Hypothesis Testing 2014 Level I Quantitative Methods IFT Notes for the CFA exam
Contents 1. Introduction... 3 2. Hypothesis Testing... 3 3. Hypothesis Tests Concerning the Mean... 10 4. Hypothesis Tests Concerning Variance... 17 5. Other Issues: Nonparametric Inference... 22 Summary... 22 Next Steps... 25 This document should be read in conjunction with the corresponding reading in the 2014 Level I CFA Program curriculum. Some of the graphs, charts, tables, examples, and figures are copyright 2013, CFA Institute. Reproduced and republished with permission from CFA Institute. All rights reserved. Required disclaimer: CFA Institute does not endorse, promote, or warrant the accuracy or quality of the products or services offered by Irfanullah Financial Training. CFA Institute, CFA, and Chartered Financial Analyst are trademarks owned by CFA Institute. Copyright Irfanullah Financial Training. All rights reserved. Page 2
1. Introduction Analysts come across many statements and ideas about how financial markets work. In this reading, we will discuss how analysts can decide whether these statements are true or false using hypothesis testing. If we can reduce an idea to a definite statement about the value of a quantity, such as the population mean, the idea becomes a statistically testable statement of hypothesis. This hypothesis can then be tested through hypothesis testing tools. Hypothesis testing is the process of making judgments about a larger group (a population) on the basis of a smaller group actually observed (a sample). The results of such a test then help us evaluate whether our hypothesis or assertion is true or false. 2. Hypothesis Testing A hypothesis is defined as a statement about one or more populations. In order to test a hypothesis, we follow the following steps: (i) Stating the hypothesis (ii) Identifying the appropriate test statistic and its probability distribution (iii) Specifying the significance level (iv) Stating the decision rule (v) Collecting the data and calculating the test statistic (vi) Making the statistical decision (vii) Making the economic or investment decision We will go through the steps of hypothesis testing using an example to illustrate the process. Suppose you are a researcher and believe that the average return on all Asian stocks was greater than 2%. In this case, you are making a statement about the population mean (µ) of all Asian stocks. The first step is stating the hypothesis. We always state two hypotheses the null hypothesis (H0) and the alternative hypothesis (Ha). The null hypothesis is the hypothesis to be tested. This Copyright Irfanullah Financial Training. All rights reserved. Page 3
is usually the hypothesis that the researcher wants to reject. If this hypothesis is false, we are led to the alternative hypothesis. The alternative hypothesis is the hypothesis accepted when the null hypothesis is rejected. This is usually the hypothesis we are trying to evaluate. An easy way to differentiate among the two is to remember that the null hypothesis always contains some form of the equal sign. Continuing with our example, you believe that the mean (µ) of Asian stocks is greater than 2%. Therefore our null and alternative hypotheses are: H0: µ 2 (Null) Ha: µ > 2 (Alternate) The value 2% is known as µ0. The next step in hypothesis testing is identifying the appropriate test statistic and its probability distribution. A test statistic is a quantity, calculated based on a sample, whose value is the basis for deciding whether or not to reject the null hypothesis. The formula for a test statistic is: Test statistic = Sample statistic Value of the parameter under H 0 Standard error of the sample statistic Continuing with our example, let us further suppose that the sample mean of 36 observations of Asian stocks is 4 and the standard deviation of the population is 4. In this case, our sample statistic will be 4, the value of the parameter under H0 will be 2 and the standard error of the sample statistic will be 0.67. Recall that the standard error is calculated using the following formula: σ x = σ n In this example, the standard error is calculated as: σ x = 4 36 = 4 6 = 0.67 Therefore, our test statistic is calculated as: Copyright Irfanullah Financial Training. All rights reserved. Page 4
