Testing the aggregation of goods and services without separability using panel data

Abstract

After Japan’s bubble economy collapsed in 1991, household values diversified, and, gradually, the budget shares on intangible services exceeded that on tangible goods, leading to a shift from demand for goods to services. In this study, we focus on the increased budget allocation to services in Japanese household expenditure and verify whether service-related items can be aggregated into a service group using Lewbel’s (Am Econ Rev 86(3):524–543, 1996) generalized composite commodity theorem (GCCT). We first accurately reclassify into all 51 items consisting of goods and services and verify whether these aggregations are justified. Next, we verify whether these 51 items can be sub-aggregated into goods or service groups. In testing the GCCT, we incorporate panel time-series analysis with cross-sectional dependence, unlike traditional GCCT tests with time-series data. We also conduct a nonparametric revealed preference test for weak separability as a benchmark against the GCCT test results. Our findings demonstrate that the utility function can be rationalized even when the data set is reclassified into 51 items, justifying the aggregation into service groups. This suggests that in the future, specifying the functional form of a service group can be developed into a traditional demand analysis, such as calculating estimates and elasticities.

Appendices

Appendix A: The generalized composite commodity theorem test for Case 3

In Appendix A, we examine whether the sub-aggregation of 51 reclassified items based on the ten major classifications is feasible. Following the original major classifications of Table 1, numbers 1–12 are classified as food; 13–15 as housing; 16–19 as fuel, light, and water charges; 20–25 as furniture and household utensils; 26–32 as clothing and footwear; 33–35 as medical care; 36–40 as transportation and communications; 41–42 as education; 43–49 as culture and recreation; and 50–51 as other expenditures. These classifications are used to verify whether the original aggregation based on the major classification is valid, even when goods and services are reclassified into 51 items. In addition, medium classifications based on the ten major classification groups are commonly used in realistic demand analyses. In Table 

Table 7 The panel cross section dependence tests in Case 3

7, ${R}_{I}$ represents the $I$-th group index, and ${\rho}_{i}$ is the relative price of the $i$-th item in Table 1. The relative prices are divided by the prices of the above-mentioned ten major groups.

Table 7 shows that for most group indices and relative prices, the null hypothesis of no cross-sectional dependence between error terms is rejected at the 5% level, with only ${\rho}_{13}$ and ${\rho}_{15}$ determined as having no cross-sectional dependence. In general, because most variables are found to have cross-sectional dependence, we incorporate cross-sectional dependence into the panel unit root and cointegration tests. For ${\rho}_{13}$ and ${\rho}_{15}$, other tests, such as the Breusch and Pagan (1980) Lagrange multiplier (LM) and Pesaran (2004) scaled LM tests, result in the adoption of cross-sectional dependence. Therefore, we apply panel unit root and cointegration tests, which also account for the cross-sectional dependence on these variables.

The left-hand side of Table 

Table 8 Generalized composite commodity theorem test results in Case 3

8 presents the results of the Bai and Ng (2004) unit root test. First, for the group indices ${R}_{I}$, the null hypothesis for the panel unit root in the common factors is not rejected, except for ${R}_{7}$. In addition, the null hypothesis for the panel unit root in idiosyncratic factors is not rejected except for ${R}_{2}$ and ${R}_{6}$. Therefore, we conclude that all group indices are I(1) variables with panel unit roots in either common or idiosyncratic factors or both factors. Second, for relative prices ${\rho}_{i}$, the null hypothesis for the panel unit root is not rejected for all the common factors. In addition, the null hypothesis for the panel unit root is not rejected for idiosyncratic factors except for ${\rho}_{12},{\rho}_{21}$, ${\rho}_{45}$, and ${\rho}_{49}$. Therefore, we conclude that all relative prices are I(1) variables with panel unit roots in either common or idiosyncratic factors or both factors.

