In [[Hierarchical Time Series|hierarchical time series]], reconciliation allows us to **perform [[Forecasting|forecasting]] at different levels of aggregation to then adjust them** so that the aggregated sums fit the top-level forecast. There advantages to forecasting the top level is that general trends will be better captured, while the advantages of low level methods is that different fine-grained series can be fed other regressors to improve its performance. ![[Hierarchical Time Series.png]] Forecast reconciliation attempts to find a way to weight low-level forecast in the aggregations so that the total final forecast is built using the information from the underlying forecasts. There are two main approaches: - **Bottom-up**, which overthrows the top level forecasts using the aggregation of the lower levels; - and **forecast reconciliation**, which uses regression methods like ordinary least squares (OLS) to find a balance between both informations. ![[Time Series OLS.png]] In this example, $S$ is the summation matrix, that aggregates each of the base forecasts to each respective rows. When using OLS, our target is to **find the matrix $P$ to weight each of the base forecasts correctly**. Optimal forecast reconciliation relies on the notion that **using bottom-up methods will rarely be optimal**: counter intuitively, having a good precision at low level does not imply having a good precision when aggregating; error does not have to be symmetrical and can accumulate.