If you want to estimate a linear regression model’s unspecified parameters, you can incorporate the ordinary least squares estimator. When you implement OLS regression, you inherit certain assumptions like linearity in the error term and coefficients. Understanding what the basic OLS regression assumptions are can help you understand if you need to use another estimation method or implement corrective measures to get accurate results. In this article, we discuss seven OLS regression assumptions so that you can learn how to minimize data variances.
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What is OLS regression?
OLS, or ordinary least squares regression, is a method that statisticians use to approximate the unspecified parameters in a linear regression model. It’s important to note that while OLS isn’t a model itself, it’s an estimator for the parameters of a linear regression model. Whenever a linear regression model accurately fulfills its assumptions, statisticians can observe coefficient estimates that are close to the actual population values.
7 OLS regression assumptions
Here’s a list of seven OLS regression assumptions:
1. The regression model has linearity in its error term and coefficients
The first OLS regression assumption refers to the estimator’s linear regression model. It’s the only assumption that refers to both the OLS estimator and the linear regression model, while all the other assumptions refer to only the OLS estimator. It’s linear because all terms follow one of two conditions. The first condition is that the terms are constant numbers. The second condition is that the terms are the multiplication of an independent variable and a parameter. The formula looks like this:
Linear regression formula: Y = β₀ + β₁X₁ + β₂X₂ + … + βₖXₖ + ε
The parameters that the OLS estimator predicts are the beta (β) symbols. The random error that the linear regression model produces is the epsilon (ε) symbol. The Xs are the independent variables that the statistician can alter.
2. The error term’s population mean is zero
The error term is the number that considers any variation in the “Y,” or the dependent variable, that the independent variables fail to show. In ideal circumstances, random chance determines the error term’s value. Otherwise, it’s challenging to create an unbiased value. To meet this expectation, a good assumption is for the error term’s population mean to equal zero. You can better understand this assumption by considering an example.
Suppose that you have a linear regression model and an OLS estimator with an average error of positive nine. This average error is a number other than zero, which means that the linear regression model fails to accurately predict the observed values. Positive average errors mean that the model under-predicts values, while negative average errors mean that the model over-predicts values. When you incorporate the constant, which is β₀, into your linear regression model, you don’t have to consider this assumption. This is because the mean of the residuals equals zero with the constant in your formula.
3. There are no correlations between the independent variables and the error term
Another important OLS regression assumption is that there are no correlations between the independent variables and the error term. If there are correlations, it’s possible to predict the error term by using the independent variable. This would mean that the error term represents a predictable random error, which violates the second assumption in this list.
Some statisticians refer to this assumption as homogeneity. Several factors may cause the opposite of endogeneity, which is called endogeneity, to occur. Some of these factors include independent variables with measurement errors or committed variable biases.
4. Each observation of the error term is independent of others
When you implement the OLS estimator in a linear regression model, you can make independent observations of the error term. A statistician may have to implement corrective measures if they notice correlations between different observations of the error term. For example, consider a linear regression model that has a positive error for one observation.
If a statistician can assume that the subsequent error is also positive, this implies a positive correlation. Similarly, if a linear regression model has a positive error for one observation and an individual can assume a negative error for the next observation, there’s a negative correlation present. If these circumstances are present, the linear regression model fails to meet this fourth OLS regression assumption. You can solve this potential problem by adding an independent variable that includes this information.
5. The error term’s variance is constant
Some statisticians refer to this OLS assumption as homoscedasticity. It states that the error term of the OLS estimator has constant variance. This means that the variance remains the same across a single observation or a range of observations. You can confirm that this assumption is true by plotting the true values versus the residuals. If you find that the spread of the residuals continues to get larger in one direction, the model fails to meet the assumption of homoscedasticity.
6. There are no independent variables that are perfect linear functions of other variables
If two variables have a coefficient of negative one or positive one, a perfect correlation is present. Some models can account for perfect correlation, but OLS regression cannot. A linear regression model with an OLS estimator that has two independent variables with perfect correlation cannot be displayed properly. It becomes necessary to remove one of the independent variables to successfully continue with the graphing process.
7. The error term adheres to a normal distribution pattern
This is the only OLS regression assumption that’s optional. Ideally, the error term adheres to a normal distribution pattern. This allows statisticians to produce reliable prediction intervals, generate accurate confidence intervals and conduct informative hypothesis testing. Even if the error term follows an atypical distribution pattern, the linear regression model can still generate unbiased approximations that have little to no variance.
I hope you find this article helpful.
