# Lags, Distributed

# Lags, Distributed

Often when we try to model statistical relationships, we tend to use contemporaneous values. For example, if we want to model changes in consumption because of a change in disposable income, we may try to run the regression Δ*y _{t}* = α + β Δ

*x*+

_{t}*ε*, where Δ

_{t}*y*

_{t}is the percentage change in consumption and Δ

*x*is the percentage change in the disposable income, α and β are the regression parameters, and ε

_{t}_{t}is the random error. The relationship works very well, but it has been documented that people demonstrate “consumption inertia”—that is, the consumption habits of consumers do not change right away in response to an increase in the disposable income. Because consumption expectations are formed by past changes in income, this class of models is called backward-looking expectations models. The best way to capture consumption inertia is to include in the regression model not only the current change in disposable income but also previous changes. If an independent variable (Δ

*x*) appears more than once, with different time lags, then the model is called a distributed lag model. Generally, with only one dependent variable and one explanatory variable, the distributed lag model is represented as

_{t}are coefficients, *x _{t-i}* are the lagged values of the explanatory variable and

*ε*is the independent white noise random error.

_{t}To demonstrate the distributed lag model empirically, with personal consumption data let *y _{t}*, be the change in U.S. personal consumption expenditure in quarter t, and

*x*be a change in U.S. personal disposable income in quarter t. The results of a regression of one lag of personal disposable income are:

_{t}*y =* .005 + .124 *x _{t} +* .142

*x*(3.52) (1.89) (2.17) (t-stats).

_{t-1}This illustrates that a 1 percent change in disposable income induces 0.124 percent of the change in the current consumption. However, last quarter’s change in disposable income has statistically significant (t-statistics in the parenthesis) influence on the change in the consumption. A 1 percent increase in disposable income would cause about 0.142 percent increase in current consumption. The implication of this finding is that changes in personal disposable income have lasting influence on the changes in consumption.

The obvious statistical question then is why estimate the model with only one lag. How many lags are appropriate? This has long been one of the problems with the distributed lag models, even in the early work of Irving Fisher (1937) in the 1930s. Researchers such as Jan Tinbergen (1949) suggest including lags until the coefficients of the lagged variables become insignificant or the signs of the coefficient become erratic. There are several problems with this kind of ad hoc specification. First, there is no guidance in terms of lag length. Second, if the sample size is small, then as the lag length increases statistical inference may be somewhat shaky, with fewer degrees of freedom. Finally, successive lags tend to have high correlations (multicollinearity), leading to smaller t-ratios and incorrect inferences.

One way to reduce the number of lags and the extent of multicollinearity is to use L. M. Koyck’s (1954) adaptive expectations model. In this type of model, in addition to the explanatory variable a lagged dependent variable is included and is represented as *y _{t} =α + β_{1}x_{t}* +

*β*+

_{2}y_{t-1}*U*, where υ is the error term. The adaptive expectation model can be illustrated by using the data on personal consumption expenditures and disposable income. The results of the model are:

_{t}*y* _{t} = .007 + .067 *x _{t} +* .107

*y*

_{t-1}

The coefficients of the above model have some interesting interpretations. The coefficient of disposable income *(x _{t})* shows short-run impact of a change in disposable income on consumption. A 1 percent change in disposable income would cause a 0.67 percent increase in consumption in the short run. The estimate 1.15 percent (0.107/(1 – 0.067)) provides the long-run impact of a change in disposable income on consumption. The results reveal that a 1 percent increase in disposable income would cause consumption to go up by 1.15 percent in the long run.

If the dependent variable is random, the lagged dependent variable may also be random, and including a random explanatory variable in the model may produce biased and inconsistent estimates. Thus, in order to use this model it is essential to verify that the lagged dependent variable is not correlated with the random errors. In addition, in the above models serial correlation in the errors cannot be tested using normal autocorrelation statistics. One of the assumptions of infinite distributed lag models such as Koyck’s is that the coefficient on the lag variables declines geometrically as the lag length increases. If the coefficients do not behave in this manner, then the above lag structure may not be suitable. In these circumstances, you need a more flexible model that would incorporate a variety of lag structures, such as Shirley Almon’s (1965) distributed lag models.

