Regression Analysis Concepts Quiz

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Questions and Answers

What is the consequence of omitting an important variable from a regression analysis?

The coefficient estimates will remain unbiased.

The standard errors will become more accurate.

The estimated coefficients will be biased and inconsistent. (correct)

All other variable coefficients will be consistent.

What is the main assumption behind the parameter stability tests?

The model must include at least three independent variables.

Data must be collected over multiple years.

Parameters are constant for the entire sample period. (correct)

Coefficient estimates are not affected by sample size.

In the Chow test, what is used to form the F-test?

Difference between the sum of squared residuals (RSS) of the regressions. (correct)

Difference between the means of the sub-periods.

The estimated variance of the error term.

The total number of observations in the sample.

What happens if an irrelevant variable is included in a regression model?

The estimators will be inefficient but unbiased. Signup and view all the answers

When creating a dummy variable, what is the purpose of setting it to zero otherwise?

To isolate the effect of a specific observation. Signup and view all the answers

What is the null hypothesis in the Goldfeld-Quandt test?

$H_0: ext{Var}( ext{ disturbances})$ are equal Signup and view all the answers

When conducting the GQ test, what is the next step after splitting the sample into two sub-samples?

Estimate the regression model on both sub-samples Signup and view all the answers

What is the formula for the GQ test statistic?

$GQ = rac{s^2_{ ext{larger}}}{s^2_{ ext{smaller}}}$ Signup and view all the answers

In White's Test, what is the purpose of running the auxiliary regression?

To obtain residuals that will test for heteroscedasticity Signup and view all the answers

What distribution does the test statistic from the GQ test follow under the null hypothesis?

F-distribution Signup and view all the answers

Why might the choice of where to split the sample in the GQ test be problematic?

The outcome of the test can vary based on the split location. Signup and view all the answers

How is the chi-squared statistic calculated in White’s test after running the auxiliary regression?

By multiplying R2 by the number of observations, T Signup and view all the answers

What is indicated by the null hypothesis in the Breusch-Godfrey Test?

There is no autocorrelation present. Signup and view all the answers

What is the consequence of ignoring autocorrelation in a regression model?

The standard error estimates become inappropriate. Signup and view all the answers

Which statement is true regarding the method to correct for autocorrelation when its form is known?

GLS procedures may introduce additional assumptions. Signup and view all the answers

What is a key characteristic of perfect multicollinearity?

Some explanatory variables are perfectly correlated. Signup and view all the answers

In the analysis of autocorrelation, what is the significance of the test statistic exceeding the critical value?

It indicates a rejection of the null hypothesis. Signup and view all the answers

What does it mean when R2 is inflated due to positively correlated residuals?

The perceived explanatory power of the model is overstated. Signup and view all the answers

What is a potential problem if near multicollinearity is present but ignored?

Standard errors of the coefficients may become high. Signup and view all the answers

Which analysis method can be used when the form of autocorrelation is unknown?

Modify the regression model based on residual analysis. Signup and view all the answers

What is the outcome if a regression model is estimated under conditions of perfect multicollinearity?

No individual coefficients can be estimated. Signup and view all the answers

What is a characteristic of regression analysis when multicollinearity is present?

Confidence intervals for parameters become wide. Signup and view all the answers

Which method is NOT commonly used to measure multicollinearity?

Standard deviation of residuals. Signup and view all the answers

What is one suggested solution to address multicollinearity?

Increase the frequency of data collection. Signup and view all the answers

What is a potential solution if the true model is a non-linear model?

Transform the data into logarithms. Signup and view all the answers

Which statistical test can be used to check for functional form mis-specification in a regression model?

Ramsey’s RESET test. Signup and view all the answers

What happens if the value of the test statistic in Ramsey’s RESET test exceeds the critical value?

Reject the null hypothesis. Signup and view all the answers

What do skewness and kurtosis measure in a distribution?

The distribution's shape characteristics. Signup and view all the answers

What is a common misconception about high correlation between one of the independent variables and the dependent variable?

It indicates multicollinearity. Signup and view all the answers

Which test formalizes checking the normality of residuals?

Bera Jarque test. Signup and view all the answers

What is the coefficient of kurtosis for a normal distribution?

3 Signup and view all the answers

Which of the following is likely a drawback of traditional solutions for multicollinearity?

They often cause more problems than they solve. Signup and view all the answers

What does the Bera Jarque test statistic W need to be transformed into?

Chi-square distribution. Signup and view all the answers

What is the purpose of including higher order terms in the auxiliary regression of Ramsey's RESET test?

To examine potential mis-specification of functional form. Signup and view all the answers

When residuals exhibit non-normality, what is a common course of action?

Use dummy variables. Signup and view all the answers

When transforming highly correlated variables into ratios, what is the intended outcome?

To reduce the number of variables without losing information. Signup and view all the answers

What is one consequence of multicollinearity that affects statistical tests?

Standard errors of coefficients increase. Signup and view all the answers

What indicates the rejection of the normality assumption in residuals?

Presence of extreme residuals. Signup and view all the answers

In the context of hypothesis testing, why is normality assumed?

It simplifies the calculation of probabilities. Signup and view all the answers

What do the coefficients of skewness and kurtosis indicate when they are jointly tested for normality?

They must equal zero for normality. Signup and view all the answers

What is the commonly used method to test for departures from normality?

