TOPIC INFO (UGC NET)
TOPIC INFO – UGC NET (Economics)
SUB-TOPIC INFO – Statistics and Econometrics (UNIT 3)
CONTENT TYPE – Detailed Notes
What’s Inside the Chapter? (After Subscription)
1. Introduction
2. The Method of Ordinary Least Squares (OLS)
3. The Classical Linear Regression Model (CLRM) Assumptions
4. The Gauss-Markov Theorem and the Property of BLUE
4.1. Linear (L)
4.2. Unbiased (U)
4.3. Best (B) Minimum Variance
4.4. Estimator (E)
4.5. Combined Statement of the Gauss-Markov Theorem
5. Consequences of Violating CLRM Assumptions for the BLUE Property
6. Goodness of FIT: The Coefficient of Determination
7. Standard Error of the Regression and Hypothesis Testing
8. Formulas
9. Conclusion
Access This Topic With Any Subscription Below:
- UGC NET Economics
- UGC NET Economics + Book Notes
Linear Regression Models and their Properties – BLUE
UGC NET ECONOMICS
Statistics and Econometrics (UNIT 3)
Introduction
Regression analysis is a statistical technique used to study the nature and strength of relationship between a dependent variable (also called the regressand, explained variable, or response variable) and one or more independent variables (also called regressors, explanatory variables, or predictors). While correlation merely measures the degree of association between variables without distinguishing cause from effect, regression goes further by expressing the dependent variable as a function of the independent variable(s), thereby enabling prediction and estimation. The term “regression” was originally coined by Sir Francis Galton in the context of his study on heredity, where he observed that the heights of children tended to “regress toward the mean” height of the population, a phenomenon he termed “regression toward mediocrity.”
The simple linear regression model, involving one explanatory variable, is expressed as:
$$Y_i = \beta_0 + \beta_1 X_i + u_i$$
where \(Y_i\) is the value of the dependent variable for the i-th observation, \(X_i\) is the value of the independent variable, \(\beta_0\) is the intercept term (the value of Y when X is zero), \(\beta_1\) is the slope coefficient (measuring the change in Y for a unit change in X), and \(u_i\) is the stochastic disturbance term or error term, which captures the combined effect of all omitted variables, measurement errors, and inherent randomness in human behaviour.
The multiple linear regression model, involving more than one explanatory variable, is expressed in general form as:
$$Y_i = \beta_0 + \beta_1 X_{1i} + \beta_2 X_{2i} + \dots + \beta_k X_{ki} + u_i$$
or in matrix notation as:
$$Y = X\beta + u$$
where Y is an \(n \times 1\) vector of observations on the dependent variable, X is an \(n \times k\) matrix of observations on the explanatory variables (including a column of ones for the intercept), \(\beta\) is a \(k \times 1\) vector of unknown parameters, and u is an \(n \times 1\) vector of disturbance terms.
A crucial distinction exists between the Population Regression Function (PRF), which represents the true, unknown relationship in the entire population, and the Sample Regression Function (SRF), which is the estimated relationship obtained from sample data and is used to estimate the unknown parameters of the PRF. The SRF is written as:
$$\hat{Y_i} = \hat{\beta_0} + \hat{\beta_1} X_i$$
where \(\hat{\beta_0}\) and \(\hat{\beta_1}\) are estimators of the true population parameters \(\beta_0\) and \(\beta_1\), and \(\hat{Y_i}\) is the estimated or fitted value of Y.