Bivariate, First-Order Vector -Autoregression Bayesian Model

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levels. In contrast, when marketing has a persistent effect, the

underlying demand process diverges over time by indenitely

accumulating demand shocks and necessitates a higher level of

safety stock to mitigate demand uncertainty. In this case, a non-

stationary demand model needs to be used instead.

To illustrate, a sales promotion may persuade a thousand

consumers to switch to a product at the promotional price during

each sale period (Dekimpe and Hanssens, 1995a). If these con-

sumers return to their previous purchasing habits once thepromotion has ended, the resulting uctuations are temporary in

nature and have no impact upon the underlying consumer

demand trend. The inventory manager must use a stationary

demand model and be careful not to overinterpret short-run

demand uctuations as an indication of future demand patterns

and set correspondingly high inventory quantities. In contrast, if

two hundred of the promotion-captured customers not only make

an initial purchase but also continue to purchase the product in

future, the demand shocks have a persistent effect and we would

see sales deviating permanently from pre-promotional levels. In

this scenario, the inventory manager is required to set a higher

safety stock level by using a non-stationary demand model in

order to buffer against persistent demand shocks.

In this paper, we assume that the inventory manager operates

with a base stock (order-up-to) policy based on a critical fractile

(e.g., Graves, 1999) and no backorders are assumed. Under this

policy, one orders a variable quantity every xed period of time so

that an inventory position is maintained at a predened base stock

level. We further assume that the underlying demand process is

autocorrelated (Urban, 2005; Charnes et al., 1995; Chen and Blue,

2010). A frequent practice is then to make inventory decisions on

the assumption that the true demand model in response to new

marketing efforts is known with certainty. An overcondent

inventory manager, believing her knowledge of the nature of

demand uctuations to be accurate, chooses either a stationary

or non-stationary demand model to estimate inventory base stock

levels. However, in the absence of a long sampling span, it is

difcult in practice to capture the long-term trend and distinguish

the correct nature of demand shocks. Often, small samples are

falsely thought to represent the properties of the statistical process

that generated them. This is known as the law of small numbers

(Camerer, 1989; Rabin, 2002).

Rather than employing one or other of the demand model

assumptions by default, one may take a step further and use a

statistical test, such as a unit root test, as a formal criterion for

making the distinction between stationary and non-stationary

demand processes (e.g., Nijs et al., 2001; Pauwels et al., 2002).

When sample size is small, it is again difcult to choose a correct

demand model, as conventional unit root tests have low statistical

power in a nite sample (Diebold and Rudebusch, 1991; DeJong

et al., 1992). To sum up, the underlying demand model cannot be

identied with certainty using both contextual expertise and a

statistical rationale in small samples. Furthermore, it may well becostly to wait for more periods to pass and obtain more data in

order to identify the trend more clearly. However, as the required

inventory levels behave much differently for one demand model

compared to the other, the incorrect demand model results in the

under or overestimation of inventory levels, leading to increased

inventory costs.

We propose an inventory policy that directly incorporates the

inherent uncertainty over stationary and non-stationary demand

models in response to new marketing efforts, by using Bayesian

model averaging. Bayesian model averaging is a complete Baye-

sian solution to average over possible models. The concept of

Bayesian model averaging was introduced by Leamer (1978), and

has recently received signicant attention in the statistics and

econometrics literature, in particular from Raftery et al. (1997),

Hoeting et al. (1999), andRaftery and Zheng (2003). We assume

that one of the stationary or non-stationary models is the true

demand model once new marketing efforts are introduced, but

that we do not know which it is. Starting from a prior about which

model is true and observing demand, we compute the posterior

probabilities that each is the true model by applying Bayes'

theorem. We then average over the inventory decisions made by

the two models, weighted by each model's posterior probability. In

this paper, structural results of the proposed inventory model arealso discussed. Specically, the Bayesian model averaging inven-

tory model estimates consistent order quantities based on a

critical fractile and provides better performance, as measured by

a logarithmic scoring rule, than using any single model.

The paper in hand relates to several studies that use a Bayesian

framework to deal with parameter uncertainty for specic

demand models (see, e.g., Azoury, 1985; Azoury and Miyaoka,

2009; Lovejoy, 1990). Using Bayes theorem, the unknown para-

meter is periodically updated based on newly obtained demand

observations. Yet these inventory models cannot address uncer-

tainty about the structure of the underlying demand generating

model, i.e., demand model uncertainty. In this paper, we use the

Bayesian framework to update the belief about candidate demand

models on the basis of past observations to explicitly account for

model uncertainty, which in our case arises from uncertainty

about the nature of demand uctuations after new marketing

efforts. While non-parametric approaches (e.g., Bookbinder and

Lordahl, 1989; Levi et al., 2007) are established to negate the need

to make assumptions about the demand model, they are limited

to independent demand processes and cannot be applied to

serially correlated demand processes. A semi-parametric approach

inLee (2014) provides consistent estimates of the critical fractile

independently of a forecasting model if the demand process

follows a stationary autoregressive demand process and the

forecasting model is within the autoregressive integrated class.

Unlike the non-parametric and semi-parametric inventory models,

our proposed Bayesian model averaging (BMA) inventory model

enables us to deal with model uncertainty in independent, serially

correlated, as well as non-stationary demand processes. As such,

our paper can be seen as a rst step towards suitably modifying

and adapting the recent developments in the Bayesian model

averaging method seen in the statistics and econometrics litera-

ture to the practical problem facing inventory management,

namely that of setting inventory levels in response to new

marketing efforts.

The interaction between two functional areas, marketing and

operations, is recurrently discussed in the literature. See, for

example,Tang (2010),Ma et al. (2013), andMarques et al. (2014).

