A Review of Multiple Linear Regression’s Uses

Multiple linear regression describes the variation of the dependent variable by using two or more independent variables. When used properly, it can improve predictions. However, if used incorrectly, it can create spurious relationships that can undermine predictions.

Typically, a multiple regression model takes the following form:

$$ Y_i=b_0+b_1X_{1,i}+b_2X_{2,i}+\ldots+b_kX_{k,i}+\epsilon_i $$

Where:

\(Y_i\) = Dependent variable.

\(b_0\) = Intercept term.

\(b_1,b_2,\ldots, b_k\) = Slope coefficients.

\(X_{1,i},X_{2,i},\ldots,X_{k,i}\) = Independent variables.

\(\epsilon_i\) = Error term.

\(n\) = Number of observations.

A regression equation has \(k\) slope coefficients and \(k+1\) regression coefficients.

The intercept term is defined as the value of the dependent variable when the independent variables are zero. On the other hand, the slope coefficient is defined as the estimated change in the dependent variable given a one-unit change in the independent variable, keeping the other independent variables constant.

Researchers can use multiple regression to test existing theories, identify relationships between variables, or forecast.

The researcher must specify the model to determine the good-fit criteria for the regression model, including an independent variable. Once the regression model has been specified, it must be estimated and analyzed to ensure it satisfies all the key assumptions.

It is equally noteworthy that a researcher can use multiple regression to test existing forecasting theories. Alternatively, multiple regression can further be used to identify relationships between variables after the model is tested and deemed acceptable for out-of-sample performance.

A single factor cannot adequately explain or forecast the complex world of investments. Due to their complexity, statistical tests and fundamental justification are critical in the exhaustive explanation of financial and economic relations.

There are several ways to use multiple regression, including:

• A company’s profitability, growth, revenue, and market share are variables that an investor is interested in. These variables can predict if a company will run into financial difficulties.
• An analyst seeking to determine how a company’s stock price and trading volume change daily can correlate them. An analyst can use linear regression to determine the relationship between variables.

Question

Which of the following most accurately detects whether the underlying assumptions of multiple linear regression models are satisfied?

1. Scatter plots.
2. Residual plots.
3. Diagnostic plots.

Solution

The correct answer is C.

Diagnostic plots for multiple regression show the prediction errors against predicted values. Therefore, they are useful in the determination of the gaps that a researcher should address in their data to improve the accuracy of their predictions.  Using diagnostic plots, a researcher can determine whether the assumptions of multiple linear regression are valid.

A is incorrect. The scatterplot is useful for detecting nonlinear relationships between dependent and independent variables.

B is incorrect. A residual plot is an effective tool for detecting violations of homoskedasticity and error independence.