College aantekeningen
Market Models Summary
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- Rijksuniversiteit Groningen (RuG)
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1MODEL SPECIFICATION
1. MODEL TYPES
A. Intended Use
- Descrip)ve: describe what happened (focus on betas)
- Predic)ve: describe what’s next (focus on y)
- Norma)ve (or Prescrip)ve): describes which decisions need to be taken based on the model
B. Level of Demand
- Product class / form sales / industry sales
- Brand sales: explain the sales of a certain brand
- Market share: explain what part of the market a brand takes up
C. Amount of behavioral details
- No details
- Some details
- Substan)al amount of details
2. MODEL ELEMENTS
The elements of a single equa)on model are:
- Dependent variable (or criterion variable)
- Independent variables (or predictor variables/explanatory variables)
- Disturbance term: it picks up what the X variables do not explain. It takes care of what was not en)rely
perfect with the rest of the model, such as random error, measurement error, missing variables,
specifica)on error, and behavior differences that are hard to model.
- MathemaGcal form of relaGonship between variables:
• Models linear in parameters and variables: linear means that every addi)onal unit of X has a
constant effect on Y. In other words, there is a fixed coefficient, Beta 1, associated with each X
variable. If X increases by 1 unit, we expect Y to increase by Beta 1. This is a linear addi)ve model
because it’s simply the sum of separate effects.
• Models linear in parameters, not in variables: we can accomodate situa)ons where certain variables
do not have a linear effect on Y by applying a transforma)on to the X variables before incorpora)ng
them into the model in their original format. Here we allow for certain parameters to not have a
linear effect.
• Models nonlinear in parameters but linearizable: The third type of model is more complex as it
involves the mul)plica)on of variables instead of addi)on (mul)plica)ve model). The effect of one
variable on Y depends on the value of another variable due to the mul)plica)on. However, it is not
linear and, to solve this, the logarithmic func)on can be used to transform the model into an addi)ve
form. One advantage of the mul)plica)ve model is that it accommodates interac)ons, which are
common in marke)ng. However, there are also disadvantages to consider.
• Models nonlinear in parameters and not linearizable
• Nonparametric models: no parameters: we let the data figure it out how the rela)onship look like
3. MODEL CRITERIA
The five criteria for a good model (John D.C. LiXle) are:
- Simple: A model should be simple and comprehensible. It should not be overly complicated, as this can
lead to more uncertainty and increase the risk of overfiZng. This can be achieved by not including too
many variables, having a simple func)onal form and using one equa)on.
- EvoluGonary: A model should be adaptable to changes in the market. It should be able to accomodate
major changes.
- Complete: A model should include all relevant variables that affect the outcome. Trade-off between
simple and complete.
- AdapGve: A model should be able to adjust to new situa)ons, like changes in the market.
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- Robust: A model should be able to give good answers under all circumstances. For example, a market
share can’t be nega)ve.
4. MODEL CHOICES
1. AccounGng for MulGple EnGGes: When dealing with heterogeneity among brands, chains, regions, etc.,
there four common op)ons for modelling:
• AggregaGon: Aggrega)on is a modeling approach that involves geZng rid of the en)ty dimension and
considering only the total number of observa)ons. This approach simplifies the model by trea)ng all
en))es as a single unit. To aggregate, you first decide upon a simple aggrega)on func)on, such as
averaging. For example, the sales at week 1 is the average sales across the 2 chains, and the price is the
average price across the 2 chains. This way, you get rid of the chain dimension, and the model looks like
the effect of average price on average sales. However, the downside is that the es)mates are not per
chain, and you have fewer observa)ons (= posi)ve side, data are not very vola)le).
• Unit-by-unit models: This approach involves es)ma)ng a separate model for each en)ty. This approach
is useful when the en))es are different and behave differently. To indicate this in the formula, an "i" is
added before the "t" to show that the parameters and prices have different values for each chain. For
example, S it is equal to a i (Alpha i) + B iPit, where "i" can be 1 or 2 for the first or second chain. To
es)mate this, you would use only the data available for the first chain, run a regression analysis, get the
parameter es)mates for that chain, and then repeat the process for the other chain. The advantages of
this approach are that you have data per chain, and the es)mates are per chain, so you can see the
effect of price for each chain separately. However, since you are es)ma)ng per chain, you have only t
number of observa)ons for each regression, so you have fewer observa)ons per parameter.
• Pooling: Pooling assumes that the en))es (e.g., different brands or chains) behave similarly and can be
analyzed together. It assumes a common effect across all en))es. In this approach, the data for both
chains is stacked underneath each other, and the same Alpha and Beta values are used for both chains.
This approach has the advantage of having many observa)ons per parameter, making the model more
efficient. However, the es)mates are not per chain, and there are no chain-specific es)mates.
• ParGal pooling: This approach accounts for some differences among the en))es while s)ll assuming
some commonality (usually the intercept has i). It is a compromise between unit-by-unit models and
pooling. Par)al pooling involves assuming that some parameters are different for different chains, while
others are assumed to be the same. For instance, the price elas)city or the reac)on to price changes for
customers who go to different chains may be assumed to be similar, but the intercepts (Alpha i) may be
different for each chain. This approach allows for a trade-off between flexibility and efficiency, as it
provides data per chain and many observa)ons per parameter for those parameters that are not chain-
specific.
