# Dynamic Regression Models

In some settings, economic agents respond not only to current values of independent variables but to past values as well. When effects persist over time, an appropriate model will include lagged variables. Example 19.1 illustrates a familiar case.

Example 19.1 A Structural Model of the Demand for Gasoline

Drivers demand gasoline not for direct consumption but as fuel for cars to provide a source of energy for transportation. Per capita demand for gasoline in any period, G/pop, is determined partly by the current price, Pg, and per capita income, Y/pop, which influence how intensively the existing stock of gasoline using "capital," K, is used and partly by the size and composition of the stock of cars and other vehicles. The capital stock is determined, in turn, by Income, Y/pop; prices of the equipment such as new and used cars, Pnc and Puc; the price of alternative modes of transportation such as public transportation, Ppf; and past prices of gasoline as they Influence forecasts of future gasoline prices. A structural model of these effects might appear as follows:

per capita demand: Gt/popt = a + fiPgt + SYt/pop, + yKt + uf, stock of vehicles: Kt = (1 - A)K,_i + /(, A = depreciation rate, investment in new vehicles: lt = 6Yt/pop, + 0E([Pg(+1] + A.i Pnc, + k2Puc, + X3Pptt expected price of gasoline: E,[Pgi+1] = w0Pg, + w1 Pgt_i + w2Pgt-2.

The capital stock is the sum of all past investments, so it is evident that not only current income and prices, but all past values, play a role in determining K. When income or the price of gasoline changes, the immediate effect will be to cause drivers to use their vehicles more or less intensively. But, over time, vehicles are added to the capital stock, and some cars are replaced with more or less efficient ones. These changes take some time, so the full impact of income and price changes will not be felt for several periods. Two episodes in the recent history have shown this effect clearly. For well over a decade following the 1973 oil shock, drivers gradually replaced their large, fuel-inefficient cars with smaller, less-fuel-intensive models. In the late 1990s in the United States, this process has visibly worked In reverse. As American drivers have become accustomed to steadily rising incomes and steadily falling real gasoline prices, the downsized, efficient coupes and sedans of the 1980s have yielded the highways to a tide of ever-larger, six- and eight-cylinder sport utility vehicles, whose size and power can reasonably be characterized as astonishing.