Module 10.1: Linear Regression

Question 1

What is the purpose of simple linear regression?

Answer 1

To explain the variation in a dependent variable in terms of the variation in a single independent variable.

Variation is interpreted as the degree to which a variable differs from its mean value.

Question 2

How is variation defined in the context of regression?

Answer 2

The degree to which a variable differs from its mean value.

Variation should not be confused with variance.

Question 3

What is the dependent variable in linear regression?

Answer 3

The variable whose variation is explained by the independent variable.

It is also referred to as the explained variable, endogenous variable, or predicted variable.

Question 4

What is the independent variable in linear regression?

Answer 4

The variable used to explain the variation of the dependent variable.

It is also referred to as the explanatory variable, exogenous variable, or predicting variable.

Question 5

Fill in the blank: The dependent variable is also referred to as the _______.

Answer 5

explained variable

Question 6

Fill in the blank: The independent variable is also referred to as the _______.

Answer 6

explanatory variable

Question 7

True or False: Variation and variance are the same concepts.

Answer 7

False

Question 8

What question does simple linear regression aim to answer?

Answer 8

“What explains fluctuations in the dependent variable?”

Question 9

What is the role of GDP in predicting stock returns?

Answer 9

GDP is the independent variable, while stock returns are the dependent variable.

Stock returns are explained by GDP.

Question 10

Fill in the blank: The dependent variable is explained by the _______.

Answer 10

independent variable

In this context, stock returns are explained by GDP.

Question 11

What is the relationship between independent and dependent variables?

Answer 11

The independent variable explains the variation in the dependent variable.

In this case, GDP explains stock returns.

Question 12

What does Y_i represent in the linear regression model?

Answer 12

ith observation of the dependent variable, y

Yi is the value we are trying to predict or explain.

Question 13

What does X_i represent in the linear regression model?

Answer 13

ith observation of the independent variable, X

Xi is the predictor variable used to explain variations in Y.

Question 14

What is b₀ in the context of linear regression?

Answer 14

regression intercept term

bo is the value of Y when X is zero.

Question 15

What does b_i denote in the linear regression model?

Answer 15

regression slope coefficient

bi indicates the change in Y for a one-unit change in X.

Question 16

What is the significance of the term e_i in the linear regression model?

Answer 16

residual for the ith observation (also referred to as the disturbance term or error term)

&i accounts for the difference between the observed and predicted values of Y.

Question 17

What is the main purpose of the regression process in this model?

Answer 17

estimates an equation for a line through a scatter plot of the data that ‘best’ explains the observed values for Y in terms of the observed values for X

This involves minimizing the sum of the squared residuals.

Question 18

Fill in the blank: In the linear regression model, Y_i = b₀ + b_i X_i + _____

Answer 18

e_i

This term represents the residual for each observation.

Question 19

What is the form of the linear equation often called the line of best fit?

Answer 19

Question 20

What does the hat ‘^’ above a variable indicate?

Answer 20

Predicted value

Question 21

What is the regression line?

Answer 21

The line that minimizes the sum of the squared differences between predicted and actual Y-values

Question 22

What is the sum of squared errors (SSE)?

Answer 22

The sum of the squared vertical distances between estimated and actual Y-values

Question 23

True or False: The regression line is the only line that can be drawn through a scatter plot of X and Y.

Answer 23

False

Question 24

Fill in the blank: The regression line minimizes the sum of the squared differences (vertical distances) between the _______ predicted by the regression equation and the actual Y-values.

Answer 24

Y-values

Question 25

What does the regression line minimize?

Answer 25

The SSE ## Footnote SSE stands for Sum of Squared Errors.

Question 26

What is another name for simple linear regression?

Answer 26

Ordinary least squares (OLS) regression

Question 27

What does the estimated slope coefficient describe?

Answer 27

The change in Y for a one-unit change in X

Question 28

How can the slope term be positive, negative, or zero?

Answer 28

It depends on the relationship between the regression variables

Question 29

How is the slope term calculated?

Answer 29

Question 30

What does the intercept term represent?

Answer 30

The line's intersection with the Y-axis at X = 0

Question 31

Can the intercept term be positive, negative, or zero?

Answer 31

Yes

Question 32

How can the intercept term be expressed using the means of Y and X?