To estimate a flexible model such as Almon, we must a priori specify the lag length to verify the changes in the size of the coefficients. If β*s* decrease at first and then increase with higher lags, *β* _{i}*s* can then be approximated by a second-degree polynomial because we have one size change. The more the turning points, the higher the degree of polynomials. We can illustrate the Almon distributed lag model by using the data on personal consumption and disposable income. First, we assume that change in consumption depends on the current change and preceding two-quarter change in disposable income. Second, we also assume that *β* _{i} can be approximated by a second-degree polynomial. The results of this estimation of are *y _{t} =*

_{t}0.006 + 0.121

*z*– 0.047

_{0}*z*– 0.064

_{1}*z*

_{2}, where

*z*s are constructed as a linear combination of

*x*(change in disposable income) series. For this example of two lags and second-degree polynomial, the

*z*s are:

From the above estimates, original *ゲ s* can be obtained and are presented as *y _{t} =* 0.006 + 0.103

*x*+ 0.121

_{t}*x*+ .001

_{t-1}*x*.

_{t-2}The above values are provided in standard econometric software programs. Although this model offers more flexibility than Koyck’s model, there are still problems with this technique. First, there is no real guidance as to the selection of lag length or the degree of polynomial. Second, the constructed *z* s are likely to exhibit multi-collinearity, which may lead to statistically insignificant coefficients due to large standard errors. Nevertheless, the distributed lag models in general are very useful in modeling issues when the dependent variable exhibits delayed reaction to changes in the independent variable.

**SEE ALSO** *Error-correction Mechanisms; Vector Autoregression*

## BIBLIOGRAPHY

Almon, Shirley. 1965. The Distributed Lag between Capital Appropriations and Expenditures. *Econometrica* 33: 178–196.

Fisher, Irving. 1937. Note on a Short-Cut Method for Calculating Distributed Lags. *International Statistical Bulletin* 29: 323–327.

Koyck, L. M. 1954. *Distributed Lags and Investment Analysis*. Amsterdam: North-Holland.

Tinbergen, Jan. 1949. Long-Term Foreign Trade Elasticities. *Metroeconomica* 1: 174–185.

*Bala G. Arshanapalli*

# Distributed Lags

# Distributed Lags

In theory, distributed lags arise when any economic cause, such as a price change or an income change, produces its effect (for example, on the quantity demanded) only after some lag in time, so that the effect is not felt all at once at a single point in time but is distributed over a period of time. Thus, when we say that the quantity of cigarettes demanded is a function of the price of cigarettes taken with a distributed lag, we mean essentially that the full effects of a change in the price of cigarettes is not felt immediately, and only after some passage of time does the quantity of cigarettes demanded show the full effect of the change in the price of cigarettes.

To consider the matter more concretely, let *q _{t}* be the quantity of cigarettes, say, demanded per unit time, and let

*p*be the price per unit of cigarettes during time period

_{t}*t*. Other things remaining constant, such as income, population, and the price of cough drops, we may express

*q*as a function of current and past prices:

_{t}In particular, let us assume the demand function *f* to be linear with constant coefficients *a, b _{0}, b_{1}*,…

Suppose the price of cigarettes has been constant for a long time at a level *p*, so that *p* = *p*_{t} = *P*_{t-1} = p_{t-2}, = …. Then the quantity of cigarettes demanded per unit time will also be a constant:

Now let the price of cigarettes change from *p* to *p* +Δ*p* in period *t* + 1 and remain at this new level indefinitely. The effect of the change in period *t* + 1 will be to change the quantity demanded from *q* to *q* + b_{0}Δ*p*. But the effects of the price change do not stop here; in the next period, *t* + 2, the quantity demanded is further altered to *q* + b_{0}Δ*p* b_{1}Δ*p* In general, after *θ* periods, the change in *q* will be Thus, the effect of the change in price on demand is distributed over time: b_{0}Δ*p* the first period, b_{0}Δ*p* + b_{1}Δ*p* the second, and so on.

The example used to illustrate the concept of a distributed lag is taken from demand analysis, but the use of distributed lags is not restricted to analysis of problems of consumer demand. The wide application of distributed lags in econometrics may be indicated by a few examples: import demand (Tinbergen 1949); hyperinflation (Cagan 1956); investment (Koyck 1954; Jorgenson 1963; Eisner I960); demand for chemical fertilizers in the United States (Griliches 1959); advertising of oranges by the two major U.S. orange grower co-operatives (Nerlove & Waugh 1961).

