Analysis of seasonal discounts in small markets using statistical methods

ELMIRA KUSHTA1, MIFTAR RAMOSAÇO2

1Department of Mathematics, Faculty of Technical Sciences, University “Ismail Qemali”, Vlore, ALBANIA

2Department of Mathematics, Faculty of Technical Sciences, University “Isamil Qemali”, Vlore, ALBANIA

Abstract: In this paper we consider the apparent changes in consumer spending in a real market (Vlora County). By calculating the

latent variable of discounts as it is done, we do not interpret the net of expenses. We found that discounts stimulate spending

behavior, which shows that the consumer buys the minimum as well as the highest prices and not with the right budget.

Essentially, the ecological steps here have functioned as data-oriented modeling that reduces subjective or am approaches to

econometric and marketing analysis. The complexity of separate actions and then its dimensions and reduced the model of

conclusion and improved analysis.

Keywords: Probit model, consumer behavior, distributions, logistic function.

Received: September 28, 2022. Revised: August 30, 2023. Accepted: September 21, 2023. Published: November 2, 2023.

1. Introduction

Consumer behavior is an opinion formation process

which is difficult to be studied quantitatively and

rather complex as evidenced in [1].

The study of concrete economic environment

unavoidably face challenges to the scholars.

Standard questionnaires that aims gathering

information from social or economic mediums

include different types of variables, non-numerical

responses, questionable answers, missing or

incomplete records etc. Usually the inquiries might

be organized and held in different moment of times.

Finally they must be included in modeling say linear

multivariate functions

In a formal approach they are assumed to act

rationality in their decision making by optimizing

some utility function, but this last cannot be

measured directly. Meanwhile the assumption of

rationality does not hold always which is clarified

by behavioral theories as discussed in [2], [3] etc.

Behavior seems to be too complex to be studied and

analyzed by deterministic methods. Even so,

researchers and scholars have outdone this

difficulty by using statistical tools and probabilities

facilitating modeling in this case as presented in

many textbooks. The second problem is related to

the tangible set of the variables included in the

models. Again, standard models belong to the

standard systems and in real ones there is a

considerable difference Mixed calculation using

econometric optimization and network dynamics

have been developed as for example in [4] and many

other applications. Generally, the consumer’s

decision making process in buying is complex but

econometrically known and measurable. But by

carefully using simple analytic tools it is possible to

avoid the complexity of the model, to control extra

errors added during calculation phase and improve

overall calculation. In our recent research in the

analysis of consumer behavior in district of Vlora,

we considered such specifics as an important step

[1], [2] etc. Aside of general models and regressions,

practical calculation have demonstrated their

capacities to describe consumer behavior in specific

systems as in [5], [6] and many others. In our recent

work [7] we applied a logistic regression to identify

the consumer profile in a specific area, the factors

affecting their behavior and other parameters

characterizing the system of consumer attitudes and

activities. Therein we have focused our attention in

the fat that the stationary of the state should be

considered in the framework of advanced analysis

elsewhere proposed in [8] and [9] with mathematical

reviews in [10]. In reference [11] a more advanced

technique have been reported dealing with

complexity in the behavioral models. From those

and other references that we are not listing here, we

acknowledged the importance of folded econometric

and mathematical details for quantitative

consideration for such systems. Detailed analysis on

those aspects are provided in many articles-guides

and statistical books as [3] or [4]. In this case the

problems could be overcomes if we adjusted

correctly the sample size or adopt a suitable

sampling method. In the case where the above step

is not suitable or even impossible, the factorial and

descriptive analyses could be used as recommended

in standard procedures to manage the sampling

error, [5] and general consideration [13]

This study is intended to evaluate a marketing aspect

as discounts for example, by specifically considering

the nature of the state of the system, the possible

presence of not-apparent factors as latent effects or

hidden variables etc.

INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,

COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING

DOI: 10.37394/232026.2023.5.15

Elmira Kushta, Miftar Ramosaço

E-ISSN: 2766-9823

171

Volume 5, 2023

2. Optimization by maximum likelihood.

Referring to the linear discriminate treatment above, we have

evidenced that each of the posed binary variables is suitable to

identify the probability of customer behavior [12].The estimate

here is a benchmark estimate that should be pushed further

with the likelihood function optimization technique through

this procedure on the candidates proposed by matching OLS or

W-OLS [7]. In most cases, we have applied specially built

routines, so the calculation part is chronologically recorded

here. Of course, calculations based on sequential forms are the

alternative behavior since most software have built-in

functions. [20] Assessing that the deviations are for accidental

reasons.

