Business Statistics. University of Chicago Booth School of Business Fall Jeffrey R. Russell

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1 Business Statistics University of Chicago Booth School of Business Fall 08 Jeffrey R. Russell There is no text book for the course. You may choose to pick up a copy of Statistics for Business and Economics by Newbold Carlson and Thorne (8 th edition) if you feel you would like an outside reference. The course web page will provide our primary communication outside of class: Homework will be given the first lecture of each week and is due the first lecture of the following week.

2 Student Evaluation Grades will be determined by 0% homework, 30% midterm, 60% final exam. Homework due at start of class. You may work in groups of size 3 or less. You may bring a 8.5x cheat sheet to the midterm (onesided) and to the final exam (two sided). Re grade requests should be made in writing within one week of the return of the midterm. It is expected that if you sign up for this section of b stats you will be available for both the midterm and final exam times. Review Sessions Thursdays 3:00 3:50 Harper 3B Thursdays 5:00 5:50 Booth 455 room 3 Review sessions don t meet 9/7,/, and /3. Support for Excel, but you can use any stat package you would like. Time split between computing and questions on material.

3 I will provide real time examples in Excel. You are free to use any software, however, to complete the homework. What is Statistics? Data Real world Models. This is the raw information. Computerization => Lots of data!. Are simplified representations of the real world data.. The allow us to concisely summarize the real world activity 3. They allow us to make predictions about what might happen 3

4 Of course no model can exactly explain real world phenomenon. There is inherent randomness in everything. This introduces challenges in how we chose which model is a good model. Our models will have a random component to capture randomness or uncertainty. Statistics is the link between the models and the data. Data Real world Statistics Models Capture Uncertainty Models make statements about how the world should look.. Estimation: We can use the observed data to figure out which models are most consistent with the data.. Testing: Conversely the right model should make accurate predictions about how the real world data should look. If the models predictions don t look like the real world data, then we can conclude that the model is not a good one. 4

5 Course structure Data Real world Statistics Models Capture Uncertainty. We begin with a discussion of summarizing real world data (means, variances, covariances, graphical summaries etc ).. We next introduce models. Since the real world is uncertain our models will involve uncertainty/probability. 3. We then combine the data with the models to discuss estimation and testing. We begin with very simple models and move to more complex models. 4. We finish with applying these ideas to the regression model..descriptive Statistics. Dotplot and Histogram. The Time Series Plot.3 The Mean and Median.4 Summation Notation.5 The Sample Variance and Standard Deviation.6 The Empirical Rule.7 Other Statistics.8 Covariance and Correlation.9 Linear Functions.0 Mean and Variance of a Linear Function. Linear Combinations. Portfolios.3 Mean and Variance of a Linear Combination.4 Statistics as Estimates (intro) 5

6 . Dotplot and Histogram Let s look at some returns data. Simple Returns The return on an asset is the percentage increase in wealth invested in the asset over a given time period. If you invest B at the beginning of the time period you get E = B+rB=(+r)B at the end of the time period, where r is the return. Clearly, given E and B we can calculate r: r = (E B)/B The Canadian Returns Data Here are 07 monthly returns on a broadbased portfolio of Canadian assets (more on portfolios later). canada Each number corresponds to a month. They are given in time order (go across rows first). Our first observation is.07. In the first month, the return was.07, in the th,.03. 6

7 The Dotplot To display the returns we ll first use a dotplot. For each number simply place a dot above the corresponding point on the number line Canada Returns The returns are centered or located at about.0 The spread or variation in the returns is huge. We also have data on countries other than Canada. Let s compare Canada with Japan. Character Dotplot. : : : : : :. : : : :. : : : : : : : : : : : : : : : : : : : : : : : : :. : : : : : : : : : : : : : : : : canada ::.. :.. :::.:: :.:. : :::.::: :::: : : :.:: :::: :::: :::: : :: : :. : japan

8 It really helps to get things on the same scale. How is Japan different from Canada?. : : : :: :.::: :.: : : :::: :::: ::: :::: :::: :::. : :::: :::: :::: ::: canada.. ::.. :.. :::.:: :.:. : :::.::: :::: : : :.:: :::: :::: :::: : :: : :. : japan Mutual Funds Data Let s use the dotplot to compare returns on some other kinds of assets. We ll look at returns on two different mutual funds, the equally weighted market, and T bills. The equally weighted market is returns on a portfolio where you spread you money out equally over a wide variety of stocks. 8

