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1、2016CFA培訓(xùn)項(xiàng)目Quantitative MethodsKEL WANG金程教育資深培訓(xùn)師地點(diǎn)： 上海北京 Summary of Readings and Framework2-94Session NO.ContentWeightingsStudy Session 1-2Ethics & Professional Standards10-15Study Session 3Quantitative Methods5-10Study Session 4Economic Analysis5-10Study Session 5-7Financial Statement Analysis1

2、5-20Study Session 8-9Corporate Finance5-15Study Session 10-12Equity Analysis15-25Study Session 13Alternative Investments5-10Study Session 14-15Fixed Income Analysis10-20Study Session 16-17Derivative Investments5-15Study Session 18Portfolio Management and Wealth Planning5-10Total:100Summary of Readin

3、gs and FrameworkSS 3Ø R10 Multiple regression and issues in regression analysisØ R11 Time-series analysisØ R12 Excerpt from Probabilistic Approaches: Scenario Analysis, Decision Trees, and Simulation3-94Ø R9 Correlation and regressionFrameworkØReview of Hypothesis Testing1.2

4、.3.4.5.Basic conceptsType I and Type II errorsTest statistic and critical value Decision ruleDefinition of p-value4-941. Basic ConceptsØHypothesis testing is the statistical assessment of a statement or idea regarding a population .Hypothesis testing procedures, based on sample statistics and p

5、robability theory, are used to determine whether a hypothesis cannot be rejected because there is insufficient evidence or if it is an unreasonable statement and should be rejected based on the empirical evidence.1. State the hypothesis2. Select the appropriate test statistic3. Specify the level of

6、significance4. State the decision rule regarding the hypothesis5. Collect the sample and calculate the sample statistics6. Make a decision regarding the hypothesis7. Make a decision based on the results of the testØ5-941. Basic ConceptsØDefinition of Null Hypothesis.lThe null hypothesis is

7、 the hypothesis to be tested.lit is the hypothesis that the researcher wants to test and is basis forthe selection of the test statistics.ØAlternative hypothesis , designated Ha , is what we conclude if there issufficient evidence to reject the null hypothesis.It is usually thealternative hypot

8、hesis that we are really trying to assess.6-941. Basic ConceptsØ One-tailed test vs. Two-tailed testl One-tailed testH0: 0H0: 0l Two-tailed testH0: =0Ha: <0Ha: >0Ha: 07-942. Type I and Type II ErrorsØØType I error: reject the null hypothesis when its actually true 拒真Type II erro

9、r: fail to reject the null hypothesis when its actually false 取偽Significance level (): the probability of making a Type I errorSignificance level =P (Type I error)Power of a test: the probability of correctly rejecting the null hypothesis when it is falsePower of a test = 1P (Type II error)All else

10、being equal, the higher significance level, the probability of Type I error is higher, the probability of Type II error is lower.The only way to simultaneously reduce the probabilities of bothType I and type II errors is to increase the size of the sample.ØØØØ8-943. Test Statisti

11、c and Critical ValueØA test statistic is a quantity, calculated on the basis of a sample, whosevalue is the basis for deciding whether to reject or not reject the null hypothesis.sample statistic -hypothesized valuetest statistic =standard error of the sample statisticØCritical valuel The

12、distribution of test statistic (z, t, ², F)l Significance level ()l One-tailed or two-tailed test9-944. Decision RuleReject H0Fail to reject H0H0 :u=u0Reject H0Fail to reject H0H0 :u>=u0Reject H0Ø We can never say “accept” H0Ø State the: is (not) significantly different from 010-94

13、5. Definition of p-ValueØp-value: the smallest significance level for which the null hypothesis can be rejectedFor one-tailed tests, the p-value is the probability that lies above the computed test statistic for upper tailed tests or below the computed test statistic for lower tailed testsFor t

