Simultaneous Versus Sequential Estimation of Multi-Stage Models of Perception, Preference, and Choice

Ruth N. Bolton, University of Alberta
Randall G. Chapman, University of Alberta
ABSTRACT - Simultaneous and sequential estimation procedures for multi-stage models of perception, preference, and choice are reviewed and discussed. The relative performance of these estimation procedures is assessed with Monte Carlo experimentation for a simple multi-stage choice model with quantitative endogenous variables. The results indicate that 2SLS estimates have about 35% less error than OLS estimates in the presence of substantial cross-equation error correlation. This result appears to hold regardless of the level of correlation in the model's exogenous variables.
[ to cite ]:
Ruth N. Bolton and Randall G. Chapman (1986) ,"Simultaneous Versus Sequential Estimation of Multi-Stage Models of Perception, Preference, and Choice", in NA - Advances in Consumer Research Volume 13, eds. Richard J. Lutz, Provo, UT : Association for Consumer Research, Pages: 463-468.

Advances in Consumer Research Volume 13, 1986      Pages 463-468

SIMULTANEOUS VERSUS SEQUENTIAL ESTIMATION OF MULTI-STAGE MODELS OF PERCEPTION, PREFERENCE, AND CHOICE

Ruth N. Bolton, University of Alberta

Randall G. Chapman, University of Alberta

ABSTRACT -

Simultaneous and sequential estimation procedures for multi-stage models of perception, preference, and choice are reviewed and discussed. The relative performance of these estimation procedures is assessed with Monte Carlo experimentation for a simple multi-stage choice model with quantitative endogenous variables. The results indicate that 2SLS estimates have about 35% less error than OLS estimates in the presence of substantial cross-equation error correlation. This result appears to hold regardless of the level of correlation in the model's exogenous variables.

INTRODUCTION

In modeling choice behavior in high-involvement settings marketing scientists have used both physical attributes and buyers' perceptions of the characteristics of the choice alternatives as explanatory determinants of choice. Either physical attributes or perceptions -- but not both -- are typically used in a single-stage choice model.

In contrast to single-stage choice models, multi-stage choice models seek to integrate physical attributes, perceptions, preferences, and choices into a comprehensive model of judgment and decision making. A popular multistage choice model has the following form:

Perception = f(Physical Attributes, Information, Other Factors)

Preference = g(Perceptions, Other Factors)

Choice = h(Preference, Situational Constraints, Other Factors).

This multi-stage choice model incorporates both physical attributes and perceptions. The model also makes a distinction between preference and choice. Preference is often defined as "unconstrained preference", where the decision maker is specifically assumed (and directed in survey research questionnaires) to ignore certain constraining factors in providing the preference judgment. However, choice is affected by these situational constraints. This multi-stage model is consistent with models of consumer behavior proposed and applied by many marketing researchers (c.f. Tybout and Hauser 1981). This multi-stage choice model may be viewed as falling within the Lens' model paradigm (Brunswik 1952) or as being derived from the information integration view of judgment and decision making (Anderson 1981).

Marketing scientists typically estimate the various parts of this multi-stage choice model using the following approach. The assumption (usually implicit) is made that the errors of the three equations are uncorrelated across equations. Under this assumption, each equation in the multi-stage model may be estimated separately, in a sequential fashion, and this procedure will yield parameter estimates with the usual desirable statistical properties, such as unbiasedness and efficiency. However, the appropriateness of this sequential estimation approach depends critically on the assumption about the uncorrelated error terms.

Cross-equation error correlations in the multi-stage model may exist for several reasons. Correlations may arise because common measuring scales are used to measure perception, preference, and choice. Omitted variables would be another factor which induces cross-equation error correlation.

If the equation errors are correlated, sequential estimation of the multi-stage choice model will yield inconsistent parameter estimates. With cross-equation error correlation, the model should be estimated with simultaneous equation techniques, using either single equation (such as two stage least squares) or system methods (such as three stage least squares). Single equation methods (also termed limited information methods) for simultaneous models will yield consistent parameter estimates. However, system methods (also termed full information methods) estimate the equations jointly, typically yielding more efficient parameter estimates.

