we obtain "standardized" Pi. Partial sums of the standardized Pi then give the proportion of the long-run, or total, impact felt by a certain time period.
Returning to the consumption regression (17.1.1), we see that the short-run multiplier, which is nothing but the short-run marginal propensity to consume (MPC), is 0.4, whereas the long-run multiplier, which is the technically, P0 is the partial derivative of Y with respect to Xt, P1 that with respect to Xt-1, P2 that with respect to Xt-2, and so forth. Symbolically, dYt/dXt-k = Pk.
Gujarati: Basic I III. Topics in Econometrics I 17. Dynamic Econometric I I © The McGraw-Hill
Econometrics, Fourth Models: Autoregressive Companies, 2004 Edition and Distributed-Lag
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long-run marginal propensity to consume, is 0.4 + 0.3 + 0.2 = 0.9. That is, following a $1 increase in income, the consumer will increase his or her level of consumption by about 40 cents in the year of increase, by another 30 cents in the next year, and by yet another 20 cents in the following year. The long-run impact of an increase of $1 in income is thus 90 cents. If we divide each fy by 0.9, we obtain, respectively, 0.44, 0.33, and 0.23, which indicate that 44 percent of the total impact of a unit change in X on Y is felt immediately, 77 percent after one year, and 100 percent by the end of the second year.
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