Classical Linear Regression Model [CLRM/CNLRM]

CNLRM (Classical Normal Linear Regression Model)

Assumption 1: Linear regression model: The regression model is linear in the parameters, as shown as

Yi = β1 + β2Xi + ui


Assumption 2: X values are fixed in repeated sampling. Values taken by the regressor X are considered fixed in repeated samples. More technically, X is assumed to be non- stochastic.


Assumption 3: Zero mean value of disturbance ui. Given the value of X, the mean, or expected, value of the random disturbance term ui is zero. Technically, the conditional mean value of ui is zero. Symbolically, we have

E(ui |Xi) = 0 .


Assumption 4: Homoscedasticity or equal variance of ui. Given the value of X, the variance of ui is the same for all observations. That is, the conditional variances of ui are identical. Symbolically, we have

var (ui |Xi)  = E[ui E(ui |Xi)]2

    = E(ui2 |Xi ) because of Assumption 3

    = σ2

where var stands for variance.


Assumption 5: No autocorrelation between the disturbances. Given any two X values, Xi and Xj ( j ), the correlation between any two ui and uj (i ≠ j ) is zero. Symbolically,

cov(ui,uj |Xi,Xj) = E{[ui E(ui)]|Xi}{[uj E(uj)]|Xj}
= E(u
i | Xi)(uj | Xj)


where i and j are two different observations and where cov means covariance.


Assumption 6: Zero covariance between ui and Xi, or E(uiXi) = 0. Formally,

cov (ui, Xi) = E [ui E(ui)][Xi E(Xi)]
      = E [u
i (Xi E(Xi))] since E(ui) = 0

      = E(uiXi) − E(Xi)E(ui) since E(Xi) is non-stochastic

      = E(uiXi) since E(ui) = 0
      = 0 by assumption


Assumption 7: The number of observations n must be greater than the number of parameters to be estimated. Alternatively, the number of observations n must be greater than the number of explanatory variables.


Assumption 8: Variability in X values. The X values in a given sample must not all be the same. Technically, var (X ) must be a finite positive number.


Assumption 9: The regression model is correctly specified. Alternatively, there is no specification bias or error in the model used in empirical analysis.


Assumption 10: There is no perfect multicollinearity. That is, there are no perfect linear relationships among the explanatory variables.