Shapiro-Wilk test. Signup and view all the answers

Study Notes

Classical Linear Regression Model Assumptions and Diagnostics

Classical linear regression models (CLRM) have assumptions for disturbance terms.
These assumptions include:
- Expected value of the error term (ε_t) is zero (E(ε_t) = 0).
- Variance of the error term is constant (Var(ε_t) = σ²).
- Covariance between any two error terms is zero (cov(ε_i, ε_j) = 0 for i ≠ j).
- The X matrix is non-stochastic or fixed in repeated samples.
- Errors are normally distributed (ε_t ~ N(0, σ²)).

Violations of CLRM Assumptions

Studying violations of assumptions, including how to test for them, their causes, and consequences.
Consequences can include incorrect coefficient estimates, inaccurate standard errors, and inappropriate test statistics.
Solutions involve addressing violations or employing alternative techniques.

Assumption 1: E(ε_t) = 0

The mean of the disturbances is assumed to be zero.
Residuals are used to test this assumption, and their mean will always be zero if there's a constant term in the regression.

Assumption 2: Var(ε_t) = σ²

Homoscedasticity - the variance of errors is constant (Var(ε_t) = σ²)
Heteroscedasticity - the variance of errors varies.
- Detection includes methods like the Goldfeld-Quandt (GQ) test and White's test.
- The GQ test involves splitting the data, calculating residual variances, and forming a ratio for the test statistic following an F distribution.
- White's test uses an auxiliary regression based on squared residuals and regressors.

Consequences of Heteroscedasticity

Using OLS with heteroscedasticity leads to unbiased coefficient estimates, but standard errors are wrong and inferences are flawed.
The degree of bias in standard errors depends on the form of heteroscedasticity.

Dealing with Heteroscedasticity

If the form of heteroscedasticity is known, generalized least squares (GLS) can be used.
A simple illustration of GLS divides the regression by a variable related to the error variance .

Autocorrelation

The CLRM assumes no pattern, or zero covariance, between errors (Cov(ε_i, ε_j) = 0).
If errors have patterns, they're autocorrelated.
- Detecting autocorrelation (formal tests, such as Durbin-Watson, and Breusch-Godfrey test)
- Durbin-Watson Test (DW) tests for first-order autocorrelation, comparing errors with prior errors; ranges from 0 to 4.
- Breusch-Godfrey test is a more general, rth-order autocorrelation test.

Consequences of Ignoring Autocorrelation

Coefficient estimates remain unbiased but are inefficient (not BLUE).
Standard errors are inappropriate and often lead to incorrect inferences, such as incorrect conclusions about variable significance.
R-squared values can be inflated in the presence of positively autocorrelated errors.

Remedies for Autocorrelation

GLS techniques can be employed if the form of autocorrelation is known.
Procedures like Cochrane-Orcutt are examples of GLS when autocorrelations are evident.
Often modify the regression to fix autocorrelation if its form cannot be identified.

Multicollinearity

High correlations between explanatory variables.
Perfect multicollinearity renders coefficient estimation impossible.
Near multicollinearity impacts coefficient standard errors (making them large) and sensitivity of the regression to specification changes.
R-squared is often high but individual variables become less significant when multicollinearity is present.

Measuring Multicollinearity

Method 1: Assessing the correlations between variables using a correlation matrix.
Method 2: Analyzing the variance inflation factors (VIFs) to measure the effect of multicollinearity on independent variables.

Solutions to Multicollinearity

Traditional techniques like ridge regression or principal component analysis.
Some practitioners opt to ignore the issue if the model's validity is otherwise well-supported.
Drop one of the collinear variables or transform the variables into ratios, or seek more data.

Incorrect Functional Form

If the relationship between variables is not linear.
Ramsey's RESET test can be used to identify non-linearity.
This test adds higher powers of fitted values to an auxiliary regression to assess if the linearity assumption is valid by examining the R squared from the auxiliary regression.

Testing Normality

Normality assumption implies errors are normally distributed.
Bera-Jarque test is used, assessing skewness (b₁) and kurtosis (b₂); a normal distribution has zero skewness and a kurtosis of 3 (b₂ = 3). A jointly zero result confirms theoretical normality.
The test statistic is a function of these coefficients, and a large value suggests non-normality.

Solutions for Non-Normality

Switch to a non-parametric method if normality tests produce rejection.
Identify and consider transformations to handle non-normality or errors that are too extreme and use dummy variables for identified extreme errors.

Omission of an Important Variable or Inclusion of an Irrelevant Variable

Omitting relevant variables leads to biased coefficient estimates in other variables.
Including irrelevant variables increases the number of variables without improving analysis effectiveness.

Parameter Stability Test

Assesses whether parameters in a model remain constant over the entire sample or just parts of the sample.
Chow test is a common technique for analyzing parameter stability, essentially comparing restricted and unrestricted models.
This approach performs a separate regression for the whole period, and each sub-part. An F ratio compares the restricted to unrestricted model.
If the statistic exceeds the critical value, you reject the null hypothesis that the parameters are unchanging.

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Description

Test your understanding of key concepts in regression analysis. This quiz covers essential topics like variable omission, parameter stability tests, and the implications of including irrelevant variables. Assess your knowledge with specific questions about various statistical tests used in regression.