Abstract
Age, period and cohort (APC) variables are included in a demand system that is used to estimate Norwegian purchases of nonalcoholic beverages. To take account of censoring, a two-step method is used. In the first step, the probabilities of purchasing milk, carbonated soft drinks and other soft drinks are estimated by probit models. The APC variables are highly significant. Older cohorts have higher probabilities of purchasing milk and lower probabilities of purchasing carbonated soft drinks than younger cohorts. In the second step, the probability density functions and the cumulative density function are used to correct for censoring. In the corrected demand system, there are positive cohort and negative age effects for milk. These effects suggest that the replacement of older by younger cohorts, in an increasingly older population, will result in reduced per capita purchases of milk. For carbonated soft drinks, there are no cohort or negative age effects, while there are positive age but no cohort effects for other soft drinks.
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Notes
A quadratic AIDS model that allows for nonlinear Engel curves was also considered. However, the focus of this article is on the inclusion of APC variables in a demand system. A linear version of the AIDS model was used to facilitate the estimation of an already complex demand system.
Demographic variables could also have been included in the system. However, the inclusion of demographic variables resulted in serious multicolinearity problems.
In this method, one equation models the decision to purchase or not, and a second equation models the expected purchase conditional on a positive purchase. Let \(y_{it}\) be the purchased quantity of good \(i, \mathbf{x}_{\mathbf{it}}\) the vector of covariates in the second-step equation and \(\mathbf{z}_{\mathbf{it}}\) the vector of covariates in the first-step equation. There is a positive purchase if \(\mathbf{{z}^{\prime }}_{\mathbf{it}} {\varvec{\uppsi }}_\mathbf{i} +v_{it} >0,\) where \(v_{it}\) is an error term. The expected mean purchase conditional on a positive purchase is \(E(y_{it} |\mathbf{x}_{\mathbf{it}} ,\mathbf{z}_{\mathbf{it}} ;v_{it} >-\mathbf{{z}^{\prime }}_{\mathbf{it}} {\varvec{\uppsi }}_\mathbf{i} )=f(\mathbf{x}_{\mathbf{it}} ,{\varvec{\beta }}_\mathbf{i} )+\delta _i {\phi (\mathbf{{z}^{\prime }}_{\mathbf{it}} {\varvec{\uppsi }}_\mathbf{i} )}/{\Phi (\mathbf{{z}^{\prime }}_{\mathbf{it}} {\varvec{\uppsi } }_\mathbf{i} )}\), where \(\phi (\cdot )\) is the probability density function and \(\Phi (\cdot )\) is the cumulative distribution function of the standard normal distribution. The expected purchase conditional on nonpurchase is zero, or \(E(y_{it} |\mathbf{x}_{\mathbf{it}} ,\mathbf{z}_{\mathbf{it}} ;v_{it} \le -\mathbf{{z}^{\prime }}_{\mathbf{it}} {\varvec{\uppsi }}_\mathbf{i} )=0,\) and we have the unconditional (on zero/nonzero purchase) mean purchase \(E(y_{it} |\mathbf{x}_{\mathbf{it}} ,\mathbf{z}_{\mathbf{it}} )=\Phi (\mathbf{{z}^{\prime }}_{\mathbf{it}} {\varvec{\uppsi }}_\mathbf{i} )f(\mathbf{x}_{\mathbf{it}} ,{\varvec{\beta }}_\mathbf{i} )+\delta _i \phi (\mathbf{{z}^{\prime }}_{\mathbf{it}} {\varvec{\uppsi }}_\mathbf{i} ).\)
The first step is modeled by univariate probit models, which imply that the decision to purchase one beverage is assumed to be independent of the decisions to purchase other goods. To take account of correlations between the purchase decisions, a potentially better way would be to estimate a multivariate probit model. However, this model is computationally difficult because of the numerical integration of the multivariate normal. The computational difficulty increases because of the bootstrap used in the estimation. Puhani (2000) recommended estimating the model in two steps rather than using maximum likelihood procedures when there are many variables, because of potentially strong colinearity. When using the two-step procedure, the uncertainty in the first step has to be provided for using the bootstrap.
We follow the cohorts for 16 years. As pointed out by one referee, it may be of interest to follow the cohorts over a longer period. However, 16 years is a longer period than used in some other studies. For example, Aristei et al. (2008) used data over the 1997–2002 period.
An alternative specification is a conditional demand system that only includes nonalcoholic beverages. Given available data a two-stage model, with a second-stage demand system for nonalcoholic beverages, could have been estimated. Using the formulas provided in Carpentier and Guyomard (2001), the demand elasticities of the two stages could have been used to calculate elasticities that could have been compared with the demand elasticities in our model. However, this comparison is not pursued in this article. For a related discussion, see Alston et al. (2000), who discussed asymmetric versus conditional demand systems without being able to conclude which specification is the preferred one.
As pointed out by one referee, an alternative price measure could be price indices. We tried to estimate the model using monthly consumer price indices for the different beverage groups. The estimation broke down, probably because of a high degree of multicolinearity among the prices. One possible explanation for this multicolinearity is that the consumer price indices do not take account of regional variability.
Unit prices are constructed by dividing expenditure by quantity for positive purchases. As pointed out by one referee, these unit prices will be downward biased if nonpurchase decisions are caused by too high prices. In our case, the effect of including (quality-corrected) unit prices were small in the probit equations used to model the purchase versus nonpurchase decisions, so any potential bias in the estimated demand functions is likely to be small. Furthermore, Cox and Wohlgenant (1986) used a cross section for 1 year. To avoid outliers, they deleted observations with prices more than five standard deviations from the average (about 2 % of the observations). We have a pooled dataset for 16 years. To avoid outliers, we replace the values below the 0.01 quantile, in each year, with the 0.01 quantile and values above the 0.99 quantile, in each year, with the 0.99 quantile.
