The literature on pooling crosssectional and time series data^{5} suggests that the random disturbance e_{it} may be composed of two or more additive components, be crosssectionally heteroscedastic and be timewise autoregressive.
This clearly happened in our case. We shall therefore make the following assumptions
e_{it} = a_{i} + e*_{it} 
(a) 
where a_{i} and e*_{it} are normally and independently distributed random variables of which we assume
E(a_{i}) = 0, E(a_{i} ^{2}) = s_{a}^{2 },^{ }E(a_{i} a_{j}) = 0 for j ¹ i 
(b) 
e*_{it} = l_{i}^{(1)} e*_{it1} + l_{i}^{(2)} e*_{it2} + e**_{it } 
(c) 

We further assume, that the random disturbances e**_{it }are normally distributed and crosssectionally as well as timewise independent. We do, however, assume that they are crosssectionally heteroscedastic, so that for each i
(e**_{it}) = 0, E(e**_{it }^{2}) = s^{2}_{i}, E(e**_{it }e**_{js }) = 0 for s ¹ t 
(d) 

It It is to be noted that the autoregression coefficients l_{i}^{(1)} and l_{i}^{(2) }are assumed to be specific to each sector. As a result, the random disturbance would be heteroscedastic, even if e**_{it }was not.
Assuming
l_{i}^{(1)} +
l_{i}^{(2)} < 1, the variance of e*_{it} will be
var(e*_{it }) = s^{2}_{i }/[1  (l_{i}^{(1)} + l_{i}^{(2)}) ] 


It is to be expected, therefore, that the removal of serial correlation in sectors will considerably diminish the crosssectional heteroscedasticity. In estimating our production functions we have chosen the standard procedure of removing the sectoral error components by introducing sectoral dummy variables — i. e. allowing for differentiated intercepts of regression lines — and removing the autocorrelation and heteroscedasticity by transforming variables. 