Generalized Correlation Measures of Causality and Forecasts of the VIX Using Non-Linear Models

The paper features an analysis of causal relations between the VIX, S&P500, and the realised volatility (RV) of the S&P500 sampled at 5 minute intervals, plus the application of an Artificial Neural Network (ANN) model to forecast the VIX. Causal relations are analysed using the recently developed concept of general correlation Zheng et al. (2012) and Vinod (2017). The neural network analysis is performed using the Group Method of Data Handling (GMDH) approach. The results suggest that causality runs from lagged daily RV and lagged continuously compounded return on the S&P500 index to the VIX. Out of sample tests suggest an ANN model can successfully predict the VIX using lagged RV and lagged S&P500 Index continuously compounded returns as inputs.


Introduction
This paper features an analysis of the causal relationships between the VIX and the volatility of the S&P500, as revealed by estimates of the realised volatility (RV) of the S&P500 index, sampled at 5 minute intervals, as calculated by His account was general in the sense that it applied to everything that required an explanation, including artistic production and human action. He mentioned the: material cause: "that out of which" it is made, the ecient cause: the source of the objects principle of change or stability, the formal cause: the essence of the object. and the nal cause: the end/goal of the object, or what the object is good for.
This treatment is far more encompassing than the customary treatment of causality in economics and nance. The modern treatment has been reduced to an analysis of correlation and statistical modelling.  (2000)), and provides an indication of both stock market uncertainty and a variance risk premium, which is also the expected premium from selling stock market variance in a swap contract. The VIX is  There has been a great deal of work on derivatives related to the VIX. This is not the concern of this paper but the relevant ground is covered in Alexander et al. (2015). The fact that the VIX provides an estimate of the variance risk premium has been used to explore its relationship with stock market returns. V ariance Risk P remium t = V RP t ≡ Implied V olatility t −Realised V olatility t  The current paper is concerned with the relationship between the VIX, implied volatility, and S&P500 index continuously compounded returns, but the focus is on an investigation of the causal path. It seeks to explore whether there is a stronger causal link between the VIX, to RV and stock returns, or in the reverse direction, from RV, and stock returns to the VIX. The GMC analysis used in the paper suggests that the latter is the stronger causal path.

Data sample
We analyse the relationship between the VIX, the S&P500 Index, and the realised volatility of the S&P500 index sampled at 5 minute intervals, using daily data from January 3 2000 to December 12 2017, a total of 4,504 observations.
The data for the VIX and S&P500 are obtained from Yahoo nance, whilst the realised volatility estimates are from the Oxford Man Realised Library (see: https://realized.oxford-man.ox.ac.uk).
In this paper, unlike the literature that uses the variance risk premium to forecast returns, we reverse the assumed direction of causality, based on our GMC analysis, and predict the VIX on the basis of market returns and realised volatility.

The approach taken by Bollerslev et al. (2009) and Baekart and Horova
(2014), is constructed on theoretical grounds and is not subjected to any tests of causal direction. A key feature of the current paper is to test, in practice, whether the causal direction runs from the VIX to returns on the S&P500 and estimates of daily RV, or, as we will subsequently demonstrate, in the reverse direction.
Given that we will be using regression analysis we require that our data sets are stationary. We know that price levels are non-stationary, and so we we use the continuously compounded returns on the S&P500 index. The results of Augmented Dickey Fuller tests shown in Table 1, strongly reject the null of non-stationarity for both the VIX and RV5MIN series, so we can combine them with the continuously componded returns for the S&P500 Index in regression analysis, without the worry of estimating spurious regression.    Note: ***, **, * denotes signicance at 1%, 5%, and 10% and pronounced skewness in RV5MIN. We also did some preliminary regression and quantile regression analysis of the relationships between our three base series to explore

Preliminary Regression Analysis
We estimated an OLS regression of the VIX regressed on the continuously compounded S&P500 return 'SPRET. The results are shown in Table 3. The slope coecient is insignicant and the R squared is a miniscule 0.000158. The Ramsey Reset test suggests that the relationship is non-linear and that the regression is misspecied.
A QQplot of the residuals from this regression shown in Figure 3 also suggests that a linear specication is inappropriate.
To further explore the relationship between the sample variables we empoyed quantile regression analysis.
Where ρ τ () is the tilted absolute value function as shown in Figure 2.4, which gives the τ th sample quantile with its solution. Taking the directional derivatives of the objective function with respect to ξ (from left to right) shows that After dening the unconditional quantiles as an optimization problem, it is easy to dene conditional quantiles similarly. Taking the least squares regression model as a base to proceed, for a random sample, y 1 , y 2 , . . . , y n , we solve which gives the sample mean, an estimate of the unconditional population mean, EY. Replacing the scalar µ by a parametric function µ(x, β) and then gives an estimate of the conditional expectation function E(Y|x).
Proceeding the same way for quantile regression, to obtain an estimate of the conditional median function, the scalar ξ in the rst equation is replaced by the parametric function ξ(x t , β) and τ is set to 1/2 . The estimates of the other conditional quantile functions are obtained by replacing absolute values by ρ τ () and solving    Table 4.
whenever E(Y 2 ) < ∞ and E(X 2 ) < ∞. Note that E(V ar(X | Y )) is the expected conditional variance of X given Y , and therefore E(V ar(X | Y )/V ar(X) can be interpreted as the explained variance of X by Y. Thus, we can write: The explained variance of Y given X can similarly be dened. This leads This pair of GMC measures has some attractive properties. It should be noted that the two measures are identical when (X, Y ) is a bivariate normal random vector.
Vinod (2017) takes this measure in expression (2) and reminds the reader that it can be viewed as kernel causality, GM C(Y | X) is the coecient of determination R 2 of the Nadaraya-Watson nonparametric Kernel regression: where g(X) is a nonparametric, unspecied (nonlinear) function. Interchanging X and Y , we obtain the other GM C(X | Y ) dened as the R 2 of the Kernel regression: Vinod (2017) denes δ = GM C(X | Y ) − GM C(X | Y ) as the dierence of two population R 2 values. When δ < 0, we know that X better predicts Y than vice versa. Hence, we dene that X kernel causes Y provided the true unknown δ < 0. Its estimate δ can be readily computed by means of regression.
which suggests that X t can be better predicted using the histories of both X t and Y t than using the history of X t alone. Similarly would say X t Granger-causes Y t if: In the same way (6) is equivalent to: They add that when both (5) and (6) are true, there is a feedback system.
, they say Y is more inuential than X.
We explore the relationship between the VIX, the lagged continuously compounded return on the S&P500 Index, (LSPRET), and the lagged daily realised volatility on the S&P500, sampled at 5 minute intervals within the day (LRV5MIN). Once we have have established causal directions between these variables, we use them to construct our ANN model. The ANN model is discussed in the next section.