The purpose of this paper is to compare simultaneous and sequential estimation procedures for multi-stage models of perception, preference, and choice. We explore the conditions under which it is necessary to use a more complicated simultaneous estimation procedure, rather than using the simpler single-equation-at-a-time sequential approach. This investigation focuses on the extent to which sequential estimation procedures may be deficient in the presence of cross-equation error correlations.

In the next section, a multi-stage model of the choice process is formulated. It is shown that some aspects of the model specification depend on the choice modeling context. In subsequent sections, appropriate estimation procedures are described for alternative model specifications. Simultaneous equation estimation techniques are discussed in detail. Monte Carlo experiments are conducted to compare the statistical properties of parameter estimates generated by sequential versus simultaneous estimation procedures. Some concluding remarks complete the paper.

MODEL SPECIFICATION OF THE MULTI-STAGE CHOICE MODEL

The multi-stage choice model has two components: a structural sub-model and a measurement sub-model. The structural sub-model describes the links among the theoretical constructs; the measurement sub-model describes the relationships between constructs and their measures. The structural sub-model focuses on three aspects of decision making: perceptions, preference, and choice. Hence, perceptions, preference, and choice are considered to be endogenous (dependent) variables. The remaining variables are exogenous (independent).

A decision maker's perceptions, P, of an alternative in his/her unconstrained choice set are modeled as a function of the objectively-verifiable physical attributes of the alternative, informational cues in the environment, and other factors. The decision maker's preference, BR, for an alternative in his/her unconstrained choice set is viewed as a function of his/her perceptions of the alternative as well as other factors. Choice, C, is considered to be a function of preference and other factors, subject to situational constraints. (The "other factors" may include both individual- and situation-specific aspects of the choice process.)

The general form of this multi-stage model my be written in the following fashion:

Pj = f(W,X) + elj, for j = 1,2,...,J    (1)

PR = g(P,X) + e2      (2)

C = h(PR,X,Z) + e3      (3)

where:

j = a perceptual dimension (j = 1,2,...,J) which characterizes the alternatives

W = a vector of objectively-verifiable physical attributes of the unconstrained set of alternatives and the information cues available in the environment

X = a vector of other factors (e.g., socio-demographic and other individual-specific variables)

Z = a vector of situational constraints which mediate the choice process

em = a stochastic disturbance term which reflects model specification and measurement errors for the dependent variable in equation m.

In equations (1)-(3), all variables have an implicit subscript denoting the alternative in the consideration set. This subscript has been suppressed for notational convenience. In equation (1), the subscript j denotes a perceptual dimension of an alternative. For example, "quality" of a college may be a relevant perceptual dimension in college choice modeling.

Throughout this paper, we make a number of assumptions about this model. First, it will be assumed that the functional forms of the equations -- the f(.), g(.), and h(.) functions -- are linear in the parameters. However, this assumption is relaxed when the principles discussed in this paper are extended to (non-linear) discrete choice model forms. Second, the disturbance terms have been defined to include both model specification and measurement error in the endogenous variables. Our parameterization of this multi-stage choice model in equations (1)-(3) does not distinguish between these two types of errors. Third, it is assumed that:

E[em|W,X,Z] = 0,        for m=1, 2, and 3   (4)

That is, equation (4) specifies that the equation error terms are uncorrelated with the exogenous variables. This is a standard econometric modeling assumption.

The structural sub-model also specifies the nature of the variance-covariance matrix of the equation disturbances. These disturbance terms may be correlated or uncorrelated across equations. Cross-equation correlations among the disturbances may arise from either model specification error, or due to the nature of the measurement error in the dependent variables.

Model specification error will be reflected in correlation among the equation errors when a common exogenous variable is omitted from all three equations. Alternative-specific indicator variables are often used to attempt to account for such omitted variables. Measurement error in perceptions, preference, and choice will also be reflected in the equation disturbances. Hence, the methodology used to measure the endogenous variables may induce correlation into the equation errors. Correlated errors may arise when the three endogenous variables are measured with similar scales, or when all the endogenous variables are measured at the same point in time rather than in stages. In the first instance, the use of a common scaling procedure may lead to a to on source of error in the measures of perception, preference, and choice. In the second instance, halo effects are more likely to occur when all measures are collected simultaneously. Furthermore, the causal links among measures of perception, preference, and choice become complex when all the measures are taken at one point in time.

