Linear regression is a statistical method used to model the relationship between a dependent variable and one or more independent variables. It is used to predict the value of the dependent variable based on the values of the independent variables. Linear regression is widely used in various fields such as economics, finance, social sciences, and many more. Ordinary Least Squares (OLS) is a method used to estimate the parameters of a linear regression model.
What is Linear Regression?
Linear regression is a statistical technique used to model the relationship between a dependent variable and one or more independent variables. It is called linear regression because it assumes that the relationship between the variables is linear. Linear regression models are of two types: Simple linear regression and Multiple linear regression. Simple linear regression models have only one independent variable, whereas multiple linear regression models have more than one independent variable.
What is Ordinary Least Squares (OLS)?
Ordinary Least Squares (OLS) is a method used to estimate the parameters of a linear regression model. OLS estimates the parameters by minimizing the sum of squared errors between the observed values and the predicted values. The OLS estimator is given by:
$\hat{\beta} = (X^{T}X)^{-1}X^{T}Y$
Where $\hat{\beta}$ is the estimator of the parameters, $X$ is the design matrix of the independent variables, $Y$ is the vector of the dependent variable, and $(X^{T}X)^{-1}$ is the inverse of the product of the transpose of $X$ and $X$.
Unbiasedness of OLS
An estimator is unbiased if its expected value is equal to the true value of the parameter being estimated. The OLS estimator is unbiased if the following conditions are met:
- The regression model is correctly specified.
- The errors are homoscedastic (have constant variance).
- The errors are normally distributed.
- The errors are independent of the independent variables.
If these conditions are met, then the OLS estimator is unbiased.
Consistency of OLS
An estimator is consistent if it converges to the true value of the parameter being estimated as the sample size increases. The OLS estimator is consistent if the following conditions are met:
- The regression model is correctly specified.
- The errors are homoscedastic (have constant variance).
- The errors are normally distributed.
- The errors are independent of the independent variables.
- The number of observations in the sample increases.
If these conditions are met, then the OLS estimator is consistent.
Efficiency of OLS
An estimator is efficient if it has the smallest variance among all unbiased estimators. The OLS estimator is efficient if the following conditions are met:
- The regression model is correctly specified.
- The errors are homoscedastic (have constant variance).
- The errors are normally distributed.
- The errors are independent of the independent variables.
If these conditions are met, then the OLS estimator is the most efficient among all unbiased estimators.
Optimality of OLS
The OLS estimator is also optimal in the sense that it minimizes the variance of the errors. This means that OLS is the best estimator in terms of minimizing the distance between the observed values and the predicted values.
BLUE: Best Linear Unbiased Estimator
The OLS estimator is also the Best Linear Unbiased Estimator (BLUE). This means that among all linear unbiased estimators, OLS has the smallest variance. Therefore, OLS is not only unbiased, consistent, and efficient but also the best (most efficient) estimator.
Assumptions of OLS
OLS has certain assumptions that need to be met for it to work correctly. These assumptions are:
- Linearity: The relationship between the dependent variable and the independent variables is linear.
- Independence: The errors are independent of the independent variables.
- Homoscedasticity: The errors have constant variance.
- Normality: The errors are normally distributed.
- No multicollinearity: The independent variables are not highly correlated with each other.
If these assumptions are not met, then the OLS estimator may not be reliable, and the results may be biased.
Multicollinearity
Multicollinearity is a condition where two or more independent variables are highly correlated with each other. This can lead to problems in OLS estimation as it can inflate the standard errors of the estimated coefficients. To avoid multicollinearity, one can either remove one of the correlated variables or use techniques such as Principal Component Analysis (PCA) to reduce the number of variables.
Heteroskedasticity
Heteroskedasticity is a condition where the errors have non-constant variance. This can lead to biased and inefficient OLS estimates. To correct for heteroskedasticity, one can use techniques such as Weighted Least Squares (WLS) or Generalized Least Squares (GLS).
Autocorrelation
Autocorrelation is a condition where the errors are correlated with each other. This can lead to biased and inefficient OLS estimates. To correct for autocorrelation, one can use techniques such as Cochrane-Orcutt Estimation or Newey-West Estimation.
Conclusion
In this article, we have discussed the properties of OLS as an estimator in linear regression. We have seen that OLS is unbiased, consistent, efficient, optimal, and the Best Linear Unbiased Estimator (BLUE). We have also discussed the assumptions of OLS and how violations of these assumptions can lead to biased and inefficient estimates. Finally, we have discussed some techniques to correct for violations of these assumptions.