9 Data on 4 different kinds of returns: Dreyfus growth fund Putman income fund (note that each dot is now points) Equally weighted market T bills (the risk free asset)..:.. ::.:: : : :::::.:. ::::::: ::: ::::::::::: :::::::::::::: ::::::::::::::::..... ::::::::::::::::::: drefus Each dot represents points.. ::..::: :::: ::::: : :::::: :.::::::.::...::::::::::: Putnminc. : : :.: : : :: : : :: :.: ::.:::.::.:...:.::::::::::::::.::::::::::::::::::: :.. :.:::::::::::::::::::::.:.: eqmrkt Each dot represents 7 points : :: :: :: :: :: :: tbill Beer Data nbeerm: the number of beers male mba students claim they can drink without getting drunk nbeerf: same for females Character Dotplot We call a point like this an outlier. : : : :.. : : : :.. :. : : :.: : : : nbeerm.... : : nbeerf Generally the males claim more, their numbers are centered or located at larger values. 9

10 The Histogram Sometimes the dotplot can look rather jumpy. The histogram gives us a smoother picture of the data. The height of each bar tells us how many observations are in the corresponding interval. nbeerf women have number of beers between.5 and women have number of beers in the interval (.5,4.5) Histogram for nbeerf <= >4.5 nbeerf The choice of bin width determines the degree of resolution. A large bin With is very smooth (flat) with no detail. A small bin has more detail, but maybe too much. Here is the histogram of the Canadian returns. Frequency canada Here it is again with the bin width half the size of the one above. Frequency canada 0. There is no special way of determining the bin width. I use my eye. 0

11 . The Time Series Plot We just looked at two data kinds of data. the returns data the number of beers For the returns data, each number corresponds to a month. For the beers data, each number corresponds to a person. The returns data has an important feature the beer data does not. It has an order! There is a first one, a second one, and... A sequence of observations taken over time is called a time series. We could have daily data (closing price stock price) quarterly data (some macro data, like GDP) Monthly data (real estate index prices) Daily data (like daily stock returns) Trade by trade transaction prices. For time series data, the time series plot is an important way to look at our data.

12 The returns data, time series plot of the Canadian returns: Time series plot of canada On the vertical axis we have the return. canada On the horizontal axis we have time Do you see a pattern? date Its also interesting to plot multiple time series on the same plot for comparison.

13 Chicago Household income and Home Prices INCNEW HOME_PRICE.3 Summarizing a Single Numeric Variable We have looked at graphs. Suppose we are now interested in having numerical summaries of the data rather than graphical representations. Two important features of any numeric variable are: ) What is a typical or average value? ) How spread out or variable are the values? 3

14 Monthly returns on Canadian portfolio and Japanese portfolio. They seem to be centered roughly at the same place but Japan has more spread. How can we summarize this? =AVERAGE(nbeerf) 4. =AVERAGE(nbeerm) In the picture, I think of the mean as the center of the data. Character Dotplot.... : : nbeerf. 4. : : : :.. : : : :.. :. : : :. : : : : nbeerm

15 Let s compare the means of the Canadian and Japanese returns. AVERAGE canada AVERAGE japan This is a big difference. It was hard to see this difference in the dotplots because the difference is small compared to the variation. The median is another measure of central tendency Arrange the data in ascending order. After this ranking, the middle number in the data set is the median. If there are an even number of data points then the median is the average of the two center numbers. Example : Median=7 Example 4 5 8: Median=4.5 5

16 Unlike the mean, the median is not affected by outliers. Consider the two data sets: : Median=7 Mean= : Median=7 Mean=4 Mean and Median Comparison The median is often nice to report when there are extreme values in the data Think about US household income (05). Mean=$7,000 Bill Gates Median = $58,476 Income (000 s) 6

17 .4 Summation Notation We are going to be doing a lot of adding up and averaging. To talk about this in general we need some special notation. First of all, we denote a general set of numbers by: x, x, x 3,x n the first number the last number of data points, or the sample size. In general the average of the numbers x is: x x x n n mean x xi n n i We often use the numbers x. x symbol to denote the mean of the We call it x bar. 7