The measurement sub-models for the endogenous and exogenous variables will depend on the model context. To focus on specific model estimation issues, this paper considers alternative measurement sub-models for the endogenous variables. However, it is assumed that the exogenous variables are measured without error. The exogenous variables will typically include both objectively-verifiable physical attributes and individual-specific socio-demographic variables. Both types of variables are likely to have little measurement error, at least compared to the measurement error inherent in perceptions, preference, and choice.

Two features of the measurement sub-model for the endogenous variables are important it determining the appropriate estimation procedure for the multi-stage model. One of these features, the correlations among the measurement errors of the endogenous variables, has been discussed above. Another important feature of the measurement model is whether perceptions, preference, and choice are represented by quantitative or qualitative (discrete choice) measures. Our preliminary exploration of the estimation issues in multi-stage choice models focuses primarily on the quantitative dependent variable case.

MODEL ESTIMATION PROCEDURES FOR QUANTITATIVE DEPENDENT VARIABLES

This section describes estimation procedures for the multi-stage choice model when the measures of the endogenous variables are quantitative. The appropriate procedure depends on whether the equation disturbances are uncorrelated or correlated.

Uncorrelated Errors Across Equations

Consider the multi-stage model in which disturbances are uncorrelated across equations. That is, assume that:

E[e1je2] = 0 ,        for j = 1,2,...,J

E[e1je3] = 0 ,        for j = 1,2,...,J

E[e2e3 = 0.

This scenario is likely to occur when the endogenous variables are measured with different scales and at different points in time. It is also likely to occur when perceptions, preference, and choice are based on somewhat different sets of alternatives due to the presence of mediating constraints.

If the covariances of the error terms are restricted to be equal to zero, the multi-stage model has a recursive structure (Johnston 1972, pp. 377-9). A recursive structure implies that the disturbances of equations (2) and (3) are uncorrelated (in the probability limit) with the endogenous variables on the right-hand side of those respective equations. In this situation, all three equations are appropriately estimated with a straightforward application of OLS to each equation. The OLS parameter estimates will be consistent and asymptotically efficient, if the model has a recursive structure.

Correlated Errors Across Equations

Consider the multi-stage model in which the disturbances may be correlated across equations. Hence, the variance-covariance matrix of the disturbances is unrestricted. This scenario is likely to occur in "one-shot" studies (i.e., when all measures are collected simultaneously), studies which use the same measurement methodology for all endogenous variables, and studies where perceptions, preference, and choice are based on the same set of alternatives.

If the disturbances are correlated across equations, the multi-stage model has a simultaneous structure. The disturbances of equations (2) and (3) are correlated (in the probability limit) with the endogenous variables on the right-hand side of the respective equations. Hence, the direct application of OLS will yield parameter estimates which are biased and inconsistent. The asymptotic bias will be greatest when there are a few exogenous variables with high intercorrelations (Johnston 1972, p. 411).

To present an alternative estimation procedure, it is useful to express the structure model in its reduced form:

Pj = f(W,X) + e1j, for j-1,2,...,J      (5)

PR = g'(W,X) + e2       (6)

C = h'(W,X,Z) + e3       (7)

where e2 and e3 are stochastic disturbance terms. Note that e2 is a function of e1j and that e3 is a function of e1j,e2,e3. Hence, the stochastic disturbances in equations (5)-(7) are correlated across equations.

Identification conditions may be used to verify whether it is possible to recover the structural parameters from the reduced form model. These conditions will depend on the functional form of the model (Maddala 1977, pp. 220-31). Assuming a linear functional form, the structural parameters of this multi-stage choice model may be shown to be identified.

The structural parameters of the simultaneous multi-stage model can be estimated with a single equation method, such as two stage least squares, or system methods, such as three stage least squares or full information maximum likelihood methods. These estimation procedures yield parameter estimates with different statistical properties.