󰇛󰇜

󰇛󰇜

 (1)

The likelihood estimator has the form

󰇛󰇜

ln 󰇡 1

σ22π󰇢wi

2e-wix-μ2

2σ2=-ln[σ22π-1

2σ2x-μ2wi󰇠 (2)

󰇛󰇜󰇛󰇜

󰇛󰇜(3)







 





 





(4)





󰇝󰇞󰇛󰆒󰇜

󰇛󰆒󰇜 (5)

Again the best regression (optimal decomposition coefficients)

is found if their infinitesimal change does not change the cost

function.

 

























....2

exp1

....2

exp1



(6)

We have tried to follow the answers in the form "expenses of

category x are in group I " where we have divided the

expenditure classes into subgroups. Accordingly, we obtained

m=3 by grouping the expenditure variants in 3 classes. In this

attempt, we have not found a better answer. [18]The following

two-valued case considers the classical case with binary

responses.

3. Identification of the local optimal

model

The treatment here is local in the sense that the treatment is

done unilaterally. [23]We have fixed a variable or several of

them as responses, and we try to find the best model i.e. the

causal variables that lead to those responses. In the analysis of

the importance of the model and the results obtained, we find

that there is evidence of causality in the decision-making to

spend more on basic goods than on all others.

Figure 1: Logistic regression for different sets of variables.

Black circles log regression for variables X1:X6 with pink

dots, regression for X1:X3. The blue and red dots give the 95%

confidence interval.

In figure 1 we have presented the result as well as the margin

of error of the algorithm. [15]We note that in this

approximation:

 The probability of the outcome (outcome) extends

well in the space of allowed values [0,1]

 There is a defect at the edges as it does not respond to

a complete sigmoid but only the central part.

This behavior indicates that the domain of determination is

incomplete at the edges and in this case a good strategy is seen

to include a continuous variable in the predictors X since they

are all categorical. Likewise, the removal of some variables

causes us to change this level of inaccuracy by suggesting the

strategy of tests for each one, we have concluded linearly as

can be seen in table 1, where the values of Eigen-significance

p, error (deviation) and statistics of factual assessments are

presented. It is found that these parameters vary from variable

to variable and have significant values that question the

completeness of the model.

INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,

COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING

DOI: 10.37394/232026.2023.5.15

Elmira Kushta, Miftar Ramosaço

E-ISSN: 2766-9823

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Volume 5, 2023

Table 1: Variables of the model

The state of the system is characterized by average purchasing

expenses, number of visits in the market. [23]Therefore the

aim of the analysis is focused in the evolution of those

quantities if sales or discounts are applied. [7] We realized it

by analyzing the distribution of the response quantity measured

in those two states respectively, and applying a probity model

to estimate the utility underlining the dynamics observed.

4. Obtaining Meaningfulness of Variable Type

and Units.

It resulted that for all response variablesrelative counterparts were

characterized by smooth distribution, Table 2. For predictor variables we

adopted a descriptive approach by classifying direct values in categories.

Table2: Variables of the model

Predictor Variables

Value

Variable

Type

set I

Set II

Family type

categorical

1-5

Education level

categorical

1-5

1-6

Age

categorical

1-6

1-4

Employment Status

categorical

1-3

1-2

Income Type

categorical

1-3

1-9

Gender

categorical

1-2

Total Consummator

Budged

numerical

Real Value

Real

Value

Initiallyvariable.

Value

Real/Proportion.

Categorical value 1-

Representativevariables.

Real/Proportional

ProposedVariable. Real/Proportio

nal

categorical

Value :1/0

Expenses for:

Common expenditure

alimentary goods

Basic expenditure

dominant:

1/0

clothes

subsistence

Alcoholicdrinks and

cigarettes

Extra expenditures

health*

Transport

Communication

(mob Phone calls)

Culture and safety

expenses

Qualitative Life

Expenditure

Quality life and

luxuryex_penditures

dominant

Education

auxiliary services

Family expenses

luxury goods

Luxury_Expenditure

Restaurant_Expenses

We asked for variables to be appropriate for modeling in

logistic, MIMIC and other form if they were found in a

stationary state. We managed the measurement realized in the

sample where an individual appears as a list of records of

different type and different meaning. [19]To include all of

them in an deterministic model we must unify their

measurement method. Hence categorical variables were

transformed using z-score method in continuous variables.

 







. In another step we produced new variable binary

by using levels of expenses.











