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The authors propose new Bayesian models to obtain individual-level and time-varying regression coefficients in longitudinal data involving a single observation per response unit at each time period. An application to explore the association between customer satisfaction and shareholder value is included in the paper. The Bayesian models allow the flexibility of incorporating industry and firm factors in the context of the application to help explain variations of the regression coefficients. Results from the analysis indicate that the effect of customer satisfaction on shareholder value is not homogeneous over time. The proposed methodology provides a powerful tool to explore the relationship between two important business concepts.

Fong, Ebbes and DeSarbo [

Some scholars have recently started to investigate whether the effect of customer satisfaction on shareholder value is homogeneous across all firms/indus- tries. Several studies reported that the effect of customer satisfaction on shareholder value is heterogeneous across all firms/industries. The authors of those studies performed regression analysis assuming individual-level regression coefficients and their results indicated that there were substantive differences on the coefficients from industry to industry [

The proposed Bayesian models provide individual-level and time-varying regression coefficients which also allow the incorporation of firmographic variables to help explain variations in the coefficients. Graphically speaking, when fitting an aggregate-level regression model, as shown in

and firm factors to help explain variations of the association.

In short, this study aims to develop new heterogeneous Bayesian regression models to explore the relationship between two important business concepts, namely, customer satisfaction and shareholder value. The findings can be useful to firms in plotting their own marketing strategies. The remainder of this paper is organized as follows. Section 2 describes how the data are collected and the operationalization of the measurements that are used in the study. Section 3 presents a traditional regression analysis following the practice of previous literature. Section 4 presents the proposed Bayesian models in details. Section 5 summarizes the model results as well as findings. Section 6 concludes with a discussion and possible extensions of the proposed models.

We collect a longitudinal data set (from 1998 to 2007) from multiple archival sources to perform our empirical study. For the measure of customer satisfaction (SAT), we use the American Customer Satisfaction Index provided by the ACSI database, which has been successfully employed by a growing body of marketing researchers (e.g., [

Consistent with the literature, we select Tobin’s q [

In addition, we include a number of firm and industry factors to help explain the variation in the customer satisfaction and shareholder value association. For firm factors, following Morgan and Rego’ work in 2006 [

The descriptive statistics for the variables in our data set for each of the nine years are presented in

Following the literature (e.g., [

Q i , t = β 0 + β 1 S A T i , t − 1 + β 2 A S i , t − 1 + β 3 A D i , t − 1 + + β 4 R D i , t − 1 + β 5 D G i , t − 1 + β 6 H H I i , t − 1 + ε i , t (1)

where

・ Q_{i,t} (logarithm of Tobin’s q) denotes the shareholder value for firm i , i = 1 , ⋯ , N , in year t , t = 1 , ⋯ , T ,

・ SAT_{i,t}_{−1} denotes customer satisfaction measured by American Customer Satisfaction Index (ACSI) for firm i in year t − 1,

Year (t) Variable | 1999 | 2000 | 2001 | 2002 | 2003 | 2004 | 2005 | 2006 | 2007 | |
---|---|---|---|---|---|---|---|---|---|---|

q (t) | Mean | 1.89 | 1.65 | 1.50 | 1.32 | 1.41 | 1.42 | 1.34 | 1.50 | 1.51 |

SD | 1.59 | 1.35 | 1.16 | 1.01 | 1.00 | .98 | .93 | 1.21 | 1.59 | |

SAT (t − 1) | Mean | 76.15 | 75.64 | 76.69 | 76.37 | 77.00 | 77.33 | 77.27 | 76.89 | 77.83 |

SD | 6.28 | 6.88 | 6.97 | 7.07 | 6.24 | 6.09 | 6.24 | 7.04 | 6.84 | |

AD (t − 1) | Mean | 0.05 | 0.04 | 0.04 | 0.04 | 0.04 | 0.04 | 0.04 | 0.04 | 0.05 |