Articial Neural Net Models
There are a variety of approaches to neural net modelling. A simple neural network model with linear input, D hidden units and activation function g, can be written as: However, we choose to apply a nonlinear neural net modelling approach, using the GMDH shell program (http:www.gmdhshell.com). This program is built around an approximation called the 'Group Method of Data Handling'.
This approach is used in such elds as data mining, prediction, complex systems modelling, optimization and pattern recognition. The algorithms feature an inductive procedure that performs a sifting and ordering of gradually complicated polynomial models, and the selection of the best solution by external criterion.
A GMDH model with multiple inputs and one output is a subset of components of the base function: where f are elementary functions dependent on dierent inputs, a are unknown coecients, and m is the number of base function components.
In general, the connection between input-output variables can be approximated by Volterra functional series, the discrete analague of which is the Kolmogorov-Gabor polynomial: where by using a heuristic and peceptron type of approach. He demonstrated that a second-order polynomial (Ivakhnenko polynomial: y = a 0 + a 1 x i + a 2 x j + a 3 x i x j + a 4 x 2 i + a 5 x 2 j ) can reconstruct the entire Kolmorogorov-Gabor polynomial using an iterative peceptron-type procedure.  Table 5.
We use the R 'generalCorr' package to undertake the analysis shown in Table   5. The output matrix is seen to report the 'cause' along columns and 'response' along the rows. The value of 0.7821467 in the R.H.S. of the second row of Table   5 is larger than the value 0.608359 in the second column, third row of Table 5.
These are our two generalised measures of correlation, when we rst condition the VIX on LRV5MIN, in the second row of Table 5, and LRV5MIN on the VIX in the third row of Table 5. This suggests that causality runs from LRV5MIN, the lagged daily value of the realised volatility of the S&P500 index, sample at 5 minute intervals.
We also test the signicance of the dierence between these two generalised measures of correlation. Vinod suggests a heuristic test of the dierence between two dependent correlation values. Vinod (2017) suggests a test based on a suggestion by Fisher (1922), of a variance stabilizing and normalizing transformation for the correlation coecient, r, dened by the formula: r = tanh(z), involving a hyperbolic tangent: The application of the above test suggests a highly signicant dierence between the values of the two correlation statistics in Table 5.
We also analyse the relationship between the VIX and the lagged daily continuously compounded return on the S&P500 index, LSPRET. The results are shown in Table 6 an suggest that lagged value of the continously compounded return on the S&P500 index, LSPRET, drives the VIX. This is because the generalised correlation measure of the VIX conditioned on LSPRET is 0.5519368, whilst the generalised correlation measure of LSPRET conditioned on the VIX is only 0.153411. Once again, these two measures are signicantly dierent.
Given that we have successfully established that causal direction runs from LSPRET and LRV5MIN to the VIX, and our preliminary regression and quantile

ANN model
Our neural network analysis is run on 80 per cent of the observations in our sample, and then its out-of-sample forecasting performance is analysed on the remaining 20 per cent, of the total sample of 4,504 observations. The ANN regression analysis produces a complex non-linear model which is shown in Table   7.
A plot of the ANN model t is shown in Figure 6. The model appears to be a good t, within the estimation period and in the 20 per cent of the sample used as a hold out forecast period. This is conrmed by the diagnostics for the Figure 6: ANN regression model t  Table 9: Residual diagnostic plots ANN model, reported in Table 8. The mean absolute error is smaller in the forecasts with a value of 3.14658, than it is when the model is being tted, with a value of 3.16466. Similarly, the R 2 is higher in the forecast hold out sample, with a value of 75 percent, than in the model tting stage, in which it has a value of almost 74 per cent.
The diagnostic plots of the behaviour of the residuals, shown in Figure 7, also appears to show acceptable behaviour. Most of the residuals plot within the error bands, the residual histogram is approximately normal, though there is some evidence of persistence in the autocorrelations suggestive of ARCH eects.

Conclusion
The paper featured an analysis of causal relations between the VIX and The results suggest an ANN model can be used successfully to predict the VIX using lagged RV and lagged S&P500 Index continously compounded returns as inputs.