 5.0,0

5.0,1

)(:

ii R

ExpencesTotal

iExpence

RvelExpencesLe

This last is suitable for logistic and probit modeling

5. Minimal modeling testing

In constructing models we analyses all minimal model of the

form logit(x)=a+bx and keeps for further consideration

variables that showed good statistics of the minimal fit. In the

table below are displayed.

Findings

By using such steps we concluded that the profile of the

costumer for the economic medium analyses is















lBudgetLeve

HoldererGenderHose

seholderAgeGroupHo

IncomeType

Emplymnet

evelEducationL

FamilyType

ofileCostumer Pr

The most appropriate reference for variables measured or

their natural units are as follows.

Cause Variables: categorical Likert type 3 or 5 values

Response variables: Binary, measuring the prevalence of

an expense type

Epoch Type Values Expectations

Gender categorical 1,2 Possibly differenced effects

Age categorical 1,2,3,4 Possibly differenced effects

Average visit on the market categorical [1-30] informative variable

Total visits nominal [1-30] informative variable

Average expenses nominal

Real Value

Depended variable

Total expenses nominal

Real Value

Depended variable

Having Market Cards binary [1,0] increase chance of contacts

Having contacted by phone binary [1,0] increase chance of contacts

Total Visits nominal [1-30] Informative variable

Average visits nominal [1-30] Informative variable

Average expenses nominal

Real Value

Depended variable

Total expenses nominal

Real Value

Depended variable

after

discount

announce

ments

before

discounts

announce

ments

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Figure 2: Distribution of visits average expenses. Normal trading period. Small

picture shows log-log representation to better picture of the fit.

After realizing the fit to some expected common

distributions, we observe that the parametric q-distribution had

best statistics of fitting. The fitted curves are mostly q-

lognormal within alpha=0.05 restriction, whereas q-Gaussian

has lower statistics, but is much less sensitive to the binning

assize. Accepting functions of the type (2) as best fitted

distributions, one can admit that the processes underlying the

expenditures dynamics are q-multiplicative, hence very

complicated. From the fitted q-lognormal we obtained the

parameter q=1.0001 which report a nearly stationary lognormal

if multiplicative processes are determinant. In particular it does

not give the opportunity to measure the level of non-stationary

as the difference q-1=0.0001 is too small. But in first equation

of (3) we see that q-addition involve additive and

multiplicative property, so for mixed processes it seems to be

more significant. For this reason we prefer q-Gaussian for

analysis of such behavior. Parameters q and adjusted R-

Squared are [1.6531 0.9661] for q-Gaussian and [1.0001

0.9742] for the q-lognormal fitted. Therefore q-Gaussian tells

that q~5/3 that is in the boundary of definition for variances

 





q35





(7)

Next we considered the data for market visits and average

expenditures after sales were applied. We obtain that the

expenditure’s distribution were found in a more stable states.

The statistics for q-distributions fitted to the frequencies of

consumer visits at the market again support the q-lognormal as

best fitted function, but again by changing bin size we observe

that q parameter in q-Gaussian changed only slowly whereas

for q-lognormal it jumps from the value 1 with high margin.

Therefore we consider q-Gaussians for further analysis. Q-

Parameters estimated and R2 for this case are found [1.6525

0.9778] for q-Gaussian and [1.0000 0.9974] for q-lognormal.

We see that the stationary parameter q is nearly the same for

the two series (before and after sales) but as we explained

above the observation time for the second is much lower. So

we accept that the state after discounts is more stable .

Figure 3: Distribution of average expenses (per visit) after trade discount

announced.

We observe that announcing price discounts cause the

number of visits to be higher and more averaged, that impose a

more relaxed state which in turn could be related to a more

stationary behavior or more common, typical consumer

activities. In this sense we can this state for statistical analysis

of the market, estimation of the representative values for

expenditures and forecasting. We can simulate scenarios e.g.,

arrival of individuals with given set of properties using the

distribution found here as PDF of the values for quantities

discussed here. Moreover, we can approach this behavior as a

response to a natural utility optimization and dominant factors

to be typical ones, hence the disturbing terms could be

neglected. Remembering that the time of post discount study

has been as few as one third of the prior discount spanning

period, the relaxation effect could be even more important and

hence we can expect an intensive change of the probability of

choice induced by discount announcement. This preliminary

finding will support a more intensive change in cumulative

probability CDF and therefore, the probit model could be more

ascribing. In statistical aspect the result as of herein stated that

the overall state is more stationary after applying discounts and

we conclude that the first direct result of such marketing

options in the relaxation of the system itself, and brining the

distribution in the zone where the mean and variance are

measurable and well defined. In this sense, marketing studies

and other analysis on the consumer behavior are likely to be

more realistic in the period of sales. It is very important result

for the market studies in the case of small market area as local

districts and limited capacities for large inquiry that impose

instability of the state of the system.

