Abstract
As indicated in the discussion of the model building process in Section 5.1, the step which logically follows specification is called parameterization. And first of all, data are needed in order to be able to determine model parameters. Specification forces the decision maker to be explicit about which variables influence other variables and in which way. At the same time, specification will point to which data concerning what variables are to be collected.1 Sometimes they are available or can be obtained without much effort. In other cases specific measurement instruments have to be developed, or the data exist but are more difficult to obtain.
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References
See also the study of side benefits in Section 3.2.
We will not give much attention to data collection methods. The interested reader is referred to, for example, Kollat, Blackwell and Robeson (1972, Ch. 4).
See Osgood, Suci and Tannenbaum (1957).
Boyd and Massy (1972, pp. 247–254) suggested the use of Bayesian analysis in this respect.
This relates to the concept of face validity to be taken up in Chapter 12.
The following text is based on Leeflang (1977a). Other authors such as Montgomery and Urban (1970) prefer the label marketing decision information system.
See also Montgomery and Urban (1969, pp. 17–26 and 1970), and Bosman (1977a).
Other equations in the full model were discussed in Chapter 5 (equations (5.5) to (5.9)), and in Chapter 7 (equations (7.3) to (7.10)). For a more complete discussion, see also Leeflang (1977a).
The reader will recognize that (11.1) is the same as (7.1).
To be defined in Chapter 12.
To be defined in Chapter 12.
See also (8.5) and Table 11.2.
Koerts and Abrahamse (1969, p. 6) add that the model should also be linear in the random variables.
For examples, see Sections 5.3.1.1. to 5.3.1.3.
For examples, see Section 5.3.1.4.
Kmenta (1971, p. 202) also adds that they should have finite variance.
See, for example, Cramer (1957, pp. 424–434).
For the details of the differentiation of S with respect to the vector ß, we refer to any of the econometric textbooks listed above. For example, Koerts and Abrahamse (1969, pp. 19–21).
Since X has rank k, X′X is nonsingular, and therefore its inverse (X′X)-1 exists.
First derived by Aitken (1935).
A number of other procedures to estimate the variances of the disturbances terms are proposed by Theil (1965a), Koerts (1967), Koerts and Abrahamse (1969, pp. 42–50), Leeflang and Van Praag (1971, pp. 70–74).
Alternative assumptions can of course be made about the structure of the variance-covariance matrix of the disturbances. For more details, we refer to Kmenta (1971), Maddala (1971), Nerlove (1971), Bass and Wittink (1975), Moriarty (1975), and Van Duyn, Leeflang, Maas (1978).
See Kmenta (1971, pp. 510–511).
Because the n sets of equations in (11.48) do not seem to be related, one refers to this structure as ‘seemingly unrelated regressions’. See Zellner (1962).
For a more detailed treatment, see Zellner (1962) or Kmenta (1971, pp. 517–519). See also Leeflang (1974, pp. 124–127), and Leeflang (1977d).
See also equation (6.5).
In a case where a system of equations is exactly identified, less complicated estimation procedures such as Indirect Least Squares (ILS) can be applied. See for example, r Wonnacott and Wonnacott (1970, pp. 161–163).
In fact, the order condition is a necessary but not always sufficient condition for iden-tifiability. Sufficiency also requires the rank condition to be satisfied. For an extensive treatment of the identification problem see Fisher (1966).
See Johnston (1972, pp. 408–420).
The value of k maximizing R 2 will be a maximum likelihood estimate. See Goldfeld and Quandt (1972, pp. 57–58).
See the footnote relative to Table 10.3 for a reference to the troll system. The method is also implemented in the BMD07R program. See Parsons (1975).
For an excellent survey of optimization techniques, see Wilde and Beightler (1967).
How (11.83) was arrived at is described in Naert and Bultez (1975, pp. 1107–1109). Since the model is derived by linking transition probabilities in a Markov chain to the number of outlets of brand 1 and those of competing brands (brand 2), the example would perhaps better fit in Section 11.3. This is of little consequence, however, since our main purpose in the current section is to point to some of the peculiarities of nonlinear estimation.
For a discussion of the Newton-Raphson method see Wilde and Beightler (1967, pp. 22–24). The SUMT computer package has a number of options with regard to the minimization technique. The Newton-Raphson method is one of these. The computer program of SUMT is described in Mylander, Holmes and McCormick (1971).
Estimates become more reliable, that is, they have smaller variances as the number of degree of freedom available for estimation is larger. See Section 12.2.
For a more rigorous derivation of this stochastic relation see Leeflang (1974, pp. 123–124).
For an extensive discussion of such models we refer to Lee, Judge and Zellner (1970).
To avoid this, a more robust specification of the transition probabilities is needed. Colard (1975), for example, applied the attraction model which has the additional advantage of allowing for interaction between the marketing instruments.
On this point see Section 9.2, where this question is also discussed.
Subjective estimation is to be discussed in Section 11.5.
Supplying inventories to wholesalers, retailers in the pipeline (= channel) from producer to final consumer.
Figure 9.3 may serve as an illustration of such an adjustment, applied to the distribution of interpurchase time.