The causes of distributed lags in economic relationships are as varied as the variables entering such relationships. To illustrate, the cigarette ex-ample may be used again. When the price is increased, we expect, other things being equal and if cigarettes are not an inferior good, that the demand will fall off. *How* it falls off, that is, the time path that demand follows, is not discussed in the ordinary static theory of demand. At first, a price rise may induce many people to stop smoking, but the tenacity of the habit being what it is, some will return to it. Others may either temporarily or permanently reduce consumption. It is also conceivable that those who do not regard the price increase as permanent, and who do not react at first, may come eventually to believe in the permanence of the price change as it persists and be willing to go through the painful process of adjustment. Finally, some who are nonsmokers, but who might have begun smoking when prices were lower (for example, adolescents), may now never begin. Thus, we see that there are three general types of factors that cause the effects of the price change to be distributed over time: (1) habit persistence, (2) expectational rigidities, and (3) a semitechnological factor related to the age distribution of the population and the conventional time at which a person begins to smoke.

We could just as well deal with an income change or with the effects of a change in the price of a factor of production upon its employment in a certain branch of manufacturing. The exact causes and pattern of the distribution of lag would, of course, depend on the particular circumstances studied. The pattern (and causes) could be different for changes in one direction than for those in another. Indeed, as F. M. Fisher has pointed out (1962, p. 29), there is no need in theory even to assume, as we implicitly have, “that when a decision is made at given time, *t*, the events of a certain fixed period previous are given a certain weight, regardless of what those events were or of what happened before or since their occurrence.” One can imagine the effect if a government report condemning smoking were issued just after the price of cigarettes declined. Nonetheless, for most practical purposes of econometrics, drastic simplifications must be made and the results can be regarded only as approximate.

**History and forms.** Irving Fisher (1925) was the first to use and discuss the concept of a distributed lag. In a later paper (1937, p. 323), he stated that the basic problem in applying the theory of distributed lags “is to find the ’best’ distribution of lag, by which is meant the distribution such that … the total combined effect [of the lagged values of the variables taken with a distributed lag has] … the highest possible correlation with the actual statistical series … with which we wish to compare it.” Thus, we wish to find the distribution of lag that maximizes the explanation of “effect” by “cause” in a statistical sense. There are unsophisticated, almost mechanical, approaches to this problem; and there are more sophisticated approaches involving considerable use of economic theory to develop underlying models of dynamic adjustment that in turn generate the distributed lags observed. There are also both more sophisticated and less sophisticated approaches to estimation.

No assumption may be made about the form of the distribution of lag, i.e., the relationships among the *b’s* in an equation such as (2). This is the approach adopted by Tinbergen (1949) and Alt (1942). The procedure is then to estimate an equation such as (2), for example, by least squares. Since the number of observations is limited, only a finite number of lag terms may be included. Indeed, the coefficients of the lagged values may quickly become quite erratic because of the presence of strong serial correlation in most economic time series. Such difficulties led Alt, Irving Fisher (1925; 1937), and others to suggest that the parameters of a relationship such as (2) be constrained by specifying the form of the distribution of lag. Fisher (1937) suggests, for example, that all *b’s*, after a certain point *θ*, are proportional to an arithmetical progression up to a certain maximal lag, *θ’.* Thus,

Fisher (1925) earlier had suggested that the weights of the distribution might plausibly be assumed to follow a logarithmic normal probability density function (with appropriate modification for the measurement of time in discrete units). Alt recommends a number of exponential forms. Koyck (1954) suggests weights that decline geometrically. Theil and Stern (1960) consider weights proportional to what are essentially approximations to the densities given by a gamma distribution with mean 2. Solow (1960) presents perhaps the ultimate flexibility and sophistication obtainable by a purely formal approach in suggesting the two-parameter family of unimodal lag distributions generated by the Pascal distribution. Thus, the weights are

This family provides the discrete analogue of the exponential and gamma distributions. It represents a natural generalization of the geometrically declining weights of Koyck and may be thought of as *r* simple Koyck lags cascaded in series. The case *r* = 1 is the original Koyck lag. The distribution of lag is skewed; the larger the value of λ and the smaller the value of *r*, the greater is the degree of skewness. Both the center of gravity and the spread of the distribution increase with λ and *r.* This distribution offers a considerable gain in flexibility as compared with earlier suggestions.

