The plot below gives a time series plot for this dataset. Let y t= the annual number of worldwide earthquakes with magnitude greater than 7 on the Richter scale for n = 100 years ( earthquakes.txt data obtained from ). An autoregressive model is when a value from a time series is regressed on previous values from that same time series. To emphasize that we have measured values over time, we use " t" as a subscript rather than the usual " i," i.e., \(y_t\) means \(y\) measured in time period \(t\). As an example, we might have y a measure of global temperature, with measurements observed each year. Let us first consider the problem in which we have a y-variable measured as a time series. Usually the measurements are made at evenly spaced times - for example, monthly or yearly. We'll explore this further in this section and the next.Ī time series is a sequence of measurements of the same variable(s) made over time. This phenomenon is known as autocorrelation (or serial correlation) and can sometimes be detected by plotting the model residuals versus time. In such a circumstance, the random errors in the model are often positively correlated over time, so that each random error is more likely to be similar to the previous random error that it would be if the random errors were independent of one another. One common way for the "independence" condition in a multiple linear regression model to fail is when the sample data have been collected over time and the regression model fails to effectively capture any time trends.
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