Correlation

Two variables are said to be in correlation if a change in one of the variables results in a change in the

other variable. Correlation analysis is used to determine the degree of this association between two

variables. For example, portfolio managers can use a correlation analysis to measure the degree of

relationship between the returns of two stocks. However, correlation analysis cannot measure a
nonlinear relationship between two variables, if such a nonlinear relationship exists.

The Pearson correlation coef�cient, also known as the sample correlation coef�cient, is an important

component of correlation analysis. The sample correlation coef�cient can range from -1.0 to +1.0. It is

typically rare for the value to be exactly equal to -1.0, 0, or +1.0. However, these values represent

speci�c meanings that are worthy of discussion.

-1

A correlation coef�cient that is equal to -1.0 indicates a perfectly negative relationship between two

variables. For example, variables representing the price and demands of a product will have negative

correlation. For most products, when the price of the product goes up, the demand for the product

goes down.

0

A correlation coef�cient that is equal to 0 indicates that there is no linear relationship between two

variables. A change in one variable is predicted to have no impact on the other variable.

+1

A correlation coef�cient that is equal to +1.0 indicates a perfectly positive relationship between two

variables. For example, variables representing the price and supply of a product will have positive

correlation. If supply can be adjusted, for most products, when the price of the product goes up, the

supply that will be provided to the marketplace goes up.

There is a t-test that can be used to determine if a correlation coef�cient is signi�cantly different from
zero. Thus, if this test shows that there is evidence that the correlation coef�cient is signi�cantly

different from zero, the research can conclude that there is a statistically signi�cant correlation (either

positive or negative) between the two variables.

Linear Regression

Download: Video Transcript (PDF 78.08KB) (media/transcripts/SU_W4L1.pdf?

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Linear regression is one form of bivariate regression, meaning that it involves two variables. The

simple regression model tries to �t a straight line to a set of data points consisting of one independent

variable (known as the x value) and one dependent variable (known as the y value). The model’s

parameters are denoted by Greek letters such as α_0,α_1,and ε.

There are a number of assumptions that must be met in order for a simple regression model to be a

reasonable choice for modeling the relationship between two variables. Probably the easiest

assumption to understand is that the relationship between the independent variable and the

dependent variable needs to be a linear relationship. For example, if we plotted lots of values that were

created by using the equation y = x2, and then attempted to predict a straight line that approximated

those points, you would have very little success in trying to model data points that had much variability

in terms of the values of the independent variable. While a linear model might work over a very small
range of x values, it would not work well over a wider range of values.

Other assumption violations which can cause unpredictable results include the assumption that the

errors (between the straight line estimate and the actual data) are independent (meaning that the

straight line estimates things just about equally well for all data points), the assumption that the

variance within the data is constant, and the assumption that all of the errors are approximately

normally distributed.

01:54

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The supplemental material entitled “Linear Regression Models” shows you both the most general form

of the model (which will include an error term) and the form of the model which is most often used

(which does not explicitly include the error term, but understand that there will be errors that must be

analyzed).

Additional Materials

View a Pdf Transcript of Linear Regression Models 

(media/week4/SUO_BUS3059%20W4%20L1%20Linear%20Regression%20Models.pdf?

_&d2lSessionVal=8n6sZsSYlBq8timxeXfki15Qm&ou=84108)



Multiple Regression

A multiple or multivariate regression is an extension of a bivariate regression, having at least two

independent variables. This method of regression is used when one independent variable is

inadequate to explain the relationship between the dependent variable and the independent variable.

While employing this method, it is important to use a correct model for analyzing data because a
model that is missing necessary independent variables can lead to results that are ambiguous.

Similarly, a model that has additional (unnecessary) independent variables can behave erratically,

especially if the additional independent variable is moderately or highly correlated with some other

independent variable. For this reason, care must be taken to examine the results of a multivariate

regression, to ensure that the model represents the relationship between the dependent variable and

the independent variables as thoroughly and as effectively as possible.

Statistical software packages are sometimes used in order to analyze a multivariate regression, and

the output from such a statistical analysis is useful for examining the degree to which the model is

complete or burdened with unnecessary independent variables. However, a multiple regression can be

alsoestimated with Microsoft Excel.

In order to measure the overall �t of a regression, an F test is utilized. Data from the analysis of

variance (ANOVA) table (the output of the regression) can be used to calculate the F test. An adjusted

coef�cient of determination (adjusted R2) can also be used to measure the overall �t of a regression. In
addition, the regression should be tested for nonlinearity. To measure non-quantitative variables, you

can use dummy variables in a regression.

Linear Regression Models

Simple Linear Regression Model

The following is an example of a linear regression model.

yi = α0 + (α1xi) + εi

yi
This letter represents the dependent variable. It is the variable measured, predicted, or otherwise
monitored by the researcher. It is expected to be affected by the manipulation of the independent
variable.

α0
This letter represents the intercept and it is one of the two regression coefficients. Intercept is the
value for the linear function when it crosses the Y-axis or the estimate of y when x is 0.

α1
This letter represents the slope and it is one of the two regression coefficients. Slope is the change
in y for a 1-unit change in x.

xi
This letter represents the independent variable. It is the variable manipulated by the researcher,
thereby causing an effect or change on the dependent variable.

εi
This letter represents an error term. The error term is needed to account for errors in the
measurement of y and those that might be caused due to other variables impacting y. However, it
is not possible to observe this error. Therefore, an assumption can be made about the error term
being a random variable with a normal distribution having a mean of 0 and a standard deviation of
sigma.

Fitted Regression Model

The following is an example of a fitted regression model.
yi = a0 +a1xi

yi Estimated value of yi
a0 Estimated intercept

2 [Document Title]
[Parent Lecture Name]

a1 Estimated slope
xi Any given value of x

Given a value of x, the fitted regression model can be used to predict a value of y.
When the estimated value of yi is subtracted from the observed value of yi, a
value known as residual is calculated. The residual can be used to estimate
sigma.

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