Testing functional forms of market share models using the Box-Cox transformation and the Lagrange multiplier approach

95

Testing functional forms of market share models using the Box-Cox transformation and the Lagrange multiplier approach Dipak Naufel

C. JAIN * J. VILCASSIM *

The most commonly used functional forms in measuring market response functions are the linear and log-linear (double-log) specifications. Although the two models are mutually non-nested, they are both nested within the class of Box-Cox regression models. This enables one to test the statistical validity of these two models using nested tests, the power characteristics of which are better established relative to non-nested hypotheses tests, at least in large samples. In this paper, an application of the Lagrange multiplier (LM) test to determine the validity of linear, log-linear, and attraction-type formulations of market share models is illustrated using marketing data. The test is easy to compute and involves running only one extra linear regression. A Monte Carlo simulation is performed to study the properties of the test for samples of varying size and different levels of error variance. The simulation results indicate that the LM test should not be used with samples of less than 100 observations. We also compare the performance of the LM test to that of the P, test developed by MacKinnon, White, and Davidson for non-nested models. The results show that the Pa test has a lower probability of a type 1 error for all sample sizes and different error levels. The power of the LM test, however, is greater when the error variance of the true mode1 is high, given a fixed sample size.

1. Introduction Regression analysis is a widely used technique in marketing research. The range of applications includes theory testing, parameter estimation, and forecasting. The construction of a regression model usually starts with the specification of the dependent (or criterion) variable and the independent variables, followed by the specification of the functional form that shows how the dependent variable is related to the independent variables. In marketing research, the linear and log-linear (or double log) specifications are the most commonly used functional forms (see, e.g., Assumus, Farley and Lehmann, 1984; Brodie and de Kluyver, 1984; Ghosh, Neslin and Shoemaker, 1984; Brodie and de Kluyver, 1987; Tellis 1988, among others). These specifications have a number of advantages including: (a) ease of estimation as the models are linear in the parameters, (b) direct and easy interpretation of the regression coefficients, and (c) the empirical results obtained often have good descriptive or face validity. Often, for a given set of data, however, the results obtained from both models have good descriptive validity and choosing between them on this basis is a difficult task. Recourse to theory may provide an answer to the problem of selecting an appropriate functional form. ’ In other instances, previous studies may support the use of one functional * J.L. Kellogg Graduate School of Management, Northwestern University, Evanston, IL 60208, U.S.A. ’ In this paper, we focus on the issue of specifying an appropriate functional form, and do not address the problem of selecting the set of regressors. Intern. J. of Research in Marketing 6 (1989) 95-107 North-Holland

0167-8116/89/$3.50

0 1989, Elsevier Science Publishers B.V. (North-Holland)

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form over the other. Often, however, such a precise guide may not be available, and statistical procedures have to be used in determining the appropriateness of the model specification. In the case of the linear and log-linear models, the R2 values are not directly comparable as the dependent variables are different in the two cases (one in levels and the other in logarithms). Further, a simple direct comparison of the R2 values (or likelihood values) from two models has no firm basis in statistical inference, even if they have the same dependent variable. Another important drawback is that the use of the R2 value will result in failing to reject one of the models when neither one is correct. One solution to this problem is to test formally linearity or log-linearity against a more general functional form that contains the two models as special cases. Specifically, both models can be nested within a Box-Cox regression formulation and hence their validity can be empirically determined using any test of nested hypotheses. Godfrey and Wickens (1981) have proposed the use of the Lagrange multiplier (LM) test (Breusch and Pagan, 1980) for this purpose. This test can either accept or reject both models, or select one of them. The main advantage of the LM test over other tests of nested hypotheses (e.g., the likelihood ratio or Wald tests) is its computational ease. A potential limitation of the test is that, in small samples, the regularity conditions under which the test statistic is derived may be violated (Rao, 1973). As a consequence, the test may yield a probability of a type 1 error that is larger than the specified size (Davidson and MacKinnon, 1985). As the linear and log-linear models are mutually non-nested, one could alternatively use a test of non-nested hypotheses to determine the validity of the two specifications (Pesaran and Deaton, 1978; Davidson and MacKinnon, 1981; Rust and Schmittlein, 1985). One difficulty is that the use of a non-nested test is not straightforward as the dependent variable is different in the two cases. For example, a non-nested test based on a comparison of likelihood values (e.g., the information criterion test proposed by Akaike, 1973) would not be appropriate. MacKinnon, White and Davidson (1983) propose a test, called the extended P test (or P, test), that can be used to determine the validity of the linear and log-linear models. This test, like the LM test, is relatively simple, but its power characteristics are not well established. Further, it is not known how the power of the P, test compares with that of the LM test for samples of different sizes. In this paper, we investigate the size and power characteristics of the LM test for samples of varying sizes using a Monte Carlo simulation. If the large sample properties of the LM test also hold in small samples, the test will be helpful to researchers in marketing working with data sets with a limited number of observations. Our objective is to provide some guidelines as to an appropriate sample size that is needed to implement the LM test properly. Like all statistical tests, however, the LM test only allows one to determine the adequacy of certain specifications (i.e., linear and log-linear) and therefore provides only partial information about model specification. We also compare the performance of the LM test to the non-nested P, test and examine the conditions under which one test performs better than the other. The rest of this paper is organized as follows. In Section 2, we discuss the Box-Cox regression model and show that the linear and log-linear models are special cases of this generalized regression model. In Section 3, we describe the LM and the P, test and show how they can be easily computed from two artificial linear regressions. We provide an illustrative example in Section 4 using marketing data. In Section 5, we conduct Monte Carlo experiments to study the properties of the LM and P, tests for samples of varying size and different levels of error variance. Section 6 concludes with a discussion of the results and implications of the two tests for researchers in marketing.

