# Econometric Estimation of Halo Effect: Single Vs Simultaneous Equation Models

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Albert C. Bemmaor and Joel C. Huber (1978) ,"Econometric Estimation of Halo Effect: Single Vs Simultaneous Equation Models", in NA - Advances in Consumer Research Volume 05, eds. Kent Hunt, Ann Abor, MI : Association for Consumer Research, Pages: 477-480.

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http://acrwebsite.org/volumes/9469/volumes/v05/NA-05

This paper analyzes a single equation model of belief as a function of preference and compares it to a simultaneous equation model. Using statistical tests, this study shows that the single equation model is well specified whereas the simultaneous equation model leads to statistically significant differences in the estimates.

The problem of measurement of halo effect has been extensively dealt with in the marketing literature. In a recent paper, Huber and James present a review of the different operational definitions of halo (Huber and James, 1976). Beckwith and Lehmann use econometric methodology to assess the halo effect (Beckwith and Lehmann, 1975). They analyze individual data on attitudes and beliefs towards television shows. After estimating a single equation model of the effect of beliefs on attitude, they use a simultaneous equation model to eliminate the "intertangled effects" of attitude and beliefs. Their study has given rise to criticism and further research in this area (Wittink and Ness, 1976; Johansson, MacLachlan and Yalch, 1976). The purpose of this paper is to test the Beckwith and Lehmann single equation model of the effect of beliefs on attitude for specification errors, and evaluate the implications of a misspecified model. In the first section, the Beckwith and Lehmann model is described. The second section briefly presents the specification error tests. The third section summarizes the results of these tests. In the last section, we assess the effect of a misspecified model on the parameter estimates.

BECKWITH AND LEHMANN SIMULTANEOUS EQUATION MODEL BETWEEN ATTITUDE AND BELIEFS

The Beckwith and Lehmann simultaneous attitude-belief model can be represented as follows:

The belief equations can be represented as follows:

In this paper, we focus our attention on equation (2), the question being: Is (2) well specified? Is it sufficient to estimate halo, or do we need to add (1)? To some extent, we are asking the problem in a direction opposite from that as posed by Beckwith and Lehmann since they consider whether equation (2) should be added to equation (I) in order to estimate the importance weights w.. However, here, we are not concerned with the estimation of importance weights, but rather concerned with the estimation of halo. We will attempt to show that equation (2) is well specified and therefore is adequate to estimate halo without (1).

SPECIFICATION ERROR TESTS

The tests which are used have recently been developed by Ramsey (Ramsey, 1969; Ramsey, 1974). These tests detect the following kinds of specification errors: omitted variable, incorrect functional form, simultaneous equation problem, heteroskedasticity and normality of the disturbances. Consider the model:

y = Xb + m (3)

where y is an Nxl vector, X is an NxK matrix of rank K, b is a Kxl vector of coefficients and m is an Nxl vector of disturbances. Ramsey denotes the Full Ideal Conditions assumed when using ordinary least squares estimation method on the model above as: i) the disturbance terms are normally distributed with mean zero and constant variance, and are independent, ii) they are independent of the explanatory variables, iii) the matrix X'X is nonsingular.

In the case of misspecification of (3), these Full Ideal Conditions are violated. The tests consist in analyzing the residuals of the model in order to detect such errors. The five errors considered are broken down into three groups. Group A errors consist of omitted variable, incorrect functional form and the simultaneous equation problem. These errors lead to failure to fulfill the assumption that the mean of the disturbances is zero. Group B errors are those which lead to the violation of the assumption of constant variance of the disturbances, an example of which is heteroskedasticity. Group C errors are those in which the disturbances are not normally distributed so that the distribution of the regression coefficients is affected.

Three tests are used for the analysis: RESET (having a central F distribution), BAMSET (based on a central chi-square distribution) and WSET (whose critical bounds have been defined by Shapiro and Wilk (Shapiro and Wilk, 1965)). RESET is designed to detect Group A errors, BAMSET is used for the detection of Group B errors and WSET for the detection of Group C errors. Further details on these tests are provided by Ramsey (Ramsey, 1974). Ramsey and Gilbert suggest that RESET and BAMSET seem to be independent under BLUS residuals. On the other hand, BAMSET and WSET are not independently distributed under BLUS residuals but do appear so under OLS residuals (Ramsey and Gilbert, 1969).

