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Posted: April 24th, 2023

Economics and Finance Subject Group Financial Econometrics

Portsmouth Business School
Economics and Finance Subject Group

Financial Econometrics (Part 2)

Aim:

To introduce students to stationary and non-stationary time series models (ARMA and ARIMA, respectively). To examine stochastic volatility via ARCH and GARCH modelling. To examine time series features and stochastic volatility properties simultaneously in a single equation model. To demonstrate the application of the above type(s) of modelling to financial data. To use these models for estimation, inference and forecasting.

Syllabus Outline:

Univariate linear stochastic models: ARMA and ARIMA modelling, estimation and forecasting using ARMA and ARIMA models, application to macroeconomic and/or financial data. Modelling stochastic volatility: ARCH and GARCH processes. ARCH in mean model, Asymmetric ARCH models, TARCH and EGARCH model. AR(I)MA models with (G)ARCH errors. Applications to financial data.

You are expected to read the relevant material in advance.
The computer software for this course is Eviews (5.0 or above), which is available on the student network. For most of the exercises you will be required to collect data from the BANKSCOPE, DATASTREAM databases.

Indicative Reading:
Brookes, C., Introductory Econometrics for Finance Cambridge University Press 2008
HARRIS, R and Sollis, R., Applied Time Series Modelling and Forecasting Wiley 2003
PINDYCK, R & RUBENFELD D Econometric Models & Economic Forecasts McGraw & Hill 3rd Edition
Harvey, A. C. Time Series Models, Harvester Wheatsheaf 1993
Mills, T. C. Time Series Techniques for Economists, Cambridge 1990
Mills, T. C. The Econometric Modelling of Financial Time Series, 1993 Cambridge.
Hamilton, J. D. Time Series Analysis, 1994 Princeton
Cuthbertson K, Nitzsche D., Quantitative Financial Ecnomics, 2004 Wiley

Assignment
This will involve answering a set of short questions based on your analysis of a data-set. The assignment is worth 25% of the total assessment. In addition to the above assignment there will be a final unseen examination (worth 50% of the final marks).

Professor Shabbar Jaffry (R4.01)

Financial Econometrics (part 2)

Assessed Coursework
Submission Deadline: 27 March 2020 (23:55 unit Moodle page)

This coursework (individual not a group work) counts for 25% of your overall assessment for this unit. Please submit your completed answers (a maximum of 7 A4 pages excluding the apprendix) to the unit Moodle page. Please also submit all your Eviews workings copied in the appendix with your coursework.

This coursework requires you to collect 4 sectors of the economy (e.g. real estate, banking, general retail etc.) daily (daily observations excluding weekends and holidays) indices (for London exchange market or for other market if the data is available) for the allocated country from the Eikon. The length of each series should not be less than 10 years. In answering the questions below, you may wish to consult the help option in Eviews.

You are expected to have collected data for the assignment by 15th February and estimations by Mid-March.

1. Choose one of the four series on which to conduct your analysis. For your chosen series, in Eviews use the Genr option to calculate (i) the log of the series, e.g. e=log(your chosen series), and (ii) the daily log returns (e.g. r=e-e(-1)).

i. Examine the descriptive statistics for both e and r. What do you conclude about the distributions of e and r? Is e normally distributed? Is r normally distributed? Explain why/why not?
ii. Obtain the correlograms, and examine the autocorrelations and partial autocorrelations for both e and r. What do you conclude about the behaviour of e and r? Are they stationary/non-stationary?
iii. Are your conclusions about stationary/non-stationary of e and r confirmed by appropriate unit root tests?

2. Estimate and select an appropriate ARMA (p,q) model for e. In selecting your preferred model, explain how you use the information provided by:

i. The estimated coefficients (and their t-statistics);
ii. Ljung-Box Q-statistics for autocorrelation in the residuals; and
iii. AIC and SBC information criteria for choosing between alternative models.
iv. Carry out forecast of e for 100 observations based on your chosen model and comment on your results.

In the following section (v), use 100 less observations from the full sample (e.g. if the full sample was 2000 then only use 1900) available to you.

v. Carry out forecast (out of sample) of e for 100 observations based on your chosen model and comment on your results.

3. Now estimate an AR (1) model for the log return r.

i. Test for the presence of ARCH effects in the residuals of this regression.
ii. Select and estimate an appropriate GARCH (p,q) model for the conditional variance of the residuals of this regression. Justify your choice of selected model.
iii. Now extend your selected model in (ii) to include EGARCH and TGARCH effects. Interpret the estimated EGARCH and TGARCH coefficients.

Discuss and explain any two from the following list:
(a) Long-run and short-run and ECM models
(b) Spurious regressions, cointegration, Engle-Granger and dynamic modelling approaches
(c) The limitations of using a single equation approach to testing for cointegration when there are more than two variables in the model. Short-run dynamic models

Please provide a word count for this question AND ensure that your answer does not exceed 1000 words. Marks will be deducted for overlong essays.

4. Estimate a VAR (p) model for the log of four selected series.

i. Test for Granger causality between pair of these series (take log of these series first).
ii. Test for cointegration between four series (chosen in part 4) using Johansen method.

