Money Management Using the Kelly Criterion

Transcription

1 Money Management Using the Kelly Criterion An Application of the Kelly Criterion on an Intraday Trading Strategy Based on the Swedish Stock Market Index OMXS30 Mårten Hagman Student Spring 2014 Master thesis I, 15 ECTS Master s program in Economics, 60/120 ECTS Supervisor: Christian Lundström

2 Abstract This paper highlights the importance of money management. Firstly, we show how an intraday trading strategy based on Swedish stock market index OMXS30 can be developed given market ineffectiveness. Secondly, we maximize the profitability of this technical trading strategy by applying the Kelly money management criterion. We estimate a concave relationship to find the optimal leverage factor. The main finding is that the profitability can be increased substantially for the given trading strategy. Key words: Breakout filter, Day trading, Future contracts, Kelly criterion, Market inefficiency, Money management, Optimal leverage factor, Relative strength index

3 Acknowledgements I would like to thank my study companions for the support and the great company, enabling me to get through the long days of studies. I would also like to express my big gratitude towards Christian Lundström for all the inspiration he has given me and also for guiding me, with his expertise, through the financial jungle.

4 Table of content 1. Introduction Background Structure Money management Previous studies The Kelly criterion Application of Kelly criterion on future contracts Technical analysis The efficient market hypothesis RSI Daily breakout filter The trading strategy Empirical results Data Profitability test Model Results Applying the Kelly criterion on the trading strategy Comparing the optimized θ to θ= Using a high leverage factor Test of robustness Overview of results from the test period Conclusions References Appendix A. How to practically obtain θ B. Relative strength index C. Frequency histogram of returns... 30

5 1. Introduction 1.1 Background What do a risk neutral gambler and an investor have in common? Both want to make as much money as possible. By using money management this paper shows how the capital or wealth of an investor can be maximized. Money management can, as we partly will demonstrate, be seen as a field that in many aspects ties gambling theory together with financial theory. The main goal of money management is to maximize the capital growth by applying an optimal leverage factor without risk of ruin. We define the concept of ruin as substantial losses of capital that limits the amount we can reinvest. If ruin occurs it thereby causes a constraint in possible capital growth. In his paper we test the theory Kelly criterion. This theory is suggested by many high achieving investors as a way to handle the money management. Among these investors we find John Maynard Keynes, Warren Buffet and Bill Gross (Lundström, 2014). The contribution of this paper to the field of money management is the application of Kelly criterion on an intraday trading strategy based on the Swedish stock market index OMXS30. 1 By involving money management when studying a trading strategy we highlight a flaw of many similar studies. What can be noticed when observing previous studies of investment strategies is that they are often simplified. Many studies are limited to investing the same amount for every trade and not allowing for reinvestments (e.g. Wang and Yu, 2000). In this study we allow for reinvestments of capital gains and thereby, unlike earlier simplified studies, we present a more reality based study. Using reinvestments, the capital grows or shrinks at an exponential speed (Kelly, 1956). Given that an agent is utility maximizing and risk neutral, it should be of importance that we maximize the profit in the long run without risk of ruin. 2 Under the assumption that one possesses an investment strategy that generates a positive average return, determining the size 1 The index corresponds to the 30 most traded stocks listed on the Stockholm stock exchange, corresponding to 65% of the trading volume and 45% of the market value. 2 Risk of ruin can be defined as the probability of losing sufficient trading or gambling money to the point where continuing a trading strategy or game is no longer considered an option to recover losses. 1

