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Questions and Answers
What is the purpose of constructing a confidence interval in statistical testing?
What is the purpose of constructing a confidence interval in statistical testing?
In the hypothesis test stated, what does the null hypothesis H0 : βj = aj signify?
In the hypothesis test stated, what does the null hypothesis H0 : βj = aj signify?
What does the t-statistic measure in the context of the hypothesis test?
What does the t-statistic measure in the context of the hypothesis test?
What does the p-value represent in statistical testing?
What does the p-value represent in statistical testing?
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In the context of constructing confidence intervals, what does the term tα/2 refer to?
In the context of constructing confidence intervals, what does the term tα/2 refer to?
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What does the F statistic help to determine in a regression model?
What does the F statistic help to determine in a regression model?
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Which of the following correctly describes the process of imposing restrictions in regression?
Which of the following correctly describes the process of imposing restrictions in regression?
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In a regression model, if the restrictions are such that only one exclusion is being tested, what is the relationship between F and t?
In a regression model, if the restrictions are such that only one exclusion is being tested, what is the relationship between F and t?
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What is the purpose of comparing the sum of squared residuals (SSR) between the unrestricted and restricted models?
What is the purpose of comparing the sum of squared residuals (SSR) between the unrestricted and restricted models?
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What would the restricted model be if the null hypothesis states H0: β1 = 1 and β3 = 0 for the voting model?
What would the restricted model be if the null hypothesis states H0: β1 = 1 and β3 = 0 for the voting model?
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What is the marginal effect of age on ln(wage) for male workers?
What is the marginal effect of age on ln(wage) for male workers?
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Which coefficients contribute to the marginal effect of age for female workers?
Which coefficients contribute to the marginal effect of age for female workers?
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What is the simplified version of the coefficients $β_1 + β_7$?
What is the simplified version of the coefficients $β_1 + β_7$?
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In the equation presented, which coefficient represents the effect of years worked?
In the equation presented, which coefficient represents the effect of years worked?
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Which of the following is true about the coefficient $β_8$?
Which of the following is true about the coefficient $β_8$?
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What does the coefficient $−0.00093$ represent in the context of the equation?
What does the coefficient $−0.00093$ represent in the context of the equation?
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If age increases by one unit, how is the wage affected for male workers?
If age increases by one unit, how is the wage affected for male workers?
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Which term is associated with the combined effect of age and tenure in the model?
Which term is associated with the combined effect of age and tenure in the model?
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What is the condition for the error term in the Classical Linear Model?
What is the condition for the error term in the Classical Linear Model?
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What does Assumption MLR.6 enable in hypothesis testing?
What does Assumption MLR.6 enable in hypothesis testing?
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Which statement regarding sample sizes is true according to the CLM assumptions?
Which statement regarding sample sizes is true according to the CLM assumptions?
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What is the distribution of the normalized estimators $eta_j$ in the Classical Linear Model?
What is the distribution of the normalized estimators $eta_j$ in the Classical Linear Model?
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In the context of the CML, what is the variance estimator $ ilde{eta}^2$ related to?
In the context of the CML, what is the variance estimator $ ilde{eta}^2$ related to?
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What does the expression $Var(etâ) = rac{σ^2 (X'X)^{-1}}{N-K-1}$ represent?
What does the expression $Var(etâ) = rac{σ^2 (X'X)^{-1}}{N-K-1}$ represent?
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The formula $ ilde{eta}^2 = rac{û_i^2}{N-K-1}$ implies that σ̂² is based on which components?
The formula $ ilde{eta}^2 = rac{û_i^2}{N-K-1}$ implies that σ̂² is based on which components?
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What ensures the independency of population errors from the explanatory variables in CLM?
What ensures the independency of population errors from the explanatory variables in CLM?
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What does the critical value F6,514 = 2.116206 indicate about the null hypothesis?
What does the critical value F6,514 = 2.116206 indicate about the null hypothesis?
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In the context of restricted OLS estimation, what does the notation Rβ̃ = r signify?
In the context of restricted OLS estimation, what does the notation Rβ̃ = r signify?
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What is the mean of the restricted OLS estimator when Rβ̂ = r?
What is the mean of the restricted OLS estimator when Rβ̂ = r?
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Which statement best describes the variance of the restricted OLS estimator?
Which statement best describes the variance of the restricted OLS estimator?