Professor Shabbar Jaffry

Financial Econometrics

Phase 1 – Analysis
By observation of variables e and r it can be inferred that both variables are not normally distributed. The premise of concluding that variable e and r are normally distributed is given through the Assessment of skewness, kurtosis and the Jarque-Bera statistic. Variable e has a skewness (a measure of asymmetry of a distribution) of -0.656623, implicating that the log of Consumer Goods has a moderately negative skew with a longer left hand tail. To determine normality of the log Consumer Goods distribution (variable e), skewness has to equal a value of zero. With a skewness value of -0.656623, variable e can be said to amply or very loosely mirror a normally distributed series and more so is inferred to be non-normally distributed due to its closeness to 1 rather than zero.
To further the investigation of normality, the kurtosis (measure of tallness or flatness of a distribution) value of variable e is reserved and taken into consideration. With a kurtosis value of 2.525799 variable e is said to be platykurtic as most values fall below the mean of 7.621370. However, it can be inferred that within comparison to the drawn conclusion of the skewness value, the kurtosis value of variable e withal roughly mirrors a normally distributed series given its proximity to a mesokurtic or normal distributed kurtosis value of 3. Imperatively, in determining the normality of variable e, the Jarque-Bera statistic has been implemented to conclusively ascertain whether variable e is normally distributed in principle to the test statistics inclusion of skewness and kurtosis (Eq 1.1). The Jarque-Bera statistic value of 318.0095 exceeds the critical chi-square value of 5.9915 to 2 degrees of freedom at the 5% level of significance, therefore we reject the null hypothesis of normal distribution. Equally, given that the probability value of variable e’s Jarque-Bera statistic equals a value of zero, the conjecture is made that at a 5 % level of significance the null hypothesis is rejected for normal distribution. The overall assumption with regard to the log of Consumer Goods as a sector within Switzerland is that the series is not normally distributed.
The skewness value of variable r equals -0.265504 which highlights that the daily log return of Consumer Goods similarly closely mirrors normal distribution. However, through the utilisation of the variable r’s kurtosis value of 8.189209 the inference is drawn that variable r is leptokurtic and as such is not normally distributed given that the kurtosis value of variable r deviates away from 3. More importantly, drawing on the Jarque-Bera statistic, it can be construed that the null hypothesis of normal distribution is rejected at a 5% level of significance as the Jarque-Bera statistic of 4437.495 exceeds that of the critical chi-square value of 5.9915 at 2 degrees of freedom. The conclusion corresponds similarly to that of variable e, as such that variable r is not normally distributed.
Paramount to the assessment of stationarity of variables e and r, it is viable to inspect the graphical illustrations of both variables and more appropriately is the examination of the autocorrelation (AC) and partial autocorrelation (PAC) of both respective variables. Through the graphical analysis of Figure 2 the inference is drawn that variable e is non-stationary on the basis that the log series of Consumer Goods’ data points do not intermittently revert around the mean of 7.621370. More importantly by means of analysing Figure 5 it is apparent that the AC values of variable e follow a trend within the dataset as there is a very slow decay of the AC values towards zero as lags lengthen. Subsequently, the log series of Consumer Goods can be said to follow a random walk with a trend. Lag 1 of the PAC of variable e, however exhibits statistical significance, but immediately becomes statistically insignificant as lags lengthen, appropriately it is indicative that variable e is non-stationary. However, given that non- stationary data tends to produce spurious and inaccurate conclusions with regard to forecasting under financial modelling, the Consumer Goods sector may potentially be an unattractive sector to invest in. In approach to this, first difference correlogram of variable e has been implemented, where Figure 6 closely resembles a white nose time series. As such given that the autocorrelation and partial autocorrelation values of the first difference of variable e hover around zero, stationarity of variable e can be assumed.
The levels correlogram output of variable r interestingly inhibits the same exact AC and PAC values as the 1st difference of variable e. Stationarity of the daily log return of Consumer Goods (variable r) can therefore be immediately determined and is furthermore reinforced by means of graphical observation of Figure 5 where it is noticeable that the variable r’s data point oscillate around the mean of 0.000341 which reinforces the notion of stationarity. Since the judgement that variable r is a stationary time series has been brought forward, the implementation of transformation or a first difference operator will not be necessary.
To ensure that the conclusions of non-stationarity and stationarity are precise both the Augmented Dicky Fuller (ADF) and Phillips-Perron (PP) tests have been exacted. By means of the enlisted tests, the acceptances or rejections of the null hypothesis are made at a 5% level of significance which in succession empirically confirms the assumptions of non-stationarity or stationarity of variable e and r.

Null hypothesis: Ho: δ = 0 (there is a unit root or the time series is non-stationary)

Alternative hypothesis: Ha: δ < 0 (the time series is stationary)

Variable E (Levels, Trend & Intercept and Schwarz Criterion)
Augmented Dicky Fuller: -2.923230 > -3.410954 Phillips-Perron: -2.970056 > -3.410953
We do not reject the null hypothesis and the conjecture of non-stationarity is made
Variable E (First Difference, Intercept and Schwarz Criterion)
Augmented Dicky Fuller: -48.10728 < -2.862082 Phillips-Perron: -63.72634 < -2.86282
We reject the null hypothesis and the conjecture of stationarity is made through the acceptance of the alternative hypothesis
Variable R (Levels, Intercept and Schwarz Criterion)
Augmented Dicky Fuller: -48.10728 < -2.862082 Phillips-Perron: -63.72634 < -2.86282
We reject the null hypothesis and the conjecture of stationarity is made through the acceptance of the alternative hypothesis
Appropriately, given the rigorous analysis and testing of variables e and r, the conclusions of stationary and non-stationarity are indeed consistent with the appropriate results of the ADF and PP unit root tests.
Phase 2 – ARMA Model Selection
Given the selection of an appropriate ARMA (p,q) model, it is pertinent to investigate which relevant model computes the lowest Akaiki and Schwarz information criterions. Intriguingly however, as displayed on Table 17, the ARMA(7, 8) and AR(6) models respectively have the lowest Akaiki and Schwarz information criterions. Appropriately, in response to this, for the purpose of selecting an appropriate ARMA (p,q) model, both models have been taken forward to investigate their suitability in regard to quality of modelling data and predicating aligned volatility in sample or out of sample.
Upon inspecting the estimation output of both rival models, it the evident that ARMA(7, 8) has more statistically insignificant variables (Table 18) in comparison to AR(6) (Table 19). The precedence of establishing statistical insignificance lies with probability values that are greater than the 5 % level of significance. On that basis alone it can be concluded that AR(6) would be a preferable model to be taken forward. However, in light of the computation of adjusted R –squared (R ̅2 ) ARMA(7, 8) outranks AR(6), indicating that ARMA(7, 8) is a better fit of data, despite the fact it does so marginally. To a greater extent it is prudent to further test rival models to ensure the appropriate model selection is achieved.
By means of further testing the acceptability of both AR(6) and ARMA(7,8), the application of the Ljung-Box test has been exacted to examine the presence of autocorrelation. The estimated results for both rival models indicated that the ARMA(7,8) is more superior given that AR(6) displayed a period of lack of fit where the alternative hypothesis was accepted. As such the given implication can be made that the AR(6) model of residuals does not align with the white noise process as there is a potential systematic or predictable behaviour of residuals.
However, given the estimated result it is to be taken into consideration that the Schwarz information criterion is most parsimonious, highlighting the simplest model with the smallest amount of variables, but greatest explanatory power (Davidson & Mackinnon, 2004, p. 696). By that measure alone the preference with regard to the appropriate ARMA model is AR(6).
Even though the acceptance of AR(6) has come from a theoretical standpoint by means of forecasting in sample and out of sample it is observable that the ARMA(7,8) as can be seen from Figures 10 to 15 highlight that ARMA(7,8) is a better model with regard to measuring or predicating volatility. Table 22 and 23 detail that Root Mean Square Error (RMSE), Mean Absolute Errors (MSE), Mean Absolute Percentage Errors (MAPE) and Theil Inequality coefficient (THEIL) were lowest for ARMA (7,8). However with the adjustment of the sample size the AR(6) model was deemed more favourable given the low values of the Akaike and Schwarz criterions. Perhaps a selection of other models enlisted in Table 16 ought to be forecasted to see which model graphically predicts the behaviour or movement of the log of Consumer Goods more adeuqately.