6 of the investment is essential. It can analogously be seen as a bet size decision of a blackjack player who is exploiting an edge he or she has in the game (Anderson and Faff, 2004). To practically perform a strategy like the one we present in this paper a financial instrument is needed. Future contracts are a financial instrument that well suits for active trading. There are a several reasons why future contracts are a facilitating instrument for this study. At first trading futures is very common and frequently used for agents acting on the financial markets. For instance, trading futures is a multi-billion US dollar industry (Lundström, 2014). Some primary reasons behind the wide prevalence are the low transaction costs and the liquid market of futures. Future contracts do also have the advantage that it is just as easy to make long trades as short trades. Its flexibility to make long and short trades also offers an opportunity for people who want to hedge against risk. Moreover, future contracts also offer the investor to set their own risk by allowing for different leverage factors. When trading futures, determining the investment size can equally be seen as determining what leverage factor to use (Sewell, 2011). When referring to investment size when observing future contracts, we from now on refer to it as leverage factor and denote it by the Greek letter. Thus, setting is the same as not involving money management. By being able to adjust the leverage factor without any additional transaction costs, futures offer a great flexibility. For an example of how is obtained practically, see appendix A. Since we have chosen to use future contracts of OMXS30 we now need a trading strategy that gives us empirical returns and a way to determine the optimal leverage factor. The optimal leverage factor is the one which gives us the highest growth rate of capital. The method of optimizing is the Kelly criterion and is explained in chapter 2. The trading strategy that is used as a base for the Kelly criterion is applied on futures traded on an intraday basis; meaning that we buy and sell the future contract the same day. Trading strategies applied for future contracts on intraday basis has earlier been proven successful by Holmberg et al. (2013) and Lundström (2013). Both studies base their trading strategies on intraday momentum in price movements. Important to clarify is that strategies like Holmberg et al. (2013), Lundström (2013) and the one used for this paper rely on the assumption of market ineffectiveness. The concept market ineffectiveness is explained in chapter 3. The trading strategy we use involves two tools in technical analysis, the relative strength index (RSI) and a breakout filter. The RSI was developed by Wilder (1978) and has a huge advantage that it is developed during a time when data snooping was not possible. Data 2

7 snooping refers to testing multiple trading tools at the same time and just present the tools producing profits. This has the implication that the tools showing profitable return might be random (Sullivan et al, 1999). The breakout filter has the benefit of capturing momentum in price developments. These trading tools are more closely explained in section 3.2 and 3.3. The main purpose of this paper is to study whether it is possible to optimize an already profitable day trading strategy by applying the Kelly criterion. The paper also aims to prove the consequences of being too aggressive or too timid when investing. To avoid the study of becoming too complex we have limited the study as follows. At first, there are many aspects of optimizing a trading strategy; hence we limit the study by just applying the Kelly criterion for one type of trade (long trades) for the given strategy. Secondly, compared to other studies we just apply it on one type of future contract (OMXS30 futures). Thirdly, we do also not consider the aspect of stop losses when we trade; the use of stop losses is briefly explained in section Structure This paper can be seen as a two parts study. The first part concerns the investment strategy and the second part is the optimization of that strategy through Kelly criterion. The outline of the remaining parts of the paper is as follows. Chapter 2 presents some of the earlier studies of applications of the Kelly criterion and also clarifies more in depth what the Kelly criterion is and how it works. Chapter 3 presents some concepts about the nature of financial markets, the technical trading tools and the strategy used for this paper. Further, Chapter 4 shows the empirical results by containing a profitability test of the investment strategy and later the solution to the optimization problem through the Kelly criterion. Finally, Chapter 5 sums up the paper in a concluding discussion. 2. Money management 2.1 Previous studies Kelly (1956) shed new light to gambling theory by showing that when reinvesting the capital one includes the aspect of money growing or shrinking at an exponential rate. In that way Kelly s sort of thinking differs from the classical gambling theory. Kelly presented his criterion through a lottery with positive expected value and two outcomes (we present a similar example in section 2.2) where the capital gains is reinvested. Thorp (1969) developed Kelly s original gambling strategy by expanding it to several Casino games such as 3