SD | 0.05 | 0.05 | 0.04 | 0.04 | 0.04 | 0.04 | 0.04 | 0.05 | 0.07 | |

RD (t − 1) | Mean | 0.06 | 0.05 | 0.05 | 0.06 | 0.07 | 0.07 | 0.07 | 0.06 | 0.07 |

SD | 0.09 | 0.07 | 0.07 | 0.08 | 0.10 | 0.10 | 0.10 | 0.08 | 0.10 | |

AS (t − 1) | Mean | 9.65 | 9.76 | 9.83 | 9.92 | 9.96 | 10.01 | 10.05 | 10.11 | 10.19 |

SD | 1.44 | 1.45 | 1.47 | 1.46 | 1.46 | 1.49 | 1.55 | 1.54 | 1.55 | |

DG (t − 1) | Mean | 0.07 | 0.07 | 0.02 | −0.01 | 0.01 | 0.03 | 0.05 | −0.02 | 0.07 |

SD | 0.14 | 0.08 | 0.12 | 0.09 | 0.16 | 0.26 | 0.22 | 0.14 | 0.12 | |

HHI (t − 1) | Mean | 0.19 | 0.20 | 0.20 | 0.21 | 0.22 | 0.23 | 0.23 | 0.24 | 0.24 |

SD | 0.15 | 0.16 | 0.16 | 0.16 | 0.16 | 0.18 | 0.19 | 0.21 | 0.21 |

・ AS_{i,t}_{−1} denotes firm asset for firm i in year t − 1,

・ AD_{i,t}_{−1} denotes firm i’s relative advertising intensity in year t − 1,

・ RD_{i,t}_{−1} denotes firm i’s relative Research and Development intensity in year t − 1,

・ DG_{i,t}_{−1} denotes demand growth for firm i in year t − 1,

・ HHI_{i,t}_{−1} denotes market concentration for firm i in year t − 1,

・ β_{0} denotes intercept,

・ β 1 , ⋯ , β 6 denote corresponding regression coefficients,

・ ɛ_{i,}_{t} denotes the error term which follows a Normal distribution.

^{2} ranging from 0.289 to 0.476 and estimates of the regression coefficient of customer satisfaction ranging from 0.132 to 0.225. These results further confirm an overall positive relationship between shareholder value and customer satisfaction.

To investigate the dynamic relationship between customer satisfaction and shareholder value, we first consider a Bayesian version of the aggregate-level regression model but in a more general form, assuming that:

Q i , t = β i , t , 0 + β i , t , 1 S A T i , t − 1 + ε i , t , (2)

Coefficient Estimate | SE | t value | |
---|---|---|---|

Intercept | 0.108*** | 0.026 | 4.148 |

SAT | 0.182*** | 0.029 | 6.225 |

AD | 0.009 | 0.029 | 0.298 |

RD | −0.090*** | 0.030 | −2.981 |

AS | −0.339*** | 0.031 | −11.093 |

DG | 0.031 | 0.027 | 1.157 |

HHI | 0.231*** | 0.029 | 7.984 |

R^{2} | 0.372 | ||

F-statistic | 61.380*** | ||

df | (6, 623) |

***p < 0.01.

and for j = 0, 1,

β i , t , j = Δ j , 0 + Δ j , 1 A S i , t − 1 + Δ j , 2 A D i , t − 1 + Δ j , 3 R D i , t − 1 (3)

where Δ j , 1 , ⋯ , Δ j , 5 denote the impact coefficients of the various firm and industry factors on the customer satisfaction and shareholder value association and Δ j , 0 _{ }is the intercept. Note that, when δ i , t , j = 0 , Equations (2) and (3) can be combined to yield an aggregate-level regression model with common regression coefficients across all firms.