Phase 3 – ARCH, GARCH, EGARCH & TGARCH

The computed AR (1) process is equated as follows:
Y_t= β_0+β_1 X_(t-1)+u_t (3.1)
Y_t= 0.00342-〖0.012078X〗_(t-1)+u_t
The computed GARCH (1) process is equated as follows:
Y_t= β_0+β_1 X_(t-1)+u_t (3.1)
Y_t^2= β_0+β_t^2 X_(t-1)+u_t (3.4)
Y_t^2= 0.0000805+0.302640X_(t-1)^2+ +u_t

The model of Y_t^2 under AR1 suggests that given the positive value of〖 β〗_1, high volatility in previous periods will follow high volatility in the current period. This exemplifies that volatility clustering is present within the model. Furthermore, reinforcing the notion of volatility clustering in the model it is apparent that the associated probability value of 〖 β〗_1 is highly significant at a probability value of zero. The inference can therefore be drawn that volatility clustering is indeed present. Through the graphical analysis of Figure 1 it is indeed evident that periods of high volatility are followed by further periods of high volatility. Fittingly, by visual confirmation it appropriate or justified as such to run a computational test to compute whether the ARCH effect is present. With the implementation of the ARCH test (Table 25), it is clear that the Prob. Chi-Square (1) estimation of 0.000 does not exceed the 5% level of statistical significance. Therefore it deduction is made that the null hypothesis is not to be rejected in favour Ha as such that an ARCH effect is present. Accordingly, the data series is said to accord with conditional heteroscedasticity or autocorrelation, where past similarities arise in the log of daily returns of Consumer Goods. The importance of testing for an ARCH affect within the log daily return of the sector of Consumer Goods is indicative of the affect volatility. Given that the chosen sector is highlighted by Consumer Goods, or as such consumer consumption. Volatility could signal uncertainty, hence leading to huge losses or gains based on the position one may take in the Consumer Goods sector.
Given the presence of the ARCH effect the GARCH model can be estimated with conditional variance where
ϑ_t^2= β_0+〖β_1μ〗_(t-1)^2+〖β_σ〗_(t-1)^2 (3.1)
σ_t^2= μ_t^2- ε_t
In the selection of an appropriate GARCH model, GARCH (p,q) model is tested firstly where σ_t^2 is denoted by the conditional variance given that it is one-period ahead estimated for the variance calculated based on any past information which is assumed relevant. The generalized autoregressive conditional heteroscedasticity (GARCH) model is vastly important in modelling asset returns and inflation. As a financial process, the GARCH model sets the precedence of real word context with respects to predicting various financial rates and prices. As the time series investigated, the log of Consumer Goods exhibits volatility clustering and for that reason has an ARCH effect due to market instability, modelling volatility given the uncertainty is essential.
Given the investigation of selecting an appropriate GARCH Model, Table 36 highlights that there are two GARCH models which are to be considered for the selection of appropriateness. The rival models are GARCH (1,1) and GARCH (2,2), GARCH(1,1) is seen to be most appropriate in forecasting volatility, but according to the Akaiki and Schwarz information criterions GARCH(2,2) is contends to be favourable. As a standard measure of analysis this reports adheres that the probability values of both GARCH (1,1) and GARCH (2,2) are all significant, however
The GARCH (1,1) model in hindsight has been reinforced by many researchers as the most appropriate model to approximate volatility (Bera, 1993) (Bollerslev, 1984) given that the first lag of variance is adequate enough in detecting the volatility clustering in specified data. However, the supposition of the assumption that the first lag captures volatility adequately does not detail whether an increase in lags would more sufficiently detect volatility with a greater degree of accuracy. Intuitively, (Brooks & Burke, 2003) outline that unspecified GARCH models have the potential to influence the forecast accuracy due to high prediction errors. Accordingly, the GARCH (2,2) model can been taken forward as the most appropriate model given that the subsequent increase in lags did not cause a disparagement of the models statistical significance.
However, given the presence of negative constraints in GARCH (2, 2) it is imperative that the selection of the most appropriate model within our stationary time series r is given to GARCH(1,1) which accords with non-negativity constrains within the estimation output. This report cognises the drawbacks of the GARCH model and accordingly tests EGARCH and TGARCH to compare estimated results. The GARCH(1,1) model takes the approach where negative and positive shocks have the same impact on the conditional variance. As a response to this, the introduction the Exponential GARCH process (EGARCH) has been exacted where negative shocks have a more renounced impact on volatility. More so given the introduction of EGARCH model conditional variance will always be positive or non-negative constraints will not be viable. Not withholding an alternative model, for comparative reasoning’s this report examines threshold Autoregressive conditional heteroscedasticity (TGARCH) which adopts that unforeseen changes in market return will have different effects on the volatility of stock return.
In view of taking GARCH(1,1) as the preferred model, the precedence is set where identical lags are applied for TGARCH and EGARCH.

