Probabilities & Statistics Revision
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1 Probabilities & Statistics Revision Christopher Ting Christopher Ting : christopherting@smu.edu.sg : : LKCSB 5036 January 6, 2017 Christopher Ting QF 302 Week 1 January 6, /42
2 Table of Contents 1 Introduction 2 Mean, Variance, and Covariance 3 Statistical Distribution 4 Sample Statistics and Estimations 5 Hypothesis Testing and Inference 6 Desirable Properties of Estimators Christopher Ting QF 302 Week 1 January 6, /42
3 Learning Objectives Discuss time-series and cross-sectional data and their differences Understand the difference between price/index level and return Recall the basics of probability concepts needed in statistical inference mean, variance, covariance, correlation independence normal, chi-square, Student t, and F distributions Recall the basics of statistical concepts sample mean, sample variance unbiased estimators law of large number, central limit theorem Discuss and develop the framework of hypothesis tests Christopher Ting QF 302 Week 1 January 6, /42
4 Introduction Quotable Economists like to deal with things that can be counted, quantified and computerised. There is nothing wrong in this, but it is a short step from this position to the serious error of believing that quantifiable variables are the only things that really matter. And they seem surprised and disappointed when their prescriptions for economic growth did not work in country after country. Dr Goh Keng Swee, November 1972 Market data are the outcomes of lots of people s decisions that involve theirs or their clients money. These quantifiable variables must be taken seriously. But the econometric models and financial investment theories by which the data are analyzed are not the things that really matter to practitioners. Don t be surprised and disappointed that the models don t work time after time. Christopher Ting QF 302 Week 1 January 6, /42
5 Introduction Straits Times Index and Major Events US Real Estate Bubble Real Estate Bubble Dotcom Bubble Subprime Crisis Black Monday Desert Storm Asian Financial Crisis 911 Terrorists Attack US Invasion of Iraq SAR Epidemic Christopher Ting QF 302 Week 1 January 6, /42
6 Introduction Daily Return on Straits Times Index Christopher Ting QF 302 Week 1 January 6, /42
7 Introduction Histogram of Daily Return on Straits Times Index Christopher Ting QF 302 Week 1 January 6, /42
8 Introduction Investment and Financial Data Analysis Investment is the process of laying out funds in financial instruments and assets with the expectation of a profit. Before making an investment, financial data analysis is a crucial step. To scout for profitable opportunities (risk-adjusted), investment companies such as real-estate investment trusts, exchange-traded funds, mutual funds, and hedge funds perform in-depth analysis on all tradable financial securities and assets. We don t start with models. We start with data. We don t have any preconceived notions. We look for things that can be replicated thousands of times. James Simons Christopher Ting QF 302 Week 1 January 6, /42
9 Introduction Time Series Data Historical observations of a financial variable Prices Trading volume Financial indices Economic indices Insiders trading activities Investment companies trading activities Analysts forecasts Corporate earnings Order flows Money flows Christopher Ting QF 302 Week 1 January 6, /42
10 Introduction Cross-Sectional Data Portfolios constructed based on securities or assets characteristics at a given time Firm characteristics (e.g. market capitalization, growth versus value) Risk profiles Price characteristics (e.g. 52-week high versus 52-week low) Physical characteristics (e.g. agricultural, metals) Industry Emerging versus developed markets Country of domicile or geographic location Funds trading strategies (e.g. convertible arbitrage, event driven) Christopher Ting QF 302 Week 1 January 6, /42
11 Mean, Variance, and Covariance Expectation and Variance of Return ℵ Simple return over one period (eg. 5 minutes, one day, one week, one month) R t = P t P t 1 P t 1 ℵ If P t is observed at t, then the resulting R t is said to be ex post return. ℵ If only P t 1 is known but P t is not observed yet, then R t is said to be ex ante return. ℵ The ex ante return R t is a random variable. ℵ Expected value of R t : ℵ Variance of R t : µ := E ( R t ), t σ 2 := V ( R t ) = E ( (Rt µ ) 2 ) = E ( R 2 t ) µ 2, t Christopher Ting QF 302 Week 1 January 6, /42
