Information Based Model Averaging and Internal Meta–analysis in Seemingly Unrelated Regressions With an Application to a Demand System
Henry L. Bryant and George C. Davis
January 2003
Paper prepared for presentation at the American Agricultural Economics Association Annual Meeting, Montreal, Canada, July 27-30, 2003
Contact Information: George C. Davis, Texas A&M University, 302 Blocker Bldg. College Station, Tx 77843-2124, Ph: (979) 845-3788. E-mail:
[email protected]. Copyright © 2003 by Henry L. Bryant and George C. Davis. All rights reserved. Readers may make verbatim copies of this document for non-commercial purposes by any means, provided that this copyright notice appears on all such copies.
Information Based Model Averaging and Internal Meta–analysis in Seemingly Unrelated Regressions With an Application to a Demand System
Abstract
This paper presents an information based model averaging and internal meta-analysis procedure that is easily applied to a large model space. In the application, the procedure is used to investigate the efficacy of some recently contested commodity promotion programs. The investigated model space consists of 576 demand systems. The internal meta-analysis indicates that theoretical restrictions and evaluation points are more important than alternative functional forms and explanatory variables in determining the elasticity values. The model averaging weights strongly support the theoretically consistent classical demand systems without promotion. The weighted or meta-price and meta-expenditure elasticities are presented and discussed. keywords: Model specification, Model Averaging, Meta-analysis, Meta-elasticities.
JEL code: C5, D0
1 Information Based Model Averaging and Internal Meta–analysis in Seemingly Unrelated Regressions With an Application to a Demand System
Economic theory is usually silent on how to fully specify an empirical model, especially in terms of the functional form and all the explanatory variables. This problem is usually addressed by what Pagan labels the ‘test-test-test’ approach, which is a classical pairwise hypothesis testing procedure. In this approach, the researcher begins by conducting a battery of tests on some initial model. If the tests indicate the initial model is unacceptable, the model is respecified, the test battery is readministered, and the new model is reevaluated. If the new model is also deemed unacceptable, this process continues until a single acceptable model is located. A major limitation of this yes/no search procedure is that the inherent model uncertainty is not accounted for in the statistical analysis (Chatfield; Draper). Consequently, the variance of some quantity of interest (e.g., a parameter estimate) will be understated because the final model is treated as though it is ‘true’, when in fact there is a positive probability that the chosen model is ‘false’. An alternative to the yes/no search procedure is to explicitly account for the model uncertainty with a Bayesian model averaging procedure. Conceptually, Bayesian model averaging is straightforward and a natural extension of standard Bayesian analysis: a universe of models is specified, the posterior odds ratio is computed for each model, and a weighted average of the models and/or the quantity of interest is constructed using the posterior odds ratios as weights (see e.g., Griffiths; Hoeting, et al.; Moulton; Raferty, Madigan, and Hoeting). Bayesian model averaging is conceptually appealing because it overcomes several of the limitations of the yes/no model search procedure: (i) no single model is selected as ‘the correct model’; (ii) because the weights are based on
2 posterior odds, less likely models are given smaller weights than more likely models; and (iii) the averaged model or quantity of interest is likely robust to alternative types of misspecifications.1 However, while conceptually appealing, Bayesian model averaging also suffers from shortcomings. It is well known that Bayesian model averaging can be sensitive to how the priors are set. More importantly, for most applied economists the results are the focus of the modeling exercise, not the technique. Yet, sound inferences may require sophisticated techniques, but the analyst often will have to decide ex ante if the expected benefit of a more sophisticated technique outweighs the expected cost. Often the answer is conceptually yes but computationally no. So it is with Bayesian model averaging; for many, it is more appealing conceptually than computationally. This conflict implies an appealing alternative would be a model averaging procedure that simultaneously satisfies the conceptual properties of the Bayesian approach but also eases its computational burden. The goal of this paper is to demonstrate an information based model averaging procedure that retains the appealing properties of the Bayesian approach but overcomes its computational burden. The general idea is quite simple and attributable to Buckland, Burnham, and Augustin but it is extended here in two directions. Schwarz has proven that the Bayesian posterior probability can be approximated by his Bayesian information criterion (BIC). The BIC can be used to form an approximation to the Bayes factor in the posterior odds ratio. The posterior odds ratios generated from the BIC statistics can then be used to form weights for model averaging. A BIC based model averaging procedure is a natural extension to the recommendations of Hansen and Granger, King, and White, who make compelling arguments for a BIC based model selection criterion over the classical pairwise testing procedure. Because the BIC is relatively easy to calculate for each model, the model averaging procedure is also computationally relatively simple. Buckland, Burnham, and Augustin
1
Of course, if one insists on choosing a single model then that is also possible within the Bayesian framework by choosing the model with the largest posterior odds ratio.
3 implement the procedure within a single regression framework, but here it is easily extended to a seemingly unrelated regression framework. The second extension is more substantial and not considered by Buckland, Burnham, and Augustin. While model averaging accounts for model uncertainty, a practical question of great scientific interest is, which of the modeling assumptions have the largest impact on the estimates of the quantity of interest? For example, does changing the functional form have a larger impact on an elasticity estimate than adding an additional variable to a specific functional form? This is an extremely important question in applied econometric work because specification searches usually explore only one dimension of a model (e.g., included and excluded variables) while holding other dimensions constant (e.g., functional form). By only exploring a single dimension, it is impossible to know how altering the explored dimension compares with altering another dimension. A researcher may spend a great deal of time exploring a model dimension to which the results are rather insensitive. This would generate a false sense of robustness when in fact the results may be much more fragile in an unexplored dimension. An important byproduct of the model averaging procedure is that the researcher will implicitly generate a data set that is well suited for an internal meta-analysis.2 Meta-analysis is a concise way to quantitatively summarize the impact that different modeling decisions have on the results (Stanley). In the present context, the model search procedure moves along a planned route that acts as an experimental design with the different classes of model assumptions acting as factors and the quantity of interest generated by each model representing the response. After estimating all the models within the planned model space, the internal meta-analysis helps the analyst determine which assumptions or
2
Meta-analysis usually concentrates on measuring the importance of different assumptions or factors across studies (Stanley). However, the same concept applies within a broader single study that quantifies the different assumptions made with each model and then conducts an internal meta-analysis. The internal meta-analysis is conceptually very similar to the response surface analysis found in Monte Carlo studies.