TGARCH (Table 38)

Our estimated result for TGARCH (Table 38) highlights that an insignificant constant is present within the data set under the mean equation which is statically insignificant. On the other hand under the variance equation it evident that the constant is statistically significant and therefore will have to remain in the estimation output.

All coefficients within the TGARCH estimation output are statistically significant which outlines that the parameters within the given model are supposed to be present and fit the model well when it concerned with predicating volatility. The highly significant coefficient 0.845836 of GARCH(-1) implies persistent volatility clustering. Given the leverage effect the TGARCH model outlines that Good news will lead to higher return, hence it is associated with higher variance (Lim & Sek, 2013).The remainder of variables yet significant are not as tenacious given their estimated coefficients. For comparative purposes the TGARCH model is assumed the best fitted model by way of predicating conditional volatility. This TGARCH model accordingly has the lowest AIC and BIC criterions (Table 38)

EGARCH (Table 37)

The Arch effect(C2), Leverage effect(C3) and GARCH effect(C4) are all statistically significant as their respective probability values are less than the 5% level of significance. The C5 coefficients is positive according to (Olbryś, 2013) outlines that negative news has a greater impact on volatility. Given the estimated results it is apparent that the market or volatility is more responsive to negative news than bad.
Phase 4 – VAR estimation

In taking the log of 4 chosen sectors/series, the VAR is estimated at lags 1 to 24 given the precedence of the series being daily data. Given the lag length criterion, it is evident that Akaike information criterion (AIC) is to be selected given that it is the most negative criterion even though the Schwarz information criterion (BIC) is more parsimonious. Accordingly, by virtue of choosing up to 24 lags as the initial selection Table 42 illustrates that the 18 lag is more favourable by way of the AIC and Final prediction error (FPE) under the lag length option.
Z Financials
The Null hypothesis: Health Care or Industrial Engineering or Consumer Goods sector (Lag 1 to Lag 18) does not impact Financial Sector
The Alternative hypothesis: Health Care or Industrial Engineering or Consumer Goods sector (Lag 1 to Lag 18) does impact Financial Sector
Utilising table 42 it is evident that null hypothesis is rejected for all 3 sectors (Health Care, Industrial Engineering and Consumer Goods sector) and the alternative hypothesis is accepted. Therefore the granger causality test highlights that the given sectors can aid in forecasting for one another
Health Care
The Null hypothesis: Financial or Industrial Engineering or Consumer Goods sector (Lag 1 to Lag 18) does not impact Heath Care Sector
The Alternative hypothesis: Financial or Industrial Engineering or Consumer Goods sector (Lag 1 to Lag 18) does impact Heath Care Sector
Utilising table 42 it is evident that null hypothesis is rejected for all 3 sectors (Financial or Industrial Engineering or Consumer Goods sector) and the alternative hypothesis is accepted. Therefore the granger causality test highlights that the given sectors can aid in forecasting for one another

Consumer Goods
The Null hypothesis: Financial or Industrial Engineering or Heath Care sector (Lag 1 to Lag 18) does not impact Consumer Goods sector
The Alternative hypothesis: Financial or Industrial Engineering or Heath Care sector (Lag 1 to Lag 18) does impact Consumer Goods sector
Utilising table 42 it is evident that null hypothesis is rejected for 2 sectors (Financial and Industrial Engineering sector) and the null hypothesis is accepted. Therefore the granger causality test highlights that the 2 given sectors can aid in forecasting for one another
The null is however not rejected under the Health Care sector where it can be assumed that the Heath care sector does not impact the Consumer goods sector.
This can be said to accord with economic theory as the consumption of goods does impact the health care sector, but the health care sector does not impact the consumer goods sector. This is an indirect relation between Consumer goods and Health care sector
Industrial Engineering
The Null hypothesis: Financial or Consumer Goods or Heath Care sector (Lag 1 to Lag 18) does not impact Industrial Engineering sector
The Alternative hypothesis: Financial or Consumer Goods or Heath Care sector (Lag 1 to Lag 18) does impact Industrial Engineering sector
Utilising table 42 it is evident that null hypothesis is rejected for all 3 sectors (Financial or Consumer Goods or Heath Care sector) and the null hypothesis is accepted. Therefore the granger causality test highlights that the given sectors can aid in forecasting for one another
Johansson Cointegration Test
The Johansson cointegration test is implemented to test whether series are cointegrated with one another. Given that non-stationary data leads onto spurious results and conclusions, the presence of cointegration between variables eliminates spurious elements within estimation. Accordingly, within the estimation of cointegration the estimated results on Table 44 highlight that the null hypothesis of no cointegration is not rejected between the 4 chosen sectors for one cointegrated equation to three cointegrated equations. To reinforce the notion of no rejection of the null hypothesis of no cointegration. The Johansson Cointegration test computes the Maximum Eigenvalue test, which similarly does not reject the null hypothesis.
By virtue of what was deliberated, the presence of no integration highlights that the estimation given four series used for testing leads to spurious results and conclusions. With regard to volatility it can be stated that given the absence of cointegration, the four series cannot appropriately model one another and therefore predict behaviour or concurrent volatility. More so given the four sector or series, the arbitrage or advantageous position will not be present for any arbitrageurs or investors who are looking for a competitive edge with the aim of maximising returns.

