## VIX as Predictor Variable of Stock Market Returns

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This document studies the power of CBOE Volatility Index (VIX) to predict stock market returns of Standard & Poor’s 500 (S&P 500), a good proxy of the USA economy. VIX measures the market’s expectation of S&P 500 volatility, however many authors call it fear index because the value of the index spikes in moments of high selling in financial markets. Even though VIX is a common indicator in technical analysis and market sentiment (Baker & Wurgler, 2007), this study fits better to fundamental analysis of the USA economy as a whole.

The download includes the complete master thesis in docx and pdf. The documents are well structured and correctly formatted. Additionally, it includes the complete project in R programming language, it is used to gather the data from internet, process the data, create all regression models and plot the images present in the management project.

- Date: 2019-02-24
- Author: Raúl Bartolomé Castro
- Tutor: Dr. Steve Wu.
- Program: Distance Learning MBA
- Module: Management Project

## Description

Table of Contents

4.2 VIX: CBOE Volatility Index 11

4.3 DGS3M0 and DGS10: USA Treasuries 12

4.4 DAAA and DBAA: Moody’s Seasoned Corporate Bond Yield 12

4.5 CPIAUCSL: Consumer Price Index for All Urban Consumers 13

4.6 SPY and SPYdiv: SPDR S&P 500 ETF 14

5.1.1 GSPCret_excp: GSPC in excess of periorized DGS3MO 17

5.1.1 GSPCret_exca: GSPC in excess of DGS3MO 18

5.2.1 VIX: CBOE Volatility Index 19

5.2.2 TB3M: Real 3-Month Treasury Bill 20

5.2.5 SPYDY: SPY Dividend Yield 24

7 Regressions Results and Discussion 27

7.3 Contribution in Multivariate Models 34

11.1 Fundamental Analysis, Technical Analysis and Market Sentiment 45

11.2.2 Periorized Simple Return 46

11.2.3 Continuously Compounded (CC) Return 46

List of Figures

Figure 1. Main events for GSPC 10

Figure 2. GSPC: S&P 500 index 10

Figure 3. Main events for VIX 11

Figure 4. VIX: CBOE volatility index 11

Figure 5. DGS3M0 and DGS10: USA treasuries 12

Figure 6. DAAA and DBAA: Moody’s seasoned corporate bond yield 13

Figure 7. CPIAUCSL: consumer price index for all urban consumers 13

Figure 8. SPY: SPDR S&P 500 ETF 14

Figure 9. SPYdiv: dividend of SPY 14

Figure 10. Lagged GSPC 16

Figure 11. Histograms of GSPCret_excp 18

Figure 12. Histograms of GSPCret_exca 18

Figure 13. Histograms of VIX 19

Figure 14. CPI time series 20

Figure 15. Histograms of TB3M 21

Figure 16. Histograms of CS 22

Figure 17. Histograms of TS 23

Figure 18. SPYdivttm time series 24

Figure 19. SPYDY time series 24

Figure 20. Histograms of SPYDY 25

Figure 21. Model 238. GSPCret_excp against VIX 29

Figure 22. Model 77. GSPCret_excp against TB3Mretp 31

Figure 23. Model 133. GSPCret_excp against TB3M 33

Figure 24. Model 122 36

Figure 25. Model 165 38

List of Tables

Table 1. Literature review summary 6

Table 2. Input variables 9

Table 3. Regression variables observations and data range 15

Table 4. Variables for regressions models 16

Table 5. Statistics of GSPCret_excp 17

Table 6. Statistics of GSPCret_exca 18

Table 7. Statistics of VIX 19

Table 8. Statistics of TB3M 21

Table 9. Statistics of CS 22

Table 10. Statistics of TS 23

Table 11. Statistics of SPYDY 25

Table 12. Regression models 26

Table 13. Per-se evaluation summary 28

Table 14. Relative evaluation, daily observations 30

Table 15. Relative evaluation, monthly observations 31

Table 16. Relative evaluation, quarterly observations 32

Table 17. Multivariate models, daily observations 34

Table 18. Multivariate models, monthly observations 35

Table 19. Multivariate models, quarterly observations 37

# Introduction

This document studies the power of CBOE Volatility Index (VIX) to predict stock market returns of Standard & Poor’s 500 (S&P 500), a good proxy of the USA economy. VIX measures the market’s expectation of S&P 500 volatility, however many authors call it fear index because the value of the index spikes in moments of high selling in financial markets^{[1]}. Even though VIX is a common indicator in technical analysis and market sentiment (Baker & Wurgler, 2007), this study fits better to fundamental analysis of the USA economy as a whole.

