2 Constructing a composite index of leading indicators[2]
The idea of using leading economic indicators in business cycle forecasting was originally developed by (Mitchell and Burns 1938) at the National Bureau of Economic Research (NBER). A composite index of leading indicators is the (generally) weighted average of several component series. Composite indexes are constructed because they tend to be more reliable as a cyclical indicator than any of its components taken individually. This is partly because much of the independent measurement error and other noise in the component series is smoothed out in an index.
The algebraic construction of a composite index of leading indicators involves two main steps: (i) standardisation and weighting of the individual component series, and (ii) standardisation and cumulation of the composite value. Both steps are explained in more detail in this section, while the selection of variables to include in an index of New Zealand employment is discussed in section 3.
2.1 Standardisation and weighting of individual component series
The first step in the construction of a (quarterly) composite index of leading indicators is to calculate quarter-to-quarter symmetrical percentage changes, 
, for each individual component j of the composite index
(1) 
where 
denotes component j at time t. For series that contain zero or negative values, or that are already in index or percentage form, the following formula is used instead
(2) 
For series that are a difference of two series (such as the interest spread) or a ratio, equation (3) is used
(3) 
The next step is to standardise the transformed series to prevent a volatile component from dominating changes in the composite index. Each transformed component variable is standardised by dividing it by its historical average without regard to sign, i.e.,
(4) 
where T denotes the number of observations.
The transformed and standardised series, 
, are then combined into a composite variable using equation (5)
(5) 
where 
is the raw composite value and 
the individual weight of each series normalised to sum to one. One way to aggregate the components is to assume equal weights. However, a number of alternative weighting schemes are available to better reflect the relationship and the importance of each component with the reference series, employment in our case. (Choosing the weights is discussed further in the next section.)
2.2 Standardisation and cumulation of the composite value
The purpose of the second step, standardisation and cumulation of the composite value, is to transform the raw composite value so that it has the same historical average (without regard to the sign) as the reference series. The standardised composite value, 
, is obtained as
(6) 
where 
is the symmetrical percentage change of the reference series (employment). The standardised composite value is then transformed into an index using the formula in equation (7)
(7) 
where 
is the standardised composite index.
Notes
- [2]The section partly follows (Zarnowitz and Boschan 1975). See also (Green and Beckman 1993).
