In this paper we consider daily financial data from various sources (stock market indices, foreign exchange rates and bonds) and analyze their multi-scaling properties by estimating the parameters of a Markov-switching multifractal model (MSM) with Lognormal volatility components. In order to see how well estimated models capture the temporal dependency of the empirical data, we estimate and compare (generalized) Hurst exponents for both empirical data and simulated MSM models. In general, the Lognormal MSM models generate "apparent" long memory in good agreement with empirical scaling provided one uses sufficiently many volatility components. In comparison with a Binomial MSM specification [7], results are almost identical. This suggests that a parsimonious discrete specification is flexible enough and the gain from adopting the continuous Lognormal distribution is very limited.