For ease of presentation, we rewrite the model (HBRM1) specification in matrix notations. Let X_{it} be a column vector with one as the first element and the ith firm’s customer satisfaction score (SAT) at time t − 1 as the second element:

Q i t = X ′ i t β i t + ε i t , (4)

β i t = Δ Z i t + δ i t , (5)

where Z_{it} is a K × 1 vector of firm and industry factor values at time t − 1 with the first element set at 1, and Δ is a J × K matrix of impact coefficients. The error terms ɛ_{it} and δ_{it} are independent and normally distributed with ɛ_{it}~ N(0, σ^{2}) and δ_{it}~N_{J}(0, Σ) . In particular, we let Σ = σ^{2}C which is commonly assumed in Bayesian dynamic linear models [

σ − 2 ~ Gamma ( p , q ) , (6)

C − 1 ~ W J ( ν , V ) , (7)

vec ( Δ ) ~ N J K ( 0 , γ ) , (8)

where Gamma(p,q) represents a Gamma distribution with mean pq and variance pq^{2}, W_{J}(ν,V) denotes a Wishart distribution with mean νV, and vec(Δ) converts Δ into a vector by stacking the rows of Δ on top of one another.

With proper priors, the joint posterior distribution is proper and one can obtain posterior estimates of various parameters of interest. An efficient Gibbs sampler is used to generate random deviates of the parameters iteratively and recursively from the full conditional distributions as listed below. Appendix B provides details of the derivation.

・ p(β_{it}|all others) is a multivariate Normal distribution, for i = 1 , ⋯ , N and t = 1 , ⋯ , T .

・ p(η|all others) is a multivariate Normal distribution, where η = vec(Δ).

・ p( σ − 2 |all others) is a Gamma distribution.

・ p( C − 1 |all others) is a Wishart distribution.

In this model we allow impacts of firm and industry factors on the association between customer satisfaction and shareholder value to vary over time. Also, the error variances may vary over time. Therefore we proposed the following heterogeneous Bayesian regression model (HBRM2):

Q i t = X ′ i t β i t + e i t , (9)

β i t = Δ t Z i t + f i t , (10)

where Δ_{t} is a J × K matrix of impact coefficients at time t. The error term e_{it} follows N(0, σ t 2 ) independently and the error term fit follows N_{J}(0, Σ_{t}) independently. Again, we let Σ_{t} = σ t 2 C t , where C_{t} is a scale-free matrix.

As observations are taken over time, we assume the prior distribution of the time varying impact coefficients and variances at time t depends on the prior of the parameters at t ? 1 as well as the previous observed data:

・ At Year 1 (t = 1), similar to HBRM1, we assume the following proper priors for the parameters:

σ 1 − 2 ~ Gamma ( p ′ , q ′ ) , (11)

C 1 − 1 ~ W J ( b , A ) , (12)

vec ( Δ 1 ) ~ N J K ( η 0 , Θ 0 ) , (13)

・ However, at subsequent years (t > 1), we use the posterior distribution of the parameters from time t ? 1 as the prior distribution of the relevant parameters at time t. Specifically, let D i , t = { Q i t , X i t , Z i t , D i , t − 1 } be the information known at time t for firm i, i = 1 , ⋯ , N and D_{i,}_{0} be the information available at time zero, t hen the prior distribution of ( σ t − 2 , C t − 1 , Δ t ) , is:

π ( σ t − 2 , C t − 1 , Δ t ) = p t − 1 ( σ t − 2 , C t − 1 , Δ t | D i , t − 1 , i = 1 , ⋯ , N ) , (14)

where p_{t−}_{1} represents the posterior distribution of the parameters at time t − 1. An advantage of this prior specification is that we only make a prior assumption at time t = 1 without the need of introducing further subjective prior input afterwards. Note that such derived priors are informative priors. In the special case where these parameters are not time varying, this model becomes HBRM1.

We develop an MCMC algorithm to simulate random deviates of the parameters iteratively and recursively from the full conditional distributions as listed below. Details of the derivation are provided in Appendix C.

・ p(β_{it}|all others) is a multivariate Normal distribution, for i = 1 , ⋯ , N and t = 1 , ⋯ , T .