2. Dynamic Effects: Dynamic effects refer to the impact of changes over )me on the variables in the model,
for example the impact of past promo)onal ac)vi)es on current sales. Customers build up a memory of
past adver)sing, and this memory affects their response to current adver)sing. The stock of adver)sing in
the mind of customers is the dynamic part of the model.
There are three models for dealing with dynamic effects:
• The current effects model: The current effects model implies that only the adver)sing and detailing
ac)vi)es on a given day have an impact on brand sales, with no lagged effects.
• The direct lag model: the direct lag model assumes a more realis)c scenario where the effects of
adver)sing and detailing ac)vi)es are delayed, poten)ally taking effect the next day or several days
later. This model allows for different lag orders for each variable, reflec)ng the direct impact of these
ac)vi)es with varying )me delays.
• The parGal adjustment model: The par)al adjustment model is a distributed lag model that assumes
that the best predic)on of sales today is the sales of yesterday: so a por)on of sales of yesterday +
some adjustments. The model uses the lag operator to account for the delayed effects of adver)sing
and detailing ac)vi)es. The model is complex, with infinite lag effects, but the effects shrink over )me
due to the variable Lamda, which reduces the impact of lags. The model can be simplified by using the
power series, which shows that the effects of adver)sing and detailing decrease as they move further
into the past.
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3. Role of Third Variable (z) as a Mediator/Moderator: The role of a third variable, denoted as "z," can have
various effects in a sta)s)cal model:
• Spurious causal effect: The spurious causal effect refers to a non-causal rela)onship between variables
X and Y, both of which depend on a third variable Z. This situa)on arises when X and Y appear to be
related, but there is no direct causal link between them. Instead, they are influenced by a common
factor, Z. If variable X is affec)ng variable Y only when variable Z is not controlled for, then the effect of
X and Y is spurious. When the third variable Z is taken into account, the effect of X on Y should
disappear. To assess spurious effects, one can test whether the rela)onship between X and Y
disappears when controlling for Z. This can be done through regression analysis, examining the
residuals, and conduc)ng separate analyses based on different levels of Z to determine the existence of
a true causal rela)onship.
• InteracGon effect (moderator): Interac)ons or moderator effects occur when a third variable Z affects
the effect (strength) of x on y. To measure the modera)ng effect, an addi)onal column is added to the
model, which includes X and Z. The dependent variable is then regressed on X, Z, and the mul)plica)on
of the two. The beta coefficient of X*Z represents the modera)ng effect, indica)ng how much the effect
of X on Y changes based on the value of Z. To properly account for the modera)ng effect, both X and Z
should be included in the model, along with their main effects.
• Indirect effect (mediator): The indirect or mediator effect involves a variable Z that reflects the process,
mechanism, or stage between input/predictor variable X and outcome/criterion variable Y. This
represents the indirect effect of X on Y through Z. A mediator effect occurs when the effect of X on Y is
indirectly transmiXed through another variable. In a conceptual model for media)on, a regression of Y
on X is conducted to determine if the effect is direct or indirect through the mediator. If the effect
vanishes upon including the mediator, it indicates full media)on. If the effect is s)ll present but weaker,
it signifies par)al media)on. If the effect does not exist, it means there is no media)on.
5. MODEL ASSUMPTIONS
AssumpGons on the disturbance term:
1. E(εt)=0 for all t: Mean zero. The error term accounts for the varia)on in the dependent variable that
the independent variables do not explain. Random chance should determine the values of the error
term. For your model to be unbiased, the average value (mean) of the error term must equal zero.
2. εt ~ N(σ2): Normality. The error term should be normally distributed
3. Cov(εt,εs) = 0 for t ≠ s: Indipendence. The error term of today should be independent of the error term
of tomorrow or any other )me. For instance, if the error for one observa)on is posi)ve and that
systema)cally increases the probability that the following error is posi)ve, that is a posi)ve correla)on
(problem of autocorrela)on). There should be no paXern in the residuals. This independence is
essen)al because, if there were a systema)c rela)onship between today’s error term and tomorrow’s,
you would want to incorporate this into your model rather than trea)ng it as an error term. If you know
of such a rela)onship, you should control for it.
4. Var(εt) = σ2 for all t: HomoscedasGcity. The variance of the errors should be consistent for all
observa)ons. In other words, the variance does not change for each observa)on or for a range of
observa)ons. In prac)ce, this implies that you can predict, for example, sales during periods with and
without price promo)ons equally well using a single model. Your model should not predict periods with
promo)ons beXer than those without. The error in predic)ng the actual outcome should not
systema)cally change its value over )me. The opposite of homoscedas)city is heteroscedas)city, and if
it exists, it would mean that you cannot trust the outcome of your es)mated model, where the error
term's uncertainty systema)cally changes over )me.
Other assumpGons:
5. Cov(xit,xjt) = 0 for all i ≠ j: Indipendent x variables. The independent variables should not only be
independent from the error term but also independent from each other (if not, mul)collinearity issue).
6. Parameters are constant over Gme. The parameters es)mated in your model should be constant over
)me. This assump)on is inferred from the absence of a )me variable in the parameters' formula.
Therefore, the parameters should remain constant over )me.
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