14、wo-tailed tests, the p-value is the probability that lies above the positive value of the computed test statistic plus the probability that lies below the negative value of the computed test statisticp- value decision rule:l Reject H0 if p-value<l Fail to reject H0 if p-value>ØØØ

15、;11-94FrameworkØCorrelation and Regression1.2.3.4.5.6.7.8.9.Scatter PlotsCovariance and Correlation Interpretations of Correlation Coefficients Significance Test of the Correlation Limitations to Correlation AnalysisThe Basics of Simple Linear Regression Interpretation of regression coefficient

16、sStandard Error of Estimate & Coefficient of Determination (R2)Analysis of Variance (ANOVA)10.Regression coefficient confidence interval 11.Hypothesis Testing about the Regression Coefficient 12.Predicted Value of the Dependent Variable13.Limitations of Regression Analysis12-941. Scatter Plots&#

17、216; A scatter plots is a graph that shows the relationship betweenthe observations for two data series in two dimensions.13-942. Covariance and CorrelationØCovariance:l Covariance measures how one random variable moves with anotherrandom variable. -It captures the linear relationship.nCov( X ,

18、Y ) = å( Xi - X )(Yi - Y ) /(n -1)i=1ll Covariance ranges from negative infito positive infir = Cov( X ,Y )ØCorrelation:sx syl Correlation measures the linear relationship between two randomvariablesl Correlation has no units, ranges from 1 to +114-943. Interpretations of Correlation Coeff

19、icientsØ The correlation coefficient is a measure of linear association.Ø It is a simple number with no unit of measurement attached, so the correlation coefficient is much easier to explain than the covariance.15-94Correlation coefficientInterpretationr = +1perfect positive correlation0 &

20、lt; r < +1positive linear correlationr = 0no linear correlation1 < r < 0negative linear correlationr = 1perfect negative correlation3. Interpretations of Correlation Coefficients16-944. Significance Test of the CorrelationØ Test whether the correlation between the population of two var

21、iables is equal to zero.Ø H0: =0Ø t-testrn-2 ,t =df= n-21-r2Ø Two-tailed testØ Decision rule: reject H0if +t critical <t, or t<- t critical17-944. Significance Test of the CorrelationExample:The covariance between X and Y is 16. The standard deviation of X is 4 and thestand

22、ard deviation of Y is 8. The sample size is 20. Test the significance of the correlation coefficient at the 5% significance level.Answer : The sample correlation coefficientr = 16/(4×8) = 0.520 - 2The t-statistic can be computed ast = 0.5´= 2.451- 0.25The critical t-value for =5%, two-tail

23、ed test with df=18 is 2.101.Since the test statistic of 2.45 is larger than the critical value of 2.101, wehave sufficient evidence to reject the null hypothesis. So we can say that thecorrelation coefficient between X and Y is significantly different from zero.18-945. Limitations to Correlation Ana

24、lysisØ Outliers (異常值)l Outliers represent a few extreme values for sample observations. Relative to the rest of the sample data, the value of an outlier may be extraordinarily large or small.l Outlier can result in apparent statistical evidence that a significant relationship exists when, in fa

25、ct, there is none, or that there is no relationship when, in fact, there is arelationship.19-945. Limitations to Correlation AnalysisØ Spurious correlationlSpurious correlation refers to the appearance of a causal linear relationship when, in fact, there is no relation. Certain data items may b

26、e highly correlated purely by chance. That is to say, there is no economic explanation for the relationship, which would be considered a spurious correlation.1) correlation between two variables that reflects chance relationships in a particular data set, 2) correlation induced by a calculation that

27、 mixes each of two variables with a third (two variables that are uncorrelated may be correlated if divided by a third variable.), and 3) correlation between two variables arising not from a directrelation between them but from their relation to a third variable. (height may belpositively correlated

28、 with the extent of a's vocabulary)20-945. Limitations to Correlation AnalysisØ Nonlinear relationshipsl Correlation only measures the linear relationship between two variables, so it dose not capture strong nonlinear relationships between variables.l For example, two variables could have a