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2. Box-Cox regression model Consider the following regression model due to Box and Cox (1964):

Y,(V=Po+ 5 P,xkt(~)+ Lz,,+% k=l

t=l,2 ,..., T,

(1)

/=I

where (+1)/h, Y(h) = In Y, 1 Y- 1,

h#O, A#l, A+O, A= 1,

for any variable Y. In equation (l), y, is the tth observation of the dependent variable, and Xkr, k = 1, 2,. . . , K, and Z,,, I = 1, 2,. . . , L, are the independent variables. The stochastic term E, is assumed to be independently and normally distributed with mean zero and constant variance a*. When X = 1, we get the linear model: K y,=p,*+

L

c bkXkt+

CaIZ/t+%

k=l

I=1

K

wherep,*=l+&-- c Pk. k=l

When X + 0, we get the log-linear model: * K

L

ln Y, = PO + C Pk ln Xk, + C aA, + 5. k=l

I=1

The appropriateness of the linear and log-linear models can be determined by testing separately the validity of the two null hypotheses HP : h = 1 and Hi : A = 0. 3 The alternative

hypothesis in each case is Hc : X # 1 and H; : X # 0, respectively. Such a test will lead to one of the following conclusions: (i) reject both HP and Hi, (ii) fail to reject both HP and Hi, (iii) reject H:, but fail to reject Hy, and (iv) fail to reject HP, but reject H!. In cases (iii) and (iv), the decision of which model to choose is clear. If we reject h = 1 but fail to reject h = 0, we choose the log-linear model as being more consistent with the data. If we fail to reject h = 1 and reject A = 0, we choose the linear model as the more preferred specification. In the first case in which both HP and Hi are rejected, the test should be interpreted as indicating that both the linear and log-linear models are inappropriate. It does not necessarily imply that one should instead choose the unconstrained Box-Cox model (equation (1)). We discuss this issue further in a later section. In the second case when both HF and H; are not rejected, the test is inconclusive and cannot be used to make an inference about the validity of either model. When the sample size T is large, one can use the likelihood ratio test, the Wald test or the Lagrange multiplier test as the three tests are asymptotically equivalent (Engle, 1984). However, both the likelihood ratio test and the Wald test involve estimating a difficult non-linear ’ We allow for the use of dummy variables (the Z’s) in the log-linear model as this is common practice in marketing research. We note that the Z’s need not only be dummy variables. s Alternatively, instead of testing hypotheses, one can, as suggested by Box and Cox, construct a confidence interval for X and determine if the values 0 and 1 are included in the computed interval. The difficulty is in obtaining consistent estimates of A in all situations as it involves estimating a non-linear model.

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unrestricted regression model given in equation (1). By contrast, the Lagrange multiplier test involves estimation of the parameters of only the restricted models (equations (2) and (3))-both of which are linear. Hence, due to computational ease the LM test is more appealing. In the next section, we briefly review the LM test and show how it can be applied to determine the validity of the linear and log-linear models. For a more thorough discussion of the test and its applications, see Breusch and Pagan (1980).

