Week12-ECMT1020_Ch6-8-1
Classical assumptions for least squares (LS) include:
Regressor: xi is nonstochastic.
Model Specification: Correctly specified as yi = β1 + β2xi + ui.
Error Term: ui has mean zero, E[ui] = 0.
Variance Condition: N^{-1} Σ(xi − x̄)² ≠ 0.
Homoskedasticity: E[u²i] = σ²; error term is independent across observations.
Normal Distribution: ui follows a normal distribution.
Model misspecification relates to the failure to include relevant variables or the inclusion of irrelevant ones.
Scenario: True model: yi = β1 + β2xi + β3wi + ui, but regressor wi is omitted leading to:
Estimated model: yi = β1 + β2xi + ei, where ei = β3wi + ui.
OLS estimator of β2 (denoted by β2-hat) is biased and its expectation is:
*Swap e with w and expand it, we will have also an OLS estimate included the third coefficient
Recall that an estimator is unbiased if its expectation is equal to its population counterpart:
Conditions for unbiasedness:
β3 = 0 (wi has no effect).
cov(xi, wi) = 0 (no correlation).
—> These two condition forces the B2 estimator above to be unbiased.
we can view the estimator of variable w as the slope coefficient that have marginal effect of x and w on dependend variable.
Omitted Variable Bias:
Direction of the omitted variable bias depends on the signs of B3 and fitted-cov(xi, wi)
Positive bias if both β3 and cov(xi, wi) are positive.
Negative bias if β3 is negative while cov(xi, wi) is positive.
R² in mispecified models will be lower than in correct model.
Alternative estimations like instrumental variable (IV) estimation are suggested.
When irrelevant regressors are included in a model:
True model: yi = β1 + β2xi + ui, estimate with zi included may yield:
If ui has mean zero, OLS remains unbiased as long as xi and zi are non-stochastic.
Irrelevant regressor doesnn’t change mmuch of the original regressor. It caused the regression model to be a little inefficient, T-statistic decreases while vaariance increases, p-vaue increases.
Definition: Proxy variables are used to approximate omitted ones. having properties similar to those of missing variable.
Example: For school quality: tuition, student-teacher ratios, reputation.
True model with proxy:
Original: yi = β1 + β2xi + β3wi + ui;
Use proxy:
yi = β1 + β3α1 + β2xi + β3α2w∗i + ui.
Result: Estimates reflect β3α2 rather than β3, R² remains unchanged.
Revised classical assumptions:
xi is randomly selected from the same distribution.
Model correctly specified with condition: E[ui|xi] = 0.
Unbiasedness requires no perfect multicollinearity and E[ui|xi] = 0.
OLS estimator remains unbiased, BLUE (Best Linear Unbiased Estimator).
As sample increases, OLS estimators converge to true parameters without needing normality of error terms.
Scenario: Observed regressor x* as contaminated version of true x, due to measurement error ϵi.
Model: yi = β1 + β2(xi + ϵi) + ui gives:
OLS estimator cβ2 converges to false estimate.
Limit of cβ2 approaches:
β2 - β2 * (σ²ϵ / (σ² + σ²ϵ)) depending on sign of β2.
No bias in OLS when y* replaces yi; results in increased variance but remains unbiased and consistent.
Permanent vs actual income with errors leads to biased estimates:
OLS estimator cβ2 converges to adjusted estimation if measurement errors exist in both dependent and independent variables.
Measurement errors, when combined with omitted variables, result in inconsistent OLS estimates.
IV estimation recommended; utilizes external, correlated variables to ensure accurate parameter estimation.
Classical assumptions for least squares (LS) include:
Regressor: xi is nonstochastic.
Model Specification: Correctly specified as yi = β1 + β2xi + ui.
Error Term: ui has mean zero, E[ui] = 0.
Variance Condition: N^{-1} Σ(xi − x̄)² ≠ 0.
Homoskedasticity: E[u²i] = σ²; error term is independent across observations.
Normal Distribution: ui follows a normal distribution.
Model misspecification relates to the failure to include relevant variables or the inclusion of irrelevant ones.
Scenario: True model: yi = β1 + β2xi + β3wi + ui, but regressor wi is omitted leading to:
Estimated model: yi = β1 + β2xi + ei, where ei = β3wi + ui.
OLS estimator of β2 (denoted by β2-hat) is biased and its expectation is:
*Swap e with w and expand it, we will have also an OLS estimate included the third coefficient
Recall that an estimator is unbiased if its expectation is equal to its population counterpart:
Conditions for unbiasedness:
β3 = 0 (wi has no effect).
cov(xi, wi) = 0 (no correlation).
—> These two condition forces the B2 estimator above to be unbiased.
we can view the estimator of variable w as the slope coefficient that have marginal effect of x and w on dependend variable.
Omitted Variable Bias:
Direction of the omitted variable bias depends on the signs of B3 and fitted-cov(xi, wi)
Positive bias if both β3 and cov(xi, wi) are positive.
Negative bias if β3 is negative while cov(xi, wi) is positive.
R² in mispecified models will be lower than in correct model.
Alternative estimations like instrumental variable (IV) estimation are suggested.
When irrelevant regressors are included in a model:
True model: yi = β1 + β2xi + ui, estimate with zi included may yield:
If ui has mean zero, OLS remains unbiased as long as xi and zi are non-stochastic.
Irrelevant regressor doesnn’t change mmuch of the original regressor. It caused the regression model to be a little inefficient, T-statistic decreases while vaariance increases, p-vaue increases.
Definition: Proxy variables are used to approximate omitted ones. having properties similar to those of missing variable.
Example: For school quality: tuition, student-teacher ratios, reputation.
True model with proxy:
Original: yi = β1 + β2xi + β3wi + ui;
Use proxy:
yi = β1 + β3α1 + β2xi + β3α2w∗i + ui.
Result: Estimates reflect β3α2 rather than β3, R² remains unchanged.
Revised classical assumptions:
xi is randomly selected from the same distribution.
Model correctly specified with condition: E[ui|xi] = 0.
Unbiasedness requires no perfect multicollinearity and E[ui|xi] = 0.
OLS estimator remains unbiased, BLUE (Best Linear Unbiased Estimator).
As sample increases, OLS estimators converge to true parameters without needing normality of error terms.
Scenario: Observed regressor x* as contaminated version of true x, due to measurement error ϵi.
Model: yi = β1 + β2(xi + ϵi) + ui gives:
OLS estimator cβ2 converges to false estimate.
Limit of cβ2 approaches:
β2 - β2 * (σ²ϵ / (σ² + σ²ϵ)) depending on sign of β2.
No bias in OLS when y* replaces yi; results in increased variance but remains unbiased and consistent.
Permanent vs actual income with errors leads to biased estimates:
OLS estimator cβ2 converges to adjusted estimation if measurement errors exist in both dependent and independent variables.
Measurement errors, when combined with omitted variables, result in inconsistent OLS estimates.
IV estimation recommended; utilizes external, correlated variables to ensure accurate parameter estimation.