A more complete review of the literature up to 1958 and some additional suggestions are contained in Nerlove (1958*a*, pp. 1–25).

Rather than directly specify the form of the lag, an alternative approach is to develop a dynamic model that leads to a distributed lag or lags in the observed relationships. This may be done at varying levels of generality, either by dealing with broad classes of causes in an attempt to derive models of wide applicability or by attempting to isolate dynamic features of a particular problem and show that these lead to certain forms of lag structure. Brown (1952) and Friedman (1957) develop different dynamic models designed to explain the behavior of total consumption expenditures. As Nerlove (1958*a*; 1958*c*) shows, both models lead essentially to a Koyck distribution of lag in the relation between income and consumption. The two models are examples of the two general classes of models leading to distributed lags extensively discussed by Nerlove (1958a; 1958b; 1958c): expectational models and dynamic adjustment models. Both classes in their simplest forms lead to Koyck distributions of lag but have quite different implications for estimation and are not at all the same in more complicated cases. Friedman’s model is a member of the class of expectational models; and Brown’s model is a member of the class of dynamic adjustment. The two models, however, were designed to be quite general and to apply to a variety of problems. Examples taken from the area of agricultural supply analysis are used here to illustrate each model (Nerlove 1958*b*, pp. 25–26).

*An expectational model of supply response.* Suppose that the quantity supplied in year *t, x _{t}*, is a linear function not of the current price

*p*but of the price expected for year

_{t}*t*in year

*t*– 1, (for example, many agricultural crops are planted or planned far ahead):

where *u _{t}* is a stochastic residual. Suppose further that price forecasts are corrected each period by farmers in proportion to the error made:

where *β* is called the coefficient of expectations. A similar model is used explicitly by Cagan (1956) to generate expectations of changes in the general price level during hyperinflations. After some manipulation, it may be shown that

A *dynamic adjustment model of supply response.* In contrast, suppose that the desired or equilibrium quantity supplied, *x _{t}**, is linearly related to the price at the time of decision,

*p*:

_{t-1}However, for a variety of technological and economic reasons (including plain habit), only a fraction, *γ*, of adjustment occurs each period: the two relations, both equations imply:

hence

Apart from the residual terms, which differ in the two relations, both equations imply:

where *α* equals *β* for the expectational model, and *α* equals *γ* for the adjustment model. Equation (12) represents supply as a function of price taken with a Koyck distribution lag. In multiple-equation systems, however, the expectational and dynamic adjustment models do not lead to such similar results; in particular, in expectational models one can make use of the fact that the distribution of lag for the same variable in different equations should be the same (Nerlove 1958*a*, pp. 31–39).

Examples of further extension toward developing distributed lag relationships incidental to a more fundamental dynamic model of behavior are given by Jorgenson (1963) and Muth (1961). Jorgenson gives a model of investment behavior based on a theory of the demand for capital goods over time and on a theory of the relation between such demand and its translation into realized investment, which in turn rests on a distribution of times-to-completion of new investment projects. This model results in a case of a very general distribution of lag. Jorgenson calls the general case the “rational power series distribution”; the Pascal distribution discussed above is a member of this class. Jorgenson’s distribution of lag, although general in form, results in fact from a highly particularized model of dynamic adjustment in the investment decision-and-realization process.

Muth elaborates on a model of expectation formation. In the highly simplified case of a single market with a nonstochastic demand function in which the quantity supplied is a linear function of expected price plus a residual generated by shocks of a permanent and transitory nature, the “rational expectations” that Muth develops can be shown to satisfy (7). Hence, the model leads to a distributed lag of the Koyck form. However, in more realistic models the rational expectations are no longer so simple.

