Hypothesis Testing

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Hypothesis Testing
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What is Hypothesis Testing?

• Sample information can be used to obtain point
  estimates or confidence intervals about
  population parameters
• Alternatively, sample information can be used to
  test the validity of conjectures about these
  parameters
• Are private banks more profitable than
  state-owned banks in the EU countries?
• Are returns on a stock different before and after
  a stock split?
• Is there a larger variability in real estate
  prices in Champaign than in Urbana?

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What is Hypothesis Testing?

• A hypothesis is a statement about a population
  parameter from one or more populations
• Statistically testable hypotheses are formulated
  based on theories that are used to make
  predictions
• A hypothesis test is a procedure that
• States the hypothesis to be tested
• Uses sample information and formulates a decision
  rule
• Based on the outcome of the decision rule the
  hypothesis is statistically validated or rejected

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Steps in Hypothesis Testing

• The following steps are followed in a hypothesis
  test
• State the hypothesis
• Identify the appropriate test statistics and its
  probability distribution
• Specify the significance level
• State the decision rule
• Collect the data and calculate the test statistic
• Make the statistical decision
• Evaluate whether the statistical decision implies
  a corresponding financial decision

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Stating the Testable Hypotheses

• A hypothesis test always includes two hypotheses
• Null Hypothesis (H0) The null hypothesis is the
  hypothesis to be tested
• E.g., The average debt-equity ratio for US
  industrial firms is 20
• Alternative Hypothesis (H1) The alternative
  hypothesis is the one accepted if the null
  hypothesis is rejected
• E.g., The average debt-equity ratio for US
  industrial firms is different than 20

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Stating the Testable Hypotheses

• Note The null hypothesis is a statement that is
  considered true unless the sample used in the
  hypothesis testing provides evidence that it is
  false
• Hypothesis tests for a population parameter ? in
  relation to a possible value ?0 can be formulated
  as follows
• H0 ? ?0 vs. H1 ? ? ?0
• H0 ? ? ?0 vs. H1 ? gt ?0
• H0 ? ? ?0 vs. H1 ? lt ?0

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Stating the Testable Hypotheses

• The first formulation is a two-sided test while
  the other two are one-sided tests
• In each formulation the null and the alternative
  account for all possible values of the population
  parameter
• Regardless of the formulation, the test is always
  conducted at the point of equality, ? ?0

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Stating the Testable Hypotheses

• How do we state the null and alternative
  hypotheses?
• Example Suppose that theory tells us that growth
  funds outperform value funds
• H0 Growth funds perform worse or equal to value
  funds
• H1 Growth funds perform better than value funds
• We formulate the alternative hypothesis as the
  statement that the condition is true and test the
  validity of the null that the statement is false

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Identifying the Test Statistic and its
Probability Distribution

• The decision rule for the hypothesis test is
  based on a test statistic
• The test statistic is a quantity calculated from
  sample information that typically has the
  following form
• (Sample Statistic Value of Parameter under
  H0)/St. Error of Sample Statistic

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Identifying the Test Statistic and its
Probability Distribution

• Example Suppose that we want to test the null
  hypothesis that the mean return on the SP 500
  index during the past five years is less or equal
  than 10 vs. the alternative that it is greater
• Drawing a sample and calculating the sample mean,
  we know
• If population distribution is normal with known
  variance, sample mean follows normal distribution
  and we use the standardized variable Z as our
  test statistic

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Identifying the Test Statistic and its
Probability Distribution

• If in the above case, the population variance is
  unknown, but the sample is large, we again use Z
  as our test statistic
• If the population variance is unknown or sample
  size is small, we use the variable t as out test
  statistic
• If, for example, the variance of SP 500 returns
  is unknown, we will use the variable t, known as
  the t-statistic

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Specifying the Significance Level

• To reject or not the null hypothesis, the
  t-statistic is compared to a pre-specified value
• The selected value is based on a pre-determined
  level of significance
• Note that the null hypothesis can be either true
  or false
• However, there are four possible outcomes when a
  hypothesis is tested

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Specifying the Significance Level

• A false null hypothesis is rejected, which is a
  correct decision
• A true null hypothesis is rejected (this is
  called a Type I error)
• A false null hypothesis is not rejected (this is
  called a Type II error)
• A true null hypothesis is not rejected, which is
  again a correct decision

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Specifying the Significance Level

• The probability of Type I error in a hypothesis
  test is called the level of significance of the
  test
• Conducting a hypothesis test, we want the chance
  of type I error to be as low as possible
• E.g., A level of significance of 5 implies a 5
  chance of type I error
• Note As we decrease the chance of a type I
  error, we increase the chance of a type II error

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Specifying the Significance Level

• Lowering the chance of type I error implies that
  the null will be rejected less often, including
  when it is false (type II error)
• To lower the probabilities of both errors we need
  to increase the sample size
• The power of a test is the probability of
  correctly rejecting a false null hypothesis (The
  power of a test is 1 P(type II error))
• Conventional significant levels when testing
  hypotheses are 10, 5, 1

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Specifying the Significance Level

• Example
• If we reject the null hypothesis at the 10
  significance level, we have some evidence that
  the alternative is true
• If we reject the null hypothesis at the 5
  significance level, we have strong evidence that
  the alternative is true
• If we reject the null hypothesis at the 1
  significance level, we have very strong evidence
  that the alternative is true

