Factors that influence the volatility of stock prices

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ОБРАЗОВАТЕЛЬНОЕ УЧРЕЖДЕНИЕ

ВЫСШЕГО ПРОФЕССИОНАЛЬНОГО ОБРАЗОВАНИЯ

«НАЦИОНАЛЬНЫЙ ИССЛЕДОВАТЕЛЬСКИЙ УНИВЕРСИТЕТ

«ВЫСШАЯ ШКОЛА ЭКОНОМИКИ»

Международный институт экономики и финансов

Выпускная квалификационная работа - БАКАЛАВРСКАЯ РАБОТА

по направлению подготовки 38.03.01 «Экономика»

образовательная программа «Программа двух дипломов по экономике

НИУ ВШЭ и Лондонского университета»

Factors that influence the volatility of stock prices

Baron A.A.

Москва 2018



Abstract

Historically, volatility was predicted by inertia-type models such as (G)ARCH or its variations. GARCH does not allow to include different kind of factors or variables to predict volatility. It is empirically proved, that a lot of variables do not predict returns. But it does not mean, that these variables can't predict volatility. The main motivation of this paper is to check existence of factors that predict volatility. For these purpose, we will use different machine learning techniques.



Introduction

“Volatility is crucial aspect of risk management, but needs a higher level of understanding” said Nick Smith, Allianz Global Investor ...After recent financial crisis much more often people started to use the concept of volatility. However it should not only be used more frequent, but it needs a clear understanding of why we need it.

Despite risk and volatility are not the same, both are parts of successful investment. If volatility behaves in forecastable way, then it is useful for risk management. However volatility forecast ability changes across the horizon and the model assumed.

The volatility risk can be managed using financial instruments whose prices rely on volatility of a given financial asset, for example caps and floors for interest rates. The recent example of bad risk management was the recession of 2008. Moreover there exist a volatility based risk management which approach to factor allocation is to scale exposure by factor volatility. While it increase risk-return ratios of momentum, it decreases them for value and size. It should be noted that factors have different drivers, for which different risk management system should be constructed. (https://www.factorresearch.com/research-factors-and-v..)

Secondly, volatility is important in determining asset prices. The key reason is that higher volatility risk require higher discount rates. In the work of Ravi Bansal and Amir Yaron it is shown that volatility implies positive risk premium. Moreover volatility is negatively correlated with consumption. To show these effect they developed a DCAPM, the update version of traditional CAPM containing three sources of risk : cash flow risk, discount rate ris and volatility risk. They show that asset pricing is strongly misleading when ignoring volatility risk from the model. Thus the result shows that volatility is crucial when calculating expected returns and macroeconomic fluctuations.

One more thing why volatility is important is trading strategies. Investors now see volatility not only as a potential loss, but also as a potential gain. The most obvious way for profiting from volatility is derivative contracts. For volatility trading strategies it is vital to realize the difference between implies volatility, which is not yet realized but is expected from the options market, and the actual volatility( difference of prices). While both types are important, mostly volatility trading relies on implied volatility. Main volatility strategies are as follows:

* Trading implied volatility against itself

* Trading implied volatility against actual volatility, as a vega play

* Trading implied volatility against actual volatility, as a gamma play

* Trading implied volatility between options on different products(relative value, volarb)

* Trading impied volatility between options on the same product

* Trading impied volatility across the term structure

The last, but not the least reason of volatility importance is crisis analyzing. In the work Ilkunur Zer, Federal Reserve board “Learning from history: Volatility and Financial crisis” the effects of volatility on financial crisis are analyzed using a cross-country database. Volatility doesn't directly predicts the banking crisis, however the type of volatility(high or low) does. According to Mishky instability hypothesis, in periods of low volatility agents take more risk (taking on more and more credits, bubbling economy). This gives the appearance of crisis more likely, especially for poorly regulated markets. While high volatility is associated with high uncertainty and political instability, reducing investment, output and other macroeconomic indicators. This also give a rise to financial crisis. However currency crisis unlike financial is not influenced by volatility of either type.

There exist many variables explained by different financial economists that predict the future stock's returns. According to Chinco et al. (2017) for instance, Banz 1981 is said to have used the market cap for the predictions, Cohen and Frazzini 2008 used customer earnings surprises while Titman and Jegadeesh used lagged returns. For the best results regarding the variables, the researchers intended to solve two distinct problems, i.e., identification and estimation. They first had to identify a subset of candidate predictors and then estimate the quality of those predictors. Since financial volatility is crucial and may impact the economy as a whole, it is primarily essential to understand the economic drivers of volatility. The purpose of my paper is to determine the variables, that predict volatility. We will consider methods of feature selection such as Lasso/Ridge/Elastic net and volatility prediction using XGBoost/GARCH. This work is based on methodology of D.Malakhov on volatility prediction. Malakhov D.I. (2017) Sign, volatility and returns: differences and commonalities in predictability and stability of factors. Manuscript.

