Trend Estimation - Fitting A Trend: Least-squares

Fitting A Trend: Least-squares

Given a set of data and the desire to produce some kind of model of those data, there are a variety of functions that can be chosen for the fit. If there is no prior understanding of the data, then the simplest function to fit is a straight line with the data plotted vertically and values of time (t = 1, 2, 3, ...) plotted horizontally.

Once it has been decided to fit a straight line, there are various ways to do so, but the most usual choice is a least-squares fit. This method minimises the sum of the squared errors in the data series, denoted the y variable.

Given a set of points in time, and data values observed for those points in time, values of and are chosen so that

is minimised. Here at + b is the trend line, so the sum of squared deviations from the trend line is what is being minimised. This can always be done in closed form since this is a case of simple linear regression.

For the rest of this article, “trend” will mean the slope of the least squares line, since this is a common convention.

Read more about this topic:  Trend Estimation

Famous quotes containing the word fitting:

    The most fitting monuments this nation can build are schoolhouses and homes for those who do the work of the world. It is no answer to say that they are accustomed to rags and hunger. In this world of plenty every human being has a right to food, clothes, decent shelter, and the rudiments of education.
    Elizabeth Cady Stanton (1815–1902)