・ p(η_{t}|all others) is a multivariate Normal distribution, where η t = vec ( Δ t ) , for t = t = 1 , ⋯ , T .

・ p( σ t − 2 |all others) is a Gamma distribution, for t = 1 , ⋯ , T .

・ p( C t − 1 |all others) is a Wishart distribution for t = 1 but a non-standard probability distribution for t = 2 , ⋯ , T .

We can generate random deviates directly from the full conditional distributions for β_{it}, η_{t}, σ t − 2 , t = 1 , ⋯ , T , and C 1 − 1 . For C t − 1 (t > 1), the corresponding full conditional distributions are not standard probability densities, so we use the Metropolis-Hasting algorithm to generate the random deviates. More sampling details are described in Appendix C.

We use an uninformative prior in our HBRM1 analysis by specifying p = 3, q = 1, γ = 10I_{JK}, V = I_{J}, and ν = J + 10. Then, we use results from HBRM1 to specify priors at time t = 1 for HBRM2. To assess the effect of customer satisfaction on the shareholder value, we compute the posterior probabilities of the SAT coefficients (β_{i,t,1}) being positive (cf., [

of the impact differs over time which indicates the dynamic influence of HHI on the association. Similarly, asset (AS) has a positive impact but advertising intensity (AD) has a negative impact on the association. However, there is not sufficient evidence in support of an impact of R&D intensity (RD) or demand growth (DG) on the association.

Finally, we compute log marginal likelihoods to compare the two Bayesian models. The values for HBRM1 and HBRM2 are −624.95 and −618.85 respectively. This result suggests that HBRM2 is preferred over HBRM1 for the data that are being investigated in the study.

In this paper we propose new Bayesian models to investigate the dynamic and heterogeneous link between customer satisfaction and shareholder value. Our results suggest that customer satisfaction does not have a homogeneous positive effect on the shareholder value for all firms. Instead, the magnitude of the link varies across firms and changes over time. The inter-firm difference is in general larger than intra-firm temporal difference. In addition, we find that the association

HBRM1 | HBRM2 | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

1999 | 2000 | 2001 | 2002 | 2003 | 2004 | 2005 | 2006 | 2007 | ||

Intercept | 0.21 | 0.21 | 0.21 | 0.22 | 0.22 | 0.21 | 0.21 | 0.22 | 0.22 | 0.22 |

1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | |

AD | −0.05 | −0.05 | −0.04 | −0.04 | −0.04 | −0.04 | −0.04 | −0.04 | −0.05 | −0.05 |

0.06 | 0.05 | 0.09 | 0.08 | 0.09 | 0.08 | 0.04 | 0.04 | 0.03 | 0.02 | |

RD | 0.01 | 0.01 | 0.01 | 0.02 | 0.02 | 0.02 | 0.02 | 0.02 | 0.01 | 0.01 |

0.59 | 0.60 | 0.68 | 0.82 | 0.84 | 0.87 | 0.88 | 0.81 | 0.71 | 0.62 | |

AS | 0.04 | 0.05 | 0.05 | 0.06 | 0.06 | 0.06 | 0.06 | 0.05 | 0.05 | 0.05 |

0.99 | 0.93 | 0.94 | 0.97 | 0.98 | 0.98 | 0.99 | 0.98 | 0.98 | 0.97 | |

DG | −0.03 | −0.03 | −0.03 | −0.03 | −0.03 | −0.03 | −0.03 | −0.03 | −0.03 | −0.03 |

0.20 | 0.17 | 0.16 | 0.12 | 0.10 | 0.10 | 0.14 | 0.13 | 0.09 | 0.13 | |

HHI | 0.06 | 0.05 | 0.05 | 0.05 | 0.06 | 0.06 | 0.07 | 0.06 | 0.06 | 0.06 |

0.93 | 0.94 | 0.93 | 0.95 | 0.97 | 0.98 | 0.99 | 0.99 | 0.98 | 0.99 |

Note: Numbers in the two rows for each parameter in each year represent the posterior mean of the parameter and the corresponding posterior probability of the parameter being positive. Bold indicates probability is over 0.9 or below 0.1.

between customer satisfaction and shareholder value is consistently strengthened over time for larger firms and firms in industries with higher market concentration, but weakened for firms with high advertising intensity.