4 decisions are most important. In particular, by using some basic statistical tools, the researcher may rank the assumptions in terms of their impact on the estimated quantity of interest. With this additional information in hand, the researcher may decide to only average over certain subsets of the model space. In addition, while there is no guarantee such results would be transferable across data sets, the results may be suggestive of where future specification searches should concentrate. The next section presents the general conceptual framework for implementing the information based model averaging procedure and the internal meta-analysis. In the following section, attention turns to estimating the elasticities of a demand system as the quantity(ies) of interest. A demand system is chosen as the example because it is well known that the exact functional form, the exact variables to include, and the appropriate parameter restrictions are all debatable a priori in demand systems and these decision will in turn affect elasticity estimates. The focus on the demand elasticities as the quantities of interest is also purposeful because many economic or legal decisions come down to demand elasticity estimates. For example, recent work in anti-trust cases has turned toward using demand systems within a Nash-Bertrand equilibrium to determine the price changes from proposed mergers (Baker). Crooke, et. al show that whether or not a merger is classified as violating anti-trust laws depends critically on the underlying demand elasticity estimates. Another example, and the one of interest here, is the case of commodity checkoff programs. In commodity checkoff programs, farmers are assessed a tax for each quantity of farm product sold, such as raw milk, and these funds are used to promote or advertise a retail product, like processed milk (i.e., the “Got Milk?” advertisements). Most of the major commodities produced in the U.S. are subjected to checkoff assessments (e.g., cotton, soybeans, milk, beef, pork, and citrus). Recently, many of these programs have been challenged in court in terms of their legality and efficacy (e.g., the beef checkoff in Livestock Marketing Assoc. v. United States Dept. of Agriculture and the pork
5 checkoff in Michigan Pork Producers Assoc. v. Campaign for Family Farms, et al.). The efficacy of such programs depends critically on the advertising elasticity of demand and therefore the underlying model specification. With these recent legal cases in mind, the empirical example considers the demand for meat in the United States. The defined model space leads to 576 possible demand systems to considered and estimated. For each demand system the elasticities are generated (i.e., the quantities of interest) and evaluated at three points: the minimum, the mean, and the maximum. With 576 demand systems, this generates 1728 individual estimates for each elasticity. Before turning directly to the model averaging procedure, the internal meta-analysis is conducted to determine which assumptions or decisions are most influential for the elasticity estimates. The model average elasticities, or metaelasticities, are then calculated. An upper bound on the standard error for the meta-elasticity estimates is also presented and provides some conservative evidence of significance. The paper closes with conclusions and suggestions for future research.
An Information Based Model Averaging Procedure with Internal Meta-analysis Consider the general estimation problem facing the applied economist. An economic theory suggests some functional relationship between variables, (1)
y = f (x1, x2, ..., xJ)
where y is the variable or phenomenon the theory seeks to explain, the J variables denoted by x are identified as being the potential determinants of y, and f : ℜJ→ ℜ. Most theories are very vague about the two major specification requirements for estimating such a relationship: (i) the functional form for f is not specified beyond stating that it is within a class of functions with certain properties (e.g., signs on partial derivatives, restrictions on functions of the partial derivatives, etc.); (ii) the J
6 variables are not uniquely identified beyond the statement that some are expected to be more important than others. If I is the number of possible functional forms, this leads to a total of I × 2J possible models from which to choose; a number that quickly becomes very large. A model Ms can be formally defined by the triple 〈fi, Xj, Rk〉, where fi ∈ F, Xj ⊆ X, and Rk ∈ R. F is a set of functional forms, X is a set of potential explanatory variables, and R is a set of parameter restrictions.3 The s index refers to a unique triple since the same variables can be used in different functional forms with the same or different restrictions. Consequently, M = {M1, M2,…,MS} denotes the model space to be considered. In most cases, the real interest lies in some output generated from the model (e.g., a parameter estimate) and this quantity of interest will be common to all models. Let η denote this quantity of interest and because each estimate of η will depend on the associated model, the estimate of the quantity of interest can be denoted as ηˆ = ηˆ ( M s ), or ηˆ s , where the circumflex denotes estimate. In generating the S estimates from different models, the researcher has implicitly created a design matrix that corresponds to the S × 1 vector of estimates. If this design matrix is made explicit, it can be used to help shed light on which assumptions are most influential in the analysis. To make the design matrix explicit let card(F) = I, card(X) = J, and card(R) = K. For each element of each set F, X, and R define a dummy variable d that is equal to one if the element is chosen in a model specification and zero otherwise. The model indexing set s now can be defined intuitively as a function of these dummy variables or s = s(d1, d2, ..., dI, dI+1, ..., dJ, dJ+1, ..., dI+J+K). For example, suppose there are two functional forms (I = 2), three potential regressors (J = 3), and two restriction matrices (K = 2). Let the first model be defined as using the first functional form, the 3
This generality allows for restrictions in addition to zero restrictions.