8 Blackjack, Baccarat, Roulette and Wheel of fortune. Further, Thorp (1969) also suggested the application of the Kelly criterion to the stock market and financial derivatives by using a continuous probability distribution of the returns instead of the discrete distribution of Kelly s original lottery example. However, Thorp (1969) assumed that the distribution of returns followed a log normal distribution, which can be questioned when applying it to the stock market. For instance Cont (2001) shows that financial returns follow a distribution with heavy tales. This would imply that Thorp (1969) underestimated the risk by using a log normal distribution. After Thorp (1969) widened the use of Kelly criterion it has been suggested by a several authors during the years. Gehm (1983) applied the Kelly criterion to the commodity market when trading futures and later on also Balsara (1992) used the Kelly criterion for the same purpose. Rotando and Thorp (1992) showed it was possible to apply the Kelly criterion on the SAP500 index. Rotando and Thorp (1992) divided the period between the years into one year intervals and optimized the investment size over the time period based on the average return of 5.8% per annum. The findings of Rotando and Thorp (1992) suggest that one should invest 117% of the capital each year, hence borrow 17% additional capital to invest each year. However, the result of Rotando and Thorp (1992) was not corrected for transaction costs. Further, Tharp (1997) uses the Kelly criterion to highlight the importance of money management rather than looking for trading strategies which aim to find the right time to enter and exit the market. Vince (1990) develops the Kelly criterion by correcting the optimization problem for risk. Vince (1990) states, that the optimal bet fraction ( ) should be corrected for the largest loss in a series of returns. According to Vince the optimal leverage factor (Vince criterion) should be described as the optimal fraction divided by the largest loss. Andersson and Faff (2004) use the optimal -technique like Vince (1990) for speculative future contracts. One of the key findings of Andersson and Faff (2004) is the importance of reinvesting when evaluating the returns for future traders. They highlight reinvestments both in the sense that it is the scenario in real investment situations and also by showing increased returns. For instance they achieve an impressive return of 2000% when trading corn futures. However, according to Andersson and Faff (2004), the Kelly criterion was not able to turn a losing strategy into profit. 4

9 Another concept in the discussion of money management is stop losses. According to Andersson and Faff (2004) stop losses can also be a useful vehicle when applying the Vince criterion to limit the largest observed loss. A successful application of stop losses minimizes the losses that especially can occur when the market experience positive momentum in a downwards direction. Lundström (2014) shows, by trading crude oil and SAP500 future contracts, that it is possible to optimize the Kelly criterion by adding stop losses to money management. Lundström (2014) show that the use of stop losses improves profitability to 8.33 respectively times the profit obtained by the Vince criterion. Although there are different developments of Kelly s original criterion (i.e. Vince criterion and stop losses) we only apply the Kelly approach of money management for this paper. We mainly have the purpose to examine whether it is possible to optimize an already profitable trading strategy with money management. We choose the Kelly criterion since it offers the most simplified solution of the optimization problem. We motivate the simplified approach since the paper already is performed in two parts (the trading strategy part and the money management part). Hence, we limit the framework of the paper to just use the Kelly approach. 2.2 The Kelly criterion As mentioned, one method to optimize the leverage factor when trading futures is to apply the Kelly criterion. In the original paper Kelly (1956) highlighted how a gambler could maximize his profit when given the opportunity to reinvest his winnings. We now demonstrate this mathematically in a theoretical example of a lottery inspired by Rotando and Thorp (1992). Assume a lottery that is designed exactly the same way each time it is drawn. The lottery has two outcomes; one can either win or fail the same amount as invested. Further, the lottery has a positive expected value and the probabilities of winning or failing on a single bet is known. Now assume an arbitrary agent which is participating in this lottery each draw. The agent is risk neutral, profit maximizing and has no restriction of how much he is allowed to bet each draw. Since the agent is reinvesting his winnings we can optimize his long term wealth by optimizing his growth rate of capital. By reinvesting the winnings the agent gets the opportunity to make the capital grow at an exponential rate during draws. The Kelly criterion does now imply that even if the agent has positive expected value and no betting constraints he should not be interested in betting too big fraction of his capital. This is since losing a big bet limits his reinvestment possibilities, thus result in a lower value of his capital after draws. The agent is also not interested in betting too small since it simply leads 5

10 to small winnings (Kelly 1956). The problem the agent is facing can be solved by applying the Kelly criterion, which tells him what fraction of his capital he should bet each time to optimize the expected value of his wealth after games. Since the agent is allowed to bet as much as he wants we describe his bet size as a fraction of his capital, denoted by. Further, if the initial wealth of the agent is denoted as his wealth after a number of bets ( ), can be described by the following equation: In the equation above corresponds to the number of successful bets and number of bets that fails where:. We follow Kelly (1956), and set the probability of going broke equal to zero by defining the bet fraction as:. The total growth of wealth after bets can be described as: The agent is interested in finding the fraction that optimizes the expected growth rate per trial, i.e. geometric mean. We therefore raise the equation by 1 divided by number of trials and describe growth rate per trial as follows: In next step the logarithms used on the function and also expectations. 3 This permits us to describe and rewrite the expected growth rate per trial in the following manner: We can denote the expected probability of success and fail as follows: 3 The use of logarithm facilitates for solving the optimization problem. 6