12 Mean, Variance, and Covariance Covariance and Correlation Consider the return on M1 stock R X,t, and on Starhub R Y,t. The respective mean and variance are µ X, σ X 2 for M1 and µ Y, σ Y 2 for Starhub. The covariance between R X,t and R Y,t is σ XY := C ( ) ( (RX,t )( ) R X,t, R Y,t = E µ X RY,t µ Y ) The correlation between R X,t and R Y,t is = E ( R X,t R Y,t ) µx µ Y. ρ XY σ XY σ X σ Y Correlation is interpretable as a normalized covariance such that the value falls between 1 and 1. Christopher Ting QF 302 Week 1 January 6, /42
13 Mean, Variance, and Covariance Properties of Expectation and Variance Operators Theorem 1.1 Let a, b and c be constant. Let X and Y be two random variables, with mean µ X and µ Y, respectively. Also, the corresponding variances are σ 2 X and σ2 Y. Then, E ( ax + by + c ) = a E ( X ) + b E ( Y ) + c (1) V ( X ) = E ( X 2) µ 2 X (2) V ( ax + b ) = a 2 V ( X ) (3) V ( ax + by + c ) = a 2 V ( X ) + b 2 V ( Y ) + 2ab C ( X, Y ) (4) Christopher Ting QF 302 Week 1 January 6, /42
14 Mean, Variance, and Covariance An Example and a Question The covariance between the return on gold and the return on silver is The volatility of return on gold is 60%, and the volatility of return on silver is 30%. What is the correlation between gold return and silver return? Answer: How should the notion of co-volatility be defined? Christopher Ting QF 302 Week 1 January 6, /42
15 Mean, Variance, and Covariance Independence Two random variables X and Y are independent when their joint probability density function f(x, y) is the product of their respective marginal probability density functions f X (x) and f Y (y): f(x, y) = f X (x) f Y (y) The conditional pdf of x given y is f(x y). Independence implies f(x y) = f(x, y) f Y (y) = f X (x) Similarly, f(y x) = f Y (y) A pair of random variables that have normal distribution are independent if and only their correlation is zero. Christopher Ting QF 302 Week 1 January 6, /42
16 Mean, Variance, and Covariance Independently and Identically Distributed Suppose the random variables X t for t = 1, 2,..., n are i.i.d. Law of Large Number (LLN) 1 lim n n n t=1 X t P µ = E ( ) X t Central Limit Theorem (CLT) For sufficiently large sample size n, given µ and σ, Y := 1 n t=1 n X t µ σ n 1 n n X t = µ + σ Y n t=1 d N ) (µ, σ2 n (5) Christopher Ting QF 302 Week 1 January 6, /42
17 Mean, Variance, and Covariance QF302 Model 0 Applying Theorem 13, it can be demonstrated easily that From CLT s (5), we write E ( Y ) = 0, V ( Y ) = 1 µ = 1 n n X t σ Y n t=1 In reality, µ and σ are unknown. We replace µ by X n+1, σ/ n by a constant ς, and let Y := Z. Given {X i } n i=1, we compute the sample mean (aka average). X := 1 n X t, we now introduce Model 0: n t=1 X n+1 = X + ςz. (6) Given the dateset {X i } n i=1, a forecast of X n+1 is the sample mean! ( ) {Xi E X n+1 } n i=1 = X. (7) Christopher Ting QF 302 Week 1 January 6, /42
18 Statistical Distribution Normal Distribution A very common assumption of finance is that the returns are normally distributed. r d N ( µ, σ 2). The probability density function f(r) is ( 1 f(r) = exp 1 ( ) ) r µ 2 2πσ 2 2 σ The mean and variance are E(r) = V(r) = r f(r) dr = µ (r µ) 2 f(r) dr = σ 2 Christopher Ting QF 302 Week 1 January 6, /42
19 For convenience, define Statistical Distribution Standard Normal Distribution z := r t µ, f(z) = 1 e z2 /2 σ 2π The probability density function f(z) is the well known bell-shaped curve with mean 0 and variance 1 Christopher Ting QF 302 Week 1 January 6, /42
20 Statistical Distribution Cumulative Distribution Function F (x) What is the probability that z < 1.645? F ( 1.645) := P(z < 1.645) = Thus there is 5% probability that r t < µ 1.645σ f(z) dz = F (x) Christopher Ting QF 302 Week 1 January 6, /42
21 Statistical Distribution Chi-Square and Student t Distributions If random variable Z d N ( 0, 1 ), then V := Z 2 d χ 2 1. More generally, n i=1 Z 2 i d χ 2 n. If X d N(0, 1), and V d χ 2 n, and both X, V are stochastically independent, then the following statistic follows Student s t distribution with n degrees of freedom, i.e., X V n d t n. Christopher Ting QF 302 Week 1 January 6, /42
22 Statistical Distribution Probability Density Function of t Statistics ℸ The pdf of t statistics have fatter tails compared to standard normal. ℸ When the number of degrees of freedom is, Student s t pdf becomes standard normal pdf. Christopher Ting QF 302 Week 1 January 6, /42
23 Statistical Distribution F Distributions If U d χ 2 n 1, V d χ 2 n 2, and both U and V are stochastically independent, then the following statistic follows F distribution with degrees of freedom n 1 and n 2. U n 1 V n 2 d F n1, n 2 The CDFs are (source: Christopher Ting QF 302 Week 1 January 6, /42