Test statistic= 4-2 0.67 = 3 We now need to determine what probability distribution this test statistic follows. There are generally four distributions for test statistics: The t-distribution The z-distribution The chi-square (χ 2 ) distribution The F-distribution In our example, we can conduct a z-test based on the central limit theorem because our sample has many observations. Hence, we can assume that it follows the standard normal distribution. The third step is specifying the significance level. When the test statistic has been calculated, we can either reject the null hypothesis or fail to reject the null hypothesis. This decision is based on comparing the calculated value of the test statistic to a specified possible value(s). The specified possible value(s) that we compare to the test statistic are based on the level of significance. Continuing with our example of Asian stocks, suppose we want to test our hypothesis at the 5% significance level, so the value that corresponds to the significance level is 1.645. Graphically, this is shown below: This is a one-tailed test because we are trying to assess whether the population mean is greater than 2% or not. Hence, we are only interested in the right tail of the distribution. If we were trying to assess whether the population mean is less than 2% we would have been interested in the left tail and the value corresponding to the significance level would have been -1.65. Copyright Irfanullah Financial Training. All rights reserved. Page 5
The fourth step is stating the decision rule. In order to test the null hypothesis, we compare the test statistic with the critical value calculated in the step before that relates to the desired level of significance. If we find that the calculated value of the test statistic is greater than the critical value, we reject the null hypothesis. This means that the result is statistically significant. If the test statistic is less than the critical value, we fail to reject the null hypothesis. This means that the result is not statistically significant. The critical value is also known as the rejection point for the test statistic. Graphically, this point separates the acceptance and rejection regions for a set of values of the test statistic. This is shown below: The region to the left of the test statistic is the acceptance region. This represents the set of values for which we do not reject (accept) the null hypothesis. The region to the right of the test statistic is known as the rejection region. The fifth step is collecting the data and calculating the test statistic. The quality of data used in conducting the test is important. We should be careful to check for measurement errors in the data as well as any instances of sample selection bias and time-period bias. Sample selection bias refers to the bias introduced by systematically excluding some members of the population according to a particular attribute. Time-period bias refers to the possibility that when we use a time-series sample, our statistical conclusion may be sensitive to the starting and ending dates of the sample. Copyright Irfanullah Financial Training. All rights reserved. Page 6
The sixth step is making the statistical decision. In our example, because the test statistic z = 3 is greater than the critical value of 1.645, we reject the null hypothesis in favor of the alternative hypothesis that the average return on all Asian stocks is greater than 2%. The seventh and final step is making the economic or investment decision. This takes into account not only the statistical decision (step 6) but also all pertinent economic issues. Based on these results, an investor might want to invest in Asian stocks. Therefore, a range of nonstatistical considerations, such as the investor s tolerance for risk, transaction costs and impact on existing portfolio, might also enter the decision-making process. Let us now consider the left tail. Suppose we believed that the average return on all Asian stocks was less than 2%. Our sample size is of 36 observations with a sample mean of -3. The standard deviation of the population is 4. In this case, our null and alternative hypotheses would have been: H0: µ 2 (Null) Ha: µ < 2 (Alternate) The standard error of the sample will be unchanged at 0.67: σ x = σ n = 4 36 = 0.67 The test statistic will be: Test statistic = Sample statistic Value of the parameter under H 0 Standard error of the sample statistic = 3 2 0.67 = 7.5 The critical values corresponding to a 5% level of significance will be -1.65. When we consider the left tail of the distribution, our decision rule is then as follows: Reject the null hypothesis if the test statistic is less than the critical value and vice versa. Since our calculated test statistic of - 7.5 is less than the critical value of -1.65, we reject the null hypothesis. Copyright Irfanullah Financial Training. All rights reserved. Page 7
Let us now discuss how two-tailed hypothesis tests are conducted. In a two-tailed test, we reject the null in favor of the alternative if the evidence indicates that the population parameter is either smaller or larger than the value of the parameter under H0. Suppose we believe that the average return on all Asian stocks was not 0%. We take a sample of 36 observations with a sample mean of 1 and a population standard deviation of 4. In this case our null and alternative hypotheses will be: H0: µ = 0 (Null) Ha: µ 0 (Alternate) The standard error of the sample will be unchanged at 0.67: σ x = σ n = 4 36 = 0.67 The test statistic will be: Test statistic = Sample statistic Value of the parameter under H 0 Standard error of the sample statistic = 1 0 0.67 = 1.5 In a two-tailed test, two critical values exist one positive and one negative. For a two-sided test at the 5% level of significance, we calculate the z-values that correspond to 0.05/2 = 0.025 level of significance. These are +1.96 and -1.96. Therefore, we reject the null hypothesis if we find that the test statistic is less than -1.96 or greater than +1.96. We fail to reject the null hypothesis if -1.96 test statistic +1.96. Graphically, this can be shown as: Copyright Irfanullah Financial Training. All rights reserved. Page 8
