Linear regression

In statistics, linear regression is an approach for modeling the relationship between a scalar dependent variable y and one or more explanatory variables (or independent variables) denoted X. The case of one explanatory variable is called simple linear regression. For more than one explanatory variable, the process is called multiple linear regression.[1] (This term should be distinguished from multivariate linear regression, where multiple correlated dependent variables are predicted, rather than a single scalar variable.)[2]

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 y given the value of X is assumed to be an affine function of X; less commonly, the median or some other quantile of the conditional distribution of y given X is expressed as a linear function of X. Like all forms of regression analysis, linear regression focuses on the conditional probability distribution of y given X, rather than on the joint probability distribution of y and X, 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:

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 loss 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.

Introduction to linear regression

Example of simple linear regression, which has one independent variable
Example of a cubic polynomial regression, which is a type of linear regression.

Given a data set \{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 yi and the p-vector of regressors xi is linear. This relationship is modeled through a disturbance term or error variable εi — an unobserved random variable that adds noise to the linear relationship between the dependent variable and regressors. Thus the model takes the form

y_i = \beta_1 x_{i1} + \cdots + \beta_p x_{ip} + \varepsilon_i = \mathbf{x}^{\rm T}_i\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 vector form as

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

where

\mathbf{y} = \begin{pmatrix} y_1 \\ y_2 \\ \vdots \\ y_n \end{pmatrix}, \quad
\mathbf{X} = \begin{pmatrix} \mathbf{x}^{\rm T}_1 \\ \mathbf{x}^{\rm T}_2 \\ \vdots \\ \mathbf{x}^{\rm T}_n \end{pmatrix} = \begin{pmatrix} x_{11} & \cdots & x_{1p} \\ x_{21} & \cdots & x_{2p} \\ \vdots & \ddots & \vdots \\ x_{n1} & \cdots & x_{np} \end{pmatrix},
\boldsymbol\beta = \begin{pmatrix} \beta_1 \\ \beta_2 \\ \vdots \\ \beta_p \end{pmatrix}, \quad \boldsymbol\varepsilon = \begin{pmatrix} \varepsilon_1 \\ \varepsilon_2 \\ \vdots \\ \varepsilon_n \end{pmatrix}.

Some remarks on terminology and general use:

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

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

h_i = \mathbf{t}^{\rm T}_i\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. Some methods are general enough that they can relax multiple assumptions at once, and in other cases this can be achieved by combining different extensions. 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):

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

Interpretation

The sets in the Anscombe's quartet have the same linear regression line but are themselves very different.

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 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] A commonality analysis may be helpful in disentangling the shared and unique impacts of correlated independent 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 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. 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. The difference between multivariate linear regression and multivariable linear regression should be emphasized as it causes much confusion and misunderstanding in the literature.

General linear models

The general linear model considers the situation when the response variable Y is not a scalar but a vector. Conditional linearity of E(y|x) = Bx is still assumed, with a matrix B replacing the vector β of the classical linear regression model. Multivariate analogues of OLS and GLS have been developed. The term "general linear models" is equivalent to "multivariate linear models". It should be noted the difference of "multivariate linear models" and "multivariable linear models," where the former is the same as "general linear models" and the latter is the same as "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 a response variable y that is bounded or discrete. This is used, for example:

Generalized linear models allow for an arbitrary link function g that relates the mean of the response variable to the predictors, i.e. E(y) = g(β′x). The link function is often related to the distribution of the response, and in particular it typically has the effect of transforming between the (-\infty,\infty) range of the linear predictor and the range of the response variable.

Some common examples of GLMs are:

Single index models allow some degree of nonlinearity in the relationship between x and y, while preserving the central role of the linear predictor β′x 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.[11]

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

Estimation methods

Comparison of the Theil–Sen estimator (black) and simple linear regression (blue) for a set of points with outliers.

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.

Least-squares estimation and related techniques

  • Ordinary least squares (OLS) is the simplest and thus most common estimator. It is conceptually simple and computationally straightforward. OLS estimates are commonly used to analyze both experimental and observational data.