Little (1975b, p. 659) refers to calibration as the overall effort to finding a set of values for the input parameters to make the model describe a particular application. Estimating from historical data, subjective estimation, and tracking are all part of the calibration process. In that sense, good tracking will be a necessary but not sufficient condition for good calibration.
We should observe that as far as Urban (1974) is concerned, tracking refers to comparing forecasted and actual values on a new set of data, that is, observations that were not used in estimating and fitting. Little (1975b) does not make that distinction, that is, tracking refers to comparing predicted and realized values, without reference to which set of data.
Although, as was indicated in Section 9.3, many definitions of attitudinal variables exist, this does not imply that there are equally many measurement instruments.
Werck (1968) observes, that one of the striking characteristics of the psychometrics of attitudes is the relative independence of measurement techniques from the conceptualizations.
There are, however, exceptions such as Parson’s (1975) study of time-varying advertising elasticities over the product life cycle. The preceding limitations of econometric methods are studied by some marketing staff members of the faculty of economics at the University of Groningen. See, Bosman (1975, 1977b), Leeflang (1977c), Reuyl (1977).
Sometimes qualitative factors or changes can be represented by dummy variables, as was the case in Schultz’ (1971) study of competition between two airlines and referred to in Section 11.2.3. The use of dummy variables is, however, limited.
Little replaces the more generally employed term econometric model by statistical model.
See in particular Sections 8.3, 11.2.2 and 11.2.4.
That is the problem of multicollinearity to be discussed in Section 12.3.3.
The most notable examples in this respect are large simultaneous equations system of national economies. For an example in marketing see Tsurumi and Tsunami (1973).
See also the discussion of side benefits of model building in Section 3.2.
See Kotier (1971, p. 584).
See, for example, Hampton, Moore and Thomas (1973, p. 4).
The example is taken from Kotier (1971, p. 585).
This part of the discussion closely follows Naert (1975b, pp. 140–143).
No brand index has been added to be consistent with the notation adopted by Little (1970).
Little (1970, p. B-47) separates long run and short run affects. We will not do so since this would only complicate the exposition, without adding to its substance.
In fact Little (1970, p. B-976) asks what market share is at the start of the period, and he then asks what advertising will maintain that share.
Since (11.94) is intrinsically nonlinear, the methods presented in Section 11.2.4 will be applicable.
On cross-impact subjective estimation, see Tydeman and Mitchell (1977) and the references contained therein.
How to use subjective probability distributions once they have been obtained has of course a much longer history, and is of particular relevance in Bayesian decision theory. For a general introductory text see Schlaifer (1969) and Raiffa and Schlaifer (1961) at a more advanced level. For an early application in marketing see Green (1963). For an interesting real life application of probability assessment see the study by Schussel (1967) on the forecasting of sales of Polaroid film to retail dealers.
An interesting introductory survey containing a substantial number of references is Hampton, Moore and Thomas (1973).
See Savage (1954), and De Finetti (1964).
Some other methods are discussed in Smith (1967), and Hampton, Moore and Thomas (1973).
Slightly adapted since Winkler’s study related to Bernouilli processes. For a full questionnaire related to the four techniques see Winkler (1967a, pp. 795–801).
Other examples related to the lognormal and Weibull distributions are given by Kotier (1971, pp. 589–591).
Edwards and Philipps (1966) show that providing monetary rewards induces people to learn more quickly.
For other examples of scoring rules see Roberts (1965), Winkler (1967b, 1967c), and Staël Von Holstein (1970).
The terminology follows Winkler (1968).
The same weighting schemes could of course also be applied to point estimates.
If there are no ties (11.103) and (11.104) are the same. In case there are ties (11.103) is the correct formula.
This is applied by Brown and Helmer (1964) but in another context.
Morris (1974, 1977) has provided the basis for a normative theory of expert use based on the tools of Bayesian inference. His approach looks very promising but, as Morris (1977, p. 693) indicates himself, its efficiency for practical problems has yet to be fully established.
The calculations themselves are easy but the underlying mechanism is more complex.
For a detailed treatment of natural conjugate distributions the interested reader is referred to Raiffa and Schlaifer (1961), and to Winkler (1968, pp. B-64 — B-69) for its application to pooling subjectively estimated probability distributions.
We again employ Winkler’s (1968) terminology.
A variant of the method could be to display the various assessments with the identity of the assessors.
Although Little (1975b, p. 659) observes that individuals working closely with a product often make surprisingly similar response estimates.
See, for example Dalkey and Helmer (1962), Brown and Helmer (1964), Dalkey (1967, 1969a, 1969b), Helmer (1966), and Brown (1968). See also Chambers, Mullick Smith (1971), and Keay (1972).
We closely follow Dalkey and Helmer (1962).
Equations (11.107) and (11.108) are the same as equations (11.7) and (11.15) respectively.
Prior information can be subjective, but it can also be objective, such as estimates obtained in other studies.
For the details see Theil (1963) and Horowitz (1970, pp. 440–443 and pp. 448–450). The reader interested in Bayesian inference in econometrics is referred to Zellner (1971).
As we have argued in Section 11.5.2.1 subjective estimates of response coefficients should normally be obtained indirectly.
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© 1978 H. E. Stenfert Kroese B. V.
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Naert, P.A., Leeflang, P.S.H. (1978). Parameterization. In: Building Implementable Marketing Models. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-6586-4_11
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