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What does the notation Var [β̂ur] - Var [β̂r] signify in the context of efficiency?
What does the notation Var [β̂ur] - Var [β̂r] signify in the context of efficiency?
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What is the main purpose of using restricted OLS estimators in regression analysis?
What is the main purpose of using restricted OLS estimators in regression analysis?
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In the expression for the variance of the restricted OLS estimator, what do the terms I and B represent?
In the expression for the variance of the restricted OLS estimator, what do the terms I and B represent?
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How does the restricted OLS estimator relate to the unrestricted OLS estimator in terms of efficiency?
How does the restricted OLS estimator relate to the unrestricted OLS estimator in terms of efficiency?
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What does the p-value represent in hypothesis testing?
What does the p-value represent in hypothesis testing?
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When conducting a one-sided test, how should the p-value be adjusted?
When conducting a one-sided test, how should the p-value be adjusted?
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Which equation is used to form the t statistic when testing if $eta_1$ is equal to another parameter?
Which equation is used to form the t statistic when testing if $eta_1$ is equal to another parameter?
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What is necessary to calculate the standard error for the difference between two estimates, $eta_1$ and $eta_2$?
What is necessary to calculate the standard error for the difference between two estimates, $eta_1$ and $eta_2$?
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What output is automatically provided by software like Stata and R for hypothesis testing?
What output is automatically provided by software like Stata and R for hypothesis testing?
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What does the term $se(\hat{\beta}_1 - \hat{\beta}_2)$ refer to?
What does the term $se(\hat{\beta}_1 - \hat{\beta}_2)$ refer to?
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Which of the following would not be considered during the calculation of $Var(\hat{\beta}_1 - \hat{\beta}_2)$?
Which of the following would not be considered during the calculation of $Var(\hat{\beta}_1 - \hat{\beta}_2)$?
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What is required to use the equation for $se(\hat{\beta}_1 - \hat{\beta}_2)$ effectively in testing linear combinations?
What is required to use the equation for $se(\hat{\beta}_1 - \hat{\beta}_2)$ effectively in testing linear combinations?
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Study Notes
Unit 5: Inference
- Inference in econometrics involves quantifying uncertainty in estimates.
- Estimates are noisy, not the true value of the parameters.
- The estimated parameter (β) depends on the random sample.
- Different random samples yield different estimates (β).
- Need to assess the precision of estimates.
Motivation
- Estimation is prone to noise.
- True parameters (β) are unknown.
- Estimates are noisy approximations.
- Estimates vary with different samples.
- How precise do estimates need to be?
Example
- True model: wage = β educ + u
- β = 0.1
- Noisy sampling process: educ ~ N(0, 1) but u ~ N(0, 10)
- Results in noisy estimate.
Example: Sample 1
- n = 100
- β^ = -0.367, SE = 0.047
- Data is noisy, but shows negative relationship.
Example: Sample 2
- n = 100
- β^ = 0.570, SE = 0.050
- Data is noisy, but shows positive relationship.
Example: Four Possible Samples
- Various samples and estimations of β are shown.
- Data looks noisy, indicates a negative relationship in sample 1, and shows a positive relationship in sample 2
- Results differ across samples.
Hypothesis Testing
- Method to quantify uncertainty:
- Establish a threshold for acceptable uncertainty.
- Example: Test if β = 0, given sample data.
- Inferring from data; crucial for inference.
The Classical Linear Model (CLM)
- Additional assumption (beyond Gauss-Markov) for hypothesis testing:
- Population error (u) is independent of explanatory variables (X).
- Normally distributed (u ~ Normal(0,σ²)).
- This assumption allows for calculations of t-statistics and F-statistics.
- The CLM assumption is y|X ~ Ν(β , σ²).
- Useful in large sample sizes.
Normal Sampling Distributions
- Conditional on sample values of the independent variables, β^ is normally distributed.
- β^ ~ N(β, Var(β))
- Var(β) = σ²(X'X)⁻¹.
- Normalized estimator is distributed N(0,1) (β/sd(β)).
Inference: The t-Test
- Under CLM assumptions, t-statistic follows a t-student distribution.
- Standardized estimate is essential in hypothesis tests.
- tβ = (β - β₀)/se(β) ~ t(N-K-1)
- Estimate of population parameter (variance) is crucial: o² estimated by ô² .
