Multiple Regression

Question 1

When do we use single variable linear regression?

Answer 1

to investigate the relationship between a dependent variable and one independent variable.

Question 2

When do we use multiple regression?

Answer 2

to investigate the relationship between a dependent variable and multiple independent variables.

Question 3

Gross relationship =

Answer 3

Gross Relationship: A single variable linear regression model determines the gross effect of an independent variable on a dependent variable. For example, the gross effect of house size on selling price is the average change in selling price when house size increases by one square foot. Since no other independent variables are included in the model, the coefficient for house size may pick up the effect of other factors related to selling price.

Question 4

Net relationship =

Answer 4

Net Relationship: A multiple regression model determines the net effect of an independent variable on a dependent variable. The net effect controls for all other factors (independent variables) included in the regression model. For example, in a regression model including both distance and house size as independent variables, the coefficient for house size controls for distance. That is, the regression determines the average change in selling price if a house’s size increases by one square foot but its distance from Boston does not change. Coefficients in multiple regression are net with respect to variables included in the model and gross with respect to variables that are omitted from the model.

Question 5

Forecasting in Excel

Answer 5

we can use Excel’s SUMPRODUCT function, =SUMPRODUCT(array1, [array2], [array3],…), to calculate a forecast from Excel’s regression output. The SUMPRODUCT function multiplies each value of the first array by the corresponding value in the second array and returns the sum of all those products.

Question 6

Which model would we use to predict the price of a house that is 2,700 square feet?

Answer 6

SellingPrice=13,490.45+255.36(HouseSize)

Since we have data about just one independent variable, we should use a single variable regression model. This is a single variable linear regression model, in which house size is the only independent variable.

Question 7

Use the single variable regression model with house size as the independent variable to predict the selling price of a house that is 2,700 square feet.

Answer 7

The expected selling price of a 2,700 square foot home is B2+B3*2700=$702,972.54. Intercept Coefficient + House Size Coefficient X Predicted Size.

Question 8

Suppose we want to forecast selling price based on house size and distance from Boston. Which equation should we use to forecast the price of a house that is 2,700 square feet and 15 miles from Boston?

Answer 8

SellingPrice=194,986.59+244.54(HouseSize)–10,840.04(DistancefromBoston)
Since we have data about two independent variables, house size and distance from Boston, we should use the multiple regression model with those two variables.

Question 9

ŷ =a+bx shows

Answer 9

The structure of the single variable linear regression line

Question 10

A coefficient in a single variable linear regression characterizes …

Answer 10

the gross relationship between the dependent variable and the independent variables.

Question 11

In single variable regression, to measure the predictive power of a single independent variable we use:

Answer 11

R2: the percentage of the variation in the dependent variable explained by the independent variable

Question 12

In multiple regression, to measure the predictive power of a single independent variable we use

Answer 12

Adjusted R2

Question 13

What insights do residual plots give to SINGLE vs MULTIPLE regression?

Answer 13

the residual plot for the single variable regression gives us insight into the gross relationship

the residual plot for multiple regression gives insight into the net relationship

Question 14

How do we Test for Significance of Variables?

Answer 14

We should also analyze the p-values of the independent variables to determine whether there is a significant relationship between the variables in the model. If the p-value of each of the independent variables is less than 0.05, we conclude that there is sufficient evidence to say that we are 95% confident that there is a significant linear relationship between the dependent and independent variables.

Question 15

What are residuals?

Answer 15

The residuals are the difference between the historically observed values and the values predicted by the regression model.

Question 16

When does multicollinearity occurs?

Answer 16

Multicollinearity occurs when two independent variables are so highly correlated that it is difficult for the regression model to separate the effect each variable has on the dependent variable.

Question 17

How many dummy variables should we use?

Answer 17

we should always use one fewer dummy variables than the number of options in a category.

Question 18

How can you reduce multicollinearity?

Answer 18

We may be able to reduce multicollinearity by either increasing the sample size or removing one (or more) of the collinear variables.

Question 19

How do you indicate that multicollinearity occured?

Answer 19

Indications of multicollinearity include seeing an independent variable’s p-value increase when one or more other independent variables are added to a regression model.

Question 20

When is Dummy variable = 1?

Answer 20

A dummy variable is equal to 1 when the variable of interest fits a certain criterion. For example, a dummy variable for “Saturday” would equal 1 for observations relating to Saturdays and 0 for observations related to all other days of the week.

Question 21

FreeIndependent, Group, Rate, SpecialEvent, TotalRewards, VIP, and Wholesale

Quantitative OR Qualitative?

Answer 21

Quantitative

FreeIndependent, Group, SpecialEvent, VIP, and Wholesale are quantitative because they represent the number of rooms registered on a given day to a particular type of guest.

Question 22

2010, 2012, Christmas, NewYears, MemorialDay, PayDay, NewYears, SuperBowl, and Thanksgiving

Quantitative OR Qualitative?

Answer 22

Qualitative

For example, the variable Christmas tells us whether the observation occurs on Christmas or not. Likewise, 2010 tells us whether the observation occurs during 2010 or not.

Question 23

If we want to investigate the relationship between a dependent variable and one independent variable, what do we use?

Answer 23

single variable linear regression

Question 24

If we want to investigate the relationship between a dependent variable and multiple independent variables, what do we use?

Answer 24

multiple regression

Question 25

What’s different when forecasting for single vs multiple regression?

Answer 25

Forecasting with a multiple regression equation is very similar to forecasting with a single variable linear model. However, instead of entering only one value for a single independent variable, we input a value for each of the independent variables.

Question 26

What happens to R2 when independent variables are added to regression?

Answer 26

R2 never decreases when independent variables are added to a regression. it is important to multiply it by an adjustment factor when assessing and comparing the fit of a multiple regression model. This adjustment factor compensates for the increase in R2 that results solely from increasing the number of independent variables.

Question 27

How to test whether the relationship between the independent and dependent variables is linear and significant?

Answer 27

by analyzing the regression’s residual plots and the p-values associated with each independent variable’s coefficient.