We contribute to this type of literature focusing on the issue of

marketing efforts and ordering decisions. In particular, our work is

linked to the literature that addresses the classical single-period

inventory problem with advertising, where advertising stimulates

the demand. Khouja and Robbins (2003) assume that the meandemand is both increasing and concave in advertising expenditure

(i.e., the returns of advertising have a diminishing effect on sales)

and demand variance is also a function of advertising expenditure.

They obtain the optimal advertising expenditure and ordering

quantity that maximizes the expected prot or the probability of

achieving a target prot. Their model assumes that the demand

process is independent and the effect of advertising on the under-

lying demand is known. Lee and Hsu (2011) and Guler (2014)

recently extend this model to the distribution-free newsboy pro-

blem. In contrast, our model considers autocorrelated demand and

the effect of marketing actions on the mean and variance of

demand is characterized by the autocorrelation parameter. In most

practical situations, we shall indeed observe autocorrelation in the

demand process, especially when new marketing efforts are made

Y.S. Lee / Int. J. Production Economics 158 (2014) 278289 279


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(Dekimpe and Hanssens, 1995b). Also we do not assume that the

effect of marketing is known with certainty. Instead, we assume

that the total long-run impact of marketing is either temporary or

permanent and the managers learn about whether the observed

movements in demand are of a temporary versus a permanent

nature as more demand observations become available.

This paper is organized as follows: Section 2presents a single-

item inventory model. We analyze the key distinction in the

required inventory level when the demand shocks observed afternew marketing efforts are temporary or permanent. In Section 3,

we illustrate that it is difcult in practice to correctly interpret the

nature of demand shocks using both the context expertise and

statistical rationale.Section 4derives the inventory model resulting

from Bayesian model averaging that deals with the uncertainty

about the nature of the demand shocks, and discusses its structural

results.Section 5examines the performance of the BMA inventory

model using numerical studies.Section 6provides a conclusion. The

Appendix contains the proof of the asymptotic analysis inSection 4.

2. A single-item inventory model

This paper considers the problem of setting an inventory level

of a consumer product for which new marketing efforts are made.

Assume that att0, new marketing efforts (single or multiple) are

introduced and these efforts induce a series of unexpected move-

ments (shocks) in the following periods. The total long-run impact

of these movements on underlying demand is assumed to be

either temporary or permanent in nature. The pre-expenditure

demand level is denoted by Y0. For every period tZ0 new

demand observations become available, and the inventory man-

ager updates the demand forecast and inventory decisions accord-

ing to a base stock (order-up-to) policy based on a critical fractile.

In the following, we describe the demand process and inven-

tory control policy in more detail and highlight the key distinction

in the required inventory level between the cases where the

effects of the marketing efforts are temporary or permanent.

Given the well-established dynamic nature of marketing effectresponse in the marketing literature, we adopt the univariate rst-

order autoregressive demand process used in Dekimpe and

Hanssens (1995b) and make a distinction between stationary

and non-stationary demand processes arising from new marketing

efforts. We then show a multivariate extension to endogenize the

marketing efforts (such as measured by marketing expenditure) in

the demand model.

2.1. Demand process

The demand process is a univariate rst-order autoregressive

(AR(1)) demand and is given as follows:

Yt cYt 1ut; 1where c is a nonnegative constant, is an autocorrelationparameter and ut are random shocks that are independent and

identically distributed from N0; 2u. For ease of exposition, wehave omitted any deterministic components from the model

except a constant term c. In general, deterministic trends and/or

seasonal dummy variables can be added to (1) (Dekimpe and

Hanssens, 1999). Specically, we can describe the mean of a time

seriestas the sum of the level componentCt, trend componentTtand seasonal component St. For example, we can write

t Ct Tt St where Tt t, St si 1iDti. In this case, is a

linear trend parameter, t is a time index, is a seasonal trendparameter, andDtiare the seasonal dummies. InSection 6, we discuss

the generalization of this demand process by including higher auto-

regressive lags (AR(p)).

Applying successive backward substitutions (Box et al., 1994),

we can rewrite(1) as in innite moving average form:

Yt c=1 utut 12

ut 2: 2

Present demand Yt is therefore explained as a weighted sum of

present and previous random shocks, u t; ut 1; ut 2;.

It is the unknown value ofthat denes the nature of demandshocks, either temporary or permanent due to the marketing

efforts. If jjo1, the effect of past uctuations wanes andeventually becomes negligible. The observed uctuations are

temporary in nature and do not alter the underlying long-term

demand trend. In this case, the demand process is stationary as it

reverts to a long-term xed mean, EYt c=1, and it has anite variance, VarYt 2u=1

2. Also note that the speed of

mean reversion is faster for the low value of . When 0, theunderlying process is simply independent and identically

distributed.

If 1,(2) becomes

Yt ctut ut 1ut 2:

In this case, the impact of observed shocks is permanent and

indenitely accumulated over time. The long-term mean does not

exist and is unbounded. The variance depends on tand increaseswith time without bound, i.e., VarYt 2ut. Such a demand seriesis non-stationary and contains a unit root. It moves freely in one

direction or another without any reversion to a xed trend.

2.2. Inventory control policy

The inventory manager operates with a base stock policy based

on a critical fractile (e.g., Graves, 1999), in which the inventory

position is brought to a prescribed level at the beginning of each

period to meet a desired consumer service level K. No backorders

are assumed. Therefore, (1 K) is the probability of stockout

during a replenishment cycle, denoted by L. Note thatKis dened

by K s=s h where h and s are the proportional holding and

shortage costs, respectively, following the well-known newsven-

dor result (Porteus, 2002).

LetQnbe the required base stock level to meet a desired service

level Kat tn, i.e., n historical demands are observed since new

marketing efforts are introduced:

QnLnz

Ln; 3

where nL and n

L are the mean and standard deviation of LTD,

respectively. z is a constant chosen to meet the desired service

levelK, i.e.,z 1Kwhere is a standard normal distributionfunction. The base stock level in each period is therefore set to

support the expected demand for replenishment lead time nL plus

a safety stock SSLnzLn to mitigate the risk of stockouts due to

demand uncertainty, measured by nL .

