Title LM07 Estimation and Inference IFT Notes.pdf ✅

LM07 Estimation and Inference 2025 Level I Notes © IFT. All rights reserved 1 LM07 Estimation and Inference 1. Introduction...........................................................................................................................................................2 2. Sampling Methods...............................................................................................................................................2 3. Central Limit Theorem and Inference .........................................................................................................5 4. Bootstrapping and Empirical Sampling Distributions ..........................................................................7 Summary......................................................................................................................................................................9 Required disclaimer: IFT is a CFA Institute Prep Provider. Only CFA Institute Prep Providers are permitted to make use of CFA Institute copyrighted materials which are the building blocks of the exam. We are also required to create / use updated materials every year and this is validated by CFA Institute. Our products and services substantially cover the relevant curriculum and exam and this is validated by CFA Institute. In our advertising, any statement about the numbers of questions in our products and services relates to unique, original, proprietary questions. CFA Institute Prep Providers are forbidden from including CFA Institute official mock exam questions or any questions other than the end of reading questions within their products and services. CFA Institute does not endorse, promote, review or warrant the accuracy or quality of the product and services offered by IFT. CFA Institute®, CFA® and “Chartered Financial Analyst®” are trademarks owned by CFA Institute. © Copyright CFA Institute Version 1.0
LM07 Estimation and Inference 2025 Level I Notes © IFT. All rights reserved 2 1. Introduction This learning module covers: • Sampling methods – simple random, stratified random, cluster, convenience, and judgmental sampling • Central limit theorem, and standard error of the sample mean • Resampling – bootstrap and jackknife techniques A sample is a subset of a population. We can study a sample to infer conclusions about the population itself. For example, if all the stocks trading in the US are considered a population, then indices such as the S&P 500 are samples. We can look at the performance of the S&P 500 and draw conclusions about how all stocks in the US are performing. This process is known as sampling and estimation. 2. Sampling Methods There are various methods for obtaining information on a population through samples. The information we obtain usually concerns a parameter, a quantity used to describe a population. To estimate a parameter, we use sample statistics. A statistic is a quantity used to describe a sample. There are two reasons why sampling is used: • Time saving: In many cases it will be very time consuming to examine every member of the population. • Monetary saving: In some cases, examining every member of the population becomes economically inefficient. There are two types of sampling methods: • Probability sampling: Every member of the population has an equal chance of being selected. Therefore, the sample created is representative of the population. • Non-probability sampling: Every member of the population may not have an equal chance of being selected. This is because sampling depends on factors such as the sampler’s judgement or the convenience to access data. Therefore, the sample created may not be representative of the population. All else equal, the probability sampling method is more accurate and reliable as compared to the non-probability sampling method. In the subsequent sections, we will discuss the following sampling methods: • Probability sampling o Simple random sampling o Systematic sampling o Stratified random sampling
LM07 Estimation and Inference 2025 Level I Notes © IFT. All rights reserved 3 o Cluster sampling • Non-probability sampling o Convenience sampling o Judgement sampling Simple Random Sampling Simple random sampling is the process of selecting a sample from a larger population in such a way that each member of the population has the same probability of being included in the sample. Sampling distribution If we draw samples of the same size several times and calculate the sample statistic, the sample statistic will be different each time. The distribution of values of the sample statistic is called a sampling distribution. For example, say you select 100 stocks from a universe of 10,000 stocks and calculate the average annual returns of these 100 stocks. Let’s say you get an average return of 15%. You repeat this process with a second sample of 100 stocks. This time, you get an average return of 14%. You keep repeating this process and each time you get a different average return. The distribution of these sample average returns is called a sampling distribution. Sampling error Sampling error is the difference between a sample statistic and the corresponding population parameter. The sampling error of the mean is given by: Sampling error of the mean = x̅ − μ For example, let’s say you want to estimate the average returns of 10,000 stocks. You draw a sample of 100 stocks and calculate the average return of these 100 stocks as 15%. However, the actual average of the 10,000 stocks was 12%. Then the sampling error = 15% - 12% = 3%. Systematic sampling: In this technique, we select every kth member of the population until we have a sample of the desired size. Samples created using this technique should be approximately random. Instructor’s Note: Researchers calculate the sampling interval ‘k’ by dividing the entire population size by the desired sample size. Stratified Random Sampling In stratified random sampling, the population is divided into subgroups based on one or more distinguishing characteristics. Samples are then drawn from each subgroup, with
LM07 Estimation and Inference 2025 Level I Notes © IFT. All rights reserved 4 sample size proportional to the size of the subgroup relative to the population. Finally, samples from each subgroup are pooled together to form a stratified random sample. The advantage of stratified random sampling is that the sample will have the same distribution of key characteristics as the overall population. This can help reduce the sampling error. Stratified random sampling therefore produces more precise parameter estimates than simple random sampling For example, you divide the universe of 10,000 stocks as per their market capitalization such that you have 5,000 large cap stocks, 3,000 mid cap stocks, and 2,000 small cap stocks. In stratified random sampling, to select a total sample of 100 stocks, you will randomly select 50 large cap stocks, 30 mid cap stocks, and 20 small cap stocks and pool all these samples together to form a stratified random sample. Example Paul wants to categorize publicly listed stocks for his research project. He first divides the stocks into 15 industries. Then from each industry, he categorizes companies into three groups: small, medium, large. Finally, he divides these into value versus growth stocks. How many cells or strata does the sampling plan entail? A. 20 B. 45 C. 90 Solution: C is correct. This is an application of the multiplication rule of counting. The total number of cells is the product of 15, 3, and 2. Thus the answer is 90. Cluster Sampling Cluster sampling is similar to stratified random sampling as it also requires the population to be divided into subpopulation groups, called clusters. Each cluster is essentially a mini- representation of the entire population. Then some random clusters are chosen as a whole for sampling. Instructor’s Note: Clusters are generally based on natural groups separating the population. For example, you might be able to divide your data into natural groupings like city blocks, voting districts, or school districts. The main difference between cluster sampling and stratified random sampling is that in cluster sampling, the whole cluster is selected; and not all clusters are included in the sample. In stratified random sampling, however, only a few members from each stratum are selected; but all strata are included in the sample. The difference between simple random sampling, stratified random sampling, and cluster sampling is illustrated in the figure below: