Multiple linear regression with the Ordinary Least Squares (OLS) method is one of the statistical techniques used to assess the influence of two or more independent variables on a dependent variable. The OLS method is carried out by minimizing the sum of squared errors between the model’s predictions and the actual values of the dependent variable.

The equation for multiple linear regression involves a dependent variable (Y) and two or more independent variables (X_{1}, X_{2}, …, X_{n}) used to predict Y. The general equation for multiple linear regression using the OLS method is as follows:

Y = β_{0} + β_{1}X_{1} + β_{2}X_{2} + … + β_{n}X_{n} + ε

Where:

Y = Dependent variable.

X_{1}, X_{2}, …, X_{n} = Independent variables.

β_{0} = Intercept.

β_{1}, β_{2}, …, β_{n} = Regression coefficient estimates. ε = Error term.

In this equation for multiple linear regression, researchers need to evaluate how well this model fits the data. Some commonly used evaluation metrics include R-squared (coefficient of determination) and hypothesis tests on the coefficients (F-statistics and T-statistics).

**Assumptions Required for Multiple Linear Regression Using OLS**

Assumptions in multiple linear regression with the OLS (Ordinary Least Squares) method are essential to ensure that the regression model yields consistent and unbiased estimates. Commonly performed assumption tests include homoscedasticity, normality of residuals, multicollinearity, and linearity. An additional autocorrelation test needs to be conducted in time series data.

The homoscedasticity test aims to confirm that the residual variances remain constant across all levels of independent variables. Several diagnostic tools can be employed for the homoscedasticity test, including the Breusch-Pagan test.

The normality of residuals test is conducted to verify that residuals follow a normal distribution. Several tests are available to detect the normality of residuals, and one of them is the Shapiro-Wilk test.

Meanwhile, the multicollinearity test ensures no strong linear relationships among independent variables. You can calculate the correlation matrix between independent variables or conduct the Variance Inflation Factor (VIF) analysis to detect multicollinearity.

Furthermore, the autocorrelation test is carried out to confirm the absence of systematic correlation patterns between residuals at previous and current time points. The Durbin-Watson test can be used to detect autocorrelation in time series data.

**When assumptions are not met**

If the assumptions of multiple linear regression are not met, it can affect the validity and reliability of the regression analysis results. In such circumstances, achieving the Best Linear Unbiased Estimator (BLUE) may not be possible.

If there is evidence of heteroskedasticity, researchers can attempt data transformation. Additionally, researchers may explore alternative methods, such as Generalized Least Squares (GLS).

Researchers can also try data transformation if residuals do not follow a normal distribution. Moreover, if there are indications of multicollinearity (strong relationships among independent variables), researchers may consider removing one of the variables. Furthermore, researchers can collect additional data or modify the model to fit the data better.

If the unmet assumptions are sufficiently serious, seeking a more appropriate alternative model or conducting a more suitable analysis may be necessary.

Those are the key points regarding the assumptions required in multiple linear regression analysis using the OLS method. I hope this information is helpful and adds value to your knowledge. Stay tuned for updates in the following article. Thank you.