SmartBook #7

What is used to evaluate how well the sample regression equation fits the data?

-The coefficient of determination, R² -The standard error of the estimate

In order to avoid the possibility of R2 creating a false impression, virtually all software packages include adjusted R2. Unlike R2, adjusted R2 explicitly accounts for what?

-The number of predictor variables k -The sample size n

If we include as many dummy variables as there are categories, then their sum will equal '____'

1

In the presence of correlated observations, the OLS estimators are unbiased, but their estimated standard errors are inappropriate. Which of the following could happen as a result?

All of the answers are correct The model looks better than it really is with a spuriously high R2 The t test may suggest that the predictor variables are individually and jointly significant when this is not true The F test may suggest that the predictor variables are individually and jointly significant when this is not true

The assumption of constant variability of observations often breaks down in studies with cross-sectional data. Consider the model y = β0 + β1x + ɛ, where y is a household's consumption expenditure and x is its disposable income. It may be unreasonable to assume that the variability of consumption is the same across a cross-section of household incomes. This violation is called:

Changing variability

If the value of the response variable is uniquely determined by the values of the predictor variables, we say that the relationship between the variables is: (Choose the correct response)

Deterministic

The example of momentum p is the product of the mass m and the velocity v of an object; that is, p = mv, is an example of a '____' relationship.

Deterministic

What is a good solution when confronted with multicollinearity?

Drop one of the collinear variables Obtain more data because the sample correlation may get weaker

If the linear regression model includes an intercept, the number of dummy variables representing a categorical variable should be one less than the number of categories of the variable. This solution helps avoid which problem?

Dummy variable trap

True or false: R2 can decrease as we add more predictor variables to the linear regression model

False

The detection methods for multicollinearity are fairly rigid - there are lists of exact steps to take to detect multicollinearity.

False. The detection methods for multicollinearity are mostly informal, not rigid.

In the case of a dummy variable categorizing a person's gender, we can define 1 for male and 0 for female. In this case, what would the reference category be?

Female

What is a measure that summarizes how well the sample regression equation fits the data?

Goodness-of-fit

For the linear regression model, y = β0 + β1x1 + β2x2 + . . . + βkxk + ɛ, which of the following are the competing hypotheses used for a test of joint significance? Choose both the correct test for the null and alternative hypotheses.

H0:β1=β2=... =βk=0 HA:At least one βi≠0

The detection methods for multicollinearity are mostly informal. Which of the following indicate a potential multicollinearity issue?

High R2 and significant F statistic coupled with insignificant predictor variables

Consider the following linear regression model, which links the response variable y with k predictor variables x1, x2,..., xk:y=β0+β1x1+β2x2+... +βkxk+ε. If, for example, the slope coefficient β1 equals zero, then the predictor variable x1 does what and implies what? Choose all that answer the 'does what?' and 'Implies what?' questions!

Implying that x1 does not influence y Drops out of the equation

A simple linear regression model and is represented as y = β0 + β1x1 + ɛ,; What do β0and β1 (the Greek letters read as betas) represent? (They must be shown in the correct order!)

Intercept, slope

The variance inflation factor (VIF) is another measure that can detect a high correlation between three or more predictor variables even if no pair of predictor variables has a particularly high correlation. What is the smallest possible value of VIF? (absence of multicollinearity).

One

A dummy variable, also referred to as an indicator or a binary variable, takes on numerical values of 1 or 0 to describe two categories of a categorical variable. For a predictor variable that is a dummy variable, it is common to refer to the category that assumes a value of 0 as: Please select all that apply.

Reference category Benchmark category

Which of the following are detection methods for multicollinearity, as discussed in this section? Select all that apply!

Sometimes researchers examine the correlations between the predictor variables to detect severe multicollinearity. Seemingly wrong signs of the estimated regression coefficients may also indicate multicollinearity. The presence of a high R2R2coupled with individually insignificant predictor variables can indicate multicollinearity.

In the presence of changing variability, the estimated standard errors of the OLS estimators are inappropriate. What does this imply about using standard testing?

