Simple Regression Analysis

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) $\mu_i$, some are about the relationship between error term $\mu_i$ 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 ($\mu_i$) is a random real number i.e. $\mu_i$ 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 $\mu$ is zero, i.e. $E(\mu_i)=0$ i.e. the mean value of $\mu_i$ is conditional upon the given $X_i$ is zero. It means that for each value of variable $X_i$, $\mu$ may take various values, some of them greater than zero and some smaller than zero. Considering all possible values of $\mu$ for any particular value of $X$, we have zero mean value of disturbance term $\mu_i$.
  • The variance of $\mu_i$ is constant i.e. for the given value of X, the variance of $\mu_i$ is the same for all observations. $E(\mu_i^2)=\sigma^2$. The variance of disturbance term ($\mu_i$) about its mean is at all values of X will show the same dispersion about their mean.
  • The variable $\mu_i$ has a normal distribution i.e. $\mu_i\sim N(0,\sigma_{\mu}^2$. The value of $\mu$ (for each $X_i$) has a bell-shaped symmetrical distribution.
  • The random term of different observation ($\mu_i,\mu_j$) are independent i..e $E(\mu_i,\mu_j)=0$, 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.
  • $\mu_i$ and $X_i$ have zero covariance between them i.e. $\mu$ is independent of the explanatory variable or $E(\mu_i X_i)=0$ i.e. $Cov(\mu_i, X_i)=0$. The disturbance term $\mu$ and explanatory variable X are uncorrelated. The $\mu$’s and $X$’s do not tend to vary together as their covariance is zero. This assumption is automatically fulfilled if 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. $y_i=\beta_1+\beta_2x_i +\mu_i$
Homoscedasticity: Regression Model Assumptions

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Simple Linear Regression Model (SLRM)

A simple linear regression model (SLRM) is based on a single independent (explanatory) variable and it fits a straight line such that the sum of squared residuals of the regression model (or vertical distances between the fitted line and points of the data set) as small as possible. The model (usually known as a statistical or probabilistic model) is

\begin{align*}
y_i &= \alpha + \beta x_i +\varepsilon_i\\
\text{OR} \quad y_i&=b_0 + b_1 x_i + \varepsilon_i\\
\text{OR} \quad y_i&=\beta_0 + \beta x_i + \varepsilon_i
\end{align*}
where $y$ is the dependent variable, $x$ is the independent variable. In the regression context, $y$ is the regressand, and $x$ is the regressor. The epsilon ($\varepsilon$) is unobservable, denoting random error or the disturbance term of a regression model. $\varepsilon$ (random error) has some specific importance for its inclusion in the regression model:

Importance of Error Term in Simple Linear Regression Model

  1. Random error ($\varepsilon$) captures the effect on the dependent variable of all variables which are not included in the model under study, because the variable not included in the model may or may not be observable.
  2. Random error ($\varepsilon$) captures any specification error related to the assumed linear-functional form.
  3. Random error ($\varepsilon$) captures the effect of unpredictable random components present in the dependent variable.

We can say that $\varepsilon$ is the variation in variable$y$ not explained (unexplained) by the independent variable $x$ included in the model.

In the above equation or model $\hat{\beta_0}, \hat{\beta_1}$ are the parameters of the model and our main objective is to obtain the estimates of their numerical values i.e. $\hat{\beta_0}$ and $\hat{\beta_1}$, where $\beta_0$ is the intercept (regression constant), it passes through the ($\overline{x}, \overline{y}$) i.e. center of mass of the data points and $\beta_1$ is the slope or regression coefficient of the model and slope is the correlation between variable x and y corrected by the ratio of standard deviations of these variables. The subscript i denotes the ith value of the variable in the model.
\[y=\beta_0 + \beta_1 x_1\]
This is a mathematical model as all the variation in $y$ is due solely to change in $x$. There are no other factors affecting the dependent variable. If this is true then all the pairs $(x, y)$ will fall on a straight line if plotted on a two-dimensional plane. However, for observed values, the plot may or may not be a straight line. A dimensional diagram with points plotted in pair form is called a scatter diagram.

scatter diagram Simple Linear Regression ModelSee Assumptions about Simple Linear Regression Model

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Interpreting Regression Coefficients in Simple Regression

How are the regression coefficients interpreted in simple regression?

The simple regression model is

Simple Regression Coefficients

The formula for Regression Coefficients in Simple Regression Models is:

$$b = \frac{n\Sigma XY – \Sigma X \Sigma Y}{n \Sigma X^2 – (\Sigma X)^2}$$

$$a = \bar{Y} – b \bar{X}$$

The basic or unstandardized regression coefficient is interpreted as the predicted change in $Y$ (i.e., the dependent variable abbreviated as DV) given a one-unit change in $X$ (i.e., the independent variable abbreviated as IV). It is in the same units as the dependent variable.

  • Note that there is another form of the regression coefficient that is important: the standardized regression coefficient. The standardized coefficient varies from –1.00 to +1.00 just like a simple correlation coefficient;
  • If the regression coefficient is in standardized units, then in simple regression the regression coefficient is the same thing as the correlation coefficient.

How to interpret the Regression Coefficients in Multiple Linear Regression Models

How to Perform Linear Regression Analysis in R Language

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