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Questions and Answers
What does the assumption E(ut) = 0 imply in a classical linear regression model?
What does the assumption E(ut) = 0 imply in a classical linear regression model?
What is the potential consequence of violating the assumption var(ut) = σ² < ∞?
What is the potential consequence of violating the assumption var(ut) = σ² < ∞?
Which of the following tests specifically addresses autocorrelation in regression residuals?
Which of the following tests specifically addresses autocorrelation in regression residuals?
What is one possible effect of ignoring a violation of the assumption cov(ut, xt) = 0?
What is one possible effect of ignoring a violation of the assumption cov(ut, xt) = 0?
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Which statement accurately defines the normality assumption in a regression model?
Which statement accurately defines the normality assumption in a regression model?
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Which of the following is a step in testing for heteroscedasticity in regression residuals?
Which of the following is a step in testing for heteroscedasticity in regression residuals?
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What might be an advantage of using a dynamic model in econometrics?
What might be an advantage of using a dynamic model in econometrics?
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What does the Breusch-Godfrey test specifically diagnose?
What does the Breusch-Godfrey test specifically diagnose?
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What conclusion can be drawn when both the F - and χ 2 tests indicate no evidence of heteroscedasticity?
What conclusion can be drawn when both the F - and χ 2 tests indicate no evidence of heteroscedasticity?
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Which regression option needs to be selected in EViews to obtain heteroscedasticity-robust standard errors?
Which regression option needs to be selected in EViews to obtain heteroscedasticity-robust standard errors?
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What effect do heteroskedasticity-consistent standard errors typically have on the parameter estimates?
What effect do heteroskedasticity-consistent standard errors typically have on the parameter estimates?
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What does Assumption 3 of the CLRM state regarding the error terms?
What does Assumption 3 of the CLRM state regarding the error terms?
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What is the implication of positive autocorrelation in residuals?
What is the implication of positive autocorrelation in residuals?
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What does the Durbin-Watson (DW) test specifically test for?
What does the Durbin-Watson (DW) test specifically test for?
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In the context of autocorrelation, what does a plot showing no pattern in residuals indicate?
In the context of autocorrelation, what does a plot showing no pattern in residuals indicate?
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Which of the following indicates evidence of negative autocorrelation?
Which of the following indicates evidence of negative autocorrelation?
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What happens to the p-values when comparing heteroscedasticity-robust standard errors to ordinary standard errors?
What happens to the p-values when comparing heteroscedasticity-robust standard errors to ordinary standard errors?
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When testing for autocorrelation, what is plotted to assess the relationship between current and previous residuals?
When testing for autocorrelation, what is plotted to assess the relationship between current and previous residuals?
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What effect does a high p-value have on the evidence of heteroscedasticity?
What effect does a high p-value have on the evidence of heteroscedasticity?
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What does the test statistic formula for DW primarily assess?
What does the test statistic formula for DW primarily assess?
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What happens to OLS estimators in the presence of heteroscedasticity?
What happens to OLS estimators in the presence of heteroscedasticity?
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When heteroscedasticity is ignored, what is most likely affected?
When heteroscedasticity is ignored, what is most likely affected?
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What does a significance test statistic of TR² = 28 indicate regarding the null hypothesis?
What does a significance test statistic of TR² = 28 indicate regarding the null hypothesis?
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Which method can be used to adjust for known heteroscedasticity in a regression model?
Which method can be used to adjust for known heteroscedasticity in a regression model?
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What effect does heteroscedasticity typically have on the slope standard errors when its variance is positively related to an explanatory variable?
What effect does heteroscedasticity typically have on the slope standard errors when its variance is positively related to an explanatory variable?
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What does the use of heteroscedasticity-consistent standard error estimates do for hypothesis testing?
What does the use of heteroscedasticity-consistent standard error estimates do for hypothesis testing?
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What is the consequence of using OLS under conditions of heteroscedasticity?
What is the consequence of using OLS under conditions of heteroscedasticity?
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What is one method suggested for transforming variables to deal with heteroscedasticity?
What is one method suggested for transforming variables to deal with heteroscedasticity?
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How does the variance of errors relate when applying GLS with the specific form var(ut) = σ²zt²?
How does the variance of errors relate when applying GLS with the specific form var(ut) = σ²zt²?
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Which option best describes the relationship between heteroscedasticity and OLS standard errors for the intercept?
Which option best describes the relationship between heteroscedasticity and OLS standard errors for the intercept?