The principle of this quantitative analysis is the evaluation of regression model fit of hundreds of models to understand the long-term relationship between macroeconomic variables. S&P 500 lagged to explanatory variables conforms the response variable. VIX, USA treasuries, Moody’s seasoned corporate bond yield, consumer price index (CPI) and S&P 500 dividend yield constitute the sources for explanatory variables. Many other relevant macroeconomics variables such as gross domestic product (GDP), oil prices or price earnings are excluded to avoid an over utilization of explanatory variables that would lead to regressions model fits with low statistical significance.

This dissertation articulates a solid empirical evaluation of VIX as predictor of stock market returns paying special attention to the method used to calculate the variables for the regression’s models. The approach bridges the articles focused in volatility and generic market returns where the variables are calculated as annualized returns (McMillan, 2016; Bekaert & Hoerova, 2014) and capital asset pricing modeling (CAPAM) studies where the variables are calculated as periorized returns (Daróczi et al., 2013, p. 44; Kempthorne et al., 2013).

The conclusions of this paper bring clarity over the past relationship of S&P 500, VIX and other macroeconomic variables that may be used by investors and mangers to take better future financial decisions. Nevertheless, assuming that the asset prices may reflect a fair value at any given time as capture in a weak-form of efficient market hypothesis (EMH), investors and managers should be cautious taking decisions from historical data.

This work is organized as follows:

- Literature review,
- hypothesis definition,
- explanation and acquisition of the input variable,
- explanation and calculation of variables used for the regression’s models,
- list of regressions models and
- summary and conclusions.

A sensible expansion of this study is a microeconomic analysis for those companies with volatility index provided by CBOE, such as VIX on Apple (VXAPLSM). It is advisable to use CAPAM theory as framework, 3-months USA treasuries as risk free asset and confirm the predictability power of VXAPLSM by comparing the regressions models fit.

# Literature Review

Exists an extensive body of literature related to market volatility and market returns. Next table depicts some of the most important articles where market volatility or VIX plays a central role in stock market returns forecasting.

Table 1. Literature review summary

The study of volatility and how it might impact to stock market returns has captured high interest in the financial literature, already in 1976 Finisher Black in his article “Studies of stock price volatility changes” starts to pay attention to the subject. French et al. in 1987 expand the study and find evidences of positive relationship between market returns in excess of treasure bills and volatility, something that is also observed in this document. They paper models volatility using the methods autoregressive conditional heteroskedasticity (ARCH) and autoregressive integrated moving average (ARIMA), VIX was not a possibility because it appears in 1990. The authors calculate markets returns as annual continuously compounded (CC)^{[2]}, the standard methods used in articles where volatility and stock markets returns coexists.

Exists a solid literature that explains how to calculate VIX (Carr & Madan, 1998; Demeterfi et al., 1999; Britten‐Jones & Neuberger, 2002 and Bollerslev et al., 2009) and the CBOE also provides a very comprehensive paper (CBOE, 2018). VIX is a model-free, as opposed to Black-Scholes-Merton, that calculates the expected volatility of S&P 500 using options derivatives with an expiration date of 30 days. Nevertheless, the calculation method brings little value to this study because we are more interested in the utilization of VIX as predictor variable rather than the calculation in itself.

