2.6.1 Translog cost function
When the selected functional form is Translog and the duality theorem is exploited in order to take advantage of the facilitating characteristics of the cost function, the estimation procedure reduces to a system of linear equations. These may be subject to the restrictions imposed by the assumptions of homotheticity and, in certain cases, linear homogeneity and separability. Indeed, in order to obtain an estimate of the AES between two inputs, one needs to estimate the parameters of the demand functions.
The most common estimation technique involves appending a stochastic additive error to each cost share equation as follows:
Si = bi+ biiln(PK)+
n
X
j=1
bi jln(Pj)+ bKq+ bittln(q)+ i with i, j= 1, ..., n (2.29)
The disturbance terms represent both random errors in the cost-minimizing behaviour and random influence of omitted variables. Since the sum of the four share equations equals one, the disturbance covariance matrix is singular and non-diagonal. The approach used in the literature to overcome this problem is to drop arbitrarily one equation from the system so that the resulting vector of disturbances is composed of identically and independent normally distributed error terms with mean zero and a non-singular covariance matrix.
This allows for the correlation between contemporaneous errors of different equations
to be nonzero. In order to obtain consistent and asymptotically efficient estimates, the chosen estimation technique must be invariant to the equation deleted. Two possible and asymptotically equivalent procedures have been employed in the literature: an iterative24 version of the Seemingly Unrelated Equations (ISUE) regression by Zellner (1962) and the Full Information Maximum Likelihood (FIML) estimation procedure. In all Translog studies, ISUE or FIML estimators are applied with the price homogeneity and symmetry parameter restrictions imposed (see Table 2.1, column Ec).
Few papers included the cost function in the system of estimated equation.25 This allows the authors to test for constant returns and Hicks neutrality. However, it implies the estimation of a cost function composed of a large number of coefficients which may lead to multicollinearity. As a consequence, standard errors may be large and coefficients difficult to interpret.
Eight out of the thirty-one studies which used the Translog function worked on time series-cross sectional data of pooled countries or sectors (see Table 2.1, column Str). Different models were adopted: the basic one estimates a system of equations where the parameters are assumed to be the same for each country (sector); an intermediate model where the bi parameters are country-specific (sector-specific); the more complex one where all the parameters are allowed to vary across countries (sectors), and thus it implies estimating a system of equation for each country considered. Griffin and Gregory (1976) compared the goodness of fit of the three models using the R2 statistic and found the second to explain better the data. However, they argue that, as long as the parameter estimates of the slopes do not vary noticeably, the first model should be preferred because in this way the inter-country mean variation is not eliminated. Pindyck and Rotemberg (1983) compared the same models through a χ2 test and found that different intercepts across countries should be allowed. Finally, Fuss (1977), Ozatalay et al. (1979), Hesse and Tarkka (1986), Iqbal (1986), Garofalo and Malhotra (1988), and Roy et al. (2006) introduced country dummy variables in the cost shares and tested for their significance.
Three special cases are represented by Arnberg and Bjorner (2007), Haller and Hyland (2014), and Khiabani and Hasani (2010) who, in their micro-estimation, used a fixed effect estimator to account for the panel nature of the data and Christopoulos (2000) who specified a dynamic model based on first differences after testing for unit roots.
24Until the estimated coefficients and residuals covariance matrix converge.
25Norsworthy and Malmquist (1983), Nguyen and Streitwieser (1999), Burki and Khan (2004), Khiabani and Hasani (2010), Haller and Hyland (2014), Zha and Ding (2014), Zha and Zhou (2014).
2.6.2 CES production function
Five different estimation techniques have been employed with a CES model and three of them involve the resolution of a cost minimization problem.
According to Prywes (1986) and Chang (1994), who used a three-level nested structure, a cost minimizing procedure needs to be employed at each of the three nests (K, E; KE, L;
KEL, M) starting with the inner one. For instance, the inner nest minimization can be specified as follows:
where r and f are the rental cost of capital and the price of energy, and qKE is the intermediate output. Solving the minimization problem, a FOC is derived:
K As they assumed exogenous prices, in the next step they equated the unit cost of qKE, that is PKE, to the Lagrangian multiplier or shadow price. Finally, adding a disturbance term, they estimated the logarithm of equation (2.32). This procedure was repeated in the two upper levels of production using each time, as one of the inputs, the intermediate input computed from the estimated coefficients of the previous level. The elasticities of substitution are, therefore, represented by the coefficients attached to the logarithm of the ratio between prices and the share parameters can be derived from the constant term. This method can be used also with increasing or decreasing returns to scale but with the limit that it would not be possible to disentangle the share parameter from the scale parameter.
Differently, Kemfert (1998) and Koesler and Schymura (2015) employed a direct method by estimating three non-linear equations for each nested structure selected.
Recently, van der Werf (2008) followed an indirect method closely related to the first one presented. He minimized a cost function at each nest and found input demand equations.
However, since he considered factor-specific technology parameters, his conditional input demand equations were under-identified. Hence, he took first differences and after few algebraic steps, he ended up with four equations for each nested structure to which he added an error term. He, then, employed a fixed effect estimator where the within variation was due to dummy variables constructed for each country-industry combination.
Baccianti (2013), who also had to face the problem of under-identification, proposed a new approach based on a panel normalization procedure to identify the input demand equations for twenty-seven countries. He estimated the normalized equations using a generalized method of moment estimator with a variance-covariance matrix robust to heteroskedasticity and autocorrelation.