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Stating the Decision Rule

• The decision rule compares the calculated test
  statistic with specific cutoffs from the tables
  of the statistics distribution
• Example Suppose that the test statistic that we
  use is the Z-statistic (Z variable) and that we
  use a 5 significance level
• If the hypothesis test is H0 ? ?0 vs. H1 ? ?
  ?0 then the two rejection values are Z0.025 1.96
  and - Z0.025 -1.96
• We would reject the null if Z lt -1.96 or Z gt 1.96

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Collecting Data, Calculating Test Statistic and
Making a Decision

• In collecting a sample, it is important to avoid
  problems of sample selection bias, such as
  survivorship bias
• Example If we want to test a hypothesis
  regarding bank performance and we choose in our
  sample only the banks that exist in the last
  quarter, we do not include the banks that have
  failed
• Banks still in existence must have performed
  better and, thus, there will be some bias in our
  sample

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Hypothesis Tests and Financial Decisions

• Deciding to reject or not the null hypothesis
  implies making a statistical decision
• Does this always translate into a corresponding
  financial decision?
• Example Suppose we find support through a test
  for the hypothesis that on average stocks provide
  higher returns than bonds

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Hypothesis Tests and Financial Decisions

• Does this statistical decision have a financial
  meaning, as well?
• From a financial or investment perspective we may
  also want to understand what are the risks of
  investing in these two types of assets
• Finally, we define the p-value as the smallest
  level of significance at which we can reject the
  null hypothesis

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Hypothesis Test for a Single Mean(Normal
Distribution, Variance Unknown)
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Example of Hypothesis Test for a Single Mean

• Suppose that the controller of a firm monitors
  the firms payments from its customers through
  days receivables
• The firm has tried to maintain an average of 45
  days in receivables
• A recent random sample of 50 accounts has shown a
  mean of 49 days and a standard deviation of 8
  days
• Can we reject the hypothesis that the average
  days in receivables for this firm has increased?

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Example of Hypothesis Test for a Single Mean

• The testable hypotheses are stated as follows
• H0 ? ? 45
• H0 ? gt 45
• The test can be conducted at the 5 and 1 levels
  of significance
• Since the population variance is unknown, we use
  the t-statistic, which is

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Example of Hypothesis Test for a Single Mean

• The cutoffs for the t-distribution with 49
  degrees of freedom at the 5 and 1 level of
  significance are 1.677 and 2.405, respectively
• Given that our t-statistic is greater than both
  cutoffs, the null hypothesis is rejected both at
  the 5 and 1 levels
• This implies that there has been a statistically
  significant increase in the days receivables for
  this firm

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Hypothesis Test for Difference Between Population
Means

• We often want to test the hypothesis that the
  population means differ between two groups
• Examples
• Is the average debt-equity ratio higher for
  mature compared to young firms?
• Do average stock returns differ by decade?
• Do community banks on average lend more to small
  businesses than larger banking institutions?
• Do average corporate defaults differ by industry?

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Hypothesis Test for Difference Between Population
Means

• Taking samples from the two populations, we can
  formulate the following hypotheses
• H0 ?1 ?2 vs. H1 ?1 ? ?2
• H0 ?1 ? ?2 vs. H1 ?1 gt ?2
• H0 ?1 ? ?2 vs. H1 ?1 lt ?2
• Two cases (assuming samples are independent)
• Populations are assumed normally distributed,
  variances are unknown, but equal
• Populations are assumed normally distributed,
  variances are unknown, but unequal

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Hypothesis Test for Difference Between Population
Means

• When population variances are assumed to be
  equal, the t-statistic is as follows
• where
• and the degrees of freedom are n1 n2 -2

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Hypothesis Test for Difference Between Population
Means

• When population variances cannot be assumed to be
  equal, the t-statistic is as follows
• and the degrees of freedom are

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Example of Hypothesis Test for DifferencesBetween
Population Means

• Suppose that we observe monthly returns on the
  SP 500 from the 1970s and the 1980s (equal
  samples 120 observations)
• For the 1970s, the mean monthly return is 0.58
  and the standard deviation is 4.598
• For the 1980s, the mean monthly return is 1.47
  and the standard deviation is 4.738
• We want to test whether the two population means
  are equal, assuming that they are both normally
  distributed and that variances are not known

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Example of Hypothesis Test for DifferencesBetween
Population Means

• The hypothesis test is formulated as follows
• H0 ?70 ?80 vs. H1 ?70 ? ?80
• Suppose we are interested in testing the above
  hypothesis at the 5 and 1 levels of
  significance
• Assuming the two samples are independent, the
  degrees of freedom are 238

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Example of Hypothesis Test for DifferencesBetween
Population Means

• Plugging the relevant information into the
  formulas for the estimator of the common
  population variance, s2, and the t-statistic, we
  find that t -1.477
• The cutoff of the t-distribution for this
  two-sided test are
• At the 5 level, we reject the null if t lt -1.972
  or t gt 1.972
• At the 1 level, we reject the null if t lt -2.601
  or t gt 2.601
• Given our t-statistic of 1.477, we cannot reject
  the null hypothesis at either the 5 or the 1
  significance level