Appendix
Phase 1 – Analysis

Figure 1 – E Histogram

Figure 2- E Graph

Figure 3 – E Differenced Graph

Figure 4- R Histogram

Figure 5- R Graph

Figure 6 – R Differenced Graph

Equation 1- Variable E Jarque-Bera Test
Jarque-Bera (JB)test= n/6 [S^2 〖(K-3)〗^2/4] (1.1)
N is the sample size, S represents skewness, and K represents kurtosis

Jarque-Bera (JB)test= 3915/6 [(〖-0.656623) 〗^2 (2.525799-3)^2/4]=318.0095 (1.2)

Jarque-Bera (JB)test= 318.0095>5.9915

Eq (1.2) exceeds the critical chi-square value for 2 degrees of freedom at the 5% level of significance, we reject the null hypothesis of normal distribution.

Equation 2- Variable R Jarque-Bera Test
Jarque-Bera (JB)test= 3915/6 [(〖-0.265504) 〗^2 (8.189209-3)^2/4]=4437.475 (1.3)

Jarque-Bera (JB)test=4437.475>5.9915

Eq (1.3) exceeds the critical chi-square value for 2 degrees of freedom at the 5% level of significance, we reject the null hypothesis of normal distribution.

Figure 7 – E Correlogram Levels

Figure 8- E Correlogram 1st Difference

Figure 9- R Correlogram Levels

Table 1- Level, Intercept and Schwarz Criterion (Augmented Dickey-Fuller Variable E)
Null Hypothesis: E_LOG_CONSUMER_GOODS has a unit root
Exogenous: Constant
Lag Length: 2 (Automatic – based on SIC, maxlag=30)

t-Statistic Prob.*

Augmented Dickey-Fuller test statistic -1.882675 0.3408
Test critical values: 1% level -3.431836
5% level -2.862082
10% level -2.567102

*MacKinnon (1996) one-sided p-values.

Augmented Dickey-Fuller Test Equation
Dependent Variable: D(E_LOG_CONSUMER_GOODS)
Method: Least Squares
Date: 03/22/19 Time: 06:22
Sample (adjusted): 3/18/2004 3/15/2019
Included observations: 3912 after adjustments

Variable Coefficient Std. Error t-Statistic Prob.

E_LOG_CONSUMER_GOODS(-1) -0.000940 0.000499 -1.882675 0.0598
D(E_LOG_CONSUMER_GOODS(-1)) -0.013114 0.015941 -0.822679 0.4107
D(E_LOG_CONSUMER_GOODS(-2)) -0.078131 0.015940 -4.901592 0.0000
C 0.007530 0.003809 1.977235 0.0481

R-squared 0.007153 Mean dependent var 0.000336
Adjusted R-squared 0.006391 S.D. dependent var 0.010797
S.E. of regression 0.010763 Akaike info criterion -6.224428
Sum squared resid 0.452691 Schwarz criterion -6.218015
Log likelihood 12178.98 Hannan-Quinn criter. -6.222153
F-statistic 9.385049 Durbin-Watson stat 2.003950
Prob(F-statistic) 0.000004

Table 2- Level, Intercept & Intercept and Schwarz Criterion (Augmented Dickey-Fuller Variable E)

Null Hypothesis: E_LOG_CONSUMER_GOODS has a unit root
Exogenous: Constant, Linear Trend
Lag Length: 2 (Automatic – based on SIC, maxlag=30)

t-Statistic Prob.*

Augmented Dickey-Fuller test statistic -2.923230 0.1551
Test critical values: 1% level -3.960385
5% level -3.410954
10% level -3.127286

*MacKinnon (1996) one-sided p-values.

Augmented Dickey-Fuller Test Equation
Dependent Variable: D(E_LOG_CONSUMER_GOODS)
Method: Least Squares
Date: 03/22/19 Time: 06:23
Sample (adjusted): 3/18/2004 3/15/2019
Included observations: 3912 after adjustments

Variable Coefficient Std. Error t-Statistic Prob.

E_LOG_CONSUMER_GOODS(-1) -0.004072 0.001393 -2.923230 0.0035
D(E_LOG_CONSUMER_GOODS(-1)) -0.011584 0.015944 -0.726542 0.4676
D(E_LOG_CONSUMER_GOODS(-2)) -0.076643 0.015942 -4.807626 0.0000
C 0.029395 0.009845 2.985940 0.0028
@TREND(“3/15/2004”) 1.02E-06 4.25E-07 2.408291 0.0161

R-squared 0.008625 Mean dependent var 0.000336
Adjusted R-squared 0.007610 S.D. dependent var 0.010797
S.E. of regression 0.010756 Akaike info criterion -6.225400
Sum squared resid 0.452020 Schwarz criterion -6.217384
Log likelihood 12181.88 Hannan-Quinn criter. -6.222556
F-statistic 8.497398 Durbin-Watson stat 2.003645
Prob(F-statistic) 0.000001

Table 3- Level, None and Schwarz Criterion (Augmented Dickey-Fuller Variable E)

Null Hypothesis: E_LOG_CONSUMER_GOODS has a unit root
Exogenous: None
Lag Length: 2 (Automatic – based on SIC, maxlag=30)

t-Statistic Prob.*

Augmented Dickey-Fuller test statistic 2.045296 0.9907
Test critical values: 1% level -2.565553
5% level -1.940905
10% level -1.616645

*MacKinnon (1996) one-sided p-values.

Augmented Dickey-Fuller Test Equation
Dependent Variable: D(E_LOG_CONSUMER_GOODS)
Method: Least Squares
Date: 03/22/19 Time: 06:30
Sample (adjusted): 3/18/2004 3/15/2019
Included observations: 3912 after adjustments

Variable Coefficient Std. Error t-Statistic Prob.