Machine learning is a process of analyzing data in order to automate analytical model building. It comes from the idea that systems can learn from data and make decisions almost without human intervention. Firstly, it is important to understand the concept of machine learning. The foundation for it is generalization from experience. In other words, generalization means being able to fulfill new, unknown tasks after having receiving a learning dataset. Machine learning started as a separate branch of science in 1990-s and now is developing faster and faster during the last decade. It is effective at lots of various tasks and is widely used in financial services, marketing and sales, health care e.t.c. So why not to use it in more complex decisions, if these tools can be applied successfully? The answer can be found in the work of Jon Kleinberg, Himabindu Lakkaraju, Jure Leskovec, Jens Ludwig, Sendhil Mullainathan, called “Human decisions and machine predictions”. In this research the data was taken on 758027 defendants who were arrested and the machine learning was used to predict crime risk, specifically gradient-boosted decision trees. The goal of the paper was to realize whether machine learning predictions are helpful for understanding and improving judges decisions. This question is a source for important econometric challenges. To solve it in the study was developed an empirical approach to have meaningful comparison between human decisions and machine predictions. The main lesson from the study we can learn is that such prediction problems needs a combination of machine learning techniques and several methods, central for economic toolkit. In the work bail decision lies on machine learning's main strong point - maximizing prediction quality, while avoiding its weakness - not guaranteeing causal or consistent estimates. One more obstacle in machine learning is basic selection problem, more precisely, “labels” can be missing in non-random way. This “selective labels” problem usually make it difficult to get a meaningful comparison of human judgments and machine predictions, while in practice this problem is often ignored. In this work researches come to conclusion that machine learning needs to be integrated with methods for analyzing decision-making. While it may seem that main target of machine learning is increase prediction accuracy, in behavioral applications predictions, separately, has little value. In fact, they become useful only when the role is identified in decision making and the hypothesis about potential gains are proposed. The result of the study suggests that machine learning is a useful tool for overcoming prediction problems, when integrated into economic framework. One more thing to my mind is vital to point out is that machine learning is beneficial for prediction accuracy and at the same time it provides close to zero marginal costs. Nowadays days machine learning is widely used and there is a huge variety of world companies and associations that benefited in different fields because of it. For example, in 2010 in The Wall Street journal was published an article about the firm Rebellion Research, which used machine learning to predict financial crisis. More surprisingly, in 2014 machine learning algorithm was integrated even into Art industry. It was used to study some art paintings, and moreover this algorithm helped to reveal influences between artists, which were not recognized before. The reason for such increasing interest is unobserved abilities of machine learning. It can create thousands of models a week, while a human can produce us with just a few. There is no doubt, that machine learning can sometimes produce confusing results due to econometric problems, which might arise, but nonetheless it is a wonderful perspective for human development



Review of Corresponding Literature

This paper shall, therefore, review the corresponding literature for the coursework discussing the volatility predictors present. It will evaluate the literature on stock forecasting by forecasters and the strategies for good return forecasts.

Some academic scholars argue that future stock earnings are not predictable on the basis of presently existing information. While such arguments include Malkiel 1973 random walk model that is consistent with the efficiency of the market, so is a foreseeable return process, insofar as predictableness portrays consistency with exposure to time-varying aggregate risk. On the contrary, others would argue that available information is enough to predict volatility of future stocks; however, there exist certain levels of predictability which are realistic that should be anticipated (Ghysels et al., 2006). Suggestions exist that variables capturing risks that vary with time are prime candidates for understanding and forecasting volatility implying that return predictors from existing literature such as foreign exchange, interest rate disparities and valuation ratios for equities qualify as promising volatility predictors as well (Makiel, 1995 ).

According to the coursework literature, stock return predictability is investigated by the aid of various variables. The variables include the return on a broad stock market index in excess of risk-free interest rate from one end of a certain period to another. They also include a variable used to predict equity premium, for example, the dividend-price ratio, and lastly is the zero disturbance mean. From the perspective of the combined variables, a predictive model can be formed; nonetheless, only a limited level of stock return predictability should be expected (Amendola and Storti, 2008). Models that claim a large part of stock volatility suggest that either there exist massive market inefficiencies or the asset pricing models are grossly not accurate. The coursework provides that such return predictability may be too good and require viewing with appropriate suspicion.

