Stats notes

I/ The OLS regression model can be represented as:

𝑌𝑖=𝛽0+𝛽1𝑋𝑖+𝜀𝑖
​
Where:
𝑌𝑖 is the observed value of the dependent variable.
𝑋𝑖 is the observed value of the independent variable.
𝛽0 and 𝛽1 are the intercept and slope coefficients, respectively.
𝜀𝑖 is the error term.


A.R-squared (R²):
R-squared is always between 0 and 1.
R-squared measures the proportion of the variance in the dependent variable that is explained by the independent variables in the model.
A higher R-squared indicates that more variance in the dependent variable is explained by the independent variables, suggesting a better fit of the model to the data.
However, R-squared alone doesn't determine the quality of the model. It's important to consider the context and purpose of the analysis.
There isn't a universally agreed-upon threshold for what constitutes a "good" R-squared, but values closer to 1.0 are generally preferred.
For example, an R-squared of 0.70 means that 70% of the variance in the dependent variable is explained by the independent variables in the model.
Limitations of R-squared: While R-squared is a useful measure of goodness of fit,it doesn't indicate the causal relationship between the independent and dependent variables, nor does it indicate whether the model is correctly specified. 
Additionally, R-squared can be inflated by adding more independent variables to the model, even if those variables are not truly related to the dependent variable.

B.Adjusted R-squared:
Adjusted R-squared is similar to R-squared but penalizes for the inclusion of additional independent variables in the model. It accounts for the degrees of freedom used by each independent variable.
Adjusted R-squared is often preferred when comparing models with different numbers of predictors.
A higher adjusted R-squared indicates a better fit of the model, similar to R-squared.
Again, there isn't a strict threshold for what constitutes a "good" adjusted R-squared, but values closer to 1.0 are generally preferred.


C.P-value associated with coefficients:
The p-value associated with a coefficient indicates the probability of observing the estimated coefficient (or one more extreme) if the null hypothesis is true (i.e., if the true population coefficient is zero).
A lower p-value suggests that the coefficient is statistically significant, meaning that it's unlikely to have occurred by chance alone.
The conventional threshold for statistical significance is often set at 0.05 (5%). Coefficients with p-values less than 0.05 are typically considered statistically significant.
However, the interpretation of p-values should be considered alongside other factors, such as effect size and practical significance.