the Poisson regression model and the INAR model reduce to the same model
in the absence of temporal dependence. Therefore the differences in modelling arise from the ways in
which temporal dependence are included in the model, as a latent stationary process in Poisson regression
models and as an observation driven binomial operator in INAR models.
3
3.1
MCMC algorithm
Likelihood of INAR(p) model with explanatory variables
For n ≥ 1, let x = (x1−p , x2−p , . . . , xn ) denote observations from the integer-valued stochastic process
{Xt }. As in Neal and Subba Rao (2007) and Enciso-Mora et al. (2008), statistical inference is greatly
facilitated by the use of data augmentation. For t ∈ Z and j = 1, 2, . . . , p, let Yt,j ≡ αt,j ◦ Xt,j . For t ≥ 1,
Pp
let yt = (yt,1 , yt,2 , . . . , yt,p ), y = (y1 , y2 , . . . , yn ) and zt = (z1 , z2 , . . . , zt ). Note that zt = xt − j=1 yt,j .
For t = 1, 2, . . . , n, let xt = (x1−p , x2−p , . . . , xt ) and let w = (w1−p , w2−p , . . . , wn ). Note that each of
Yt,1 , Yt,2 , . . . , Yt,p and Zt are independent given (γ, δ, xt−1 ). Thus for t = 1, 2, . . . , n,
 z
o
n¡
¢ yt,i
Pp
xt−i
 λt t e−λt × Qp
xt−i −yt,i
(1
−
α
)
α
zt + i=1 yt,i = xt ,
t,i
t,i
i=1
zt !
yt,i
ft (yt , zt |x, w, δ, γ) ∝
(3.1)
 0
otherwise.
Therefore
f (y, zn |x, w, δ, γ)
=
n
Y
t=1
n
Y
ft (yt , zt |x, w, δ, γ)
p
λzt t −λt Y
e
×
∝
z!
t=1 t
i=1
subject to the constraint zt +
Pp
i=1
½µ
¶
¾
xt−i yt,i
xt−i −yt,i
αt,i (1 − αt,i )
,
yt,i
(3.2)
yt,i = xt for t = 1, 2, . . . , n. For i = 0, 1, . . . , r, γi and δj,i (j =
1, 2, . . . , p) are real-valued. Therefore in the absence of conjugate priors, we assign independent Gaussian
priors to each of γi (i = 0, 1, . . . , r) and δj,i (j = 1, 2, . . . , p; i = 0, 1, . . . , r). For simplicity of exposition
we take π(γi ), π(δj,i ) ∼ N (µ, σP2 ) and throughout this paper take µ = 0 and σP = 1. As mentioned
in Section 2, we assume that there is a maximum (fixed) model order p ≥ 1 and a (unknown) subset
A ⊆ {1, 2, . . . , p} such that for j ∈ A, αt,j 6= 0 and for j 6∈ A, αt,j = 0, and hence, yt,j = 0. Therefore we
4
shall make inference about A with similar analysis for real-valued AR processes considered in Troughton
and Godsill (1997). The prior chosen for A is
π(A) ∝
√ −|A|
n
A ⊆ {1, 2, . . . , p}.
(3.3)
The prior on A penalises models with larger values of |A| and is motivated by the BIC-penalisation,
Schwarz (1978), which has been used effectively in Enciso-Mora et al. (2008). Furthermore, we will not
assume that all of the explanatory variables are necessary and will perform explanatory variable selection.
That is, we assume that there exists R ⊆ {1, 2, . . . , r} such that for i ∈ R, γi 6= 0 and δj,i 6= 0 (j ∈ A)
and for i 6∈ R, γi = 0 and δj,i = 0 (j ∈ A). Finally, we impose a BIC-penalisation prior on R,
π(R) ∝
√ −|R|
n
R ⊆ {1, 2, . . . , r}.
(3.4)
Note that we assume that there is an intercept term, so γ0 6= 0 and δj,0 6= 0 (j ∈ A).
3.2
Details of the MCMC algorithm
The MCMC algorithm naturally separates into four parts, parameter updates γ and δ, updating (y, zn ),
updating R and updating A. We shall describe each part of the MCMC algorithm in turn with the latter
two parts requiring reversible jump MCMC, Green (1995). The algorithm is then run for the desired
number of iterations to obtain a sample from the posterior distribution of (A, R, γ, δ).
For j ∈ A and i ∈ R, we update, one at a time, each γi and δj,i using random walk Metropolis. For
′
2
example, we propose a new value for δj,i , δj,i
∼ N (δj,i , σδ;j,i
), where σδ;j,i is chosen based upon pilot runs
of the algorithm such that approximately 40% of proposed moves are accepted, see Gelman et al. (1995).
The acceptance probability is straightforward to calculate using (3.2) and the prior for δj,i .
For t = 1, 2, . . . , n, we update one set of components (yt , zt ) at a time. We use an efficient independent
sampler which is essentically identical to that introduced in Neal and Subba Rao (2007). For j ∈ A,
P
P
′
′
′
and accept the
. Set zt′ = xt − j∈A yt,j
∼ Bin(xt−j , αt,j ), resampling if xt ≤ j∈A yt,j
sample yt,j
proposed values (yt′ , zt′ ) with probability
¾
½
zt ! zt′ −zt
.
min 1, ′ λt
zt !
Explanatory variable selection is performed as follows. Select K uniformly at random from {1, 2, . . . , r}
and propose to change the inclusion/exclusion status of explanatory variable K. That is, if K ∈ R,
we propose that K 6∈ R and if K 6∈ R, we propose that K ∈ R. We shall describe the inclusion of a
5
explanatory variable move in detail with the exclusion of a explanatory variable move simply being the
reverse procedure. In order to use reversible jump MCMC it is necessary to define a bijection between
the parameters in the current model and the parameters in the proposed model. This can be done by
extending the state space of one or both of the models to include auxiliary variables. Therefore for
2
2
j ∈ A, let Sj ∼ N (0, σR
) and let S̃ ∼ N (0, σR
), where σR is chosen to produce efficient between model
moves and σR = 1 is found to be more than adequate for the examples considered in this paper. We set
′
δj,K
= Sj (j ∈ A) and γK = S̃ with all other parameters remaining unchanged. Therefore the Jacobian
for the transformation from (δ, γ, S, S̃) to (δ ′ , γ ′ ) is 1. The proposed move affects {αt,j } and {λt }, so
the acceptance probability for the proposed move is the minimum of 1 and


µ ′ ¶yt,j µ
n µ ′ ¶zt
′ ¶xt−j −yt,j 
Y
α
1
−
α
′ Y
λt
t,j
t,j
eλt −λt

 λt
α
1
−
α
t,j
t,j
t=1
j∈A
!
Ã
¶
µ ′ 2
′
′
′ 2
Y σR
(δj,K
)2
(δj,K
)2
1
σR
(γK
)
(γK )
× √ ×
−
×
−
.
exp
exp
2
2
σP
2σR
2σP2
σP
2σR
2σP2
n
(3.5)
j∈A
The latter term is the ratio of the proposal and prior for the parameters added into the model as a result
√
of including the explanatory variable and the multiplier 1/ n comes from the prior on R. For the case
σR = σP = 1 the latter term is identical to 1.
Order determination for INAR models has been considered in detail in Enciso-Mora et al. (2008). The
procedure used here is based upon Enciso-Mora et al. (2008), Section 3 adjusted to take account of the
inclusion of explanatory variables and the form of A. It was noted that from equations (3.2) and (3.3) of
Enciso-Mora et al. (2008) a certain function of the parameters, corresponding to E[Xt ] was well estimated
from the data and this was used to devise an efficient order switching reversible jump MCMC algorithm.
This idea is the motivation for the proposed moves in the intercept terms given below. Select L uniformly
at random from {1, 2, . . . , p} and propose to change the inclusion/exclusion status of the lag L term in
the AR processes. That is, if L ∈ A, we propose that L 6∈ A and if L 6∈ A, we propose that L ∈ A.
We shall describe the inclusion of the lag L term move in detail with the exclusion of the lag L term
move simply being the reverse procedure. Select M uniformly from A, if A = ∅ an alternative scheme is
required which we will describe later. We propose to ’split’ the AR terms, so that the AR contribution
from the M lag term in the current model is divided between the L lag and M lag terms in the proposed
′
model. In Enciso-Mora et al. (2008), this is done by drawing U ∼ U [0, 1] and setting αL
= U αM and
′
′
′
αM
= (1 − U )αM . We propose δL,0
and δM,0
based upon the approach used in Enciso-Mora et al. (2008).
6
Specifically, draw U ∼ U [0, 1] and set
1
′ )
1 + exp(δL,0
1
′
1 + exp(δM,0
)
1
1 + exp(δM,0 )
1
(1 − U )
.
1 + exp(δM,0 )
= U
(3.6)
=
(3.7)
2
For j ∈ R, let Vj ∼ N (0, σO
), where σO can be tuned to produce efficient between model moves (σO = 1
′
′
was found to be adequate for the examples in this paper) and propose δL,j
= Vj and δM,j
= δM,j − Vj .
The above defines the proposed changes in parameter values which necessitates changes to the augmented
data. For t = 1, 2, . . . , n, we propose
′
yt,M
′
yt,L
Ã
′
αt,M
∼ Bin yt,M , ′
′
αt,M + αt,L
!
′
= yt,M − yt,M
,
(3.8)
(3.9)
which splits yt,M based upon the proposed parameter values. This scheme is again similar to EncisoMora et al. (2008), Section 3.2. The transformation form (δ, y, U, V) to (δ ′ , y′ ) factorises with only
the transformation given by (3.6) and (3.7) having a Jacobian other than 1. The Jacobian, J, for the
′
′
) is given by
transformation from (δM,0 , U ) to (δM,0
, δL,0
J=
exp(δM,0 ){1 + exp(δM,0 )}
1
×
.
(1 − U )U
{U + exp(δM,0 )}{1 − U + exp(δM,0 )}
(3.10)
√
Increasing the AR order of the model by 1 leads to the ratio of the priors for the two models being 1/ n.
Therefore the acceptance probability for the proposed move is the minimum of 1 and
!
Ã
µ
¶
′
′
Y σO
(δj,L
)2
(δj,L
)2
1
1
2
′
2
√ ×J ×
exp
exp
−
(δ
− (δj,M ) )
2
σP
2σO
2σP2
2σP2 j,M
n
j∈A
¡xt−M ¢ ′
¢ ′ y′
¡
′
′
′
′
′
n
(αt,M )yt,M (1 − αt,M
)xt−M −yt,M × xyt−L
(αt,L ) t,L (1 − αt,L
)xt−L −yt,L
′
′
Y
yt,M
t,L
,
×
¡
¡xt−M ¢
¢
′
′
xt−M −yt,M × yt,M (ω )yt,M (1 − ω )yt,M −yt,M
yt,M (1 − α
t,M )
t
t
t=1
yt,M (αt,M )
y′
(3.11)
t,M
′
′
′
where ωt = αt,M
/(αt,M
+ αt,L
). The reverse move where the lag L term is excluded from the AR model
is dealt with by reversing the above procedure.
The inclusion of an AR lag term when A = ∅ requires a slightly different procedure. In this case we
adjust the noise term to compensate for the inclusion of an AR term, see Enciso-Mora et al. (2008). Let
′
2
U ∼ U [0, 1] and set 1/(1+exp(δL,0
)) = U and exp(γ0′ ) = (1−U ) exp(γ0 ). For j ∈ R, let Vj ∼ N (0, σO
) and
′
′
′
′
.
propose δL,j
= Vj . Then change the data augmentation by yt,L
∼ Bin(xt ∧ xt−1 , αt,L
) and zt′ = xt − yt,L
Finally, note that the algorithm described above generally performs well. However, the performance
of the algorithm can be improved by normalizing the explanatory variables, so that for i = 1, 2, . . . , r,
7
1
n
Pn
i=1
wt,i = 0 and
1
n
Pn
i=1
2
wt,i
= 1. For the examples, considered in Section 4 normalization was not
necessary. Since normalization is a linear operation the algorithm can be used to obtain parameter estimates for the normalized explanatory variables which can then be transformed into parameter estimates
associated with the original explanatory variables.
4
Data analysis
We apply the MCMC algorithm of Section 3 to two real-life data sets. The first data set is taken from
Zeger (1988) and consists of the monthly total number of polio cases in the USA from January 1970
to December 1983. The second data set is taken from Zhu and Joe (2006) and consists of the monthly
number of claims of short-term disability benefits made by injured logging industry workers to the British
Columbia Workers’ Compensation Board from January 1985 to December 1994.
4.1
Polio data
The polio data has been studied by Zeger (1988) and Davis et al. (2000) using Poisson regression models
and therefore provides an ideal opportunity to compare INAR models with Poisson regression models.
Therefore we follow these earlier papers in taking
wt =
µ
1,
t′
, cos
1000
µ
2πt′
12
¶
, sin
µ
2πt′
12
¶
, cos
µ
2πt′
6
¶
, sin
µ
2πt′
6
¶¶
,
where t′ = t − 73. Thus there is assumed to be in a linear trend (with intercept January 1976) and two
periods, one of 6 months and the other of 12 months. Furthermore, for a full comparison with Davis et
al. (2000) we assume that the model is INAR(1) model and all the explanatory variables are included in
the model. Therefore for this data we do not consider either order or model selection.
We ran the MCMC algorithm to obtain a sample of size 100,000 from the posterior distribution of the
parameters following a burn-in of 10,000 iterations. The results are presented in Table 1 below. It should
be noted that the estimates of γ are similar to those obtained for the Poisson regression model in Davis
et al. (2000) and in even closer agreement to those of Zeger (1988). This is to be expected given the high
value of the intercept for δ1,0 which implies αt,1 is small for most t. That is, the observations are driven
8
by the explanatory variables and the Poisson innovations rather than the AR dependence.
Explanatory variable
γ
δ1
Intercept
0.079(0.088)
2.893(0.776)
Trend
¡ ¢
cos 2πt
¡ 12 ¢
sin 2πt
¡ 12 ¢
cos 2πt
¡ 6 ¢
sin 2πt
6
−3.242(1.298)
0.022(2.091)
−0.492(0.129)
0.462(0.907)
−0.100(0.113)
0.820(0.846)
0.207(0.119)
0.532(0.850)
−0.299(0.120)
1.962(0.864)
Table 1. Means (standard deviations) of parameter coefficients for the INAR(1) model.
Finally, to test the models appropriateness we studied the predictive capabilities of the model. At each
iteration of the MCMC algorithm we simulated a realization for the number of monthly cases for each
month in 1983 given the parameter estimates and the true number of cases for the previous month. We
then compared the sample means from the posterior distributions of the number of monthly cases for each
month in 1983 with the actual monthly figures. For the Poisson regression model Monte Carlo estimation
of {Wt } was performed using the parameter estimates obtained in Davis et al. (2000) since the data is
informative about the process {Wt }. Since the driving force is the explanatory variables rather than
autoregressive dependence, the results are unsurprisingly rather similar for the two models.
Jan
Feb
March
April
May
June
July
Aug
Sept
Oct
Nov
Dec
0
1
0
0
0
1
2
1
0
1
3
6
0.965
0.400
0.408
0.328
0.529
0.795
1.015
1.060
0.894
0.896
1.262
1.484
Poisson reg. 0.553
0.245
0.203
0.241
0.398
0.581
0.676
0.684
0.640
0.742
1.158
1.533
Actual
IN AR(1)
Table 2. Actual monthly total number of cases for 1983 compared with mean predictions.
4.2
Benefit claims data
Exploratory data analysis gives support for an annual period being included in the model. Further
analysis reveals that July has the highest number of claims and the summer months (May to November)
produce significantly more claims than the winter months (December to April). Finally, there is evidence
to suggest that the number of claims might be decreasing over time. Therefore we take the following
explanatory variable vector
¶
µ
¶ ¶
µ
µ
2πt
2πt
t
, sin
, st ,
, cos
wt = 1,
120
12
12
9
(4.1)
where st = 1 for the summer months (May to November inclusive) and st = 0 otherwise. The explanatory
variables are normalized before applying the MCMC algorithm.
Since the data consists of only 120 timepoints we assume a maximum AR order of 3. The MCMC
algorithm was run to obtain a sample of size 100,000 from the posterior distribution of the parameters
following a burn-in of 10,000 iterations. Every sample from the posterior distribution contained a summer
effect and only 51 samples did not contain the first order AR term. There was also some support for
the inclusion of the second order AR term (49,869 samples) and the linear trend (26,575 samples). Only
8,357 samples contained the third order AR term and less than 2,000 samples contained either the cosine
or sine term. Therefore there is no evidence to support a periodic term in the model in addition to the
summer effect. Only six of the models were obtained in more than 1,000 samples and their details are
given in the table below.
AR orders Explanatory variables Number of samples
1, 2
St
35, 311
1
St
31, 416
1
t, St
14, 321
1, 2
t, St
9, 134
1, 2, 3
St
3, 973
1, 3
St
2, 562
Table 3. Models with the highest posterior probabilities.
The contribution from the second and third order AR terms are small even when they are included in
the model. Therefore we focus upon the parameter estimates for the noise term and the first order AR
term only.
Explanatory variable
Intercept
Trend
Summer
γ
δ1
0.8806 (0.2522)
0.5476 (0.7939)
−0.4255 (0.2890)
0.5887 (0.8264)
0.7152 (0.2511)
−0.0107 (0.7662)
Table 4. Mean (standard deviation) of parameter coefficients.
Thus the summer effect is only really present in the Poisson innovation term. This is confirmed by
allowing the explanatory variables to be present in some terms but not in others. Interestingly, the linear
trend is decreasing in the noise term, whilst increasing in the AR(1) term. This leads to similar size
observations over the length of the period but with higher correlation between later observations.
10
References
[1] Brännäs, K. (1995) Estimation and testing in integer-valued AR(1) models. Umeå Economic Studies.
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