In econometrics

Econometrics

Econometrics has been defined as "the application of mathematics and statistical methods to economic data" and described as the branch of economics "that aims to give empirical content to economic relations." More precisely, it is "the quantitative analysis of actual economic phenomena based on...

, a dynamic factor (also known as a diffusion index) is a series which measures the co-movement of many time series

Time series

In statistics, signal processing, econometrics and mathematical finance, a time series is a sequence of data points, measured typically at successive times spaced at uniform time intervals. Examples of time series are the daily closing value of the Dow Jones index or the annual flow volume of the...

. It is used in certain macroeconomic models.

Formally


where is the vector

Vector space

A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

 of lagged factors of the variables

Variable (mathematics)

In mathematics, a variable is a value that may change within the scope of a given problem or set of operations. In contrast, a constant is a value that remains unchanged, though often unknown or undetermined. The concepts of constants and variables are fundamental to many areas of mathematics and...

 in the matrix

Matrix (mathematics)

In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...

  (T is the number of observations and N is the number of variables), are the factor loadings, and is the factor error

Error

The word error entails different meanings and usages relative to how it is conceptually applied. The concrete meaning of the Latin word "error" is "wandering" or "straying". Unlike an illusion, an error or a mistake can sometimes be dispelled through knowledge...

.

History

Diffusion indexes were originally designed to help identify business cycle turning points.

Example

A diffusion index of monthly employment levels across industries measures the degree to which a growth in employment levels in a population is made up of growth in all industries versus sharp growth in just a few industries. In one published data series on that design, the diffusion index is computed from a panel of discrete time series by assigning a value of 0 to an observation if it is lower than its analog in the previous month, 50 if it is at the same level, and 100 if it has increased. The average of these component values for a given period over the time period is a diffusion index. Relative to the equation above, the underlying factors are drawn from the values {0, 50, 100} based on employment changes, and the diffusion index works out to be the percentage of these employment counts that increased in the previous month. Some researchers have reported that a diffusion index of monthly manufacturing-sector employment is a leading indicator of turning points in the business cycle.

Literature

• Forni, Mario & Lippi, Marco, 2001. "The Generalized Dynamic Factor Model: Representation Theory", Econometric Theory, vol. 17(6), pages 1113-41.
• Getz, Patricia M. and Mark Ulmer. "Diffusion indexes: an economic barometer", Monthly Labor Review
  Monthly Labor Review
  The Monthly Labor Review is published by the U.S. Bureau of Labor Statistics. Issues often focus on a particular topic. Researchers outside of the BLS are welcome to submit articles.- History :...
  
  , April 1990, Vol. 113, No. 4, pp. 13-22.
• Stock, James H & Watson, Mark W, 2002. "Macroeconomic Forecasting Using Diffusion Indexes", Journal of Business & Economic Statistics, vol. 20(2), pages 147-62.