    The OLS method minimizes the sum of squared residuals, and leads to a closed-form expression for the estimated value of the unknown parameter β:

    \hat{\boldsymbol\beta} = (\mathbf{X}^{\rm T}\mathbf{X})^{-1} \mathbf{X}^{\rm T}\mathbf{y} = \big(\,{\textstyle\sum} \mathbf{x}_i \mathbf{x}^{\rm T}_i \,\big)^{-1} \big(\,{\textstyle\sum} \mathbf{x}_i y_i \,\big).

    The estimator is unbiased and consistent if the errors have finite variance and are uncorrelated with the regressors[12]

    \operatorname{E}[\,\mathbf{x}_i\varepsilon_i\,] = 0.

    It is also efficient under the assumption that the errors have finite variance and are homoscedastic, meaning that E[εi2|xi] does not depend on i. The condition that the errors are uncorrelated with the regressors will generally be satisfied in an experiment, but in the case of observational data, it is difficult to exclude the possibility of an omitted covariate z that is related to both the observed covariates and the response variable. The existence of such a covariate will generally lead to a correlation between the regressors and the response variable, and hence to an inconsistent estimator of β. The condition of homoscedasticity can fail with either experimental or observational data. If the goal is either inference or predictive modeling, the performance of OLS estimates can be poor if multicollinearity is present, unless the sample size is large.

    In simple linear regression, where there is only one regressor (with a constant), the OLS coefficient estimates have a simple form that is closely related to the correlation coefficient between the covariate and the response.
  • Generalized least squares (GLS) is an extension of the OLS method, that allows efficient estimation of β when either heteroscedasticity, or correlations, or both are present among the error terms of the model, as long as the form of heteroscedasticity and correlation is known independently of the data. To handle heteroscedasticity when the error terms are uncorrelated with each other, GLS minimizes a weighted analogue to the sum of squared residuals from OLS regression, where the weight for the ith case is inversely proportional to var(εi). This special case of GLS is called "weighted least squares". The GLS solution to estimation problem is
    \hat{\boldsymbol\beta} = (\mathbf{X}^{\rm T}\boldsymbol\Omega^{-1}\mathbf{X})^{-1}\mathbf{X}^{\rm T}\boldsymbol\Omega^{-1}\mathbf{y},
    where Ω is the covariance matrix of the errors. GLS can be viewed as applying a linear transformation to the data so that the assumptions of OLS are met for the transformed data. For GLS to be applied, the covariance structure of the errors must be known up to a multiplicative constant.
  • Percentage least squares focuses on reducing percentage errors, which is useful in the field of forecasting or time series analysis. It is also useful in situations where the dependent variable has a wide range without constant variance, as here the larger residuals at the upper end of the range would dominate if OLS were used. When the percentage or relative error is normally distributed, least squares percentage regression provides maximum likelihood estimates. Percentage regression is linked to a multiplicative error model, whereas OLS is linked to models containing an additive error term.[13]
  • Iteratively reweighted least squares (IRLS) is used when heteroscedasticity, or correlations, or both are present among the error terms of the model, but where little is known about the covariance structure of the errors independently of the data.[14] In the first iteration, OLS, or GLS with a provisional covariance structure is carried out, and the residuals are obtained from the fit. Based on the residuals, an improved estimate of the covariance structure of the errors can usually be obtained. A subsequent GLS iteration is then performed using this estimate of the error structure to define the weights. The process can be iterated to convergence, but in many cases, only one iteration is sufficient to achieve an efficient estimate of β.[15][16]
  • Instrumental variables regression (IV) can be performed when the regressors are correlated with the errors. In this case, we need the existence of some auxiliary instrumental variables zi such that E[ziεi] = 0. If Z is the matrix of instruments, then the estimator can be given in closed form as
    \hat{\boldsymbol\beta} = (\mathbf{X}^{\rm T}\mathbf{Z}(\mathbf{Z}^{\rm T}\mathbf{Z})^{-1}\mathbf{Z}^{\rm T}\mathbf{X})^{-1}\mathbf{X}^{\rm T}\mathbf{Z}(\mathbf{Z}^{\rm T}\mathbf{Z})^{-1}\mathbf{Z}^{\rm T}\mathbf{y}.
  • Optimal instruments regression is an extension of classical IV regression to the situation where E[εi|zi] = 0.
  • Total least squares (TLS)[17] is an approach to least squares estimation of the linear regression model that treats the covariates and response variable in a more geometrically symmetric manner than OLS. It is one approach to handling the "errors in variables" problem, and is also sometimes used even when the covariates are assumed to be error-free.