Methodologically, there are several advantageous features of the proposed models. First, our models provide individual level, time-varying estimates of the association. Without separating the part-worth into aggregate time dependent and static individual level components as in Liechty et al.’s model [

Future work may include extensions of the method to handle different types of data such as panel choice data and extensions to include more complicated structure like an autoregressive structure in the model. It would also be desirable to obtain extensions that allow variable selection within the models.

Fong, D.K.H., Chen, Q., Chen, Z. and Wang, R. (2017) An Application of Heterogeneous Bayesian Regression Models with Time Varying Coefficients to Explore the Relationship between Customer Satisfaction and Shareholder Value. Open Journal of Statistics, 7, 36-53. https://doi.org/10.4236/ojs.2017.71004

・ For i = 1 , ⋯ , N and t = 1 , ⋯ , T :

p ( β i t | allothers ) ∝ exp { − 1 2 [ β ′ i t ( σ − 2 X i t X ′ i t + Σ − 1 ) β i t − 2 β ′ i t ( σ − 2 X i t Q i t + Σ − 1 Δ Z i t ) ] } . (15)

This expression is proportional to a Normal density, N J ( β ¯ i t , Σ i t ) , where

Σ i t = ( X i t X ′ i t σ − 2 + Σ − 1 ) − 1 , (16)

β ¯ i t = ( X i t X ′ i t σ − 2 + Σ − 1 ) − 1 ( σ − 2 X i t Q i t + Σ − 1 Δ Z i t ) . (17)

#Math_66# (18)

where ⊗ is the Kronecker product. This expression is proportional to a normal density, N J K ( η ¯ , Θ ) where

Θ = [ γ − 1 + ∑ t = 1 T ∑ i = 1 N ( I J ⊗ Z ′ i t ) ′ ( σ 2 C ) − 1 ( I J ⊗ Z ′ i t ) ] − 1 , (19)

η ¯ = Θ ∑ t = 1 T ∑ i = 1 N ( I J ⊗ Z ′ i t ) ′ ( σ 2 C ) − 1 β i t = Θ ∑ t = 1 T ∑ i = 1 N ( I J ⊗ Z i t ) ( σ 2 C ) − 1 β i t . (20)

・ p ( C − 1 | allothers )

∝ | σ 2 C | − N T 2 exp { − 1 2 [ ∑ t = 1 T ∑ i = 1 N ( β i t − Δ Z i t ) ′ ( σ 2 C ) − 1 ( β i t − Δ Z i t ) ] } × | C − 1 | ν − J − 1 2 2 ν J 2 | V | ν 2 Γ J ( ν 2 ) exp { − t r ( V − 1 C − 1 ) 2 } (21)

#Math_73# (22)

∝ | C − 1 | N T + ν − J − 1 2 exp { − 1 2 t r [ C − 1 ( V − 1 + σ − 2 ∑ t = 1 T ∑ i = 1 N ( β i t − Δ Z i t ) ( β i t − Δ Z i t ) ′ ) ] } . (23)

This expression is proportional to a Wishart density, W J ( v ¯ , V ¯ ) , where

v ¯ = N T + ν (24)

V ¯ = [ V − 1 + σ − 2 ∑ t = 1 T ∑ i = 1 N ( β i t − Δ Z i t ) ( β i t − Δ Z i t ) ′ ] − 1 . (25)

・ p ( σ − 2 | allothers )

#Math_79# (26)

∝ ( σ − 2 ) p − 1 + N T ( 1 + J ) 2 exp { − σ − 2 q − 1 2 σ − 2 ∑ i = 1 N ∑ t = 1 T ( Q i t − X ′ i t β i t ) 2 − 1 2 σ − 2 ∑ i = 1 N ∑ t = 1 T ( β i t − Δ Z i t ) ′ C − 1 ( β i t − Δ Z i t ) } (27)

