Linear regression facts for kids
Linear regression is a way to explain the relationship between a dependent variable and one or more explanatory variables using a straight line. It is a special case of regression analysis.
Linear regression was the first type of regression analysis to be studied rigorously. This is because models which depend linearly on their unknown parameters are easier to fit than models which are non-linearly related to their parameters. Another advantage of linear regression is that the statistical properties of the resulting estimators are easier to determine.
Linear regression has many practical uses. Most applications fall into one of the following two broad categories:
- Linear regression can be used to fit a predictive model to a set of observed values (data). This is useful, if the goal is prediction, forecasting or reduction. After developing such a model, if an additional value of X is then given without its accompanying value of y, the fitted model can be used to make a predicted value of y (written as
). - Given a variable y and a number of variables X1, ..., Xp that may be related to y, linear regression analysis can be applied to quantify the strength of the relationship between y and the Xj, to assess which Xj has no relationship with y at all, and to identify which subsets of the Xj contain redundant information about y.
Linear regression models try to make the vertical distance between the line and the data points (that is, the residuals) as small as possible. This is called "fitting the line to the data." Often, linear regression models try to minimize the sum of the squares of the residuals (least squares), but other ways of fitting exist. They include minimizing the "lack of fit" in some other norm (as with least absolute deviations regression), or minimizing a penalized version of the least squares loss function as in ridge regression. The least squares approach can also be used to fit models that are not linear. As outlined above, the terms "least squares" and "linear model" are closely linked, but they are not synonyms.
Contents
Usage
Economics
Linear regression is the main analytical tool in economics. For example, it is used to guess consumption spending, fixed investment spending, inventory investment, purchases of a country's exports, spending on imports, the demand to hold liquid assets, labor demand and labor supply.
Related pages
Images for kids
-
In linear regression, the observations (red) are assumed to be the result of random deviations (green) from an underlying relationship (blue) between a dependent variable (y) and an independent variable (x).
-
To check for violations of the assumptions of linearity, constant variance, and independence of errors within a linear regression model, the residuals are typically plotted against the predicted values (or each of the individual predictors). An apparently random scatter of points about the horizontal midline at 0 is ideal, but cannot rule out certain kinds of violations such as autocorrelation in the errors or their correlation with one or more covariates.
-
Francis Galton's 1886 illustration of the correlation between the heights of adults and their parents. The observation that adult children's heights tended to deviate less from the mean height than their parents suggested the concept of "regression toward the mean", giving regression its name. The "locus of horizontal tangential points" passing through the leftmost and rightmost points on the ellipse (which is a level curve of the bivariate normal distribution estimated from the data) is the OLS estimate of the regression of parents' heights on children's heights, while the "locus of vertical tangential points" is the OLS estimate of the regression of children's heights on parent's heights. The major axis of the ellipse is the TLS estimate.
See also
In Spanish: Regresión lineal para niños