7 first regressor, and the first restriction matrix. Then the first model is denoted M1 and s =1 = s(1, 0, 1, 0, 0, 1, 0). For simplicity, let dI be the 1 × I vector of dummy variables for the functional forms, dJ the 1 × J vector of dummy variables for the regressors, and dK the 1 × K vector of dummy variables for the parameter restrictions so that a model may be denoted Ms and s = s(dI, dJ, dK). Though the dummy design variables will uniquely identify a specific model, they may not uniquely identify the quantity of interest estimate because each estimate of η may also depend on where in the data space it is evaluated. Let E define the set of evaluation points and let L indicate the number of alternative evaluation points (i.e., card(E) = L). An evaluation point may be any point in the data set but it also may be some other relevant point that is a function of the data, such as the mean. Similar to above, create dummy variables to correspond one-to-one with the elements of E. If a specific point of evaluation is chosen then the associated dummy variable will equal one and zero otherwise. Let dL be the 1 × L vector of dummy variables for the evaluation points. The design matrix for the internal meta-analysis can then be denoted as D = (dI, dJ, dK, dL). To continue the example above, suppose three evaluation points are considered for each quantity of interest, the minimum, the mean, and the maximum, so E has these three elements. Suppose now the quantity of interest from the first model is evaluated at the minimum of the data. In this case, the first row of the design matrix is D1 = (1, 0, 1, 1, 0, 1, 0, 1, 0, 0). If the quantity of interest from the first model was also evaluated at the mean then the second row in the design matrix would be D2 = (1, 0, 1, 1, 0, 1, 0, 0, 1, 0) and for the first model evaluated at the minimum the third row is D3 = (1, 0, 1, 1, 0, 1, 0, 0, 0, 1). Once the design matrix D is explicitly generated, the internal meta-analysis proceeds by applying standard statistical procedures to the relationship between the quantity of interest and the underlying modeling assumptions as represented by the design matrix, or ηˆ = g ( D), whereηˆ is the S × 1 vector
8 of the estimates of the quantity of interest. Most will recognize this as analogous to a factorial design experiment with four factors or class variables: (i) the functional form with I levels, (ii) the focal variable with J levels, (iii) the theoretical restriction with K levels, and (iv) the evaluation point with L levels. The internal meta-analysis will quantify and therefore help the researcher rank the impact of different assumptions on the quantity of interest. In addition, this information may prove useful in deciding the model space over which to average the quantity of interest. With the estimates of the quantity of interest in hand and the internal meta-analysis completed, attention can turn to the model averaging of the quantity of interest. The weighted or meta-estimate of the quantity of interest can be written as (2) ηˆ meta = ∑s∈U wsηˆ s where ws is the weight from model s, ηˆ s is the estimate of η from model s and U ⊆ S. Given U is a subset of the entire model space indexing set S, the researcher can average over whatever model space is deemed appropriate with weights scaled such that
∑
s∈U
ws = 1.
The key to obtaining the meta-estimate (2) is defining the weights. An intuitive weighting scheme is one where the model that more closely resembles the underlying data generating process receives a larger weight. One way to obtain this reasonable requirement is with a Jeffrey-Bayes posterior probability criterion. However, a standard Bayesian posterior probability criterion can be difficult to implement for a large model space due to the need to specify priors for the parameters in each model and the computational complexities. Fortunately, Schwarz has proven that the Bayesian posterior probability can be approximated by the Bayesian information criterion (BIC). The BIC is
9 very simple to implement in practice and has the attractive property that it is model consistent (Nishi).4 The weighting scheme can then be based on the BIC. More specifically, the BIC based model selection criterion is defined as (3)
I = –2 log(L) + p log(n).
where L is the value of the likelihood function and p log(n) is the penalty function, with p representing the number of parameters in the model and n the number of observations. Using (3), model s is preferred to model t if Is ≤ It. This comparison can be made by considering the ratio (4)
exp(− I s / 2) Ls exp(− p s log(n)) = . exp(− I t / 2) Lt exp(− pt log(n))
The motivation for (4) is well established within the Bayesian literature as it gives Schwarz’s approximation to the Bayes factor in the posterior odds ratio when the BIC is used. Because the posterior odds ratio can be used to form the weights for model averaging, the obvious candidate for the weighting is (5)
ws =
P( M s ) exp(− I s / 2) ∑r∈U P(M r ) exp(−I r / 2)
r, s ∈ U ⊆ S ,
where P(Ms) denotes the prior probability that model Ms is ‘true’. Within a Bayesian context the weights have the natural interpretation of being the posterior probabilities of a particular model (see e.g., Moulton). With U ⊆ S possible models, the model with the largest value of (5) is considered the best model within the set U. Substituting (5) into (2) gives the formula for averaging over models to obtain the meta-estimate of η.
4
Akaike’s information criterion (AIC) could be implemented as an alternative but Nishi has demonstrated it is not model consistent. For some this may be of no concern but it should be recognized that the modeling philosophies underlying the BIC and AIC are quite different. Bergman, Burnham, and Augustin provide a short discussion of the modeling philosophy underlying the AIC and BIC and Chow provides a more extensive discussion.
10 In addition to obtaining the meta-estimate ηˆ meta in (2), there also will be interest it its variance. Let
δˆt = ηˆ t − ηˆ define the estimate of the specification bias associated with model t. Buckland, Burnham, and Augustin derive an upper bound on the variance of ηˆ meta as (6) var (ηˆ meta ) ≤ ∑s∈U ws2 [var(ηˆ s | δˆs ) + δˆs2 ]. As an upper bound, this will give a conservative estimate of the significance of ηˆ meta . An Empirical Illustration with Meat Demand in the United States To illustrate the procedures discussed above, the demand for meat in the United States is analyzed. The interest in the demand for meat stems from two sources. First, as indicated in the introduction, beef and pork consumption are promoted through assessments paid by farmers called checkoff programs. Both the beef and pork checkoff programs have been recently contested in court and there is much interest in determining the actual impact that advertising will have on the demand for meat in the U.S. Second, even before the recent interest in the effectiveness of checkoff programs, there has been great interest in explaining the shift in consumption of meat away from pork and beef and toward chicken, which occurred between the 1970s and early 1990s. Not surprisingly, these issues have led to scientific debates about the model specification issue with respect to the appropriate functional form and explanatory variables (e.g., Alston and Chalfant 1991, 1993; Buse; Davis; Kinnucan, et al.; LaFrance). For example, in terms of functional form, several studies have compared alternative functional forms, in particular the Rotterdam Model versus the AIDS model (e.g., Alston and Chalfant (1991); Alston and Chalfant (1993); La France). Using a pairwise hypothesis testing framework, Alston and Chalfant found that the Rotterdam model was favored over the AIDS model. Alternatively, LaFrance found within his pairwise hypothesis testing framework that the AIDS model
11 was preferred over the Rotterdam model. Alternatively, several studies have used pairwise testing procedures to test the impact of additional explanatory variables within a specific functional form. Brester and Schroeder tested for significance of generic and branded advertising in the demand for meat within a Rotterdam model and found that branded advertising was significantly different from zero but that generic advertising was not. Kinnucan, et al. tested for the significance of a health information index and generic advertising within a Rotterdam model of meat demand and found the health index to be more significant relative to the generic advertising variable. While the research has correctly focused on the model specification issue, the individual studies have limited the model space explorations to one dimension: either the functional form or additional explanatory variables in isolation. Consequently, the conclusions about functional form could be sensitive to the inclusion and exclusion of other determinants of demand, such as advertising or health factors, and vice-versa. Furthermore, the only procedure that has been implemented is the yes/no pairwise hypothesis testing approach, which does not explicitly account for model uncertainty in the analysis and therefore will overstate significance levels. All of this suggests that some new light may be shed on this issue by employing the model averaging and internal meta-analysis procedure proposed here.
