what makes one estimator better than another estimator is jargon term for method of estimating

What makes one estimator better than another

Estimator is jargon term for method of estimating

Estimate

• The estimator produces an estimate.• The estimate is the number.• The estimator is the method.

What makes one estimator better than another

A better estimator is more likely to be close to the true line.

How close is our regression line to the true line?

• To answer, we must make assumptions.• Assumption 1 is right in the question

above. It’s that there is a true line that we’re trying to find.

• Assumptions are needed to assess an estimator.

• To see where we’re going with the assumptions…

True line demo review

• Yi = α + βXi + ei

• (spreadsheet)

Least squares demo review

Errors’ expected value is 0.

– Assumption 2• Why we draw our regression line

through the middle of the points’ pattern• Implies that the least squares estimator

is unbiased• Estimator = Method

Bias

• Unbiased means aimed at target.– Bias demo

• The expected value of the least squares slope is the true slope.

• Same for intercept.

All errors have the same variance

– Assumption 3• Why you give each point equal

consideration

Errors not correlated with each other

– Assumption 4• Correlated means a linear relationship

that lets you predict one error once you know another error.

• Serial correlation would be if one error helps you anticipate the direction of the next error.

Errors not correlated with each other

• Why you predict on the regression line rather than above or below it.

Normal distribution for errors

– Assumption 5• Normal distribution results from the

accumulation of small disturbances. Random walk with small steps.

• Normal distribution demos show how tight the normal distribution is.

Normal distribution for errors

• Least squares is best.– Unbiased– Least variance -- most efficient -- of any

estimator that is unbiased • Efficiency demos

• Can do hypothesis testing.

1A spreadsheet adds …

• Standard error of coefficient for the slope

• T-statistic– Coefficient ÷ its Standard error

• R-squared• Standard error of the regression

Standard error of coefficient

• Shows how near the estimated coefficient might be to the true coefficient.

t

• A unitless number with a known distribution, if the assumptions about the errors are true.

• Used here to test the hypothesis that the true slope parameter is 0.

R2

• Between 0 and 1. Demo• Least squares maximizes this.• Correlation coefficient r is square root.

1Sum of squares of residuals

Sum of squares of Y Y

Standard error of the regression

• “s”• Should be called standard residual

– But it isn’t

s

• Root-mean-square average size of the residuals

• s2 is an estimate of 2

s2 sum of squares of residuals

Number of observations 2

S2 and 2

S2

Sum of squares of residuals

Divided by

N-2

2

Expected value of sum of squares of the errors

Divided by

N