The coefficients have been confirmed as different from zero

within 90% confidence, whereas the free coefficient seems to

not pass the test

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Figure 1: Probit regression for Increasing Expenditures after discounts

Therefore the utility of the attractiveness is obtained by

probit regression as follows

 







ctPhoneConta*0.0002 Group Age* 0.0962

rGenderBuye*0.1367- 0.0698Y*

(8)

From relation (3) we observe that the gender of buyers

(F=1,M=2) is mostly decisive in the increasing number of

visits in the market after sales, and usually male buyers are not

more frequent in the market after discounts have been applied.

The phone contact has a slight effect on it. The age group has

comparable role to the gender of consumer. In Figure 4 is seen

that the probability for more visits in the market is high for

almost all the values of utility function (8) and only few values

are less than 0.5. In this sense, for nearly all consumers’

specifics, the marketing strategy (prices discounts) has been

found attractive for peoples that respond by increasing the

number visits in the market. Thus is the intermediate change on

the consumer behavior. In the second stage, the final behavior

is considered.

Now the response variable is the change in expenditure

measured by the natural function “is greater than” e.g., in

absence of the marketing stimulus. It is possible that the buyer,

under budget constraint, would respond to the discount

spending the same quantity of money and therefore just buying

some more goods, so this variable is meaningful and not

trivially known. Here the independent variables include even

average expenditures before sales and registered cards

consumer. The first is expected to give information about

which consumer category has increased the expenses, and the

second could inform the role of being a formal consumer.

Performing probit regression we obtain the utility function

odOpen0.0396Peri )0.0013Call(CardHolder* 0.136-

ensesAverageExp 0.854 itAverageVis*0.0199

)C.AgeGroup*(0.0133-C.Gender * 0.093 -0.916*y





(9)

In (7), the statistical significance is acceptable for all

variables except Tel,Call and Age.Group (the p.Value is high,

~0.3) so we put it in parentheses. Notice that their coefficients

are small and so this does not affect the estimation of the utility

so we kept them in the equation (7). Now we make use of the

binary outcome expression using continues probability. The

switching value for utility is

   

08.0 uPforeXExpencesBeterExpencesAfP

(9)

Figure 5: Full model probit regression

By using equation (8) we can realized conditions that (9) is

fulfilled and therefore it is possible to forecast what happen

with purchasing behavior for an individual with values

721 ....}{ nnniConsumer 

(10)

This is done by just putting values (10) in equation (9). We

observe that rational or cognitive issues weight more in the

utility value as seen from the coefficients for female byer that

usually behave as major house holdings buyers, common

average expenses that indicate the level of budget in the

purpose, the time of effectiveness for sales. The expected

psychic parameters as telephone call have less effect in shifting

the utility. By randomly selecting consumer (predictor) values

according to the population considered, we see that the value

of (9) is usually reached, therefore the conclusion herein seem

to be a global tendency in the area studied. Remember that

those findings are general characteristics because the

distribution in this state has been acknowledged as being

stationary.

6. Conclusions

The harmonization of statistical techniques is an effective

instrument in the analysis of econometric and psychometric

systems. The study of changes as an argument of psychosocial

elements with economic and financial ones requires high

mathematical rigor for the influence of the utilization of

different sectors of statistical analysis is a finding of value.

From the modeling side, in the reviewed systems and others of

these categories, the harmonization of statistics, numerical and

analytical scaled factors are defined in the creation of a reliable

calculation scheme. In the case of making purchases

conditional on the customer's behavior being rational, he

weighs the alternatives in terms of his use. In particular, to

conclude that we consider the consumer towards discounts was

characterized by the increase of the expenses themselves, not

only of the volumes of the items bought. From a marketing

point of view, this work suggests that the concessions have

studied and programmed the program of carrying out their

responsibilities not only to make the depreciation of a stock,

but also for themselves the profit. We arrived at a more

described finding of the system, in a more natural form of its

other places.

INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,

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DOI: 10.37394/232026.2023.5.15

Elmira Kushta, Miftar Ramosaço

E-ISSN: 2766-9823

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Volume 5, 2023

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No funding was received for conducting this study.

Conflict of Interest

The authors have no conflicts of interest to declare

that are relevant to the content of this article.

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INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,

COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING

DOI: 10.37394/232026.2023.5.15

Elmira Kushta, Miftar Ramosaço

E-ISSN: 2766-9823

176

Volume 5, 2023