3. The LM test Consider a sample of size T drawn from some distribution which is known except for a finite number J of unknown parameters, 8 = (8,, &, . . . , 19,). Let r(0) be the associated log-likelihood function. The hypothesis to be tested is specified as a set of P ( < J) restrictions: h,(8)=0,

p=l,2 )...) P.

(4)

The objective in estimating 8 is to maximize Z(0) subject to (4). Silvey (1959) approaches this by setting up the Lagrangean function w + i ~~~~~~~~ p=l

(5)

where p.,, p= 1, 2 ,..., P, are the Lagrange multipliers. Differentiating (5) with respect to 8 and p = (pr, p2,. . . , pp) yields 8 and fi as solutions to i3+ti/i=o, h,(#)=O,

(6)

p=1,2 )...) P,

(7)

where B is the J x 1 vector { aZ(e)/%“}, j = 1, 2,. . . , J, and k is the J X P matrix {ah,($)/ ad,}, p=1,2 ,..., P, j=l,2,..., J. The basic idea underlying the test is that when the null hypothesis is true, the restricted estimates will tend to be near the unrestricted maximum likelihood estimates so that b will be close to the zero vector. Using this idea and certain regularity conditions, it can be shown that tLM = &-lb - x2(P),

(8)

where 2 is a consistent estimator of the information matrix based on the restricted estimator 8. This form of the test is commonly referred to as the Score version (Rao, 1973). By comparing the value of tLM from (8) to the critical value x:-J P), one can test the validity of the set of restrictions at a significance level (Y. In the form given by (8), computing the test statistic may be cumbersome. It can be simplified by noting that, under certain regularity conditions (see Rao, 1973), the information matrix can be obtained from the log-likelihood function as follows: g= E[

-cPqQ/ae ae’] = E{[al(e^)/ae][al(e^)/ae]‘}

(9)

and plim(g^-’ [ -

tPr(tf)/ae ae’] } = I,,

(10) where IT is a TX T identity matrix, T being the number of observations. The LM statistic tLM can then be expressed as

tLM = - [ ar(@/ae]’ { a2r(e^)/ae ae’ > -‘[ ar(d)/ae] .

wj

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Asymptotically, the LM test statistic can be written as

03) where I, is the log-likelihocd for the t th observation. Let w,, = a1,( #)/ae,, t = 1, 2,. . . , T, j=l,2,..., J, and let W( 0) be the T x J matrix with typical element wrj. Define L’ = (1, I,..., 1, 1) to be a TX 1 vector with each element equal to unity. Then, equation (13) can be expressed in matrix form as

tLM = L’W(B)[ w(e)‘w(8)] -‘w(e)‘l. The expression for 6 LM given in equation (14) can be further simplified as follows: Consider the regression Y = X’p + E. Then, 6 = (X’X))‘X’Y and p = Xl/?. Examining equationA (14), we note that [ W( 6) ’ W( &)I-’ W( 6)‘~ is the OLS estimator of the regression of I on W( 19). Hence, equation (14) can be written as tLM = L’w(&)~

where p = [ w(B)‘w(B)]-‘w(G)‘&.

05) The regression of the vector L on W(B) yields 9. This regression has no conceptual basis. It is an artificial regression that is performed merely to simplify the computation of tLM. To further simplify equation (15) we note that W( J)P is equal to the vector of predicted values (;) obtained from the artificial regression of 1 on W( 9). Hence, T

Hence, the LM test statistic is equal to the sum of the predicted values from the artificial regression of I on W( e^). Computing the LM test statistic therefore reduces to determining the sum of the predicted values from one extra linear regression, and is computationally quite simple. In the following section, we provide an illustration of the LM test in choosing among competing functional forms for an aggregate market share model using the IRI consumer panel data on the purchases of coffee. Next we briefly describe the P, test for non-nested hypotheses which can also be use to test for the adequacy of the linear and log-linear specifications. The PE test This test developed by MacKinnon, White and Davidson (1983) involves treating each specification (linear and log-linear) in turn as the model under the null hypothesis and testing it against the other specification. Hence, to test the validity of the linear model, we have H,:

K=Pcl+ IF Pkxkt+%I-fr(Xr> PI? k=l

k=l

(17)

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where the eir (i = 0, 1) the random error term is assumed independently and identically distributed (i.i.d.) normal with mean zero and variance 02. To test the validity of the linear model, MacKinnon et al. propose the following artifical composite regression: T-l=co[gt-ln(fl)] +&,+E~,

