### Learning Outcomes

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.

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

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%.