Various research work provides that volatility predictors can be learned away over time. They suggest that strength of the evidence of return predictability derived from time series or cross-sectional regressions is apt to weaken as the knowledge of such patterns becomes more prevalent. A credible mechanism is that investor's attempts to exploit predictive patterns result in self-destruction as new cash flows out of overvalued assets or into undervalued assets. Studies such as those of Timmerman 2018 emphasize the difference between out-sample and in-sample volatility predictability. The initial imposes the constraint that only information made available at a given time is essential in generating the predictions while the later utilizes full sample information to approximate model parameters and hence could not have been exploited by investors in real-time.

According to Farmer, Schmidt & Timmermann (2018), there are four variables considerably helpful in determining predictability. First and foremost is the lagged dividend yield which is defined as the dividend over the most recent period dividend divided by the stock price at the close of a given time. The predictor has been used in several studies to predict volatility by scholars such as Campbell, Fama and French and Stambaugh. The other variable is the yield on stocks that Ang and Bekaert (2007) and Campbell (1987) well utilizes to predict stock returns. The third is the term-spread which is defined as the variance in the yield on short and long-term stocks while the last one is the realized variance measure well-defined as the realized variance over the previous short period like two to three months. The predictors can be used in different frequencies, i.e., daily, monthly, or quarterly.

Daily predictor variables are seemingly persistent at the daily frequency causing estimation challenges and inference with daily data. Therefore, on economic grounds, return predictability tends to be very weak on the daily horizon and as a result, there exist no great economic benefits to investors who exploit daily return predictability. All the four predictors stated above are highly persistent; however, more concerns would be directed when dealing with daily return regressions as they tend to be more persistent at the daily frequency as compared to the monthly and quarterly frequencies. To account for persistence in the repressors, allowance for volatility dynamics in returns should be done in addition to the incorporation of constant return predictability from a time-varying variable.

Economic scholars agree that stock markets are not efficient and that volatility measures in the returns are much far to be ascribed to just new information (Rapach & Zhou, 2013). In essence, markets are not efficient as they don't implement new info in the right way. Investors always tend to overreact to information and oversell respectively overbuy stocks when new info becomes accessible. If markets are efficient, then they could not let investors earn above more than the average returns without accepting risks above average.

Commodity indices are other crucial volatility predictors. By the use of various commodity indices as predictors at the same time, better volatility prediction may be achieved as financial variables have a close relationship with them. The use of commodities as predictors has been effective and still has a lot of potentials that requires examination since the correlation between commodities and stocks may in general rise over time due to new trading patterns of commodities. Use of added financial as well as macroeconomic variables is another volatility predictor and is almost similar to the out-of-sample prediction (Christiansen, 2012). The out-of-sample prediction tends to provide more accurate results as compared to the in-sample and hence, investors are likely to use the former since they are interested in predicting the future and not only look back in-sample.

Regularization via Elastic Net/L1/L2

Elastic Net is a technique, that is used for variable selection and regularization purposes. Generally, we expect our model to be both accurate in sense of prediction of target variable, and parsimonious: we prefer simpler models, with less number of predictors. In case of large number of predictors (in my case, it is 385 variables) OLS will be inappropriate, as it will always choose models with larger number of variables and does not punish for overparametrization. In addition, OLS is not applicable in my case, because of multicollinearity issue. To overcome the problems above regularization(shrinkage) techniques should be used. Two main types of such techniques, which address over-fitting and variable selection are L1(Lasso) & L2(Ridge). Lasso and Ridge impose a penalty in order to reduce the value of coefficients. These methods require to select a tuning parameter л, which is crucial and determine the degree of shrinkage. The difference between them is the way they treat penalty term: Ridge use sum of squared coefficients and Lasso - sum of absolute coefficients.

Lasso shrinks the less important coefficients to zero, so reduces set of potential parameters, makes model more interpretable and provides the variable selection (in contrast, Ridge shrinks such coefficients close, but not equal to zero, so number of parameters is not reduced). However, in set of my 357 potential predictors, it is expected to be groups of variables with high correlation, so Lasso will be inappropriate here, as it would choose only one variable from the group and miss other variables. Tibshirani (1996) has provided an empirical evidence of better prediction performance of the Ridge over the Lasso in situation when there is high correlation between predictors and number of observations exceeds number of parameters ( 8831 observations > 357 variables ).

Elastic net is a combination of Lasso & Ridge and allows for both variable and grouped selection in high-dimensional data. It can be said, that Lasso and Ridge are extreme cases of elastic net: when б =1, it is same as Lasso, and when б=0, is same as Ridge. The elastic net estimator is defined as or equivalently , subject to , for some , where .

For high-dimension data, the model selected should have the oracle property. Oracle property states, that estimator must be consistent in parameter estimation and variable selection. Usual Lasso does not satisfy oracle property, so adaptive Lasso estimator was introduced by Zou(2006), which use special adaptive data-driven weights to make estimator consistent.

