ybarT

ybarT is a notation used in statistics and related fields to denote the mean of a variable y over a subset or time horizon indexed by T. It expresses an average value computed from the data points encompassed by T and is read as “y-bar sub T.”

Formally, for a finite index set T with elements t, the mean is defined as ȳ_T = (1/|T|) ∑_{t∈T} y_t. If T = {1, ..., T}, this reduces to the conventional sample mean ȳ_T = (1/T) ∑_{t=1}^T y_t. The exact interpretation of ȳ_T depends on how T is specified in a given context.

In time series and econometrics, ȳ_T is commonly used to describe the average value over a rolling window or a specified time span. For example, a moving average might be written as ȳ_{t−k+1:t}, indicating the mean of the most recent k observations. In cross-sectional data, ȳ_T can denote the average outcome across a group or category indexed by T.

Notes and caveats: the overline indicates averaging, but the precise meaning of T must be defined in each analysis. ȳ_T is distinct from conditional expectation E[y|T] in probability theory, which denotes the expected value of y given another event or sigma-algebra, not merely an average over a subset.

Related concepts include the sample mean, population mean, and moving averages, as well as general aggregation techniques used to summarize data across subsets. In software and statistical practice, ȳ_T is computed by aggregating y over the indices in T.