18 Let s look summation in more detail. n i x i means, for each value of i from to n add to the sum the value indicated, in this case x i. add in this value for each i Example: Suppose we have data x and y with n=4. Think of each row as an observation of both x and y. To make things concrete, think of each row as corresponding to a year and let x and y be annual returns on two different assets. x y year In year asset x had return 7%. In year 4 asset y had return 3%. 8

19 The average 4 xi x i 4 4 y yi i But we can do lots of other things with summation notation: x y year xi x i The Sample Variance and Standard Deviation Character Dotplot of x and y: x y The y numbers are more spread out than the x numbers. We want a numerical measure of variation or spread. 9

20 To examine the variation of the data x, we look at: x i x A lot of variation means these numbers are big. In our example, x x x y y y Character Dotplot x x y y

21 Now we need an overall measure of how big the differences are. We can t just sum them because the negative distances cancel out the positive ones. We average the squared distances. n n i ( x x) i For technical reasons (we ll see why later) the sample variance of the data x is defined to be: Sample Variance: s x n ( xi x) n i With large n, there is little difference between dividing by n and (n ). Think of it as the average squared distance from the mean (center).

22 What is the smallest value a variance can be? What are the units of the variance?? It will be helpful to have a measure of spread which is in the original units. The sample standard deviation is: Sample Standard Deviation: s x s x The units of the standard deviation are the same as those of the original data.

23 Let s try these formulas on our example. s y n i n i y y x x x y y y s y The sample standard deviation for the y data is bigger than that of the x data. This numerically captures the fact that y has more variation about it s mean than x. s s x x n i n i x x.0.0 3

24 Returns Example the standard deviations Character Dotplot measure the fact that there is more spread in the Japanese. returns : : : :: :.::: :.: : : :::: :::: ::: :::: :::: :::. : :::: :::: :::: ::: canada.. ::.. :.. :::.:: :.:. : :::.::: :::: : : :.:: :::: :::: :::: : :: : :. : japan Summary measures for selected variables canada japan Count Mean Standard deviation The Empirical Rule We now have the two summaries x s x where the data is The mean is pretty easy to understand. What are the units? how spread out, or variable the data is We know that the bigger s x is, the more variable the data is, but how do we interpret the number? What is a big s x, what is a small one? What are the units of s x? 4

25 The empirical rule will help us understand s x and relate the summaries back to the dotplot (or histogram). Empirical rule: For mound shaped data : Approximately 68% of the data is in the interval ( xs, xs ) xs x x x Approximately 95% of the data is in the interval ( xs, xs ) xs x x x Let s see this with the Canadian returns. x x s x x s x s x The empirical rule says that roughly 95% of the observations are between the dashed line and 68% between the dotted. Looks reasonable. Percent x s x canada x s x 5

26 Comparing Mutual Funds Let s use means and sd s to compare mutual funds. For 9 different assets we compute the mean and sd. Then plot mean vs sd. The assets are: #C - R Drefus (growth) #C- R30 Fidelity Trend fund (growth) #c3- R55 Keystone Speculative fund (max capital gain) #c4- R9 Putnam Income Fund (income) #c5- R99 Scudder Income #c6- R9 Windsor Fund (growth) #c7- equally weighted market #c8- value weighted market #c9- tbill rate # sample period monthly returns : Summary measures for selected variables drefus fidel keystne Putinc scudinc windsr eqmrkt valmrkt tbill Count Mean Standard deviation

27 It is considered good to have a large mean return and a small standard deviation. 0.0 eqmrkt How does this graph relate back to the dotplots we saw before?? Mean tbill windsor drefus valmrkt Putnminc keystne fidel scudinc Standard deviation Let s compare some countries: Based on monthly returns usa singapor france honkong Mean 0.0 canada belgium australi germany finland italy japan Standard deviation 7

28 .7 Other statistics We computed the mean by averaging the data. We computed the variance by essentially taking the average of the squared deviations of the data from the mean. We can take averages of higher powers too like the 3 rd or 4 th power. In doing so, we learn other interesting summary statistics about the data. Consider the S&P500 daily returns 3000 Histogram Frequency Frequency What do you notice about the symmetry? r 8

