Assumptions of Classical Linear Regression Model

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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.

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