2.3. Base stock levels under stationary and non-stationary demand

processes

In this section, we demonstrate that the required base stock

levels behave much differently for the case of stationary demand

than for non-stationary demand, which is driven by the nature of

random shocks observed after new marketing efforts. We demon-

strate that the nature of demand signals can affect the base stock

levelQnin two ways: rst, by directly altering the expected LTD nL

(mean effect); and second, by altering the safety stock level

requirement SSLnzLn (variance effect).

When the demand process follows a rst-order autoregressive

process, as described in(1), the LTD, which is the sum of demand

Y.S. Lee / Int. J. Production Economics 158 (2014) 278289280


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over periods n 1; n2;; n L, is expressed as follows:

Yn 1Yn 2 Yn L

cYnun 1 c1 2

Ynun 1 un 2

c1 2

L 1

L

YnL 1

un 1 L 2

un 2 un L:

4

The expected LTD is then found by setting the expected values of

the future random terms in(4), un 1; un 2;; un L, to zero:

Ln c1 1 1 2

L 1

Yn2

L

c L 1

i 0

i

j 0

j" #

Yn L

i 1

i: 5

First, note that the expected LTD is dependent on the current

demand state in period n, i.e., Yn. When jjo1, the currentdemand state is discounted and its effect on Yn L eventually becomes

negligible as L increases, i.e., EYn L c12

L 1

LYn . Therefore, future demands Yn L, especially for large L, tendto revert back to the long-term mean.

When 1, the expected LTD becomes

Ln c L 1

i 0

1i" #LYn L L 12

cYn : 6Under a non-stationary process, the current demand state Ynis an

anchor for all future demands and in each period a constant drift c

is added to this anchor point during the replenishment lead time,

i.e., EYn L cL Yn. Therefore, the expected LTD under a non-

stationary process may be signicantly different from under a

stationary process, depending on the value of, and the differenceincreases as lead time increases.

Next, we demonstrate how the safety stock level under non-

stationary demand will be more conservative than under station-

ary demand; that is to say, an inventory manager will hold higher

levels of safety stock under the non-stationary process to achieve

the same service level as in the stationary process. Using (4), we

can express the total LTD uncertainty, denoted by EnL , as a linear

combination of future random terms un 1; un 2;

; un L:

ELn un L1un L 11 2

L 1

un 1:

Since the random errorsutare assumed to be uncorrelated in (1),

the standard deviation of LTD is given by

Ln u

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi11 2 1

2L

12

q

u

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

L 1

i 0

i

j 0

j !2vuut : 7

For a given lead time, the standard deviation of LTD is an

increasing function of the autocorrelation parameter jj. When1 and the underlying demand process is non-stationary, (7)

becomes

Ln u

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

L 1

i 0

1 i2

s u

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiLL 12L 1

6

r : 8

This implies that the relationship between nL and L becomes

convex for the non-stationary demand process (Graves, 1999).

Since the required safety stock level is proportional toLnfor a pre-specied service level, i.e., SSLnz

Ln where z

1K and is a

constant, we require dramatically more safety stock for non-

stationary demand compared with fast mean reverting-

stationary demand. The difference in safety stock requirements

is larger for longer lead times. We note that, unlike the expected

LTD, the standard deviation of LTD is independent of the current

demand state Yn. For a given demand process and a xed lead

time, the required safety stock is constant over different periods.

2.4. A multivariate extension

In practice, companies respond to perceived changes in the

market environment or in their goals by adjusting their marketing

mix. In the empirical study conducted by Dekimpe and Hanssens

(1995a), the adjustments in the marketing mix are shown to be

either temporary, in that the company abandons the change in

favor of the previous level after some periods, or permanent, in

that there is no return to the previous level. It is therefore ofimportance to examine to what extent the persistence of demand

can be related to an unexpected change in marketing activity. For

this purpose, we use vector autoregressive (VAR) models to model

the dynamic interactions between performance variable (in our

case demand) and marketing variables (such as measured in

marketing expenditure) (Dekimpe and Hanssens, 1995b). Both

demand and marketing actions are endogenous so that they are

explained by their own past and by the past of the other

endogenous variables. Specically, the vector autoregressive

model estimates the baseline of each endogenous variable and

forecasts its future values on the basis of the dynamic interactions

of all jointly endogenous variables.

For ease of exposition, consider the bivariate, rst-order vector

autoregressive model:

Y1;t

Y2;t

" #

c1

c2

" #

11 1221 22

" # Y1;t 1

Y2;t 1

" #

u1;t

u2;t

" #

whereY1;tand Y2;tdenote the underlying demand and marketing

action at time t, respectively, and u1;t and u2;tare white noises in

the respective processes. Equivalently, we write the vector auto-

regressive model in the following form:

Yt C Yt 1Ut; 9

whereYt,CandUtare 2 1 vectors and is a 2 2 matrix. This

specication directly captures purchase reinforcement (11),delayed response (12), performance feedback (21) and inertia

in marketing decision making (22) (Dekimpe and Hanssens,1999). When12 21 0, the underlying demand and marketingaction are independent of each other.

Using (9), we can express the mean and variance of demand

and marketing mix for the lead time as

Ln L 1

i 0

i

j 0

j

AC L

i 1

iYn

and

Ln

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

L 1

i 0

i

j 0

j

A

!2vuut uwhere A

0 1

1 0 and

u is a vector of standard deviations of

u1;t; u2;t0.

When dealing with evolving variables, Yt in(9) is replaced by

YtYtYt 1. In this case, the mean and variance of demand

and marketing mix for the lead time are

Ln L L 1

2 C Yn

and

Ln

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiLL 12L 1

6

r u:

Since the results from the univariate model can be easily

extended to the multivariate model using (3), the rest of this

paper focuses on the exposition of the univariate model.



