2016 Wiley. Study Session 2: Ethical and Professional Standards Application

Transcription

1 6 Wley Study Sesson : Ethcal and Professonal Standards Applcaton

2 LESSON : CORRECTION ANALYSIS Readng 9: Correlaton and Regresson LOS 9a: Calculate and nterpret a sample covarance and a sample correlaton coeffcent and nterpret a scatter plot. Vol, pp 56 6 Two of the most popular methods for examnng how two sets of data are related are scatter plots and correlaton analyss. Scatter Plots A scatter plot s a graph that llustrates the relatonshp between observatons of two data seres n two dmensons. See Example -. Example -: Scatter Plot The followng table lsts average observatons of annual money supply growth and nflaton rates for 6 countres over the perod 99 to. Illustrate the data on a scatter plot and comment on the relatonshp. Country Money Supply Growth Rate (X ) Inflaton Rate (Y ) A B C D E.5.85 F Fgure -: Scatter Plot Inflaton Rate (%) Money Supply Growth Rate (%) Note that each observaton n the scatter plot s represented as a pont, and the ponts are not connected. The scatter plot does not show whch pont relates to whch country; t just plots the observatons of both data seres as pars. The data plotted n Fgure - suggests a farly strong lnear relatonshp wth a postve slope for the countres n our sample over the sample perod. 6 Wley 35

3 Correlaton Analyss Correlaton analyss expresses the relatonshp between two data seres n a sngle number. The correlaton coeffcent measures how closely two data seres are related. More formally, t measures the strength and drecton of the lnear relatonshp between two random varables. The correlaton coeffcent can have a maxmum value of + and a mnmum value of. A correlaton coeffcent greater than means that when one varable ncreases (decreases) the other tends to ncrease (decrease) as well. A correlaton coeffcent less than means that when one varable ncreases (decreases) the other tends to decrease (ncrease). A correlaton coeffcent of ndcates that no lnear relaton exsts between the two varables. Fgures -, -3, and - llustrate the scatter plots for data sets wth dfferent correlatons. Fgure -: Scatter Plot of Varables wth Correlaton of + Varable Y Varable X Analyss: Note that all the ponts on the scatter plot llustratng the relatonshp between the two varables le along a straght lne. The slope (gradent) of the lne equals +.6, whch means that whenever the ndependent varable (X) ncreases by unt, the dependent varable (Y) ncreases by.6 unts. If the slope of the lne (on whch all the data ponts le) were dfferent (from +.6), but postve, the correlaton between the two varables would equal + as long as the ponts le on a straght lne Wley

4 Fgure -3: Scatter Plot of Varables wth Correlaton of Varable Y Varable X Analyss: Note that all the ponts on the scatter plot llustratng the relatonshp between the two varables le along a straght lne. The slope (gradent) of the lne equals.6, whch means that whenever the ndependent varable (X) ncreases by unt, the dependent varable (Y) decreases by.6 unts. If the slope of the lne (on whch all the data ponts le) were dfferent (from.6) but negatve, the correlaton between the two varables would equal as long as all the ponts le on a straght lne. Fgure -: Scatter Plot of Varables wth Correlaton of Varable Y Varable X Analyss: Note that the two varables exhbt no lnear relaton. The value of the ndependent varable (X) tells us nothng about the value of the dependent varable (Y). 6 Wley 37

5 Calculatng and Interpretng the Correlaton Coeffcent In order to calculate the correlaton coeffcent, we frst need to calculate covarance. Covarance s a smlar concept to varance. The dfference les n the fact that varance measures how a random varable vares wth tself, whle covarance measures how a random varable vares wth another random varable. Propertes of Covarance Covarance s symmetrc, that s, Cov(X, Y) = Cov(Y, X). The covarance of X wth tself, Cov(X, X), equals the varance of X, Var(X). Interpretng the Covarance Bascally, covarance measures the nature of the relatonshp between two varables. When the covarance between two varables s negatve, t means that they tend to move n opposte drectons. When the covarance between two varables s postve, t means that they tend to move n the same drecton. The covarance between two varables equals zero f they are not related. Sample covarance s calculated as: Sample covarance = Cov( XY, ) = ( X X)( Y Y)/( n ) n = where: n = sample sze X = th observaton of Varable X X = mean observaton of Varable X Y = th observaton of Varable Y Y = mean observaton of Varable Y The numercal value of sample covarance s not very meanngful as t s presented n terms of unts squared, and can range from negatve nfnty to postve nfnty. To crcumvent these problems, the covarance s standardzed by dvdng t by the product of the standard devatons of the two varables. Ths standardzed measure s known as the sample correlaton coeffcent (denoted by r) and s easy to nterpret as t always les between and +, and has no unt of measurement attached. See Example -. Cov( XY, ) Sample correlatoncoeffcent = r = sxsy 38 6 Wley