Bollerslev et al. (2009, 2013) and Bekaert & Hoerova (2014) are pioneers to recognize that VIX captures Implied Volatility (IV) and risk premium and explore the decomposition of the variable. The risk premium is what gives to VIX the pseudonym of fear index (Whaley, 2000 and Fernandes et al., 2013), that is the willingness of traders and investors to purchase options in declining markets to hedge long positions. Bekaert & Hoerova (2014) split VIX in an elegant equation by the sum of variance premium (VP) and realized variance (RV):

Where the root square of VIX is equal to VP plus the expectation of RV over the next 22 days. The authors conduct a profound evaluation of models to forecast RV, concluding that the best estimators are a combination of VIX, continuous and discontinuous jumps of RV as defined per Corsi et al. (2010).

One of the conclusions of Bollerslev et al. (2009) and Bekaert & Hoerova (2014) is that VP is better market returns indicator than VIX. This opens an additional research line for this investigation, to evaluate VP calculated as periorized simple return. Albeit, this is excluded because it would excessively bend this research to volatility modeling rather than stock market returns prediction.

Academics and professionals have developed a rich pallet of articles related to volatility modeling, where sometimes VIX is included. Kozyreva (2007) using the Nordic market index, compares the calculation method of VIX and generalize ARCH, concluding that GARCH is more accurate to capture volatility. She acknowledges the presence of significant differences during disruptive events in the market (“jumps”). Fernandes et al. (2013) perform a thorough statistical examination of VIX using heterogeneous autoregressive (HAR) processes for modeling and forecasting purposes. They conclude that VIX does not depend of other macroeconomic variables like oil features or interest rates. Chow et al. (2014) demonstrate that VIX does not truly capture volatility and define a generalized VIX method that is more effective. Thorlie et al. (2015) and Seul-Ki et al. (2017) are some of the authors that expand the state of the art methodologies to model volatility using GARCH, asymmetric power ARCH and HAR for S&P 500 and Korea Stock Price Index (KOSPI). Some of these volatility per-se studies deviate from the main goal of this dissertation, consequently they will not play a relevant role. Though, they provide a good foundation for a holistic study of volatility and market returns that could expand this paper.

Comparing the performance of VIX with other well know predictor variables, is a sensible method to judge up to what extend VIX is effective in its predictions. Consequently, part of the literature research consists to identify articles related to market returns studies with diverse variables. Anilowski et al. (2007) investigate the influence of earnings and earnings announcements to the performance of the stock market. Bollerslev et al. (2009) include in their article macroeconomic variables such as price to earnings ratio (P/E), price to dividends ratio (P/D), consumption-wealth ratio (CAY), term spread (TS), default spread (DFSP) and risk-free rate (RREL). Ferreira & Santa-Clara (2011) using the method sum-of-the-parts (SOP) for forecasting stock markets utilizes three variables: P/E ratio, P/D ratio and earnings ratio growth. Bekaert et al. (2013) and Bekaert & Hoerova (2014) are influencing articles in this dissertation, they consider VIX, TS, credit spread (CS), consumer price index (CPI), industrial price index (IPI), P/D ratio and other macroeconomic variables. McMillan (2016) conducts a very extensive evaluation of tenths of variables, among them P/E ratio, P/D ratio, earnings ratio, GDP, CPI, IPI, TSMP, market capitalization, etc. The benchmarking variables for this work are real 3-month treasury bills (TB3M), CS, TS and SPY dividend yield (SPYDY).