E_LOG_CONSUMER_GOODS(-1) 4.62E-05 2.26E-05 2.045296 0.0409
D(E_LOG_CONSUMER_GOODS(-1)) -0.013109 0.015947 -0.822027 0.4111
D(E_LOG_CONSUMER_GOODS(-2)) -0.078081 0.015946 -4.896634 0.0000

R-squared 0.006160 Mean dependent var 0.000336
Adjusted R-squared 0.005651 S.D. dependent var 0.010797
S.E. of regression 0.010767 Akaike info criterion -6.223940
Sum squared resid 0.453143 Schwarz criterion -6.219130
Log likelihood 12177.03 Hannan-Quinn criter. -6.222233
Durbin-Watson stat 2.003927

Table 4 – Levels, Intercept and Phillips–Perron (Variable E)
Null Hypothesis: E_LOG_CONSUMER_GOODS has a unit root
Exogenous: Constant
Bandwidth: 14 (Newey-West automatic) using Bartlett kernel

Adj. t-Stat Prob.*

Phillips-Perron test statistic -1.949096 0.3099
Test critical values: 1% level -3.431836
5% level -2.862082
10% level -2.567102

*MacKinnon (1996) one-sided p-values.

Residual variance (no correction) 0.000116
HAC corrected variance (Bartlett kernel) 9.15E-05

Phillips-Perron Test Equation
Dependent Variable: D(E_LOG_CONSUMER_GOODS)
Method: Least Squares
Date: 03/22/19 Time: 06:32
Sample (adjusted): 3/16/2004 3/15/2019
Included observations: 3914 after adjustments

Variable Coefficient Std. Error t-Statistic Prob.

E_LOG_CONSUMER_GOODS(-1) -0.000968 0.000500 -1.937314 0.0528
C 0.007721 0.003813 2.024729 0.0430

R-squared 0.000958 Mean dependent var 0.000341
Adjusted R-squared 0.000703 S.D. dependent var 0.010797
S.E. of regression 0.010793 Akaike info criterion -6.219349
Sum squared resid 0.455695 Schwarz criterion -6.216144
Log likelihood 12173.27 Hannan-Quinn criter. -6.218211
F-statistic 3.753185 Durbin-Watson stat 2.023683
Prob(F-statistic) 0.052779

Table 5- Levels, Intercept & Trend and Phillips–Perron (Variable E)
Null Hypothesis: E_LOG_CONSUMER_GOODS has a unit root
Exogenous: Constant, Linear Trend
Bandwidth: 10 (Newey-West automatic) using Bartlett kernel

Adj. t-Stat Prob.*

Phillips-Perron test statistic -2.970056 0.1408
Test critical values: 1% level -3.960384
5% level -3.410953
10% level -3.127286

*MacKinnon (1996) one-sided p-values.

Residual variance (no correction) 0.000116
HAC corrected variance (Bartlett kernel) 9.64E-05

Phillips-Perron Test Equation
Dependent Variable: D(E_LOG_CONSUMER_GOODS)
Method: Least Squares
Date: 03/22/19 Time: 06:35
Sample (adjusted): 3/16/2004 3/15/2019
Included observations: 3914 after adjustments

Variable Coefficient Std. Error t-Statistic Prob.

E_LOG_CONSUMER_GOODS(-1) -0.004415 0.001393 -3.169282 0.0015
C 0.031781 0.009845 3.228041 0.0013
@TREND(“3/15/2004”) 1.13E-06 4.26E-07 2.650349 0.0081

R-squared 0.002750 Mean dependent var 0.000341
Adjusted R-squared 0.002240 S.D. dependent var 0.010797
S.E. of regression 0.010785 Akaike info criterion -6.220632
Sum squared resid 0.454878 Schwarz criterion -6.215825
Log likelihood 12176.78 Hannan-Quinn criter. -6.218926
F-statistic 5.391658 Durbin-Watson stat 2.020342
Prob(F-statistic) 0.004588

Table 6 – Levels, None and Phillips–Perron (Variable E)
Null Hypothesis: E_LOG_CONSUMER_GOODS has a unit root
Exogenous: None
Bandwidth: 13 (Newey-West automatic) using Bartlett kernel

Adj. t-Stat Prob.*

Phillips-Perron test statistic 2.123614 0.9924
Test critical values: 1% level -2.565553
5% level -1.940905
10% level -1.616645

*MacKinnon (1996) one-sided p-values.

Residual variance (no correction) 0.000117
HAC corrected variance (Bartlett kernel) 9.28E-05

Phillips-Perron Test Equation
Dependent Variable: D(E_LOG_CONSUMER_GOODS)
Method: Least Squares
Date: 03/22/19 Time: 06:38
Sample (adjusted): 3/16/2004 3/15/2019
Included observations: 3914 after adjustments

Variable Coefficient Std. Error t-Statistic Prob.

E_LOG_CONSUMER_GOODS(-1) 4.27E-05 2.26E-05 1.885711 0.0594

R-squared -0.000088 Mean dependent var 0.000341
Adjusted R-squared -0.000088 S.D. dependent var 0.010797
S.E. of regression 0.010797 Akaike info criterion -6.218812
Sum squared resid 0.456172 Schwarz criterion -6.217210
Log likelihood 12171.22 Hannan-Quinn criter. -6.218244
Durbin-Watson stat 2.023610

Table 7- 1st Difference, Intercept and Schwarz (Augmented Dickey-Fuller Variable E)
Null Hypothesis: D(E_LOG_CONSUMER_GOODS) has a unit root
Exogenous: Constant
Lag Length: 1 (Automatic – based on SIC, maxlag=30)

t-Statistic Prob.*

Augmented Dickey-Fuller test statistic -48.10728 0.0001
Test critical values: 1% level -3.431836
5% level -2.862082
10% level -2.567102

*MacKinnon (1996) one-sided p-values.

Augmented Dickey-Fuller Test Equation
Dependent Variable: D(E_LOG_CONSUMER_GOODS,2)
Method: Least Squares
Date: 03/22/19 Time: 06:40
Sample (adjusted): 3/18/2004 3/15/2019
Included observations: 3912 after adjustments

Variable Coefficient Std. Error t-Statistic Prob.