Testing Parameters: A Typical Example
- Test null hypotheses (Ho : β = 0) of the significance of a variable (x).
- Reject Ho if β is statistically significant, if x has an effect on y.
One-sided Alternatives
- Hypothesis tests need an alternative hypothesis and a significance level.
- One-sided: β > 0 or β < 0.
- Two-sided: β ≠ 0
- Critical value (c) depends on significance level (α).
One-sided vs. Two-sided
- t-distribution is symmetrical, testing H₁: β < 0 is straightforward.
- Critical value is just the negative value of c
- Two-sided test: critical values are at α/2.
- A rejection rule is used to accept or reject the null hypothesis.
Testing other Hypotheses
- Test if β; differs from other values (e.g., Ho: β₁ = a₁)
- Formula for t-statistic
Initial Example Continued: Large Sample Size
- Larger sample size (n = 10000) has less noisy data.
- Estimates are close to true value (β = 0.1).
Confidence Intervals
- Alternative to hypothesis testing—confidence intervals based on critical values.
- (1 - α)% confidence interval = β ± tα/2·se(β)
- Value c is the (1- α/2) percentile of the t distribution.
Computing p-values for t-tests
- Finding the smallest significance level at which to reject a null hypothesis.
- The p-value is the probability that we would observe a t-statistic, if the null hypothesis was true.
- P-value is calculated by using a relevant t-distribution, given degrees of freedom.
- Computer packages compute p-values; divide by 2 for one-sided tests.
OLS Estimates by Stata
- Output with coefficient estimates, standard errors, t-statistics, p-values, and confidence intervals for OLS method.
- Interpret regression results, particularly significant variables.
Testing Linear Combinations
- Test if a linear combination of coefficients equals a constant (e.g., ß₁ - ß₂ = 0).
- Formula for t-statistic uses variance and covariances of estimates as a measure of variability.
- In Stata, use the "test" command for hypotheses tests.
Multiple Linear Restrictions
- Jointly test multiple hypotheses (e.g., all coefficients).
- Evaluate a group of parameters using F-test, not just evaluating each individually.
- Need to estimate both restricted and unrestricted models.
The F-Statistic
- Measure of overall increase in residual sum of squares (SSR) when moving from unrestricted to restricted model
- q = number of restrictions or numerator degrees of freedom
- N-K-1= df_unrestricted
The R² Form of the F-statistic
- Alternative formula for F statistic using R².
- Computations in relation of SSR given R²
Overall Significance
- Test hypotheses about all model coefficients (Ho :β1=β2=…=βk = 0)
- F-statistic related to R² measure of fit
General Linear Restrictions
- Basic F-statistic formula for any set of linear restrictions.
- p-values are found in the appropriate percentile of relevant F-distribution
Example
- Illustrative applications and use-cases of hypothesis testing with regression models.
- Example voting model (with campaign expenditues).
- Specific restriction on coefficient estimates tested.
Interpretation of t-test and F-test
- Example code in R
- Example exercise
Chow Test for Structural Breaks
- Method to test for structural shifts in coefficients over time or in specific data segments.
- Determine if effect of regressors differs significantly between (e.g.) different time periods or groups in data.
- Requires estimating restricted and unrestricted models.
- Evaluating F-statistics over time to establish if structural breaks are present.
Exercise: Marginal Effects and F-test
- Exercise using real world data
- Calculate and interpret marginal effects, particularly for wage determination
- F-test for linear restrictions
- Example applications in wage regression.
Appendix A: Restricted OLS Estimation
- Summary of t-tests and F-tests within the framework of restricted regressions.
- Basic formulas and notation for restricted estimators are provided
Moments of the OLS Estimator
- Calculation and interpretation of mean of restricted OLS estimators
- Exploiting normality; estimation of variance of the restricted estimator.
Moreover
- Formula for variation of the restricted estimator.
- Properties of SSR. SSRr > SSRur
The F-Test
- Test hypotheses using framework of 'restricted regression', formulas for univariate test statistic and modified statistic are provided.
Special Case: The t-test
- Special F-test case that helps with testing hypotheses about just one coefficient.
- Useful for analyzing and assessing significance of individual variables by using a univariate t-test.
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Description
Test your knowledge on fundamental concepts in statistical testing and regression analysis. This quiz covers topics such as confidence intervals, null hypotheses, t-statistics, and the F statistic. Ideal for students of statistics and data analysis.