Based on these results, we conduct an individual cointegration test between each relative price ${\rho}_{i}$ and the group index ${R}_{I}$ to verify whether the GCCT is satisfied in each group. The right-hand side of Table 8 presents the results of the Gengenbach et al. (2006) cointegration test. Among the ten major groups, the applicability of GCCT is rejected in transportation and communications and culture and recreation groups. In the transportation and communications group, we failed to reject, even at the 10% level, the null hypothesis for no panel cointegration relationship between each relative price ${\rho}_{i}$ and group index ${R}_{I}$ for the common factor for public transportation (i.e., item 36). By contrast, for the idiosyncratic factor, we reject the null hypothesis for no panel cointegration between each relative price ${\rho}_{i}$ and group index ${R}_{I}$. Therefore, the cointegration relationship between the group index and the relative price for public transportation has not been established, and the GCCT is not satisfied. In the culture and recreation group, the null hypothesis for no panel cointegration relationship between each relative price ${\rho}_{i}$ and group index ${R}_{I}$ for the common factor is not rejected, even at the 10% level for recreational goods services (i.e., item 47). However, for the idiosyncratic factor, we reject the null hypothesis for no panel cointegration between each relative price ${\rho}_{i}$ and the group index ${R}_{I}$. Therefore, the cointegration relationship between the group index and the relative price of recreational goods services has not been established, and the GCCT is not satisfied. In the remaining groups, the null hypothesis for no panel cointegration between each relative price ${\rho}_{i}$ and group index ${R}_{I}$ for both common and idiosyncratic factors is rejected. In other words, the cointegration relationship between each relative price and the group index is established. The GCCT is satisfied in these groups, and aggregation is justified for each group. Thus, two out of ten groups had no cointegration relationship.

The first row of each group in Table 8 shows the results of the GCCT family-wise cointegration test. In this test, we employ Pedroni’s (2004) cointegration test based on Engel–Granger for both common and idiosyncratic factors. In Case 3, where 51 reclassified items are aggregated into ten groups, the null hypothesis for no cointegration relationship is rejected for both common and idiosyncratic factors. Although the individual GCCT tests failed to establish a cointegration relationship between the two groups, the family-wise GCCT test established a cointegration relationship. Thus, we conclude that the GCCT is satisfied. One would first suspect the possibility of a spurious relationship problem, where the results of the family-wise test suggest a false cointegration relationship, as a reason for the differences between these results. In addition, the two groups of transportation and communication and culture and recreation have seen their budget shares for services increasing in recent years, and it cannot be ruled out that their preferences may change. However, since this requires careful discussion, it would be treated as just one possibility. On the other hand, the GCCT test was performed with the null hypothesis of no panel cointegration; however, the GCCT may be satisfied if the null hypothesis for having a panel cointegration was tested.

Appendix B: Nonparametric weak separability test

The general method of constructing nonparametric revealed preference tests was first introduced by Afrait (1967) and developed by Varian (1982, 1983) and Diewert and Parkan (1985). Subsequently, Fleissig and Whitney (2003) proposed a methodology using linear programming. Furthermore, Hjertstrand et al. (2016) formulated a mixed-integer linear programming (MILP) problem for a new testing procedure for weak separability, as proposed by Cherchye et al. (2015). In this test, the dataset satisfies weak separability if there exists a feasible solution to the optimization problem (7)–(15). However, if there is no feasible solution to the problem, the dataset violates weak separability. We calculated in Python using the MIP package with solver CBC to solve the optimization problems.

We suppose the $n$ goods and services at T time periods. Let ${\mathbf{x}}_{t}=\left({x}_{1t},{x}_{2t},\dots ,{x}_{nt}\right)$ denote the observed quantity vector at time $t\in T$, and ${\mathbf{p}}_{t}=\left({r}_{1t},{r}_{2t},\dots ,{r}_{nt}\right)$ the corresponding price vector at time $t\in T$. Next, we suppose the data set is split into two subgroups ${\mathbf{w}}_{t}=\left({x}_{1t},{x}_{2t},\dots ,{x}_{gt}\right)$ with prices ${\mathbf{z}}_{t}=\left({r}_{1t},{r}_{2t},\dots ,{r}_{gt}\right)$ and ${\mathbf{q}}_{t}=\left({x}_{g+1t},{x}_{g+2t},\dots ,{x}_{nt}\right)$ with prices ${\mathbf{v}}_{t}=\left({r}_{g+1t},{r}_{g+2t},\dots ,{r}_{nt}\right)$, defined a data set as D = ${\{{\mathbf{z}}_{\mathbf{t}}, {\mathbf{v}}_{\mathbf{t}};{\mathbf{w}}_{\mathbf{t}}, {\mathbf{q}}_{\mathbf{t}}\}}_{t\in T}$. The utility function $U\left(\mathbf{x}\right)$ is weakly separable in the partition of two subgroups if it can be written as $U\left(\mathbf{x}\right)=U\left(\mathbf{w},\mathbf{q}\right)=u\left(\mathbf{w},V\left(\mathbf{q}\right)\right)$ with a macro-function $u$ and sub-utility function $V$. The data set D can be rationalized by a weakly separable utility function, if there exist well-behaved utility functions $u$ and $V$, such that for all observations $t\in T$.

