Economics 105: Statistics• Any questions?• GH #19 due Friday.• Introduce Modeling Exercise group project

Modeling Exercise examples• What is the effect of your roommate’s SAT

scores on your grades? The effect of studying?

• Do police reduce crime?

• Does more education increase wages?

• What is the effect of school start time on academic achievement?

• Does movie violence increase violent crime?

Endogenous Explanatory Variable• Causes of endogenous explanatory variables

include …• Wrong functional form• Omitted variable bias … occurs if both the

1. Omitted variable theoretically determines Y2. Omitted variable is correlated with an included X

• Errors-in-variables (aka, measurement error)• Sample selection bias• Simultaneity bias (Y also determines X)

Stochastic Linear Models•Assumptions• (1)

– Simple regression vs. Multiple regression – Linear function, plus error– Variation in Y is caused by , the error (as well as X)

• (2) – Sources of error

• Idiosyncratic, “white noise”• Measurement error on Y• Omitted relevant explanatory variables … why?

Stochastic Linear Models•Assumptions• (3)

– Homoskedasticity•(4)

– No autocorrelation• (5)

– Errors and the explanatory variable are uncorrelated

• (6)– Errors are normally distributed

Y

X

E[Y] = 0+ 1X

Stochastic Linear Models• Assumptions so far imply• • • Need to estimate population intercept & slope• Take a sample of data & obtain the sample regression line

•

The sample regression line equation provides an estimate of the population regression line

Sample Regression Equation (Prediction Line)

Estimate of the regression

intercept

Estimate of the regression slope

Estimated (or predicted) Y value for observation i

Value of X for observation i

The individual random error terms ei have a mean of zero

Other notation:

chosen in samplenot chosen in sample

estimated error for X3

(residual)

Y

X

Observed Value of Y for X3

Predicted Value of Y for X3

X3

ε3

Sample Regression Equation

e3

Sample Regression Equation• Residual, ei, is the prediction error

• Positive errors• Negative errors

Y

X

Derivation of OLS Estimators•

•Select to minimize SSE• Set first partial derivatives = 0

• Results are

OLS Estimators

OLS Example: Scatterplot

OLS calculations “by hand”

File is in P:\Economics\Eco 105 (Statistics)\lec_simple reg.xls

Sample regression line

OLS Residuals (Excel output)

Interpretation of OLS Parameters

• For one-unit change in X, the average value of Y changes by 1 units

• intercept

• The effect of X on Y (from regressing “Y on X”)

Properties of OLS Estimator

• Gauss-Markov Theorem• Under assumptions (1) - (5) [don’t need normality

of errors], is B.L.U.E. of

• Unbiased estimator• Efficiency of an estimator

• Intuition for when var is smaller• We won’t know , so we’ll need to estimate it