Expected Value of a Discrete Random Variable (A.K.A. Mean)

Formula Sheet

Chapter 2

Expected Value of a Discrete Random Variable (a.k.a. mean):

The expected value of a function of a discrete random variable X:

The variance of a discrete random variable X:

but also,

Standard Deviation

Rules of Expectation: where a and c are constants and X is a random variable

Rules of Variance:

Conditional Probability:

where f(X,Y) is the joint probability

Covariance:

but also

where

Correlation:

The sum of two random variables: Let X and Y be two random variables and a and b are constants:

E(aX + bY) = aE(X) + bE(Y)

Var(aX + bY) = a2Var(X) + b2Var(Y) + 2abCov(X,Y)

Sample Statistics: assume a sample of T observation on Xt

Sample Mean

Sample Variance

Sample Standard Deviation

Sample Covariance

Sample Correlation

Chapter 3 and 4 Formulas The Method of Least Squares

For the linear model : , the Least Squares estimator is comprised of the following 2 formulas (where T is the size of the sample):

Note that b2 could also be calculated using one of the other 3 formulas:

where

These estimators have the following means and variances:

and

and

where s2 is the variance of the error term and is assumed constant.

The estimated line is: and a residual is

The estimator of s2 is so that

and where a “hat” means that has been used in place of s2 in the variance formulas.

Chapter 5 and 6 Equations

From Section 5.3: The Least squares predictor uses the estimated model to make a prediction at xo.

The estimated variance of the prediction error is:

R-squared is:

where

So that

Chapter 7 and 8

For the multiple regression model:

Adjusted R-squared:

F-statistic:

Let SSER be the sum of squared residuals from the Restricted Model

Let SSEU be the sum of squared residuals from the Unrestricted Model.

Let J be the number of “restrictions” that are placed on the Unrestricted

model in constructing the Restricted model. Let T be the number of observations in the data set. Let k be the number of RHS variables plus one for intercept in the

Unrestricted model.

Chapter 11

Goldfeld Quandt statistic:

This statistic has an F distribution with t1-k degrees of freedom in the numerator and t2-k degrees of freedom in the denominator

The variance for b2 when the error term is heteroskedastic is:

White standard errors use this Var(b2) formula, using as an estimate for .

Chapter 12

Durbin-Watson test statistic is calculated as:

and