11 Since it is a binomial lottery with a positive expected value it follows that, where. Substituting and into the expression now allows us to describe the expected growth rate as: The agent is looking to optimize the expected growth rate per trial, hence we take the derivative with respect to the fraction,. Recall that. This results in: In order to confirm that the solution to the optimization problem gives us a maximum of we show that: Under the assumption that we have a maximum of growth rate of wealth when. By applying the Kelly criterion the agent can optimize his expected growth rate of wealth by betting the following each lottery game: where is the percentage of the capital which gives the highest expected growth rate. By using the assumptions in the example we show in figure 1 how different values of probability of success gives different concave relationships with maximums that are implying that the higher value of w the bigger fraction of our wealth should be played in each game. 7

12 Growth rate per game 0 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 0,45 0,5 0,55 0,6 0,65 0,7 0,75 0,8 0,85 0,9 0,95 0,2 0,1 0-0,1-0,2 w=0.9 w=0.8 w=0,7 w=0.6 w=0.5-0,3-0,4 Bet fraction Figure 1: Showing how different bet fractions of the wealth correspond to different values of growth rate per trade. 2.3 Application of Kelly criterion on future contracts We now demonstrate how the Kelly criterion can be applied on a series of returns obtained from a trading strategy. We use the term Terminal Wealth Relative (TWR) as a wealth measure, which tells us how much capital we have relative to the starting capital after trades. TWR is defined as: To clarify, TWR can be seen as the total return during a series of trades (e.g. a TWR of 2.3 corresponds to a total return of 130%). If we set the wealth after a trade can be described as:, where the term is the leverage factor and the return of trade. The term can be regarded in a similar manner as the term in the example in section 2.2. By calculating this for a series of reinvestments we can now write: 4 8

13 Like the case for the agent in section 2.2 we are looking for the growth rate per trade (i.e. the geometric mean): By taking the logarithm of the expression we can express the growth rate per trade series of successive trades as: in a 5 To empirically find the optimized we need to obtain empirical values of a series of returns. To make the study applicable on reality we will divide the time period into two parts; the first period will be used to obtain the optimal value of that maximizes the growth rate per trade and then apply that leverage factor on the second part of the sample. We need to do this since it is not reasonable that we know the optimal in advance, thus we use the first part to just observe the behavior of the returns and the second part of the returns to use the optimized obtained from the first part to perform the trading strategy practically. 3. Technical analysis 3.1 The efficient market hypothesis Eugene Fama (1965) developed the Efficient Market Hypotheses (EMH). According to EMH all information are available to the investors. Thereby every financial asset has its correct price which also is referred to as the equilibrium price. Assuming that all investors act rational on available information of already happened events, but also events that can be expected in the future everything is included in the current market price. Given that all information about that underlying asset is known and used correctly by the investors, it is exactly 50% for additional information to be either good or bad. Further, under EMH, all agents on the market are always trying to find systematical patterns in the price development and take advantage of these. This leads to prices randomly moves around the equilibrium price. Under the 4 5 9

14 assumption that the prices of financial assets behave as described by EMH, price movement of these assets can be described as a random walk. However, there are studies and theories that contradict EMH. Shiller (2001) explains various biases among investors causing that the market price can often deviate significantly from its equilibrium price. For instance, representative heuristics is a phenomenon when people tend to overreact to news resulting that the price deviates from the long term trend due to good or bad news and then slowly adjust back to its equilibrium price. Further, Shiller also states that people often seem to neglect history; leading to that people continues to make wrong choices of investment by not learning from the past. Besides this, Shiller means that some investors in general believe that they possess magical or quasi magical thinking. These investors are experiencing that a certain behavior leads to profit but in fact the profit from the magical thinking investors is due to the stochastic nature of financial market or a general economic growth during the period they performs that certain strategy. Important to know is that even if Shiller describes a lot of concepts in behavioral finance that can imply market ineffectiveness he states that relevance of these concepts should be taken with caution due to the complexity of human behavior. Shiller also states that markets can be seen as impressively efficient in certain respects. Previous studies within the field of EMH have come up with different results. For instance Malkiel (1996) mean that the market is very close to be efficient while studies like Jegadeesh and Titman (1993) succeed to build a profitable investment strategy relying on market ineffectiveness. Lim & Brooks (2011) can verify some of aspects of both Fama s and Shiller s views on financial markets by showing that the market effectiveness varies over time. This suggests the essence of using long data series when studying objects depending on market efficiency. Technical analysis relies on the assumption that historical price patterns repeat themselves, enabling an investor to predict future development of prices (Pring, 1991). Thus, becoming familiar with previous price pattern an agent achieves the potential to find profitable investments. If a market is not entirely efficient it consequently implies that prices temporarily deviates from its equilibrium and might follow certain patterns due to information asymmetries and actions from irrational investors. We now take advantage of these price patterns and formulate an investment strategy. We combine two tools (described in section 3.2 and 3.3) from technical analysis to develop a trading strategy on the Swedish stock market index OMXS30. 10

