# Linear regression

Last updated

In statistics, linear regression is a linear approach for modelling the relationship between a scalar response and one or more explanatory variables (also known as dependent and independent variables). The case of one explanatory variable is called simple linear regression ; for more than one, the process is called multiple linear regression. [1] This term is distinct from multivariate linear regression, where multiple correlated dependent variables are predicted, rather than a single scalar variable. [2]

## Contents

In linear regression, the relationships are modeled using linear predictor functions whose unknown model parameters are estimated from the data. Such models are called linear models. [3] Most commonly, the conditional mean of the response given the values of the explanatory variables (or predictors) is assumed to be an affine function of those values; less commonly, the conditional median or some other quantile is used. Like all forms of regression analysis, linear regression focuses on the conditional probability distribution of the response given the values of the predictors, rather than on the joint probability distribution of all of these variables, which is the domain of multivariate analysis.

Linear regression was the first type of regression analysis to be studied rigorously, and to be used extensively in practical applications. [4] 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 and because 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:

• If the goal is prediction, forecasting, or error reduction,[ clarification needed ] linear regression can be used to fit a predictive model to an observed data set of values of the response and explanatory variables. After developing such a model, if additional values of the explanatory variables are collected without an accompanying response value, the fitted model can be used to make a prediction of the response.
• If the goal is to explain variation in the response variable that can be attributed to variation in the explanatory variables, linear regression analysis can be applied to quantify the strength of the relationship between the response and the explanatory variables, and in particular to determine whether some explanatory variables may have no linear relationship with the response at all, or to identify which subsets of explanatory variables may contain redundant information about the response.

Linear regression models are often fitted using the least squares approach, but they may also be fitted in other ways, such as by minimizing the "lack of fit" in some other norm (as with least absolute deviations regression), or by minimizing a penalized version of the least squares cost function as in ridge regression (L2-norm penalty) and lasso (L1-norm penalty). Conversely, the least squares approach can be used to fit models that are not linear models. Thus, although the terms "least squares" and "linear model" are closely linked, they are not synonymous.

## Formulation

Given a data set ${\displaystyle \{y_{i},\,x_{i1},\ldots ,x_{ip}\}_{i=1}^{n}}$ of n statistical units, a linear regression model assumes that the relationship between the dependent variable y and the p-vector of regressors x is linear. This relationship is modeled through a disturbance term or error variableε — an unobserved random variable that adds "noise" to the linear relationship between the dependent variable and regressors. Thus the model takes the form

${\displaystyle y_{i}=\beta _{0}+\beta _{1}x_{i1}+\cdots +\beta _{p}x_{ip}+\varepsilon _{i}=\mathbf {x} _{i}^{\mathsf {T}}{\boldsymbol {\beta }}+\varepsilon _{i},\qquad i=1,\ldots ,n,}$

where T denotes the transpose, so that xiTβ is the inner product between vectors xi and β.

Often these n equations are stacked together and written in matrix notation as

${\displaystyle \mathbf {y} =X{\boldsymbol {\beta }}+{\boldsymbol {\varepsilon }},\,}$

where

${\displaystyle \mathbf {y} ={\begin{pmatrix}y_{1}\\y_{2}\\\vdots \\y_{n}\end{pmatrix}},\quad }$
${\displaystyle X={\begin{pmatrix}\mathbf {x} _{1}^{\mathsf {T}}\\\mathbf {x} _{2}^{\mathsf {T}}\\\vdots \\\mathbf {x} _{n}^{\mathsf {T}}\end{pmatrix}}={\begin{pmatrix}1&x_{11}&\cdots &x_{1p}\\1&x_{21}&\cdots &x_{2p}\\\vdots &\vdots &\ddots &\vdots \\1&x_{n1}&\cdots &x_{np}\end{pmatrix}},}$
${\displaystyle {\boldsymbol {\beta }}={\begin{pmatrix}\beta _{0}\\\beta _{1}\\\beta _{2}\\\vdots \\\beta _{p}\end{pmatrix}},\quad {\boldsymbol {\varepsilon }}={\begin{pmatrix}\varepsilon _{1}\\\varepsilon _{2}\\\vdots \\\varepsilon _{n}\end{pmatrix}}.}$

### Notation and terminology

• ${\displaystyle \mathbf {y} }$ is a vector of observed values ${\displaystyle y_{i}\ (i=1,\ldots ,n)}$ of the variable called the regressand, endogenous variable, response variable, measured variable, criterion variable, or dependent variable . This variable is also sometimes known as the predicted variable, but this should not be confused with predicted values, which are denoted ${\displaystyle {\hat {y}}}$. The decision as to which variable in a data set is modeled as the dependent variable and which are modeled as the independent variables may be based on a presumption that the value of one of the variables is caused by, or directly influenced by the other variables. Alternatively, there may be an operational reason to model one of the variables in terms of the others, in which case there need be no presumption of causality.
• ${\displaystyle X}$ may be seen as a matrix of row-vectors ${\displaystyle X_{i}}$ or of n-dimensional column-vectors ${\displaystyle X_{j}}$, which are known as regressors, exogenous variables, explanatory variables, covariates, input variables, predictor variables, or independent variables (not to be confused with the concept of independent random variables). The matrix ${\displaystyle X}$ is sometimes called the design matrix.
• Usually a constant is included as one of the regressors. In particular, ${\displaystyle \mathbf {x} _{i0}=1}$ for ${\displaystyle i=1,\ldots ,n}$. The corresponding element of β is called the intercept . Many statistical inference procedures for linear models require an intercept to be present, so it is often included even if theoretical considerations suggest that its value should be zero.
• Sometimes one of the regressors can be a non-linear function of another regressor or of the data, as in polynomial regression and segmented regression. The model remains linear as long as it is linear in the parameter vector β.
• The values xij may be viewed as either observed values of random variables Xj or as fixed values chosen prior to observing the dependent variable. Both interpretations may be appropriate in different cases, and they generally lead to the same estimation procedures; however different approaches to asymptotic analysis are used in these two situations.
• ${\displaystyle {\boldsymbol {\beta }}}$ is a ${\displaystyle (p+1)}$-dimensional parameter vector, where ${\displaystyle \beta _{0}}$ is the intercept term (if one is included in the model—otherwise ${\displaystyle {\boldsymbol {\beta }}}$ is p-dimensional). Its elements are known as effects or regression coefficients (although the latter term is sometimes reserved for the estimated effects). In simple linear regression, p=1, and the coefficient is known as regression slope. Statistical estimation and inference in linear regression focuses on β. The elements of this parameter vector are interpreted as the partial derivatives of the dependent variable with respect to the various independent variables.
• ${\displaystyle {\boldsymbol {\varepsilon }}}$ is a vector of values ${\displaystyle \varepsilon _{i}}$. This part of the model is called the error term, disturbance term, or sometimes noise (in contrast with the "signal" provided by the rest of the model). This variable captures all other factors which influence the dependent variable y other than the regressors x. The relationship between the error term and the regressors, for example their correlation, is a crucial consideration in formulating a linear regression model, as it will determine the appropriate estimation method.