# Hypothesis Definition

The hypothesis definition pivots around model testing to confirm whether VIX is an affective indicator of stock market returns. Consequently, the null hypothesis (H0) assumes that VIX is not an effective indicator of S&P 500 returns and the alternative hypothesis (H1) assumes that VIX is an effective indication of S&P 500 returns. We evaluate the hypothesis from three different angles:

First approach consists to build linear regression models with VIX and the response variable, to obtain the coefficient values ( and fit the model with the sample data, to obtain adjusted R^{2}. The t-statistic tests the null hypothesis, the alternative hypothesis is accepted if is different from 0 and the chance to get a sample outside of the alternative hypothesis is lower than 1% (t-statistic significance). Adjusted R^{2} defines the proportion of the variance of the response variable captured by the explanatory variable. The value ranks from +1 to -1, where a value closed to unity means a high effective indicator.

Previous technique provides a good per-se evaluation; albeit, it does not indicate how good is compared with other indicators. To fill this gap, the second evaluation of the hypothesis is done in relative terms, comparing the effectiveness of VIX with other well-established indicators such as TB3M, CS, TP and SPYDY. Adjusted R^{2 }is the main parameter for the comparison but the t-statistic significance is also taken into account.

Finally, the hypothesis is tested evaluating the effectiveness of VIX to improve the predictability of stock markets returns in multivariate regression models. This is capture by a relative comparison of four explanatory variables versus models with the same four variables plus VIX.

# Input Variables

The foundation of this dissertation is the input data used for the statistical inference. Next are the nine considered variables:

- GSPC: daily closing value of S&P 500 index.
- VIX: daily closing value of CBOE volatility index.
- DGS3M0: USA 3-month treasury bill.
- DGS10: USA 10-year treasury note.
- DAAA: Moody’s seasoned Aaa corporate bond yield.
- DBAA: Moody’s seasoned Baa corporate bond yield.
- SPY: daily closing value of SPDR S&P 500 ETF.
- SPYdiv: dividend of SPDR S&P 500 ETF.
- CPIAUSCL: monthly consumer price index for all urban consumers.

Table 2 provides the key quantitative characteristics of the data. All variables are secondary data obtained from reputable sources. The variables GSPC, VIX, SPY and SPYdiv are downloaded from Yahoo! Finance. DGS3M0, DGS10, DAAA, DBAA and CPIAUSCL are obtained from Economical Data from Federal Reserve Bank of St. Louis (FRED).

In order to achieve the highest statistical significance, the time series should cover the longest period of time. The starting date of the data set is defined by SPY, the variable with the most recent creation, that was launched by State Street Global Advisors (SSgA) on January 1993. The ending date is defined by the date of the last data acquisition for this study, limited by the most recent date, in this case CPIAUSCL on October 2018.

It is a common practice in market returns studies to consider various observation periods (Bollerslev et al., 2009; Bekaert & Hoerova, 2014), we target three different observations period: daily, monthly and quarterly. For the variables GSPC, VIX, DGS3MO, DGS10, DAAA, DBAAA and SPY this is done by downloading data with daily frequency and with decimation process build monthly and quarterly data sets. Numerically, this is translated to 6720, 311 and 104 observations.

The decimation process is excluded for CPIAUSCL and SPYdiv because they participate in intermediate steps to create the regressions variables. SPYdiv is the dividends distribution of SPY that is provided every quarter granting 104 observations. CPIAUSCL is monthly information provided by the FRED, it is issued since 1947 that implies 861 observations.

## GSPC: S&P 500 Index

S&P 500 index, ticker symbols ^GSPC, INX and $SPX, is a stock market index that tracks the 500 large USA companies listed in New York Stock Exchange (NYSE) and Nasdaq Stock Market (NASDAQ). The index is built by assigning a weight to each company based on the market capitalization, calculated as the market value of their outstanding shares. S&P 500 captures many industries and covers approximately 80% of the available market capitalization (Standard & Poor’s, 2018) consequently it is a good proxy for USA stock market analysis.

Figure 1 shows the most distinctive traits of GSPC: the dot com bubble in 2001, the starting of the financial crisis and Lehman bankruptcy in 2008 and the longest bull market period of 9 years stating in 2009 present a clear pattern. Other events such as the world trade center attack in 1998 or the rubble devaluation are not too significant.