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3. Uncertainty about the effect of new marketing efforts on

product demand

In this section, we discuss the inherent uncertainty in deter-

mining the nature of demand uctuations arising from new

marketing efforts, which leads to demand model uncertainty.

As illustrated inSection 2(due to both mean and variance effect),

stationary and non-stationary demand processes require much

different base stock levels and it is crucial for efcient inventoryplanning that we distinguish between the two processes. An

incorrect demand model assumption leads to under or overestim-

ated base stock levels and subsequently increased inventory costs.

Let us imagine that a company launches a new advertising

campaign for a product at t0 and observes the effect this has on

product demand over a sustained period. Fig. 1 plots simulated

autoregressive demand processes (as given by (1)) to illustrate

instances where the demand shocks resulting from this marketing

action are temporary or permanent in nature. We have constructed

these series using the same random process, i.e., the same simu-

lated errors utare used. The only difference being the value of the

autocorrelation parameter. If the marketing effect is found to betemporaryo1, the demand process tends to revert to the long-term mean and follows the dotted line. The higher values of the

autocorrelation parameter correspond to the slower rate of the

mean reversion rate. When the demand is highly autocorrelated

and the mean reversion rate is slow, we call the underlying demand

process to be nearly non-stationary. In contrast, if advertising has a

permanent effect (1), the demand process diverges over timewithout mean reversion and follows the solid line. Note that the

gure is constructed based on one simulated random process,

therefore illustrates one possible scenario.

As observed inFig. 1, the stationary demand process with the

slow mean reversion rate follows a similar path to the non-

stationary demand process for up to 60 periods. It would therefore

have been very difcult to interpret whether or not the observed

movements in demand during these periods were temporary

uctuations around a xed mean. This difculty is also known as

the law of small numbers (Camerer, 1989; Rabin, 2002). How-

ever, observe that the consequence of an incorrect demand model

by under or overinterpreting short-run uctuations as indicators

of future demand patterns is very signicant in the long run. As we

review more data, the distinction between the two processes

becomes clearer.

Rather than employing one or other demand model by sub-

jective judgment, one may use a statistical test such as a unit root

test as a formal criterion for differentiating stationary models from

non-stationary models (e.g.,Nijs et al., 2001; Pauwels et al., 2002).

One popular unit root test, a DickeyFuller test (Dickey and Fuller,

1979), hypothesizes whether or not there is a unit root in the rst-

order autoregressive process. The test is based on the t-ratio

DFn 1

s ; 10

where is the ordinary least squares estimator of the autocorre-lation coefcient ands is its standard error. IfDFnrrwhererisa critical value, the null hypothesis of a unit root is rejected and a

stationary demand model is chosen. Otherwise, we assume a non-

stationary demand model.

This formal statistical test is also known to have low power

against nearly non-stationary processes in small samples (DeJong

et al., 1992; Diebold and Rudebusch, 1991). Fig. 2 illustrates

the empirical power of the DickeyFuller test against the station-

ary alternative with a 5% signicance level. We observe that the

test has particularly low statistical power when the value of isvery close to unity and the sample size is small. For example, when

0.9 and the sample size is 40, the likelihood of specifyingthe correct stationary model is as low as 10% and there is an

approximate 90% probability ofnding a seemingly non-stationarypattern (Type II error). When the underlying process is non-

stationary, the test correctly detects a unit root with a 95%

probability, which is consistent with the chosen signicance level.

4. Bayesian model averaging inventory model

We propose an inventory model that explicitly acknowledges

uncertainty over stationary and non-stationary demand models as

a result of new marketing efforts, using Bayesian model averaging.

We rst introduce the general concept of Bayesian model aver-

aging and apply this to make robust inventory decisions where

there is uncertainty about the persistence effect of marketing

efforts on the product demand.

4.1. Bayesian model averaging

Bayesian model averaging is an overall concept that directly

incorporates model uncertainty in the estimation process. Con-

sider a set of m models M fM1;; Mmg. We know that one of

these models is the true model, but do not know which one it is.

We have prior beliefs about the probability that the ith model is

Time

Simulateddeman

d

0 20 40 60 80 100

200

250

300

=1

=0.9

=0.8

=0.7

Fig. 1. Simulated rst-order autoregressive demand processes where 1 (solid

line), 0.9, 0.8, 0.7 (dotted lines), and u 6.

20 40 60 80 100 120 140 160

0

20

40

60

80

100

Sample size

Likelihoodofcorrectmodelsp

ecification(%)

=0.9

=0.8

=0.7

Fig. 2. Empirical power of the DickeyFuller test against the stationary alternative

with a 5% signicance level u 1.



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the true model, which is denoted by PrMi. Given the observed

data D, we then update our beliefs to compute the posterior

probabilities:

PrMijD PrDjMiPrMi

mj 1 PrDjMjPrMj;

where

PrDjMi Z

PrDji;MiPrijMidi

is the integrated likelihood of model Mi, i is the vector ofparameters of model Mi, PrijMi is the prior density ofi undermodel Mi, and PrDji;Mi is the likelihood.

Denote as the observable to be predicted; this could be afuture observation (in our case LTD) or the utility of a course of

action. Each model implies an estimate of. The estimates offrom all models are then weighted by their posterior probabilities

to compute the posterior distribution of given the data D:

PrjD m

i 1

PrjMi; DPrMijD: 11

4.2. Bayesian model averaging (BMA) inventory model

The model uncertainty we study in this paper results from

uncertainty about the nature of demand uctuations resulting

from new marketing efforts. The two natural candidate demand

models are, therefore, the stationary model MS jjo1 and thenon-stationary model MNS(1).

