Hypothesis Testing

Quantitative Methods

Hypothesis Tests for Finance

Learning Outcome Statement:

explain hypothesis testing and its components, including statistical significance, Type I and Type II errors, and the power of a test.

Summary:

Hypothesis testing is a statistical method used to make decisions about a population based on sample data. It involves formulating a null hypothesis (H0) and an alternative hypothesis (Ha), choosing an appropriate test statistic and its distribution, specifying a significance level, and making a decision based on the calculated test statistic compared to critical values. The process aims to control the risks of Type I and Type II errors, which are false positives and false negatives, respectively.

Key Concepts:

Null and Alternative Hypotheses

The null hypothesis (H0) is a statement assumed to be true unless evidence suggests otherwise, while the alternative hypothesis (Ha) is considered if the null is rejected. These hypotheses are mutually exclusive and collectively exhaustive.

Test Statistic and Distribution

A test statistic is calculated from sample data and compared against a theoretical distribution (e.g., t-distribution, Chi-square) to determine whether to reject H0.

Significance Level and Type I Error

The significance level (alpha, α) is the probability of rejecting a true null hypothesis (Type I error). Common levels are 5% or 1%, reflecting the risk of a false positive the researcher is willing to accept.

Type II Error and Power of a Test

Type II error (beta, β) is the risk of failing to reject a false null hypothesis. The power of a test (1 - β) is the probability of correctly rejecting a false null hypothesis, indicating the test's sensitivity.

Formulas:

Test Statistic for Single Mean

t = \frac{\bar{X} - \mu_0}{s / \sqrt{n}}

Calculates the t-statistic for testing a single population mean when the population standard deviation is unknown.

Variables:

$\bar{X}$ :: sample mean
$\mu_0$ :: hypothesized population mean
$s$ :: sample standard deviation
$n$ :: sample size

Units: unitless

Test Statistic for Difference in Means

t = \frac{(\bar{X}_{d1} - \bar{X}_{d2}) - (\mu_{d1} - \mu_{d2})}{\sqrt{\frac{s_p^2}{n_{d1}} + \frac{s_p^2}{n_{d2}}}}

Used to compare the means of two independent samples, assuming equal variances.

Variables:

$\bar{X}_{d1}, \bar{X}_{d2}$ :: sample means of two independent samples
$\mu_{d1}, \mu_{d2}$ :: hypothesized population means of the two samples
$s_p$ :: pooled standard deviation
$n_{d1}, n_{d2}$ :: sample sizes of the two samples

Units: unitless

Chi-Square Test Statistic for Variance

\chi^2 = \frac{(n-1)s^2}{\sigma_0^2}

Used to test a single population variance against a hypothesized value.

Variables:

$n$ :: sample size
$s$ :: sample standard deviation
$\sigma_0^2$ :: hypothesized population variance

Units: unitless

Tests of Return and Risk in Finance

Learning Outcome Statement:

construct hypothesis tests and determine their statistical significance, the associated Type I and Type II errors, and power of the test given a significance level

Summary:

This LOS covers various hypothesis testing methods used in finance to assess the risk and return characteristics of financial instruments or portfolios. It includes tests for single means and variances, differences between means (both dependent and independent samples), and equality of variances. The tests utilize t-distributions, chi-square distributions, and F-distributions, depending on the nature of the data and the hypothesis being tested.

Key Concepts:

Test of a Single Mean

Used to determine if the sample mean significantly differs from a known or hypothesized population mean using a t-distribution when the population standard deviation is unknown.

Test of a Single Variance

Used to assess if the sample variance significantly differs from a hypothesized variance using a chi-square distribution.

Test Concerning Differences between Means with Independent Samples

Compares the means of two independent samples to determine if they significantly differ, using a t-distribution and assuming equal or unequal variances.

Test Concerning Differences between Means with Dependent Samples

Compares the means of two related samples (paired comparisons) to determine if they significantly differ, using a t-distribution and accounting for the dependency in the data.

Test Concerning the Equality of Two Variances

Used to compare the variances of two independent samples to determine if they are significantly different, using an F-distribution.

Formulas:

t-statistic for a Single Mean

t = \frac{\bar{X} - \mu_0}{s / \sqrt{n}}

Calculates the t-statistic to compare a sample mean against a hypothesized population mean.

Variables:

$\bar{X}$ :: sample mean
$\mu_0$ :: hypothesized population mean
$s$ :: sample standard deviation
$n$ :: sample size

Units: unitless

Chi-square statistic for a Single Variance

\chi^2 = \frac{(n - 1) s^2}{\sigma_0^2}

Calculates the chi-square statistic to compare a sample variance against a hypothesized population variance.

Variables:

$n$ :: sample size
$s$ :: sample standard deviation
$\sigma_0^2$ :: hypothesized population variance

Units: unitless

t-statistic for Differences between Independent Means

t = \frac{(\bar{X}_1 - \bar{X}_2) - (\mu_1 - \mu_2)}{\sqrt{\frac{s_p^2}{n_1} + \frac{s_p^2}{n_2}}}

Calculates the t-statistic to compare the means of two independent samples, assuming equal variances.

Variables:

$\bar{X}_1$ :: mean of sample 1
$\bar{X}_2$ :: mean of sample 2
$\mu_1$ :: mean of population 1
$\mu_2$ :: mean of population 2
$s_p^2$ :: pooled variance
$n_1$ :: size of sample 1
$n_2$ :: size of sample 2

Units: unitless

F-statistic for Equality of Two Variances

F = \frac{s_1^2}{s_2^2}

Calculates the F-statistic to compare the variances of two independent samples.

Variables:

$s_1^2$ :: variance of sample 1
$s_2^2$ :: variance of sample 2

Units: unitless

Parametric versus Nonparametric Tests

Learning Outcome Statement:

compare and contrast parametric and nonparametric tests, and describe situations where each is the more appropriate type of test

Summary:

Parametric tests are used when the data meet certain assumptions regarding distribution and involve parameters like mean and variance. Nonparametric tests, on the other hand, are used when these assumptions do not hold, when data are ranked or ordinal, or when the hypothesis does not concern a parameter. Nonparametric tests make minimal assumptions and can handle data that are not suitable for parametric tests.

Key Concepts:

Parametric Tests

Parametric tests are statistical tests that make specific assumptions about the population distribution and involve parameters such as mean and variance. They are used when the sample size is large enough and the population distribution is known to be normal.

Nonparametric Tests

Nonparametric tests do not assume a specific population distribution and are used in situations where parametric tests are not suitable. These include cases with small sample sizes, unknown distributions, ranked data, or when testing hypotheses that do not involve parameters.

Situations for Nonparametric Tests

Nonparametric tests are particularly useful when the data do not meet distributional assumptions, contain outliers, are ranked or use an ordinal scale, or when the hypothesis does not concern a parameter.

Power of Tests

While nonparametric tests are more flexible in terms of assumptions, parametric tests generally have more power, meaning they are more likely to reject a false null hypothesis when it is indeed false.

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