Statistics Unit 2: Single Regression Model

<p>Independently and Identically Distributed. (B)</p> Signup and view all the answers

<p>The control group must only consist of similar individuals from the population. (C)</p> Signup and view all the answers

<p>Preventing reverse causality can be complicated. (A)</p> Signup and view all the answers

<p>The entities are chosen from the same population and are independently distributed. (A)</p> Signup and view all the answers

<p>The potential outcome for a different treatment scenario. (C)</p> Signup and view all the answers

<p>Ordinary Least Squares (OLS) (C)</p> Signup and view all the answers

<p>The differences between observed and predicted values squared (C)</p> Signup and view all the answers

<p>It causes the slope estimate variance to increase. (B)</p> Signup and view all the answers

<p>The variance of the slope estimate decreases. (A)</p> Signup and view all the answers

<p>The residuals observed from the OLS regression. (C)</p> Signup and view all the answers

<p>$\sigma^2 = \frac{1}{N-K-1} \sum_{i=1}^{N} u_i^2$ (C)</p> Signup and view all the answers

<p>The degrees of freedom adjustment. (A)</p> Signup and view all the answers

<p>The expected value of the dependent variable when the independent variable is zero (B)</p> Signup and view all the answers

<p>The error term must include systematic variation (B)</p> Signup and view all the answers

<p>The disturbance capturing unobserved factors (D)</p> Signup and view all the answers

<p>To ensure that the OLS estimator is unbiased (D)</p> Signup and view all the answers

<p>The relationship defined by β0 and β1 · x (D)</p> Signup and view all the answers

<p>It minimizes the sum of squared residuals. (C)</p> Signup and view all the answers

<p>The estimated relationship is confined to a single value for x. (C)</p> Signup and view all the answers

<p>To capture the influence of factors not included in the model (C)</p> Signup and view all the answers

<p>Detecting a consistent error in predicting y based on x (D)</p> Signup and view all the answers

<p>Health expenditure is the independent variable; life expectancy is dependent. (B)</p> Signup and view all the answers

<p>The fraction of total sum of squares explained by the model (D)</p> Signup and view all the answers

<p>R-squared is a measure of goodness of fit that ranges from zero to one. (D)</p> Signup and view all the answers

<p>It usually increases with the addition of independent variables. (D)</p> Signup and view all the answers

<p>R-squared does not provide information about the number of variables in a model. (D)</p> Signup and view all the answers

<p>The ratio of the total variance explained to the total variance of the actual values. (C)</p> Signup and view all the answers

<p>The OLS estimators will be biased. (D)</p> Signup and view all the answers

<p>Their expected values equal the population parameters. (B)</p> Signup and view all the answers

<p>The mean of the error term is zero given any value of x. (D)</p> Signup and view all the answers

<p>To ensure that the variances used are finite. (B)</p> Signup and view all the answers

<p>The value of x is centered around its mean. (A)</p> Signup and view all the answers

<p>β̂1 = β1 + 1/Sx2 Σ (xi − x̄)ui. (C)</p> Signup and view all the answers

<p>It implies that the error term does not affect the relationship. (B)</p> Signup and view all the answers

<p>They represent the true population parameters. (C)</p> Signup and view all the answers

<p>The model cannot be estimated. (A)</p> Signup and view all the answers

The effect of an explanatory variable (x) on the distribution of the dependent variable (y), holding other relevant factors constant. It's the expected difference in y from a controlled experiment where x is manipulated randomly.

A measure of how much the expected value of y changes when x changes slightly (holding all else constant).

Assumptions needed to ensure regression estimates reflect the causal effect of x on y and not other factors.

A relationship between x and y where a change in x causes a proportional change in y.

Data points (x, y) collected using random selection, ensuring that observations are independent and identically distributed.

Independent and identically distributed. A crucial assumption for valid inferences in regression analyses.

A group in an experiment that does not receive the treatment (being studied).

When the relationship between variables is incorrectly estimated because y may influence x (rather than the other way around).

A linear model showing the relationship between two variables (x and y). y = β0 + β1x + u, where β0 is the intercept, β1 is the slope, and u is the error term.

A line that represents the average value of a dependent variable for given values of an independent variable in a population.

A term in a linear regression model that accounts for factors besides the independent variable that affect the dependent variable.

Simple Linear Regression

Average lifespan for a population. Often used in public health studies.

The variable used to predict or explain the dependent variable (y).

Spending on healthcare services per capita in a population.

The variable being predicted or explained in the linear regression model.

A method used to estimate the parameters (β0, β1) in a linear regression model. It minimizes the sum of squared errors.

A method to estimate regression parameters by minimizing the sum of squared residuals between observed and modeled values.

Conditions, like linearity, independence, and zero mean error, necessary to make valid causal inferences from simple linear regression.

Differences between observed values and predicted values in a regression model.

Data collected from many different subjects at a single point in time.

The effect of a change in the independent variable on the dependent variable, assuming other factors don't change.

Differences in results obtained from different samples.

The variable that we want to explain or predict based on other variables.

independent and identically distributed data.

The relationship between variables is linear.

A measure of the variability of the unobserved errors in a regression model, reflecting the extent to which actual values deviate from the predicted values.

The difference between the observed dependent variable value (yᵢ) and the predicted value in a regression model.

An estimator of the error variance that, on average, equals the true error variance.

The number of independent pieces of information available to estimate a parameter.

An estimate of the error variance in a regression model, calculated from the residuals.

R-squared is a statistical measure representing the proportion of variance in the dependent variable that's predictable from the independent variables in a regression model. It's the fraction of the total variation in the data explained by the regression.

R² = (SSR/SST), where SSR is the sum of squares due to regression and SST is the total sum of squares.

R² values range from 0 to 1.

A higher R² indicates a better fit of the regression model to the data, which means the model explains more of the variability in the dependent variable.

Adding more independent variables to a model will typically increase R². This doesn't necessarily mean the model is better, as it could be overfitting the data, or that the additional variables do not contribute substantively.

Sample variation in the independent variable (x) is not zero.

The expected value of the error term (u), given the independent variable (x), is zero.

OLS estimators (β̂0 and β̂1) have expected values equal to their true population parameters (β0 and β1).

OLS estimators are biased if any assumptions (SLR.1 to SLR.4) are violated.

The OLS estimator for the slope (β̂1) is calculated by dividing the sum of products of (xi − x̄) and yi by the sum of squares of (xi − x̄).

The sum of deviations from the sample mean (x̄) is zero: Σ(xi − x̄) = 0.

Technical assumption requiring the fourth moments of x and y to be finite.

The proof of unbiasedness involves rewriting β̂1, and showing, through mathematical steps, that the expected value of β̂1 equals β1, conditional on x.

OLS estimates are unbiased under assumptions SLR.1 to SLR.4. The proof relies on the conditions (assumptions).

If there's a correlation between the error term (u) and the independent variable (x), the zero conditional mean assumption is violated. This can happen if omitted variable bias is present.