29、 nonlinear relationshipsuch as Y= (1-X) 3 and the correlation coefficient would be close to zero, which is a limitation of correlation analysis.21-946. The Basics of Simple Linear RegressionØLinear regression allows you to use one variable to make predictions about another, test hypotheses abou

30、t the relation between two variables, and quantify the strength of the relationship between the two variables.Linear regression assumes a linear relation between the dependent andthe independent variables.ØlThe dependent variable is the variable whose variation is explained by the independent v

31、ariable. The dependent variable is also refer to as the explained variable, the endogenous variable, or the predicted variable.The independent variable is the variable whose variation is used to explain the variation of the dependent variable. The independent variable is also refer to as the explana

32、tory variable, the exogenous variable, or the predicting variable.ylx22-946. The Basics of Simple Linear RegressionØ The simple linear regression mYi = b0 + b1 Xi + ei ,i = 1,., nØ Where,Yi = ith observation of the dependent variable, Y Xi = ith observation of the independent variable, X b

33、0 = regression intercept termb1 = regression slope coefficienti= the residual for the ith observation (also referred to as the disturbance term or error term)23-946. The Basics of Simple Linear RegressionØThe assumptions of the linear regressionl A linear relationship exists between X and Yl X

34、is not random, and the condition that X is uncorrelated with the error term can substitute the condition that X is not random.l The expected value of the error term is zero (i.e., E(i)=0 )l The variance of the error term is constant (i.e., the error terms arehomoskedastic)l The error term is uncorre

35、lated across observations (i.e., E(ij)=0for all ij)l The error term is normally distributed.24-947. Interpretation of regression coefficientsØInterpretation of regression coefficientsl The estimated intercept coefficient ( b ) is interpreted as the value0of Y when X is equal to zero.l The estim

36、ated slope coefficient ( b) defines the sensitivity of Y to1ba change in X .The estimated slope coefficient (covariance divided by variance of X.) equals1n - Y )å( X i - X )(YiCov( X ,Y )Var ( X )b= Y - b Xb = i=1101n å( X- X )2ii=1ØExamplel An estimated slope coefficient of 2 would i

37、ndicate that the dependent variable will change two units for every 1 unit change in the independent variable.l The intercept term of 2% can be interpreted to mean that the independent variable is zero, the dependent variable is 2%.25-94An example: calculate a regression coefficientØBouvier Co.

38、 is a Canadian company that sells forestry products to several Pacific Rim customers. Bouviers sales are very sensitive to exchange rates. The following table shows recent annual sales (in millions of Canadian dollars) and the average exchange rate for the year (expressed as the units of foreign cur

39、rency needed to buy one Canadian dollar).Year iXi = Exchange RateYi= Sales1234560.400.360.420.310.330.34202516303530ØCalculate the intercept and coefficient for an estimated linear regression with the exchange rate as the independent variable and sales as the dependent variable.26-94An answer:

40、calculate a regression coefficientThe following table provides several useful calculations:Year iXi = Exchange RateYi= Sales(X -X)2(Y -Y)2(X -X)(Y -Y)iiii123456Sum0.40.360.420.310.330.342.1620250.001600.00360.00250.00090.00040.009361100168116250-0.240-0.6-0.2-0.27-0.08-1.3927-94Answer: calculate a r

41、egression coefficientØThe sample mean of the exchange rate is:X = å Xi / n = 2.16 / 6 = 0.36i=1The sample mean of sales is:Y = åYi / n = 156 / 6 = 26i=1nØnØWe want to estimate a regression equation of the form Yi= b0+ b1Xi+i.The estimates of the slope coefficient and the int

42、ercept aren()()åY -YX -Xiib= i=1=1n ()å2X -Xii=1= Y - b1X = 26 - (-15b0ØSo the regression equation is Yi = 81.6 154.444Xi28-948. Standard Error of Estimate & Coefficient of Determination (R2)ØStandard Error of Estimate (SEE) measures the degree of variability of the actual Y-

