Testing intra-daily seasonality using Maximum Entropy Density

The objective of this paper is to test for intra-daily seasonality using Maximum Entropy Density (MED). Specifically, this paper attempts to investigate seasonal patterns over weekdays and through the hours of a given trading day. The MED estimation is essentially a data driven technique which produces a density function. A comparison of MEDs across different time segments of the data allows one to test for the existence of intra-daily seasonality. More importantly with regard to comparisons, an MED presents a richer source of information compared to a singular metric such as the mean or variance. In other words, such a comparison enables one to measure different facets of a distribution. In particular, different moments of the distribution. Repeated patterns in one or more moments between time segments over a period implies the presence of intra-daily seasonality. MED estimation techniques assume that a random variable is independent and identically distributed (iid). This condition ensures that consistent estimators are produced. However, return data has a correlation structure and as such the observations are not iid. It is proposed that this correlation structure be filtered out prior to the construction of the MED using an ARMA(1,1) GARCH(1,1) model. The overall methodology of this paper is as follows. A data series consisting of returns is segmented into weekdays. For example, all the Mondays are extracted from the data. Each Monday segment is referred to as a block. Note that there is a time discontinuity between two consecutive Monday blocks. This has important implications with regard to filtering the correlation structure. An ARMA(1,1) GARCH(1,1) model is only implemented at the block level since time is continuous within the block. The residuals are standardised and checked for autocorrelation. A MED is computed on the residuals of each block. The first four moment constraints are used for the MED construction. As such, there are four MED parameters i.e. λi where i ∈ 1, 2, 3, 4. Subsequently, the mean values for each λi are computed over all blocks corresponding to a given weekday. These values represent the final MED parameters for a given weekday. For example, the λi values are averaged over all Monday blocks to get the overall MED parameters for the Monday segment. Testing for intra-daily seasonality is done in two parts. Firstly in order to verify the structure of the resulting MED, tests are conducted to assess if the resulting mean values are significantly different from zero. Secondly in order to check for intra-daily seasonality, tests are conducted to assess if the resulting mean values are significantly different across the weekdays. Significant differences in the mean values of λi across weekdays indicates that the distribution of returns changes during the week. This pattern over a period of time corresponds to intradaily seasonality. This process repeated to check for intra-daily seasonality across different time intervals over a trading day. The results indicate that one of the mean values of the MED parameters for Wednesday is significantly different from the rest of the weekdays. Similarly, one of the mean values of the MED parameters is significant for the 12p.m. 2p.m. interval.