# **5. Demand system estimation**

The challenge of estimating demand functions is to maintain empirical applications coherent with microeconomic theory, especially with respect to assumptions and restrictions from demand and consumer behavior theories. Since the seminal work of Stone in 1954, who developed the Linear Expenditure System (LES), the evolution of empirical demand models progressed significantly. The highest coherence with demand theory has been achieved so far by considering the non-linearity of the Engel’s curves and including in the demand function a quadratic term of the relationship between total expenditure and prices. The quadratic model that follows the demand and consumer theories most closely is the Banks et al. (1997) QUAIDS, which is employed in this paper.

Although QUAIDS can effectively deal with the issues related to the coherence with microeconomic theory, a remaining issue is the potential endogeneity of prices and expenditure when using survey data and unit values as proxies for market prices. That is, expenditure and/or unit values can be correlated with the errors, resulting in inconsistent estimations and in biased shapes of the Engel curves. This endogeneity can be caused by a variety of factors, such as measurement errors, unobserved commodity quality and unobserved prices and expenditure shocks (Blundell and Robin, 1999).

In Section 3, we discussed the two main approaches that are used in the literature to address this endogeneity bias when market prices are not available – i.e. Deatons’ and CW’s approaches. Both approaches assume that the variation in unit values that cannot be explained by the household characteristics is caused by quality differences, correcting the unit values. Moreover, Deaton's method necessitates weak separability restrictions, which overstate quantity responses over quality responses, and it uses the geographical clustering structure of data assuming that households in the same cluster share the same market price. The CW's approach has the advantage of being easier to implement, but it does not deal with measurement error problems.

However, if there is correlation between prices/expenditure and the error terms, such correlation can be accounted for with instrumental variable and augmented regression techniques (Hausman 1978; Holly and Sargan 1982). A novel approach based on instrumental variables (IV) methods for reducing the endogeneity bias in demand systems has been developed by Lecocq and Robin (2015). This method shares with Deaton's and CW's methods the implicit assumption that price changes have no impact on quality, but it has significant advantages: i) it relies on fewer assumptions, ii) it does not require a clustered structure of the data, iii) it is computationally efficient, and iv) can potentially rely exclusively on the data available in the household surveys. For these reasons, in this paper we apply the Lecocq and Robin’s method.

In order to estimate consistent expenditure, own- and cross-price elasticities we estimate the following demand system in vector notation, which is the Banks et al. (1997) quadratic extension of the Deaton and Muellbauer’s (1980) AIDS model: where _a_ and _b_ are non-linear price aggregator functions defined as: _w_<sup>_h_</sup> _i_ is the share of the expenditure of good _i_ = 1, . . . ,10 (see table 3 for a list and description of the goods) on the total expenditure _x_<sup>_h_</sup> of household _h_ = 1, . . . , _H_ ; _p_ is a vector of prices; _u_ is the error term; and α _,_ β _,_ γ _,_ θ are the parameters to be estimated. These parameters satisfy the theoretical restrictions of additivity, homogeneity, symmetry, which are defined, respectively: 15