Following Zou(2006) discussion, due to certain conditions naпve elastic net estimator will be inconsistent, hence does not satisfy oracle property. The solution is the adaptive elastic net, which can be considered as combination of adaptive lasso and elastic net. The adaptive lasso achieves oracle property, and elastic net manages collinearity. Firstly, we compute elastic net estimator and construct adaptive weights, as , where г is a positive constant. Then, optimization problem should be solved, to get adaptive elastic net estimator:

For adaptive elastic net, we should find л1(n), л2(n) ,г (using cross-validation).

After using Lasso and its variations, we are expected to determine the set of best predictors for volatility and reduce overfitting. We will consider models with different dependent variable: depending on the proxy for volatility. The relationship between response variable and explanatory predictors is captured by usual linear regression model:

where = 1315 observations, k = 356 predictors and

are regression coefficients. In performing Lasso, Elastic net and Ridge “glmnet” and “biglasso” packages of R were used. tuning parameter for our model, that control the degree of penalty. To choose right is crucial: if it too large, all coefficients will go to zero, and if it too small, our model will not differ from usual OLS, which is not suitable for our case. In order to choose right , cross-validation should be used. It allows to find , that gives the smallest cross-validated error of a model or , that gives the model with cross-validated error, that is within one standard error of minimum one.

Least Angle Regression

Least Angle Regression is a model selection method, introduced by Efron, Hastie, Johnstone, Tibshirani (2004). Least Angle Regression is used, because it is less greedy, than forward stagewise, allows to explain simillar results of LASSO and Stagewise and to reduce computational difficulties: faster than LASSO and Stagewise. LARS relates to forward regression method, in which we select the variable with the most strong explanotary power, obtain linear regreesion from it to dependent variable, and project other explanotary variables orthogonally to selected one, repeatedly. This method has disadvantages of being over greedy and eliminating variables which are correlated with selected predictors. Forward stagewise regression is aimed to reduce greediness by adding variables partially. It adjust weights of explanotary variables by epsilon at each step, in certain direction, however problem of inefficiency arises. Lasso and Stagewise are basic forms of LARS. LARS requires m steps, where m equal to number of covariates. The algorithm of LARS: at first stage we find predictor, with the strongest correlation with residuals: it means, that variable has the least angle with residuals. We continue moving in direction of that predictor, until other predictor is equally correlated with the residuals. Then, move in direction, such that both predictors are equally correlated with the residuals, continue until third predictor become equally correlated and etc. LARS advantage is that it requires number of steps, which is not more than number of variables, as at each step new variable is added. LARS is exected to give similar to LASSO and Stagewise results.

Data set

As potential predictors for volatility, it has been chosen 356 financial and macroeconomics variables. Our dependent variable ( - volatility. Volatility is unobservable, so the following proxies were used: Squared return, Absolute return and Range-Based proxy. Range-based proxy is equal to: , where . As foundation for proxy, I chose return on SPDR® S&P 500 Exchange Traded Fund from the start of 1993 year till nowadays, with a weekly frequency. As another proxy, I have also used sample variance for each week, using daily SPY returns.

The initial dataset of potential predictors contained:

1)Various indexes: intercontinental exchange completes acquisition of Bank of America Merrill Lynch's (ICE BofAML) indexes on corporate bonds of different credit rating, Russel 1000, Russel 2000(small-cap), Russel 2500, Russel top 200,Russel 3000(measure performance of 3000 largest US companies) value, growth and price indexes, Wilshire 5000 & Wilshire US real estate securities indexes, Dow Jones index and CBOE volatility indexes(Chicago Board Options Exchange VIX) & Cleveland Financial stress index.

2)Yield on different securities: Treasury Inflation-Indexed notes of different maturity, Treasury Bills rate, SWAP rates, fitted yields of zero-coupon bonds(1,2,3,5,10 years), ICE BofAML US Corporate bonds yield, Moody's Seasoned Aaa/Bbb Corporate Bond Minus Federal Funds Rate, Overnight Repurchase Agreements rate and Reverse, return on a diversified portfolio of small stocks minus the return on a diversified portfolio of big stocks(SMB), difference between the returns on diversified portfolios of stocks with robust and weak profitability, difference between the returns on diversified portfolios of high and low B/M stocks and difference between the returns on diversified portfolios of low and high investment stocks, Certificate of deposits rate

3)Commercial paper rate: financial & nonfinancial(AA rating, 20-30-60-90 days)

4)Prices on oil and precious metals: Conventional Gasoline Prices: New York Harbor/ U.S. Gulf Coast, Gold Fixing Price 3:00 P.M. (London time) in London Bullion Market, Kerosene-Type Jet Fuel Prices: U.S. Gulf Coast, Propane Prices: Mont Belvieu, Reformulated Gasoline Blendstock for Oxygenate Blending (RBOB) Prices, Regular Gasoline(Los Angeles),Diesel fuel prices, Crude oil prices Brent.

