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Ordinary Least Squares (OLS) is the nearly common estimation method for linear models—and that'southward truthful for a good reason. Every bit long as your model satisfies the OLS assumptions for linear regression, yous can rest easy knowing that you lot're getting the best possible estimates.

Regression is a powerful assay that tin can analyze multiple variables simultaneously to respond circuitous research questions. However, if you don't satisfy the OLS assumptions, yous might not be able to trust the results.

In this post, I embrace the OLS linear regression assumptions, why they're essential, and assistance you determine whether your model satisfies the assumptions.

What Does OLS Estimate and What are Good Estimates?

Showtime, a bit of context.

Regression analysis is like other inferential methodologies. Our goal is to draw a random sample from a population and use it to approximate the properties of that population.

In regression analysis, the coefficients in the regression equation are estimates of the actual population parameters. Nosotros want these coefficient estimates to be the all-time possible estimates!

Suppose you request an guess—say for the cost of a service that you are considering. How would you ascertain a reasonable estimate?

1. The estimates should tend to be correct on target. They should non be systematically as well high or likewise low. In other words, they should exist unbiased or correct on average.
2. Recognizing that estimates are almost never exactly right, you want to minimize the discrepancy betwixt the estimated value and bodily value. Large differences are bad!

These two properties are exactly what we need for our coefficient estimates!

When your linear regression model satisfies the OLS assumptions, the process generates unbiased coefficient estimates that tend to be relatively close to the true population values (minimum variance). In fact, the Gauss-Markov theorem states that OLS produces estimates that are ameliorate than estimates from all other linear model interpretation methods when the assumptions hold true.

For more than data near the implications of this theorem on OLS estimates, read my post: The Gauss-Markov Theorem and BLUE OLS Coefficient Estimates.

The Seven Classical OLS Assumptions

Like many statistical analyses, ordinary to the lowest degree squares (OLS) regression has underlying assumptions. When these classical assumptions for linear regression are truthful, ordinary least squares produces the best estimates. Yet, if some of these assumptions are not true, you might need to employ remedial measures or apply other estimation methods to ameliorate the results.

Many of these assumptions describe properties of the error term. Unfortunately, the error term is a population value that nosotros'll never know. Instead, we'll employ the next best thing that is available—the residuals. Residuals are the sample gauge of the error for each observation.

Residuals = Observed value – the fitted value

When it comes to checking OLS assumptions, assessing the residuals is crucial!

There are seven classical OLS assumptions for linear regression. The start half dozen are mandatory to produce the all-time estimates. While the quality of the estimates does not depend on the seventh supposition, analysts often evaluate it for other important reasons that I'll cover.

OLS Assumption 1: The regression model is linear in the coefficients and the error term

This supposition addresses the functional grade of the model. In statistics, a regression model is linear when all terms in the model are either the constant or a parameter multiplied by an contained variable. You build the model equation only by adding the terms together. These rules constrain the model to one type:

In the equation, the betas (βs) are the parameters that OLS estimates. Epsilon (ε) is the random error.

In fact, the defining feature of linear regression is this functional form of the parameters rather than the power to model curvature. Linear models can model curvature by including nonlinear variables such as polynomials and transforming exponential functions.

To satisfy this assumption, the correctly specified model must fit the linear pattern.

Related posts: The Departure Between Linear and Nonlinear Regression and How to Specify a Regression Model

OLS Supposition 2: The error term has a population mean of naught

The error term accounts for the variation in the dependent variable that the contained variables do not explain. Random chance should determine the values of the fault term. For your model to be unbiased, the average value of the error term must equal nothing.

Suppose the average error is +vii. This non-zero boilerplate error indicates that our model systematically underpredicts the observed values. Statisticians refer to systematic error similar this as bias, and it signifies that our model is inadequate because it is not right on average.

Stated another fashion, nosotros want the expected value of the error to equal zippo. If the expected value is +7 rather than zero, function of the fault term is predictable, and we should add that information to the regression model itself. We want merely random fault left for the fault term.

You don't need to worry almost this assumption when you lot include the constant in your regression model considering it forces the mean of the residuals to equal zero. For more information nigh this supposition, read my mail near the regression constant.

OLS Assumption 3: All independent variables are uncorrelated with the error term

If an independent variable is correlated with the mistake term, we can use the independent variable to predict the fault term, which violates the notion that the error term represents unpredictable random mistake. We need to notice a way to incorporate that information into the regression model itself.

This assumption is also referred to as exogeneity. When this type of correlation exists, there is endogeneity. Violations of this supposition can occur because there is simultaneity between the independent and dependent variables, omitted variable bias, or measurement error in the independent variables.

Violating this supposition biases the coefficient estimate. To understand why this bias occurs, go on in mind that the error term always explains some of the variability in the dependent variable. However, when an independent variable correlates with the mistake term, OLS incorrectly attributes some of the variance that the error term actually explains to the contained variable instead. For more information near violating this assumption, read my post nearly confounding variables and omitted variable bias.

Related post: What are Contained and Dependent Variables?

