Regression Model Assumptions

Linear Regression Model Assumptions

The linear regression model (LRM) is based on certain statistical assumptions, some of which are related to the distribution of a random variable (error term) $u_i$, some are about the relationship between error term $u_i$ and the explanatory variables (Independent variables, $X$‘s) and some are related to the independent variable themselves. The linear regression model assumptions can be classified into two categories

1. Stochastic Assumption
2. None Stochastic Assumptions

These linear regression model assumptions (or assumptions about the ordinary least square method: OLS) are extremely critical to interpreting the regression coefficients.

• The error term ($u_i$) is a random real number i.e. $u_i$ may assume any positive, negative, or zero value upon chance. Each value has a certain probability, therefore, the error term is a random variable.
• The mean value of $u$ is zero, i.e. $E(u_i)=0$ i.e. the mean value of $u_i$ is conditional upon the given $X_i$ is zero. It means that for each value of variable $X_i$, $u$ may take various values, some of them greater than zero and some smaller than zero. Considering all possible values of $u$ for any particular value of $X$, we have zero mean value of disturbance term $u_i$.
• The variance of $u_i$ is constant i.e. for the given value of $X$, the variance of $u_i$ is the same for all observations. $E(u_i^2)=\sigma^2$. The variance of disturbance term ($u_i$) about its mean is at all values of $X$ will show the same dispersion about their mean.
• The variable $u_i$ has a normal distribution i.e. $u_i\sim N(0,\sigma_{u}^2$. The value of $u$ (for each $X_i$) has a bell-shaped symmetrical distribution.
• The random terms of different observations ($u_i,u_j$) are independent i..e $E(u_i,u_j)=0$, i.e. there is no autocorrelation between the disturbances. It means that the random term assumed in one period does not depend on the values in any other period.
• $u_i$ and $X_i$ have zero covariance between them i.e. $u$ is independent of the explanatory variable or $E(u_i X_i)=0$ i.e. $Cov(u_i, X_i)=0$. The disturbance term $u$ and explanatory variable $X$ are uncorrelated. The $u$’s and $X$’s do not tend to vary together as their covariance is zero. This assumption is automatically fulfilled if the $X$ variable is nonrandom or non-stochastic or if the mean of the random term is zero.
• All the explanatory variables are measured without error. It means that we will assume that the regressors are error-free while $y$ (dependent variable) may or may not include errors in measurements.
• The number of observations $n$ must be greater than the number of parameters to be estimated or the number of observations must be greater than the number of explanatory (independent) variables.
• The should be variability in the $X$ values. That is $X$ values in a given sample must not be the same. Statistically, $Var(X)$ must be a finite positive number.
• The regression model must be correctly specified, meaning there is no specification bias or error in the model used in empirical analysis.
• There is no perfect or near-perfect multicollinearity or collinearity among the two or more explanatory (independent) variables.
• Values taken by the regressors $X$ are considered to be fixed in repeating sampling i.e. $X$ is assumed to be non-stochastic. Regression analysis is conditional on the given values of the regressor(s) $X$.
• The linear regression model is linear in the parameters, e.g. $y_i=\beta_1+\beta_2x_i +u_i$

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