# Regression Analysis in Finance

## Best-Fit Linear Model


### Learning Outcomes

By the end of this section, you will be able to:
1. Calculate the slope and y-intercept for a linear regression model using technology.
2. Interpret and apply the slope and y-intercepts.

### Calculate the Slope and y-Intercept for a Linear Regression Model Using Technology

Once a correlation has been deemed as significant, a best-fit linear regression model is developed. The goal in the regression analysis is to determine the coefficients a and b in the following regression equation:
The slope (b) and y-intercept (a) can be calculated using the following formulas:
These formulas can be quite cumbersome, especially for a significant number of data pairs, and thus technology is often used (such as Excel, a calculator, R statistical software, etc.).
Using Excel: To calculate the slope and y-intercept of the linear model, start by entering the (x, y) data in two columns in Excel. Then the Excel commands =SLOPE and =INTERCEPT can be used to calculate the slope and intercept, respectively.
The following data set will be used as an example: the monthly amount spent on advertising and the monthly revenue for a Fortune 500 company for 12 months (data is shown in ).
To calculate the slope of the regression model, use the Excel command
It’s important to note that this Excel command expects that the y-data range is entered first and the x-data range is entered second. Since revenue depends on amount spent on advertising, revenue is considered the y-variable and amount spent on advertising is considered the x-variable. Notice the y-data is contained in cells C2 through C13 and the x-data is contained in cells B2 through B13. Thus the Excel command for slope would be entered as
In the same way, the Excel command to calculate the y-intercept of the regression model is
For the data set shown in the above table, the Excel command would be
The results are shown in , where
Based on this, the regression equation can be written as
where x represents the amount spent on advertising (in thousands of dollars) and y represents the amount of revenue (in thousands of dollars).

### Using a Financial Calculator

The financial calculator provides the slope and y-intercept for the linear regression model once the (x, y) data is inputted into the calculator.
Follow the steps in  for calculating the slope and y-intercept for the data set of amounts spent on advertising and revenue shown previously.
From the statistical results shown on the calculator display, the slope b is 61.8 and the y-intercept a is 9,367.7.
Based on this, the regression equation can be written as

### Interpret and Apply the Slope and y-Intercept

The slope of the line, b, describes how changes in the variables are related. It is important to interpret the slope of the line in the context of the situation represented by the data. You should be able to write a sentence interpreting the slope in plain English.

### Interpretation of the Slope

The slope of the best-fit line tells us how the dependent variable (y) changes for every one unit increase in the independent (x) variable, on average.
In the previous example, the linear regression model for the monthly amount spent on advertising and the monthly revenue for a Fortune 500 company for 12 months was generated as follows:
Since the slope was determined to be 61.8, the company can interpret this to mean that for every $1,000 dollars spent on advertising, on average, this will result in an increase in revenues of $61,800.
The intercept of the regression equation is the corresponding y-value when.

### Interpretation of the Intercept

The intercept of the best-fit line tells us the expected mean value of y in the case where the x-variable is equal to zero.
However, in many scenarios it may not make sense to have the x-variable equal zero, and in these cases, the intercept does not have any meaning in the context of the problem. In other examples, the x-value of zero is outside the range of the x-data that was collected. In this case, we should not assign any interpretation to the y-intercept.
In the previous example, the range of data collected for the x-variable was from $49 to $153 spent per month on advertising. Since this interval does not include an x-value of zero, we would not provide any interpretation for the intercept.

### 

Once a correlation has been deemed significant, a linear regression model is developed. The goal in the regression analysis is to determine the coefficients a and b in the following regression equation: . Typically some technology, such as Excel, R statistical tool, or a calculator, is used to generate the coefficients a and b since manual calculations are cumbersome.