09)

where f: is the estimated value of f, (the linear model), g, is a vector of derivatives of f, with respect to p evaluated at their estimated values from the linear model, g, is the estimate of g, from the log-linear model, and c,, and b, are unknown regression parameters. MacKinnon et al. show that the OLS estimate of co (to) is asymptotically normally distributed with mean 0 and unit variance when Ho is true. Hence, the validity of the linear regression specification can be determined by estimating equation (19) and testing whether c^ = 0. Likewise, to test the validity of the log-linear model, we interchange H, and H, and estimate the composite regression: ln Y,-~i=cl[.kxp(&)] +G,h+u,,

(20) where G, is the vector of derivates of g, with respect to y evaluated at their estimated values from the log-linear models. Again, c^r is asymptotically N(0, 1) and the statistical validity of the log-linear model can be determined in a similar manner. 4. Illustrative example The issue of specifying the proper functional form for an aggregate market share model has recently been the focus of some debate in the marketing literature (see Brodie and de Kluyver, 1984; Ghosh et al., 1984; Naert and Weverbergh, 1981; Naert and Weverberg, 1985). Three types of functional forms-viz., linear, log-linear, and attraction-type models-have been used. As the attraction-type models are ex-ante logically consistent (i.e., the predicted market share lie between 0 and 1 and they sum to one) while the linear and log-linear models are not, some authors (e.g., Naert and Bultez, 1973) argue that they are the theoretically superior. However, the empirical evidence regarding the performance of each type has been mixed. For example, Brodie and de Kluyver (1984) claim that, empirically, the linear and log-linear models perform as well as or better than attraction-type models, despite the theoretical superiority of the latter. This is in contrast to the findings of Naert and Weverbergh (1981) favoring the attraction-type models. We propose that the LM test be used in selecting among these three models: 4 mit = POi +

5 5 PkjXkjt +

‘it?

(21)

j=l k=l K In m,t =

YOi + 5 C Ykj j=l k=l

In

xkjt + uit,

mit = [ SOi( k=l fi X”. k’~)uit]/[~180j(

filx~$t)ujt]~

(24

(23)

4 Some authors (e.g., Brodie and de Kluyver, 1984) have expressed the marketing mix variables for the brand under consideration relative to that of the competition: for example, instead of having the competitors’ variables expressed’separately, a relative expression of the form [X, /(Z, + iX,)] has been used. An implicit assumption here is that the effects on brand i due to actions by competitors j, k, are equal. This is a tenuous assumption and may only be acceptable when there is a very limited number of observations.

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where m,, denotes the market share of brand i in period t; Xkj, is the value of the k th decision variable for brand j in period t; n is the total number of brands; E,~ and u,( are i.i.d. normal random variables; and the u;,‘s are i.i.d. lognormal random variables. To test the validity of the attraction specification using the LM test, we use the log-centering linearization procedure proposed by Nakanishi and Cooper (1974), which reduces the estimating equation (23) to

where wit = In{ u;,/u”.,}, and Ez., and u”., are the geometric means of ml,, m,,, . . . , mn,, and uir, u2,, . . ., vnr, respectively. The error terms wit are i.i.d. normally distributed under the assumption that the oil, i = 1, 2,. . . , n, in equation (23) are i.i.d. log-normally distributed. Hence, the attraction formulation can be reduced to a double-log specification with a log-centered dependent variable. We note, however, that due to this log-centering of the dependent variable, equation (24) is not nested within the same Box-Cox model as equations (21) and (22). Further, equation (24) implies across-equation restrictions for the a,k parameters. 5 Therefore, it must be estimated as a system of equations subject to these parameter restrictions. In our empirical analysis, however, we estimate the model equation-by-equation only, consistent with the development of the LM test described previously. Hence, caution is warranted in making any direct comparison between the attraction-type and the linear or log-linear models. In Table 1, we report the results of the LM test for three brands of instant decaffeinated freeze dried coffee. 6 The market share of each brand for each week was expressed as a function of six explanatory variables, viz., the weekly prices of the three brands (Pl, P2, P3), and dummy variables (FDl, FD2, FD3) representing whether the brands were featured in newspaper advertising during that week. In all these cases, the lagged dependent variable was not significant and was dropped from the estimating equation. ‘A total of 96 weekly observations were used in the estimation. It is clear from Table 1 that using the face validity of the results to discriminate among the model is a difficult task. The R* values are about the same (though not directly comparable), the coefficients (where significant) are of the proper signs, i.e., the own-price terms are negative while the cross-price terms are positive, where as for the feature variables, the own-feature coefficients are positive while the cross-feature coefficients are negative. The results of the LM tests are unambiguous. For brands 2 and 3, the linear model is rejected at a significance level of (Y = 0.05 as the computed values of the test statistic 40.84 and 29.28 respectively exceed the critical value of 3.84 (x&(l)). For brand 1, the linear model is rejected at a significance level of CY = 0.10 as the test statistic value of 3.22 exceeds 2.71 (~&~(l)). The log-linear model is also rejected in all three cases. For brands 1, 2, and 3, the computed test values of 33.24, 7.10, and 14.27, respectively, exceed the critical value of 3.84 (x&(l)). For the attraction model, the test values of 1.67, 2.18, and 2.19 for brands 1, 2, and 3 are all less than 3.84, and hence we fail to reject the attraction specification. ’ These restrictions imply, for example, in a three-brand situation; the price effect of brand 3 on the share of brand 1 is equal to that of brand 3 on the share of brand 2. Clearly, this is a restrictive aspect of the attraction model. We thank an anonymous reviewer for bringing this to our attention. 6 Studies on market structure analysis (e.g., Urban et al., 1984) have shown that instant decaffeinated freeze dried coffee is a separate submarket, and hence can be studied in isolation. We identified three brands belonging to this submarket. ’ The LM test would be valid in the presence of lagged variables if the error terms are independently distributed.