29 Skewness Skewness another sample statistic that measures the degree of symmetry in the data. Technically, it is computed as: Skew N i N i 3 r r 3 s Where s is the sample standard deviations of the returns and r is the sample average of the returns. When we take the 3 rd power, we retain the sign. When we cube a data point, we make extreme values even more extreme. The skew examines whether there are more extreme value in the right tail or in the left tail. A distribution with a longer right tail will have a positive skew. A distribution with a longer left tail will have a negative skew. Hence, we look at the sign of the skew. 9

30 Longer left tail 3000 Histogram 500 Frequency Frequency More r The skew for this dataset is:.6483 =skew(data) Remember the household income data we spoke about before? That data would have a positive skew because the right tail is longer than the left tail. A perfectly symmetric distribution will have a skew of zero. 30

31 Kurtosis Kurtosis measures how much weight there is in the tails of the distribution relative to the middle (we call this a measure of the fatness of the tails). How is this different from the variance? Let s look at two distributions with the SAME variance, but different relative fatness of the tails Chart Title Series Series Both distributions have a variance of. The Blue distribution has more weight in the middle AND more weight in the tails! 3

32 The kurtosis is defined as: Kurt N i N i r r 4 s 4 The Blue distribution has a kurtosis of 4.5, the red distribution has a kurtosis of 3. The larger the kurtosis the more peaked and fatter the tails of the distribution. The Normal or Gaussian distribution (more to come later) has a Kurtosis of 3. Sometimes we report the Excess Kurtosis. Excess Kurtosis = Kurtosis 3. Hence if the data were normal, the excess kurtosis should be about zero. Distributions with positive excess kurtosis have fatter tails than the Normal. Most returns data have positive excess kurtosis. This is called leptokurtic. A distribution with negative excess kurtosis is called plattykurtic this is rare. 3

33 .8 Covariance and Correlation The mean and sd help us summarize a bunch of numbers which are measurements of just one thing. A fundamental and totally different question is how one thing relates to another. We just used a scatterplot to look at two things: the mean and sd of different assets. In this section of the notes we look at scatterplots and how correlation can be used to summarize them. Example nbeer weight i nbeer # beers without getting drunk vs weight weight Now we think of each pair of numbers as an observation. Each pair corresponds to a person. Each person has two numbers associated with him/her, #beers and weight. 33

34 Example Are returns on a mutual fund related to market returns? 0.4 Each point corresponds to a month. windsor valmrkt In general we have observations ( x, y ) i i the ith observation is a pair of numbers and each point on the plot corresponds to an observation. Our data looks like: x y i The plot enables us to see the relationship between x and y

35 In both examples it does look like there is a relationship. Even more, the relationship looks linear in that it looks like we could draw a line through the plot to capture the pattern. Let me first introduce covariance and correlation then we will consider the interpretation and use of each. The sample covariance between x and y is: s n x x y y n ( )( ) xy i i i The sample correlation between x and y is: r xy s xy ss x y so the correlation is just the covariance divided by the two standard deviations. 35

36 We ll get some intuition about these formulas, but first let s see them in action. How do they summarize data for us? Correlation FACTS: r xy The closer r is to the stronger the linear relationship is with a positive slope. That is, the largest values of y tend to be associated with largest value of x (and vice versa). The closer r is to the stronger the linear relationship is with a negative slope. That is, large values of y tend to be associated with small value of x (and vice versa). The correlations corresponding to the two scatterplots we looked at are: Correlation of valmrkt and windsor = 0.93 Correlation of nbeer and weight = 0.69 The larger correlation between valmrkt and windsor indicates that the linear relationship is stronger. Let s look at some more examples. 36

37 Correlation of y and x = 0.09 y x 3 3 ycorrelation of y and x = x Correlation of y3 and x3 = y x3 3 3 Correlation of y4 and x4 = y x4 3 37

38 What s the correlation here??? Correlation of x and y = 0.00 (basically 0) Correlation only measures the linear relationship. 38

39 Example: The Countries Data Which countries go up and down together? I have data on 3 countries. That would be a lot of plots!!! 0. canada usa 0. To summarize we can compute all pairwise correlations. australi belgium canada finalnd france germany honkong italy belgium 0.89 canada finalnd why is this blank? france germany honkong italy japan usa singapor japan usa usa 0.46 singapor This got wrapped around 39

