Abstract: In order to circumvent the heavy-tailed problem in estimating the conditional autoregressive range model(CARR), the lognormal distribution is considered. Under conditions that the innovations have a finite 12th moment, which allows the model to have a unit root,we show that the quasi-maximum likelihood estimator which uses the lognormal distribution as the likelihood is locally consistent and asymptotically normal by the properties of the M-estimator and functional central limit theorem for martingale.Meanwhile the efficiency of the estimator can also be improved by the heavier tail of lognormal distribution than the exponential likelihood methods currently used in the literature.

Key words: CARR, M-estimator, heavy-tail, lognormal distribution