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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 parameters associated with distributed lags is by ordinary least squares, assuming a fixed maximum lag $p$ , assuming independent and identically distributed errors, and imposing no structure on the relationship of the lagged covariates with each other. However, multicollinearity among the lagged covariates often arises, leading to poor variance estimation; by applying constraints on the shape of the lag structure. For example, the parameters might be specified such that they lie on a function of their lag numbers, examples of such functions are simple polynomials or splines.

Structured estimation

Structured 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. If the estimated coefficient of the lagged dependent variable is $\lambda$ and the estimated coefficient of the current value of the independent variable is b, then the estimated short-run (same-period) effect of a unit change in the independent variable is b while the long-run (cumulative) effect of a sustained unit change in the independent variable is $b/(1-\lambda )$ .

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.

[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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 ==Unstructured estimation==
-The simplest way to estimate parameters associated with distributed lags is by [[ordinary least squares]], assuming a fixed maximum lag <math> p </math>, [[independent and identically distributed]] errors and imposing no structure on the relationship of the lagged covariates with each other.  However, [[multicollinearity]] among the lagged covariates often arises, leading to poor variance estimation; by applying constraints on the shape of the lag structure.  For example, the parameters might be specified such that they lie on a function of their lag numbers, examples of such functions are simple polynomials or [[splines]].
+The simplest way to estimate parameters associated with distributed lags is by [[ordinary least squares]], assuming a fixed maximum lag <math> p </math>, assuming [[independent and identically distributed]] errors, and imposing no structure on the relationship of the lagged covariates with each other.  However, [[multicollinearity]] among the lagged covariates often arises, leading to poor variance estimation; by applying constraints on the shape of the lag structure.  For example, the parameters might be specified such that they lie on a function of their lag numbers, examples of such functions are simple polynomials or [[splines]].
 ==Structured estimation==