Respuesta :
Let us consider the problem in which we have a y-variable and x-variables all measured as a time series. As an example, we might have y as the monthly highway accidents on an interstate highway and x as the monthly amount of travel on the interstate, with measurements observed for 120 consecutive months.
The difficulty that often arises in this context is that the errors may be correlated with each other. In other words, we have autocorrelation or dependency between the errors.
We may consider situations in which the error at one specific time is linearly related to the error at the previous time. That is, the errors themselves follow a simple linear regression model.
Our model for the errors of the original Y versus X regression is an autoregressive model for the errors, specifically AR(1) in this case. One reason why the errors might have an autoregressive structure is that the Y and X variables at time t may be (and most likely are) related to the Y and X measurements at time t – 1. These relationships are being absorbed into the error term of our multiple linear regression model that only relates Y and X measurements made at concurrent times. Notice that the autoregressive model for the errors is a violation of the assumption that we have independent errors and this creates theoretical difficulties for ordinary least squares estimates of the beta coefficients.
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