6 n X = Sample varance = s = ( X X) /( n ) Sample standard devaton = s X = s X Example -: Calculatng the Correlaton Coeffcent Usng the money supply growth and nflaton data from 99 to for the 6 countres n Example -, calculate the covarance and the correlaton coeffcent. Soluton: Country Money Supply Growth Rate (X ) Inflaton Rate (Y ) Cross Product ( X X)( Y Y) Squared Devatons ( X X) Squared Devatons ( Y Y) A B C D E F Sum Average..78 Covarance.77 Varance Std. Dev (s) Illustratons of Calculatons Covarance = Sum of cross products / n =.3735/5 =.77 Var (X) = Sum of squared devatons from the sample mean / n =.9/5 =.98 Var (Y) = Sum of squared devatons from the sample mean / n =.39/5 =.69 Cov( XY, ).77 Correlatoncoeffcent = r = = =.9573 or 95.73% sxsy (.3373)(.87) The correlaton coeffcent of.9573 suggests that over the perod, a strong lnear relatonshp exsts between the money supply growth rate and the nflaton rate for the countres n the sample. Note that computed correlaton coeffcents are only vald f the means and varances of X and Y, as well as the covarance of X and Y, are fnte and constant. 6 Wley 39

7 LOS 9b: Descrbe lmtatons to correlaton analyss. Vol, pp 6 65 Lmtatons of Correlaton Analyss It s mportant to remember that the correlaton s a measure of lnear assocaton. Two varables can be connected through a very strong nonlnear relaton and stll exhbt low correlaton. For example, the equaton Y = + 3X represents a lnear relatonshp. However, two varables may be perfectly lnked by a nonlnear equaton, for example, Y = (5 + X) but ther correlaton coeffcent may stll be close to. Correlaton may be an unrelable measure when there are outlers n one or both of the seres. Outlers are a small number of observatons that are markedly numercally dfferent from the rest of the observatons n the sample. Analysts must evaluate whether outlers represent relevant nformaton about the assocaton between the varables (news) and therefore, should be ncluded n the analyss, or whether they do not contan nformaton relevant to the analyss (nose) and should be excluded. Correlaton does not mply causaton. Even f two varables exhbt hgh correlaton, t does not mean that certan values of one varable brng about the occurrence of certan values of the other. Correlatons may be spurous n that they may hghlght relatonshps that are msleadng. For example, a study may hghlght a statstcally sgnfcant relatonshp between the number of snowy days n December and stock market performance. Ths relatonshp obvously has no economc explanaton. The term spurous correlaton s used to refer to relatonshps where: Correlaton reflects chance relatonshps n a data set. Correlaton s nduced by a calculaton that mxes the two varables wth a thrd. Correlaton between two varables arses from both the varables beng drectly related to a thrd varable. LOS 9c: Formulate a test of the hypothess that the populaton correlaton coeffcent equals zero and determne whether the hypothess s rejected at a gven level of sgnfcance. Vol, pp Testng the Sgnfcance of the Correlaton Coeffcent Hypothess tests allow us to evaluate whether apparent relatonshps between varables are caused by chance. If the relatonshp s not the result of chance, the parameters of the relatonshp can be used to make predctons about one varable based on the other. Let s go back to Example -, where we calculated that the correlaton coeffcent between the money supply growth rate and nflaton rate was Ths number seems pretty hgh, but s t statstcally dfferent from zero? In order to use the t test, we assume that the two populatons are normally dstrbuted. ρ represents the populaton correlaton. To test whether the correlaton between two varables s sgnfcantly dfferent from zero the hypotheses are structured as follows: H : ρ = H a : ρ Note: Ths would be a two taled t test wth n degrees of freedom. 6 Wley

8 The test statstc s calculated as: r n Test-stat = t = r where: n = Number of observatons r = Sample correlaton The decson rule for the test s that we reject H f t stat > +t crt or f t stat < t crt From the expresson for the test statstc above, notce that the value of sample correlaton, r, requred to reject the null hypothess, decreases as sample sze, n, ncreases: As n ncreases, the degrees of freedom also ncrease, whch results n the absolute crtcal value for the test (t crt ) fallng and the rejecton regon for the hypothess test ncreasng n sze. The absolute value of the numerator (n calculatng the test statstc) ncreases wth hgher values of n, whch results n hgher t values. Ths ncreases the lkelhood of the test statstc exceedng the absolute value of t crt and therefore, ncreases the chances of rejectng the null hypothess. See Example -3. Example -3: Testng the Correlaton between Money Supply Growth and Inflaton Based on the data provded n Example -, we determned that the correlaton coeffcent between money supply growth and nflaton durng the perod 99 to for the sx countres studed was Test the null hypothess that the true populaton correlaton coeffcent equals at the 5% sgnfcant level. Soluton: Test statstc = = Degrees of freedom = 6 = The crtcal t values for a two taled test at the 5% sgnfcance level (.5% n each tal) and degrees of freedom are.776 and Snce the test statstc (6.63) s greater than the upper crtcal value (+.776) we can reject the null hypothess of no correlaton at the 5% sgnfcance level. 6 Wley