D(E_LOG_CONSUMER_GOODS(-1)) -1.091278 0.022684 -48.10728 0.0000
D(E_LOG_CONSUMER_GOODS(-1),2) 0.078126 0.015945 4.899718 0.0000
C 0.000367 0.000172 2.132693 0.0330

R-squared 0.509111 Mean dependent var -3.15E-06
Adjusted R-squared 0.508860 S.D. dependent var 0.015363
S.E. of regression 0.010766 Akaike info criterion -6.224033
Sum squared resid 0.453101 Schwarz criterion -6.219223
Log likelihood 12177.21 Hannan-Quinn criter. -6.222326
F-statistic 2027.053 Durbin-Watson stat 2.003937
Prob(F-statistic) 0.000000

Table 8- 1st Difference, Trend & Intercept and Schwarz (Augmented Dickey-Fuller Variable E)

Null Hypothesis: D(E_LOG_CONSUMER_GOODS) has a unit root
Exogenous: Constant, Linear Trend
Lag Length: 1 (Automatic – based on SIC, maxlag=30)

t-Statistic Prob.*

Augmented Dickey-Fuller test statistic -48.11439 0.0000
Test critical values: 1% level -3.960385
5% level -3.410954
10% level -3.127286

*MacKinnon (1996) one-sided p-values.

Augmented Dickey-Fuller Test Equation
Dependent Variable: D(E_LOG_CONSUMER_GOODS,2)
Method: Least Squares
Date: 03/22/19 Time: 06:41
Sample (adjusted): 3/18/2004 3/15/2019
Included observations: 3912 after adjustments

Variable Coefficient Std. Error t-Statistic Prob.

D(E_LOG_CONSUMER_GOODS(-1)) -1.091666 0.022689 -48.11439 0.0000
D(E_LOG_CONSUMER_GOODS(-1),2) 0.078326 0.015947 4.911668 0.0000
C 0.000635 0.000345 1.841241 0.0657
@TREND(“3/15/2004”) -1.37E-07 1.52E-07 -0.895466 0.3706

R-squared 0.509212 Mean dependent var -3.15E-06
Adjusted R-squared 0.508835 S.D. dependent var 0.015363
S.E. of regression 0.010767 Akaike info criterion -6.223727
Sum squared resid 0.453008 Schwarz criterion -6.217314
Log likelihood 12177.61 Hannan-Quinn criter. -6.221451
F-statistic 1351.568 Durbin-Watson stat 2.003984
Prob(F-statistic) 0.000000

Table 9 – 1st Difference, None and Schwarz (Augmented Dickey-Fuller Variable E)

Null Hypothesis: D(E_LOG_CONSUMER_GOODS) has a unit root
Exogenous: None
Lag Length: 1 (Automatic – based on SIC, maxlag=30)

t-Statistic Prob.*

Augmented Dickey-Fuller test statistic -48.03819 0.0001
Test critical values: 1% level -2.565553
5% level -1.940905
10% level -1.616645

*MacKinnon (1996) one-sided p-values.

Augmented Dickey-Fuller Test Equation
Dependent Variable: D(E_LOG_CONSUMER_GOODS,2)
Method: Least Squares
Date: 03/22/19 Time: 06:43
Sample (adjusted): 3/18/2004 3/15/2019
Included observations: 3912 after adjustments

Variable Coefficient Std. Error t-Statistic Prob.

D(E_LOG_CONSUMER_GOODS(-1)) -1.089114 0.022672 -48.03819 0.0000
D(E_LOG_CONSUMER_GOODS(-1),2) 0.077043 0.015944 4.832053 0.0000

R-squared 0.508540 Mean dependent var -3.15E-06
Adjusted R-squared 0.508414 S.D. dependent var 0.015363
S.E. of regression 0.010771 Akaike info criterion -6.223381
Sum squared resid 0.453628 Schwarz criterion -6.220175
Log likelihood 12174.93 Hannan-Quinn criter. -6.222243
Durbin-Watson stat 2.003716

Table 10 – Levels, Intercept and Augmented Dickey-Fuller (Variable R)

Null Hypothesis: R_LOG_DAILY_RETURN has a unit root
Exogenous: Constant
Lag Length: 1 (Automatic – based on SIC, maxlag=30)

t-Statistic Prob.*

Augmented Dickey-Fuller test statistic -48.10728 0.0001
Test critical values: 1% level -3.431836
5% level -2.862082
10% level -2.567102

*MacKinnon (1996) one-sided p-values.

Augmented Dickey-Fuller Test Equation
Dependent Variable: D(R_LOG_DAILY_RETURN)
Method: Least Squares
Date: 03/23/19 Time: 14:23
Sample (adjusted): 3/18/2004 3/15/2019
Included observations: 3912 after adjustments

Variable Coefficient Std. Error t-Statistic Prob.

R_LOG_DAILY_RETURN(-1) -1.091278 0.022684 -48.10728 0.0000
D(R_LOG_DAILY_RETURN(-1)) 0.078126 0.015945 4.899718 0.0000
C 0.000367 0.000172 2.132693 0.0330

Table 11- Levels, Intercept & Intercept, and Augmented Dickey-Fuller (Variable R)

Null Hypothesis: R_LOG_DAILY_RETURN has a unit root
Exogenous: Constant, Linear Trend
Lag Length: 1 (Automatic – based on SIC, maxlag=30)

t-Statistic Prob.*

Augmented Dickey-Fuller test statistic -48.11439 0.0000
Test critical values: 1% level -3.960385
5% level -3.410954
10% level -3.127286

*MacKinnon (1996) one-sided p-values.

Variable Coefficient Std. Error t-Statistic Prob.

R_LOG_DAILY_RETURN(-1) -1.091666 0.022689 -48.11439 0.0000
D(R_LOG_DAILY_RETURN(-1)) 0.078326 0.015947 4.911668 0.0000
C 0.000635 0.000345 1.841241 0.0657
@TREND(“3/15/2004”) -1.37E-07 1.52E-07 -0.895466 0.3706

Table 12 -Levels, None and Augmented Dickey-Fuller (Variable R)

Null Hypothesis: R_LOG_DAILY_RETURN has a unit root
Exogenous: None
Lag Length: 1 (Automatic – based on SIC, maxlag=30)

t-Statistic Prob.*

Augmented Dickey-Fuller test statistic -48.03819 0.0001
Test critical values: 1% level -2.565553
5% level -1.940905
10% level -1.616645

*MacKinnon (1996) one-sided p-values.