5)LIBOR with maturity of 1,2 weeks 1,3,4,6,9,12 months in different currencies: US dollar, Australian dollar, Canadian dollar, Japanese Yen, Swiss franc, British pound & Euro.

6)Exchange rate: China-EU, China-Japan, China-USA, EU-Australia, EU-Canada, EU-Russia, EU-USA, USA-UAE (United Arab Emirates ), USA-Australia, USA-Brazil, USA-Canada, USA-India, USA-Mexico, USA-Russia.

7)Fama-French 25 Portfolios Formed on Size and Book-to-Market

8)Some key factors, which probably effect volatility: primary credit rate, TED(Treasury Euro-Dollar) spread, Bank Prime loan spread, risk-free rate, overnight bank funding volume and interest rate on required reserves.

9)OECD based Recession Indicators full data set is in appendix

Initially, all data was in daily frequency, from 1990 year(then it was cut from 1993 in order to fit with dependent variable) and was than aggregated into weekly data on the following basis:

For indexes, exchange rates and prices, I took log difference between beginning of the week(Tuesday) and end of the week (see R code in appendix for further details)

For yields of securities and commercial papers, I use compounding to aggregate daily yield into weekly one.

LIBOR was aggregated by taking average rate during the week.

Indicators were aggregated by taking maximum during the week. ( 1 or 0 )

All explanatory variables has lags of t-1 and earlier, compared to response variable: volatility proxy.

Main Results:

In the following section, I describe the results obtained from regularization techniques, described above. Coefficients obtained from each regularization method show that variable is significant in prediction of volatility if they not equal to zero.

If we consider elastic net/Lasso/Ridge, cross-validation method was used to find appropriate parameter: . We have conducted elastic net procedures for alpha sequence from 0 to 1 with step 0.1 . The set of most significant variables is going to be selected by comparing all the models with different alphas on basis of their loss functions. In order to minimize errors and for best regularization results that minimize cross-validation error should be selected. How we get it? - through cross-validation procedures. Cross-validation is out-of-sample testing where we compute erros for each level of . To demonstrate it, we can look at the plot of cross validation in case of alpha = 1(Lasso) and absolute returns proxy as a response variable. for such case is 0.00115056 and it considers 16 variables as significant. The cross-validating error in this case is 0.00146994.



Red line is cross-validation curve, there are also error bags - upper and lower standard deviation curves along the л sequence, and vertical line indicate the selected л.

To start, we have conducted these methods to the model with absolute returns as dependent variable.

Ridge regression, even though does not assign zero coefficients to predictors, has estimated only several not too small betas (bigger than 0.0001). Small minus Big Fama French portfolio return has small and negative coefficient, so rise in return of SMB will lead to lower volatility. The same situation is with Wilshire US Micro-Cap Total Market Index, Reformulated Gasoline Blendstock for Oxygenate Blending (RBOB) Prices and ICE BofAML US High Yield BB Total Return Index Value. ICE BofAML Euro High Yield Index Total Return Index Value has the highest, but also negative coefficient. Ridge regression encourage intercept to be small, and its small significance decrease more by increase in alpha. Also with increase in alpha, beta coefficients of variables increase and there are more and more non-zero coefficients. In Lasso case, there are new significant variables, such that: 1-Year Swap Rate, 3-Month Commercial Paper Minus Federal Funds Rate, 5-Year Breakeven Inflation Rate and CBOE DJIA Volatility Index. Models with alpha 0.8 and 0.9 choose even more number of variables than Lasso: Fama French portfolio “BIGLoPRIOR”( which we will find as one of the most important variables in prediction volatility in XGBoost method described later) , Repurchase Agreements: Mortgage-Backed Securities Purchased by the Federal Reserve in the Temporary Open Market Operations and 10-Year 2-1/8% Treasury Inflation-Indexed Note.

The most reasonable way, will be to choose parameters, selected by elastic net as it is a trade -off between lasso and ridge. But, before it, we should consider loss function for models with different alpha and select one with smallest loss. There is a table of loss for all alpha values:

Alpha	Loss
0	0.2774631
0.1	0.2774631
0.2	0.2672045
0.3	0.2653557
0.4	0.263717
0.5	0.2603938
0.6	0.2621397
0.7	0.2757193
0.8	0.2536584
0.9	0.2602541
1	0.2672045

As we could see, loss value decrease as alpha rise from 0 to 0.5. So ridge is worse than elastic net. However, we could also see than minimum loss function obtained when alpha = 0.8 . So, we choose it as the most appropriate for prediction volatility. It includes the following variables as significant:

1) Intercept

2) 1-Month London Interbank Offered Rate (LIBOR) in Japanese Yen

3) 3-Month Commercial Paper Minus Federal Funds Rate

4) Small minus Big portfolio return

5) TED Spread

6) Contributions to the Cleveland Financial Stress Index: Credit Markets

7) 2-Year swap rate

8) ICE BofAML US Corporate 5-7 Year Effective Yield

9) ICE BofAML US Corp 15+yr Total Return Index Value

10) Reformulated Gasoline Blendstock for Oxygenate Blending (RBOB) Prices

11) Venezuela / U.S. Foreign Exchange Rate

12) Wilshire US Large-Cap Value Total Market Index

Wilshire US Micro-Cap Total Market Index
We get 14 predictors out of 356 variables. Bellow is a table of coefficients estimated: : only non-zero variables are presented. Other variables have zero coefficients for all alpha values, so are insignificant

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ALPHA:	0(Ridge)	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	1(Lasso
INTERCEPT	0.01153	0.01153	0.01085	0.01072	0.01060	0.01051	0.01053	0.01139	0.01044	0.01085
JPY1MTD156N	0.0000..	0	0	0	0	0	0	0	-0.00017	0
CPFF	0.0000..	0	0.00064	0.00077	0.00088	0.00109	0.00104	0	0.00122	0.00064
SMB	-0.000143	-0.00014	-0.001822	-0.002127	-0.002398	-0.00283	-0.00263	-0.00046	-0.00311	-0.00182
TEDRATE	0.00214	0.00214	0.00175	0.00160	0.001450	0.0010393	0.001222	0.0021897	0.0007641	0.001752567
CMRKTSD678FRBCLE	0.0000..	0	0.0000755	0.0000897	0.000103	0.0001251	0.0001175	0.0000046	0.000155	0.00007558
DSWP1	0.0000..	0	-0.000142	-0.000191	-0.000214	0	0.00008	0	0	-0.00014229
DSWP2	0.0000..	0	0	0	-0.0000262	-0.000344	-0.000216	0	-0.000460	0
BAMLC3A0C57YEY	0.0000..	0	0.0000340	0	0.0000801	0.0001175	0.0000975	0	0.0001625	0.000034086
BAMLC3A0C57Y	0.0001894	0.0001894	0.0003461	0.0003698	0.000389	0.0003373	0.0003393	0.000241	0.0003604	0.00034611
BAMLH0A2HYBSYTW	0.000141	0.000141	0.0001402	0.0001387	0.0001374	0.0001388	0.0001396	0.0001412	0.0001335	0.00014021
RPMBSD	0.0000..	0	0	0	0	0	0	0	-0.0000105	0
DTP10J19	0.0000..	0	0	0	0	0.0003765	0.0002149	0	0.0008276	0
T5YIE	0.0000..	0	-0.000271	-0.000325	-0.0003704	-0.0004365	-0.000385	0	-0.0006598	-0.00027174
EFFRSD	0.01518	0.0151803	0.0226196	0.0242763	0.025912	0.030950	0.0279550	0.0161065	0.0392495	0.0226196
BIGLoPRIOR	0.0000..	0	0	0	0	0	0	0	0.0001605	0
VXDCLS	0.0000..	0	0.0024735	0.0031793	0.0038073	0.0048662	0.0043568	0	0.0068523	0.00247357
BAMLHE00EHYITRIV	-0.042199	-0.04219	-0.064234	-0.068222	-0.0718708	-0.0784876	-0.075503	-0.0462721	-0.0977860	-0.064234
BAMLCC8A015PYTRIV	0.0000..	0	0	0	0	-0.0027	0	0	-0.012628	0
BAMLEMCLLCRPIUSTRIV	0.0000..	0	0	0	0	0	0	-0.0034807	0	0
BAMLHYH0A1BBTRIV	-0.036709	-0.036709	-0.026835	-0.021549	-0.016466	-0.00303	-0.010165	-0.0383969	0	-0.0268358
DRGASLA	-0.000112	-0.0001127	-0.0066303	-0.0076674	-0.0086338	-0.01046	-0.0095609	-0.001652	-0.0130172	-0.00663039
DEXVZUS	0.0000..	0	0	0	0	0.0000291	0	0	0.0005995	0
BAMLEM1RAAA2ALCRPIUSTRIV	0.0000..	0	-0.0416	-0.058615	-0.074662	-0.103074	-0.090456	0	-0.115968	-0.041620
WILLLRGCAPVAL	0.0000..	0	0	0	0	0	0	0	-0.00581	0
RPONTSYD	0.0000..	0	0	0	0	0	0	0	0.000148	0
WILLMICROCAPPR	-0.002286	-0.002286	-0.001123	-0.00038	0	0	0	-0.003465	-0.003105	-0.0011232