9 From the addtonal examples n the CFA Program Currculum (Examples 3- and -) you should understand the takeaways lsted below. If you understand the math behnd the computaton of the test statstc, and the determnaton of the rejecton regon for hypothess tests, you should be able to dgest the followng ponts qute comfortably: All other factors constant, a false null hypothess (H : ρ = ) s more lkely to be rejected as we ncrease the sample sze due to () lower and lower absolute values of t crt and () hgher absolute values of t test stats. The smaller the sze of the sample, the greater the value of sample correlaton requred to reject the null hypothess of zero correlaton (n order to make the value of the test statstc suffcently large so that t exceeds the absolute value of t crt at the gven level of sgnfcance). When the relaton between two varables s very strong, a false null hypothess (H : ρ = ) may be rejected wth a relatvely small sample sze (as r would be suffcently large to push the test statstc beyond the absolute value of t crt ). Note that ths s the case n Example -3. Wth large sample szes, even relatvely small correlaton coeffcents can be sgnfcantly dfferent from zero (as a hgh value of n ncreases the absolute value of the test statstc and reduces the absolute value of the crtcal value for the hypothess test). Uses of Correlaton Analyss Correlaton analyss s used for: Investment analyss (e.g., evaluatng the accuracy of nflaton forecasts n order to apply the forecasts n predctng asset prces). Identfyng approprate benchmarks n the evaluaton of portfolo manager performance. Identfyng approprate avenues for effectve dversfcaton of nvestment portfolos. Evaluatng the approprateness of usng other measures (e.g., net ncome) as proxes for cash flow n fnancal statement analyss. LESSON : LINEAR REGRESSION LOS 9d: Dstngush between the dependent and ndependent varables n a lnear regresson. Vol, pp 76 8 Another way to look at smple lnear regresson s that t ams to explan the varaton n the dependent varable n terms of the varaton n the ndependent varable. Note that varaton refers to the extent that a varable devates from ts mean value. Do not confuse varaton wth varance. Lnear Regresson wth One Independent Varable Lnear regresson s used to summarze the relatonshp between two varables that are lnearly related. It s used to make predctons about a dependent varable, Y (also known as the explaned varable, endogenous varable, and predcted varable) usng an ndependent varable, X (also known as the explanatory varable, exogenous varable, and predctng varable), to test hypotheses regardng the relaton between the two varables, and to evaluate the strength of the relatonshp between them. The dependent varable s the varable whose varaton we are seekng to explan, whle the ndependent varable s the varable that s used to explan the varaton n the dependent varable. 6 Wley

10 The followng lnear regresson model descrbes the relaton between the dependent and the ndependent varables. Regresson modelequaton = Y = b + bx +ε, =,., n where: b and b are the regresson coeffcents. b s the slope coeffcent. b s the ntercept term. ε s the error term that represents the varaton n the dependent varable that s not explaned by the ndependent varable. Based on ths model, the regresson process estmates the lne of best ft for the data n the sample. The regresson lne takes the followng form: Regresson lneequaton = Yˆ = bˆ + bx ˆ, =,..., n Lnear regresson computes the lne of best ft that mnmzes the sum of the squared regresson resduals (the squared vertcal dstances between actual observatons of the dependent varable and the regresson lne). What ths means s that t looks to obtan estmates, ˆb and ˆb, for b and b respectvely, that mnmze the sum of the squared dfferences between the actual values of Y, Y, and the predcted values of Y, Y ˆ, accordng to the regresson equaton ( Y ˆ ˆ ˆ = b + bx ). Therefore, lnear regresson looks to mnmze the expresson: n = [ Y (ˆ b + bx ˆ )] Hats over the symbols for regresson coeffcents ndcate estmated values. Note that t s these estmates that are used to conduct hypothess tests and to make predctons about the dependent varable. where: Y = Actual value of the dependent varable bˆ bˆ + X =Predcted value of dependent varable The sum of the squared dfferences between actual and predcted values of Y s known as the sum of squared errors, or SSE. 6 Wley 3