econometrics Econometrics is an application of statistical methods to economic data in order to give empirical content to economic relationships. M. Hashem Pesaran (1987). "Econometrics", '' The New Palgrave: A Dictionary of Economics'', v. 2, p. 8 p. 8 ...

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

time series In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. ...

. It is used in certain

macroeconomic model A macroeconomic model is an analytical tool designed to describe the operation of the problems of economy of a country or a region. These models are usually designed to examine the comparative statics and dynamics of aggregate quantities such a ...

s. A diffusion index is intended to indicate * the changes of the fraction of economic data

which increase or decrease over the selected time interval, * an increase or decrease in future economic activity, * provide some correlation to the business sentiment of companies. Formally :

X_=\Lambda_F_+e_,

where

F_=(f^_,\dots,f^_)

is the

vector Vector most often refers to: * Euclidean vector, a quantity with a magnitude and a direction * Disease vector, an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematics a ...

of lagged factors of the

variables Variable may refer to: Computer science * Variable (computer science), a symbolic name associated with a value and whose associated value may be changed Mathematics * Variable (mathematics), a symbol that represents a quantity in a mathemat ...

in the

T \times N

matrix Matrix (: matrices or matrixes) or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the m ...

X_

(T is the number of observations and N is the number of variables),

\Lambda_

are the factor loadings, and

e_

is the factor

error An error (from the Latin , meaning 'to wander'Oxford English Dictionary, s.v. “error (n.), Etymology,” September 2023, .) is an inaccurate or incorrect action, thought, or judgement. In statistics, "error" refers to the difference between t ...

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

f_t

are drawn from the values based on employment changes, and the diffusion index

X_{t}

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.Getz and Ulmer, pp 13-22.

References

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 The ''Monthly Labor Review'' (''MLR'') is published by the U.S. Bureau of Labor Statistics The Bureau of Labor Statistics (BLS) is a unit of the United States Department of Labor. It is the principal fact-finding agency for the government o ...

'', 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. Time series