Quants: Single Variable Linear Regression Analysis

• This is the “least squares” method.

• Situational Example:

• “If a country’s broad stock index (X) appreciates 5%, by how much will the value of that index’s largest air transportation stock change?”

• You will apply this type of modeling in the equity section, when looking at the CAPM approach to asset valuation.

• Simple Regression can be expressed as follows:

  Yi = b0 + b1Xi + ei

  • b0 = the y-intercept.  Yi = b0, when observation Xi = 0 (zero)
  • b1 = the slope coefficient – the change in Y per unit change in X.
  • Xi = the observation for the independent variable.  Ex: the value of a stock index.
  • Yi = the observed dependent variable for Xi.
  • ei = the error term or the part of the dependent value not explained by the independent variable; the expected value of the error term is zero and this is one of the standard assumptions for simple regression.
  • Note: the y-intercept and slope coefficient are the known as the model’s parameters.

• A hat “^” is typically used when referring to predicted values and the subscript “i” typically refers to actual observations.

Six Assumptions of Simple Linear Regression

1. Y and X must have a liner relationship.
2. X is not random.
3. The expected value of e is 0 (zero).
4. The e term does not exhibit heteroskedasticity, meaning that the error term’s variance is the same for all observations.
5. The error term is uncorrelated across all observations (or no serial correlation).
6. The error term has a normal distribution.

A violation of one or more of these assumptions threatens the validity of your model’s conclusions.