15 3.2 RSI RSI is a Key Performance Indicator which is based on recent price movements of a financial asset. Its application is to identify whether a financial asset is overbought or oversold. The implication is that an overbought asset can be expected to fall in price whilst an oversold asset can be expected a rise in price. The most common period to measure RSI is to observe the last 14 days, but there are several versions of the length of this period. RSI can assume values in the range between 0 and 100, where a value below 30 indicates oversold and a value above 70 indicates overbought. According to Wilder (1978) RSI is calculated as follows: For further information regarding RSI and RS, see appendix B. 3.3 Daily breakout filter Opening range breakout aims to identify entry courses for long as well as short trades (Crabel 1990). The opening range often refers to the first 30 minutes after the stock market has opened for the day; it is also the most volatile period of the day. Under the opening period a highest and a lowest price is observed, these prices is then used as Pivot levels for long and short trades. A Pivot level is defined as a level for an investment decision, based on previous price patterns (Pring, 1991). Besides from just observing the opening range it is also possible to use previous days extreme values to determine the Pivot level. The basic idea of this strategy is to avail oneself for that the market often moves in same direction under certain time periods, also referred to as momentum in price development. This phenomenon is supported by Jegadeesh and Titman (1993), Holmberg et al. (2013) and Lundström (2013). The daily breakout filter aims to use these short term trends to make profitable investments by making a long (short) trade if the high (low) Pivot level are intersected from below (above). The positive momentum implies that that when the Pivot is intersected from below (above) the price can be expected to keep rising (descending). The application of this kind of strategy is often combined with other technical trading tools and rarely used solely for an investment decision (Crabel, 1990). 11

16 3.4 The trading strategy Both the RSI and the breakout filter can be seen to be deficient to solely provide a basis for an investment decision. But if they are used together we have the opportunity to find out when the market is oversold (from RSI<30) and at the same time experience a positive momentum (from the high Pivot level to be intersected from below). We apply these two trading tools such that an entry for a long trade occurs if the following conditions are fulfilled day t: where the indexation refers to any point in time during day t and price indexation corresponds to the highest price during day t-1. This means that an entry is made during day t if RSI the previous day, t-1 ( ), is below 30 at the same time as the price at some point during day t ( ) reaches the highest price of day t-1 ( ). The Pivot level of the strategy is set at the highest price from the previous day, meaning that we make an entry at day t at a price corresponding to ( ). Since we are day trading we sell at the end of day t for closing price ( ). See Figure 2 for an example. Figure 2: Intraday price movements from a day where RSI the previous day (t-1) is below 30 and we are looking to find an entry for a long trade. 12

17 What also needs to be considered is that, in some rare cases, there might occur price jumps between days so that the lowest price at day t ( ) is higher than the highest price at day t-1 ( ). This causes that the entry price is not included in the range of. To adjust for this when calculating returns we set the entry price to if a day with a price jump occurs. We have to be aware of that it is not possible to know, in advance, when the lowest price occurs during a day. However, we choose to involve these days anyway, since a day when a price jump occurs often corresponds to a day of positive momentum. This implies that excluding these days could potentially have an impact of the returns of the trading strategy. Further, unlike other studies, for instance Andersson and Faff (2004), we make the study more applied to reality by adding transaction costs. When observing internet brokers like Capital spreads or Interactive brokers one can notice that the fee of trading futures varies, depending on what kind of future contract one observes. Based on the fees on the mentioned brokers we set the transaction cost arbitrarily to 0.04%. The return of the trading strategy is calculated as follows: The equation that is used for calculating the return for a specific trade depends on if condition is fulfilled or not. If hold equation (4) is applied. Otherwise if a price jump occurs between days, such that holds equation (5) is applied. As stated in the introduction, futures can just as easily be used for short as long trades. We have chosen to just use long trades. It might be reasonable to believe that statistical properties of returns differ in mean and variance between long and short trades. Different statistical properties would mean that we would need two different optimization problems to solve in next section. Also, if we would pool long and short returns to the same sample and optimize the leverage factors based on the pooled sample we might get a biased solution of the optimization problem. The consequences might be a too high leverage factor for one type of trade and too low leverage factor for the other one. The effects of using a too low/high leverage factor are illustrated in next chapter. 13

