Question

Pick a minimum of 20 observations on any subject. This will include a dependent variable plus two independent variables that you may think are either negatively or positively correlated with the dependent variable. List the observed data (include the source). Then do the following:

a. State before doing any calculations whether you think they are positively or negatively correlated. What is your rationale?

Example: I test for a correlation between the quantity of coffee that people buy (Y) with the price of coffee (X1) and the household income (X2).   

I hypothesize that there is a negative correlation between quantity and price because people like to buy goods at lower rather than higher prices. I also hypothesize that there is also a positive correlation between the quantity of coffee and household income because people can buy more coffee when their income increases.

b. Draw a graph of each of the two independent variables with the dependent variable either by hand or by using Excel. (Do this by inserting an XY/Scatter chart.)

c. Use Excel to do the necessary regression. Give the values for the y-intercept, b1 and b2. Write out the equation. Also show R-square, the F-statistic and its p-value and the t-statistics with their respective p-values.

d. Test for multicollinearity using the rule that the two independent variables are multicollinear if their correlation coefficient is .70 or greater (implying r-square is .49 or greater). If they are multicolliear, give a brief statement on why do you think that is the case.

e.Pretend that this was an assignment from your manager and communicate your findings to the manager in 100 words or less. You should assume in preparing this memo:

• Your manager is not familiar with statistical terminology. So if you use terms like     R-        square or t-statistic, you need to explain what they mean.

I ONLY NEED HELP WITH (E). THE REST IS JUST FOR REFERENCE. Thank you!!!

Answer 1

Answer: In a regression model, Y is known as the dependent variable, whose value depends upon some Xs, which are independent variables. In this case, the quantity of coffee bought depends on the price and the household income. Here, Y is the amount of coffee bought and X1 = price of coffee and X2 = household income.

In order to see whether there is a relationship between the given Y and Xs, we fit a regression model. This is a statistical model that helps us know whether a set of given variables affect the changes in a particular dependent variable and if so, what is the change that occurs in the dependent variable with 1 unit change in the independent variable/s. In order to fit this model, we use the following steps:

a. Draw the scatter plot of X vs Y. This helps us to see whether there is a linear relationship between the variables. Because we can perform linear regression only when there is a linear relationship between the dependent variable and the independent variable/s.

b. Fit the regression model. The regression model in this case will be given as

y = o + 1X1 + 2X2. Here, o is the y-intercept. It means that when the value of X1 and X2 is 0, this is the value of Y. It is the initial amount of coffee purchased, irrespective of the price or the household income.

Now, X1 = price of coffee. 1 is the amount by which the value of Y changes when there is a change of 1 unit in the value of X1. Thus, in this case, if the price of coffee increases by 1 unit, the amount of coffee purchases is affected by 1 units.

X2 = household income. 2 is the amount by which the value of Y changes when there is a change of 1 unit in the value of X2. Thus, in this case, if the household income increases by 1 unit, the amount of coffee purchased is affected by 2 units.

Now, in order for these independent variables to affect the dependent variable, 1 and 2 must not be equal to 0. In order to test this, we use the t-test for the coefficients. We hypothesize that the coefficients are 0 and use t-test to see if our hypothesis is true or not. If any one of the is 0, then we conclude that there is no relationship between Y and the given X. We conclude this on the basis of a p-value. This is the probability that for a given hypothesis, the t-stat obtained lies outside the acceptable range. For most tests, we use a 5% significance level for p-value. That is, if the p-value obtained is less than 0.05, then we conclude that is not 0, otherwise it's 0. The p-value comes as an output of the t-test for coefficients.

Also, since this is a model, it can be used to predict the values for the dependent variable for any simulated value of the independent variable. Thus, in order to do so, the model must be robust and accurate. This estimate is given through the value of R2.

R-squared (R2) is a statistical measure that represents the proportion of the variance for a dependent variable that's explained by an independent variable or variables in a regression model. Whereas correlation explains the strength of the relationship between an independent and dependent variable, R-squared explains to what extent the variance of one variable explains the variance of the second variable. So, if the R2 of a model is 0.50, then approximately half of the observed variation can be explained by the model's inputs.

For a model to be a good fit, R2 must be high.

There's also a concept of multicollinearity. This usually happens when the independent variables are related to each other/correlated. In this case, there's always a bias if we take just the variables. We also need to consider their interaction effect. Thus, in this case, for example, if price of coffee and household income were related, we would introduce a third variable Coffee*Household Income, which would find the interaction effect of these variable on the amount of coffee purchased.