# Assumptions of Classical Linear Regression Model

## Following are the assumptions underlying when we use method of Least Squares

Assumption 1: Linear regression model

The regression model is linear in the parameters i.e.,

$$Y_i \ =\ \beta_1 +\ \beta_2X_i\ +\ u_i$$

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 value of disturbance $$u_i$$

Given the value of $$X$$, the mean, or expected, value of the random disturbance term $$u_i$$ is zero. Technically, the conditional mean value $$u_i$$ is Zero. Symbolically, we have

$$E(u_i|X_i) = 0$$

Assumption 4: Homoscedasticity or equal variance of $$u_i$$

Given the value of $$X$$, the variance of $$u_i$$ is the same for all observations i.e., the conditional variances of $$u_i$$ are identified. Symbolically, we have

$$Var (u_i|X_i) = E[u_i\ -\ E(u_i|X_i)]^2$$

$$=E(u_i^2\ |X_i)$$ ( Because of assumption 3)

$$= \sigma^2$$

Assumption 5: No autocorrelation between the disturbances.

Given any two $$X$$ values, $$X_i$$ and $$X_j(i\neq j)$$, the correlation between any two $$u_i$$ and $$u_j\ (i\neq j)$$ is zero. Symbolically,

$$cov(u_i\ , u_j\ |\ X_i,\ X_j)$$ = $$E{[u_i-E(u_i)] |X_i }{[u_j-E(u_j – E(u_j)]|X_j}$$

$$= E(u_i|X_i)(u_j|X_j)$$

$$=0$$

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

Assumption 6 : Zero covariance between $$u_i$$ and $$X_i$$, or $$E(u_iX_i)=0$$. Formally,

$$cov(u_i,X_i)=$$ $$E[u_i-E(u_i)][X_i-E(X_i)]$$

$$=E[u_i(X_i-E(X_i)]$$ since $$E(u_i)=0$$

$$=E(u_iX_i)\ -\ E(X_i)E(u_i)$$ since $$E(X_i)$$ is non-stochastic

$$=E(u_iX_i)$$ since $$E(u_i)=0$$

$$=0$$ By Assumption

Assumption 7: The number of observations n must be grater 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.