Given n demand observations, let Qn;S and Qn;NS be the base

stock levels that are based on the stationary and non-stationary

demand models, respectively. When the stationary model is

applied, we derive Qn;Susing(3), (5) and (7) as

Qn;S c L 1

i 0

i

j 0

j" #

Yn L

i 1

i z u

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

L 1

i 0

i

j 0

j !2vuu

t : 12

When the non-stationary model is applied, we derive Qn;NSusing(3), (6) and (8)as

Qn;NS c L 1

i 0

1 i

" #LYnz u

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

L 1

i 0

1i2

s : 13

Since Qn represents the K% quantile of LTD, the Bayesian model

averaging base stock level is calculated using (11) (Raftery and

Zheng, 2003):

Qn;BMA Qn;SPrMSjD Qn;NSPrMNSjD: 14

This implies that the proposed Bayesian model averaging (BMA)

inventory model weighs each of the two models by their posterior

probabilities to estimate the base stock level.

In order to employ the BMA inventory model, we need to

specify the model priors PrMiand the parameter priors PrijMi.Specication of PrMi, the prior distribution over competing

models, can be based on the subjective knowledge of the modeler.

When there is little prior information about the relative plausi-

bility of the models under consideration, the assumption that all

models are equally likely a priori is a reasonable neutral choice,

i.e., PrMS PrMNS 0:5. We assume equal model priors in our

simulation studies inSection 5.

The BMA inventory model not only updates beliefs about

model assumption but also updates model parameter values. For

this we need to set the parameter priors PrijMi and it is usefulto express the rst-order autoregressive demand model in(1) in a

vector form:

Y cX1u X u

whereYis a vector of observed demandYn; Yn 1;; Y10 andX1is

a vector of lagged demand, Yn 1 ; Yn 2;; Y00 since the introduc-

tion of the new marketing efforts at t0. We consider the (2 1)

individual parameters c;0 and u2 to be unknown. We then

use the standard normal gamma conjugate class of priors (Raftery

et al., 1997),

N; 2uV

and

2uX2:

Therefore,, , the 2 2 matrix V, and the 2 1 vector arehyperparameters to be selected. Based on the chosen priors, we

can calculate the required marginal likelihood of model Mianalytically, followingMadigan and Raftery (1994):

PrYi;Vi; Xi; Mi

n

2

=2

n=2

2

jIXiViX

ti j

1=2 YXii

t I XiViXti

1Y Xii n=2:

Next we discuss the choice of the hyperparameters (Raftery et al.,

1997). For the stationary model, we choose S c; 0, where c isthe ordinary least squares estimate of c. The prior covariance

matrix for Sis equal to

CovS 2uVS

2u

s2Y 0

0 2s 2X1

!;

wheres2Ydenotes the sample variance ofY,s2

X1denotes the sample

variance ofX1, andis another hyperparameter to be selected. Theprior variance ofcis chosen conservatively and that of is chosento reect increasing precision about this parameter as the variance

ofX1 increases. For the non-stationary model, we need to impose

the unit root restriction, i.e., 1. Therefore,NS c; 1 and

CovNS 2

uVNS 2

u

s2Y 0

0 0 !

:

In the simulation study that follows, we choose the remaining

hyperparameters , and as suggested by Raftery et al. (1997),i.e., we set2.58,0.28, and2.85.

Next, we show that the posterior probability of the true model

tends to 1 as the sample size goes innity (Theorem 1). This

implies that the proposed BMA inventory model estimates con-

sistent order quantities based on a critical fractile (Corollary 1).

Also,Theorem 2shows that in the presence of model uncertainty,

using the BMA inventory model provides better performance in

estimating inventory levels, as measured by a logarithmic scoring

rule, compared to by using any single modelMSorMNS.Theorem 3

(Theorem 4) shows the relative size of inventory levels made by

the stationary, BMA and non-stationary models when the desiredservice level is higher (lower) than 50%. See the Appendix for

proofs.

Theorem 1. If MS is the true model and PrMS PrMNS 0:5,

limn-1PrMSjD 1 and vice versa.

Corollary 1. If MS is the true model and PrMS PrMNS 0:5,

limn-1Qn;BMA Qn;Sand vice versa.

Theorem 2. Assume that Qn is the critical fractile of the true demand

model. Then, we have Elog fPrQnjDgrElog fPrQn;BMAjDgr

Elog fPrQn;jjDg where j S; NS.

Theorem 3. Assume c;; Yn40. For a desired service level KZ0:5,

Qn;SoQn;BMAoQn;NS.



7/12

Theorem 4. Assume c;; Yn40. For a desired service level Ko0:5,Qn;SoQn;BMAoQn;NS if

cL 1i 0 f1i i

j 0j

g YnL Li 1

j

uffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL 1i 0f1i

2 ij 0j

2g

q 4 11K:Otherwise, Qn;NSoQn;BMAoQn;S.

5. Numerical results

5.1. Experimental design

In this section we investigate the nite sample performance of

the BMA inventory model using Monte Carlo experiments, where

there is uncertainty about the persistence of new marketing efforts

on customer demand. We benchmark its performance against four

alternative inventory models:

1. The stationary inventory model (S): This approach employs a

stationary demand model by default and ignores demand

model uncertainty. Given n demand observations, it estimates

base stock levels Qn;S using (12) with point estimates of the

parameter vector, c; ; 2

u0 where c and are the uncon-

strained least squares estimators, and 2u is the estimatedvariance of the demand model residuals. Since the underlying

process is assumed to be stationary, these model parameters

are estimated using the level of the demand variable rather

than its rst difference.