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How can the residuals help identify heteroscedasticity?
How can the residuals help identify heteroscedasticity?
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What happens when a researcher applies OLS yet the errors are inversely related to an explanatory variable?
What happens when a researcher applies OLS yet the errors are inversely related to an explanatory variable?
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What is a common problem faced when trying to identify the exact cause of heteroscedasticity?
What is a common problem faced when trying to identify the exact cause of heteroscedasticity?
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What distribution does the LM test statistic follow in the context of regression diagnostic tests?
What distribution does the LM test statistic follow in the context of regression diagnostic tests?
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What is one reason why R-squared values may be meaningless when the regression does not include a constant term?
What is one reason why R-squared values may be meaningless when the regression does not include a constant term?
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Which test is a commonly used method for detecting heteroscedasticity in regression?
Which test is a commonly used method for detecting heteroscedasticity in regression?
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Under the null hypothesis of heteroscedasticity tests like Goldfeld-Quandt, what is assumed about the variances?
Under the null hypothesis of heteroscedasticity tests like Goldfeld-Quandt, what is assumed about the variances?
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What does the Wald test statistic follow in terms of distribution?
What does the Wald test statistic follow in terms of distribution?
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In the Goldfeld-Quandt test, how are the two residual variances calculated?
In the Goldfeld-Quandt test, how are the two residual variances calculated?
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What is a potential drawback of the Goldfeld-Quandt test?
What is a potential drawback of the Goldfeld-Quandt test?
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What phenomenon is observed when the variance of the errors changes over time?
What phenomenon is observed when the variance of the errors changes over time?
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What is the consequence of forcing a regression line through the origin by omitting the constant term?
What is the consequence of forcing a regression line through the origin by omitting the constant term?
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Under the assumptions detailed for regression analysis, what is homoscedasticity?
Under the assumptions detailed for regression analysis, what is homoscedasticity?
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What happens to the equivalence of the LM and Wald tests as the sample size increases?
What happens to the equivalence of the LM and Wald tests as the sample size increases?
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Which of the following is NOT a reason for using diagnostic tests in regression models?
Which of the following is NOT a reason for using diagnostic tests in regression models?
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In the context of residual analysis for heteroscedasticity, what kind of plot is generally used?
In the context of residual analysis for heteroscedasticity, what kind of plot is generally used?
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Which of the following statements is true regarding the implications of heteroscedasticity in a regression model?
Which of the following statements is true regarding the implications of heteroscedasticity in a regression model?
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What does a Durbin-Watson (DW) statistic value less than the lower critical value indicate?
What does a Durbin-Watson (DW) statistic value less than the lower critical value indicate?
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Which of the following is NOT a condition for the Durbin-Watson test to be valid?
Which of the following is NOT a condition for the Durbin-Watson test to be valid?
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If the DW statistic value is equal to 4, what does that suggest about the residuals?
If the DW statistic value is equal to 4, what does that suggest about the residuals?
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What would be the implication if the DW statistic is found between the upper and lower critical values?
What would be the implication if the DW statistic is found between the upper and lower critical values?
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What does the numerator of the DW test statistic help identify in regression errors?
What does the numerator of the DW test statistic help identify in regression errors?
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Which of these statistics follows an irregular distribution, making it difficult to classify autocorrelation?
Which of these statistics follows an irregular distribution, making it difficult to classify autocorrelation?
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What is the acceptable value range for the DW statistic to conclude no autocorrelation exists?
What is the acceptable value range for the DW statistic to conclude no autocorrelation exists?
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What does the Breusch-Godfrey test assess in comparison to the Durbin-Watson test?
What does the Breusch-Godfrey test assess in comparison to the Durbin-Watson test?
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In the example given, what conclusion can be drawn if the DW statistic value is 0?
In the example given, what conclusion can be drawn if the DW statistic value is 0?
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Which term is used to refer to the presence of errors in regression that are correlated across time periods?
Which term is used to refer to the presence of errors in regression that are correlated across time periods?
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What does a positive autocorrelation in the errors indicate about the model's residuals?
What does a positive autocorrelation in the errors indicate about the model's residuals?
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What is the general hypothesis test structure used in Breusch-Godfrey test for autocorrelation?
What is the general hypothesis test structure used in Breusch-Godfrey test for autocorrelation?
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Why must the conditions for using the DW test be strictly adhered to?
Why must the conditions for using the DW test be strictly adhered to?
