SCATTER PLOTS A scatter plot is a graph that shows the relationship between the observations for two data series in two dimensions.
SAMPLE COVARIANCE Recall that covariance is the weighted average of the cross-product of each variable’s departure from its mean. SAMPLE COVARIANCE Focus On: Calculations Lending rates and current borrower burden are generally believed to be related.
CORRELATION COEFFICIENT The correlation coefficient measures the extent and direction of a linear association between two variables. LIMITATIONS OF CORRELATION ANALYSIS Focus On: Outliers Outliers are small numbers of observations with extreme values vis-a-vis the rest of the sample.
SPURIOUS CORRELATION Spurious correlation is estimated correlation that arises because of the estimating process, not because of a fundamental underlying linear association. THE BASICS OF LINEAR REGRESSION Linear regression allows us to describe one variable as a linear function of another variable.
ASSUMPTIONS UNDERLYING LINEAR REGRESSION 1.The relationship between the dependent variable, Y, and the independent variable, X, is linear in the parameters b 0 and b 1.
STANDARD ERROR OF THE ESTIMATE The standard error of the estimate gives us a measure of the goodness of fit for the relationship. COEFFICIENT OF DETERMINATION The coefficient of determination is the portion of variation in the dependent variable explained by variation in the independent variable(s). PREDICTION AND LINEAR REGRESSION Focus On: Calculating Predicted Values Continuing with our example, we can calculate predicted values for our dependent variable given our estimated regression model and values for our independent variable.
PREDICTION AND LINEAR REGRESSION Focus On: Calculations Just as we can estimate a confidence interval for our coefficients, we can also estimate a confidence interval for our predicted (forecast) values. ANALYSIS OF VARIANCE Known as ANOVA, this process enables us to divide the total variability in the dependent variable into components attributable to different sources. ANALYSIS OF VARIANCE Focus On: Calculations For our example, with a single independent variable, we can test the overall significance of the estimated relationship. LIMITATIONS OF REGRESSION ANALYSIS 1.Parameter instability occurs when regression relationships change over time.

SUMMARY We are often interested in knowing the extent of the relationship between two or more financial variables. Regression Analysis Once a linear relationship is defined, the independent variable can be used to forecast the dependent variable.
Chapter 12 Linear Regression and Correlation General Objectives: In this chapter we consider the situation in which the mean value of a random variable.
Scatter plots are formed by using the data from two different series to plot coordinates along the x- and y-axis, where one element of the data series forms the x-coordinate and the other the y-coordinate. Sample covariance is calculated by using the same process as sample variance; however, rather than squaring the deviation of each observation from its mean, we take the product of two different variables’ deviations from their respective means. The following data cover the debt-to-income ratio for 10 borrowers and the interest rate they are being charged on five-year loans. If the sample covariance is denoted as s x,y, then the sample correlation coefficient is the sample covariance divided by each sample standard deviation or Continuing with our example, the sample correlation coefficient is then From this result, we can conclude that there is a strong linear relationship between the debt-to-income ratio of the borrowers and the loan rate they are charged. The independent variable (X i ) is the variable you are using to explain changes in the dependent variable (Y i ), the variable you are attempting to explain. Total variation = Unexplained variation + Explained variation; therefore, we can calculate it two ways. ANOVA allows us to estimate the usefulness of an independent variable or variables in explaining the variation in the dependent variable.
We can assess this relationship in several ways, including -correlation, which measures the degree to which two variables move together, and -linear regression, which describes at a more fundamental level the nature of any linear relationship between two variables. Correlation Analysis correlation analysis expresses the relationship between two data series using a single number. 2 Correlation & Regression In two independent samples t-test, differences between the means of independent variable groups on the.
Null Hypothesis The analysis of business and economic processes makes extensive use of relationships between variables. Multiple Regression Model Multiple regression enables us to determine the simultaneous effect of several independent.

Simple Regression Simple regression analysis is a statistical tool That gives us the ability to estimate. Furthermore, we can conclude that the relationship has a positive sign, indicating that an increase in the debt-to-income ratio is associated with a higher loan rate. The linear regression estimation process chooses parameter estimates to minimize the sum of the squared departures of the predicted values from the observed values.
1.Square the correlation coefficient when we have one dependent and one independent variable.
We can combine hypothesis testing from the prior chapter with linear regression and correlation to test beliefs about the nature and extent of relationships between two or more variables. 3.Correlation between two variables arising not from a direct relationship between them but from their relationship to a third variable. 2.We can use the above relationship to determine the unexplained portion of the total variation as the sum of the squared prediction errors divided by the total variation in the dependent variable when we have more than one independent variable.
Economically ? A unit increase in the debt-to-income ratio leads to a 0.7774 unit increase in the loan rate. 3.Violation of the underlying assumptions makes hypothesis tests and prediction intervals invalid, and we may not be certain as to whether the assumptions have been violated. In other words, an increase of 1% in the debt-to-income ratio leads to a 77.74 basis point increase in the loan rate charged. Statistically ? at least one b is non-zero Economically ? the specified relationship has valid explanatory power 20 Pred.

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