2. The non-stationary inventory model (NS): This approach

employs a non-stationary demand model by default and also

ignores model uncertainty. It imposes a unit root restriction

( 1) and estimates base stock levels Qn;NS using (13) withpoint estimates of the parameter vector, c; 2u

0 where c is the

constrained least squares estimator, and 2u is the estimatedvariance of the demand model residuals. Since the underlying

process is assumed to be non-stationary, these model para-

meters are estimated using the rst difference of the demand

variable.3. The pre-testing inventory model(PT): Rather than employing one

or other demand model by default, this model uses a unit root

test as a formal criterion for selecting a demand model. We use

the DickeyFuller test, which is based on the t-ratio DFn as

dened in(10). The pre-testing inventory model estimates base

stock levels by Qn;PT Qn;SIDFnrr Qn;NSIDFn4r, where r

is a critical value and I is an indicator function. We select the

value ofrfor a 5% signicance level.

4. The Bayesian inventory model (B): This approach uses a Bayesian

framework to allow for parameter uncertainty in the stationary

demand model. It uses a Bayesian formulation to update

parameters of the stationary demand model based on the

normal gamma conjugate class of priors from standard Baye-

sian theory (Rossi et al., 2005). Given n demand observations,the Bayesian inventory model estimates base stock levels Qn;Busing(12) with the updated parameter values ~c; ~; ~2u

0 using

Bayes' theorem. It acknowledges errors arising from parameter

estimation in the stationary inventory model, yet disregards

demand model uncertainty.

Note that the BMA inventory model nests the Bayesian model

that updates base stock levels with respect to parameter uncer-

tainty. The BMA model learns about not only the correct model

specication (stationary versus non-stationary), but also the

parameter values of each model as the sample size grows.

A simple averaging of the stationary and non-stationary models

by setting the posterior probabilities PrMSjD PrMSjD 0:5 in

(14)was also considered, but the results of this are omitted as the

process underperforms that of the BMA inventory model. In

addition, we considered using different prior probabilities in the

BMA inventory model, such as by setting the prior probabilities

based on the p-values of the DickeyFuller unit root test. The

results of this are also omitted as the process underperforms that

by setting equal prior probabilities (i.e., PrMS PrMNS 0:5).

As discussed in Section 3, we focus on the nearly non-

stationary demand process, for which it is difcult to identify

the correct nature of observed demand shocks. The nearly non-stationary process is designed with a set of high-value autocorre-

lation parameters, Af0:70; 0:75; 0:80; 0:85; 0:90; 0:95; 1g. Recallthat the value of this autocorrelation parameter determines the

nature of demand process (o1 and 1 correspond to astationary and non-stationary process, respectively). In the simu-

lations, this value is treated to be unknown and thus there is

0.70 0.75 0.80 0.85 0.90 0.95 1.00

5

0

5

10

Autocorrelation parameter,

Averagebias(%)

S

NS

PT

B

BMA

Fig. 3. Average single-period bias in base stock levels (%) compared to the full-

information policy.

0.70 0.75 0.80 0.85 0.90 0.95 1.00

0

20

40

60

80

100

Autocorrelation parameter,

Averagecost

error(%)

S

NS

PT

B

BMA

Fig. 4. Average single-period cost error (%) compared to the full-information policy.



8/12

uncertainty about the persistence of new marketing efforts on

demand. The performance of each inventory model is evaluated in

terms of how far the estimated base stock levels and one-period

inventory costs are from the optimal value Qnn and its cost CQn

n,

which are calculated using the true demand model, i.e.,cand areknown with certainty (full-information policy).

The parameters kept constant for all simulations are u 4,K0.95 andL 10. Ifo1, the constantcin the demand model is

set by c EYt1 with EYt 100. This ensures that thedemand series with different values of revert to the same meanEYt. When1, we simply set c0.

Given these parameters, we rst generate a demand path for

n30 consecutive time periods. For each sample path, we com-

pute the base stock level Qn using each inventory model for the

upcoming replenishment time (from t n to t n L). We then

calculate the bias of the estimated base stock level QnQn

n=Qn

n

and cost error C Qn CQn

n=CQn

n of each inventory model. We

report the average bias of base stock level and average cost error,

which are calculated over 1000 sample paths. We also present a

sensitivity analysis in Section 5.3 to examine the impact of the

assumed parameters, such as sample size, lead time and service

level, on inventory cost error.

5.2. Results

Figs. 3 and 4 compare the BMA inventory model to the alter-

native models (stationary (S), non-stationary (NS), pre-testing

(PT), or Bayesian (B)). The average bias in base stock levels and

average cost error of each inventory model are plotted as a function

of the autocorrelation parameter . The observed bias and costerrors are calculated in reference to the full-information case and

therefore are due to parameter and/or model uncertainty about the

demand process when estimating inventory policies.

As shown in Fig. 3, the stationary inventory model (S) under-

estimates base stock levels. This is due to the downward bias in safety

stock levels as the least squares estimators of are biased towardzero, especially for the value of close to unity (Shaman and Stine,1988). This inventory model (S) therefore assumes the unnecessarily

stronger reversion of demand and set a lower level of safety stock. We

also observe that the bias created in the stationary inventory model isreduced by the Bayesian approach (B), which integrates uncertainty

over parameter values into the base stock level calculation.

The non-stationary inventory model (NS) overestimates safety

stock levels for the underlying stationary demand processes

o1. This implies that the non-stationary inventory modelgenerates unnecessary safety stocks by assuming higher demand

uncertainty, especially when the underlying stationary demand

process has relatively low autocorrelation. The pre-testing inven-

tory model (PT) noticeably dominates the non-stationary inven-

tory model, as it avoids safety stock overestimation by incorrectly

selecting the non-stationary demand model when the unit root

test correctly rejects the null of the non-stationary process.

Distortions of the pre-testing inventory model still exist as a result

of the low power of the DickeyFuller test in small samples.