24 Statistical Distribution Summary F n1, n 2 F n1, n 2 = χ2 n 1 /n 1 χ 2 n 2 /n 2 Student s t n lim t n Z n Chi square χ 2 n V := n i=1 Z 2 i d χ 2 n Standard normal Z Z = r µ σ Christopher Ting QF 302 Week 1 January 6, /42
25 Statistical Distribution Other Important Facts of Statistics χ 2 n lim n n 1 F n1, = χ2 n 1 n 1 F 1,n2 = χ 2 n 2 t 2 n = F 1,n = χ 2 n F 1, = t 2 = Z 2 = χ 2 1 Christopher Ting QF 302 Week 1 January 6, /42
26 Sample Statistics and Estimations Most Popular Statistic: Sample Mean Given a set of past observations {R 1, R 2,..., R n }, how should one estimate µ and σ 2? Treat the observed value as a particular outcome or realization of the random variable R t at each t: R = 1 n The sample mean R itself is a random variable with mean and variance, assuming identical distribution, n t=1 R t E ( R ) = 1 n n E ( ) R t = µ t=1 V ( R ) = 1 n n 2 V ( ) nσ 2 R t = t=1 n 2 = σ2 n For the sample variance, independence is also assumed. Christopher Ting QF 302 Week 1 January 6, /42
27 Sample Statistics and Estimations Estimation of Variance and t Statistic Unbiased sample variance is s 2 = 1 n 1 n ( Rt R ) 2 The ratio of the sample variance with population variance Application ) R d N (µ, σ2 n = t=1 V := (n 1) s2 σ 2 or ( ) n R µ σ V n 1 = R µ σ 2 n ) ( n R µ σ d χ 2 n 1 = = s 2 σ 2 n ( R µ ) σ n ( R µ ) s d N(0, 1) (8) d t n 1 (9) Christopher Ting QF 302 Week 1 January 6, /42
28 Sample Statistics and Estimations Case Study of Temasek s Performance The past observations of Temasek s portfolio value are Source: Temasek Review 2011 Compute the annual portfolio returns. Compute the unbiased sample mean of annual portfolio return. Compute the unbiased sample variance of annual portfolio return. If µ = 7%, compute the t statistic. Christopher Ting QF 302 Week 1 January 6, /42
29 Hypothesis Testing and Inference Hypothesis and Test Statistic Suppose a theory predicts that the population mean µ is, say, 7%. The null hypothesis is H 0 = 7%. The alternative hypothesis is H A 7%. A statistical test of the hypothesis is a decision rule that either rejects or does not reject the null H 0. The set of sample values that lead to the rejection of H 0 is called the critical region. The statistical rule on H 0 : µ = 7%, H A ) : µ 7%, is that if the ( n R µ t-distributed test statistic t n 1 = s falls within the critical region defined as {t n 1 < a or t n 1 > +a}, a > 0, then H 0 is rejected. Otherwise H 0 cannot be rejected. s Note that is known as the of the sample n mean. Christopher Ting QF 302 Week 1 January 6, /42
30 Hypothesis Testing and Inference Illustration of Critical Regions The critical regions correspond to the shaded areas. The sum of the shaded area is the probability of rejecting H 0 when it is true. This probability is known as the significance level. Christopher Ting QF 302 Week 1 January 6, /42
31 Hypothesis Testing and Inference Two-Tail versus One-Tail If the hypotheses are, say, H 0 : µ = 7% and H A : µ 7%, then the decision rule is based on two-tail test. The critical region comprises of the left and right tails of the t n 1 pdf. When the theory rules out, say, µ > 7%, the hypotheses become H 0 : µ = 7% and H A : µ < 7%, then the decision rule is based on one-tail test. The critical region is only the left side, for when µ < 7%, then tn 1 will become larger. Thus at the one-tail 5% significance level, the critical region is {t n 1,95% > 1.671} for n = 61, where {t n 1,95% > 1.671} is the 95-th percentile of the t distribution. Christopher Ting QF 302 Week 1 January 6, /42
32 Hypothesis Testing and Inference P Value P value is the observed significance level, which is the probability of getting a value of the t statistic that is extreme or more extreme than the observed value of t statistic. Example H 0 = 7% against H 1 7% n = 25 R = 10% s = 5% t = 25(10 7)/5 = 3 The P value is P = P ( t 24 > 3 ) Christopher Ting QF 302 Week 1 January 6, /42
33 Hypothesis Testing and Inference Type I and Type II Errors If H 0 is true but is rejected, Type I error is committed. If H 0 is false but is accepted, Type II error is committed. Reality Result of the Test H 0 is true H 0 is false Reject H 0 Type I error Correct inference Do not reject H 0 Correct inference Type II error Christopher Ting QF 302 Week 1 January 6, /42