Fitting a linear model to a given data set usually requires estimating the regression coefficients ${\displaystyle {\boldsymbol {\beta }}}$ such that the error term ${\displaystyle {\boldsymbol {\varepsilon }}=\mathbf {y} -X{\boldsymbol {\beta }}}$ is minimized. For example, it is common to use the sum of squared errors ${\displaystyle ||{\boldsymbol {\varepsilon }}||_{2}^{2}}$ as a measure of ${\displaystyle {\boldsymbol {\varepsilon }}}$ for minimization.

### Example

Consider a situation where a small ball is being tossed up in the air and then we measure its heights of ascent hi at various moments in time ti. Physics tells us that, ignoring the drag, the relationship can be modeled as

${\displaystyle h_{i}=\beta _{1}t_{i}+\beta _{2}t_{i}^{2}+\varepsilon _{i},}$

where β1 determines the initial velocity of the ball, β2 is proportional to the standard gravity, and εi is due to measurement errors. Linear regression can be used to estimate the values of β1 and β2 from the measured data. This model is non-linear in the time variable, but it is linear in the parameters β1 and β2; if we take regressors xi = (xi1, xi2)  = (ti, ti2), the model takes on the standard form

${\displaystyle h_{i}=\mathbf {x} _{i}^{\mathsf {T}}{\boldsymbol {\beta }}+\varepsilon _{i}.}$

### Assumptions

Standard linear regression models with standard estimation techniques make a number of assumptions about the predictor variables, the response variables and their relationship. Numerous extensions have been developed that allow each of these assumptions to be relaxed (i.e. reduced to a weaker form), and in some cases eliminated entirely. Generally these extensions make the estimation procedure more complex and time-consuming, and may also require more data in order to produce an equally precise model.

The following are the major assumptions made by standard linear regression models with standard estimation techniques (e.g. ordinary least squares):

• Weak exogeneity. This essentially means that the predictor variables x can be treated as fixed values, rather than random variables. This means, for example, that the predictor variables are assumed to be error-free—that is, not contaminated with measurement errors. Although this assumption is not realistic in many settings, dropping it leads to significantly more difficult errors-in-variables models.
• Linearity. This means that the mean of the response variable is a linear combination of the parameters (regression coefficients) and the predictor variables. Note that this assumption is much less restrictive than it may at first seem. Because the predictor variables are treated as fixed values (see above), linearity is really only a restriction on the parameters. The predictor variables themselves can be arbitrarily transformed, and in fact multiple copies of the same underlying predictor variable can be added, each one transformed differently. This technique is used, for example, in polynomial regression, which uses linear regression to fit the response variable as an arbitrary polynomial function (up to a given degree) of a predictor variable. With this much flexibility, models such as polynomial regression often have "too much power", in that they tend to overfit the data. As a result, some kind of regularization must typically be used to prevent unreasonable solutions coming out of the estimation process. Common examples are ridge regression and lasso regression. Bayesian linear regression can also be used, which by its nature is more or less immune to the problem of overfitting. (In fact, ridge regression and lasso regression can both be viewed as special cases of Bayesian linear regression, with particular types of prior distributions placed on the regression coefficients.)
• Constant variance (a.k.a. homoscedasticity ). This means that the variance of the errors does not depend on the values of the predictor variables. Thus the variability of the responses for given fixed values of the predictors is the same regardless of how large or small the responses are. This is often not the case, as a variable whose mean is large will typically have a greater variance than one whose mean is small. For example, a person whose income is predicted to be $100,000 may easily have an actual income of$80,000 or $120,000—i.e., a standard deviation of around$20,000—while another person with a predicted income of $10,000 is unlikely to have the same$20,000 standard deviation, since that would imply their actual income could vary anywhere between −$10,000 and$30,000. (In fact, as this shows, in many cases—often the same cases where the assumption of normally distributed errors fails—the variance or standard deviation should be predicted to be proportional to the mean, rather than constant.) The absence of homoscedasticity is called heteroscedasticity. In order to check this assumption, a plot of residuals versus predicted values (or the values of each individual predictor) can be examined for a "fanning effect" (i.e., increasing or decreasing vertical spread as one moves left to right on the plot). A plot of the absolute or squared residuals versus the predicted values (or each predictor) can also be examined for a trend or curvature. Formal tests can also be used; see Heteroscedasticity. The presence of heteroscedasticity will result in an overall "average" estimate of variance being used instead of one that takes into account the true variance structure. This leads to less precise (but in the case of ordinary least squares, not biased) parameter estimates and biased standard errors, resulting in misleading tests and interval estimates. The mean squared error for the model will also be wrong. Various estimation techniques including weighted least squares and the use of heteroscedasticity-consistent standard errors can handle heteroscedasticity in a quite general way. Bayesian linear regression techniques can also be used when the variance is assumed to be a function of the mean. It is also possible in some cases to fix the problem by applying a transformation to the response variable (e.g., fitting the logarithm of the response variable using a linear regression model, which implies that the response variable itself has a log-normal distribution rather than a normal distribution).
• Independence of errors. This assumes that the errors of the response variables are uncorrelated with each other. (Actual statistical independence is a stronger condition than mere lack of correlation and is often not needed, although it can be exploited if it is known to hold.) Some methods such as generalized least squares are capable of handling correlated errors, although they typically require significantly more data unless some sort of regularization is used to bias the model towards assuming uncorrelated errors. Bayesian linear regression is a general way of handling this issue.
• Lack of perfect multicollinearity in the predictors. For standard least squares estimation methods, the design matrix X must have full column rank p; otherwise perfect multicollinearity exists in the predictor variables, meaning a linear relationship exists between two or more predictor variables. This can be caused by accidentally duplicating a variable in the data, using a linear transformation of a variable along with the original (e.g., the same temperature measurements expressed in Fahrenheit and Celsius), or including a linear combination of multiple variables in the model, such as their mean. It can also happen if there is too little data available compared to the number of parameters to be estimated (e.g., fewer data points than regression coefficients). Near violations of this assumption, where predictors are highly but not perfectly correlated, can reduce the precision of parameter estimates (see Variance inflation factor). In the case of perfect multicollinearity, the parameter vector β will be non-identifiable—it has no unique solution. In such a case, only some of the parameters can be identified (i.e., their values can only be estimated within some linear subspace of the full parameter space Rp). See partial least squares regression. Methods for fitting linear models with multicollinearity have been developed, [5] [6] [7] [8] some of which require additional assumptions such as "effect sparsity"—that a large fraction of the effects are exactly zero. Note that the more computationally expensive iterated algorithms for parameter estimation, such as those used in generalized linear models, do not suffer from this problem.