The BMA inventory model (BMA) sets the base stock levels in

between the non-stationary model (NS) and the stationary model

(S) as Theorem 3predicts. Fig. 3 suggests that both the Bayesian

inventory model (B) and BMA inventory models (BMA) are close to

the full-information case on average. For lower values of, theBayesian inventory model outperforms the BMA model, but for close to 1, the reverse is true.

Fig. 4 presents the average cost errors of the ve inventory

models. It is consistent with the results in Fig. 3 as the smaller

20 40 60 80 100 120 140

0

20

60

100

140

=0.7

Sample size, n

Averagecosterror(%

)

PT B BMA

20 40 60 80 100 120 140

0

20

60

100

140

=0.8

Sample size, n

Averagecosterror(%

)

PT B BMA

20 40 60 80 100 120 140

0

20

60

100

140

=0.9

Sample size, n

Averagecosterror(%)

PT B BMA

20 40 60 80 100 120 140

0

20

60

100

140

=1

Sample size, n

Averagecosterror(%)

PT B BMA

Fig. 5. Impact of sample size n on average cost error (%) compared to the full-information policy.



9/12

absolute value of bias corresponds to the lower cost error. This

gure demonstrates the robustness of the BMA inventory model

against the nature of demand shocks, i.e., different values of. Itsmaximum value loss is 45% in all cases examined. The perfor-

mance edge of the BMA model is attributable to incorporating

model uncertainty directly into the estimation process. In contrast,

the performance of alternative models can deteriorate signicantly

depending on the value of. For the stationary demand process

with small values of , the stationary and Bayesian inventorymodels perform well. However, their performance deteriorates as increases, as they tend to underestimate demand uncertaintyand stockouts are frequent. In contrast, the performance of the

non-stationary and pre-testing inventory models is poor for small

values of because of overestimated demand uncertainty, result-ing in high holding costs; however their performance improves as

increases. This is consistent with the ndings in Fig. 3, as theconstant upward and downward biases in the base stock estimates

increase overall inventory costs.

5.3. Sensitivity analysis

We examine the effectiveness of the BMA inventory model

across values of sample size, lead time and service level. Forbrevity, we use the Bayesian and pre-testing inventory models as

benchmarks, as they tend to be more robust than the stationary

and non-stationary models as indicated in Figs. 3 and 4.

Fig. 5shows the cost error of the BMA model as a function of

sample size compared with the Bayesian and pre-testing models.

There are four panels for different values of{0.7,0.8,0.9,1}. Eachpanel, therefore, represents a different mean reverting rate in the

demand process. We observe that the cost error of all three

inventory models decreases to zero as the sample size increases.

The cost error of the BMA inventory model converges to zero as

the posterior distributions of the model probabilities and model

parameters are more accurately updated with a larger sample size

(Corollary1). Similarly, the model parameter values in the Baye-

sian inventory model are better updated in larger samples. As

discussed in Section 3, the DickeyFuller test has a higher

statistical power and correctly distinguishes between the station-

ary and non-stationary processes as the sample size increases.

Subsequently, the performance of the pre-testing model improveswith sample size.

Fig. 5also suggests that cost savings from the BMA inventory

model may be substantial for smaller samples no50. The BMA

approach is therefore useful in improving the efciency of an

inventory system where there is only a limited amount of available

historical demand data and it is prohibitively costly to wait for

time to pass in order to make a more informed decision.

Fig. 6 illustrates that the cost errors of the inventory models

increase as the lead time increases. This is because errors in

forecasting LTD, resulting from parameter uncertainty and model

uncertainty, accumulate over lead time. A reduction in lead time

will enable cost performance to be less sensitive to the additional

uncertainty. If a reduction is not achievable in practice, the BMA

inventory model is preferred among those considered due to its

robustness. Fig. 7 illustrates the impact of service level K on

inventory costs. This explains the general tendency observed for

cost error to be greater at higher service levels. When Kis high, an

inventory manager may benet from substantial cost savings by

using the BMA inventory model.

We also ran simulations to test for the impact of demand

volatility, measured byu, on cost sub-optimality. We found littleeffect on cost error as a function of u for any of the threeinventory models.

5 10 15 20

0

50

100

150

200

=0.7

Lead time, L

Averagecosterror(%

)

PT B BMA

5 10 15 20

0

50

100

150

200

=0.8

Lead time, L

Averagecosterror(%

)

PT B BMA

5 10 15 20

0

50

100

150

200

=0.9

Lead time, L

Averagecosterror(%)

PT B BMA

5 10 15 20

0

50

100

150

200

=1

Lead time, L

Averagecosterror(%)

PT B BMA

Fig. 6. Impact of lead time L on average cost error (%) compared to the full-information policy.



10/12

6. Conclusion

In practice, new marketing efforts introduce random uctua-

tions in the underlying demand process. False interpretation of the

nature of demand signals leads to under or overestimation of

inventory levels, and subsequently increased inventory costs. In

this paper, we discussed the inherent uncertainty in determining

the nature of demand uctuations, i.e., demand model, when a

reversion to a long-term trend in the underlying demand process

is very gradual and there is insufcient historical data since new

marketing efforts are introduced. Under these conditions, there is

a high probability of nding a spurious non-stationary pattern

using both contextual expertise and prevailing statistical ratio-

nales. We therefore face the risk of choosing an incorrect demand

model when making inventory decisions.

We proposed an inventory model based on Bayesian modelaveraging to better address uncertainty about the demand model

when estimating required base stock levels based on a critical

fractile. In particular, we consider the case where demand

becomes either stationary or non-stationary in response to new

marketing efforts. We show that the proposed BMA inventory

model estimates consistent base stock levels and it provides better

predictive performance, as measured by a logarithmic scoring rule,

than by using any single demand model (either stationary or non-

stationary model). The simulation results conrm the structural

results and suggest that the BMA inventory model carries impress-

ively low risk and is the most robust option among the feasible

inventory models we considered. The BMA model can bring a

signicant improvement to an inventory system, particularly for a

system characterized by highly autocorrelated demand, small

sample sizes, long replenishment lead times and high service

levels.