To apply the two stage least squares (2SLS) procedure to this multi-stage choice model, the following two steps wouLd be required (Johnston 1972, pp. 380-4) :

Step 1: Equations (5) and (6) of the reduced form model are estimated with OLS. Instrumental variables for perceptions (P*) and preference (PR*) are formed from the predicted values.

Step 2: The instrumental variables replace the endogenous variables.on the right-hand side of the structural model as follows:

PR = g(P*,X) + e2       (8)

C = h(PR*,X,Z) + e3.       (9)

Equations (8) and (9) are estimated with OLS. Note that the estimated standard errors from the second stage must be adjusted to obtain the correct values at the second staRe (Maddala 1977, p. 239).

Since equation (5) of the reduced form model is identical to equation (1) of the structural model, stage one yields consistent estimates for the structural coefficients of equation (1). The disturbances in equations (8) and (9) are uncorrelated (in the probability limit) with the variables on the right-hand side of the equations because the instrumental variables do not have a stochastic component. Hence, stage two will yield consistent estimates of the structural coefficients of equations (2) and (3). If the instrumental variables are "poor" surrogates for perceptions and preferences, the sampling variance for the parameter estimates may be very large. In these circumstances, Johnston (1972, p. 211) points out that this may be a high price to pay for consistency.

2SLS estimates will be less biased than OLS estimates in finite samples. Monte Carlo studies indicate that multicollinearity may produce a greater increase in bias in 2SLS estimates than in OLS estimates, but the results are mixed on this point. I;hen bias and standard deviation are combined in a measure of mean square error, the results are also mixed. However, 2SLS is more likely to lead to correct inferences about the true values of structural coefficients because the estimated standard errors are centered around an unbiased expectation. It is important to note that the differences between the estimation procedures are not very large, and vary considerably across data sets (c.f., Johnston 1972, pp. 410-7). In addition, OLS estimates tend to predict well and be more robust than many simultaneous methods in the presence of model specification error (Maddala 1977, p. 231).

As an aside, the reduced form version of the multi-stage choice model consists of "seemingly unrelated" equations. Such equations are characterized by error terms which are correlated across equations and by sets of exogenous variables which differ across equations. Under these conditions, Zellner's system method (1962) will provide asymptotically more efficient estimates than OLS. Hence, Zellner's method of seemingly unrelated regressions could be used in stage one to estimate the reduced form model and form the instrumental variables.

If the structural model is characterized by error terms which are correlated across equations, and by at least one equation which is over-identified, system estimation methods will provide asymptotically more efficient estimates of simultaneous equation coefficients (Zellner and Theil 1962). System estimation procedures include full information maximum likelihood techniques and three stage least squares.

In full information maximum likelihood (FIML), it is assumed that the disturbances of equations (1)-(3) are normally distributed and serially uncorrelated, with an unrestricted covariance matrix. Then, the joint likelihood function is maximized with respect to the structural parameters (i.e., structural coefficients and error covariances). FIML is complex (involving the solution of non-linear equations) and computationally expensive. As a result, FIML is less frequently used than other simultaneous equation methods.

In three stage least squares (3SLS), the first two stages are those of 2SLS. In the third stage, generalized least squares is applied to equations (1), (8), and (9) using the covariance matrix of the estimated equation errors (Johnston 1972. p. 398).

Although system methods are asymptotically more efficient than 2SLS, they have a number of disadvantages (Theil 1971, pp. 528-9). System methods require a much more detailed specification of the equation system than 2SLS does. With 2SLS, if a coefficient in a given equation is postulated to be equal to zero when in fact it is not, the coefficient estimates for that equation will be biased. In system methods, the same model specification error will bias the coefficient estimates for all equations. Thus, model specification error -- a particularly common problem in marketing modeling efforts -- will have its influence felt throughout all equations when using a system estimation method, even if the specification error is only in a single equation!

Whether system methods really outperform the much simpler OLS estimates in real marketing data remains an unresolved issue. Chapman and Palda (1984) compared 3SLS and OLS estimates for a 10-equation model of "sales response" in an electoral marketplace. They found that the theoretically preferable system estimation technique (3SLS) yielded approximately the same pattern of parameter estimates as the simpler OLS (equation-by equation) approach.