The above figure also illustrates the relationship between confidence intervals and hypothesis tests. The 5% level of significance in the hypothesis tests corresponds to a 95% confidence interval. When the hypothesized value of the population parameter (in this case µ0) under the null hypothesis is outside the corresponding confidence interval, the null hypothesis is rejected. When the hypothesized value of the population parameter is inside the corresponding confidence interval, the null hypothesis is not rejected. We could use confidence intervals to test hypotheses; practitioners, however, usually do not. Computing a test statistic is more efficient. Furthermore, only when we compute a test statistic can we obtain a p-value. The p-value (also known as probability value) is an alternative approach to hypothesis testing. The p-value is the smallest level of significance at which the null hypothesis can be rejected. The p-value is analogous to the test-statistic. High test-statistic means low p-value and low test-statistic denotes high p-value. The smaller the p-value, the stronger is the evidence against the null hypothesis and in favor of the alternative hypothesis. We can use the p-values in the hypothesis testing framework presented earlier as an alternative to using rejection points. If the p-value is less than our specified level of significance, we reject the null hypothesis. Otherwise, we do not reject the null hypothesis. When we conduct hypothesis testing, there are four possible outcomes: (i) (ii) (iii) (iv) We reject a false null hypothesis. This is a correct decision. We reject a true null hypothesis. This is called a Type I error. We do not reject a false null hypothesis. This is called a Type II error. We do not reject a true null hypothesis. This is a correct decision. Decision H0 True H0 False Do not reject H0 Correct Decision Type II Error Reject H0 (accept Ha) Type I Error Correct Decision The probability of a Type I error in testing a hypothesis is denoted by the Greek letter alpha, α. This probability is also known as the level of significance of the test. For example, a level of Copyright Irfanullah Financial Training. All rights reserved. Page 9
significance of 0.05 for a test means that there is a 5% probability of rejecting a true null hypothesis. The probability of a Type II error is denoted by the Greek letter, β. All else equal, if we decrease the probability of a Type I error by specifying a smaller significance level, we increase the probability of making a Type II error because we will reject the null less frequently, including when it is false. The only way to reduce the probabilities of both types of errors simultaneously is to increase the sample size. The power of a test is the probability of correctly rejecting the null i.e. the probability of rejecting the null when it is false. When more than one test statistic is available to conduct a hypothesis test, we should prefer the most powerful. To summarize, α is the probability of a Type I error and β is the probability of a Type II error. The power of a test is equal to 1 P (Type II error). 3. Hypothesis Tests Concerning the Mean Hypothesis tests concerning the mean are very common and are usually concerned with the following: Tests concerning a single mean Tests concerning differences between means Tests concerning mean differences 3.1 Tests Concerning a Single Mean One of the decisions we need to make in hypothesis testing is deciding which test statistic and which corresponding probability distribution to use. We generally choose between a t-test and a z-test. The t-test is a hypothesis test using a statistic (t-statistic) that follows a t-distribution. As discussed in the previous reading, this is defined by a single parameter known as the degrees of freedom (df). Copyright Irfanullah Financial Training. All rights reserved. Page 10