Standard t or F tests are not valid as they are based on these estimated standard errors.

When we assess a linear regression model, there are several tests we can use. What is the test called that determines whether the predictor variables x1, x2,..., xk have a joint statistical influence on y?

Test of joint significance

The nonzero slope coefficient test is used for a renowned financial application referred to as the capital asset pricing model (CAPM).The model y = α + βx + ɛ, is essentially a simple linear regression model that uses α and β, in place of the usual β0 and β1, to represent the intercept and the slope coefficients, respectively. Which of the following is true about the slope coefficient α, called the stock's alpha? Select that apply!

The CAPM theory predicts α to be zero Abnormal returns are positive when α > 0 Abnormal returns are negative when α < 0.

In order to select the preferred model, we examine several goodness-of-fit measures: Select all goodness-of-fit measures examined!

The coefficient of determination The adjusted coefficient of determination The standard error of the estimate

We can use residual plots to gauge changing variability.The residuals are generally plotted against each predictor variable xj Which of the following indicates there is no violation?

The residuals are randomly dispersed across the values of xj

We can plot the residuals sequentially over time to look for correlated observations. If there is no violation, then what would you see?

The residuals should show no pattern around the horizontal axis.

Instead of se2,we generally report the standard deviation of the residual, denoted se, more commonly referred to as

The standard error of the estimate

We use analysis of variance (ANOVA) in the context of the linear regression model to derive R2.We denote the total variation in y as Σ(yi−y ̄)2, which is the numerator in the formula for the variance of y. What is this total variation called?

Total sum of squares

An important first step before running a regression model is to compile a comprehensive list of potential predictor variables. How can we reduce the list to a smaller list of predictor variables?

Use the adjusted R2 criterion to reduce the list

A crucial assumption in a linear regression model is that the error term is not correlated with the predictor variables. In general, when does this assumption break down?

When important predictor variables are excluded.

We can plot the residuals sequentially over time to look for correlated observations. How are violations indicated?

When positive residuals and negative residuals alternate over a few periods, sometimes positive or negative for a couple of periods.

The nonzero slope coefficient test is used for a renowned financial application referred to as the capital asset pricing model (CAPM).The model y = α + βx + ɛ, is essentially a simple linear regression model that uses α and β, in place of the usual β0 and β1, to represent the intercept and the slope coefficients, respectively. Which of the following is true about the slope coefficient β, called the stock's beta? Select that apply!

When β equals 1, any change in the market return leads to an identical change in the given stock return. Measures how sensitive the stock's return is to changes in the level of the overall market A stock for which β > 1 is considered more "aggressive" or riskier than the market

As mentioned earlier, in the presence of changing variability, the OLS estimators are unbiased, but their estimated standard errors are inappropriate. Since the t and the F tests are no longer valid, which are suggested strategies to correct the standard errors?

White's robust standard errors

If one or more of the relevant predictor variables are excluded, then the resulting OLS estimators are biased. The extent of the bias depends on the degree of the '____" between the included and the excluded predictor variables.

correlation

We can use residual plots to gauge changing variability. The residuals are generally plotted against each predictor variable xj. There is a violation if the variability increases or '____' over the values of xj.

decreases

When comparing models with the same response variable, we prefer the model with a smaller se. A smaller se implies that there is '____' dispersion of the observed values from the predicted values.

less

When confronted with multicollinearity, the best approach may be to do '____' if the estimated model yields a high R2,

nothing

It is important that we include all "____" predictor variables in the regression model.

relevant

If a linear regression model uses only one predictor variable, then the model is referred to as a '____' linear regression model

simple

In the presence of changing variability, the OLS estimators are '____', but their estimated standard errors are inappropriate.

unbiased

In the presence of correlated observations, the OLS estimators are "____" but their estimated standard errors are inappropriate.

unbiased

Suppose the competing hypotheses in testing for individual significance are H0: βj = 0 versus HA: βj ≠ 0. What would rejecting the null hypothesis imply?

xj is significant in explaining y