43、values relative to the estimated Y-values from a regression equation.SEE will be low (relative to total variability) if the relationship is very strong and high if the relationship is weak.The SEE gauges the “fit” of the regression line. The smaller thestandard error, the better the fit.The SEE is t

44、he standard deviation of the error terms in the regression.ØØØ29-948. Standard Error of Estimate & Coefficient Determination (R2)Ø The Coefficient Determination (R2) is defined as the percentage of the total variation in the dependent variable explained by theindependent vari

45、able.Ø Example: R2 of 0.63 indicates that the variation of the independentvariable explains 63% of the variation in the dependent variable.30-949. ANOVA TableØ ANOVA TableSSEØ Standard error of estimateSEE =MSEn - 2Ø Coefficient of determination (R²)= SSR = 1- SSEØR2SST

46、SST= explained variation =1- unexplained variationtotal variationtotal variationØ For simple linear regression, R²is equal to the squared correlation coefficient (i.e., R²= r²)31-94dfSSMSSRegressionk=1RSSMSR=RSS/kErrorn-2SSEMSE=SSE/(n-2)Totaln-1SST-Example:ØAn analyst ran a

47、regression and got the following result:ØØØØØFill in the blanks of the ANOVA Table. What is the standard error of estimate?What is the result of the slope coefficient significance test? What is the result of the sample correlation?What is the 95% confidence interval of the s

48、lope coefficient?32-94ANOVA TabledfSSMSSRegression18000?Error?2000?Total51?-Coefficientt-statisticp-valueIntercept-0.5-0.910.18Slope212.00<0.00110. Regression coefficient confidence intervalØRegression coefficient confidence intervalb± t s 查表所得1cb1ØIf the confidence interval at the

49、 desired level of significance dose not include zero, the null is rejected, and the coefficient is said to be statistically differentfrom zero.s bs Øis the standard error of the regression coefficient. As SEE rises,alsob11increases, and the confidence interval widens because SEE measures thevar

50、iability of the data about the regression line, and the more variable the data,the less confidence there is in the regression mto estimate a coefficient.33-9411. Hypothesis Testing about the Regression CoefficientØSignificance test for a regression coefficientl H0: b1=The hypothesized valueb- b

51、t =l 11df=n-2sb1l Decision rule: reject H0 if +t critical <t, or t<- tcriticall Rejection of the null means that the slope coefficient is differentfrom the hypothesized value of b34-9412. Predicted Value of the Dependent VariableØPredicted values are values of the dependent variable based

52、 on the estimated regression coefficients and a prediction about the value of the independent variable.Point estimateØY = b + b X'01ØConfidence interval estimateY ± (t ´ s )cftcs f= the critical t-value with df=n2= the standard error of the forecast1( X ' - X )2( X '

53、- X )21s f = SEE ´1+n= SEE ´1+nå( X(n -1)s2- X )2Xi35-9413. Limitations of Regression AnalysisØRegression relations change over timel This means that the estimation equation based on data from a specific time period may not be relevant for forecasts or predictions in another time

54、 period. This is referred to as parameter instability.The usefulness will be limited if others are also aware of and act on the relationship.Regression assumptions are violatedl For example, the regression assumptions are violated if the data isheteroskedastic (non-constant variance of the error ter

55、ms) or exhibits autocorrelation (error terms are not independent).ØØ36-94Summary of Readings and FrameworkSS 3Ø R9 Correlation and regressionØ R11 Time-series analysisØ R12 Excerpt from Probabilistic Approaches: Scenario Analysis, Decision Trees, and Simulation37-94Ø R10 Multiple regression and issues in regression analysisFrameworkØMultiple Regression1.2.3.4.5.6.7.8.9.The Basics of Multiple Regression Interpreting the Multiple Regression ResultsHypothesis Testing about the Regression Coefficient Regression Coefficient F-t