Maximum-likelihood estimation and related techniques

Other estimation techniques

Further discussion

In statistics and numerical analysis, the problem of numerical methods for linear least squares is an important one because linear regression models are one of the most important types of model, both as formal statistical models and for exploration of data sets. The majority of statistical computer packages contain facilities for regression analysis that make use of linear least squares computations. Hence it is appropriate that considerable effort has been devoted to the task of ensuring that these computations are undertaken efficiently and with due regard to numerical precision.

Individual statistical analyses are seldom undertaken in isolation, but rather are part of a sequence of investigatory steps. Some of the topics involved in considering numerical methods for linear least squares relate to this point. Thus important topics can be

Fitting of linear models by least squares often, but not always, arises in the context of statistical analysis. It can therefore be important that considerations of computational efficiency for such problems extend to all of the auxiliary quantities required for such analyses, and are not restricted to the formal solution of the linear least squares problem.

Matrix calculations, like any others, are affected by rounding errors. An early summary of these effects, regarding the choice of computational methods for matrix inversion, was provided by Wilkinson.[26]

Using linear algebra

It follows that one can find a "best" approximation of another function by minimizing the area between two functions, a continuous function f on [a,b] and a function g\in W where W is a subspace of C[a,b]:

\text{Area }= \int_{a}^{b}\left\vert f(x) - g(x)\right\vert \, dx,

all within the subspace W. Due to the frequent difficulty of evaluating integrands involving absolute value, one can instead define

\int_{a}^{b} [ f(x) - g(x) ] ^2\, dx

as an adequate criterion for obtaining the least squares approximation, function g, of f with respect to the inner product space W.

As such, \lVert f-g \rVert ^2 or, equivalently, \lVert f-g \rVert, can thus be written in vector form:

\int_{a}^{b} [ f(x)-g(x) ]^2\, dx = \left\langle f-g , f-g\right\rangle = \lVert f-g\rVert ^2.

In other words, the least squares approximation of f is the function g\in \text{ subspace } W closest to f in terms of the inner product \left \langle f,g \right \rangle. Furthermore, this can be applied with a theorem:

Let f be continuous on [ a,b ], and let W be a finite-dimensional subspace of C[a,b]. The least squares approximating function of f with respect to W is given by
g = \left \langle f,\vec w_1 \right \rangle \vec w_1 + \left \langle f,\vec w_2 \right \rangle \vec w_2 + \dots + \left \langle f,\vec w_n \right \rangle \vec w_n,
where B = \{\vec w_1 , \vec w_2 , \dots , \vec w_n \} is an orthonormal basis for W.

Applications of linear regression

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

Main article: Trend estimation

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, suppose we have a regression model in which cigarette smoking is the independent variable of interest, and the dependent variable is lifespan measured in years. Researchers might include socio-economic status as an additional independent variable, to ensure that any observed effect of smoking on lifespan is not due to some effect of education or income. 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

Main article: Econometrics

Linear regression is the predominant empirical tool in economics. For example, it is used to predict consumption spending,[27] fixed investment spending, inventory investment, purchases of a country's exports,[28] spending on imports,[28] the demand to hold liquid assets,[29] labor demand,[30] and labor supply.[30]

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.[31]

