Statistics Chapter on Data Types and Analysis

Data collected over a period of time. It shows how a variable changes over time.

How often data is collected.

Data collected at a single point in time on multiple subjects.

A statistical method used to analyze how a variable changes over time, often in relation to other variables.

Data that has both time series and cross-sectional dimensions.

Data that can take on any value within a range.

Data that can only take on specific values.

A measurement of the performance of a set of stocks.

A matrix showing the variances of individual portfolio returns on the main diagonal and covariances between returns as off-diagonal elements.

The element in the variance-covariance matrix representing the covariance between returns of stock i and stock j.

Calculating V involves using actual returns (R), adjusting for mean returns (ri), and then a formula: V = (R’R) / (T-1).

The variance of returns for a specific portfolio. Calculated using the weights (w) and variance-covariance matrix (V).

The formula for portfolio variance is VP = w'Vw, where ' denotes transpose. Weights (w) are multiplied by the variance-covariance matrix (V) and by their transposed value

A matrix representing correlations between different return series.

The matrix has 1's on the diagonal and correlations between returns on the off-diagonal; symmetrical around the main diagonal.

Portfolio variance can also be calculated using the correlation matrix (C) and standard deviations (S) of returns: VP = w'SCSw

The slope of the regression line, representing the change in the dependent variable for every unit change in the independent variable.

The point where the regression line intersects the y-axis, representing the value of the dependent variable when the independent variable is zero.

A measure of the spread of the data points around the regression line. It indicates how well the regression line fits the data.

A measure of the uncertainty in the estimated intercept. A higher SE indicates a less precise estimate of the intercept value.

A measure of the uncertainty in the estimated slope. A lower SE indicates a more precise estimate of the relationship between the dependent and independent variables.

The sum of the squared differences between the observed values and the predicted values. It measures how well the regression line fits the data.

The regression coefficients (α and β) are calculated using the formula: α = ȳ - βx̄ and β = (∑xtyt - T * x̄ * ȳ) / (∑xt² - T * x̄²), where T is the number of observations.

Regression results are typically presented in the form of an equation: yˆt = αˆ + βˆxt, where αˆ is the estimated intercept, βˆ is the estimated slope, and yˆt is the predicted value for the dependent variable.

A number representing the relationship between two variables in a regression equation. It shows how much the dependent variable changes for every unit change in the independent variable.

A measure of the reliability of a regression coefficient. It indicates how much the estimated coefficient is likely to vary from the true population parameter.

A statistical method to determine if there is enough evidence to reject a claim about a population parameter.

A statement about the population parameter that we want to disprove. It's typically the status quo.

A statement that contradicts the null hypothesis and suggests a different value for the population parameter. It's what we want to prove.

A hypothesis test where we're interested in whether the population parameter is different from a specific value, allowing for deviations in both directions.

A hypothesis test where we're only interested in whether the population parameter is greater than or less than a specific value, not both.

A range of values that is likely to contain the true population parameter with a specified level of confidence.

A method to determine if the estimated coefficient is significantly different from a hypothesized value.

Builds a range of values around the estimated coefficient, within which the true coefficient is likely to lie.

Concluding that the estimated coefficient is significantly different from the hypothesized value.

Range of values where the test statistic lies, indicating the estimated coefficient is not significantly different from the hypothesized value.

Calculate the upper and lower bounds of the confidence interval using the estimated coefficient, critical value, and standard error of the coefficient.

Determined based on the confidence level and degrees of freedom, representing the cut-off point for rejecting or accepting the null hypothesis.

Measures the uncertainty or variability in the estimated coefficient.

Assumes the true coefficient can be either greater than or less than the hypothesized value.

The area under the distribution where the test statistic would lead to rejection of the null hypothesis. It defines the range of values that are considered statistically significant.

The probability of rejecting the null hypothesis when it is actually true. It sets the threshold for statistical significance (e.g., 0.05 means a 5% chance of incorrect rejection).

A probability distribution used for hypothesis testing when the sample size is small or the population standard deviation is unknown. It is bell-shaped and symmetrical.

A parameter in the t-distribution that reflects the number of independent observations available to estimate the population variance. Commonly calculated as the number of observations minus 2.

The value that separates the rejection region from the non-rejection region. It is determined based on the significance level and the degrees of freedom.

A calculated value that summarizes the evidence from the sample to support or refute the null hypothesis.