Beyond these assumptions, several other statistical properties of the data strongly influence the performance of different estimation methods:

• The statistical relationship between the error terms and the regressors plays an important role in determining whether an estimation procedure has desirable sampling properties such as being unbiased and consistent.
• The arrangement, or probability distribution of the predictor variables x has a major influence on the precision of estimates of β. Sampling and design of experiments are highly developed subfields of statistics that provide guidance for collecting data in such a way to achieve a precise estimate of β.

### Interpretation

A fitted linear regression model can be used to identify the relationship between a single predictor variable xj and the response variable y when all the other predictor variables in the model are "held fixed". Specifically, the interpretation of βj is the expected change in y for a one-unit change in xj when the other covariates are held fixed—that is, the expected value of the partial derivative of y with respect to xj. This is sometimes called the unique effect of xj on y. In contrast, the marginal effect of xj on y can be assessed using a correlation coefficient or simple linear regression model relating only xj to y; this effect is the total derivative of y with respect to xj.

Care must be taken when interpreting regression results, as some of the regressors may not allow for marginal changes (such as dummy variables, or the intercept term), while others cannot be held fixed (recall the example from the introduction: it would be impossible to "hold ti fixed" and at the same time change the value of ti2).

It is possible that the unique effect can be nearly zero even when the marginal effect is large. This may imply that some other covariate captures all the information in xj, so that once that variable is in the model, there is no contribution of xj to the variation in y. Conversely, the unique effect of xj can be large while its marginal effect is nearly zero. This would happen if the other covariates explained a great deal of the variation of y, but they mainly explain variation in a way that is complementary to what is captured by xj. In this case, including the other variables in the model reduces the part of the variability of y that is unrelated to xj, thereby strengthening the apparent relationship with xj.

The meaning of the expression "held fixed" may depend on how the values of the predictor variables arise. If the experimenter directly sets the values of the predictor variables according to a study design, the comparisons of interest may literally correspond to comparisons among units whose predictor variables have been "held fixed" by the experimenter. Alternatively, the expression "held fixed" can refer to a selection that takes place in the context of data analysis. In this case, we "hold a variable fixed" by restricting our attention to the subsets of the data that happen to have a common value for the given predictor variable. This is the only interpretation of "held fixed" that can be used in an observational study.

The notion of a "unique effect" is appealing when studying a complex system where multiple interrelated components influence the response variable. In some cases, it can literally be interpreted as the causal effect of an intervention that is linked to the value of a predictor variable. However, it has been argued that in many cases multiple regression analysis fails to clarify the relationships between the predictor variables and the response variable when the predictors are correlated with each other and are not assigned following a study design. [9]

## Group effects

In a multiple linear regression model

${\displaystyle y=\beta _{0}+\beta _{1}x_{1}+\cdots +\beta _{p}x_{p}+\varepsilon ,}$

parameter ${\displaystyle \beta _{j}}$ of predictor variable ${\displaystyle x_{j}}$ represents the individual effect of ${\displaystyle x_{j}}$. It has an interpretation as the expected change in the response variable ${\displaystyle y}$ when ${\displaystyle x_{j}}$ increases by one unit with other predictor variables held constant. When ${\displaystyle x_{j}}$ is strongly correlated with other predictor variables, it is improbable that ${\displaystyle x_{j}}$ can increase by one unit with other variables held constant. In this case, the interpretation of ${\displaystyle \beta _{j}}$ becomes problematic as it is based on an improbable condition, and the effect of ${\displaystyle x_{j}}$ cannot be evaluated in isolation.

For a group of predictor variables, say, ${\displaystyle \{x_{1},x_{2},\dots ,x_{q}\}}$, a group effect ${\displaystyle \xi (\mathbf {w} )}$ is defined as a linear combination of their parameters

${\displaystyle \xi (\mathbf {w} )=w_{1}\beta _{1}+w_{2}\beta _{2}+\dots +w_{q}\beta _{q},}$

where ${\displaystyle \mathbf {w} =(w_{1},w_{2},\dots ,w_{q})^{\intercal }}$ is a weight vector satisfying ${\textstyle \sum _{j=1}^{q}|w_{j}|=1}$. Because of the constraint on ${\displaystyle {w_{j}}}$, ${\displaystyle \xi (\mathbf {w} )}$ is also referred to as a normalized group effect. A group effect ${\displaystyle \xi (\mathbf {w} )}$ has an interpretation as the expected change in ${\displaystyle y}$ when variables in the group ${\displaystyle x_{1},x_{2},\dots ,x_{q}}$ change by the amount ${\displaystyle w_{1},w_{2},\dots ,w_{q}}$, respectively, at the same time with variables not in the group held constant. It generalizes the individual effect of a variable to a group of variables in that (${\displaystyle i}$) if ${\displaystyle q=1}$, then the group effect reduces to an individual effect, and (${\displaystyle ii}$) if ${\displaystyle w_{i}=1}$ and ${\displaystyle w_{j}=0}$ for ${\displaystyle j\neq i}$, then the group effect also reduces to an individual effect. A group effect ${\displaystyle \xi (\mathbf {w} )}$ is said to be meaningful if the underlying simultaneous changes of the ${\displaystyle q}$ variables ${\displaystyle (w_{1},w_{2},\dots ,w_{q})^{\intercal }}$ is probable.

