As previously argued, the correlation between included and omitted regressors generally causes inconsistency of standard estimators for count data models. Non-linear instrumental variables estimation of an exponential model under conditional moment restrictions is one of the proposed remedies. This approach is extended here by fully exploiting the model assumptions and thereby improving efficiency of the resulting estimator. Empirical likelihood in particular has favourable properties in this setting compared with the two-step generalized method of moments, as demonstrated in a Monte Carlo experiment. The proposed method is applied to the estimation of a cigarette demand function.