Defining the Model Space A conditional demand system consisting of the demand for beef, pork, poultry, and fish is considered. The four functional forms considered here are representative of those found in the literature: the Rotterdam (ROTT) model (Theil), the first differenced AIDS (FAIDS) model (Deaton and Muellbauer), the CBS model (Keller and Driel), and the NBR model (Neves). Neves has
12 demonstrated how these demand systems are related. The total differential of the expenditure share wi can be written as (7)
dwi = wi d ln qi + wi d ln pi – wi d ln E
with qi, pi, and E representing the per capita quantity and price on the ith good and E is the total expenditure on beef, pork, poultry (chicken and turkey), and fish. The Rotterdam model (ROTT) has the form wi d ln qi = µi d ln V + ∑ j πij d ln pj
(8)
i = beef, pork, poultry, fish
where µi is the constant marginal budget share for good i, πij is the price parameter, and d ln V = (d ln E – d ln P) is the volume index and P is the Divisia price index. Defining the parameters, bi = µi – wi and γij = πij – wiwj + wiδij, δij being the Kronecker delta, the first difference AIDS (FAIDS) model using (7) and (8) is wi d ln qi = bi d ln V + ∑ j γij d ln pj + wi(d ln E – d ln pi)
(9)
i = b, p, r, f.
The CBS model has the form (10)
wi d ln qi = bi d ln V + ∑ j πij d ln pj + wi d ln Q
i = b, p, r, f.
where Q is the Divisia quantity index and the NBR model has the form (11)
wi d ln qi = µi d ln V + ∑ j γij d ln pj + wi (d ln P – d ln pi) i = b, p, r, f.
Expressed in these forms, the models only differ by the last terms, but these terms also cause the parameters to have different interpretations. The CBS model shares expenditure coefficients with the AIDS model but price coefficients with the Rotterdam model. Alternatively, the NBR model shares expenditure coefficients with the Rotterdam model but price coefficients with the AIDS model. The elasticity formulas associated with each model are given in Neves. Three classes of variables are also considered. Defining variables are the variables that define the functional form and represent the last terms in equations (9), (10), and (11). Core variables are those
13 that are included in every model and are the individual prices, total expenditure, and quarterly dummy variables. Focal variables are those of interest that may or may not be included in a particular model.5 Three types of focal variables are considered: (i) advertising, (ii) health information, and (iii) women’s labor force participation. Classical demand theory generates demand functions that depend only on prices and expenditures, however these additional variables can generated as additional arguments through more generalized theories of demand (e.g., household production theory). Advertising is captured by contemporaneous and lagged branded advertising expenditures for beef, pork, chicken, and fish, and contemporaneous and lagged generic advertising expenditures for beef and pork. There is no generic advertising for poultry and fish. Health information is captured by an index of consumers’ exposure to health information in the news media. Women’s labor force participation is captured by women’s labor force participation rate. All variables and their sources are described in more detail in the data section below. Some structure is imposed on how the focal variables enter the models. The advertising variables are included in the models on an all-or-nothing basis. For example, either generic advertising for both beef and pork are both included, or both are excluded in a particular model. Thus, generic advertising and branded advertising can be thought of as single variables when calculating the number of possible subset combinations of variables. There are six possible focal variables that may be included in a model: Women’s labor force participation rate (WPR), the health index (HI), contemporaneous generic advertising (GAD), lagged generic advertising (LGAD), branded advertising (BAD), and lagged branded advertising (LBAD). There are therefore 26 = 64 possible subset combinations of these variables; however, models that include a lagged advertising variable without including the corresponding contemporaneous advertising variable are omitted. There are 16
5
The terminology of core variables and focal variables is based on the extension to Leamer’s nomenclature by Pesaran and Timmerman.
14 models that include LGAD but not GAD (64 * ½ = 32 models include LGAD, ½ of these 32 exclude GAD). Omitting these 16 reduces the number of combinations of variables to 48. There are also 16 models that included LBAD but not BAD, however 4 of these are removed in the previous step (16 * ¼ = 4 of the models removed previously). Thus the number of models removed in this second step is 16 – 4 = 12. Removing these 12 from the 48 leaves 36 possible subset combinations of the focal variables. Finally, there are four theoretical restriction combinations that are also considered: an unrestricted model, a model with only homogeneity imposed, a model with only symmetry imposed, and a model with homogeneity and symmetry imposed, or the fully restricted model.6 Putting all this together, the model space consists of 4 (functional forms) × 36 (focal variable combinations) × 4 (theoretical restrictions) or 576 demand systems to be estimated. The elasticity estimates from each demand system are evaluated at three points: the minimum, the mean, and the maximum of the data yielding 576 × 3 = 1728 estimates for each elasticity in the demand system.
Empirical Model Based on equations (9), (10), and (11) all 576 models can be nested within the general empirical model, including the additional variables, (12)
w it Dq it = α i1 + ∑ j= 2 α ij m jt + µ i DVit + ∑ j=1π ij Dp jt + ∑ j=1θ ij Df jt 4
4
14
+ γ 1 w it (DE t − Dp it ) + γ 2 w it (DPt − Dp it ) + γ 3 w it DQ t + ε it , where w it is the average budget share between period t and t –1 for the ith meat, Dzit = ln(zit/zit-1) is the discrete log change for any variable z and εit is the disturbance term. The additional terms are the intercept, three quarter dummy variables (m2t, m3t, m4t) and the 14 focal variables in log change form
15 (Df1t, Df2t,..., Df14t).7 Note the restriction γ1 = γ2 = γ3 = 0 yields the Rotterdam model, the restriction γ1=1, γ2 = γ3 = 0 yields the first differenced AIDS model, the restriction γ1 = γ2 = 0, γ3 =1 yields the CBS model, and restriction γ1 = γ3 = 0, γ2 = 1 yields the NBR model. The exclusion of certain focal variables amounts to zero restrictions on the appropriate θ parameters. The homogeneity restriction is ∑jπij = 0 and the symmetry restriction is πij = πji. Given this structure, the 576 demand systems can be derived by placing different restrictions on the γ, θ, and π parameters. As is often pointed out in the encompassing literature, the unrestricted encompassing model (12) has no real economic interpretation or significance but is only a testing tool.