Group effects provide a means to study the collective impact of strongly correlated predictor variables in linear regression models. Individual effects of such variables are not well-defined as their parameters do not have good interpretations. Furthermore, when the sample size is not large, none of their parameters can be accurately estimated by the least squares regression due to the multicollinearity problem. Nevertheless, there are meaningful group effects that have good interpretations and can be accurately estimated by the least squares regression. A simple way to identify these meaningful group effects is to use an all positive correlations (APC) arrangement of the strongly correlated variables under which pairwise correlations among these variables are all positive, and standardize all ${\displaystyle p}$ predictor variables in the model so that they all have mean zero and length one. To illustrate this, suppose that ${\displaystyle \{x_{1},x_{2},\dots ,x_{q}\}}$ is a group of strongly correlated variables in an APC arrangement and that they are not strongly correlated with predictor variables outside the group. Let ${\displaystyle y'}$ be the centred ${\displaystyle y}$ and ${\displaystyle x_{j}'}$ be the standardized ${\displaystyle x_{j}}$. Then, the standardized linear regression model is

${\displaystyle y'=\beta _{1}'x_{1}'+\cdots +\beta _{p}'x_{p}'+\varepsilon .}$

Parameters ${\displaystyle \beta _{j}}$ in the original model, including ${\displaystyle \beta _{0}}$, are simple functions of ${\displaystyle \beta _{j}'}$ in the standardized model. The standardization of variables does not change their correlations, so ${\displaystyle \{x_{1}',x_{2}',\dots ,x_{q}'\}}$ is a group of strongly correlated variables in an APC arrangement and they are not strongly correlated with other predictor variables in the standardized model. A group effect of ${\displaystyle \{x_{1}',x_{2}',\dots ,x_{q}'\}}$ is

${\displaystyle \xi '(\mathbf {w} )=w_{1}\beta _{1}'+w_{2}\beta _{2}'+\dots +w_{q}\beta _{q}',}$

and its minimum-variance unbiased linear estimator is

${\displaystyle {\hat {\xi }}'(\mathbf {w} )=w_{1}{\hat {\beta }}_{1}'+w_{2}{\hat {\beta }}_{2}'+\dots +w_{q}{\hat {\beta }}_{q}',}$

where ${\displaystyle {\hat {\beta }}_{j}'}$ is the least squares estimator of ${\displaystyle \beta _{j}'}$. In particular, the average group effect of the ${\displaystyle q}$ standardized variables is

${\displaystyle \xi _{A}={\frac {1}{q}}(\beta _{1}'+\beta _{2}'+\dots +\beta _{q}'),}$

which has an interpretation as the expected change in ${\displaystyle y'}$ when all ${\displaystyle x_{j}'}$ in the strongly correlated group increase by ${\displaystyle (1/q)}$th of a unit at the same time with variables outside the group held constant. With strong positive correlations and in standardized units, variables in the group are approximately equal, so they are likely to increase at the same time and in similar amount. Thus, the average group effect ${\displaystyle \xi _{A}}$ is a meaningful effect. It can be accurately estimated by its minimum-variance unbiased linear estimator ${\textstyle {\hat {\xi }}_{A}={\frac {1}{q}}({\hat {\beta }}_{1}'+{\hat {\beta }}_{2}'+\dots +{\hat {\beta }}_{q}')}$, even when individually none of the ${\displaystyle \beta _{j}'}$ can be accurately estimated by ${\displaystyle {\hat {\beta }}_{j}'}$.

Not all group effects are meaningful or can be accurately estimated. For example, ${\displaystyle \beta _{1}'}$ is a special group effect with weights ${\displaystyle w_{1}=1}$ and ${\displaystyle w_{j}=0}$ for ${\displaystyle j\neq 1}$, but it cannot be accurately estimated by ${\displaystyle {\hat {\beta }}'_{1}}$. It is also not a meaningful effect. In general, for a group of ${\displaystyle q}$ strongly correlated predictor variables in an APC arrangement in the standardized model, group effects whose weight vectors ${\displaystyle \mathbf {w} }$ are at or near the centre of the simplex ${\textstyle \sum _{j=1}^{q}w_{j}=1}$ (${\displaystyle w_{j}\geq 0}$) are meaningful and can be accurately estimated by their minimum-variance unbiased linear estimators. Effects with weight vectors far away from the centre are not meaningful as such weight vectors represent simultaneous changes of the variables that violate the strong positive correlations of the standardized variables in an APC arrangement. As such, they are not probable. These effects also cannot be accurately estimated.

A group effect of the original variables ${\displaystyle \{x_{1},x_{2},\dots ,x_{q}\}}$ can be expressed as a constant times a group effect of the standardized variables ${\displaystyle \{x_{1}',x_{2}',\dots ,x_{q}'\}}$. The former is meaningful when the latter is. Thus meaningful group effects of the original variables can be found through meaningful group effects of the standardized variables. [10]

## Extensions

Numerous extensions of linear regression have been developed, which allow some or all of the assumptions underlying the basic model to be relaxed.

### Simple and multiple linear regression

The very simplest case of a single scalar predictor variable x and a single scalar response variable y is known as simple linear regression . The extension to multiple and/or vector-valued predictor variables (denoted with a capital X) is known as multiple linear regression, also known as multivariable linear regression (not to be confused with multivariate linear regression [11] ).

Multiple linear regression is a generalization of simple linear regression to the case of more than one independent variable, and a special case of general linear models, restricted to one dependent variable. The basic model for multiple linear regression is

${\displaystyle Y_{i}=\beta _{0}+\beta _{1}X_{i1}+\beta _{2}X_{i2}+\ldots +\beta _{p}X_{ip}+\epsilon _{i}}$

for each observation i = 1, ... , n.

In the formula above we consider n observations of one dependent variable and p independent variables. Thus, Yi is the ith observation of the dependent variable, Xij is ith observation of the jth independent variable, j = 1, 2, ..., p. The values βj represent parameters to be estimated, and εi is the ith independent identically distributed normal error.

In the more general multivariate linear regression, there is one equation of the above form for each of m > 1 dependent variables that share the same set of explanatory variables and hence are estimated simultaneously with each other:

${\displaystyle Y_{ij}=\beta _{0j}+\beta _{1j}X_{i1}+\beta _{2j}X_{i2}+\ldots +\beta _{pj}X_{ip}+\epsilon _{ij}}$

for all observations indexed as i = 1, ... , n and for all dependent variables indexed as j = 1, ... , m.

Nearly all real-world regression models involve multiple predictors, and basic descriptions of linear regression are often phrased in terms of the multiple regression model. Note, however, that in these cases the response variable y is still a scalar. Another term, multivariate linear regression, refers to cases where y is a vector, i.e., the same as general linear regression.

### General linear models

The general linear model considers the situation when the response variable is not a scalar (for each observation) but a vector, yi. Conditional linearity of ${\displaystyle E(\mathbf {y} \mid \mathbf {x} _{i})=\mathbf {x} _{i}^{\mathsf {T}}B}$ is still assumed, with a matrix B replacing the vector β of the classical linear regression model. Multivariate analogues of ordinary least squares (OLS) and generalized least squares (GLS) have been developed. "General linear models" are also called "multivariate linear models". These are not the same as multivariable linear models (also called "multiple linear models").

### Heteroscedastic models

Various models have been created that allow for heteroscedasticity, i.e. the errors for different response variables may have different variances. For example, weighted least squares is a method for estimating linear regression models when the response variables may have different error variances, possibly with correlated errors. (See also Weighted linear least squares, and Generalized least squares.) Heteroscedasticity-consistent standard errors is an improved method for use with uncorrelated but potentially heteroscedastic errors.