∝ ( σ − 2 ) p − 1 + N T ( 1 + J ) 2 exp { − σ − 2 [ 1 q + 1 2 ∑ i = 1 N ∑ t = 1 T ( Q i t − X ′ i t β i t ) 2 + 1 2 ∑ i = 1 N ∑ t = 1 T ( β i t − Δ Z i t ) ′ C − 1 ( β i t − Δ Z i t ) ] } (28)

This expression is proportional to a Gamma density,

Gamma ( N T ( 1 + J ) 2 + p , Q ∗ ) , where

Q ∗ = [ 1 q + 1 2 ∑ i = 1 N ∑ t = 1 T ( Q i t − X ′ i t β i t ) 2 + 1 2 ∑ i = 1 N ∑ t = 1 T ( β i t − Δ Z i t ) ′ C − 1 ( β i t − Δ Z i t ) ] − 1 . (29)

A random sample from the joint posterior distribution can be obtained by generating random deviates iteratively and recursively according to the above full conditional distributions. We have simulated 30,000 iterations, out of which the last 20,000 iterations are used for generating parameter estimates. Convergence was checked by starting the chain from multiple initial values and by an inspection of trace plots.

Appendix C: Derivation of Full Conditional Distributions for the HBRM2 ModelWe first derive the prior density π ( σ t + 1 − 2 , C t + 1 − 1 , Δ t + 1 ) , t = 1 , ⋯ , T , by computing the posterior density p t ( σ t − 2 , C t − 1 , Δ t | D i , t , i = 1 , ⋯ , N ) of the parameters from time t and then replacing σ t − 2 , C t − 1 , Δ t , by σ t + 1 − 2 , C t + 1 − 1 , Δ t + 1 , respectively, in the ex- pression. If we substitute Equation (10) into Equation (9), we will have:

Q i t = X ′ i t Δ t Z i t + X ′ i t f i t + e i t = X ′ i t Δ t Z i t + ω i t , (30)

where ω i t follows a normal distribution N ( 0 , σ t 2 ( X ′ i t C t X i t + 1 ) ) . Thus, the likelihood function of ( σ t − 2 , C t − 1 , Δ t ) is given by a product of normal densities, N ( X ′ i t Δ t Z i t , σ t 2 ( X ′ i t C t X i t + 1 ) ) , i = 1 , ⋯ , N . Since a posterior density is proportional to the product of the corresponding likelihood function and prior density, and π ( σ 1 − 2 , C 1 − 1 , Δ 1 ) = π ( σ 1 − 2 ) π ( C 1 − 1 ) π ( Δ 1 ) is a product of the individual prior densities, we have:

and in general, for t ≥ 2 ,

#Math_97# (32)

where Z i t ∗ = I J ⊗ Z ′ i t and ⊗ is the Kronecker product. Combining with the likelihood function, we obtain the following full conditional distributions:

・ For i = 1 , ⋯ , N and t = 1 , ⋯ , T :

p ( β i t | allothers )

#Math_103# (33)

∝ exp { − σ t − 2 2 [ β ′ i t ( X i t X ′ i t + C t − 1 ) β i t − 2 β ′ i t ( X i t Q i t + C t − 1 Δ t Z i t ) ] } . (34)

This expression is proportional to a normal density, N J ( β i t 0 , Ψ i t ) , where

Ψ i t = σ t 2 ( X i t X ′ i t + C t − 1 ) − 1 (35)

β i t 0 = ( X i t X ′ i t + C t − 1 ) − 1 ( X i t Q i t + C t − 1 Δ t Z i t ) . (36)