40 Understanding the Cov and Corr Formulas Why are the formulas for cov and corr capturing the relationship? To get a feeling for this, let s go back to the simple example and compute the correlation. x y First let s compute the covariance: n ( xi x)( yi y) n i (( )(.. 07) ( )( ) ( )( ) ( )( )) 3 (. 0* * (. 0) (. )*. 0 (. 0) * (. 04)) (.... ) (. 00) =.0004 x x x y y y Each of the 4 points makes a contribution to the sum. Let s see which point does what. 40

41 ( x3 x)( y3 y) (. 0)* x ( x x)( y y). 0* y (III) (II) (I) (IV) y x ( x4 x)( y4 y) (. 0)*(. 04). 008 ( x x)( y y). 0*(. 0). 000 Points in (I) have both x and y bigger than their mean so we get a positive contribution. Points in (II) have both less than their mean so we get a positive contribution. In (III) and (IV) one of x and y is less than its mean and the other is greater so we get a negative contribution. The farther out the point is, the bigger the contribution. just a few relatively small contributions Lots of positive contributions windsor valmrkt Lots of positive contributions just a few relatively small contributions 4

42 So, A positive covariance means that when one variable is above it s average the other one tends to be as well. They move up and down together. A negative covariance means that when one is up the other tends to be down. They move in opposite direction. to finish the example, r xy (. 0365)(. 083). 6 The division by the standard deviations standardizes the covariance so that the correlation is always between +/. What are the units of the covariance? What are the units of the correlation? 4

43 .9 Linear Functions In this section of the notes we study the situation where two variables are exactly linear related. That is if I give you the value of x you know exactly what the value of y must be. Example Suppose we have these temps in Celsius and Fahrenheit. cel fahr How are the F values related to the C values? F = 3 + (9/5)C 43

44 .0 Mean and Variance of a Linear Function Suppose y is a linear of function of x. How are the mean and variance (standard deviation) of y related to those of x? Let s look at our temperature example. Information on the Worksheet Summary measures for selected variables cel fahr Suppose we first multiply by (9/5) and then add 3. Count Create a new column, mul, in Excel =(9/5)*cel Let s get some intuition about what happens when we multiply and add numbers to a data set to obtain a new data set. Celsius cel Mul=(9/5)C mul F = 3 + (9/5)C fahr cel fahr mul Mean Standard deviation

45 Suppose then, y c c x 0 y c c x 0 s y c sx s c s y x Example We convert from fractions to percent by taking y 00x What is mean of y compared to x? What is the standard deviation of y compared to? We convert from $ to 000 s of $ by taking y x 000 Now what are the mean and standard deviation of y? s x 45

46 Why? You can verify on your own. y c c x x n i i n y ( c0 cxi ) n 0 i n i c n n c n 0 x i n c x i c x i 0 i s x x n x ( i ) n i s c c x c c x n y ( 0 i 0 ) n i n n i i ( c 0 cx c i 0 cx) n c ( x x) c s n i x. Linear Combinations When a variable y is linearly related to several others, we call it a linear combination. y c0 cxcx ckxk y is a linear combination of the x s. c i is the coefficient of x i. We have k data sets (notation has different meaning here). 46

47 . Portfolios and Linear Combinations Suppose you have $00 to invest. Let x be the return on asset. If x =., and you put all your money into asset you will have $0 at the end of the period. Let x be the return on asset. If x =.5, and you put all your money into asset you will have $5 at the end of the period. Suppose you put / your money into and / into. What will happen? At the end of the period you will have E=E+E where E is the end of period wealth from your investment in asset and E is the end of period wealth from your investment in asset. E=(+.)*.5*00 +(+.5)*.5*00 E E Hence the return on the total investment of $00 is E=(+r)*B=(+.5*.+.5*.5)*00 r r=.5*. +.5*.5=.5. 47

48 To generalize, let w be the fraction of your wealth you invest in asset. Let w be the fraction of your wealth you invest in asset. Let B be the wealth. The w s are called the portfolio weights. What are the restrictions on the weights? Then at the end of the period you have: E x wb x wb w xw w xw B xw xw B r p Hence the portfolio return is, r w x w x p Suppose we have k assets. The return on the ith asset is x i. Put w i fraction of your wealth into asset i.. Your portfolio is determined by the portfolio weights w i. Then the return on the portfolio is: R k wx p i i i What is return on the equally weighted portfolio? 48