### Generalized linear models

Generalized linear models (GLMs) are a framework for modeling response variables that are bounded or discrete. This is used, for example:

• when modeling positive quantities (e.g. prices or populations) that vary over a large scale—which are better described using a skewed distribution such as the log-normal distribution or Poisson distribution (although GLMs are not used for log-normal data, instead the response variable is simply transformed using the logarithm function);
• when modeling categorical data, such as the choice of a given candidate in an election (which is better described using a Bernoulli distribution/binomial distribution for binary choices, or a categorical distribution/multinomial distribution for multi-way choices), where there are a fixed number of choices that cannot be meaningfully ordered;
• when modeling ordinal data, e.g. ratings on a scale from 0 to 5, where the different outcomes can be ordered but where the quantity itself may not have any absolute meaning (e.g. a rating of 4 may not be "twice as good" in any objective sense as a rating of 2, but simply indicates that it is better than 2 or 3 but not as good as 5).

Generalized linear models allow for an arbitrary link function, g, that relates the mean of the response variable(s) to the predictors: ${\displaystyle E(Y)=g^{-1}(XB)}$. The link function is often related to the distribution of the response, and in particular it typically has the effect of transforming between the ${\displaystyle (-\infty ,\infty )}$ range of the linear predictor and the range of the response variable.

Some common examples of GLMs are:

Single index models[ clarification needed ] allow some degree of nonlinearity in the relationship between x and y, while preserving the central role of the linear predictor βx as in the classical linear regression model. Under certain conditions, simply applying OLS to data from a single-index model will consistently estimate β up to a proportionality constant. [12]

### Hierarchical linear models

Hierarchical linear models (or multilevel regression) organizes the data into a hierarchy of regressions, for example where A is regressed on B, and B is regressed on C. It is often used where the variables of interest have a natural hierarchical structure such as in educational statistics, where students are nested in classrooms, classrooms are nested in schools, and schools are nested in some administrative grouping, such as a school district. The response variable might be a measure of student achievement such as a test score, and different covariates would be collected at the classroom, school, and school district levels.

### Errors-in-variables

Errors-in-variables models (or "measurement error models") extend the traditional linear regression model to allow the predictor variables X to be observed with error. This error causes standard estimators of β to become biased. Generally, the form of bias is an attenuation, meaning that the effects are biased toward zero.

### Others

• In Dempster–Shafer theory, or a linear belief function in particular, a linear regression model may be represented as a partially swept matrix, which can be combined with similar matrices representing observations and other assumed normal distributions and state equations. The combination of swept or unswept matrices provides an alternative method for estimating linear regression models.

## Estimation methods

A large number of procedures have been developed for parameter estimation and inference in linear regression. These methods differ in computational simplicity of algorithms, presence of a closed-form solution, robustness with respect to heavy-tailed distributions, and theoretical assumptions needed to validate desirable statistical properties such as consistency and asymptotic efficiency.

Some of the more common estimation techniques for linear regression are summarized below.

Assuming that the independent variable is ${\displaystyle {\vec {x_{i}}}=\left[x_{1}^{i},x_{2}^{i},\ldots ,x_{m}^{i}\right]}$ and the model's parameters are ${\displaystyle {\vec {\beta }}=\left[\beta _{0},\beta _{1},\ldots ,\beta _{m}\right]}$, then the model's prediction would be

${\displaystyle y_{i}\approx \beta _{0}+\sum _{j=1}^{m}\beta _{j}\times x_{j}^{i}}$.

If ${\displaystyle {\vec {x_{i}}}}$ is extended to ${\displaystyle {\vec {x_{i}}}=\left[1,x_{1}^{i},x_{2}^{i},\ldots ,x_{m}^{i}\right]}$ then ${\displaystyle y_{i}}$ would become a dot product of the parameter and the independent variable, i.e.

${\displaystyle y_{i}\approx \sum _{j=0}^{m}\beta _{j}\times x_{j}^{i}={\vec {\beta }}\cdot {\vec {x_{i}}}}$.

In the least-squares setting, the optimum parameter is defined as such that minimizes the sum of mean squared loss:

${\displaystyle {\vec {\hat {\beta }}}={\underset {\vec {\beta }}{\mbox{arg min}}}\,L\left(D,{\vec {\beta }}\right)={\underset {\vec {\beta }}{\mbox{arg min}}}\sum _{i=1}^{n}\left({\vec {\beta }}\cdot {\vec {x_{i}}}-y_{i}\right)^{2}}$

Now putting the independent and dependent variables in matrices ${\displaystyle X}$ and ${\displaystyle Y}$ respectively, the loss function can be rewritten as:

{\displaystyle {\begin{aligned}L\left(D,{\vec {\beta }}\right)&=\|X{\vec {\beta }}-Y\|^{2}\\&=\left(X{\vec {\beta }}-Y\right)^{\textsf {T}}\left(X{\vec {\beta }}-Y\right)\\&=Y^{\textsf {T}}Y-Y^{\textsf {T}}X{\vec {\beta }}-{\vec {\beta }}^{\textsf {T}}X^{\textsf {T}}Y+{\vec {\beta }}^{\textsf {T}}X^{\textsf {T}}X{\vec {\beta }}\end{aligned}}}

As the loss is convex the optimum solution lies at gradient zero. The gradient of the loss function is (using Denominator layout convention):

{\displaystyle {\begin{aligned}{\frac {\partial L\left(D,{\vec {\beta }}\right)}{\partial {\vec {\beta }}}}&={\frac {\partial \left(Y^{\textsf {T}}Y-Y^{\textsf {T}}X{\vec {\beta }}-{\vec {\beta }}^{\textsf {T}}X^{\textsf {T}}Y+{\vec {\beta }}^{\textsf {T}}X^{\textsf {T}}X{\vec {\beta }}\right)}{\partial {\vec {\beta }}}}\\&=-2X^{\textsf {T}}Y+2X^{\textsf {T}}X{\vec {\beta }}\end{aligned}}}

Setting the gradient to zero produces the optimum parameter:

{\displaystyle {\begin{aligned}-2X^{\textsf {T}}Y+2X^{\textsf {T}}X{\vec {\beta }}&=0\\\Rightarrow X^{\textsf {T}}X{\vec {\beta }}&=X^{\textsf {T}}Y\\\Rightarrow {\vec {\hat {\beta }}}&=\left(X^{\textsf {T}}X\right)^{-1}X^{\textsf {T}}Y\end{aligned}}}

Note: To prove that the ${\displaystyle {\hat {\beta }}}$ obtained is indeed the local minimum, one needs to differentiate once more to obtain the Hessian matrix and show that it is positive definite. This is provided by the Gauss–Markov theorem.