GSPC is not used directly as response variable for the regression’s models, but GSPC in excess of the 3-month treasury bill. Likewise, since this is a forecasting exercise, the responsible variable is lagged against the explanatory variable for the regression fit testing, more details are given in subsequent chapters. Next figure depicts GSPC with three observation periods, the decimation process does a fair job holding the majority of the relevant information.

Figure 1. Main events for GSPC

## VIX: CBOE Volatility Index

VIX is a stock market index created by Chicago Board Options Exchange (CBOE) that measures the market’s expectation of S&P 500 volatility with a horizon of 30-days expressed in annualized percentage. VIX is calculated using options prices from S&P 500 with expiration date more than 23 days and less than 37 days and risk-free interest rates (CBOE, 2018). VIX captures the implied volatility but it does not assume the Black-Sholes-Merton model but rather a “model-free” estimator (Britten‐Jones & Neuberger, 2000; Jiang & Tian, 2005). Other approach to calculate the implied volatility is solving the Black-Sholes-Merton option price model that uses as parameters price of the derivative, strike price, drift rate, time to expiration risk free interest rate and standard deviation of the underlying security’ returns also called implied volatility (Hull, 2018, pp. 343-369) however in this study focuses on VIX rather than Black-Sholes-Merton derivative.

VIX receives his name of fear index because it spikes in periods of big uncertainty. This is well represented in Figure 3, where the top three events are the global financial crisis in 2008, the US and Europe dept downgrade in 2011 and the Russian crisis and default of long-term capital management in 1998, although other well know black swan events such as the dot com bubble in 2000 are hidden with the background noise.

VIX is the explanatory variable that captures most of the attention in this study, the hypothesis testing pivots around it and the strategy of regressions modelling as well. Figure 4 represents the variable with three observation periods. It is worth to mention that the quarterly data filters the information dramatically, the most significant information event, the global financial crisis in 2008, scores bellow US and Europe dept downgrade in 2011.

Figure 4. VIX: CBOE volatility index

## DGS3M0 and DGS10: USA Treasuries

USA Treasuries are government dept issued by USA Treasury Department used to finance government’s activities. The FRED provides data of 16 treasuries with constant maturity, next chart presents DGS3MO and DGS10. USA treasures occupies a preeminent position in macroeconomic models affecting the flow of money in the economy. It is not a surprise that they are key contributors in this document, especially DGS3MO.

DGS3MO is used in three occasions for the regression’s variables. First time to calculate the response variable as excess of GSPC and two times for explanatory variables as main constituent of real 3-months treasury bills and deductible member of TS. DGS10 only accounts one time as positive term of TS. Next figure plots DGS3MO in green and DGS10 in blue, it is easy to notice the long period of low interest rates from 2009 to 2016 to stimulate USA economy after the global financial crisis in 2008.

Figure 5. DGS3M0 and DGS10: USA treasuries

## DAAA and DBAA: Moody’s Seasoned Corporate Bond Yield

The FRED provides multitude of corporate bonds quotes, this investigation considers DAAA that is a granted to companies with extremely strong capacity to fulfill its financial commitments and DBAA that reflects that the company has an adequate capability to fulfill its financial obligations though it might face challenges in adverse economic conditions.

We create the explanatory variable CS, by subtracting DAAA from DBAA, representative of the ability of corporates to fulfil their dept obligations. Next image presents DAAA in red and DBAA in green, it is noticeable that the time series present a significant similitude to USA treasures, that is not a surprise since corporates depts are dependent of national interest rates.

Figure 6. DAAA and DBAA: Moody’s seasoned corporate bond yield

## CPIAUCSL: Consumer Price Index for All Urban Consumers

The CPIAUCSL is a measure of the average monthly change in the price for goods and services paid by urban consumers between any two time periods. It can also represent the buying habits of urban consumers. This particular index includes roughly 88 percent of the total population, accounting for wage earners, clerical workers, technical workers, self-employed, short-term workers, unemployed, retirees, and those not in the labor force (FRED, 2018).

