Linear Regression - Introduction To Linear Regression

Introduction To Linear Regression

Given a data set of n statistical units, a linear regression model assumes that the relationship between the dependent variable y_i and the p-vector of regressors x_i is linear. This relationship is modelled through a disturbance term or error variable ε_i — 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 x_iTβ is the inner product between vectors x_i and β.

Often these n equations are stacked together and written in vector form as

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

where

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

Some remarks on terminology and general use:

is called the regressand, exogenous variable, response variable, measured variable, or dependent variable (see dependent and independent variables.) 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.
are called regressors, endogenous variables, explanatory variables, covariates, input variables, predictor variables, or independent variables (see dependent and independent variables, but not to be confused with independent random variables). The matrix is sometimes called the design matrix.
- Usually a constant is included as one of the regressors. For example we can take x_i1 = 1 for i = 1, ..., 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 regressors x_ij may be viewed either as random variables, which we simply observe, or they can be considered as predetermined fixed values which we can choose. 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.
is a p-dimensional parameter vector. Its elements are also called effects, or regression coefficients. Statistical estimation and inference in linear regression focuses on β.
is called the error term, disturbance term, or noise. This variable captures all other factors which influence the dependent variable y_i other than the regressors x_i. The relationship between the error term and the regressors, for example whether they are correlated, is a crucial step in formulating a linear regression model, as it will determine the method to use for estimation.

Example. Consider a situation where a small ball is being tossed up in the air and then we measure its heights of ascent h_i at various moments in time t_i. Physics tells us that, ignoring the drag, the relationship can be modelled 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 ε_i 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 x_i = (x_i1, x_i2) = (t_i, t_i2), the model takes on the standard form

$h_i = \mathbf{x}^{\rm T}_i\boldsymbol\beta + \varepsilon_i.$

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