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Table 1 Test of linear, log-linear, and attraction-type market share models

’

Linear model Brand 1:

MS,, =0.76-1.27P,,+0.59P,,+0.34P,,+0.14FD,,-0.03FD,,-0.03FD,, (1.91)

RZ = 0.40, Brand 2:

(1.70)

(1.10)

(2.95)

( - 0.42)

+ g,,

( - 0.44)

tLM = 3.22

F-value = 9.89,

MS,, = 0.59+0.48P,,-0.72P,,-0.23P,,-0.08FD,,+0.14FD,,-0.03FD,, (1.61)

R2 = 0.20, Brand 3:

(-3.82)

(1.59)

(-2.26)

(-0.83)

F-value = 3.64,

(2.07)

(-1.98)

+ B,,

( - 0.45)

tLM = 40.84

MS,, = -0.34+0.78P,,+0.13P,,-0.10P,,-0.05FD,,-O.llFD,,+0.07FD,, (0.46) (-0.42) ( - 1.39) (- 1.08) (2.89) R* = 0.24, F-value = 4.76, tLM = 39.28

(-1.83)

+ f,,

(1.04)

Log-linear model Brand 1:

lnMSr,=-0.99-1.86ln P,,+0.69ln P,,+0.37ln P,,-0.21FD,,+O.OlFD,,-O.OSFD,,+ti,, ( - 3.09) ( - 3.52)

R* = 0.31, Brand2:

(1.25)

(0.79)

(1.91)

(0.03)

( - 0.29)

tLM = 33.24

F-value = 6.65,

1nMS,,=-1.73+2.001nP,,-2.031nP,,-0.571nP~,-0.34FD,,+0.63FD2,+0.03FD,,+~,, (2.16)

(- 3.17) (2.23)

R2 = 0.23,

F-value = 4.38,

( - 0.70)

(1.81)

(2.21)

(0.09)

5tM = 7.10

B r a n d 3 : lnMS,,=-0.68+2.94ln P,,+0,32In P,,-0.41ln (1.10) (2.87) (0.30) ( - 0.45) R* = 0.33, F-value = 7.26, ILM = 14.97

P,,-0.34FD,,-1.20FD2,+0.68FD,,+ir,, (3.71)

(- 1.59)

(1.91)

Arrraction-type model Brand 1: b ln(MS,,/MS.,)= 0.15-2.89 In P,,+1.03 In P,,+O.58 In P,,+0.36FD,,+0.19FD2,-0.27FD,, R* = 0.40, Brand 2:

(1.40) (0.34) (- 4.10) F-value = 9.99, tLM = 1.67

ln{MS,,/MS.,)=-0.60+0.97ln (-1.20)

R* = 0.16, Brand 3:

In(MS,,/MS.,) R2 = 0.26,

(1.19)

F-value = 2.85,

P,,-1.69ln (- 1.99)

(0.91)

PI,-0.37ln ( - 0.50)

(2.50)

(0.87)

+ F,,

(-1.11)

P,,-0.18FD,,+0.82FD2,-O.l9FD,,+F,, (-1.08)

(3.17)

( - 0.67)

tLM = 2.18

=0.45+1.91 In P,,+0.66ln P,,-0.21 In P,,-O.lSFD,,-l.OlFD,,+0.46FD,,+0,, (0.89) (2.29)

(0.76)

F-value = 5.44,

.$tM = 2.19

( - 0.28)

(-1.05)

( - 3.82)

(1.59)

a The figures in parentheses are the corresponding r-values. b MS., is the geometric mean of MS,,, i =l, 2, 3.

