Why least squares

Kenney and Keeping These can be rewritten in a simpler form by defining the sums of squares. Here, is the covariance and and are variances. Note that the quantities and can also be interpreted as the dot products.

In terms of the sums of squares, the regression coefficient is given by. The overall quality of the fit is then parameterized in terms of a quantity known as the correlation coefficient , defined by. Let be the vertical coordinate of the best-fit line with -coordinate , so.

Now define as an estimator for the variance in ,. Then can be given by. The standard errors for and are. Acton, F. Analysis of Straight-Line Data. New York: Dover, Bevington, P. New York: McGraw-Hill, Chatterjee, S. New York: Wiley, pp. Edwards, A. San Francisco, CA: W. Freeman, pp. Farebrother, R. New York: Springer-Verlag, It finds a straight line of best fit through a set of given data points. Financial Ratios. Financial Analysis.

Advanced Technical Analysis Concepts. Tools for Fundamental Analysis. Portfolio Management. Your Privacy Rights. To change or withdraw your consent choices for Investopedia. At any time, you can update your settings through the "EU Privacy" link at the bottom of any page. These choices will be signaled globally to our partners and will not affect browsing data. We and our partners process data to: Actively scan device characteristics for identification. I Accept Show Purposes.

Your Money. Personal Finance. Your Practice. Popular Courses. Financial Analysis How to Value a Company. Introduction to Process Modeling 4. What are some of the different statistical methods for model building? Linear least squares regression is by far the most widely used modeling method.

It is what most people mean when they say they have used "regression", "linear regression" or "least squares" to fit a model to their data. Not only is linear least squares regression the most widely used modeling method, but it has been adapted to a broad range of situations that are outside its direct scope.

It plays a strong underlying role in many other modeling methods, including the other methods discussed in this section: nonlinear least squares regression , weighted least squares regression and LOESS. In statistical terms, any function that meets these criteria would be called a "linear function". The term "linear" is used, even though the function may not be a straight line, because if the unknown parameters are considered to be variables and the explanatory variables are considered to be known coefficients corresponding to those "variables", then the problem becomes a system usually overdetermined of linear equations that can be solved for the values of the unknown parameters.

To differentiate the various meanings of the word "linear", the linear models being discussed here are often said to be "linear in the parameters" or "statistically linear". The deviations between the actual and predicted values are called errors , or residuals.

The better the line fits the data, the smaller the residuals on average. How do we find the line that best fits the data? In other words, how do we determine values of the intercept and slope for our regression line? Intuitively, if we were to manually fit a line to our data, we would try to find a line that minimizes the model errors, overall. But, when we fit a line through data, some of the errors will be positive and some will be negative.