34 Hypothesis Testing and Inference Inference The probability of committing a Type I error when H 0 is true is called the. The probability of the population t-statistic exceeding the t-statistic obtained from the test sample is known as the p value. If the p value < test significance level, reject H 0 ; otherwise H 0 cannot be rejected. In practice, the probability of Type I error is fixed and the significance level set at e.g. 10%, 5%, or 1%. Given that the null hypothesis is not true, the power of a test is the probability of not committing Type II error. Christopher Ting QF 302 Week 1 January 6, /42
35 Hypothesis Testing and Inference Confidence Interval Suppose x 1, x 2,..., x n are randomly sampled from X d N ( µ, σ 2), and an a > 0 such that P ( a t n 1 +a ) = 95% Since t n 1 is symmetrically distributed, P ( a t n 1 ) = 97.5% and P ( t n 1 +a ) = 97.5%. Then P ( a n ( Xn µ ) s a ) = 0.95 Thus the probability of µ falling within the confidence interval is 95%: ( P X n a s µ X n + a s ) = 95%. (10) n n Christopher Ting QF 302 Week 1 January 6, /42
36 Hypothesis Testing and Inference Connection with QF302 Model 0 From (10), and given the critical value a for the t distribution, the lower bound is given by and the upper bound by LB := X n UB := X n + s n a s n a In summarizing the data, it is better to give a range rather than a point estimate. Therefore, given a sample {X t } n t=1, the range is between LB and UB. Continuation from the case study of Temasek s performance (Slide 28) how would you summarize Temasek s 1-year return? Christopher Ting QF 302 Week 1 January 6, /42
37 Hypothesis Testing and Inference Application of QF302 Model 0: Forecasting Let X n+1 denote a forecast of yet-to-be-observed X n+1 based on the sample {X t } n t=1 observed up to period n. If X n+1 is assumed to be independently drawn from the same population as the sample, then the forecast that minimizes mean squared error is simply the sample mean X, i.e., X n+1 = X n. We have obtained the point forecast. In the absence of other information, the sample mean is an unbiased forecast. Christopher Ting QF 302 Week 1 January 6, /42
38 Hypothesis Testing and Inference Forecast Error Under Model 0 The standard error of the forecast (denoted as s f ) comprises the standard error of Model 0 and the standard error of the sample mean. Proof: In Slide 17 Model 0 is X n+1 = X n + ςz. Taking the variance operation on both sides and given i.i.d. assumption, V ( X n+1 ) = V ( Xn ) + ς 2, since V ( Z ) = 1. For any member of the population, the unbiased estimate of σ 2 is none other than the sample variance s 2! Therefore s f := V ( ) s 2 X n+1 = n + s2 = s n. (11) What is your forecast for Temasek s portfolio value in 2012? Christopher Ting QF 302 Week 1 January 6, /42
39 Desirable Properties of Estimators Unbiasedness A statistic ψ(x) is an unbiased estimator of θ if E ( ψ(x) ) = θ. If E ( ψ(x) ) θ, then the estimator is said to be biased, The bias is simply the difference: E ( ψ(x) ) θ. Question: Estimator ψ 1 is unbiased but, depending on the sample X, can take value far away from θ. Estimator ψ 2 is biased but ranges close to θ and has its mean slightly away from θ. Which is better? Christopher Ting QF 302 Week 1 January 6, /42
40 Desirable Properties of Estimators Consistency In practice, uunbiased estimators workable on small samples are rare. A sequence of estimators θ n (X) of θ from sample X of size n is said to be a consistent estimator if θ n P θ as n or plim θ n = θ. That is, θ n converges in probability to θ; for any arbitrary ɛ > 0, ( lim P θ n θ ) < ɛ = 1. n Is a consistent estimator necessarily unbiased? Christopher Ting QF 302 Week 1 January 6, /42
41 Desirable Properties of Estimators Efficiency, BLUE Given the dataset X, an unbiased estimator ψ (X) of θ is said to be efficient if for any other unbiased estimator ψ(x) V ( ψ (X) ) V ( ψ(x) ). If ψ is an unbiased linear estimator and it has the minimum variance in the class of unbiased linear estimators, then ψ is said to be BLUE, best linear unbiased estimator. Christopher Ting QF 302 Week 1 January 6, /42
42 Desirable Properties of Estimators Takeaways Standard normal, Student s t, chi-square, F distributions Population mean µ and variance σ 2, which are mostly unknown Unbiased sample mean X and variance s 2, given the sample QF302 Model 0: the unbiased point estimate, and a point forecast for the next outcome is the sample mean X. The range of forecast under Model 0 is from LB to UB with a confidence level of, say 95%. In other words, there is a 5% chance that the actual value may fall outside the range. Hypothesis test, confidence interval, Type 1 and Type 2 errors Unbiasedness, consistency, efficiency, BLUE Christopher Ting QF 302 Week 1 January 6, /42
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