Econometrics: Solved Questions and Mathematical Proofs

Econometrics: when the slope is zero, the intercept equals the sample mean, derived via ordinary least squares in a two-variable regression.

Compute the intercept and slope estimators for a two-variable regression, deriving b2 from sum xy, sum x^2, and means, then obtain b1 as ybar minus b2 times xbar.

Explore interpreting the slope in the regression of sales on price, construct a 95% confidence interval for the slope with price in thousands of rupees, and interpret R-squared.

Compute the OLS estimators for regressing monthly food expenditure on income, yield the line y_hat = 1.145 + 0.2641 x, and report the standard error 0.9958 and R-squared 0.9185.

Rewrite the simple linear regression with the same slope, a new intercept, and a zero-mean error by subtracting the error's expected value.

Analyze how education relates to fertility, explain the unobserved error factors, and show how correlation between education and these factors biases OLS estimates and obscures ceteris paribus effects.

Learn to estimate a simple linear regression of GPA on ACT for eight students, compute beta0 and beta1, and assess fitted values, residuals, and r-squared.

Compute r squared from the explained and total sums of squares in a simple regression with n=20, using y and y bar, and verify r squared equals 0.8455.

Derive the intercept estimator via ordinary least squares by minimizing the residual sum of squares in a simple linear regression, using first and second order conditions for B1 and B2.

Explore an optional method to solve for the intercept and slope estimators in ordinary least squares by differentiating the sum of squared residuals and deriving Beaven and B2 formulas.

Explore multiple formulas for calculating the slope estimator in simple regression, including covariance over variance, using x and y data, means, and sample adjustments.

Three key OLS results: residuals have zero mean with an intercept, fitted values equal the mean of actual values, and the regression line passes through the sample means (x-bar, y-bar).

Completing the five useful ols results, the lecture shows that the first-order condition yields sum e_i x_i equals zero and the sample covariance between the residuals and x is zero.

Explore the classical linear regression model, detailing the assumptions of linearity in parameters, non stochastic x, and zero mean of disturbances.

the lecture continues the clrm assumptions, detailing homoscedastic disturbances, no serial correlation, normal distribution of ui, correct model specification, and the necessity of positive x-variance in samples.

Explore the Gauss-Markov weight k_i, showing that sum k_i = 0, sum k_i x_i = 1, and sum k_i^2 = 1/(sum x_i^2), highlighting a linear estimator foundation.

Learn how the Gauss-Markov slope estimator is unbiased by showing its expected value equals the true parameter, decomposing it into fixed and random components under linear regression assumptions.

Derive the variance of the OLS slope estimator and prove the Gauss-Markov theorem: it has minimum variance among linear unbiased estimators under the assumptions, preparing a comparison with other methods.

Examine the relationship between R-squared and adjusted R-squared; adjusted is generally lower, but can equal R-squared when R-squared equals 1, per the stated formula.

Recognize that a high r-squared value alone does not guarantee a good model; verify coefficient signs, theoretical relevance of variables, and statistical significance when comparing models.

Complete the anova table for a three-variable regression, and perform the test of overall significance using the f statistic. Learn to compute r-squared and adjusted r-squared.

Interpret beta2 and beta3 in the money demand model, income in billions of rupees, holding the interest rate constant. With R^2 0.519 and F 15.17 (2,28), the model is significant.

Compare restricted and unrestricted models using an f-test for beta3 = 0. If f calculated exceeds f critical (72.23 > 4.00), reject the null, so model B is preferred.

Perform a joint F-test of potato consumption against cauliflower and cabbage prices in a lin-log model, and explain the interpretation of a 200 coefficient on log X1.

analyze a log-log regression of output on log labor and log capital, yielding elasticities and returns to scale; test if labor elasticity exceeds 0.2 with a 10% one-sided t-test.

Solve slope and elasticity for two forms: log inverse and Lynn log model; elasticity are -beta2/x and beta2/y respectively.

Explain testing the regression slope for sales on advertisement expenditure with a two-tailed t-test at 1%, then confirm with an F-test via F = t^2 for a single restriction.

Proves that residuals are uncorrelated with the fitted values in least squares regression. Shows zero covariance between e_i and y_hat_i by establishing E[e_i]=0 and E[e_i x2_i]=0 and E[e_i x3_i]=0.

Compute the t-test statistic for the female coefficient in regression two, testing beta3 = -3, in an econometrics model with gender, education (dummy), and age (squared), yielding |t| = 0.63.

Compute a joint f-test for beta four and beta five using unrestricted (model 3) and restricted (model 2) regressions; with n = 251 and two restrictions, the f-statistic is 35.47.

Explore how adding the elementary education dummy creates the dummy variable trap, causing perfect multicollinearity unless you drop the intercept or a dummy.

Add an interaction between college and female to the earnings model to let college differ by gender. Use dummy variables for college and gender to illustrate the change.

Learn to read operator and machine dummy variables in a production regression, estimate hourly output, and identify the least productive operator-machine combination.

Examines perfect collinearity among regressors in a four-parameter model, showing how x4 equals x2 plus x3 creates exact linear dependence and prevents unique ordinary least squares estimates.

Learn how White's test for heteroscedasticity uses an auxiliary regression of squared residuals on the original regressors, their squares, and cross terms, with chi-square inference.

Learn how to transform a regression to address heteroscedasticity by dividing by the appropriate root of the error variance, yielding a constant-variance transformed model across cases.

The lecture applies the Durbin-Watson test to a money demand regression with n=100 and k'=2 at 5% significance, finding D=1.755 and no evidence of first-order autocorrelation.

Apply the Durbin-Watson d-test for first-order autocorrelation with n = 45 and five independent variables excluding intercept; D = 2.317 falls in the indecision zone, so no conclusion on autocorrelation.

Correlation zero implies the regression of y on x has slope zero and intercept y-bar. Regression of x on y has slope zero and intercept x-bar; lines are perpendicular.

explain an econometrics regression problem by deriving coefficient relationships and variances, using var(aX)=a^2 var(X) to identify the correct part c result.

Solving a simple linear regression problem, the video shows that a zero slope estimate implies no influence of X on Y and R-squared equals zero.

Analyze the least squares regression of y on a single x, explaining how r-squared equals the squared correlation with an intercept, and differs when no intercept is included.

Assess adding a constant to x or y in regression; show b2* = b2, b1* = b1 - b2 λ, and that fitted values stay equal.

Analyze whether the covariance between residuals and the regressor is zero in simple regression estimated by OLS. It is zero with an intercept, but not without.

Explore how including an irrelevant variable affects unbiasedness and efficiency in econometric estimation. See that estimates stay unbiased but can be inefficient unless r-squared x1 with added variable is zero.

Compare the variances of beta_hat and beta_delta under X2 irrelevant or relevant, noting omitted variable bias, and conclude that the best choice is option C.