49 Example (the country data again) Let s use our country data and suppose that we had put.5 into usa and.5 into Hongkong. What would our returns have been? In Excel, create a new column, entitled port : =.5*(honkong) +.5*(usa) honkong usa port for each month, we get the portfolio return as / hongkong + / usa. How do the returns on this portfolio compare with those of hongkong and usa? It looks like the mean for my portfolio is right in between the means of usa and hongkong. Mean usa port honkong What about the sd? Standard deviation 49

50 Let s try a portfolio with three stocks. The weights must add up to one, but they can be negative, this is called going short in the asset. = -.5*(canada)+ usa +.5*(honkong) 0.0 Clearly, forming portfolios is an interesting thing to do!! Mean usa canada port honkong Standard deviation Why would we form a portfolio? Maybe the port has a nice mean and variance. It would be nice to be able to figure out what the mean and variance are of any portfolio as a function of the portfolio weights. We need some basic equations to relate means and variances of the assets to means and variances of the different possible portfolios obtained by different weights. There are some basic formulas that relate the mean sd of a linear combination to the means, variances, and covariances of the input x variables. 50

51 .3 Mean and Variance of a Linear Combination First we consider the case where we have two inputs. Suppose y c0 cxcx then, y c0 cxcx s c s c s c c s y x x x x Example =.5*(hongkong) +.5*(usa) honkong usa port for each month, we get the portfolio return as / hongkong + / usa. The mean returns on port, usa, and honkong are.074,.0346, and =.5* *.003 5

52 off diagonals are covariances Table of covariances (variances on the diagonal) hongkong usa port hongkong usa port = (.5)*(.5)* (.5)*(.5)*.00 + *(.5)*(.5)*.00 =.5* * *.00 Let s do one more: =.5*(usa)+.75*(hongkong) Table of covariances (variances on the diagonal) honkong usa port honkong usa port = (.5)*(.5)*.00 + (.75)*(.75)*.005+()*(.5)*(.75)*(.0003) 5

53 The variance of the portfolio depends upon the variances of the individual assets as well as the covariance. The covariance tells us something about the relationship between the returns on the individual assets. Why should this relationship matter? We ll look at several examples to get a feel. Example -0. y =.5x +.5 x At each point we plot the value of y. The variances and covariance are: x x x x x The variance of y is - 0 x =.5*.5* *.5*.06 +*.5*.5*( ) Why is the variance of y so much smaller than those of the x s? 53

54 Example y =.5x +.5 x At each point we plot the value of y. The variances and covariance are: x x x.5867 x x x The variance of y is.053 =.5*.5* *.5*.96 + *.5*.5*.0465 Why is the variance of y similar to those of the x s? Example y =.5x +.5 x At each point we plot the value of y. The variances and covariance are: x x x x x The variance of y is x The dashed lines are drawn at the mean of x and x =.5*.5* *.5* *.5*.5*.976 Why is the variance of y less than that of x and x? 54

55 Suppose y c c x c x c x c x then, y c c x c x c x c x k k k k s y c s x c s x... c all the possible combinations of covariance terms k s x k Example y c0 cxcx c3x3 y c0 cxcx c3x3 s c s c s c s y x x 3 x 3 c cs c cs c cs xx 3 xx 3 x x

56 Example =.*(fidel)+.4*(eqmrkt)+.5*(windsor) Table of covariances (variances on the diagonal) fidel windsor eqmrkt port fidel windsor eqmrkt port = (.)*(.)* (.4)*(.4)* (.5)*(.5)* *((.)*(.4)* (.)*(.5)*.004+(.4)*(.5)*.0099) Example By forming portfolios we can reduce variance of returns!!! This means people should hold portfolios!!!! 56

57 Cut from a Finance Textbook: 57

58 .4 Statistics as Estimates Consider the following sample averages of the daily S&P500 returns data using different time periods Sample Mean 980 s s s present The different samples give us different values of the sample mean. If you are thinking of investing in the S&P500 over the next decade, which of these should characterize the rate of return you should expect to get? 58