Data The data are quarterly observations of all variables for 1976 through 1993. Per capita beef, pork, poultry, quantities, and retail prices come from Putnam and Allhouse and USDA’s Livestock and Poultry Situation and Outlook Reports. Per capita fish quantities and prices come from Kinnucan, et al. The health information index is constructed from two series. First a series of the cumulative number of articles appearing in medical journals that discuss the link between blood serum cholesterol and heart disease was constructed and labeled negative information. Another series was constructed giving the cumulative number of articles appearing in medical journals that attack or question the link and this was labeled positive information. The negative series was then weighted by
6
Adding-up is automatically satisfied in these models. An intercept in these models is very common and is designed to capture constant changes over time not explained by the model (Barten). The focal variables are defined as f1 = women’s labor force participation rate, f2 = health information index, f3 = generic beef advertising expenditures, f4 = generic beef advertising expenditures lagged one quarter, f5 = generic pork advertising expenditures, f6 = generic pork advertising expenditures lagged one quarter, f7 = branded beef advertising expenditures, f8 = branded beef advertising expenditures lagged one quarter, f9 = branded pork advertising expenditures, f10 = branded pork advertising expenditures lagged one quarter, f11 = branded poultry advertising expenditures, f12 = branded poultry advertising expenditures lagged one quarter, f13 = branded fish advertising expenditures, f14 = branded fish advertising expenditures lagged one quarter. 7
16 the percent of total articles that are negative in order to reflect the discounting of this information implied by the positive articles. Admittedly, this is only a proxy for health information but it has been used elsewhere (e.g., Kinnucan, et. al). As in Brester and Schroeder, advertising expenditure data was obtained from Class/Brand QTR $ published by Leading National Advertisers. Beef and Pork advertising were separated into branded and generic. Poultry advertising, which includes chicken and turkey, was all branded during this time period. Fish advertising was also all branded during this time period. Finally, the women’s labor force participation rate (as percent of total employment) was obtained from the Bureau of Labor Statistics. Results and Discussion All 576 demand systems and the 1728 estimates for each elasticity were estimated in GAUSS.8 All internal meta-analysis was conducted in TSP and heteroskedasticity was accounted for by using a heteroskedasticity-consistent covariance estimator. Because the design matrix D is comprised of all dummy variables, to avoid the implied singularity or dummy variable trap, a base set of assumptions or model must be chosen. The base model for the meta-analysis is the Rotterdam model, with no focal variables, no theoretical restrictions, and the evaluation point being the mean of the data. Consequently, all results for the meta-analysis will represent deviations from the results obtained from this base model. Table 1 gives the meta-analysis results by elasticity type: own price, cross price, and expenditure. The dependent variable in each model is the stated elasticity and there are 1728 observations or estimates for each elasticity. A constant was included in every model but is not reported to save space. Reported are the coefficient estimates for each model assumption or dummy indicator 8
The well known singularity problem associated with demand systems was handled as usual by dropping one equation (the fish equation) and estimating the remaining system by an iterated seemingly unrelated regression procedure which is
17 variable. These coefficients represent the mean deviation from the elasticity estimates obtained from the base model described above. Below each coefficient estimate is reported its squared t-statistic. The squared t-statistic is reported because it gives a measure of the contribution to the explanatory power of the model associated with the variable, or as is the case here, the model assumption (Greene, p. 101). The squared t-statistics in bold correspond to the t-statistics that are significant at the .01 level. The critical values for the t-statistics (not squared) are 2.59 at the .01 level and 1.96 at the .05 level. Below each squared t-statistic is also reported the rank of the squared t-statistic in descending order. The information in table 1 can be used to determine the directional impact an assumption will make on the elasticity estimate and how important that assumption is relative to some other assumption. A glance at table 1 reveals that there is no single uniform pattern across all elasticity estimates with respect to directional impact, significance, or ranking. Different assumptions have different impacts on different elasticities. However, if one focuses on the own-price, cross-price, and expenditure elasticities as subsets, some patterns do emerge. With respect to the own price elasticities, the only uniform directional impacts are that the first difference AIDS (d1), the CBS (d2), generic advertising (d4), branded advertising (d6), and women’s labor force participation (d9) all lead to larger (less negative) elasticity estimates across all own price elasticities. The other assumptions do not have uniform directional impacts on the own price elasticities. In terms of significance, all functional form indicators are significant at the .01 level except one (first difference AIDS (d1) in Pork own price model). Many of the focal variable indicators are not significant, especially in the poultry model. All of the theoretical restriction and evaluation point indicator variables are significant except for symmetry (d11) in the Poultry own
invariant to equation deletion. The fish elasticities are not reported here but may be recovered from the other estimates. The program written in GAUSS estimates all systems in about a minute and is free upon request from the authors.
