5.1 Model formulation

Regression models under complex sampling designs make it possible to move beyond descriptive statistics and approach causal or predictive explanations, provided that the particularities of the sampling design are recognized and adjusted for. Their correct application opens the possibility of examining how sociodemographic and economic characteristics are associated with different outcomes of interest, contributing key evidence for public policy formulation.

As a starting point, it is useful to present the general structure of regression models. The most basic formulation corresponds to the simple linear regression model, which describes the relationship between a response variable and a single explanatory variable through the following expression:

\[ y = \beta_{0} + \beta_{1}x + \varepsilon \]

Here, \(y\) represents the response (dependent) variable, \(x\) the explanatory (independent) variable, \(\beta_{0}\) the model intercept, \(\beta_{1}\) the coefficient associated with the independent variable, and \(\varepsilon\) the error term, which captures the variability not explained by the model and can be interpreted as the difference between the observed value and the value estimated by the model, denoted by \(\hat{y}_k\).

In many empirical applications, and especially in the analysis of household surveys, the phenomenon of interest depends simultaneously on multiple factors. In these cases, multiple linear regression models are used, incorporating several explanatory variables:

\[ y = \beta_{0} + \beta_{1}x_{1} + \cdots + \beta_{p}x_{p} + \varepsilon, \]

where each coefficient \(\beta_{j}\) quantifies the association between the response variable and the corresponding covariate \(x_{j}\), holding the other variables included in the model constant. More compactly, the model can be expressed using matrix notation as:

\[ y_{k} = \mathbf{x}_{k}\boldsymbol{\beta} + \varepsilon_{k}, \quad k=1,\ldots,n, \]

where \(\mathbf{x}_{k} = [1, x_{1i}, \ldots, x_{pi}]\) represents the vector of covariates associated with unit \(k\), while \(\boldsymbol{\beta} = [\beta_{0}, \beta_{1}, \ldots, \beta_{p}]'\) corresponds to the vector of unknown model parameters. In this context, the expected value of the response variable conditional on the set of covariates can be expressed as:

\[ E(y \mid \mathbf{x}) = {\beta}_{0} + {\beta}_{1}x_{1} + \cdots + {\beta}_{p}x_{p} \]

Linear regression models are based on a series of theoretical assumptions that guarantee the validity of the estimates and the inferences derived from the model. First, it is assumed that the expected value of the residuals conditional on the covariates is equal to zero, that is, \(E(\varepsilon_{k} \mid \mathbf{x}_{k}) = 0\), which implies the absence of systematic bias in the estimation of the response variable. Likewise, homogeneity of the error variance is assumed, so that residual variability remains constant for all values of the covariates, that is, \(Var(\varepsilon_{k} \mid \mathbf{x}_{k}) = \sigma^2\). In addition, the errors are considered to follow a normal distribution with zero mean and constant variance, expressed as \(\varepsilon_{k} \mid \mathbf{x}_{k} \sim N(0,\sigma^2)\), and the residuals associated with different observations are assumed to be independent of one another, such that \(Cov(\varepsilon_{k},\varepsilon_{j}\mid \mathbf{x}_{k},\mathbf{x}_{j})=0\).

The validity of linear regression models depends on the fulfillment of several classical assumptions widely discussed in the specialized literature. Taken together, these assumptions allow the estimators obtained through the model to have desirable statistical properties, such as unbiasedness, efficiency, and consistency.