In hypothesis testing, analysts require the population standard deviation. In practice, however, analysts estimate the population standard deviation by calculating a sample standard deviation because the population variance is unknown. Hence, in hypothesis tests concerning the population mean of a normally distributed population with unknown variance, the theoretically correct test statistic is the t-statistic. Consider a simple example. Fund Alpha has been in existence for 20 months and has achieved a mean monthly return of 2.00% with a sample standard deviation of 5%. The expected monthly return for a fund of this nature is 1.60%. Assuming monthly returns are normally distributed, are the actual results consistent with an underlying or population mean monthly return of 1.60%? The null and alternative hypotheses for this example will be: H0: µ = 1.60 (Null) versus Ha: µ 1.60 (Alternate) Since we only have the sample standard deviation, the appropriate test statistic in this case will be the t-statistic. In order to calculate the t-statistic we use the following formula: t n 1 = X μ 0 s/ n where t n 1 = t-statistic with n-1 degrees of freedom X = the sample mean µ0 = the hypothesized value of the population mean s = the sample standard deviation n = sample size Using this formula, we see that the value of the test statistic is 0.35. The critical values at a 0.05 level of significance can be calculated from the t-distribution table. Since this is a two-tailed test, we should be careful to look at a 0.05/2 = 0.025 level of significance with df = 20 1 = 19. This gives us two values of +2.1 and -2.1. Since our test statistic of 0.35 lies between -2.1 and +2.1, i.e. the acceptance region, we do not reject the null hypothesis. Copyright Irfanullah Financial Training. All rights reserved. Page 11
Given at least approximate normality, the t-test is always called for when we deal with small samples and do not know the population variance. For large samples and unknown population variance, analysts sometimes use a z-test instead of a t-test for tests concerning a mean. The justifications for using a z-test are: In large samples, the sample mean should follow the normal distribution at least approximately, fulfilling the normality assumption of the z-test. In large samples, the difference between the rejection points for the t-test and z-test become quite small. If the population sampled is normally distributed with known variance, then the test statistic for a hypothesis test concerning a single population mean is: z = X μ 0 σ/ n If the population sampled has unknown variance and the sample is large, in place of the t-test, an alternative test statistic is: z = X μ 0 s/ n In both the equations: σ = the known population standard deviation s = the sample standard deviation µ0 = the hypothesized value of the population mean 3.2 Tests Concerning Differences between Means We often want to know whether a mean value differs between two groups. In this section, we discuss how to study the difference between the means of two independent and normally distributed populations. We can use two kinds of t-tests for a test concerning differences between means. In one case the population variances, although unknown, can be assumed to be equal. In the second case the population variances are assumed to be unknown and unequal. Copyright Irfanullah Financial Training. All rights reserved. Page 12
Given two populations, we often want to test whether the population means of the first and second populations are equal or whether one is larger than the other. To test this, we usually formulate the following hypotheses: (i) H0: µ1 - µ2 = 0 versus Ha: µ1 - µ2 0 This is used when we believe the population means are not equal. (ii) H0 : µ1 - µ2 0 versus Ha: µ1 - µ2 > 0 This is used when we believe the mean of the first population is greater than the mean of the second population. (iii) H0 : µ1 - µ2 0 versus Ha: µ1 - µ2 < 0 This is used when we believe the mean of the first population is less than the mean of the second population. where µ1 = population mean of the first population µ2 = population mean of the second population Unknown but Equal Population Variance: When we can assume that the two populations are normally distributed and that the unknown population variances are equal, the t-test based on independent random samples is given by: t = (X 1 X 2) (μ 1 μ 2 ) ( s p 2 + s 2 p ) 1/2 n 1 n 2 The number of degrees of freedom is n1 + n2 2. The term s p 2 is known as the pooled estimator of the common variance. A pooled estimate is an estimate drawn from the combination of two different samples. It is calculated by the following formula: s p 2 = (n 1 1)s 1 2 + ((n 2 1)s 2 2 n 1 + n 2 2 Unknown and Unequal Population Variance: When we can assume that the two populations are normally distributed and that the unknown population variances are unequal, an approximate t- test based on independent random samples is given by: Copyright Irfanullah Financial Training. All rights reserved. Page 13