Linear least squares methods include mainly:

• Maximum likelihood estimation can be performed when the distribution of the error terms is known to belong to a certain parametric family ƒθ of probability distributions. [14] When fθ is a normal distribution with zero mean and variance θ, the resulting estimate is identical to the OLS estimate. GLS estimates are maximum likelihood estimates when ε follows a multivariate normal distribution with a known covariance matrix.
• Ridge regression [15] [16] [17] and other forms of penalized estimation, such as Lasso regression , [5] deliberately introduce bias into the estimation of β in order to reduce the variability of the estimate. The resulting estimates generally have lower mean squared error than the OLS estimates, particularly when multicollinearity is present or when overfitting is a problem. They are generally used when the goal is to predict the value of the response variable y for values of the predictors x that have not yet been observed. These methods are not as commonly used when the goal is inference, since it is difficult to account for the bias.
• Least absolute deviation (LAD) regression is a robust estimation technique in that it is less sensitive to the presence of outliers than OLS (but is less efficient than OLS when no outliers are present). It is equivalent to maximum likelihood estimation under a Laplace distribution model for ε. [18]
• Adaptive estimation. If we assume that error terms are independent of the regressors, ${\displaystyle \varepsilon _{i}\perp \mathbf {x} _{i}}$, then the optimal estimator is the 2-step MLE, where the first step is used to non-parametrically estimate the distribution of the error term. [19]

### Other estimation techniques

• Bayesian linear regression applies the framework of Bayesian statistics to linear regression. (See also Bayesian multivariate linear regression.) In particular, the regression coefficients β are assumed to be random variables with a specified prior distribution. The prior distribution can bias the solutions for the regression coefficients, in a way similar to (but more general than) ridge regression or lasso regression. In addition, the Bayesian estimation process produces not a single point estimate for the "best" values of the regression coefficients but an entire posterior distribution, completely describing the uncertainty surrounding the quantity. This can be used to estimate the "best" coefficients using the mean, mode, median, any quantile (see quantile regression), or any other function of the posterior distribution.
• Quantile regression focuses on the conditional quantiles of y given X rather than the conditional mean of y given X. Linear quantile regression models a particular conditional quantile, for example the conditional median, as a linear function βTx of the predictors.
• Mixed models are widely used to analyze linear regression relationships involving dependent data when the dependencies have a known structure. Common applications of mixed models include analysis of data involving repeated measurements, such as longitudinal data, or data obtained from cluster sampling. They are generally fit as parametric models, using maximum likelihood or Bayesian estimation. In the case where the errors are modeled as normal random variables, there is a close connection between mixed models and generalized least squares. [20] Fixed effects estimation is an alternative approach to analyzing this type of data.
• Principal component regression (PCR) [7] [8] is used when the number of predictor variables is large, or when strong correlations exist among the predictor variables. This two-stage procedure first reduces the predictor variables using principal component analysis, and then uses the reduced variables in an OLS regression fit. While it often works well in practice, there is no general theoretical reason that the most informative linear function of the predictor variables should lie among the dominant principal components of the multivariate distribution of the predictor variables. The partial least squares regression is the extension of the PCR method which does not suffer from the mentioned deficiency.
• Least-angle regression [6] is an estimation procedure for linear regression models that was developed to handle high-dimensional covariate vectors, potentially with more covariates than observations.
• The Theil–Sen estimator is a simple robust estimation technique that chooses the slope of the fit line to be the median of the slopes of the lines through pairs of sample points. It has similar statistical efficiency properties to simple linear regression but is much less sensitive to outliers. [21]
• Other robust estimation techniques, including the α-trimmed mean approach[ citation needed ], and L-, M-, S-, and R-estimators have been introduced.[ citation needed ]

## Applications

Linear regression is widely used in biological, behavioral and social sciences to describe possible relationships between variables. It ranks as one of the most important tools used in these disciplines.

### Trend line

A trend line represents a trend, the long-term movement in time series data after other components have been accounted for. It tells whether a particular data set (say GDP, oil prices or stock prices) have increased or decreased over the period of time. A trend line could simply be drawn by eye through a set of data points, but more properly their position and slope is calculated using statistical techniques like linear regression. Trend lines typically are straight lines, although some variations use higher degree polynomials depending on the degree of curvature desired in the line.

Trend lines are sometimes used in business analytics to show changes in data over time. This has the advantage of being simple. Trend lines are often used to argue that a particular action or event (such as training, or an advertising campaign) caused observed changes at a point in time. This is a simple technique, and does not require a control group, experimental design, or a sophisticated analysis technique. However, it suffers from a lack of scientific validity in cases where other potential changes can affect the data.

### Epidemiology

Early evidence relating tobacco smoking to mortality and morbidity came from observational studies employing regression analysis. In order to reduce spurious correlations when analyzing observational data, researchers usually include several variables in their regression models in addition to the variable of primary interest. For example, in a regression model in which cigarette smoking is the independent variable of primary interest and the dependent variable is lifespan measured in years, researchers might include education and income as additional independent variables, to ensure that any observed effect of smoking on lifespan is not due to those other socio-economic factors. However, it is never possible to include all possible confounding variables in an empirical analysis. For example, a hypothetical gene might increase mortality and also cause people to smoke more. For this reason, randomized controlled trials are often able to generate more compelling evidence of causal relationships than can be obtained using regression analyses of observational data. When controlled experiments are not feasible, variants of regression analysis such as instrumental variables regression may be used to attempt to estimate causal relationships from observational data.

### Finance

The capital asset pricing model uses linear regression as well as the concept of beta for analyzing and quantifying the systematic risk of an investment. This comes directly from the beta coefficient of the linear regression model that relates the return on the investment to the return on all risky assets.

### Economics

Linear regression is the predominant empirical tool in economics. For example, it is used to predict consumption spending, [22] fixed investment spending, inventory investment, purchases of a country's exports, [23] spending on imports, [23] the demand to hold liquid assets, [24] labor demand, [25] and labor supply. [25]

### Environmental science

Linear regression finds application in a wide range of environmental science applications. In Canada, the Environmental Effects Monitoring Program uses statistical analyses on fish and benthic surveys to measure the effects of pulp mill or metal mine effluent on the aquatic ecosystem. [26]

### Machine learning

Linear regression plays an important role in the subfield of artificial intelligence known as machine learning. The linear regression algorithm is one of the fundamental supervised machine-learning algorithms due to its relative simplicity and well-known properties. [27]

## History

Least squares linear regression, as a means of finding a good rough linear fit to a set of points was performed by Legendre (1805) and Gauss (1809) for the prediction of planetary movement. Quetelet was responsible for making the procedure well-known and for using it extensively in the social sciences. [28]

## Related Research Articles

In statistics, the term linear model is used in different ways according to the context. The most common occurrence is in connection with regression models and the term is often taken as synonymous with linear regression model. However, the term is also used in time series analysis with a different meaning. In each case, the designation "linear" is used to identify a subclass of models for which substantial reduction in the complexity of the related statistical theory is possible.

