Distributed Lags and Dynamic Models

Abstract

Many economic models have lagged values of the regressors in the regression equation. For example, it takes time to build roads and highways. Therefore, the effect of this public investment on growth in GNP will show up with a lag, and this effect will probably linger on for several years. It takes time before investment in research and development pays off in new inventions which in turn take time to develop into commercial products. In studying consumption behavior, a change in income may affect consumption over several periods. This is true in the permanent income theory of consumption, where it may take the consumer several periods to determine whether the change in real disposable income was temporary or permanent. For example, is the extra consulting money earned this year going to continue next year? Also, lagged values of real disposable income appear in the regression equation because the consumer takes into account his life time earnings in trying to smooth out his consumption behavior. In turn, one’s life time income may be guessed by looking at past as well as current earnings. In other words, the regression relationship would look like $$ {Y_t} = \alpha + {\beta _0}{X_t} + {\beta _1}{X_{t - 1}} + .. + {\beta _s}{X_{t - s}} + {u_0}\quad t = 1,2,....,T $$ (6.1) where Y t denotes the t–th observation on the dependent variable Y and Xt-s denotes the (t-s)th observation on the independent variable X. α is the intercept and β 0, β1,..., β S are the current and lagged coefficients of X t . Equation (6.1) is known as a distributed lag since it distributes the effect of an increase in income on consumption over s periods. Note that the short-run effect of a unit change in X on Y is given by β 0, while the long-run effect of a unit change in X on Y is (β 0 + β 1 + .. + β s ).