OLS Assumption 4: Observations of the error term are uncorrelated with each other

One observation of the fault term should not predict the next ascertainment. For instance, if the error for ane observation is positive and that systematically increases the probability that the following error is positive, that is a positive correlation. If the subsequent mistake is more than likely to have the opposite sign, that is a negative correlation. This problem is known both as serial correlation and autocorrelation. Serial correlation is most likely to occur in time series models.

For example, if sales are unexpectedly high on one day, then they are probable to exist higher than average on the next day. This type of correlation isn't an unreasonable expectation for some subject field areas, such every bit aggrandizement rates, Gross domestic product, unemployment, and and so on.

Assess this supposition by graphing the residuals in the order that the data were collected. You want to see randomness in the plot. In the graph for a sales model, at that place is a cyclical pattern with a positive correlation.

As I've explained, if yous accept information that allows yous to predict the error term for an observation, y'all must incorporate that information into the model itself. To resolve this upshot, you lot might need to add together an independent variable to the model that captures this information. Analysts usually utilise distributed lag models, which use both current values of the dependent variable and past values of contained variables.

For the sales model above, we demand to add variables that explains the cyclical design.

Serial correlation reduces the precision of OLS estimates. Analysts tin can also use fourth dimension series analysis for time dependent effects.

An alternative method for identifying autocorrelation in the residuals is to appraise the autocorrelation function, which is a standard tool in time series assay.

Related post: Introduction to Time Series Analysis

OLS Assumption v: The mistake term has a abiding variance (no heteroscedasticity)

The variance of the errors should be consistent for all observations. In other words, the variance does not modify for each observation or for a range of observations. This preferred condition is known as homoscedasticity (same scatter). If the variance changes, we refer to that equally heteroscedasticity (different scatter).

The easiest way to check this supposition is to create a residuals versus fitted value plot. On this type of graph, heteroscedasticity appears equally a cone shape where the spread of the residuals increases in one direction. In the graph below, the spread of the residuals increases as the fitted value increases.

Heteroscedasticity reduces the precision of the estimates in OLS linear regression.

Related post: Heteroscedasticity in Regression Analysis

Annotation: When assumption 4 (no autocorrelation) and 5 (homoscedasticity) are both true, statisticians say that the error term is independent and identically distributed (IID) and refer to them every bit spherical errors.

OLS Assumption 6: No independent variable is a perfect linear office of other explanatory variables

Perfect correlation occurs when two variables have a Pearson's correlation coefficient of +ane or -one. When 1 of the variables changes, the other variable also changes by a completely stock-still proportion. The two variables move in unison.

Perfect correlation suggests that two variables are dissimilar forms of the same variable. For case, games won and games lost take a perfect negative correlation (-1). The temperature in Fahrenheit and Celsius have a perfect positive correlation (+1).

Ordinary least squares cannot distinguish one variable from the other when they are perfectly correlated. If y'all specify a model that contains contained variables with perfect correlation, your statistical software can't fit the model, and it will display an fault message. You must remove one of the variables from the model to proceed.

Perfect correlation is a prove stopper. However, your statistical software can fit OLS regression models with imperfect but potent relationships between the independent variables. If these correlations are high enough, they can cause problems. Statisticians refer to this condition every bit multicollinearity, and it reduces the precision of the estimates in OLS linear regression.

Related post: Multicollinearity in Regression Analysis: Bug, Detection, and Solutions

OLS Supposition 7: The error term is unremarkably distributed (optional)

OLS does not require that the error term follows a normal distribution to produce unbiased estimates with the minimum variance. Still, satisfying this assumption allows y'all to perform statistical hypothesis testing and generate reliable confidence intervals and prediction intervals.

The easiest way to determine whether the residuals follow a normal distribution is to assess a normal probability plot. If the residuals follow the straight line on this blazon of graph, they are ordinarily distributed. They look practiced on the plot below!

If you demand to obtain p-values for the coefficient estimates and the overall test of significance, check this assumption!

Why Yous Should Care Nearly the Classical OLS Assumptions

In a nutshell, your linear model should produce residuals that take a hateful of zero, have a constant variance, and are non correlated with themselves or other variables.

If these assumptions agree true, the OLS process creates the best possible estimates. In statistics, estimators that produce unbiased estimates that have the smallest variance are referred to every bit being "efficient." Efficiency is a statistical concept that compares the quality of the estimates calculated by dissimilar procedures while holding the sample size constant. OLS is the most efficient linear regression figurer when the assumptions hold true.

Another benefit of satisfying these assumptions is that every bit the sample size increases to infinity, the coefficient estimates converge on the actual population parameters.

If your error term too follows the normal distribution, y'all can safely use hypothesis testing to determine whether the contained variables and the entire model are statistically significant. You can besides produce reliable confidence intervals and prediction intervals.

Knowing that y'all're maximizing the value of your data past using the about efficient methodology to obtain the all-time possible estimates should set your listen at ease. Information technology's worthwhile checking these OLS assumptions! The all-time way to assess them is past using residue plots. To learn how to do this, read my mail near using residue plots!

If you're learning regression and like the approach I employ in my blog, bank check out my eBook!

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Source: https://statisticsbyjim.com/regression/ols-linear-regression-assumptions/

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