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Assumptions Of Linear Regression

By Moloy De posted Thu December 01, 2022 06:45 AM

  
Standard linear regression models with standard estimation techniques make a number of assumptions about the predictor variables, the response variables and their relationship. Numerous extensions have been developed that allow each of these assumptions to be relaxed i.e. reduced to a weaker form, and in some cases eliminated entirely. Generally these extensions make the estimation procedure more complex and time-consuming, and may also require more data in order to produce an equally precise model.


Weak exogeneity.

This essentially means that the predictor variables x can be treated as fixed values, rather than random variables. This means, for example, that the predictor variables are assumed to be error-free—that is, not contaminated with measurement errors. Although this assumption is not realistic in many settings, dropping it leads to significantly more difficult errors-in-variables models.


Linearity.

This means that the mean of the response variable is a linear combination of the parameters i.e. the regression coefficients and the predictor variables. Note that this assumption is much less restrictive than it may at first seem. Because the predictor variables are treated as fixed values, linearity is really only a restriction on the parameters. The predictor variables themselves can be arbitrarily transformed, and in fact multiple copies of the same underlying predictor variable can be added, each one transformed differently. This technique is used, for example, in polynomial regression, which uses linear regression to fit the response variable as an arbitrary polynomial function up to a given degree of a predictor variable. With this much flexibility, models such as polynomial regression often have "too much power", in that they tend to overfit the data. As a result, some kind of regularization must typically be used to prevent unreasonable solutions coming out of the estimation process. Common examples are ridge regression and lasso regression. Bayesian linear regression can also be used, which by its nature is more or less immune to the problem of overfitting. In fact, ridge regression and lasso regression can both be viewed as special cases of Bayesian linear regression, with particular types of prior distributions placed on the regression coefficients.


Constant variance or homoscedasticity.

This means that the variance of the errors does not depend on the values of the predictor variables. Thus the variability of the responses for given fixed values of the predictors is the same regardless of how large or small the responses are. This is often not the case, as a variable whose mean is large will typically have a greater variance than one whose mean is small.
The absence of homoscedasticity is called heteroscedasticity. In order to check this assumption, a plot of residuals versus predicted values or the values of each individual predictor can be examined for a "fanning effect" i.e., increasing or decreasing vertical spread as one moves left to right on the plot. A plot of the absolute or squared residuals versus the predicted values or each predictor can also be examined for a trend or curvature. Formal tests can also be used; see Heteroscedasticity. The presence of heteroscedasticity will result in an overall "average" estimate of variance being used instead of one that takes into account the true variance structure. This leads to less precise but in the case of ordinary least squares, not biased parameter estimates and biased standard errors, resulting in misleading tests and interval estimates. The mean squared error for the model will also be wrong. Various estimation techniques including weighted least squares and the use of heteroscedasticity-consistent standard errors can handle heteroscedasticity in a quite general way. Bayesian linear regression techniques can also be used when the variance is assumed to be a function of the mean. It is also possible in some cases to fix the problem by applying a transformation to the response variable e.g., fitting the logarithm of the response variable using a linear regression model, which implies that the response variable itself has a log-normal distribution rather than a normal distribution.


Independence of errors.

This assumes that the errors of the response variables are uncorrelated with each other. Actual statistical independence is a stronger condition than mere lack of correlation and is often not needed, although it can be exploited if it is known to hold. Some methods such as generalized least squares are capable of handling correlated errors, although they typically require significantly more data unless some sort of regularization is used to bias the model towards assuming uncorrelated errors. Bayesian linear regression is a general way of handling this issue.


Lack of perfect multicollinearity in the predictors.

For standard least squares estimation methods, the design matrix X must have full column rank p; otherwise perfect multicollinearity exists in the predictor variables, meaning a linear relationship exists between two or more predictor variables. This can be caused by accidentally duplicating a variable in the data, using a linear transformation of a variable along with the original e.g., the same temperature measurements expressed in Fahrenheit and Celsius, or including a linear combination of multiple variables in the model, such as their mean. It can also happen if there is too little data available compared to the number of parameters to be estimated e.g., fewer data points than regression coefficients. Near violations of this assumption, where predictors are highly but not perfectly correlated, can reduce the precision of parameter estimates. In the case of perfect multicollinearity, the parameter vector β will be non-identifiable—it has no unique solution. In such a case, only some of the parameters can be identified i.e., their values can only be estimated within some linear subspace of the full parameter space Rp.

Beyond these assumptions, several other statistical properties of the data strongly influence the performance of different estimation methods:

The statistical relationship between the error terms and the regressors plays an important role in determining whether an estimation procedure has desirable sampling properties such as being unbiased and consistent.

The arrangement, or probability distribution of the predictor variables x has a major influence on the precision of estimates of β. Sampling and design of experiments are highly developed subfields of statistics that provide guidance for collecting data in such a way to achieve a precise estimate of β.

QUESTION I: How important is it to check these assumptions before using Linear Regression?
QUESTION II: Does Normality of data affects the performance of Linear Regression?

Reference: Wikipedia Linear Regression
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