The subtraction of CPIAUCSL from DGS3MO provides the real part of DGS3M0 an important explanatory variable. For the purpose of units’ coherence, the variable needs to be converted from index to annualized percentage, the operations are explained in detail in chapter 5.

It is easy to observe in the figure that the indicator grows steadily, only during the global financial crisis in 2008 is noticeable a peak of acceleration.

Figure 7. CPIAUCSL: consumer price index for all urban consumers

## SPY and SPYdiv: SPDR S&P 500 ETF

SPDR S&P 500 is an exchanged traded found (ETF) that trades under the symbol SPY. Yahoo! Finance provides the share price (SPY) and dividends (SPYdiv) which are used for the calculation of the explanatory variable SPY dividend yield (SPYDY). Next chart shows SPY in three observations periods. It is easy to appreciate that SPY tracks GSPC with high accuracy.

Figure 8. SPY: SPDR S&P 500 ETF

GSPC cannot be used for the dividend yields calculation because as an index does not provide dividends. Fortunately, SPY provides dividends four times per annum and tracks with accuracy GSPC. Figure 9 plots SPYdiv, since its inception in 1993 the fund is providing a fast-growing stream of capital with a peak just before the global financial crisis in 2008.

Figure 9. SPYdiv: dividend of SPY

# Regression Variables

This chapter describes the 21variables used to build regressions models, all of them are derived from input variables. 6 are the response variables grouped in two types, with periorized or annualized DGS3MO. The explanatory variables are 15, grouped in VIX, TB3M, CS, TS and SPYDY. In more detail:

- GSPCret_excp: CC return of GSPC in excess of CC return of periorized DGS3MO. With lags of 1, 3 and 12 months.
- GSPCret_exca: CC return of GSPC in excess of CC return of DGS3MO. With lags of 1, 3 and 12 months.
- VIX: CBOE Volatility Index.
- VIXretp: CC return of periorized VIX.
- VIXreta: CC return of VIX.
- Real 3-month rate (TB3M): difference between DGS3MO and CPI.
- TB3Mretp: CC return of periorized TB3M minus CC return of periorized CPI.
- TB3Mreta: CC return of TB3M minus CC return of CPI.
- Credit spread (CS): difference between DBAA and DAAA.
- CSretp: CC return of periorized DBAA minus CC return of periorized DAAA.
- CSreta: CC return of DBAA minus CC return of DAAA.
- Term spread (TS): difference between DGS10 and DGS3MO.
- TSretp: CC return of periorized DGS10 minus CC return of periorized DGS3MO.
- TSreta: CC return of DGS10 minus CC return of DGS3MO.
- SPYDY: SPY dividend yield.
- SPYDYretp: CC return of periorized SPDY.
- SPYDYreta: CC return of SPDY.

This investigation evaluates three different forecasting horizons (1, 3 and 13 months), consequently, the response variables shall be lagged 1, 3 and 12 months with regards to the explanatory variables. When the explanatory and response variable are consolidated, the lagging operation slices one more year the data set^{[3]}. Table 3 summarizes the number of observations and data range.

Table 3. Regression variables observations and data range

Table 4 lists the 21 regressions variables. The column variable name gives a distinctive abbreviation of the variable. The units of the majority of variables are percentages expressed in per unit basis (per one) except VIX, TB3M, CS, TS and SPYDY that are expressed per hundred basis (percent). The table also captures a simplified view of the calculation method per each case. Next chapters provide a detail explanation per each case.

Table 4. Variables for regressions models

## Response Variables

For the purpose to test the forecasting effectiveness of the regression models, the response variables shall be lagged with regards the explanatory variable. It is also valid all the way around, the key requirement is that the variables shall be lagged each other’s. In this occasion GSPC is lagged 1, 3 and 13 months with regards to the explanatory variables.