Now we will check cases with other proxies. Let us consider case of squared returns as proxy for volatility. In this case, intercept has less weight than before(absolute returns case) and number of significant predictors is almost the same. Ridge regression does not provide with big enough coefficients: ICE BofAML US High Yield BB Total Return Index Value has negative and highest coefficient. In absolute returns case there is also such predictors, so we can conclude that it is not by chance and it really affect volatility negatively. Ridge also points out Effective federal funds rate as a predictor. Lasso includes some new variables as Wilshire US Small-Cap Value Total Market Index and Repurchase Agreements: Mortgage-Backed Securities Purchased by the Federal Reserve in the Temporary Open Market Operations. We will proceed the same procedure as before to determine the appropriate alpha.

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Alpha	Loss
0	0.002471132
0.1	0.002235313
0.2	0.002548035
0.3	0.002088046
0.4	0.002203469
0.5	0.002000437
0.6	0.00226212
0.7	0.002288374
0.8	0.002235313
0.9	0.002288374
1	0.002203469

The least loss is in elastic net model(alpha = 0.5). The loss,overall, much more smaller than in case of absolute returns case, even though models select the same amount of predictors. Elastic net model in case of squared returns as a response variable selects:

1) Intercept

2) 3-Month Commercial Paper Minus Federal Funds Rate

3) Overnight London Interbank Offered Rate (LIBOR), EURO

4) 30-Year Swap Rate

5) 4-Year Swap Rate

6) ICE BofAML US High Yield B Semi-Annual Yield to Worst

7) Repurchase Agreements: Mortgage-Backed Securities Purchased by the Federal Reserve in the Temporary Open Market Operations

8) Effective Federal Funds Rate

9) Dow Jones Utility Average

10) Sweden / U.S. Foreign Exchange Rate

11) Wilshire US Large-Cap Value Total Market Index

12) ICE BofAML AAA-A US Emerging Markets Liquid Corporate Plus Sub-Index Effective Yield

13) Propane Prices: Mont Belvieu

Table 4. “ Elastic net/Lasso/Ridge” estimated coefficients: dependent variable - Squared returns; from R-software

ALPHA	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1
INTERCEPT	0.00016	0.00009	0.00020	0.00012	0.00009	0.00015	0.00009	0.00009	0.00009	0.00009	0.00009
CPFF	0.00004	0.00021	0.00000	0.00019	0.00021	0.00016	0.00020	0.00019	0.00021	0.00019	0.00021
EURONTD156N	0.00000	0.00000	0.00000	0.00000	0.00000	-0.00002	0.00000	0.00000	0.00000	0.00000	0.00000
TEDRATE	0.00009	0.00009	0.00007	0.00003	0.00008	0.00000	0.00010	0.00010	0.00009	0.00010	0.00008
DSWP2	0.00000	-0.00001	0.00000	0.00000	-0.00001	0.00000	-0.00001	0.00000	-0.00001	0.00000	-0.00001
DSWP30	0.00000	0.00000	0.00000	0.00000	0.00000	-0.00000	0.00000	0.00000	0.00000	0.00000	0.00000
DSWP3	0.00000	0.00000	0.00000	-0.00001	-0.00001	0.00000	0.00000	0.00000	0.00000	0.00000	-0.00001
DSWP4	0.00000	0.00000	0.00000	-0.00005	0.00000	-0.00006	0.00000	0.00000	0.00000	0.00000	0.00000
BAMLC1A0C13Y	0.00001	0.00002	0.00000	0.00000	0.00002	0.00000	0.00003	0.00003	0.00002	0.00003	0.00002
BAMLH0A2HYBSYTW	0.00001	0.00001	0.00001	0.00001	0.00001	0.00001	0.00001	0.00001	0.00001	0.00001	0.00001
RPMBSD	0.00000	-0.00000	0.00000	-0.00000	-0.00000	-0.00000	0.00000	0.00000	-0.00000	0.00000	-0.00000
T5YIE	0.00000	0.00000	0.00000	0.00000	0.00000	-0.00001	0.00000	0.00000	0.00000	0.00000	0.00000
DTP10L18	0.00000	0.00000	0.00000	0.00008	0.00000	0.00011	0.00000	0.00000	0.00000	0.00000	0.00000
EFFRSD	0.00434	0.00543	0.00405	0.00764	0.00587	0.00882	0.00517	0.00497	0.00543	0.00497	0.00587
BIGLoPRIOR	0.00000	0.00000	0.00000	0.00000	0.00000	0.00001	0.00000	0.00000	0.00000	0.00000	0.00000
EVZCLS	0.00000	0.00000	0.00000	0.00000	0.00000	-0.00005	0.00000	0.00000	0.00000	0.00000	0.00000
DJUA	0.00000	-0.00361	0.00000	-0.00419	-0.00379	-0.00482	-0.00340	-0.00316	-0.00361	-0.00316	-0.00379
BAMLHE00EHYITRIV	-0.00318	-0.00892	-0.00122	-0.01107	-0.00931	-0.01264	-0.00848	-0.00800	-0.00892	-0.00800	-0.00931
BAMLCC1A013YTRIV	0.00000	0.00000	0.00000	-0.00775	0.00000	-0.01639	0.00000	0.00000	0.00000	0.00000	0.00000
BAMLHYH0A1BBTRIV	-0.01034	-0.01183	-0.00737	-0.00794	-0.01133	-0.00722	-0.01231	-0.01277	-0.01183	-0.01277	-0.01133
DPROPANEMBTX	0.00000	0.00000	0.00000	-0.00017	0.00000	-0.00048	0.00000	0.00000	0.00000	0.00000	0.00000
DEXSDUS	0.00000	0.00000	0.00000	0.00000	0.00000	-0.00041	0.00000	0.00000	0.00000	0.00000	0.00000
BAMLEM1RAAA2ALCRPIUSTRIV	0.00000	-0.01423	0.00000	-0.02524	-0.01658	-0.02744	-0.01208	-0.00990	-0.01423	-0.00990	-0.01658
WILLLRGCAPVAL	0.00000	-0.00173	0.00000	-0.00339	-0.00202	-0.00673	-0.00136	-0.00093	-0.00173	-0.00093	-0.00202
DJTA	0.00000	0.00000	0.00000	0.00000	0.00000	0.00210	0.00000	0.00000	0.00000	0.00000	0.00000