MODEL ESTIMATION PROCEDURES FOR QUALITATIVE DEPENDENT VARIABLES

This section describes estimation procedures for the multistage choice model when at least one dependent variable is qualitative (discrete choice). For the purposes of the present discussion, it will be assumed that choice is a qualitative variable, while perceptions and preference are quantitative variables.

Choice may typically be a polychotomous variable because it represents a selection from a finite set of alternatives. When choice is polychotomous, a popular model form is the multinomial logit model (McFadden 1974):

C = MNL(PR,X,Z)      (10)

where MNL is the now familiar multinomial logit model functional form. Thus, equation (10) replaces equation (3) in the specification of the multi-stage choice model.

As before, the estimation of the model consisting of equations (1), (2), and (13) depends on whether the three equations' disturbances are uncorrelated or correlated.

Uncorrelated Errors Across Equations

Consider the multi-stage model of the choice process described by equations (1), (2), and (10) in which the disturbances are uncorrelated across equations. (The conditions under which uncorrelated errors are likely to occur were described earlier for quantitative endogenous variables.) As before, the multi-stage model has a recursive structure. Hence, all three equations can appropriately be estimated with limited information estimation procedures. OLS techniques may be applied to equations (1) and (2), and maximum likelihood estimation of the parameters of equation (10) may be performed. Sequential estimation of each equation yields coefficient estimates which will be unbiased and asymptotically efficient.

In the above discussion, it has been assumed that choice is the only qualitative (discrete) endogenous variable. Preference might, of course, also be a qualitative endogenous variable. Under the assumption that the equation errors are uncorrelated, this version of the multi-stage choice model would also be estimated by limited information estimation procedures. Elliot and Hollenhorst (1981) formulate a sequential response model with four stages, each measured by a qualitative endogenous variable. The outcome at each stage is conditional on "success" at the previous stage. For this type of model, the Joint likelihood function can be maximized by maximizing the likelihood functions of each equation separately (Maddala 1983, pp. 49-51).

Limited information estimation procedures are typically utilized for multi-stage models of the choice process (e.g., Hauser, Tybout, and Koppleman 1981; Hauser and Wisniewski 1982). However, the underlying assumptions about the disturbance terms are generally not discussed, despite their impact on the statistical properties of the estimates.

Correlated Errors Across Equations

Consider the multi-stage model of the choice process described by equations (1), (2), and (10) in which the variance-covariance matrix of the disturbances is unrestricted. (The conditions under which correlated errors are likely to occur were described earlier for quantitative endogenous variables.) If the disturbances are correlated across equations, the multi-stage model has a simultaneous structure and limited information estimation procedures will yield biased and inconsistent parameter estimates.

The reduced form of the multi-stage model with quantitative endogenous variables was described by equations (5)-(7). Since choice is now defined to be a qualitative variable, equation (7) should be re-written in the form of equation (10), where choice (C) depends on all the exogenous variables (W, X, and Z).

Identification conditions for models with both quantitative and qualitative endogenous variables are somewhat different than those in the usual simultaneous equations model (Maddala 1983, pp. 117-25; Lee 1981). However, the identification problems in the multi-stage choice model are much the same as those in the usual simultaneous equations model, except that some parameters are estimable only up to a scle factor due to the presence of the qualitative endogenous variable (Maddala 1983, p. 120).

The structural parameters of the simultaneous multi-stage model can be estimated with a single equation or a system method. These estimation procedures yield parameter estimates with different statistical properties.

A two stage estimation method for the multi-stage choice model is the following (Maddala 1983, pp. 242-7):

Step 1: Equations (5) and (6) of the reduced form model are estimated with OLS. Instrumental variables for perceptions (P*) and preference (PR*) are formed from the predicted values.

Step 2: The instrumental variables replace the endogenous variables on the right-hand side of the structural model. As before, equation (2) is re-written with instrumental variables P* as shown in equation (8). Equation (10) is re-written with instrumental variable PR*. as follows:

C = MNL(PR*,X,Z)      (11)

Equation (8) can be estimated with OLS. Equation (11) can be estimated with maximum likelihood estimation techniques.