t = (X 1 X 2) (μ 1 μ 2 ) ( s 1 2 + s 2 2 ) 1/2 n 1 n 2 In this formula, we use the tables of the t-distribution using the modified degrees of freedom. The modified degrees of freedom are calculated using the following formula: df = ( s 1 2 + s 2 2 ) 2 n 1 n 2 (s 2 1 /n 1 ) 2 + (s 2 2 /n 2 ) 2 n 1 n 2 Worked Example 1 You believe the mean return on NYSE stocks was different from the mean on NSE stocks last month. To test your hypothesis you collect the following data: Sample Size (n) Sample Mean (X ) Sample Standard Deviation (s) NSE 20 2% 4 NYSE 40 3% 5 Determine whether to reject the null hypothesis at the 0.10 level of significance. Solution: The first step is to formulate the null and alternative hypotheses. Since we want to test whether the two means were equal or different, we define the hypotheses as: H0: µ1 - µ2 = 0 Ha: µ1 - µ2 0 Since the population standard deviation is unknown and we cannot assume that it is equal, we use the following formula to calculate the test statistic: t = (X 1 X 2) (μ 1 μ 2 ) (2 3) (0) ( s 1 2 + s 2 = = 0.84 2 ) 1/2 ( 42 + 52 n 1 n 2 20 40 )1/2 Next, we calculate the modified degrees of freedom: Copyright Irfanullah Financial Training. All rights reserved. Page 14
df = ( s 1 2 + s 2 2 ) 2 n 1 n 2 = (s 2 1 /n 1 ) 2 + (s 2 2 /n 2 ) 2 n 1 n 2 ( 42 + 52 20 40 )2 (4 2 /20) 2 + (52 /40) 2 20 40 = 48 For a 0.10 level of significance, we find the t-value for 0.10/2 = 0.05 using df = 48. The t-value is therefore ta/2= +1.677 and -1.677. Since our test statistic of -0.84 lies in the acceptance region, we fail to reject the null hypothesis. 3.3 Tests concerning Mean Differences In the previous section, in order to perform hypothesis tests on differences between means of two populations, we assumed that the samples were independent. What if the samples were not independent? For example, suppose you want to conduct tests on the mean monthly return on Toyota stock and mean monthly return on Honda stock. These two samples are believed to be dependent. In such situations, we conduct a t-test that is based on data arranged in paired observations. The test is also sometimes known as a paired comparisons test. Paired observations are observations that are dependent because they have something in common. A paired comparisons test is a statistical test for differences in dependent items. We will now discuss the process for conducting such a t-test. Suppose that we gather data regarding the mean monthly returns on stocks of Toyota and Honda for the last 20 months. Month Mean return of Toyota Mean monthly return of Difference in mean stock Honda stock monthly returns (di) 1 0.5% 0.4% 0.1% 2 0.7% 1.0% -0.3% 3 0.3% 0.7% -0.4% 20 0.9% 0.6% 0.3% Average 0.750% 0.600% 0.075% Here is a simplified process for conducting the hypothesis test: Step 1: Define the null and alternate hypotheses We believe the mean difference is not 0. Hence the null and alternate hypotheses are: Copyright Irfanullah Financial Training. All rights reserved. Page 15
H0: µd = µd0 versus Ha: µd µd0 µd stands for the population mean difference and µd0 stands for the hypothesized value for the population mean difference. Step 2: Calculate the test-statistic Determine the sample mean difference using: n d = 1 n d i i=0 For the data given, the sample mean difference is 0.075. Calculate the sample standard deviation. The process for calculating the sample standard deviation has been discussed in an earlier reading. The simplest method is to plug the numbers (0.1, -0.3, -0.4 0.3) into a financial calculator. The entire data set has not been provided. We ll take it as a given that the sample standard deviation is 0.150%. Use this to calculate the standard error of the mean difference as follows: s d = s d n For our data this is 0.150 / 20 = 0.03354. We now have the required data to calculate the test statistic using a t-test. This is calculated using the following formula using n - 1 degrees of freedom: t = d μ d0 s d For our data the test statistic is (0.075 0) / 0.03354 = 2.23 Step 3: Determine the critical value based on the level of significance We will use a 5% level of significance. Since this is a two-tailed test we have a probability of 2.5% (0.025) in each tail. This critical value is determined from a t-table using a one-tailed probability of 0.025 and df = 20 1 = 19. This value is 2.093. Step 4: Compare the test statistic with the critical value and make a decision In our case the test statistic (2.23) is greater than the critical value (2.093). Hence we can reject the null hypothesis. Our conclusion: the data seems to indicate that the mean difference is not 0. Copyright Irfanullah Financial Training. All rights reserved. Page 16
The hypothesis test presented above is based on the belief that the population mean difference is not equal to 0. If we believe that the population mean difference is greater than 0, the null and alternate hypotheses will be written as: H0: µd µd0 versus Ha: µd > µd0 If we believe that the population mean difference is less than 0, the null and alternate hypotheses will be written as: H0: µd µd0 versus Ha: µd < µd0 4. Hypothesis Tests Concerning Variance Variance is a widely used quantitative measure of risk in investments and so analysts should be familiar with hypothesis tests concerning variance. We discuss two kinds of tests: - Tests concerning the value of a single population variance - Tests concerning the differences between two population variances 4.1 Tests