These variables are similar to the variables used in Cox and Wohlgenant (1986), Park and Capps (1997) and Kuchler et al. (2005) to filter out the effects of quality differences. Our hypothesized effects include: larger families purchase less convenience products than smaller families, high-income households purchase more expensive brands than low-income households and older households choose different brands than younger households. In addition, there may be regional or seasonal differences: for example, related to transportation costs.
We also tried to include prices in the probit equations. Because of identification problems, the estimation of the second step did not converge and prices were excluded from the first step. However, the choice of first-step variables had only minor effects on the second-step estimates. When we excluded the cohort variables and instead included the price variables, the percentage of correct predictions did not change for any beverage. Furthermore, the second-step elasticities changed only marginally; the own-price elasticity changed from \(-\)0.59 to \(-\)0.56 for milk, from \(-\)1.28 to \(-\)1.21 for carbonated soft drinks, from \(-\)0.91 to \(-\)0.94 for other soft drinks and it did not change for other goods.
The percentage of correct predictions is, however, a questionable measure of the goodness of fit. For example, the naive predictor that every household purchases milk is very good according to this criterion.
For example, when we dropped the equation for other soft drinks instead of the equation for nondurables and services in the estimation, the own-price elasticity of milk changed from \(-\)0.59 to \(-\)0.61, the own-price elasticity of carbonated soft drinks changed from \(-\)1.28 to \(-\)1.31, the own-price elasticity for other soft drinks changed from \(-\)0.91 to \(-\)0.95 and the own-price elasticity for other nondurables and services changed from \(-\)1.02 to \(-\)1.01.
Figure 1 shows the mean annual purchase in the different age/cohort groups, while the data used in the estimation are for household purchases in a 2-week period. The insignificant cohort effects may be caused by household heterogeneity in the data or, as pointed out by a referee, older cohorts may purchase soft drinks for other people like their grandchildren.
As pointed out by one referee, reduced milk purchases is also seen among younger cohorts suggesting that there also may be some changes in taste among consumers.
The differences in numerical values between the two studies are not surprising given the differences between them. Gustavsen and Rickertsen (2003) did not include APC variables, used aggregate time-series data, had a different commodity specification and used a different sample period.
A simultaneous CI for the system can be constructed by a Bonferroni correction. Let \(C_{i}\) denote a confidence statement for coefficient \(i\). Then, the probability that all the CIs simultaneously are true is \(P(\text{ all} C_i \text{ true})=1-P(\text{ at} \text{ least} \text{ one} C_i \text{ false})\ge 1-\sum \limits _{i=1}^m {P(C_i \text{ false})} =1-\sum \limits _{i=1}^m {\left( {1-P(C_i \text{ true})} \right) } =1-(\alpha _1 +\alpha _2 +\cdot \cdot \cdot +\alpha _m ),\) where \(m\) is the number of estimated coefficients and \(\alpha \) is the confidence level. In our case, the simultaneous CI is a 70 % CI.
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Acknowledgments
The Research Council of Norway, grants 173388/I10 and 190306/I10, provided financial support for this research. The authors thank Rudy Nayga, two anonymous referees, and seminar participants at the 2009 IAAE conference and at the ERS, USDA, Washington DC for useful comments to earlier versions of this article.
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Appendices
Appendix 1
See Table 5.
Appendix 2: The Monte Carlo experiment
The exclusion of cohort variables may lead to biased parameter estimates when the data-generating process includes cohort effects. To investigate the bias of omitting cohort variables, we performed a Monte Carlo analysis.
First, system (5) was assumed to be the data generating process. The residuals were assumed to follow a multivariate normal distribution \(MN(0, \Sigma )\), where \({\Sigma }\) is the estimated covariance matrix, and 1,000 residuals were drawn for each equation from this distribution. Second, we used the estimated parameters, as reported in Table 2, the observations of the independent variables and the drawn residuals to construct 1,000 observations of the expenditure shares in system (5). Some of the constructed shares were negative and they were set to zero. Third, we estimated system (5) for each of our 1,000 drawings without the APC variables but including a (log of) trend and a (log of) age variable.
The median values of the estimated coefficients in the Monte Carlo analysis and the 99 % confidence intervals (CIs) for the corresponding coefficients in Eq. (5) are shown in Table 6.Footnote 17 The coefficients associated with the intercepts, expenditure terms and the correction parameter \(\phi \) are always outside the CIs. In the carbonated soft drink equation, several of the price parameters are also outside the CIs. Estimated coefficients outside the CIs may be explained in several ways. First, excluding the cohort variables may result in biased effects of other variables suggesting the importance of cohort variables. Second, the log-linear specification of the age and period variables may be wrong.
The median values of the estimated coefficients of the year and age variables and their associated \(t \)values are reported at the bottom of Table 6. These variables are significant with the exception of the trend variable in the carbonated soft drinks equation and the age variable in the other soft drinks equation. However, compared with the model including APC variables, the interpretation of the effects changes in the milk equation. The negative effect of age changes to a positive effect, suggesting that older people purchase more milk than younger people. Furthermore, the positive cohort effect, which suggests that older cohorts consume more milk than younger cohorts, is replaced by a strong negative trend, which suggests that there is a negative trend affecting all age groups simultaneously.
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Gustavsen, G.W., Rickertsen, K. Consumer cohorts and purchases of nonalcoholic beverages. Empir Econ 46, 427–449 (2014). https://doi.org/10.1007/s00181-013-0688-3
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DOI: https://doi.org/10.1007/s00181-013-0688-3