14.7: Use of R Statistical Analysis Tool for Regression Analysis

Learning Objectives

By the end of this section, you will be able to:

• Generate correlation coefficients using the R statistical tool.
• Generate linear regression models using the R statistical tool.

Generate Correlation Coefficients Using the R Statistical Tool

R is an open-source statistical analysis tool that is widely used in the finance industry. R is available as a free program and provides an integrated suite of functions for data analysis, graphing, and statistical programming. R provides many functions and capabilities for regression analysis.

Recall that most calculations in R are handled via functions.

The typical method for using functions in statistical applications is to first create a vector of data values. There are several ways to create vectors in R. For example, the c function is often used to combine values into a vector. For example, this R command will generate a vector called salaries, containing the data values 40,000, 50,000, 75,000, and 92,000:

      > salaries <- c(40000, 50000, 75000, 92000)
      

To calculate the correlation coefficient r, we use the R command called cor.

As an example, consider the data set in Table 14.8, which tracks the return on the S&P 500 versus return on Coca-Cola stock for a seven-month time period.

Table 14.8: Monthly Returns of Coca-Cola Stock versus Monthly Returns for the S&P 500

Month	S&P 500 Monthly Return (%)	Coca-Cola Monthly Return (%)
Jan	8	6
Feb	1	0
Mar	0	-2
Apr	2	1
May	-3	-1
Jun	7	8
Jul	4	2

Create two vectors in R, one vector for the S&P 500 returns and a second vector for Coca-Cola returns:

      > SP500 <- c(8,1,0,2,-3,7,4)

      > CocaCola <- c(6,0,-2,1,-1,8,2)
      

The R command called cor returns the correlation coefficient for the x-data vector and y-data vector:

      > cor(SP500, CocaCola)
      

Generate Linear Regression Models Using the R Statistical Tool

To create a linear model in R, assuming the correlation is significant, the command lm (for linear model) will provide the slope and y-intercept for the linear regression equation.

The format of the R command is

      lm(dependent_variable_vector ~ independent_variable_vector)
      

Notice the use of the tilde symbol as the separator between the dependent variable vector and the independent variable vector.

We use the returns on Coca-Cola stock as the dependent variable and the returns on the S&P 500 as the independent variable, and thus the R command would be

      > lm(CocaCola ~ SP500)

      Call:
      
      lm(formula = CocaCola ~ SP500)
      
      Coefficients:
      
      (Intercept)    SP500
      
       -0.3453    0.8641
      

The R output provides the value of the y-intercept as $-0.3453$ and the value of the slope as 0.8641. Based on this, the linear model would be

$$\begin{array}{rcl}\hat{y}& =& a+bx\\ \hat{y}& =& -0.3453+0.8641x\end{array}$$

14.21

where x represents the value of S&P 500 return and y represents the value of Coca-Cola stock return.

The results can also be saved as a formula and called “model” using the following R command. To obtain more detailed results for the linear regression, the summary command can be used, as follows:

      > model <- lm(CocaCola ~ SP500)
      
      > summary(model)
      
      Call:
      
      lm(formula = CocaCola ~ SP500)
      
      Residuals:
      
        1    2    3    4    5    6    7
      
      -0.5672 -0.5188 -1.6547 -0.3828 1.9375 2.2969 -1.1109
      
      Coefficients:
      
           Estimate Std. Error t value Pr(>|t|)
      
      (Intercept) -0.3453   0.7836 -0.441 0.67783
      
      SP500     0.8641   0.1734  4.984 0.00416 **
      
      ---
      
      Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
      
      Residual standard error: 1.658 on 5 degrees of freedom
      
      Multiple R-squared: 0.8325,  Adjusted R-squared: 0.7989
      
      F-statistic: 24.84 on 1 and 5 DF, p-value: 0.004161
      

In this output, the y-intercept and slope is given, as well as the residuals for each x-value. The output includes additional statistical details regarding the regression analysis.

Predicted values and prediction intervals can also be generated within R.

First, we can create a structure in R called a data frame to hold the values of the independent variable for which we want to generate a prediction. For example, we would like to generate the predicted return for Coca-Cola stock, given that the return for the S&P 500 is 6.

We use the R command called predict.

To generate a prediction for the linear regression equation called model, using the data frame where the value of the S&P 500 is 6, the R commands will be

      > a <- data.frame(SP500=6)
      
      > predict(model, a)
      
         1
      
      4.839062
      

The output from the predict command indicates that the predicted return for Coca-Cola stock will be 4.8% when the return for the S&P 500 is 6%.

We can extend this analysis to generate a 95% prediction interval for this result by using the following R command, which adds an option to the predict command to generate a prediction interval:

      > predict(model,a, interval="predict")
      
         fit    lwr   upr
      
      1 4.839062 0.05417466 9.62395
      

Thus the 95% prediction interval for Coca-Cola return is (0.05%, 9.62%) when the return for the S&P 500 is 6%.