18 price model. In terms of ranking the impacts, there again is no uniform ranking for an individual assumption, but it is the case that the theoretical restrictions (d10 and d11) and the evaluation points (d12 and d13) are usually in the top three. For the cross-price elasticities, with respect to directional impact, there is no uniformity across the individual estimates for any one assumption. The theoretical restrictions and evaluation points are all significant at the .01 level with one exception (Poultry-Beef symmetry d11), whereas the focal variables and functional forms do not show such significance uniformity. The theoretical restrictions and evaluation points tend to dominate the top ranking positions. With respect to the expenditure elasticities, there is uniformity with respect to the impact of the three functional forms and generic advertising across all expenditure elasticities. The first differenced AIDS (d1), NBR (d3), and generic advertising (d4) all lead to higher expenditure elasticity estimates for beef, pork, and poultry. Alternatively, the CBS (d2) leads to lower expenditure elasticity estimates for beef, pork, and poultry. In terms of statistical significance, the NBR (d3), generic and branded advertising lagged (d5, d7), health index (d8), women’s labor force participation (d9), and the homogeneity restriction (d10) are all significant for all expenditure elasticities, but the other indicators vary in terms of significance. Furthermore, the theoretical restrictions and evaluation points do not dominate the rankings for the expenditure elasticities as they did for the own and cross price elasticities. While table 1 provides information on ranking the individual model assumptions, it is only suggestive of how the classes of assumptions rank. Each of the individual assumptions fall under a particular assumption class: functional form (d1-d3), focal variable (d4-d9), theoretical restrictions (d10-d11), and evaluation point (d12-d13). F tests are used to determine the contribution of the classes of assumptions to the overall explanatory power of the meta-analysis models. The class F-
19 statistics and their rank are reported in table 2 for the individual elasticity estimates. The F-statistics reported in bold are significant at the .01 level. The .01 and .05 critical values for each class are respectively: functional form (3.79, 2.61), focal variables (2.81, 2.10), theoretical restriction and evaluation point (4.61, 3.00). Table 2 shows a rather consistent picture. All of the F-statistics are significant at the .01 level, with exception of the focal variable class in the Poultry own price model and the functional form class in the Pork-Beef cross price model. Of the 12 models, the theoretical restriction class is ranked first or second in 11 of the models. In 8 of the 12 models, the evaluation point class is ranked first or second. The general message from table 2 is that the theoretical restrictions and the evaluation points are more important than the functional forms and the focal variables in terms of explaining the variation in the elasticity estimates. Turning to the model averaging weights, an issue of interest is often whether or not the demand system under consideration satisfies the implied theoretical restrictions. In classical analysis, the hypothesis itself is not viewed as being a probability statement but either falls inside or outside the rejection region and one gets the yes/no answer. In Bayesian analysis, the hypothesis is regarded as a probability statement and the weights give an indication of the probability within the defined model space. Table 3 gives the model weights constructed using equation (5) across theoretical restrictions and focal variables for each functional form. That is, for each theoretical restriction there are 36 models (the focal variable combinations) for each functional form. The denominator in the weight equation (5) is then the sum across all 4 theoretical restrictions and 36 focal variable combinations for a given functional form or over 144 models (i.e. 4 × 36). The prior probabilities are assumed equal for all models so the weights are the approximate Bayes factors. Only those weights that are nonzero at the third digit are reported. As seen in table 3, across all functional forms the fully restricted
20 theoretically consistent models (symmetry and homogeneity) dominate with weights of .999, .999, .998, and .999, respectively. Table 3 also shows that the model receiving the greatest weight is model 1 for all functional forms with weights of .925, .972, .947, and .970. Model 1 corresponds to the classical demand system with only prices, expenditures, and three quarterly dummy variables and no focal variables. This result is of interest in the present legal environment. Of the 36 possible focal variable combinations, 32 have advertising included in the model in some form. Yet none of the specifications with advertising have weights greater than zero. Therefore, within the model space considered here, once model uncertainty is accounted for, advertisement funded through checkoff programs does not seem to be an important determinant of the demand for meat. While it is technically possible to model average over the entire model space (i.e., U = S), it seems counterintuitive to average over both theoretically consistent and theoretically inconsistent models, especially in light of the earlier meta-analysis results and the results in table 3. Consequently, table 4 gives the weights based on averaging over functional forms and focal variable combinations for a given theoretical restriction, again assuming all prior model probabilities are equal. As seen in table 4, the Rotterdam model with no focal variables receives the largest weight for the unrestricted and symmetry restricted models, with weights of .847 and .852, respectively. However, for the homogeneity and homogeneity and symmetry restricted models, the NBR with no focal variables has the larger weights of .876 and .882, respectively. Though not shown, the weights were also calculated across all models as well. The fully restricted NBR model 1 (no focal variables) received a weight of .89, the fully restricted NBR model 10 (health index is only focal variable) a weight of .03 and the fully restricted Rotterdam model 1 (no focal variable) a weight of .08. As in
21 table 3, these results demonstrate how misleading model specification explorations in one dimension can be while higher dimensioned model explorations may provide a clearer picture. Of course, the major point of model averaging is that there is no need to base the quantity of interest on only one model in the model space and proceed to conduct inference as if it is the correct model. Furthermore, part of the purpose of conducting the internal meta-analysis and considering the model weights is to help the researcher make an informed decision regarding the subset of models to average over. Given the fact that averaging over both theoretically consistent and inconsistent models seems counterintuitive, and the overwhelming support for the theoretically consistent systems (i.e., symmetry and homogeneity) provided by table 3, table 5 gives the Hicksian meta-price elasticities and meta-expenditure elasticities over the theoretically consistent systems. Each elasticity is evaluated at all three data points, which correspond to the minimum, mean, and maximum expenditure share. The first elasticity estimate is evaluated at the minimum expenditure share, the second elasticity estimate in parenthesis is evaluated at the mean expenditure share, and the third elasticity estimate in brackets is evaluated at the maximum expenditure share. The estimates that are at least twice their standard error estimate are given in bold. Table 5 shows that all meta-own price elasticities are negative and significant, except for the poultry own price evaluated at the minimum budget share, which is negative but insignificant. All significant cross price elasticities are positive indicating the meats are net substitutes, except for porkpoultry and poultry-pork evaluated at the minimum share. The meta-expenditure elasticities are all positive and significant, except for the poultry meta-expenditure elasticity, which is positive but insignificant. Two tendencies are apparent in table 5. First, as the point of evaluation increases from the minimum to the maximum budget share, the meta-price elasticity estimates all increase, except for
22 own price poultry. Alternatively, the meta-expenditure elasticities decrease. As Neves indicates, across these functional forms the direction of the change in the elasticity estimate with respect to the expenditure share cannot be signed a prior, so this is an empirical result not an analytic result. Second, there is still a great deal of variation in some of the elasticity estimates, depending on the evaluation point, and the evaluation point is not a dimension that falls within the standard model averaging framework.