Next image plots three lags for daily and monthly data and two lags for quarterly data. Note that one month lagged data (1m) seems overlapped with non-lagged data (0m) due to the resolution of the chart.

Previous image is intended to illustrate the lagging concept, in computational terms GSPC is not lagged but GSPC in excess of DGS3MO either periorized or annualized.

### GSPCret_excp: GSPC in excess of periorized DGS3MO

Next three equations represent the calculation method for three time series of GSPCret_excp. Each one includes the lag period, the CC return of GSPC minus the periorized CC return of DGS3MO.

The following table and illustrations provide the statistical information and graphics representation of the variables. The daily observations present a normal distribution shape with very high kurtosis. Monthly data is more normally distributed but with many observations away from the mean. The quarterly observations do not follow normal distribution with strong presence of observations in the extremes, indicative of fat tails.

Table 5. Statistics of GSPCret_excp

Figure 11. Histograms of GSPCret_excp

### GSPCret_exca: GSPC in excess of DGS3MO

The equations for GSPCret_exca follow high similitude to GSPCret_exca, the differences reside in the method used to calculate the excess return. In this occasion the excess is calculated annually rather than periorized.

The statistics and charts present a distinctive distribution with extremely high kurtosis, observations distributed around the mean, fat tails in both directions and certain negative kurtosis for monthly and quarterly time series.

Table 6. Statistics of GSPCret_exca

Figure 12. Histograms of GSPCret_exca

## Explanatory Variables

### VIX: CBOE Volatility Index

This study uses three expression of VIX as explanatory variable. The first one is the input variable VIX and the other two are CC calculated as annualized returns and periorized returns. Matematically:

Table 7 and Figure 13 depict VIX with a distinctive shape of log-normal distribution with high positive kurtosis, though a high standard deviation flattens the curve. VIXretp shows a clear log-normal distribution but the quarterly data is questionable. VIXreta is a variable with a fairly normal distribution and high kurtosis.

### TB3M: Real 3-Month Treasury Bill

In economics a meaningful variable is the short-term obligations from US-Treasure department because affects to the flows of money in the economy. In this investigation we use the real 3-Months treasury bill, that is the difference between the 3-months treasure bill and the CPI inflation.

The first step is to calculate the CPI since it is not a direct input variable. CPI is calculated as the summation average of last 12 months of CPIAUCSL expressed as simple return from last 12 months.

The time series of CPI daily data uses linear interpolation for missing observations between months. The monthly data is derived from previous equation and the quarterly data ignores the observations that are not quarterly.

Three forms of TB3M are the inputs variables for regression variables. TB3M is the difference between the DGS3MO and CPI, both variables expressed in annual percentages. TB3Mretp is the difference between DGS3MO and CPI, with both variables expressed in periorized CC returns. Finally, TB3Mreta is also the DGS3MO in excess of CPI expressed in annualized CC returns. Mathematically is represented by next equations:

Next panels depict TB3M and TB3Mretp with fairly similar frequency distribution, the difference relies on the standard deviation of the second is much lower than the first, which elevates the normal distribution bell. TB3Mreta shows observations very concentrated around the mean that implies high kurtosis.

### CS: Credit Spread

Credit spread can have different definitions, in our case is the difference between two Moody’s seasoned corporate bond yield of different maturities. Since they are related to companies, investors might interpret a positive value as a sign of positive economic outlook and negative or narrow difference, as turbulent and uncertainty economic future. We use three expressions as explanatory variables, CS as the difference between DBAA and DAAA, both variables expressed in annual percentages. CSretp expressed as difference between DBAA and DAAA, with both variables as periorized CC returns. Finally, CSreta is also the DBAA in excess of DAAA expressed in annualized CC return. Mathematically:

Next representations show CS and CSretp with a familiar log-normal distribution consistent in high skewness, in the first case the standard deviation is too high to lift the curve and in the second case all the way around. The chart depicts a fairly normal distribution for CSreta, with high kurtosis for daily observations.