Therefore, the results of the LM test indicate that of the three market share models considered, the true model underlying the data is different from the linear and log-linear specification, while the attraction-type specification is not inconsistent with the data. The example discussed above illustrates the usefulness of the LM test in model selection. However, it is based on a sample of 96 observations. The question that arises is the following: “what is the minimum sample size that is required for the results of the LM test to be valid?“. In the next section, we investigate the properties of the LM test for samples of varying size. The objective is to identify the appropriate number of observations required to implement the test properly.

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5. Monte Carlo simulation We performed three related sets of Monte Carlo experiments to study the properties (size and power) of the LM test for samples of varying size. An additional objective was to determine whether these properties depend on the nature and goodness-of-fit of the underlying or true model. Accordingly, each of the following three models was used, in turn, as the true model in the three experiments: 8 H,O:

Yt = 010 + (YlXl, + qX2t + axX3, + E,,

Hi:

ln rt=Po+P1 ln Xll+P2 ln Xzt+P3 ln X3r+~,T

Hi:

yt= [yo+yl exp(y,X,,+y,X,,+y,X,,)l-‘+u,,

E, - N(O, DE’),

(25)

u,--N(O,

d),

u,-N(o, Oi>-

(26)

(27)

In all three cases, we tested the adequacy of the linear and log-linear specifications. These experiments enabled us to determine the size and power characteristics of the LM and P, tests for different combinations of sample size, nature, and goodness-of-fit of the true model. Further, the third set of experiments (based on Hi) allowed us to determine whether the two tests are capable of rejecting both the linear and log-linear models when neither specification is correct. In generating the data from the linear model HF, we used the parameter values (Y,, = 31.15, (Ye = 12.97, a2 = 41.24, and 01~ = 11.52. These values were obtained by estimating the linear model for the data given in Churchill (1987, p. 692). Given these fixed parameter values, we generated two sets of data for the linear model by choosing two levels (low and high) for the error variance, viz., a, = 16.3 (low) and a, = 375.9 (high). The different values of a, were selected so as to produce models with varying goodness-of-fit as measured by the R2 values. When Us = 16.3, the R2 values were 0.9 or higher and, for a, = 375.9, the R2 values were between 0.5 and 0.7. The rationale was to determine the sensitivity of the LM test to different fits of the true model. The X, values, i = 1, 2, 3, were independently generated from a normal distribution, with means 75, 35, and 55 and variances 21.9, 7.1, and 16.2. The values for the means and variances were chosen so as to minimize the probability of generating negative values. Sample sizes chosen were 15, 25, 50, 100, and 200 observations. In generating the data from the log-linear model, we choose &, = 4.72, ,Bi = 0.26, p2 = 0.45, and & = 0.16. These values were chosen so that the ‘elasticities’ of Xi, X2, and X, were about the same as those from the linear model. As with the linear model, we choose two levels for the error variance (u, = 0.007 (low), uU = 0.195 (high)), to generate models of varying fit as measured by the R2 values. In generating the data from the third model, we choose after some experimentation, y0 = 0.01, yi = 0.5, y2 = 0.10, yJ = 0.02, and u, = 0.005. Only one level of error was selected. In Table 2, we report the results of 500 replications of the LM and P, tests for each combination of sample size (N) and standard error of residuals (a) for data generated from the linear and log-linear models. 9 Replicating each experiment 500 times also allowed us to generate the X values within a very wide range. Hence, the test results reflect this wide range of values of the explanatory variables. For sample sizes of 15 and 25 observations, we see that in both the linear and log-linear cases the LM test incorrectly rejects the true model with probability far * Since the attraction model can be expressed as a log-linear model, it is obtained by linearizing the following multiplicative model: y, = Xb 9 We did some limited analysis to determine the sensitivity of the LM N = 100 and the high error variance condition were essentially the log-linear cases.

was not examined separately in the simulation. The log-linear X& X5 exp( PO + u,). test to changes in parameter values. The simulation results for same as those reported in the paper in both the linear and