Summary and Conclusions Model specification uncertainty is perhaps the Achilles heel of applied econometrics. Bayesian model averaging is conceptually an appealing alternative to the pairwise testing approach for addressing the model specification problem because it explicitly takes into account the model specification uncertainty. However, though conceptually appealing, for many Bayesian model averaging is still computationally overwhelming, especially for a large model space. This paper presents an information based model averaging procedure that retains the appealing properties of the Bayesian approach but overcomes its computational burden. While the model averaging procedure accounts for the model uncertainty, the paper also addresses a parallel question of great scientific interest: which of the modeling assumptions have the largest impact on the estimates of the quantity of interest? The paper shows how the model averaging approach is implicitly well suited for conducting an internal meta-analysis to determine the relative importance of different model assumptions on some quantity of interest. In the empirical application, the demand for meat in the U.S. is considered. The model space is explored in three dimensions: 4 functional forms, 36 possible focal variable combinations, and 4 possible combinations of theoretical restrictions, which implies 576 possible demand systems. The
23 chosen quantities of interest are the meat demand price and expenditure elasticities. Each elasticity from each demand system is evaluated at three different points so there are 1728 estimates for each price and expenditure elasticity. The 1728 estimates for each elasticity therefore depend on four assumption classes: (i) the functional form, (ii) the focal variables, (iii) the theoretical restrictions, and (iv) the evaluation point. Overall and contrary to the focus of much of the model specification literature, the internal metaanalysis indicates that the theoretical restrictions and the evaluation points explain much more of the variation in the elasticity estimates than the functional form and the focal variables. Within a Bayesian context, the constructed weights can be interpreted as the approximate posterior probabilities of a particular model within the model space. The constructed weights provide overwhelming support for the classical theory of demand with the only arguments being prices, expenditures, and three quarterly dummy variables and with symmetry and homogeneity imposed. The weighted or meta-elasticity matrix is presented and discussed. Perhaps most importantly, though not exhaustive in possible advertising specifications, models that include advertising variables do not have high probabilities and therefore the impact of checkoff funded advertising on meat demand is not significant. As with most empirical research, the results may not carry over to other data sets. However, the results suggest that exploring multiple dimensions in model selection may lead to more theoretically consistent models and that dimensions that have not received a great deal of attention in the literature (e.g., the evaluations point) may be more important than dimensions that have received a great deal of attention (e.g., functional form). The modeling procedure presented here should not be viewed as a substitute for either a pure classical pairwise testing approach or a pure Bayesian approach but rather a complementary tool. As Granger, King, and White have indicated, an attractive modeling strategy
24 may be to use an information criterion to first narrow down the model space to some reasonable subset and then apply hypothesis testing or Bayesian procedures to answer additional questions.
