SIMPLE CLASSICAL LINEAR REGRESSION MODEL: SPECIFICATION
I. Statistical model
a. Data generation process
b. Reasonable approximation
II. Simple classical linear regression model (SCLRM)
a. How useful
III. SCLRM specification
a. Components of model
1. Equation
2. Variables
3. Functional form
4. Error term
b. Variables
1. Random and nonrandom variables
a. Discrete, continuous, observable, unobservable
2. Describing the behavior of random variables
a. Marginal probability distribution
1. Discrete probability distribution
a. Form, mean, variance/standard deviation
2. Continuous probability distribution
a. Form, mean, variance/standard deviation
b. Conditional probability distribution
c. Conditional mean function
c. Functional form
1. Assumption
a. Functional form is linear
d. Error term
1. Random variable
2. Probability distribution
3. Assumptions
a. Error term has mean zero
b. Error term is not correlated with the explanatory variable
1. Endogenous and exogenous variables
c. Error term has constant variance. No heteroscedasticity
1. Homoscedasticity
2. No heteroscedasticity
d. Errors are independent
1. No autocorrelation
e. Error term has a normal distribution
SIMPLE CLASSICAL LINEAR REGRESSION MODEL: ESTIMATION
I. Estimation of statistical model
a. What are the parameters
II. Types of estimates
a. Point estimate
b. Interval estimate
III. Estimator
a. Choosing an estimator
b. Notation
c. Accurate and reliable estimator
d. Behavior of an estimator
1. Sampling variability
2. Sampling error
e. Sampling distribution
1. Experimental and theoretical sampling distribution
2. Form, mean, variance/standard deviation
f. Definition of an accurate and reliable estimator
1. Small sample properties
a. Unbiasedness
b. Efficiency
2. Large sample property
a. Consistency
IV. Ordinary least squares (OLS) estimator for regression coefficients
a. Rule
b. Residual sum of squares function
c. Deriving the OLS estimator
d. Properties of the OLS estimator
1. Sampling distribution
a. Mean
b. Variance
c. Variance-covariance matrix
2. Small sample properties
a. Unbiasedness
b. Efficiency
c. Gauss-Markov theorem
3. Large sample property
a. Consistency
4. Conclusions
a. Best may not be very good
b. Sources of bias in OLS estimator
1. Confounding variables
2. Reverse causation
V. Estimator for error variance
a. Standard error of the regression
VI. Estimator for variance, standard error, and covariance of OLS estimator
VII. Interval Estimate
a. Formula
b. Margin of error of point estimate
c. Rule of thumb for constructing 95% interval estimate
SIMPLE CLASSICAL LINEAR REGRESSION MODEL: HYPOTHESIS TESTING
I. Logic of hypothesis testing
a. An intuitive example
b. Null hypothesis
c. Type I and II errors
d. Level of significance
e. Power of a test
II. Procedures for hypothesis testing
a. Level of significance approach
b. Confidence interval approach
c. P-value approach
III. Level of significance approach
a. Five steps
1. Specify the null and alternative hypothesis
2. Derive an appropriate test statistic and the sampling distribution under the null
3. Choose a level of significance and find the critical value(s) of the test statistic
4. Use the sample data to calculate the actual/calculated value of the test statistic
5. Compare the actual/calculated value of the test statistic to the critical value(s)
b. T-test
1. Testing the null hypothesis of “no effect”
2. Testing the hypothesis of a specific quantitative effect
IV. P-value as a measure of strength of evidence
a. Guidelines
1. Strong evidence
p ≤ 0.05
2. Moderate evidence
0.10 ≥ p > 0.05
3. Weak evidence
0.20 ≥ p > 0.10
4. Little or no evidence
p > 0.20
V. Two approaches to answering the question: Does X have an effect on Y?
a. Level of significance approach
b. P-value strength of evidence approach
GOODNESS OF FIT AND PREDICTION
I. How do we make a prediction
II. Measures of goodness of fit
a. Standard error of the regression
b. R2 Statistic
CONCLUSIONS
I. Draw conclusions
a. Answer questions
1. Does X have an effect on Y
a. Level of significance test evidence
b. P-value evidence
2. What is direction of effect
a. Algebraic sign
3. What is size of effect
a. Magnitude of marginal effect
II. Assess validity of conclusions
a. Internal validity
1. Is the estimate of the marginal effect unbiased
2. Is the estimate of the standard error of the marginal effect unbiased
b. External validity