**Estimation problems** . If one examines equations (8) or (11)—those that one might attempt to estimate by least-squares methods if one sought to determine both the distribution of lag and the long-run supply response—it can be seen that the presence of serial correlation in the residual terms will cause serious trouble. For example, least-squares estimates of autoregressions with serially correlated residual terms are known to be statistically inconsistent. Indeed, from (8) it appears that serially correlated residuals will be the rule. This is a subject that is too technical and complex for discussion here. The reader is referred to Koyck (1954, pp. 32–39); Griliches (1961); Klein (1958); Liviatan (1963); Malinvaud (1964); Phillips (1956); and especially Hannan (1964), who gives the most complete and fundamental discussion of this problem and its solution. While Hannan’s results refer primarily to Koyck lags, they can be generalized.

Marc Nerlove

## BIBLIOGRAPHY

Alt, Franz L. 1942 Distributed Lags. *Econometrica* 10: 113–128.

Brown, Tillman M. 1952 Habit Persistence and Lags in Consumer Behavior. *Econometrica* 20:355–371.

Cagan, Phillip 1956 The Monetary Dynamics of Hyperinflation. Pages 23–117 in Milton Friedman (editor), *Studies in the Quantity Theory of Money.* Univ. of Chicago Press.

Eisner, Robert 1960 A Distributed Lag Investment Function. *Econometrica* 28:1–29.

Fisher, Franklin M. 1962 Rigid Lags and the Estimation of “Long-run” Economic Reactions. Pages 21–47 in Franklin M. Fisher, *A Priori Information and Time Series Analysis: Essays in Economic Theory and Measurement.* Amsterdam: North-Holland Publishing.

Fisher, Irving 1925 Our Unstable Dollar and the So-called Business Cycle. *Journal of the American Statistical Association* 20:179–202.

Fisher, Irving 1937 Note on a Short-cut Method for Calculating Distributed Lags. International Statistical Institute, *Bulletin* 29, no. 3:323–328.

Friedman, Milton 1957 A *Theory of the Consumption Function.* National Bureau of Economic Research, General Series, No. 63. Princeton Univ. Press.

Griliches, Zvi 1959 Distributed Lags, Disaggregation, and Regional Demand Functions for Fertilizer. *Journal of Farm Economics* 41:90–102.

Griliches, Zvi 1961 Note on Serial Correlation Bias in Estimates of Distributed Lags. *Econometrica* 29:65–73.

Hannan, E. J. 1964 Estimation of Relationships Involving Distributed Lags. *Econometrica* 33:206–224.

Jorgenson, Dale W. 1963 Capital Theory and Investment Behavior. *American Economic Review* 53:247–259.

Klein, Lawrence R. 1958 The Estimation of Distributed Lags. *Econometrica* 26:553–565.

Koyck, Leendert M. 1954 *Distributed Lags and Investment Analysis.* Amsterdam: North-Holland Publishing.

Liviatan, Nissan 1963 Consistent Estimation of Distributed Lags. *International Economic Review* 4:44–52.

Malinvaud, Edmond 1964 Modèles à retards échelonnés. Pages 478–499 in Edmond Malinvaud, *Méthodes statistiques de l’économétrie.* Paris: Dunod.

Muth, John F. 1961 Rational Expectations and the Theory of Price Movements. *Econometrica* 29:315–335.

Nerlove, Marc 1958*a Distributed Lags and Demand Analysis for Agricultural and Other Commodities.* U.S. Dept. of Agriculture, Handbook No. 141. Washington: Government Printing Office.

Nerlove, Marc 1958*b The Dynamics of Supply: Estimation of Farmers’ Response to Price.* Studies in Historical and Political Science, Series 76, No. 2. Baltimore: Johns Hopkins Press.

Nerlove, Marc 1958*c* The Implications of Friedman’s Permanent Income Hypothesis for Demand Analysis. *Agricultural Economics Research* 10:1–14.

Nerlove, Marc; and Waugh, Frederick V. 1961 Advertising Without Supply Control: Some Implications of a Study of the Advertising of Oranges. *Journal of Farm Economics* 43:813–837.

Phillips, A. W. 1956 Some Notes on the Estimation of Time-forms of Reactions in Interdependent Dynamic Systems. *Economica* New Series 23:99–113.

Solow, Robert M. 1960 On a Family of Lag Distributions. Econometrica 28:393–406.

Theil, H.; and Stern, Robert M. 1960 A Simple Unimodal Lag Distribution. Metroeconomica 12:111–119.

Tinbergen, Jan 1949 Long-term Foreign Trade Elasticities. Metroeconomica 1:174–185.

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