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 reviewdemo Y i = α + βX i + e i (spreadsheet)

Least squares demo reviewdemo

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 demodemo 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.demos

Normal distribution for errors Least squares is best. –Unbiased –Least variance -- most efficient -- of any estimator that is unbiased Efficiency demosdemos 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.

R2R2 Between 0 and 1. DemoDemo Least squares maximizes this. Correlation coefficient r is square root.

Standard error of the regression “s” Should be called standard residual –But it isn’t

s Root-mean-square average size of the residuals s 2 is an estimate of  2

S 2 and  2 S2S2 Sum of squares of residuals Divided by N-2 22 Expected value of sum of squares of the errors Divided by N