This paper applies Bayesian model averaging to address uncer-

tainty about the nature of demand signals in the rst-order

autoregressive (AR(1)) demand process. We are able to generalize

the proposed inventory model in more complex demand processes

that include other autoregressive lags (AR(p)):

Yt c1Yt 12Yt 2pYtput;

wherec is a constant and p is the lag order of the autoregressive

process. As the stationary/evolving character of a series is still

determined by its level of integration, we can derive stationary

and non-stationary models based on whether or not the autore-

gressive polynominalL 11LpLp has a root on the

unit circle. We can then average over the inventory decisions made

by the two demand models to make more robust inventorydecisions. Note that any deterministic components of the model,

such as deterministic trends and seasonal dummy variables, can be

removed prior to applying the proposed approach and added back

again to the resultant order quantities. More generally still,

Bayesian model averaging can be applied to address other types

of demand model uncertainty when estimating base stock levels,

such as uncertainty about potential predictors in a linear regres-

sion demand model (e.g., Azoury and Miyaoka, 2009).

There are some limitations of using Bayesian model averaging.

Specically, there is a lack of consensus regarding how to use

subjective prior information and, subsequently, how to choose the

set of candidate models, how to assign prior probabilities to each

candidate model, and how to select prior distributions for para-

meters. This is a general concern in the growing body of Bayesian

0.80 0.85 0.90 0.95

0

50

100

150

=0.7

Service level, K

Averag

ecosterror(%)

PT B BMA

0.80 0.85 0.90 0.95

0

50

100

150

=0.8

Service level, K

Averag

ecosterror(%)

PT B BMA

0.80 0.85 0.90 0.95

0

50

100

150

=0.9

Service level, K

Averagecosterror(%)

PT B BMA

0.80 0.85 0.90 0.95

0

50

100

150

=1

Service level, K

Averagecosterror(%)

PT B BMA

Fig. 7. Impact of service level Kon average cost error (%) compared to the full-information policy.



11/12

model averaging literature and calls for further research. Despite

these limitations, it is often shown that Bayesian model averaging

provides better average predictive performance than any single

model in a range of practical applications (Hoeting et al., 1999). In

addition, there is the computational burden involved in the

construction of posterior distributions where there is no closed-

form solution to the marginal likelihood of the model. With the

development of computational power, the process of calculating

model likelihoods should not be an obstacle to the practicalimplementation of the BMA inventory model.

There are a number of directions in which it would be of

interest to extend the proposed Bayesian model averaging

approach to. These could include extensions to a multi-period

setting and dealing effectively with censored demand data. Finally,

it is important to examine the empirical validity of the Bayesian

model averaging model using real demand data in the future

research.

Acknowledgments

The author thanks editor Edwin Cheng and an anonymous

referee for their thoughtful, clear, and constructive feedback. She is

also thankful for many valuable comments made by Stefan

Scholtes, Danny Ralph and James Taylor.

Proofs

Proof of Theorem 1. Assume that MS is the true model. When

prior probabilities PrMS PrMNS 0:5 are used, the posterior

probability for MSis

PrMSjD PrDjMSPrMS

PrDjMSPrMS PrDjMNSPrMNS

1

1 PrDjMNS

PrDjMS

:

15

SinceMSandMNSare nested models and proper priors are used for

the model parameter, the Bayes factor ofMNSwhen compared with

the true model,MSin this case, tends in probability to zero as the

sample size tends to innity (O'Hagan and Forster, 2004):

limn-1

PrDjMNS

PrDjMS 0:

By substituting this into (15), we have limn-1PrMSjD 1.

Similarly, limn-1PrMNSjD 1 can be shown when MNS is the

true model.

Proof of Corollary 1. This immediately follows from Theorem 1

and(14).

Proof of Theorem 2. From the non-negativity of the Kullback

Leibler information divergence, Elogfmi 1 PrjMi; DPrMijDgrElogfPrjMj; Dg forj 1;; m (Madigan and Raftery, 1994).Therefore, we have Elog fPrQn;BMAjDgrElogfPrQn;jjDg for

j S;NS. Since a logarithmic scoring is proper, with a negative sign the

logarithmic scoring is minimized when reporting the true probability,

PrQnjD. Therefore, we have ElogfPrQnjDgrElogfPrQn;BMAjDg.

Proof of Theorem 3. Qn;SoQn;NS holds only if Ln;NS

Ln;S

zLn;NSLn;S40. By rearranging, we need to satisfy the condition

Ln;NSLn;S

Ln;NSLn;S

4 z:

First note that Ln;SoLn;NS and

Ln;So

Ln;NS hold since n

L andnL in

(5) and (7) are increasing functions of for non-negative values of

c;; Yn. Secondly, for KZ0:5, the value ofz 1

K is positive or

equal to zero. Therefore, the above condition holds trivially and we

have Qn;SoQn;NS. Also, 0oPrMSjDo1 and PrMNSjD 1

PrMSjD. Since Qn;BMA is dened by (14), we have Qn;SoQn;BMAoQn;NS.

Proof of Theorem 4. Qn;SoQn;NSholds only if

Ln;NSLn;S

Ln;NSLn;S4z:

For Ko0:5, the value of z is negative. Therefore, this condition

does not hold trivially. Since z 11 K and from(12) and(13), we can rewrite this condition as

cL 1i 0 f1 i i

j 0j

g YnL Li 1

j

u

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL 1

i 0f1i2 ij 0

j2g

q 4 11 K:We can then use the same argument as above to show

Qn;SoQn;BMAoQn;NS. Similarly, we can show the condition for

Qn;NSoQn;BMAoQn;S.

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Bivariate, First-Order Vector -Autoregression Bayesian Model

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