Under the Full Ideal Conditions, the ordinary least squares residuals are distributed as N(O, s^{2}M), i.e., they do not have a scalar covariance matrix. Therefore, an alternative estimator of the vector of disturbances m can be used to detect specification errors via the analysis of residuals. The BLUS residuals m(Best Linear Unbiased Scalar covariance matrix residuals) developed by Theil are such that m ~ N(O, s^{2}K) under the Full Ideal Conditions (Theil, 1965). Ramsey and Gilbert showed that for the case of omitted variable, incorrect functional form and simultaneous equation problem (Group A errors), the use of y = Xb+m, m~N(O ,s^{2}I) as the true regression model leads to a BLUS residual vector distributed as N(A'e, s^{2}I_{N-K}) where e is a nonstochastic vector the definition of which depends upon the particular misspecification, and A is an (N-K) x K matrix such that A'X = 0, A'A = I_{N-K} and AA'=I-X(X'X)^{-1}X'=M (Ramsey and Gilbert, 1969). In the case of heteroskedasticity (Group B errors), i.e. m_~ N(O, s^{2}W) the distribution of m derived from the use of y = Xb+m m_~ N(O, s^{2}W) as the true model, is N(O_{N-K},A'WA). Finally, in the case of nonnormality (Group C errors), the BLUS residuals are such that m ~ D(O, s^{2}I_{N-K}), D being any distribution other than the normal distribution. Comparing the performances of the test statistics using BLUS and OLS residuals, Ramsey and Gilbert recommend the use of BLUS residuals when the primary concern is about Group A errors (Ramsey and Gilbert, 1972). As we are mostly concerned about the problem of simultaneity between belief and attitude, BLUS residuals are used in this study.

TEST RESULTS

The data used in this study were collected by Huber and James (Huber and James, 1976). Individual analyses were made on the preferences of fourteen master students towards seventeen U. S. cities as sites for their first jobs. They were shown pairs of cities and asked in which they would prefer to live, given that work related variables such as opportunity for advancement were kept constant. A dollarmetric scale was developed for each individual and used as a measure of preference (Pessemier and Teach, 1966; Huber and James, 1976). Three attributes were chosen for analysis: size of the city, warmth of the climate and opportunities for spectator sports. These attributes were measured on ten-point Likert scales. The results of the tests for the first attribute "size of the city" are presented in Table 1. This table indicates the number of times the null hypothesis of no specification error was rejected at an a = .10. For each individual, the model is rejected when the null hypothesis of no specification error is rejected by one or more of the tests. These results are indicated in the column DECISION. For the first attribute, the null hypothesis is rejected only once (in favor of the alternative hypothesis of Group C errors). It should be noted that the null hypothesis is never rejected in favor of the hypothesis of Group A errors. Table 2 presents the test results for all three attributes. In total, the null hypothesis of no specification error is rejected only twice against the alternative hypothesis of omitted variable, incorrect functional form and simultaneous equation problem (Group A errors). Also, for Group B errors, the Beckwith and Lehmann model is misspecified in two cases only. The assumption of normality of the residuals, tested by WSET, seems to be the most critical one, as evidenced by the number of rejections of the null hypothesis. Table 3 indicates the number of rejections for each individual. Except for Individual 4, the Beck-with and Lehmann model seems to be appropriate for at least two attributes out of three. The major result of this analysis is that equation (2) appears to be well specified for most individuals. Despite the fact that the simultaneous equation model might make theoretical sense, it is contradicted by the empirical evidence with this set of data.

RAMSEY TEST RESULTS FOR THE BECKWITH AND LEHMANN MODEL (SIZE OF THE CITY)

OVERALL PERFORMANCE OF THE BECKWITH AND LEHMANN MODEL FOR ALL THREE ATTRIBUTES

INDIVIDUAL AND MODEL COMPATIBILITY FOR ALL THREE ATTRIBUTES

A Monte Carlo analysis of the small sample properties of the tests discussed here is reported in Ramsey and Gilbert (Ramsey and Gilbert, 1972). The basic conclusion is that RESET and BAMSET are reasonably powerful tests against their respective alternatives. The small sample properties of WSET have also been examined by Monte Carlo analysis (Ramsey and Gilbert, 1969). The tests are capable of differentiating between Group A, Group B and Group C errors. However, it is not possible to discriminate between errors within a given group. Hence, the rejection of the model by RESET for two individuals (see Table 2) may not necessarily be due to the simultaneous equation problem but implies the presence of Group A errors in the model. On the other hand, acceptance of the null hypothesis against the hypothesis of a given group of errors implies that all the types of errors comprised in that group are excluded from the model.

EFFECT OF A MISSPECIFIED SIMULTANEOUS EQUATION MODEL ON THE PARAMETER ESTIMATES

Summarizing the findings of their study, Beckwith and Lehmann state: "The TSLS regression results indicated that ... for the six belief equations ... the regression coefficients (remained) relatively unchanged from OLS, on average" (Beckwith and Lehmann, 1976). Hence, their model specification does not seem to have much affected the estimates of the halo effect b_{j} and of g_{j}. The TSLS estimates of b_{j} and g_{j} were calculated and compared to the OLS estimates. [In this study, for all fourteen individuals, the range of the identification statistic W_{b} is between B4.9033 and .3154. W_{b} never being equal to, the structural model is identified for each individual.] Table 4 presents these estimates for the first attribute, "size of the city". As shown by the results of the t-tests, the differences between the OLS estimates and the TSLS estimates of the halo effect b_{j} are statistically significant for most individuals. On the other hand, the estimates of the coefficient g_{j} of b_{ij}^{*} do not change much. The results of the tests of significance of the differences between the two estimates for all three attributes are summarized in Table 5. This table shows that, in contrast to the Beckwith and Lehmann results, our estimates of g_{j} do not change much, but significant differences occur in the estimates of the halo effect. This means that the TSLS halo estimates cannot be accepted.

