This can be seen in Figure 13.6 by the shape of the distributions placed on the predicted line at the expected value of the relevant value of \(Y\). While the independent variables are all fixed values they are from a probability distribution that is normally distributed. Figure 13.6 shows the case of homoscedasticity where all three distributions have the same variance around the predicted value of \(Y\) regardless of the magnitude of \(X\). If the assumption fails, then it is called heteroscedasticity. The assumption is for constant variance with respect to the magnitude of the independent variable called homoscedasticity. It is plausible that as income increases the variation around the amount purchased will also increase simply because of the flexibility provided with higher levels of income. Consider the relationship between personal income and the quantity of a good purchased as an example of a case where the variance is dependent upon the value of the independent variable, income. The meaning of this is that the variances of the independent variables are independent of the value of the variable. The error term is a random variable with a mean of zero and a constant variance.This assumption is saying in effect that \(Y\) is deterministic, the result of a fixed component “\(X\)” and a random error component “\(\epsilon\).” The independent variables, \(x_i\), are all measured without error, and are fixed numbers that are independent of the error term.Some of the failures of these assumptions can be fixed while others result in estimates that quite simply provide no insight into the questions the model is trying to answer or worse, give biased estimates. If one of these assumptions fails to be true, then it will have an effect on the quality of the estimates. These are that the \(Y\) is normally distributed, the errors are also normally distributed with a mean of zero and a constant standard deviation, and that the error terms are independent of the size of \(X\) and independent of each other.Īssumptions of the Ordinary Least Squares Regression ModelĮach of these assumptions needs a bit more explanation. \nonumber\]Īs with our earlier work with probability distributions, this model works only if certain assumptions hold.
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