Answer 32

Question 33

What coordinates does the regression line pass through?

Answer 33

The mean of the independent and dependent variables (i.e., the point Xbar , Ybar)

Question 34

What does the estimated intercept represent in regression analysis?

Answer 34

The value of the dependent variable at the point of intersection of the regression line and the vertical axis.

Question 35

When is the intercept an estimate of the dependent variable?

Answer 35

When the independent variable is zero.

Question 36

What does the estimated slope coefficient indicate?

Answer 36

The expected change in the dependent variable for a one-unit change in the independent variable.

Question 37

If the estimated slope coefficient is 2, how much is the dependent variable expected to change for a one-unit change in the independent variable?

Answer 37

By two units.

Question 38

What is the slope coefficient in a regression of excess returns on market returns called?

Answer 38

The stock's beta ## Footnote Beta is an estimate of the systematic risk of a stock.

Question 39

What does a beta less than 1 indicate about a stock's risk?

Answer 39

Less risky than the average stock ## Footnote This means the stock's returns tend to change less than the overall market.

Question 40

What does a beta of 1 signify?

Answer 40

Average level of systematic risk ## Footnote A stock with this beta moves in line with the market.

Question 41

What does a beta greater than 1 indicate?

Answer 41

More-than-average systematic risk ## Footnote This means the stock's returns are more volatile than the market.

Question 42

What must be assessed to determine the importance of an independent variable in explaining a dependent variable?

Answer 42

Statistical significance of the slope coefficient ## Footnote This involves hypothesis testing or forming a confidence interval.

Question 43

What is the first assumption of linear regression?

Answer 43

A linear relationship exists between the dependent and the independent variables.

Question 44

What does homoskedasticity refer to in linear regression?

Answer 44

The variance of the residual term is constant for all observations.

Question 45

What does it mean for the residual term to be independently distributed?

Answer 45

The residual for one observation is not correlated with that of another observation.

Question 46

How is the normality of the residual term defined in linear regression?

Answer 46

The residual term is normally distributed.

Question 47

When is a linear regression model not appropriate?

Answer 47

When the underlying relationship between X and Y is nonlinear.

Question 48

What is a method to check for linearity in a regression model?

Answer 48

Examine the model residuals in relation to the independent regression variable.

Question 49

Fill in the blank: The residual term in linear regression must be _______.

Answer 49

normally distributed.

Question 50

True or False: In linear regression, the paired x and y observations are dependent on each other.

Answer 50

False.

Question 51

What does homoskedasticity refer to?

Answer 51

The case where prediction errors all have the same variance. ## Footnote Homoskedasticity is an important assumption in regression analysis.

Question 52

What is heteroskedasticity?

Answer 52

The situation when the assumption of homoskedasticity is violated. ## Footnote Heteroskedasticity can lead to inefficient estimates in regression models.

Question 53

What is a type of heteroskedasticity related to the error term?

Answer 53

The variance of the error term changes over time.

Question 54

What does it indicate if the magnitude of errors exhibits a pattern of changing over time?

Answer 54

It indicates heteroskedasticity.

Question 55

What is the implication of large prediction errors observed in monthly sales data?

Answer 55

It suggests a lack of independence in the relationship between the variables.

Question 56

What does it mean if the residuals are not independent?

Answer 56

It means our estimates of the model parameters' variances will not be correct.

Question 57

True or False: Independence of the relationship between X and Y ensures that residuals are independent.

Answer 57

True

Question 58

What does it indicate when the residuals are normally distributed? We can conduct hypothesis testing for evaluating the goodness of fit of the model when the residual are ____________.

Answer 58

Normally distributed

Question 59

What is the central limit theorem's role regarding parameter estimates, when the residuals are not normally distributed?

Answer 59

With a large sample size, parameter estimates may be valid even when the residuals are not normally distributed. ## Footnote This theorem supports the robustness of statistical inference under certain conditions.

Question 60

What are outliers in the context of regression analysis?

Answer 60

Observations that are far from our regression line, characterized by large prediction errors or X values that are far from others. ## Footnote Outliers can skew the results and affect the accuracy of the model.

Question 61

What are the values determined by the estimated regression equation called?

Answer 61

Least squares estimates