This two stage procedure yields consistent estimates of the structural coefficients. As with 2SLS, it yields poorer estimates in the presence of multicollinearity (Maddala 1983, p. 252). Note that the estimated standard errors from the second stage must be adjusted to obtain their correct values. Lyon (1984) uses a variation of this two stage method in her study of the dynamic attitude behavior relationship.

With regard to system estimation methods, maximum likelihood methods can be applied to the multi-stage choice model with mixed quantitative and qualitative endogenous variables. (These methods were described in the previous section.) Two stage estimation methods are likely to be more popular than maximum likelihood, due to their computational simplicity.

Amemiya (1979) has suggested some alternative estimators to two stage estimaters, such as the one described above. Such estimators are more efficient than two stage estimators, but there exists a generalized instrumental variable estimator that is more efficient than the GLS estimator derived from Amemiya's method (Lee 1981; Maddala 1983).

RELATIVE PERFORMANCE OF SIMULTANEOUS AND SEQUENTIAL ESTIMATION PROCEDURES

To compare the relative performance of simultaneous (2SLS) and sequential estimation procedures, Monte Carlo experimentation on simulated data may be used. In this preliminary analysis, we focus only on quantitative endogenous variables. The corresponding analysis for a mixed system of quantitative and qualitative endogenous variables is deferred to future research efforts in this area. Our goal is to examine how cross-equation error correlations and multicollinearity among the exogenous variables influence the relative performance of OLS and 2SLS estimation procedures.

Design of the Monte Carlo Experiments

The following multi-stage choice model was used in these Monte Carlo experiments:

P = s0 + s1X + s2W e1       (12)

PR = b0 + b1P + b2X + e2       (13)

C = gO + g1PR + g2X + g3Z + e3.      (14)

These equations are the simplest possible representation of the multi-stage choice model which was originally described in equations (1)-(3): there is one perception variable and the minimum number of X, W, and Z variables. This model might be used by decision makers in the evaluation and assessment of a single alternative or action

In simulating choice model data, the distributional characteristics of the exogenous variables and the error terms must be specified, the true parameter values must be chosen, and correlation patterns (among the exogenous variables and between the error terms in the three equations) must be specified. Random draws made from the specified distributions substituted into equations (12)(14) yield measures for the endogenous variables in this model. The model's parameters may then be estimated by OLS and 2SLS procedures, and the resulting parameter estimates may be compared to the true (known) parameter values to assess the relative performance of the estimation procedures.

In these Monte Carlo experiments, all exogenous variables and error terms were drawn from normal distributions with mean zero and variance equal to one for exogenous variables and variance equal to four for the error terms. With regard to the true parameter values, all constant terms were set equal to zero and all slope parameters were set equal to one. These choices resulted in the explained variance averaging about 30%, 75%, and 85% in the P, PR, and C equations, respectively. Note that these levels of explained variance indicate that the instrumental variables generated by the reduced form equations will be "adequate" surrogates for the endogenous variables on the right-hand side of the structural equations. Hence, it should be expected that simultaneous estimation methods (such as 2SLS) achieve consistency at the expense of sampling variation.

Correlations among the exogenous variables were generated by making the W and Z variables equal to a linear combination of X and a white noise term. With an appropriate choice of weights, correlations of 0.00, 0.24, and 0.45 were obtained for the low, medium, and high correlation conditions in the experiment.

For the error terms, correlations were induced into e2 and e3 terms by making them linear combinations of the preceding error term in the model plus a white noise term. Appropriate choice of the weights and the error variances of the white noise terms yielded correlations of 0.00, 0.40-0.44, and 0.50-0.71 for the low, medium, and high cross-equation error correlation conditions in the experiment.

A factorial design was employed in the experiment. Each of the three levels of exogenous variable correlation was tested with each of the three levels of cross-equation error correlation, for a total of nine separate experiments. In each cell of this experimental design, a total of 25 replications were conducted. Each replication involved generating 250 observations, according to the model in equations (12)-(14) and the specified correlational structure of the exogenous variables and the error terms. After generating the data in each of the replications, the model was estimated using both OLS and 2SLS techniques.

