Properties of the least squares estimates...

1

• Properties of the least squares estimates...
• They are all unbiased estimators i.e., their
  expected values are equal to the parameters they
  estimate (see 1-4 on in section 2.4.1 on page 32
  dont worry about the proofs given, just get
  the results)
• Section 2.4.2 gives the variances of the
  estimators
• Since one of our main goals is to say something
  about the parameters in the regression model,
  well do this by testing hypotheses about them
  this means we have to know the distributions of
  the estimators ...

2
To get the distributions of the estimators, we
must assume normality of the error term in the
model so not just that E(e) 0 and V(e) s2
but that eN(0, s2). This assumption implies the
following important distributional result about
the estimator of the slope Notice that weve
basically standardized b1-hat by taking away its
mean and dividing by its so-called standard
error. (the standard error is the estimated
standard deviation of the estimator)... The
resulting standardized statistic has a
t-distribution with n-2 degrees of freedom. Note
that the df n-2 the number of df associated
with the error sum of squares, the numerator in
our estimator s2 .
3

• So now, we may use the distribution of the
  estimated slope to either
• test the hypothesis that H0 b1 0 or
• get a confidence interval for b1
• Lets review both of these concepts from
  elementary statistics...
• compare the value of the test statistic, T on the
  previous slide, assuming the null hypothesis is
  true, with the percentiles of the t(n-2)
  distribution to decide on whether to reject the
  null hypothesis or not. This comparison yields a
  so-called p-value and small p-values yield
  rejection while large p-values yield
  non-rejection of the null hypothesis.
• construct a 100(1 a ) confidence interval for
  the true slope b1 by the usual estimate /-
  (margin of error)
• estimate /- (value from t table)(s.e. of
  estimator)
• ...remember that the s.e. of the estimator
  standard error of estimator estimated standard
  deviation of the estimator

4

• Similarly, we may either test hypotheses or
  estimate m0 where using the unbiased least
  squares estimator
• Note that when x0 0, we have the special case
  of testing and/or estimating b0 .
• Now go back over the Hardness example and do the
  various tests dont forget to write out an
  interpretation of what the results mean...!
• Look at how R implements these tests and
  estimates, try for the Hardness data....
• do a scatterplot plot in the prediction line
  (the regression line, the mean line) can you
  plot the confidence bands around this line using
  the formula 2.23 at the top of page 37?
• do the hypothesis test for no slope and give
  your results in terms of p-value. Is Hardness
  linearly related to Temperature of the quench
  bath water?