Distributed lag

In statistics and econometrics, a distributed lag model is a model for time series data in which a regression-like equation is used to predict current values of a dependent variable based on both the current values of an explanatory variable and the lagged (past period) values of this explanatory variable.^[1]^[2].

Unstructured estimation

The simplest way to estimate an equation with a distributed lag is by ordinary least squares, putting a pre-determined number of lagged values of the independent variable on the right-hand side of the regression equation, without imposing any structure on the relationship of the values of the lagged-variable coefficients to each other. A problem that usually arises with this approach is multicollinearity: the various lagged values of the independent variable are so highly correlated with each other that their coefficients can be estimated only very imprecisely. Therefore researchers often impose structure on the shape of the distributed lag (the relation of the lagged-variable coefficients to each other). Such a structure is called a distributed lag model.

Distributed lag models come in two types: finite and infinite. Infinite distributed lags allow the value of the independent variable at a particular time to influence the dependent variable infinitely far into the future, or to put it another way, they allow the current value of the dependent variable to be influenced by values of the independent variable that occurred infinitely long ago; but beyond some lag length the effects taper off toward zero. Finite distributed lags allow for the independent variable at a particular time to influence the dependent variable for only a finite number of periods.

Finite distributed lags

The most important finite distributed lag model is the Almon lag^[3]. This model allows the data to determine the shape of the lag structure, but the researcher must specify the maximum lag length; an incorrectly specified maximum lag length can distort the shape of the estimated lag structure as well as the cumulative effect of the independent variable.

Infinite distributed lags

The most common type of infinite distributed lag model is the geometric lag, also known as the Koyck lag. In this lag structure, the weights (magnitudes of influence) of the lagged independent variable values decline exponentially with the length of the lag; while the shape of the lag structure is thus fully imposed by the choice of this technique, the rate of decline as well as the overall magnitude of effect are determined by the data. Specification of the regression equation is very straightforward: one includes as explanators (right-hand side variables in the regression) the one-period-lagged value of the dependent variable and the current value of the independent variable.

Other infinite distributed lag models have been proposed to allow the data to determine the shape of the lag structure: the rational lag^[4], the gamma lag ^[5], the polynomial inverse lag^[6]^[7], and the geometric combination lag^[8].

References

^ Jeff B. Cromwell, et. al., 1994. Multivariate Tests For Time Series Models. SAGE Publications, Inc. ISBN 0-8039-5440-9
^ Judge, George, et al., 1980. The Theory and Practice of Econometrics. Wiley Publ.
^ Almon, Shirley, "The distributed lag between capital appropriations and net expenditures," Econometrica 33, 1965, 178-196.
^ Jorgenson, Dale W., "Rational distributed lag functions," Econometrica 34, 1966, 135-149.
^ Schmidt, Peter, "A modification of the Almon distributed lag," Journal of the American Statistical Association 69, 1974, 679-681.
^ Mitchell, Douglas W., and Speaker, Paul J., "A simple, flexible distributed lag technique: the polynomial inverse lag," Journal of Econometrics 31, 1986, 329-340.
^ Gelles, Gregory M., and Mitchell, Douglas W., "An approximation theorem for the polynomial inverse lag," Economics Letters 30, 1989, 129-132.
^ Speaker, Paul J., Mitchell, Douglas W., and Gelles, Gregory M., "Geometric combination lags as flexible infinite distributed lag estimators," Journal of Economic Dynamics and Control 13, 1989, 171-185.

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[1] Jeff B. Cromwell, et. al., 1994. Multivariate Tests For Time Series Models. SAGE Publications, Inc. ISBN 0-8039-5440-9

[2] Judge, George, et al., 1980. The Theory and Practice of Econometrics. Wiley Publ.

[3] Almon, Shirley, "The distributed lag between capital appropriations and net expenditures," Econometrica 33, 1965, 178-196.

[4] Jorgenson, Dale W., "Rational distributed lag functions," Econometrica 34, 1966, 135-149.

[5] Schmidt, Peter, "A modification of the Almon distributed lag," Journal of the American Statistical Association 69, 1974, 679-681.

[6] Mitchell, Douglas W., and Speaker, Paul J., "A simple, flexible distributed lag technique: the polynomial inverse lag," Journal of Econometrics 31, 1986, 329-340.

[7] Gelles, Gregory M., and Mitchell, Douglas W., "An approximation theorem for the polynomial inverse lag," Economics Letters 30, 1989, 129-132.

[8] Speaker, Paul J., Mitchell, Douglas W., and Gelles, Gregory M., "Geometric combination lags as flexible infinite distributed lag estimators," Journal of Economic Dynamics and Control 13, 1989, 171-185.

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