・ For t = 1 , ⋯ , T :

p ( η t | allothers ) ∝ exp { − 1 2 [ ( η t − η 0 ) ′ Θ 0 − 1 ( η t − η 0 ) + ∑ i = 1 N ( ∑ s = 1 t − 1 σ t − 2 ( Q i s − X ′ i s Z i s ∗ η t ) 2 X ′ i s C t X i s + 1 + ( β i t − Z i t ∗ η t ) ′ ( σ t 2 C t ) − 1 ( β i t − Z i t ∗ η t ) ) ] } . (37)

This expression is proportional to a normal density N J K ( η t 0 , Θ t ) where

Θ t = [ Θ 0 − 1 + σ t − 2 ∑ i = 1 N ( Z ′ i t ∗ C t − 1 Z i t ∗ + ∑ s = 1 t − 1 ( X i s X ′ i s ) ⊗ ( Z i s Z ′ i s ) X ′ i s C t X i s + 1 ) ] − 1 , (38)

η t 0 = Θ t [ Θ 0 − 1 η 0 + σ t − 2 ∑ i = 1 N ( Z ′ i t ∗ C t − 1 β i t + ∑ s = 1 t − 1 Q i s ( X i s ⊗ Z i s ) X ′ i s C t X i s + 1 ) ] . (39)

・ For t = 1 , ⋯ , T :

p ( σ t − 2 | allothers ) ∝ ( σ t − 2 ) ( t − 1 ) N 2 + p ′ − 1 exp { − σ t − 2 q ′ − 1 2 σ t − 2 ∑ i = 1 N ∑ s = 1 t − 1 ( Q i s − ( X ′ i s Δ t Z i s ) 2 ) X ′ i s C t X i s + 1 } × ( σ t − 2 ) N ( J + 1 ) 2 exp { − 1 2 ∑ i = 1 N σ t − 2 [ ( Q i t − X ′ i t β i t ) 2 + ( β i t − Δ t Z i t ) ′ ( σ t 2 C t ) − 1 ( β i t − Δ t Z i t ) ] } . (40)

This expression is proportional to a gamma density,

Gamma ( 1 2 N ( J + t ) + p ′ , Q t ∗ ) , where

Q t ∗ = [ 1 2 ∑ i = 1 N ( ( Q i t − X ′ i t β i t ) 2 + ( β i t − Δ t Z i t ) ′ C t − 1 ( β i t − Δ t Z i t ) + ∑ s = 1 t − 1 ( Q i s − X ′ i s Δ t Z i s ) 2 X ′ i s C t X i s + 1 ) + 1 q ′ ] − 1 . (41)

・ For t = 1 , ⋯ , T :

p ( C t − 1 | allothers ) ∝ | C t − 1 | N + b − J − 1 2 [ ∏ s = 1 t − 1 ∏ i = 1 N ( X ′ i s C t X i s + 1 ) − 1 2 ] × exp [ − 1 2 t r ( A − 1 C t − 1 ) − σ t − 2 2 ∑ i = 1 N ( ( β i t − Δ t Z i t ) ′ C t − 1 ( β i t − Δ t Z i t ) + ∑ s = 1 t − 1 ( Q i s − X ′ i s Δ t Z i t ) 2 X ′ i s C t X i s + 1 ) ] , (42)

which is not a standard probability density except when t = 1. (In the case of t = 1, the above expression is proportional to a Wishart density.) Random deviates from the distribution can be generated using the Metropolis-Hastings algorithm with the following Wishart proposal density,

Wishart ( N + b , ( A − 1 + σ t − 2 ∑ i = 1 N ( β i t − Δ t Z i t ) ( β i t − Δ t Z i t ) ′ ) − 1 ) . Then, we will ac-

cept the proposed estimate C t ∗ − 1 with probability,

If the proposed estimate is not accepted, we will keep the current estimate of C t − 1 . A random sample from the joint posterior distribution can be obtained by generating random deviates iteratively and recursively according to the above full conditional distributions. We have simulated 40,000 iterations, out of which the last 20,000 iterations are used for generating parameter estimates. Convergence was checked by starting the chain from multiple initial values and by the inspection of trace plots.

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