25 Table 1. Individual Equation Meta–Analysis Results. Functional Form
Focal Variable
Theoretical Restriction
Evaluation Point
Variable Class Indicator Variablea Own Price Elasticities
d1
d2
d3
d4
d5
d6
d7
d8
d9
d10
d11
d12
d13
Beef Coefficient t–squaredb Rank
0.07 1311.64 4
0.00 4.01 12
0.06 783.64 5
0.00 1.72 13
–0.01 59.30 9
0.01 29.11 11
–0.02 85.23 8
–0.03 251.83 6
0.02 157.98 7
0.09 3076.99 1
–0.01 30.92 10
–0.08 1575.61 2
0.07 1407.47 3
Pork Coefficient t–squared Rank
0.00 3.73 12
0.00 11.77 11
–0.01 29.03 8
0.00 0.06 13
–0.01 23.41 9
0.01 60.47 5
–0.01 31.58 7
0.01 42.04 6
0.00 13.35 10
0.03 948.71 3
–0.01 103.13 4
–0.03 1081.82 2
0.04 1504.42 1
Poultry Coefficient t–squared Rank
0.03 28.44 4
0.01 5.88 6
0.02 8.95 5
0.00 0.57 11
0.00 0.17 12
0.01 4.99 7
0.00 1.21 10
0.00 0.01 13
0.01 4.35 8
–0.03 50.15 3
0.01 4.34 9
0.07 264.66 1
–0.04 108.54 2
Beef–Pork Coefficient t–squared Rank
–0.01 13.51 11
0.01 38.52 8
–0.02 93.68 5
0.00 0.36 13
–0.01 16.88 10
–0.01 71.83 7
–0.02 158.67 3
0.01 92.65 6
0.00 11.77 12
0.03 739.00 2
–0.05 1528.18 1
0.02 152.65 4
–0.01 27.08 9
Beef–Poultry Coefficient t–squared Rank
0.00 1.06 12
0.00 3.27 10
0.01 8.04 8
0.00 0.46 13
0.01 17.38 5
0.00 1.71 11
0.00 4.45 9
0.00 9.37 7
–0.01 23.71 4
–0.01 12.82 6
0.02 270.39 1
–0.02 134.21 3
0.02 204.91 2
Cross Price Elasticities
26 Table 1. Individual Equation Meta–Analysis Results (con’t). Functional Form
Focal Variable
Theoretical Restriction
Evaluation Point
Variable Class Indicator Variablea
d1
d2
d3
d4
d5
d6
d7
d8
d9
d10
d11
d12
d13
Pork–Beef Coefficient t–squared Rank
0.01 3.26 7
0.00 0.78 10
0.00 0.90 9
0.00 0.06 13
–0.01 5.30 6
0.00 0.09 12
–0.07 439.57 2
0.00 1.70 8
0.00 0.26 11
0.03 132.50 3
0.09 1202.58 1
–0.02 43.91 4
0.02 30.07 5
Pork–Poultry Coefficient t–squared Rank
–0.01 11.49 9
–0.01 28.77 6
0.00 3.80 11
0.00 5.53 10
0.00 2.29 12
–0.02 156.16 3
0.02 153.09 4
–0.01 21.75 8
0.00 0.06 13
–0.01 30.36 5
0.01 26.71 7
–0.03 494.56 2
0.04 630.93 1
Poultry–Beef Coefficient t–squared Rank
0.02 9.01 8
–0.01 0.94 11
0.02 16.51 5
0.00 0.68 12
0.01 2.70 10
–0.01 6.88 9
0.02 11.62 7
0.03 46.94 2
–0.02 23.48 4
–0.10 532.94 1
0.00 0.63 13
–0.02 12.58 6
0.02 24.94 3
Poultry–Pork Coefficient t–squared Rank
–0.04 202.55 4
–0.03 119.43 5
–0.01 17.70 10
0.00 0.08 13
0.01 27.20 9
0.00 4.74 12
0.02 73.34 7
–0.02 119.34 6
0.01 9.11 11
–0.04 506.61 3
–0.01 43.26 8
–0.08 1390.45 1
0.06 948.30 2
Beef Coefficient t–squared Rank
0.08 295.70 3
–0.01 3.29 12
0.09 298.10 2
0.00 0.02 13
0.05 101.82 5
–0.05 93.38 6
–0.02 13.72 10
0.02 23.45 7
–0.06 282.60 4
0.20 2852.67 1
–0.02 22.39 8
0.02 15.02 9
–0.01 9.60 11
Pork Coefficient t–squared
0.07 330.96
–0.01 9.50
0.08 431.42
0.03 59.54
0.01 19.90
0.00 0.07
0.02 35.89
0.01 6.73
–0.03 98.33
0.05 389.78
0.02 32.93
0.01 2.77
–0.01 3.67
Cross Price Elasticities
Expenditure Elasticities
27 Rank
3
9
1
5
8
13
6
10
4
2
7
12
11
28 Table 1. Individual Equation Meta–Analysis Results (con’t).
Functional Form
Focal Variable
Theoretical Restriction
Evaluation Point
Variable Class Indicator Variablea
d1
d2
d3
d4
d5
d6
d7
d8
d9
d10
d11
d12
d13
0.01 1.78 11
–0.06 45.27 6
0.07 63.77 5
0.01 0.57 13
–0.03 11.56 9
0.04 29.12 7
–0.02 5.61 10
–0.03 17.75 8
0.06 97.01 3
–0.29 2129.31 1
–0.01 0.65 12
–0.09 152.71 2
0.06 94.34 4
Expenditure Elasticities Poultry Coefficient t–squared Rank
a. Indicator variables are d1 = first differenced AIDS form, d2 = CBS form, d3 = NBR form, d4 = generic advertising, d5 = generic advertising lagged, d6 = branded advertising, d7 = branded advertising lagged, d8 = health index, d9 = women’s labor force participation rate, d10 = homogeneity restriction, d11 = symmetry restriction, d12 = minimum budget share, and d13 = maximum budget share. b. Bold indicates the t-statistic is significant at the .01 level.
29 Table 2. Design Classes Meta–Analysis Results Classa Functional Form
Focal Variable
Theoretical Restriction
Evaluation Point
Own Price Elasticities Beef F Rank
604.76 3
98 4
1585.04 2
2843.6 1
Pork F Rank
27.23 3
24.94 4
526.56 2
1502.54 1
Poultry F Rank
9.59 3
1.65 4
27.17 2
193.14 1
82.63 4
96.44 2
1205.57 1
95.88 3
Beef-Poultry F Rank
7.4 4
10.89 3
142.73 2
189.51 1
Pork–Beef F Rank
1.1 4
100.3 2
646.34 1
45.12 3
Pork–Poultry F Rank
22.22 4
40.03 2
28.19 3
667.57 1
Poultry–Beef F Rank
11.87 4
14.88 3
266.74 1
20.01 2
Poultry–Beef F
85.8
48.24
267.47
1231.15
Cross Price Elasticities Beef-Pork F Rank
30 Rank
3
Table 2. Design Classes Meta–Analysis Results (con’t) Class Functional Form
4
2
1
Focal Variable
Theoretical Restriction
Evaluation Point
Expenditure Elasticities Beef F Rank
199.97 2
102.78 3
1428.81 1
15.71 4
Pork F Rank
289.55 1
51.13 3
208.89 2
4.8 4
Poultry F 68.35 25.79 1064.96 135.3 Rank 3 4 1 2 a. Functional form class includes first differenced AIDS form indicator (d1), CBS form indicator (d2), NBR form indicator (d3). Focal Variable class includes generic advertising indicator (d4), generic advertising lagged indicator (d5), branded advertising indicator (d6), branded advertising lagged indicator (d7), health index indicator (d8), women’s labor force participation rate indicator (d9). Theoretical restriction class includes homogeneity restriction indicator (d10) and symmetry restriction indicator (d11). Evaluation point class includes minimum budget share indicator (d12) and maximum budget share indicator (d13). b. Bold indicates the F-statistic is significant at the .01 level.
31 Table 3. Model Weights by Functional Forma Theoretical Restrictions
Focal Variable Modelsb
Functional Forms Rotterdam
FAIDS
CBS
NBR
Unrestricted
Sum
0.000
0.000
0.000
0.000
Symmetry
1 Sum
0.001 0.001
0.001 0.001
0.001 0.001
0.001 0.001
Homogeneity
1 Sum
0.000 0.000
0.000 0.000
0.001 0.001
0.000 0.000
Symmetry and Homogeneity
1 10 28 Sum
0.925 0.070 0.004 0.999
0.972 0.020 0.007 0.999
0.947 0.043 0.008 0.998
0.970 0.025 0.004 0.999
1.000
1.000
1.000
1.000
TOTAL
a. Weights rounded off to third digit. b. Model 1 = no focal variables, Model 10 = focal variable is health index, and Model 28 = focal variable is women’s labor force participation.
32 Table 4. Model Weights by Theoretical Restrictionsa Functional Form
Focal Variable Modelsb
Theoretical Restrictions Unrestricted
Symmetry
Homogeneity Symmetry and Homogeneity
Rotterdam
1 10 19 28 Sum
0.847 0.077 0.001 0.018 0.943
0.852 0.053 0.001 0.021 0.927
0.062 0.006 0.000 0.000 0.068
0.073 0.005 0.000 0.000 0.078
FAIDS
1 10 Sum
0.005 0.000 0.005
0.007 0.000 0.007
0.020 0.001 0.021
0.014 0.000 0.014
CBS
1 10 28 Sum
0.028 0.001 0.001 0.030
0.022 0.001 0.001 0.024
0.000 0.000 0.000 0.000
0.000 0.000 0.000 0.000
NBR
1 10 28 Sum
0.020 0.001 0.001 0.022
0.040 0.001 0.001 0.042
0.876 0.032 0.003 0.911
0.882 0.023 0.003 0.908
1.000
1.000
1.000
1.000
TOTAL
a. Weights rounded off to third digit. b. Model 1 = no focal variables, Model 10 = focal variable is health index, Model 19 = focal variables are health index and women’s labor force participation, and Model 28 = focal variable is women’s labor force participation.
33 Table 5. Meta-Elasticity Estimatesa Quantity
Beef
Pork
Regressors Beef Price -0.531b (-0.460)c [-0.394]d
Pork Price 0.258 (0.267) [0.284]
Poultry Price 0.045 (0.091) [0.140]
Expenditure 0.844 (0.720) [0.634]
0.489 (0.552) [0.610]
-0.694 (-0.679) [-0.659]
-0.113 (-0.060) [0.000]
0.653 (0.606) [0.554]
0.163 -0.020 0.107 -0.213 (-0.092) (0.084) (0.290) (-0.178) [0.000] [0.068] [0.395] [-0.268] a. Bold indicates estimate is at least twice the standard error. b. Evaluated at the minimum budget share. c. Evaluated at the mean budget share. d. Evaluated at the maximum budget share. Poultry
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