Hjertstrand et al. (2016) and Hjertstrand and Swofford (2019) focused on some of the weak separability condition proposed by Varian (1983) and transformed it into the optimization problem. The conditions are as follows.

There exist numbers ${Q}_{t}$, ${u}_{t}$, and ${\phi }_{t}>0$ such that, for all $s,t\in T$,

$${{Q}_{t}}-{{Q}_{s}}-{{\phi}_{t}}{\mathbf{v}}_{t}\left({{\mathbf{q}}_{t}}-{{\mathbf{q}}_{s}} \right) \geq 0,$$

(4)

$$\mathrm{if }{\phi }_{t}{\mathbf{z}}_{t}\left({\mathbf{w}}_{t}-{\mathbf{w}}_{s}\right)+({Q}_{t}-{Q}_{s})\ge 0\mathrm{ then }{u}_{t}-{u}_{s}\ge 0,$$

(5)

$$\mathrm{if }{\phi }_{t}{\mathbf{z}}_{t}\left({\mathbf{w}}_{t}-{\mathbf{w}}_{s}\right)+\left({Q}_{t}-{Q}_{s}\right)>0\mathrm{ then }{u}_{t}-{u}_{s}>0.$$

(6)

They also solved the following MILP problem to implement Varian (1983)’s weak separability conditions (4)–(6), for which a formal proof was provided by Cherchye et al. (2015).

$${\text{min}}\left[\sum_{t=1}^{T}0\times {Q}_{t}+\sum_{t=1}^{T}0\times {\phi }_{t}+\sum_{t=1}^{T}0\times {u}_{t}+\sum_{t=1}^{T}\sum_{s=1}^{T}0\times {X}_{t,s}\right] s.t.$$

$${Q}_{s}-{Q}_{t}-{\phi }_{t}{\mathbf{v}}_{t}\left({\mathbf{q}}_{s}-{\mathbf{q}}_{t}\right)\le 0,$$

(7)

$${u}_{t}-{u}_{s}-{X}_{t,s}\le -\varepsilon ,$$

(8)

$$\left({X}_{t,s}-1\right)\le {u}_{t}-{u}_{s},$$

(9)

$${\phi }_{t}{\mathbf{z}}_{t}\left({\mathbf{w}}_{t}-{\mathbf{w}}_{s}\right)+\left({Q}_{t}-{Q}_{s}\right)-{X}_{t,s}{A}_{t}\le -\varepsilon ,$$

(10)

$$\left({X}_{t,s}-1\right){A}_{s}\le {\phi }_{s}{\mathbf{z}}_{s}\left({\mathbf{w}}_{t}-{\mathbf{w}}_{s}\right)+\left({Q}_{t}-{Q}_{s}\right),$$

(11)

$$0\le {R}_{t}\le 1,$$

(12)

$$0\le {u}_{t}\le 1-\varepsilon ,$$

(13)

$$\varepsilon \le {\phi }_{t}\le 1,$$

(14)

$$ {X}_{t,s} {\in} {0,1}, $$

(15)

where ${X}_{t,s}$ are binary (0–1) variables, $\varepsilon$ is a small positive number, and ${A}_{t}$ is a large, fixed number. In this study, we set $\varepsilon ={10}^{-6}$ and ${A}_{t}={{\varvec{z}}}_{t} {{\varvec{w}}}_{t}+3$. The objective function is set to zero because we are only interested in whether a feasible solution to the equalities exists. $\mathbf{q}$ goods and services are weakly separable from the remaining goods and services if there exists a feasible solution to problems (7)–(15). On the other hand, if there is no feasible solution to the problem, dataset D violates weak separability.