Hypothesis Testing

1
Hypothesis Testing

• Start with a null hypothesis, H0 ? ?k
• The Alternative hypothesis, HA ? ? ?k (two sided
  alternative)
• OR HA ? gt ?k (one sided alternative)
• OR HA ? lt ?k (one sided alternative)
• OR Some other forms that is generated by your
  economic model.

2
The Strategy of Testing

• Step 1 Assume that H0 is true.
• Step 2 Calculate the distribution of the
  estimated ? based on your assumption in step1.
• Step 3 Test to see if the sample gives a value
  for the estimated ? that is likely given the
  distribution.
• Step 4 If the estimated value for ? is very
  unlikely, we will reject H0 if it is likely we
  will fail to reject H0.

3
One-Sided Alternatives

• If we want to have only a 5 probability of
  rejecting H0 if it is really true, then we say
  our significance level is 5.
• Having picked a significance level, a, we look up
  the (1 a)th percentile in a t distribution with
  n 1 df and call this c, the critical value.
• We can reject the null hypothesis if the t
  statistic is greater than the critical value.
• If the t statistic is less than the critical
  value then we fail to reject the null.

4
One-Sided Alternatives (cont.)
H0 ? ?k HA ? gt ?k
Fail to reject
reject
(1 - a)
a
c
0
5
One-Sided Alternatives (cont.)

• Because the t distribution is symmetric, testing
  HA ? lt ?k is straightforward. The critical
  value is just the negative of before.
• We can reject the null if the t statistic lt c,
  and if the t statistic gt than c then we fail to
  reject the null.

6
Two-Sided Alternatives

• For a two-sided test, we set the critical value
  based on a/2 and reject HA ? ? ?k 0 if the
  absolute value of the t statistic gt c.

fail to reject
reject
reject
(1 - a)
a/2
a/2
-c
c
0
7
Hypothesis Testing

• A one-tailed test places the rejection region
  entirely in one tail of the distribution and a
  two-tailed test places some part of the rejection
  in both tails of the distribution.
• Unless otherwise stated, the alternative is
  assumed to be two-sided.
• A test can never establish that H0 is true, but
  it can establish that it is false.

8
The Size and Power of Tests

• Type I error Rejecting a null hypothesis when it
  is true is known as a type I error.
• Type II error Fail to reject the null hypothesis
  when it is false.
• The size of a test is the probability that we
  will reject the null hypothesis when it is true
  (same as the level of significance). We want the
  size to be low. Why?

9
The Size and Power of Tests

• The power of a test is the probability that we
  will reject the null hypothesis when it is false
  that is the probability that we will not make a
  type II error.
• What is the most powerful hypothesis tests?
• We must reject a false hypothesis increase the
  tests rejection region. If we do this, then we
  increase the size of the test reject when the
  null is true.