Classical Linear Regression Model Assumptions

Classical Linear Regression Model (CLRM) Assumptions

• The CLRM makes 7 assumptions that are critical to the valid interpretation of regression estimates.

Assumption 1 - Linearity

• The regression model is linear in parameters, not necessarily in variables.
• The model can be extended to multivariate models.
• Linearity is in parameters, not variables.

Assumption 2 - Fixed X Values

• Fixed X values or X values independent of the error term (strict exogeneity).
• Values taken by the regressor X may be considered fixed in repeated samples or sampled along with the dependent variable Y.
• In the latter case, it is assumed that the X variable(s) and the error term are independent: cov(X, u) = 0.

Assumption 2 - Why assume fixed Xs?

• Consider the example of consumption (Y) vs income (X) in lecture 3.
• Keeping the value of income X fixed, we can randomly draw the following family consumption.
• The value of X is fixed at $80, and the process can be repeated for all the X values.
• If the X variable(s) is stochastic, the resulting model is called the neo-classical linear regression model (NLRM), in contrast to the CLRM, where the X's are treated as fixed or nonrandom.

Assumption 3 - Zero Mean of the Disturbance

• Zero Mean Value of Disturbance ui: Given the value of Xi, the mean, or expected, value of the random disturbance term ui is zero.
• Symbolically, E(ui | Xi) = 0 or E(ui) = 0.
• The assumption implies that the factors not explicitly included in the model, and therefore subsumed in u, do not systematically affect the mean value of Y.
• In other words, the positive ui values cancel out the negative ui values so that their average or mean effect on Y is zero.
• E(ui | Xi) = 0 implies that E(Y | Xi) = β0 + β1Xi.

Assumption 4 - Homoscedasticity or Constant of the Variance of the Disturbance

• The variance of ui for each Xi (i.e., the conditional variance of ui) is some positive constant number equal to σ^2.

Assumption 5 - Non Autocorrelation between the Disturbances

• Equation 3.2.5 postulates that the disturbances ui and uj are uncorrelated.
• Technically, this is the assumption of no serial correlation, or no autocorrelation.
• This means that, given Xi, the deviations of any two Y values from their mean value do not exhibit patterns such as those shown in Figures (a) and (b).

Assumption 7 - Nature of the X Variables

• There can be no outliers in the values of the X variable, that is, values that are very large in relation to the rest of the data.
• The requirement that there are no outliers in the X values is to avoid the regression results being dominated by such outliers.

Assumption 9 - The Regression Model is Correctly Specified

• There is no specification bias or error in the model used in the empirical analysis.
• As discussed earlier, an econometric investigation begins with the specification of the econometric model, including what variables should be included, what is the functional form of the model, and what probabilistic assumptions are made about Y, X, and u.
• It will be shown later that the choice of wrong variables, wrong functional form, etc. will lead to questionable estimations.