The method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems by minimizing the sum of the squares of the residuals made in the results of each individual equation.

In statistics, the Gauss–Markov theorem states that the ordinary least squares (OLS) estimator has the lowest sampling variance within the class of linear unbiased estimators, if the errors in the linear regression model are uncorrelated, have equal variances and expectation value of zero. The errors do not need to be normal, nor do they need to be independent and identically distributed. The requirement that the estimator be unbiased cannot be dropped, since biased estimators exist with lower variance. See, for example, the James–Stein estimator, ridge regression, or simply any degenerate estimator.

In statistics, the logistic model is a statistical model that models the probability of an event taking place by having the log-odds for the event be a linear combination of one or more independent variables. In regression analysis, logistic regression is estimating the parameters of a logistic model. Formally, in binary logistic regression there is a single binary dependent variable, coded by an indicator variable, where the two values are labeled "0" and "1", while the independent variables can each be a binary variable or a continuous variable. The corresponding probability of the value labeled "1" can vary between 0 and 1, hence the labeling; the function that converts log-odds to probability is the logistic function, hence the name. The unit of measurement for the log-odds scale is called a logit, from logistic unit, hence the alternative names. See § Background and § Definition for formal mathematics, and § Example for a worked example.

In statistics, a generalized linear model (GLM) is a flexible generalization of ordinary linear regression. The GLM generalizes linear regression by allowing the linear model to be related to the response variable via a link function and by allowing the magnitude of the variance of each measurement to be a function of its predicted value.

In statistical modeling, regression analysis is a set of statistical processes for estimating the relationships between a dependent variable and one or more independent variables. The most common form of regression analysis is linear regression, in which one finds the line that most closely fits the data according to a specific mathematical criterion. For example, the method of ordinary least squares computes the unique line that minimizes the sum of squared differences between the true data and that line. For specific mathematical reasons, this allows the researcher to estimate the conditional expectation of the dependent variable when the independent variables take on a given set of values. Less common forms of regression use slightly different procedures to estimate alternative location parameters or estimate the conditional expectation across a broader collection of non-linear models.

The general linear model or general multivariate regression model is a compact way of simultaneously writing several multiple linear regression models. In that sense it is not a separate statistical linear model. The various multiple linear regression models may be compactly written as

In statistics, nonlinear regression is a form of regression analysis in which observational data are modeled by a function which is a nonlinear combination of the model parameters and depends on one or more independent variables. The data are fitted by a method of successive approximations.

In statistics, multicollinearity is a phenomenon in which one predictor variable in a multiple regression model can be linearly predicted from the others with a substantial degree of accuracy. In this situation, the coefficient estimates of the multiple regression may change erratically in response to small changes in the model or the data. Multicollinearity does not reduce the predictive power or reliability of the model as a whole, at least within the sample data set; it only affects calculations regarding individual predictors. That is, a multivariate regression model with collinear predictors can indicate how well the entire bundle of predictors predicts the outcome variable, but it may not give valid results about any individual predictor, or about which predictors are redundant with respect to others.

In statistics, a probit model is a type of regression where the dependent variable can take only two values, for example married or not married. The word is a portmanteau, coming from probability + unit. The purpose of the model is to estimate the probability that an observation with particular characteristics will fall into a specific one of the categories; moreover, classifying observations based on their predicted probabilities is a type of binary classification model.

In statistics, ordinary least squares (OLS) is a type of linear least squares method for choosing the unknown parameters in a linear regression model by the principle of least squares: minimizing the sum of the squares of the differences between the observed dependent variable in the input dataset and the output of the (linear) function of the independent variable.

Weighted least squares (WLS), also known as weighted linear regression, is a generalization of ordinary least squares and linear regression in which knowledge of the variance of observations is incorporated into the regression. WLS is also a specialization of generalized least squares.

In statistics, generalized least squares (GLS) is a technique for estimating the unknown parameters in a linear regression model when there is a certain degree of correlation between the residuals in a regression model. In these cases, ordinary least squares and weighted least squares can be statistically inefficient, or even give misleading inferences. GLS was first described by Alexander Aitken in 1936.

In statistics, binomial regression is a regression analysis technique in which the response has a binomial distribution: it is the number of successes in a series of independent Bernoulli trials, where each trial has probability of success . In binomial regression, the probability of a success is related to explanatory variables: the corresponding concept in ordinary regression is to relate the mean value of the unobserved response to explanatory variables.

In statistics, Bayesian multivariate linear regression is a Bayesian approach to multivariate linear regression, i.e. linear regression where the predicted outcome is a vector of correlated random variables rather than a single scalar random variable. A more general treatment of this approach can be found in the article MMSE estimator.

In statistics, the projection matrix, sometimes also called the influence matrix or hat matrix, maps the vector of response values to the vector of fitted values. It describes the influence each response value has on each fitted value. The diagonal elements of the projection matrix are the leverages, which describe the influence each response value has on the fitted value for that same observation.

In statistics, polynomial regression is a form of regression analysis in which the relationship between the independent variable x and the dependent variable y is modelled as an nth degree polynomial in x. Polynomial regression fits a nonlinear relationship between the value of x and the corresponding conditional mean of y, denoted E(y |x). Although polynomial regression fits a nonlinear model to the data, as a statistical estimation problem it is linear, in the sense that the regression function E(y | x) is linear in the unknown parameters that are estimated from the data. For this reason, polynomial regression is considered to be a special case of multiple linear regression.

Linear least squares (LLS) is the least squares approximation of linear functions to data. It is a set of formulations for solving statistical problems involved in linear regression, including variants for ordinary (unweighted), weighted, and generalized (correlated) residuals. Numerical methods for linear least squares include inverting the matrix of the normal equations and orthogonal decomposition methods.

In statistics, the class of vector generalized linear models (VGLMs) was proposed to enlarge the scope of models catered for by generalized linear models (GLMs). In particular, VGLMs allow for response variables outside the classical exponential family and for more than one parameter. Each parameter can be transformed by a link function. The VGLM framework is also large enough to naturally accommodate multiple responses; these are several independent responses each coming from a particular statistical distribution with possibly different parameter values.