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Study Notes
Classical Linear Regression Model Assumptions and Diagnostic Tests
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Five assumptions underpin the classical linear regression model (CLRM) to ensure ordinary least squares (OLS) estimation's desirable properties and validity of hypothesis tests. These include:
- Expected value of the error term (ut) is zero (E(ut) = 0).
- Variance of the error term is constant and finite (var(ut) = σ2 < ∞).
- Covariance between any two error terms is zero (cov(ui, uj) = 0).
- Covariance between the error term and explanatory variables is zero (cov(ut, xt) = 0).
- Error term follows a normal distribution (ut ~ N(0, σ2)).
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Violations of these assumptions can lead to several problems:
- Incorrect coefficient estimates (β̂s).
- Incorrect standard errors.
- Invalid test statistic distributions.
Diagnostic Tests
- Diagnostic (misspecification) tests calculate a test statistic.
- Common approaches include the Lagrange Multiplier (LM) test and Wald test, both asymptotically equivalent but with slightly different small-sample results.
- LM test statistics follow a chi-squared distribution (χ2) with degrees of freedom (m) equal to the restrictions.
- Wald test statistics follow an F-distribution with (m, T - k) degrees of freedom.
- Common approaches include the Lagrange Multiplier (LM) test and Wald test, both asymptotically equivalent but with slightly different small-sample results.
Assumption 2: Constant Variance (Homoscedasticity)
- Homoscedasticity assumes the error term's variance is constant.
- Heteroscedasticity implies varying error variances—a common violation.
- A graph of residuals against an explanatory variable can illustrate heteroscedasticity. Increasing variance with the variable in the plot clearly depicts heteroscedasticity.
Detecting Heteroscedasticity
- Visual inspection of residual plots can be unreliable.
- Formal statistical tests like the Goldfeld-Quandt test are more robust.
Goldfeld--Quandt Test
- Divides the sample into subsamples (T1, T2).
- Estimates the regression and calculates residual variances (s12, s22).
- Assumes equal error variances (H0: σ12 = σ22) and tests for heteroscedasticity using the ratio (GQ) of the variances.
- Large GQ values lead to heteroscedasticity rejection.
Consequences of Ignoring Heteroscedasticity
- OLS coefficient estimates remain unbiased and consistent but aren't the Best Linear Unbiased Estimators (BLUE).
- OLS standard errors are incorrect, resulting in misleading inferences.
- Intercept standard errors are typically underestimated.
- Slope standard errors depend on the heteroscedasticity form.
Dealing with Heteroscedasticity
- GLS (Generalized Least Squares) or WLS (Weighted Least Squares) can address known heteroscedasticity patterns.
- Data transformation (e.g., logs) may reduce the effect of heteroscedasticity.
- Robust standard error estimates account for heteroscedasticity. A software packages adjustment of the std errors.
Testing for Heteroscedasticity in EViews
- Residual plots can indicate heteroscedasticity if variability changes systematically over time.
Assumption 3: Zero Covariance (No Autocorrelation)
- The covariance between error terms across observations (or over time) is zero in the classical linear model.
- Autocorrelation (serial correlation) indicates correlated error terms.
Detecting Autocorrelation
- Visual inspection of plots (residuals against lagged residuals, or residuals over time).
- Positive autocorrelation indicates a pattern of similar signs in successive errors.
- Negative autocorrelation indicates alternating signs.
- No pattern represents no autocorrealtion
Durbin-Watson (DW) Test
- Tests for first-order autocorrelation (correlation between consecutive errors).
- DW statistic's value depends on the autocorrelation level (positive or negative), which can be inconclusive in some cases.
- Critical values (dL, dU) help determine rejection or non-rejection regions.
Breusch-Godfrey Test
- A generalized test for autocorrelation up to a specified order (r).
- Uses auxiliary regressions and calculated χ2 or F statistics to test for autocorrelation.
- Assumes no relationship between the error term and previous values of errors (H0).
Conditions for Valid DW Test
- Constant term in the regression.
- Non-stochastic regressors.
- No lagged dependent variables in the regression.
Other Key Considerations
- White's method provides heteroscedasticity-consistent standard errors for OLS and helps with appropriate inference in case of heteroscedasticity during regression analysis.
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Description
This quiz covers the five key assumptions that underpin the classical linear regression model (CLRM), essential for the validity of ordinary least squares (OLS) estimation. Understand the implications of each assumption and the potential issues arising from their violations. Test your knowledge through diagnostic tests and mis-specification assessments.
