In the paper we provide two sets of computationally convenient specification tests for the multinomial logit model. The first test is an application of the \textit{J. Hausman} [ibid. 46, 1251-1271 (1978; Zbl 0397.62043)] specification test procedure. The basic idea for the test here is to test the reverse implication of the independence from irrelevant alternatives property. The test statistic is easy to compute since it only requires computation of a quadratic form which involves the difference of the parameter estimates and the differences of the estimated covariance matrices. The second set of specification tests that we propose is based on more classical test procedures. We consider a generalization of the multinomial logit model which is called the nested logit model. Since the multinomial logit model is a special case of the more general model when a given parameter equals one, classical test procedures such as the Wald, likelihood ratio, and Lagrange multiplier tests can be used. The two sets of specification test procedures are then compared for an example where exact and approximate comparisons are possible.