Linear Regression Intercept

Introduction

Linear Regression Intercept is one of the indicators calculated by using the Linear Regression technique. Linear regression indicates the value of the Y (generally the price) when the value of X (the time series) is 0. Linear Regression Intercept is used along with the Linear Regression Slope to create the Linear Regression Line. The Linear Regression Intercept along with the Slope creates the Regression line.

Calculation of Linear Regression Intercept

Linear Regression Intercept can be calculated from the standard equation for calculating the Linear Regression. The Linear regression statistical technique is used for calculating the value of one dependent variable when one has the values of independent variable or variables M. This is in the form of a straight line which “best fits” the data points of prices available, using the least squares technique. The Linear Regression Intercept is derived by using the same calculation that is used for calculating the Linear Regression line and solving for the intercept or price value at the first point of the x where x=0.

Consider the equation for Linear Regression:

Y = a + bx

Where:

Y = the price for each date x
a = the intercept (or constant where value of x is 0)
b = slope of the regression line
x = the date for each value

From this, we can derive:

Where:

n = number of data points selected

Now if we take x to be 0 to get the price when the value of x is 0, we can solve the equation as:

We can also derive the same value if we go back to the definition that regression line starts at the intercept and its direction and gradient from that point is determined by the slope. If we take the slope to be 0, it means that that the regression line is flat and parallel to the x axis and that the price is same for all values of x or on all dates.

Lengths and timeframes

Every linear regression line will have an intercept as the intercept is basically the point where the line intersects the y axis. Therefore the intercept shows the price value at the first bar from the time series of data or the first day of the period under analysis.

Trend identification / Crossovers

Intercept indicates the first and starting value of the regression line. This calculation can be used by Technical Analysts to backtest assumptions and regression lines given a data set and compare the predicted line and its calculated intercept against the actual historical price value. This may be useful as a confirmation of assumptions and identified patterns.

In some situations, for example when using the Capital Asset pricing model, the linear regression intercept may also be used to evaluate performance of a stock or a fund with that of the index or market. This will help in indicating how much better or worse a stock or fund did than the CAPM predicted. In essence, if we breakdown the linear regression equation into two parts, one part is the alpha (or the intercept) and the second part is the price movement of the market. The price movement of the market is explained by the regression. Therefore, in one way, the alpha or intercept shows the performance that is more than the market movement.