getting this shit locked

Sample Average (mean):

found by adding up all of the observations in a sample and dividing by the number of observations

sample variance

The SV is an estimator of population variance, using (unbiased estimator) a sample of observations from that population. Formula: s² = 1/(n−1) Σᵢ₌₁ⁿ (yᵢ − ȳ)²

bias of an estimator

Bias is the difference between the expected value of an estimator and the true population parameter it is trying to estimate. If the bias is zero the estimator is unbiased, meaning that on average, over many samples, it gets the true value.

standard error

The standard error is the estimated standard deviation of an estimator. It tells us how much the estimate would vary from sample to sample, so a smaller standard error means a more precise estimate.

linear regression model

A linear regression model describes the relationship between a dependent variable y and an independent variable x as a linear function of the parameters plus an error term u. The error term captures everything other than x that affects y.

regressor

A regressor is a variable on the right hand of a regression that explains the left hand side variable. Ex: in the model y = β₀ + β₁xᵢ + u, xᵢ is the regressor.

intercept coefficient (Bo)

The intercept coefficient is the predicted value of y when x equals zero. It is where the fitted line crosses the y-axis.

slope coefficient (B1)

The slope coefficient is the change in the predicted value of y for a one unit change in x, on average. It tells us the direction and size of the relationship between x and y.

least squared estimator

The least squares estimator is the method that chooses the intercept and slope that minimize the sum of the squared residuals. It gives the line that fits the data best.

fitted value

A fitted value is the value of y that the estimated regression predicts for a given x. It is the point on the fitted line for that observation.

fitted line

The fitted line is the estimated regression line, ŷ = β̂₀ + β̂₁x. It is the line the least squares estimator draws through the data, and it passes through the sample means of x and y.

residuals

A residual is the difference between the actual value of y and the fitted value for an observation. It is the part of y that the regression does not explain. Residuals are not the same as the error term, which cannot be observed.

sums of squares (SST)

the total sum of squares, the total variation in y around its mean

sums of squares (SSE)

the explained sum of squares, the variation explained by the regression

sums of squares (SSR)

the residual sum of squares, the variation left unexplained.

R-Squared

the fraction of the sample variation in y that is explained by the regression. It is between 0 and 1, and a higher value means a better fit. It is found by dividing the explained sum of squares by the total sum of squares.

level - level

model y = β₀ + β₁x + u. You interpret the model as follows, given a 1 unit change in x there is an associated β₁ unit change in y on average.

level - log

model y = β₀ + β₁ log(x) + u. You interpret the model as follows, given a 1% change in x there is an associated β₁/100 change in y on average.

log-level

model log(y) = β₀ + β₁x + u. You interpret the model as follows, given a 1 unit change in x there is an associated (100·β₁)% change in y on average.

log-log

model log(y) = β₀ + β₁ log(x) + u. You interpret the model as follows, given a 1% change in x there is an associated β₁% change in y on average. β₁ is called an elasticity.

hypothesis testing

a way of using sample data to decide whether a claim about a population parameter is believable. We state a null hypothesis (H₀) and an alternative hypothesis (H₁), calculate a test statistic, and then either reject or fail to reject the null.

significance level

(α) is the probability of rejecting the null hypothesis when it is actually true. It is chosen before running the test, and common values are 10%, 5%, and 1%.

t-test

the test used to check a hypothesis about a single regression coefficient. The t-statistic is the estimate minus the hypothesized value, divided by the standard error. It tells us how many standard errors the estimate is away from the value in the null hypothesis.

critical value

the cutoff from the t distribution that separates the rejection region from the non-rejection region. It depends on the significance level, the degrees of freedom, and whether the test is one-tailed or two-tailed. If the t-statistic is beyond the critical value, we reject the null.

two tailed test

test has an alternative hypothesis of "not equal to" and splits the rejection region between both tails.

one tailed test

has an alternative of "greater than" or "less than" and puts the whole rejection region in one tail. A one-tailed test is only used when theory tells us the direction ahead of time.

p-value

the probability of getting a test statistic at least as extreme as the one we observed, assuming the null hypothesis is true. A small p-value is evidence against the null. We reject the null if the p-value is less than the significance level.

confidence interval

a range of values that is likely to contain the true population parameter. It is built from the estimate plus and minus the critical value times the standard error. If a 95% confidence interval does not contain zero, we reject the null that the coefficient is zero at the 5% level.

gauss Markov theorem

says that if SLR.1 through SLR.5 hold, then the least squares estimators are BLUE, the Best Linear Unbiased Estimators. Best means they have the smallest variance of all linear unbiased estimators.

SLR.1

linear in parameters, the model is wrong. nothing may work

SLR.2

random sampling, SE(B1) is wrong

SLR.3

sample variation in x, B1 does not exist

SLR.4

Zero condition mean, E(u|x) = 0, Bias, least squares is biased and invalid

SLR. 5

se(β̂₁) is wrong. Least squares is still unbiased, but the t-test is invalid.