In statistics, particularly regression analysis, the Working–Hotelling procedure, named after Holbrook Working and Harold Hotelling, is a method of simultaneous estimation in linear regression models. One of the first developments in simultaneous inference, it was devised by Working and Hotelling for the simple linear regression model in 1929. It provides a confidence region for multiple mean responses, that is, it gives the upper and lower bounds of more than one value of a dependent variable at several levels of the independent variables at a certain confidence level. The resulting confidence bands are known as the Working–Hotelling–Scheffé confidence bands.

## References

### Citations

1. David A. Freedman (2009). Statistical Models: Theory and Practice. Cambridge University Press. p. 26. A simple regression equation has on the right hand side an intercept and an explanatory variable with a slope coefficient. A multiple regression e right hand side, each with its own slope coefficient
2. Rencher, Alvin C.; Christensen, William F. (2012), "Chapter 10, Multivariate regression – Section 10.1, Introduction", Methods of Multivariate Analysis, Wiley Series in Probability and Statistics, vol. 709 (3rd ed.), John Wiley & Sons, p. 19, ISBN   9781118391679 .
3. Hilary L. Seal (1967). "The historical development of the Gauss linear model". Biometrika. 54 (1/2): 1–24. doi:10.1093/biomet/54.1-2.1. JSTOR   2333849.
4. Yan, Xin (2009), Linear Regression Analysis: Theory and Computing, World Scientific, pp. 1–2, ISBN   9789812834119, Regression analysis ... is probably one of the oldest topics in mathematical statistics dating back to about two hundred years ago. The earliest form of the linear regression was the least squares method, which was published by Legendre in 1805, and by Gauss in 1809 ... Legendre and Gauss both applied the method to the problem of determining, from astronomical observations, the orbits of bodies about the sun.
5. Tibshirani, Robert (1996). "Regression Shrinkage and Selection via the Lasso". Journal of the Royal Statistical Society, Series B. 58 (1): 267–288. JSTOR   2346178.
6. Efron, Bradley; Hastie, Trevor; Johnstone, Iain; Tibshirani, Robert (2004). "Least Angle Regression". The Annals of Statistics. 32 (2): 407–451. arXiv:. doi:10.1214/009053604000000067. JSTOR   3448465.
7. Hawkins, Douglas M. (1973). "On the Investigation of Alternative Regressions by Principal Component Analysis". Journal of the Royal Statistical Society, Series C. 22 (3): 275–286. JSTOR   2346776.
8. Jolliffe, Ian T. (1982). "A Note on the Use of Principal Components in Regression". Journal of the Royal Statistical Society, Series C. 31 (3): 300–303. JSTOR   2348005.
9. Berk, Richard A. (2007). "Regression Analysis: A Constructive Critique". Criminal Justice Review. 32 (3): 301–302. doi:10.1177/0734016807304871.
10. Tsao, Min (2022). "Group least squares regression for linear models with strongly correlated predictor variables". Annals of the Institute of Statistical Mathematics. doi:10.1007/s10463-022-00841-7.
11. Hidalgo, Bertha; Goodman, Melody (2012-11-15). "Multivariate or Multivariable Regression?". American Journal of Public Health. 103 (1): 39–40. doi:10.2105/AJPH.2012.300897. ISSN   0090-0036. PMC  . PMID   23153131.
12. Brillinger, David R. (1977). "The Identification of a Particular Nonlinear Time Series System". Biometrika. 64 (3): 509–515. doi:10.1093/biomet/64.3.509. JSTOR   2345326.
13. Galton, Francis (1886). "Regression Towards Mediocrity in Hereditary Stature". The Journal of the Anthropological Institute of Great Britain and Ireland. 15: 246–263. doi:10.2307/2841583. ISSN   0959-5295.
14. Lange, Kenneth L.; Little, Roderick J. A.; Taylor, Jeremy M. G. (1989). "Robust Statistical Modeling Using the t Distribution" (PDF). Journal of the American Statistical Association. 84 (408): 881–896. doi:10.2307/2290063. JSTOR   2290063.
15. Swindel, Benee F. (1981). "Geometry of Ridge Regression Illustrated". The American Statistician. 35 (1): 12–15. doi:10.2307/2683577. JSTOR   2683577.
16. Draper, Norman R.; van Nostrand; R. Craig (1979). "Ridge Regression and James-Stein Estimation: Review and Comments". Technometrics. 21 (4): 451–466. doi:10.2307/1268284. JSTOR   1268284.
17. Hoerl, Arthur E.; Kennard, Robert W.; Hoerl, Roger W. (1985). "Practical Use of Ridge Regression: A Challenge Met". Journal of the Royal Statistical Society, Series C. 34 (2): 114–120. JSTOR   2347363.
18. Narula, Subhash C.; Wellington, John F. (1982). "The Minimum Sum of Absolute Errors Regression: A State of the Art Survey". International Statistical Review. 50 (3): 317–326. doi:10.2307/1402501. JSTOR   1402501.
19. Stone, C. J. (1975). "Adaptive maximum likelihood estimators of a location parameter". The Annals of Statistics. 3 (2): 267–284. doi:. JSTOR   2958945.
20. Goldstein, H. (1986). "Multilevel Mixed Linear Model Analysis Using Iterative Generalized Least Squares". Biometrika. 73 (1): 43–56. doi:10.1093/biomet/73.1.43. JSTOR   2336270.
21. Theil, H. (1950). "A rank-invariant method of linear and polynomial regression analysis. I, II, III". Nederl. Akad. Wetensch., Proc. 53: 386–392, 521–525, 1397–1412. MR   0036489.; Sen, Pranab Kumar (1968). "Estimates of the regression coefficient based on Kendall's tau". Journal of the American Statistical Association . 63 (324): 1379–1389. doi:10.2307/2285891. JSTOR   2285891. MR   0258201..
22. Deaton, Angus (1992). Understanding Consumption. Oxford University Press. ISBN   978-0-19-828824-4.
23. Krugman, Paul R.; Obstfeld, M.; Melitz, Marc J. (2012). International Economics: Theory and Policy (9th global ed.). Harlow: Pearson. ISBN   9780273754091.
24. Laidler, David E. W. (1993). The Demand for Money: Theories, Evidence, and Problems (4th ed.). New York: Harper Collins. ISBN   978-0065010985.
25. Ehrenberg; Smith (2008). Modern Labor Economics (10th international ed.). London: Addison-Wesley. ISBN   9780321538963.
26. EEMP webpage Archived 2011-06-11 at the Wayback Machine
27. "Linear Regression (Machine Learning)" (PDF). University of Pittsburgh.
28. Stigler, Stephen M. (1986). . Cambridge: Harvard. ISBN   0-674-40340-1.