Performance Measures

Two measures of precision were chosen for gauging the performance of the estimation procedures, related to bias and squared error deviation:

EQUATION (15) and (16)

where cr is the estimate of parameter c obtained on replication r (r= 1,2,...,R) of the Monte Carlo experiment and c* is the true (known) parameter value. BIAS(.) measures the tendency of coefficient estimates to be, on average, above or below their true value. RMSE(.) is a measure of the precision of the parameter estimates (i.e., how "close", in an absolute sense, the estimates are to truth). RMSE(.) is equivalent to adopting a squared error loss function to account for deviations between a coefficient estimate and its true value.

The performance measures defined in equations (15) and (16) are for a single parameter. If these measures are summed over all parameters in the model in equations (12)-(14), 10 in total, we obtain BIAS and RMSE, the aggregate bias and root mean squared error, respectively. These measures of aggregate performance were used to compare OLS and 2SLS estimation procedures in these Monte Carlo experiments.

Results

The results of these Monte Carlo experiments are reported in Table 1. The aggregate BIAS and aggregate RMSE are reported in each cell of the experimental design. The key findings are as follows:

(1) 2SLS outperforms OLS in terms of both BIAS and RMSE in the presence of any degree (medium or high) of cross-equation error correlation. This is, of course, expected based on standard results in econometric theory.

(2) When there is medium cross-equation error correlation. the RMSE for 2SLS is about 10% less than the RMSE for the OLS estimates. In the high cross-equation error correlation case, the 2SLS estimates are about 35% more precise than the OLS estimates.

(3) The bias in the OLS estimates increases as the degree of cross-equation error correlation increases.

(4) OLS consistently underestimates the true values of the parameters.

(5) If there is no cross-equation error correlation, the application of 2SLS yields less precise estimates than OLS (by about 35%), as measured by RMSE. There is no apparent pattern of bias in the 2SLS estimates (in the situation where there is no cross-equation error correlation).

(6) As the collinearity of the exogenous variables increases, the precision of the estimates decreases (i.e., RMSE increases), for both OLS and 2SLS regardless of the degree of cross-equation error correlation.

As expected, the Monte Carlo experiments yield results which are consistent with the theoretical econometric results. These numerical results are particularly interesting, however, because they provide some sense of the magnitude of the potential gains in precision available by applying the theoretically correct, but more complicated, 2SLS estimation procedure.

The main finding is that the degree of cross-equation error correlation is crucial to the relative performance of 2SLS versus OLS. If the cross-equation error correlation is small (less than 0.50 in our Monte Carlo experiments), then the potential gains in precision are fairly small, about 10%. However, even small "errors" in parameter estimation may lead to incorrect inferences about the statistical significance of model parameters. In more extreme cross-equation error correlation situations, it will be necessary for the marketing scientist to use 2SLS since (a) precision will be improved by about 35% and (b) the parameter estimates will be noticeably less biased. These results hold regardless of the degree of correlation among the exogenous variables.

CONCLUDING REMARKS

Theoretical considerations suggest that simultaneous estimation procedures, rather than sequential procedures, may be more appropriate to estimate the parameters of multi-stage models of perception, preference, and choice. The rationale for this claim, plus issues associated with simultaneous estimation methods, were reviewed in this paper. Monte Carlo experiments indicate that limited information methods of estimating simultaneous multi-stage choice models dominate sequential estimation methods, if there is cross-equation error correlation.

Simultaneous estimation methods yield substantial gains in the precision of parameter estimates compared to sequentia sequential estimation methods. Also, bias is reduced. The Monte Carlo results reported in this paper provide numerical estimates of the potential gains in precision for a simple multi-stage choice model with quantitative endogenous variables.

TABLE 1

SUMMARY RESULTS OF THE MONTE CARLO EXPERIMENTS

Extensions of this work would involve a detailed study of the qualitative endogenous variable case, as well as considering how sample size, relative size of the error variances, and the ratio of exogenous to endogenous variables influence the relative performance of sequential and simultaneous estimation procedures.

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