Time Series and Curve-Fitting: How Are They Related?

Consider times series data like that displayed on the right Why shouldn't it be modeled as a regression of the dependent variable on the time variable? The regression (i.e. curve-fitting) model would analyze the data into two parts. The first is called the explained part, which is the part of the dependent variable that is captured as a function of the independent variable, which would be time in this case. For example, we might model the data in terms of a downward parabolic trend. The values of the dependent variable given by the best fitting parabola would be the explained part of the data. The unexplained part is then the difference between the observed value and the explained value, otherwise known as the residue, or the error. Why would this model be a poor model, or at least, not the best model? The reason is that regression models assume that the residues, or errors, are probabilistically independent of the independent variable, in this case time, t. In particular, this implies that residues that are next to each other in time should be independent of each other. If that were true, then we would expect, for example, that the signs of the residues (+ if they a datum is above the parabola, and − if the datum is below the parabola) should be uncorrelated, in the same way as two independent coin tosses should be uncorrelated. But we can see from the graph that if a datum is small, then the one next to it tends to be small, and if the datum is large, then the one next to it is large. There is clear evidence that the residues are not mutually independent.