18 Price 4. Empirical results 4.1 Data The data used for the active day trading strategy on OMXS30 contains daily prices between April in 2000 and April A complete time series of daily closing prices is presented in Figure Daily closing price OMXS Time Figure 3: Daily closing prices of OMXS30. The descriptive statistics of daily returns of the entire time period can be observed in Table 1. The data in Table 1 are just involving daily change in percentage between the closing prices and not including any transaction costs. The daily returns, described in Table 1, are calculated as: 14 years is not a long period when observing series of financial returns compared to other studies like Rotando and Thorp (1992) or Jegadeesh and Titman (1993). However, it should be sufficient time period to find enough trades to support the significant profit of our trading strategy. One can argue that it is more relevant to observe a shorter time period in the case of looking at a trading strategy. This is, since we are playing against active agents on the market, using too long data would include returns made against retired players, which are irrelevant 14

19 when looking at a successful trading strategy today. The descriptive statistics of the returns from the strategy described in section 3.4 is observed in Table 2. For frequency histogram of the returns from the trading strategy, see appendix C. The returns in Table 2 are calculated as described in equations (4) and (5). Table 1: Descriptive statistics of the daily returns of the OMXS30 index between April 2000 and April Observations Mean Highest value Lowest value Kurtosis Skewness % % % Table 2: Descriptive statistics of the returns from the trading strategy performed between April 2000 and April Time period Observations Mean Highest value Lowest value Kurtosis Skewness % 4.622% % % 4.622% % % 3.082% % As observed in Table 2, 122 trades are obtained from the 14 year long data sample. For the reasons explained in section 2.2 we also divide the return series of the trading strategy into two parts. As observed in Table 2 each part contains 61 observations. The amount of 61 observations can possibly cause problems for significance of a positive average return, which need to be the case for the application of the Kelly criterion. However, Rotando and Thorp (1992) only use 59 trades and Anderson and Faff (2004) have between 35 and 62 trades on different future contracts they are trading for a similar purpose as we have in this thesis. 4.2 Profitability test Model To establish that the Kelly criterion is applicable we now exercise a profitability test to verify that the returns from the trading strategy described in section 3.4 have a positive mean. Before estimating the mean we have to consider which estimation method to apply. One problem by just computing the mean and use a simple t-test is that financial data such as 15

20 return series generally suffers from heteroscedasticity (Cont, 2001). We also confirm this fact by using a Breusch-Pagan 6 test which shows that we can reject the null hypothesis of homoscedasticity ( when testing on the entire return series of 122 observations). Hence, by referring to Cont (2001) and the Breusch-Pagan test we choose to use heteroscedasticity corrected standard errors (HCSE). The technique we use to apply the HCSE when estimating the return uses a regression model as follows: where the observed returns ( ) is set as the dependent variable. The term is estimating the mean of the variable, meanwhile corresponds to deviations from that estimated mean. Due to the heteroscedasticity of the return series the variance of can be described as a function of as follows (Studenmund, 2011): where is a variable used to theoretical construct a non constant variance depending on the size of observation i Results In Table 3 we present the results from the model described by equation 6. Table 3: Results from the regression models. Time period Observations (n) Mean return ( ) Robust HCSE P-value % 0.153% % 0.247% % 0.183% As observed, the mean value of the returns is 0.363% with a P-value of when using HCSE-technique over the entire time period. The interpretation of the P-value in Table 3 is that we now have quite good evidence that the active trading strategy generates positive 6 Breusch-Pagan is a test of linear heteroscedasticity. Another alternative is White s general test for heteroscedasticity. (Greene, 2012) 16