Models with Range - Based proxy and weekly variance proxy select very large amount of variables: more than 200 predictors are selected according to different levels of alpha, so full table of coefficients see in Appendix. v

For range-based proxy models' loss is the following:

Alpha	Loss
0	2257.252
0.1	1548.473
0.2	2328.864
0.3	2221.366
0.4	1304.84
0.5	2005.689
0.6	1439.564
0.7	1668.803
0.8	2367.631
0.9	2400.224
1	2517.607

Loss is very big compared to other cases, so inference about possible predictors would not be reliable. It is consistent with the fact, that model choose to many variables as potential predictors, so that results in big penalty for overfitting. The smallest loss is in the case of alpha = 0.4

There are some variables with very big coefficients, which have attracted our attention:

1)Trade Weighted U.S. Dollar Index (coefficient = 11.424)

2)Thailand / U.S. Foreign Exchange Rate (coefficient = 8.15)

3)ICE BofAML US indices (with coefficients from -16 to 96)

4)TED Spread ( coefficient = 3.4)

The most interesting result from analysis of coefficients is volatility is very affected by different ICE BofAML indices. Its resulting coefficients are:

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-11.626016	BAMLEMPBPUBSICRPIEY
-1.42770964	BAMLEMPUPUBSLCRPIUSOAS
-48.5929859	BAMLCC7A01015YTRIV
-16.0994596	BAMLCC8A015PYTRIV
-42.1312499	BAMLCC3A057YTRIV
12.6695322	BAMLCC0A1AAATRIV
93.5374674	BAMLCC0A2AATRIV
-29.9658017	BAMLCC0A3ATRIV
-67.3080103	BAMLCC0A4BBBTRIV
96.80983291	BAMLCC4A0710YTRIV
-5.08671201	BAMLEMCLLCRPIUSTRIV
-6.22979298	BAMLHYH0A1BBTRIV
-15.8589592	BAMLHYH0A2BTRIV
-10.9445417	BAMLHYH0A3CMTRIV
-16.696839	BAMLH0A0HYM2SYTW

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Multi-assets type indices provide a broad benchmark of the market performance. We have considered broad range of indices of different regions and asset classes. So, high predictive power of these variables is consistent with concept of volatility clustering. Some, though not all, ICE BofAML indeces were captured by all types of elastic net models. However, scale of coefficients do not look realistic, so It may be caused by choosing wrong proxy: Range - Based proxy may not be an appropriate proxy for volatility.

The last type of model we should consider is model with weekly variance proxy for volatility as response variable See Appendix for full coefficient table. It also estimated very large amount of coefficients, as in case of Range - Based proxy: around 200 variables are considered to have non-zero coefficient for different alpha values. As in case before, we will put attention on variables with largest coefficients. But at first, we look for the best alpha value for model. The corresponding loss values are:

...

Alpha	Loss
0	0.000051
0.1	0.000078
0.2	0.000048
0.3	0.000078
0.4	0.000612
0.5	0.000053
0.6	0.000077

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