Concerning a Single Variance In tests concerning the variance of a single normally distributed population, we use the chisquare test statistic, denoted by χ 2. The chi-square distribution is asymmetrical and like the t- distribution, is a family of distributions. This means that a different distribution exists for each possible value of degrees of freedom, n - 1. Since the variance is a squared term, the minimum value can only be 0. Hence, the chi-square distribution is bounded below by 0. The graph below shows the shape of a chi-square distribution: The term σ 2 represents the true population variance and σ0 2 represents the hypothesized variance. There are three hypotheses that can be formulated: Copyright Irfanullah Financial Training. All rights reserved. Page 17
(i) (ii) (iii) H0 : σ 2 = σ0 2 versus Ha : σ 2 σ0 2 This is used when we believe the population variance is not equal to 0. H0 : σ 2 σ0 2 versus Ha : σ 2 < σ0 2 This is used when we believe the population variance is less than 0 or any other specified value. H0 : σ 2 σ0 2 versus Ha : σ 2 > σ0 2 This is used when we believe the population variance is greater than 0 or any other specified value. After drawing a random sample from a normally distributed population, we calculate the test statistic using the following formula using n - 1 degrees of freedom: χ 2 = (n 1)(s2 ) σ 0 2 where n = sample size s = sample variance We then determine the critical values using the level of significance and degrees of freedom. The table below is a snapshot of the chi-square distribution table which is used to calculate the critical value. Copyright Irfanullah Financial Training. All rights reserved. Page 18
Suppose our degrees of freedom are 19 and we are testing at the 0.05 level of significance. In this case, the critical value will be 10.117. This critical value is then compared with the test statistic calculated earlier and a decision is reached on whether or not the null hypothesis can be rejected. It is important to note that the chi-square test is sensitive to violations of its assumptions. If the sample is not actually random or if it does not come from a normally distributed population, inferences based on a chi-square test are likely to be faulty. Worked Example 2 Consider Fund Alpha which we discussed in an earlier example. This fund has been in existence for 20 months. During this period the standard deviation of monthly returns has been 5%. You want to test a claim by the fund manager that the standard deviation of monthly returns is less than 6%. Solution: Copyright Irfanullah Financial Training. All rights reserved. Page 19
The null and alternate hypotheses are formulated as mentioned below. Please note that the standard deviation is 6%. Since we are dealing with population variance, we will square this number to arrive at a variance of 36%: H0: σ 2 36 versus Ha: σ 2 < 36 We then calculate the value of the chi-square test statistic: 2 = (n - 1) s 2 / σ0 2 = 19 x 25/36 = 13.19 Next we determine the rejection point based on df = 19 and significance = 0.05. Using the chisquare table, we find that this number is 10.117. Since the test statistic (13.19) is higher than the rejection point (10.117) we cannot reject H0. In other words, the sample standard deviation is not small enough to validate the fund manager s claim that population standard deviation is less than 6%. 4.2 Tests Concerning the Equality (Inequality) of Two Variances In order to test the equality or inequality of two variances, we use an F-test. An F-test is the ratio of sample variances. For an F-test to be valid, it is important that the samples be independent and that the populations from which the samples are taken are normally distributed. The F-distribution, like the chi-square distribution, is a family of asymmetrical distributions bounded from below by 0. Each F-distribution is defined by two values of degrees of freedom, called the numerator and denominator degrees of freedom. As shown in the figure below, the F- distribution is skewed to the right and is truncated at zero on the left hand side. As shown in the graph, the rejection region is always in the right side tail of the distribution. When working with F-tests, there are three hypotheses that can be formulated: (i) H0 : σ1 2 = σ2 2 versus Ha : σ1 2 σ2 2 This is used when we believe the two population variances are not equal. Copyright Irfanullah Financial Training. All rights reserved. Page 20