As to the attitude equation (1), Beckwith and Lehmann summarize their results by stating: "(T)he attitudes of less than half of the respondents could be explained with positive R^{2}'s by the TSLS estimated model. We concluded that the linear attribute model is false for at least a substantial fraction of the respondents in this study" (Beckwith and Lehmann, 1975). They argue that "the removal of the halo effect, as in the TSLS estimates, reduced the diagnostic and explanative power of (the attitude equation) for many people" (Beckwith and Lehmann, 1976). It should be mentioned at this point that very small values of the multiple correlation coefficient do not indicate a "poor fit" or lack of significance of the set of explanatory variables in the structural equation. As R2 is constrained only to the interval (- 4, 1) large negative values can easily occur, but "negative values certainly do not discredit the model in any meaningful way" (Basmann, 1962; Tomek, 1970). Hence, the Beckwith and Lehmann conclusions seem to be questionable.

In order to assess the robustness of the Beckwith and Lehmann model of the effect of attitude on belief, the "average belief" variable was replaced by the "true" value for all three attributes, and the OLS estimates of the halo effect in the two alternative models were compared. The results which are not reported here show that both estimates are generally very close. On the other hand, the explanatory power of the Beckwith and Lehmann model is generally higher than that of the alternative model. Hence, the effect of the "average belief" variable Bi. is to provide an upward bias in the level of explanatory power of the belief equations. This may be partly due to the similar nature of the explanatory variable and the dependent variable.

In conclusion, a single equation model of the effect of attitude on beliefs about cities seems to be well specified according to our test results. Furthermore the specification of the model - whether a single or a simultaneous equation - seriously affects the halo estimates. A similar type of model testing procedures could be carried out in another context, such as the modeling of the relationship between sales and advertising. In general, these testing methods should prove useful to the assessment of the functional relationships between marketing variables.

REFERENCES

R. L. Basmann, Letter to the Editor, __Econometrica__, 30 (October, 1962), 824-6.

Neil E. Beckwith and Donald R. Lehmann, "The Importance of Halo Effects in Multi-Attribute Attitude Models," __Journal of Marketing Research__, 12(August, 1975), 265-275.

Neil E. Beckwith and Donald R. Lehmann, "Halo Effects in Multi-Attribute Attitude Models: An Appraisal of Some Unresolved Issues," __Journal of Marketing Research__, 13 (November, 1976), 418-21.

Joel Huber and William James, "The Marginal Value of Physical Attributes: A Dollarmetric Approach," Presented at the Seventh American Marketing Association Attitude Conference, Hilton Head, South Carolina (February, 1976).

Joel Huber and Bill James, "A Theory of Halo," Working Paper, Krannert Graduate School of Management, Purdue University (March, 1976).

Johny K. Johansson, Douglas J. MacLachlan and Richard F. Yalch, "Halo Effects in Multi-Attribute Attitude Models: Some Unresolved Issues," __Journal of Marketing Research__, 13(November, 1976) 414-7.

E. A. Pessemier and R. D. Teach, "A Single Subject Scaling Model Using Judged Distances Between Pairs of Stimuli", Paper No. 143, Institute for Research in the Behavioral, Economic and Management Sciences, Krannert Graduate School of Management, Purdue University, 1966.

J. B. Ramsey, "Tests for Specification Errors in Classical Least Squares Regression Analysis," __Journal of the Royal Statistical Society__, Ser. B, 31, 1969, 350-71.

James B. Ramsey, "Classical Model Selection Through Specification Error Tests," in __Frontiers in Econometrics__, ed. by Paul Zarembka (New York: Academic Press, 1974).

J. B. Ramsey and R. Gilbert, "A Monte Carlo Study of Some Small Sample Properties of Tests for Specification Error," Econometrics Working Paper, No. 6813, Michigan State University, East Lansing, Michigan, 1969.

J. B. Ramsey and R. Gilbert, "A Monte Carlo Study of Some Small Properties of Tests for Specification Error," __Journal of the American Statistical Association__, 67 (March 1972) 180-6.

S. S. Shapiro and M. B. Wilk, "An Analysis of Variance Test for Normality" (Complete Samples), __Biometrika__, 52, 1965, 591-611.

H. Theil, The Analysis of Disturbances in Regression Analysis, __Journal of the American Statistical Association__, 60(December, 1965) 1067-79.

W. G. Tomek, "R^{2} in TSLS and GLS Estimation," __American Journal of Agricultural Economics__, 55(November, 1973), 670.

Dick R. Wittink and Reiar Ness, "Halo Effects in Multi-Attribute Attitude Models: A Caveat", Working Paper, Graduate School of Business, Stanford University (February, 1976).

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