Y(t) = beta0 + beta1 Y(t-1) + beta2 X(t) + beta3 X(t-1) + beta4 Z(t-1) + noise
An example would be forecasting Y(t)= consumer spending at time t, where the input variables can be consumer spending in previous time periods and/or other information that is available at time t or earlier.
In economics, when Y(t) is the state of the economy at time t, there is a distinction between three types of variables (aka "indicators"): Leading, coincident, and lagging variables. Leading indicators are those that change before the economy changes (e.g. the stock market); coincident indicators change during the period when the economy changes (e.g., GDP), and lagging indicators change after the economy changes (e.g., unemployment). -- see about.com.
This distinction is especially revealing when we consider the difference between building an econometric model for the purpose of explaining vs. forecasting. For explaining, you can have both leading and coincident variables as inputs. However, if the purpose is forecasting, the inclusion of coincident variables requires one to forecast them before they can be used to forecast Y(t). An alternative is to lag those variables and include them only in leading-indicator format.
I found a neat example of a leading indicator on thefreedictionary.com: The "Leading Lipstick Indicator"
is based on the theory that a consumer turns to less-expensive indulgences, such as lipstick, when she (or he) feels less than confident about the future. Therefore, lipstick sales tend to increase during times of economic uncertainty or a recession. This term was coined by Leonard Lauder (chairman of Estee Lauder), who consistently found that during tough economic times, his lipstick sales went up. Believe it or not, the indicator has been quite a reliable signal of consumer attitudes over the years. For example, in the months following the Sept 11 terrorist attacks, lipstick sales doubled