Introduction

In the world of statistics and data analysis, the ordinary least squares (OLS) regression is a powerful tool used to model the relationship between variables. In this chapter, we will dive into the details of interpreting the results of an OLS regression when it involves a single explanatory variable. By the end of this chapter, you will have a solid grasp of how to extract meaningful insights from OLS regression outputs and make informed decisions based on the results.

Interpreting OLS Regression Results

When conducting an OLS regression with a single explanatory variable, the primary goal is to understand how changes in the explanatory variable impact the dependent variable. Let’s break down the process of interpreting OLS regression results:

1) Coefficients: In the regression equation, the coefficient of the single explanatory variable represents the estimated change in the dependent variable for a one-unit change in the explanatory variable, while holding other variables constant. If the coefficient is positive, it implies a positive relationship, and if negative, it implies a negative relationship.

Example: Suppose we have an OLS regression: $$Y={\beta }_0+{\beta }_{1}X+\epsilon$$

If ${\beta }_1$ is 0.75, it means that for every unit increase in $X$, $Y$ is expected to increase by 0.75 units.

2) Intercept (Constant): The intercept, denoted as ${\beta }_0$, represents the expected value of the dependent variable when the explanatory variable is zero. It might or might not have a practical interpretation, depending on the context.

3) R-squared $\left(R^2\right)$: $R^2$ measures the proportion of the variance in the dependent variable that is explained by the regression model. A higher $R^2$ indicates a better fit of the model to the data.

Example: If $R^2$ is 0.85, it means that 85% of the variance in $Y$ can be explained by changes in $X$.

4) Standard Error of Coefficient: The standard error (SE) measures the variability of the coefficient estimate. A smaller SE indicates more precise estimates.

5) Hypothesis Test and p-Value: Hypothesis tests are performed to determine whether the coefficient is statistically significant. The p-value associated with the coefficient indicates the probability of observing such an extreme result if the null hypothesis (no relationship) were true. A low pvalue (typically $<0.05$ ) suggests a significant relationship.