(ii) (iii) H0 : σ1 2 σ2 2 versus Ha : σ1 2 > σ2 2 This is used when we believe the variance of the first population is greater than the variance of the second population. H0 : σ1 2 σ2 2 versus Ha : σ1 2 < σ2 2 This is used when we believe the variance of the first population is less than the variance of the second population. The term σ1 2 represents the population variance of the first population and σ2 2 represents the population variance of the second population. The formula for the test statistic of the F-test is calculated by the following formula: F = s 1 2 where 2 s 1 = the sample variance of the first population with n observations 2 s 2 = the sample variance of the second population with n observations df1 = n1 1 numerator degrees of freedom df2 = n2 1 denominator degrees of freedom s 2 2 A convention is to put the larger sample variance in the numerator and the smaller sample variance in the denominator. When we follow this convention, the value of the test statistic is always greater than or equal to 1. The test statistic is then compared with the critical values found using the two degrees of freedom and the F-tables. Finally a decision is made whether to reject or not reject the null hypothesis. Worked Example 3 You are investigating whether the population variance of the Indian equity market changed after the deregulation of 1991. You collect 120 months of data before and after deregulation. Variance of returns before deregulation was 13. Variance of returns after deregulation was 18. Solution: Copyright Irfanullah Financial Training. All rights reserved. Page 21
Null and alternate hypothesis: H0: σ1 2 = σ2 2 versus HA: σ1 2 σ2 2 F-statistic: 18/13 = 1.4 df = 119 for the numerator and denominator α = 0.01 which means 0.005 in each tail. From the F-table: critical value = 1.6 Since the F-stat is less than the critical value, do not reject the null hypothesis. 5. Other Issues: Nonparametric Inference The hypothesis-testing procedures we have discussed so far have two characteristics in common: They are concerned with parameters, such as the mean and variance Their validity depends on a set of assumptions Any procedure which has either of the two characteristics is known as a parametric test. Nonparametric tests are not concerned with a parameter and/or make few assumptions about the population from which the sample comes. We use nonparametric procedures in three situations: Data does not meet distributional assumptions Data given in ranks (Example: relative size of company and use of derivatives) Hypothesis does not concern a parameter (Example: is a sample random or not) The Spearman rank correlation coefficient test is a popular nonparametric test. The coefficient is calculated based on the ranks of two variables within their respective samples. Summary (Note: This summary has been taken from the curriculum.) In this reading, we have presented the concepts and methods of statistical inference and hypothesis testing. A hypothesis is a statement about one or more populations. The steps in testing a hypothesis are as follows: 1. Stating the hypotheses. Copyright Irfanullah Financial Training. All rights reserved. Page 22
2. Identifying the appropriate test statistic and its probability distribution. 3. Specifying the significance level. 4. Stating the decision rule. 5. Collecting the data and calculating the test statistic. 6. Making the statistical decision. 7. Making the economic or investment decision. We state two hypotheses: The null hypothesis is the hypothesis to be tested; the alternative hypothesis is the hypothesis accepted when the null hypothesis is rejected. There are three ways to formulate hypotheses: 1. H0: θ = θ0 versus Ha: θ θ0 2. H0: θ θ0 versus Ha: θ > θ0 3. H0: θ θ0 versus Ha: θ < θ0 where θ0 is a hypothesized value of the population parameter and θ is the true value of the population parameter. In the above, Formulation 1 is a two-sided test and Formulations 2 and 3 are one-sided tests. When we have a suspected or hoped for condition for which we want to find supportive evidence, we frequently set up that condition as the alternative hypothesis and use a onesided test. To emphasize a neutral attitude, however, the researcher may select a not equal to alternative hypothesis and conduct a two-sided test. A test statistic is a quantity, calculated on the basis of a sample, whose value is the basis for deciding whether to reject or not reject the null hypothesis. To decide whether to reject, or not to reject, the null hypothesis, we compare the computed value of the test statistic to a critical value (rejection point) for the same test statistic. In reaching a statistical decision, we can make two possible errors: We may reject a true null hypothesis (a Type I error), or we may fail to reject a false null hypothesis (a Type II error). The level of significance of a test is the probability of a Type I error that we accept in conducting a hypothesis test. The probability of a Type I error is denoted by the Greek letter alpha, α. The standard approach to hypothesis testing involves specifying a level of significance (probability of Type I error) only. The power of a test is the probability of correctly rejecting the null (rejecting the null when it is false). Copyright Irfanullah Financial Training. All rights reserved. Page 23
