Horizontal and vertical Aggregation Bias of time series with empirical study

Aggregation bias is defined (Theil 1954) as the difference between the macro parameter and the average of the micro parameters. In this article, a new definition of time series aggregation is introduced. Imperially, a new definition of aggregation bias is introduced. Using the new definition, the time series are aggregated in two different methods, horizontal and vertical. Bias resulting from this aggregation is analyzed in both cases. Horizontal aggregation revealed unbiasedness in the slope and intercept of the trend of the series. Vertical aggregation revealed bias in estimating both parameters of the trend of the series depending on the length of the aggregation period. A new theorem "Samar theorem" is introduced to assess bias in both parameters. Proof of the Samar theorem is annexed to the paper. An Empirical study containing data of a number of time series are analyzed in the frame of the new theorem. The data is the weekly income of some Egyptian farmers working on the same farm collected through an Egyptian agrarian year. Samar theorem introduce rules to compute bias in the slope and the intercept of the trend of the series. It also shows the relation between the two biases. In addition, the effect of the length of aggregation period on the amount of bias is discussed.