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Table 2 Simulation results: Probabilities of type 1 error ’ Sample size ( Iv )

True mode1 Log-linear

Linear o = 16.3 (low)

15 25 50 100 200

0 = 375.9 (high)

0 = 0.007 (low)

LM test

P, test

LM test

P, test

LM test

P, test

(T = 0.195 (high) LM test

P, test

0.54 0.31 0.16 0.10 0.08

0.05 0.09 0.09 0.08 0.02

0.60 0.37 0.16 0.13 0.07

0.00 0.03 0.00 0.04 0.04

0.57 0.33 0.18 0.11 0.09

0.05 0.03 0.03 0.07 0.06

0.52 0.21 0.17 0.13 0.09

0.02 0.04 0.03 0.02 0.02

’ In all cases, we used a significance level of (Y = 0.05.

greater than that implied by the level of significance. When the sample size increases, the probability of a type 1 error clearly decreases in both the linear and log-linear cases. For example, when the linear model is true (Table 2) the probability of a type 1 error decreases from 0.16 when N = 50 to 0.08 when N = 200 for the low error variance. Moreover, these values are approximately the same when the error variance is increased to the higher level (0.16 for N = 50, and 0.07 for N = 200). Similar results are observed in the log-linear case for the LM test. Comparing the size of the LM test to that of the P, test, we see from Table 2 that the latter has a lower probability of a type 1 error for all sample sizes and different levels of error variance in both the linear and log-linear cases. Further, the probability of a type 1 error for the P, test shows a much lower variation and is closer to the nominal value (0.05) than the LM test for different sample sizes and error levels for each of the true models. The simulation results indicate that when considering the probability of a type 1 error, the LM test requires a sample of at least around 100 observations, while the P, test can be used with samples as small as 15 observations. Table 3 provides information regarding the power of the two tests for samples of varying size under different fits of the true model. We see from Table 3 that when the error variance of the true model is low, the LM test rejects the false null hypothesis with probability equal to one for all sample sizes in both the linear and log-linear cases. However, when the error variance of the true model (linear or log-linear) is increased, the power of the test depends on the sample size as one expects. For very small sample sizes (N = 15 or 25), the power of the test is low in both Table 3 Simulation results: Probabilities of type 2 error N

True model Linear

15 25 50 100 200

Log-linear

Logistic

o = 16.3 (low) Null model: log-linear

u = 375.9 (high) Null model: log-linear

0 = 0.007 (low) Null model: linear

LM test

P, test

L M t e s t P, t e s t

L M t e s t P, t e s t

0.00 a 0.00 0.00 0.00 0.00

0.00 a 0.00 0.00 0.00 0.00

0.36 0.39 0.23 0.08 0.01

0.00 0.00 0.00 0.00 0.00

0.89 0.88 0.76 0.48 0.10

a These are the probabilities of type 2 error using a = 0.05

0.00 0.00 0.00 0.00 0.00

LT = 0.195 (high) 0 = 0.005 Null model: Null model: Null model: linear l i n e a r log-linear L M t e s t P, t e s t ~ ~ LM test P, test L M t e s t P, t e s t 0.29 0.37 0.18 0.03 0.00