See also

Notes

  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 equation has two or more explanatory variables on the 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 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.
  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. 1 2 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. 1 2 Efron, Bradley; Hastie, Trevor; Johnstone,Iain; Tibshirani,Robert (2004). "Least Angle Regression". The Annals of Statistics 32 (2): 407–451. doi:10.1214/009053604000000067. JSTOR 3448465.
  7. 1 2 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. 1 2 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. Regression Analysis: A Constructive Critique. Sage. doi:10.1177/0734016807304871.
  10. ↑ Warne, R. T. (2011). Beyond multiple regression: Using commonality analysis to better understand R2 results. Gifted Child Quarterly, 55, 313-318. doi:10.1177/0016986211422217
  11. ↑ 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.
  12. ↑ Lai, T.L.; Robbins, H.; Wei, C.Z. (1978). "Strong consistency of least squares estimates in multiple regression". PNAS 75 (7): 3034–3036. Bibcode:1978PNAS...75.3034L. doi:10.1073/pnas.75.7.3034. JSTOR 68164.
  13. ↑ Tofallis, C (2009). "Least Squares Percentage Regression". Journal of Modern Applied Statistical Methods 7: 526–534. doi:10.2139/ssrn.1406472.
  14. ↑ del Pino, Guido (1989). "The Unifying Role of Iterative Generalized Least Squares in Statistical Algorithms". Statistical Science 4 (4): 394–403. doi:10.1214/ss/1177012408. JSTOR 2245853.
  15. ↑ Carroll, Raymond J. (1982). "Adapting for Heteroscedasticity in Linear Models". The Annals of Statistics 10 (4): 1224–1233. doi:10.1214/aos/1176345987. JSTOR 2240725.
  16. ↑ Cohen, Michael; Dalal, Siddhartha R.; Tukey,John W. (1993). "Robust, Smoothly Heterogeneous Variance Regression". Journal of the Royal Statistical Society, Series C 42 (2): 339–353. JSTOR 2986237.
  17. ↑ Nievergelt, Yves (1994). "Total Least Squares: State-of-the-Art Regression in Numerical Analysis". SIAM Review 36 (2): 258–264. doi:10.1137/1036055. JSTOR 2132463.
  18. ↑ Lange, Kenneth L.; Little, Roderick J. A.; Taylor,Jeremy M. G. (1989). "Robust Statistical Modeling Using the t Distribution". Journal of the American Statistical Association 84 (408): 881–896. doi:10.2307/2290063. JSTOR 2290063.
  19. ↑ Swindel, Benee F. (1981). "Geometry of Ridge Regression Illustrated". The American Statistician 35 (1): 12–15. doi:10.2307/2683577. JSTOR 2683577.
  20. ↑ 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.
  21. ↑ 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.
  22. ↑ 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.
  23. ↑ Stone, C. J. (1975). "Adaptive maximum likelihood estimators of a location parameter". The Annals of Statistics 3 (2): 267–284. doi:10.1214/aos/1176343056. JSTOR 2958945.
  24. ↑ 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.
  25. ↑ 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.
  26. ↑ Wilkinson, J.H. (1963) "Chapter 3: Matrix Computations", Rounding Errors in Algebraic Processes, London: Her Majesty's Stationery Office (National Physical Laboratory, Notes in Applied Science, No.32)
  27. ↑ Deaton, Angus (1992). Understanding Consumption. Oxford University Press. ISBN 0-19-828824-7.
  28. 1 2 Krugman, Paul R.; Obstfeld, M.; Melitz, Marc J. (2012). International Economics: Theory and Policy (9th global ed.). Harlow: Pearson. ISBN 9780273754091.
  29. ↑ Laidler, David E. W. (1993). "The Demand for Money: Theories, Evidence, and Problems" (4th ed.). New York: Harper Collins. ISBN 0065010981.
  30. 1 2 Ehrenberg; Smith (2008). Modern Labor Economics (10th international ed.). London: Addison-Wesley. ISBN 9780321538963.
  31. ↑ EEMP webpage

References

Further reading

External links

The Wikibook R Programming has a page on the topic of: Linear Models
Wikiversity has learning materials about Linear regression
This article is issued from Wikipedia - version of the Tuesday, February 02, 2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.