EL CUADRO DE MANDO DEL RECURSO HUMANO

We use the conditional approach put forward by Browning and Meghir (1991) to estimate a demand system of eight (presumably) adult goods in which we have as the “conditioning set” the following variables. First, male and female participation d u m m ies.S eco n d , household composition variables (number of children in different ages, number of adults and number of elderly people in the household). Finally, we include other demographic variables, i.e., education of the head of the household, labour status of the head of the household, age and age squared of both the head and his partner, number of earners in the household, tenure status of the house, and quarterly and yearly dummies. Other variables used are prices and total expenditure.

As pointed out by Browning and Meghir (1991), with this kind of setting an econometric issue arises: the need to account for the possible endogeneity of some of the right hand side variables in the (conditional) system. The potentially

We control for sample selection in both cases and check that the estimates are similar, regardless of the sample we use.

The order we use to transform the data has been decided taking into account two aspects. On one hand, the higher the order the smaller the percentage of zeros remaining. On the other hand, we need to keep sufficient periods to transform (e.g., take first differences) and instrument the potentially endogenous variables of the model.

Although the survey does not provide information on hours, we can identify whether the individual in the household works full or part time. In the context of the Spanish labour market, these dummies are sufficient indicators of the hours people work.

endogenous variables are male and female labour market variables and total expenditure. In order to identify the parameters of the model they assume that conditional on total expenditure and the labour variables, asset income and education do not enter the demand. In a panel data context, however, we also have some other instruments (mainly lagged variables) in order to achieve identification. Identification is achieved treating all demographics as exogenous and instrumenting total expenditure (and total expenditure squared) with total income and/or lags of total expenditure.

The demand system (2.16) we estimate can be written as ( 2 . 2 2 ) W j , = / ( p j, , Xj, , Zj, | 5 ) + Ej,

where Wht is the vector of commodity shares for household h in period t. Given (2 .2 2 ), a consistent instrumental variables estimator for the parameters Ô can be

obtained by minimising E'(/(S) E , where e is the vector of error terms, = Z (Z 'Z )“’Z'and Z is the matrix of instruments. The vector of parameters is obtained through an iterative procedure that minimises

(2.23) ® - f { p „ ,x „ ,z „|5 )]

Finally, the asymptotic variance-covariance matrix of this estimator is (2.24)

V(8 ) = [X G (I )GZ]"‘ X G ( I )£2(/ ® Pz )GX[X' G(I<S>P^ )GZ]"‘

where Q. is the error covariance matrix and G is the stacked matrix of derivatives in (2.23). In estimating the variance-covariance matrix we evaluate G at the estimated parameter vector and we use the estimated residual vector in order to calculate

In the demand system we estimate we impose homogeneity, adding-up

and symmetry/^ Moreover, the price index used to deflate total expenditure depends on estimated parameters and is common across all equations, giving rise to an additional set of within and cross-equation restrictions. Following Browning and Meghir (1991) and considering the size of the data set, we have used a two- steps strategy to estimate the demand system. We first estimate the model without imposing symmetry but imposing the cross-equations restrictions, i.e. that we have the same a(p,z) and b(p) deflating total expenditure in every equation, and the within-equation restriction of homogeneity. Adding up is imposed by letting out of the estimated system the last equation. Then, we estimate the system equation by equation and compute the price index a{p) and b(p) using the estimated parameters and re-estimate the model.^^ We iterate until this process converges.^^ This procedure provides consistent parameter estimates for the model without imposing symmetry. The covariance matrix for this IV estimator is given in (2.24) and takes into account the cross-equation restrictions. The second step is then to estimate again the system imposing symmetry using a minimum distance procedure. The symmetry restricted parameters are obtained by minimising where ô is the unrestricted parameter

c*

vector, V is the variance-covariance matrix of o and K is the restrictions matrix. The minimised value of follows a Chi-squared distribution with degrees of freedom equal to the number of restrictions. An estimate of the covariance matrix of the restricted estimator is .

As pointed out by Arellano and Bond (1991), an estimator that uses lags as instruments, under the assumption of white noise errors, would loose its

W e test both for symmetry and homogeneity.

As pointed out in Browning and Meghir (1991), if the value o f a(p,z) and b(p) were known for each household, then we would have a linear estimation problem. Thus as a first approximation to a(p,z) we compute household specific Stone price indices, defined by logP^^ = S log where Wiht is the budget share of good i in period t for household h.

We observe important differences in the parameters obtained in the first and last iterations indicating that using the Stone price index approximation with no iteration might not be acceptable. Pashardes (1993) points out that the results using the Stone approximation (without iteration) can result in biased parameter estimates, particularly when the AIDS is applied to microdata.

consistency if the errors were serially correlated. Therefore, we have carried out some tests for the validity of the instruments, i.e., tests of the lack of serial correlation. In this chapter we consider three tests: a direct test on the first order serial correlation coefficient of the residuals, a Sargan test of over-identifying restrictions and a Hausman specification test, Hausman (1978). In appendix A, we present a formal description of the test for first order serial correlation, while the other two tests are presented in Arellano and Bond (1991).

In the above stochastic specification (equation 2.24) the vector of error terms, denoted by e.^^, can be decomposed as 6 , where is

assumed to have zero expectation and to be independently distributed over time and across households, whereas p is the unobservable individual effect.^"^ Even with these assumptions, a problem that may arise in the specification is the correlation between variables and individual effects.^^ Under infrequency of purchase, total expenditure is correlated with the mixed error. If the purchase policy is time invariant then the fact of having infrequency of purchase can be considered as a fixed effect, and therefore total expenditure will be correlated with the time invariant part of the error term (see Meghir and Robin, 1992). When dealing with data that suffers from infrequency it is crucial to analyse the characteristics of the unobserved individual heterogeneity. Investigating the nature of our data we can detect that due to infrequency of purchase we might have a measurement error problem both in the RHS (the commodity shares) and in the LHS (total expenditure). Therefore total expenditure and the error term are currently correlated via the individual effect, but total expenditure can be correlated with the error term due to other reasons. If the only reason for the correlation were the presence of individual unobserved heterogeneity then within- groups estimation would be sufficient to get consistent estimates. If that is not the case, then we should still instrument total expenditure (apart from removing the

The of independence over time can be easily relaxed, as we do in the empirical work. In any case, for deciding the instrument set, we test for the presence of autocorrelation (see Arellano and Bond, 1991).

The assumption of independence between individual effects and regressors has little or no interest in economic models.

individual effects). To check to what extent the infrequency of purchase is leading to correlation between total expenditure and the error term we propose the following test. Using the information provided in our data set, we calculate empirical purchase probabilities approximating them as the number of times a household has a positive purchase over eight (as eight is the number of quarters a household is tracked in the survey). With these estimates at hand, we follow Meghir and Robin (1992) in order to correct the data to avoid the measurement error problem. The weighting scheme (using the empirical probabilities) eliminates the measurement error problem by accounting for its precise structure. With these new data, we re-estimate the model and check if the results are similar. If they are, then our assumption about the nature of the individual effects is correct (under a time invariant purchase policy, which is not a strong assumption, since the length of the observed period for each household is eight quarters). But even so, total expenditure might still be correlated with the error term in which case we still need to instrument this variable (by using total income and lags of total expenditure). Finally, there are several ways to control for the individual effects: we can model them as proposed by Mundlak (1978) or Chamberlain (1982) or take some transformation to remove them (see, for instance, Anderson and Hsiao, 1980 or Arellano and Dover, 1995).^^