OLS Implementation for Econometrics 25117

The predicted value of Y (average test score) when X (share of subsidized meals) is equal to 0.

The predicted change in Y (average test score) for a one unit change in X (share of subsidized meals).

The statistical test used to assess whether there is a statistically significant relationship between the independent variable (X) and the dependent variable (Y).

The percentage of the total variation in the dependent variable (Y) that is explained by the independent variable (X).

A statistical relationship between two variables where a change in one variable is associated with a change in the other variable. The relationship can be positive, negative, or linear.

A measure of the strength and direction of the linear relationship between two variables. Ranges from -1 (perfect negative correlation) to +1 (perfect positive correlation).

A statistical technique used to estimate the relationship between a dependent variable (Y) and one or more independent variables (X). It involves creating a linear equation that best fits the data points.

A hypothesis that there is no relationship between the independent variable (X) and the dependent variable (Y).

The standard error (SE) of an estimator measures how much the estimator varies from sample to sample.

A t-statistic is a measure of how many standard errors the sample estimate is away from the hypothesized value.

The sampling distribution of an estimator is the distribution of the estimator's values when repeated samples are taken from the population.

The variance of the sampling distribution is a measure of the spread of the estimator's values.

The slope of the population regression line represents the average change in the dependent variable for a one-unit change in the independent variable.

The OLS estimator (β̂1) is a statistical estimate of the true slope of the regression line (β1).

The Central Limit Theorem states that the sampling distribution of the mean approaches a normal distribution as the sample size increases.

A statistical test used to determine if a coefficient in a regression model is significantly different from zero.

The p-value represents the probability of observing a result as extreme as the one obtained, assuming the null hypothesis is true.

The standard error of the regression (SER) measures the typical deviation of the observed values from the regression line.

A confidence interval (CI) is a range of values that is likely to contain the true population parameter. In regression, it provides a range of plausible values for the coefficient.

A regression slope is the change in the dependent variable (Y) for every one-unit change in the independent variable (X).

If a confidence interval for a coefficient does not include zero, it implies that the coefficient is significantly different from zero.

The process of rejecting the null hypothesis if the p-value is less than a predetermined significance level.

A statistical model that describes the relationship between a dependent variable (Yi) and one or more independent variables (Xi) using a linear equation. It assumes the relationship can be represented by a straight line.

An assumption in linear regression stating that the average value of the error term (ui) is zero for any given value of the independent variable (Xi). This implies that the independent variable doesn't systematically affect the error term.

An assumption that data points are independent of each other and drawn from the same population distribution. This ensures that observations are not influenced by previous ones.

An assumption in linear regression stating that the variance of the error term (ui) is constant across all values of the independent variable (Xi). It means the spread of the error term is consistent.

A theorem in statistics that proves the OLS estimators are the Best Linear Unbiased Estimators (BLUE) under certain assumptions. BLUE estimators are unbiased and have the smallest variance among all linear unbiased estimators.

The method used to estimate the parameters (β0 and β1) in a linear regression model by minimizing the sum of squared residuals. Residuals are the differences between the actual observed values and the predicted values from the regression line.

A linear function of the dependent variables (Yi) used to estimate the regression coefficients.

The weights used in linear regression models to calculate the OLS estimators. They are chosen to minimize the variance of the estimated coefficients.

The Least Absolute Deviations (LAD) estimator is an alternative to OLS that minimizes the sum of absolute differences between the actual values and the predicted values. It is less sensitive to outliers than OLS.

The term "treatment" refers to the policy intervention or change being studied in a research setting. In this context, the treatment is the 10% increase in the share of subsidized school meals.

Omitted variable bias occurs when a relevant variable is not included in the regression model. This can lead to misleading estimates of the causal effect of the included variables.

The correlation between the error term (ui) and the independent variable (Xi) indicates omitted variable bias. If corr(ui, Xi) > 0, it means that the error term is systematically related to the independent variable, indicating a missed factor.

In this case, the correlation between the error term (ui) and the share of subsidized meals (Xi) is likely positive, indicating that districts with a low share of subsidized meals have other resources that provide more learning opportunities for children. This constitutes a potential omitted variable bias.

Assumptions about the variability of errors in a regression model. Homoskedasticity means constant error variance across all values of the independent variable. Heteroskedasticity means varying error variance.

A technique used to estimate standard errors in regression analysis when heteroskedasticity is present. It helps to obtain valid inferences even when error variances are not constant.

The assumption that the variance of the error term (ui) is constant across all values of the independent variable (X).

The violation of the homoskedasticity assumption, meaning the variance of the error term (ui) changes with the values of the independent variable (X).

The variance of the regression coefficient (β̂1) is calculated by taking the sum of squared deviations of observed values from their mean, multiplied by the variance of the error term (ui) at each value of X, all divided by the sum of squared deviations of X from its mean.

The robust standard error of the regression coefficient (β̂1) is calculated as the square root of the robust variance of β̂1, taking into account the varying error variances.

A specific type of robust standard error that adjusts for heteroskedasticity, also known as Eicker–Huber–White standard error.

If heteroskedasticity is present and you use robust standard errors, your inferences are still valid. If heteroskedasticity is present but you use the homoskedasticity-only formula, your standard errors will be wrong.

The estimator for the variance of β̂1 obtained under the assumption of homoskedasticity may be inaccurate if the true error variances are not constant.

When the sample size (n) increases, the variance of the regression coefficient (β̂1) converges to a specific value that depends on the error variances and the deviations of X from its mean.