0.98 0.98 0.93 0.94 0.60

0.22 0.15 0.04 0.00 0.00

0.98 0.88 0.82 0.47 0.19

0.37 0.47 oq49 0.45 0.31

0.76 0.55 0.26 0.05 0.00

D. C. Jairt N.J. Vilcussim / Funcrioml forms of marker share models

105

cases. However, as the sample size is increased, the power of the test approaches one. For N = 200, the power is equal or close to one in both the linear and log-linear cases, while for N = 100, the power is 0.92 when the true model is linear and 0.97 when the log-linear model is the correct one. When N = 50, the power values are 0.77 and 0.82, respectively. Hence, we see that for a sample size of around 100 observations or more, the test has high power to discriminate between the linear and log-linear models. A comparison of the probabilities of a type 2 error between the LM and P, tests shows that when the error variance is low, the two tests produce identical results in both the linear and log-linear cases (Table 3). However, when the error variance is increased, the power of the P, test falls dramatically, even for sample sizes of 100 observations. When N = 100, the power of the P, test is equal to 0.52 when the linear model is true and equal to 0.06 when the log-linear model is the correct one. Hence, for a given sample size the power of the LM test is greater than that of a P, test when the error variance of the true model is high. In Table 4 we summarize these results for the size and power of the two tests using an ANOVA procedure. The main effects considered are the nature of the true model (linear, log-linear), sample size (N = 15, 25, 50, 100, 200), error variance (high, low) and the particular test used (LM, P, test). We see that the probability of a type 1 error depends on the size of the sample and the particular test used, and is not significantly influenced by the nature of the true model and the error variance. The probability of a type 2 error is significantly influenced by the level of error of variance in addition to the sample size and the test used. As before, the nature of the true model does not significantly influence the power of the tests. In the previous discussion, the true model was either the linear or log-linear specification. In Table 3, we also report the power of the LM and P, tests when the correct specification is the non-linear model (logistic-type) given by equation (27) and the null hypothesis is the linear or log-linear model. For N 3 50, the LM test has very high power when the null hypothesis is the linear model. When the null is the log-linear model, the power of the test is poor. Even for very large sample size (e.g., N = 200) the maximum power attained is 0.69. This might be due to the reason that under certain conditions a logistic-type formulation can be approximated by some type of logarithmic function and in such a situation the LM test fails to discriminate between the two specifications. ‘O In Section 2, we stated that if both the linear and log-linear models are rejected, one should not necessarily choose the extended Box-Cox regression as the proper specification. The example discussed illustrates this point. Finally, the results from the P, test are mixed when compared to the LM test. When the (false) null model is linear, the LM test has a higher power for all sample sizes. However, when ” Our model in equation (27) can be written as

[ 1 [ 1

1 ; = Yo + Yl exp

3

C ,=I

Y,+I&

Consider the special case when y,, = 1. then 3

1 --l=y,exp CX+~T, K I=1

Taking the natural logarithm on both sides, we get the semi-log form 3

In Y,‘=Y;+ C u,+lK,. !=I

where y,’ = In

y,

and y,’ = l/y, - 1.

D.C. Jain, N.J. Vilcassim / Functional forms

106

of

market share models

Table 4 ANOVA results Source of variation

Type 1 error MSa

Fb

Pr>F=

Type 2 error MS

F

Pr > F

True model Sample size Error variance Test used Residual

0.0009 0.0719 0.0031 0.4223 0.0106

0.08 6.78 0.29 39.83 -

0.77 0.00 d 0.60 0.00 d

0.0084 0.1253 1.9981 0.6401 0.0471

0.18 2.67 42.42 13.59

0.68 0.05 0.00 d 0.99

a MS = mean square. b F = computed F-statistic. ’ Pr > F at a significance level of 0.05. d Probability value less than 0.0001.

the null model is log-linear, the P, test has greater power for sample sizes of 50 observations or more. Further, when N = 100, the power of the P, test equals 0.95 and is equal to one for very large sample sizes. 6. Discussion and conclusions We have illustrated a simple but reasonably powerful means of testing the adequacy of the linear and log-linear specification of regression models using marketing data when the number of observations exceed one hundred. We have studied the properties of the test for samples of varying size using Monte Carlo simulation. Based on the results of the simulation study, we can state that whenever the sample size is equal to or greater than 100 observations, the LM test has sufficient power to discriminate between the linear and log-linear formulations of a regression model. With an increase in the availability of consumer and store scanner panel data, samples of such magnitude will not be uncommon. As these two functional forms are used very often, this test would be of interest to researchers in marketing. It is easy to compute and involves running only one extra linear regression. In choosing between the linear and log-linear specification, an alternative to the LM test is a test of non-nested hypotheses. We chose the P, test, proposed by MacKinnon et al. (1983) for comparative purposes because of its ease of execution. Other non-nested tests are not easily extendable for choosing between models with different dependent variables. The results from the simulation study indicate that while the P, test performs better than the LM test with respect to the probability of a type 1 error, its power is considerably lower when the error variance of the true model is high. When the error variance of the true model is low, the power of the two tests are identical. As it is not possible to know in practice what the true error level is, we recommend the LM test over the P, test when the number of observations is equal to or exceeds 100. In this study, our objective was to keep the number of variables fixed in the model and determine the adequacy of its functional form. We have not addressed the issues of selecting the appropriate independent variables in model formulation. We leave this for a future study. Acknowledgments

\ . The authors would like to thank Professor Yu-Min Chen of the University of Texas at Dallas and four anonymous reviewers for many helpful suggestions. The usual disclaimer applies.

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/ Functional forms of market share models

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Support for this research was provided by the Summer research fund of the Kellogg Graduate School of Management, Northwestern University. We thank IRI for making available the data used in a part of the analysis.

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