### TS: Term Spread

Term spread or yield curve is the difference between 3-months treasury bills and 10-years treasury notes. It is interpreted similarly to CS, a positive value of TS as a sign stable economic future and a narrow difference or inversion as worsening economic conditions. In our study, 3 forms of TS are the inputs variables for the regressions. TS is the difference between the DGS10 and DGS3MO, both variables expressed in annual percentages. TSretp is the difference between DGS10 and DGS3MO, with both variables expressed in periorized CC returns. Finally, TSreta is also the DGS10 in exceeds of DGS3MO expressed in annualized CC return. Mathematically is represented by next equations:

Table 10 and Figure 17 show TS and TSretp with a distinctive random distribution, the exception is TSrep with daily observation that matches a log-normal shape. TSreta presents an evident normal distribution with extreme kurtosis.

### SPYDY: SPY Dividend Yield

Dividend paid by S&P 500 and dividend per annum divided by share price, are common indicators of the USA economy. High dividends represent that companies are doing high benefits and the contrary in opposite case.

SPDY is calculated as the summation of the past 12 consecutive months of SPYdiv (trailing 12 months – TTM) divided by SPY. GSPC is not used for the dividend yield calculation because indices do not provide dividends, however SPDR S&P 500 is an ETF that trades under the symbol SPY provides dividends and tracks the GSPC index with high accuracy.

The first step is to calculate the TTM of SPYdiv, this is done by the summation of the yearly dividend distribution, a total of four times per annum. Next expression shows the mathematical calculation:

Next figure plots the variable SPYdivttm, it has similitude to SPYdiv but without spikes.

Figure 18. SPYdivttm time series

Finally, SPYDY is the ratio between SPYdivttm and SPY expressed in percentage:

Next image shows SPYDY time series. It presents a clear peak around the global financial crisis in 2008, that can be interpreted as the GSPC prices crashed but the dividend paid still are distributed. This is a characteristic of a lagging indicator rather than a leading indicator, which is what we are trying to identify.

The last set of explanatory variables for this work are based on SPYDY. Following a systematical cadence of this research, the first one is SPYDY in percentage terms and the other two are CC returns of the periorized and annualized variable.

Next information presents some similarities to some of the previous cases. SPYDY and SPYDYretp shape a log-normal distribution with two different standard deviations though quarterly data embodies an imperfect normal distribution. SPYDYreta presents more distinctive normal distribution with high kurtosis in daily data.

Figure 20. Histograms of SPYDY

# Regressions Models

This research contemplates 336 regression models structured in response and explanatory variables, observations periods, forecasting horizons and model type. In more detail:

- 2 sets of 168 regressions per each response variable: GSPCret_excp and GSPCret_exca.
- 3 observation periods: daily, monthly and quarterly.
- 3 time horizons: 1, 3 and 12 months, except with quarterly data with only 2 horizons of 3 and 12 months.
- The first 15 model types are intended to study regression fits with one explanatory variable. The models with VIX are used for hypothesis testing in absolute terms or per-se.
- The model types 16, 17 and 18 evaluate the predictability power of well know multivariate models. They act as benchmarking reference.
- Model type 19, 20 and 21 study the additive contribution of VIX to previous models. They are used for hypothesis testing in relative terms.

Next table summarize all regressions models:

# Regressions Results and Discussion

The regressions results discussion follows a repetitive patter along three chapters. Firstly, identification of regressions models with best linear regressions fit with Ordinary Least Squares (OLS). The main parameter criteria is adjusted R^{2}, an indicator that captures how good the explanatory variable represents the variance of the response variable. The indicator ranks from +100% to -100%, the sign represents whether the relationship is proportional or inversely propositional. A regression model with high predictability power should score a high value in adjusted R^{2} in absolute terms.

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