A decision rule consists of determining the rejection points (critical values) with which to compare the test statistic to decide whether to reject or not to reject the null hypothesis. When we reject the null hypothesis, the result is said to be statistically significant. The (1 α) confidence interval represents the range of values of the test statistic for which the null hypothesis will not be rejected at an α significance level. The statistical decision consists of rejecting or not rejecting the null hypothesis. The economic decision takes into consideration all economic issues pertinent to the decision. The p-value is the smallest level of significance at which the null hypothesis can be rejected. The smaller the p-value, the stronger the evidence against the null hypothesis and in favor of the alternative hypothesis. The p-value approach to hypothesis testing does not involve setting a significance level; rather it involves computing a p-value for the test statistic and allowing the consumer of the research to interpret its significance. For hypothesis tests concerning the population mean of a normally distributed population with unknown (known) variance, the theoretically correct test statistic is the t-statistic (zstatistic). In the unknown variance case, given large samples (generally, samples of 30 or more observations), the z-statistic may be used in place of the t-statistic because of the force of the central limit theorem. The t-distribution is a symmetrical distribution defined by a single parameter: degrees of freedom. Compared to the standard normal distribution, the t-distribution has fatter tails. When we want to test whether the observed difference between two means is statistically significant, we must first decide whether the samples are independent or dependent (related). If the samples are independent, we conduct tests concerning differences between means. If the samples are dependent, we conduct tests of mean differences (paired comparisons tests). When we conduct a test of the difference between two population means from normally distributed populations with unknown variances, if we can assume the variances are equal, we use a t-test based on pooling the observations of the two samples to estimate the common (but unknown) variance. This test is based on an assumption of independent samples. When we conduct a test of the difference between two population means from normally distributed populations with unknown variances, if we cannot assume that the variances are equal, we use an approximate t-test using modified degrees of freedom given by a formula. This test is based on an assumption of independent samples. Copyright Irfanullah Financial Training. All rights reserved. Page 24
In tests concerning two means based on two samples that are not independent, we often can arrange the data in paired observations and conduct a test of mean differences (a paired comparisons test). When the samples are from normally distributed populations with unknown variances, the appropriate test statistic is a t-statistic. The denominator of the t- statistic, the standard error of the mean differences, takes account of correlation between the samples. In tests concerning the variance of a single, normally distributed population, the test statistic is chi-square (χ 2 ) with n 1 degrees of freedom, where n is sample size. For tests concerning differences between the variances of two normally distributed populations based on two random, independent samples, the appropriate test statistic is based on an F-test (the ratio of the sample variances). The F-statistic is defined by the numerator and denominator degrees of freedom. The numerator degrees of freedom (number of observations in the sample minus 1) is the divisor used in calculating the sample variance in the numerator. The denominator degrees of freedom (number of observations in the sample minus 1) is the divisor used in calculating the sample variance in the denominator. In forming an F-test, a convention is to use the larger of the two ratios, s1 2 / s2 2 or s2 2 / s1 2, as the actual test statistic. A parametric test is a hypothesis test concerning a parameter or a hypothesis test based on specific distributional assumptions. In contrast, a nonparametric test either is not concerned with a parameter or makes minimal assumptions about the population from which the sample comes. A nonparametric test is primarily used in three situations: when data do not meet distributional assumptions, when data are given in ranks, or when the hypothesis we are addressing does not concern a parameter. The Spearman rank correlation coefficient is calculated on the ranks of two variables within their respective samples. Next Steps Work through the examples presented in the curriculum. Copyright Irfanullah Financial Training. All rights reserved. Page 25
Solve the practice problems in the curriculum. Solve the IFT Practice Questions associated with this reading. Review the learning outcomes presented in the curriculum. Make sure that you can perform the implied actions. Copyright Irfanullah Financial Training. All rights reserved. Page 26