Augmented Dickey-Fuller Test Equation
Dependent Variable: D(R_LOG_DAILY_RETURN)
Method: Least Squares
Date: 03/23/19 Time: 14:24
Sample (adjusted): 3/18/2004 3/15/2019
Included observations: 3912 after adjustments

Variable Coefficient Std. Error t-Statistic Prob.

R_LOG_DAILY_RETURN(-1) -1.089114 0.022672 -48.03819 0.0000
D(R_LOG_DAILY_RETURN(-1)) 0.077043 0.015944 4.832053 0.0000

Table 13-Levels, Intercept and Phillips–Perron (Variable R)
Null Hypothesis: R_LOG_DAILY_RETURN has a unit root
Exogenous: Constant
Bandwidth: 13 (Newey-West automatic) using Bartlett kernel

Adj. t-Stat Prob.*

Phillips-Perron test statistic -63.72634 0.0001
Test critical values: 1% level -3.431836
5% level -2.862082
10% level -2.567102

*MacKinnon (1996) one-sided p-values.

Residual variance (no correction) 0.000117
HAC corrected variance (Bartlett kernel) 9.47E-05

Phillips-Perron Test Equation
Dependent Variable: D(R_LOG_DAILY_RETURN)
Method: Least Squares
Date: 03/23/19 Time: 14:25
Sample (adjusted): 3/17/2004 3/15/2019
Included observations: 3913 after adjustments

Variable Coefficient Std. Error t-Statistic Prob.

R_LOG_DAILY_RETURN(-1) -1.012078 0.015987 -63.30497 0.0000
C 0.000342 0.000173 1.979794 0.0478

R-squared 0.506095 Mean dependent var -5.00E-06
Adjusted R-squared 0.505968 S.D. dependent var 0.015361
S.E. of regression 0.010797 Akaike info criterion -6.218616
Sum squared resid 0.455912 Schwarz criterion -6.215411
Log likelihood 12168.72 Hannan-Quinn criter. -6.217479
F-statistic 4007.520 Durbin-Watson stat 2.002003
Prob(F-statistic) 0.000000

Table 14- Levels, Trend & Intercept and Phillips–Perron (Variable R)

Null Hypothesis: R_LOG_DAILY_RETURN has a unit root
Exogenous: Constant, Linear Trend
Bandwidth: 14 (Newey-West automatic) using Bartlett kernel

Adj. t-Stat Prob.*

Phillips-Perron test statistic -63.78683 0.0000
Test critical values: 1% level -3.960384
5% level -3.410954
10% level -3.127286

*MacKinnon (1996) one-sided p-values.

Residual variance (no correction) 0.000116
HAC corrected variance (Bartlett kernel) 9.32E-05

Variable Coefficient Std. Error t-Statistic Prob.

R_LOG_DAILY_RETURN(-1) -1.012258 0.015989 -63.30809 0.0000
C 0.000593 0.000346 1.714739 0.0865
@TREND(“3/15/2004”) -1.28E-07 1.53E-07 -0.837407 0.4024

R-squared 0.506183 Mean dependent var -5.00E-06
Adjusted R-squared 0.505931 S.D. dependent var 0.015361
S.E. of regression 0.010797 Akaike info criterion -6.218285
Sum squared resid 0.455830 Schwarz criterion -6.213476
Log likelihood 12169.07 Hannan-Quinn criter. -6.216578
F-statistic 2003.957 Durbin-Watson stat 2.002030
Prob(F-statistic) 0.000000

Table 15- Levels, None and Phillips–Perron (Variable R)

Null Hypothesis: R_LOG_DAILY_RETURN has a unit root
Exogenous: None
Bandwidth: 11 (Newey-West automatic) using Bartlett kernel

Adj. t-Stat Prob.*

Phillips-Perron test statistic -63.57691 0.0001
Test critical values: 1% level -2.565553
5% level -1.940905
10% level -1.616645

*MacKinnon (1996) one-sided p-values.

Residual variance (no correction) 0.000117
HAC corrected variance (Bartlett kernel) 9.72E-05

Phillips-Perron Test Equation
Dependent Variable: D(R_LOG_DAILY_RETURN)
Method: Least Squares
Date: 03/23/19 Time: 14:26
Sample (adjusted): 3/17/2004 3/15/2019
Included observations: 3913 after adjustments

Variable Coefficient Std. Error t-Statistic Prob.

R_LOG_DAILY_RETURN(-1) -1.011073 0.015985 -63.25042 0.0000

R-squared 0.505600 Mean dependent var -5.00E-06
Adjusted R-squared 0.505600 S.D. dependent var 0.015361
S.E. of regression 0.010801 Akaike info criterion -6.218126
Sum squared resid 0.456369 Schwarz criterion -6.216523
Log likelihood 12166.76 Hannan-Quinn criter. -6.217557
Durbin-Watson stat 2.001850

Phase 2 – ARMA Model Selection
Table 16- ARMA Selection
Model Akaiki Information Criterion (AIC) Schwarz Criterion (SIC) Hannan-Quinn Criterion (HQIC)

AR1 -6.219349 -6.216144 -6.218211
AR2 -6.219022 -6.214214 -6.217316
AR3 -6.224428 -6.218015 -6.222153
AR4 -6.224425 -6.216407 -6.221579
AR5 -6.224610 -6.214986 -6.221194
AR6 -6.230524 -6.219295* -6.226539*
AR7 -6.230938 -6.218101 -6.226382
AR8 -6.231050 -6.216606 -6.225924
MA1 -0.626199 -0.621392 -0.624493
MA2 -1.778273 -1.771864 -1.775999
MA3 -2.634505 -2.626494 -2.631662
MA4 -3.293786 -3.284173 -3.290375
MA5 -3.739172 -3.727957 -3.735192
MA6 -4.123248 -4.110430 -4.118699
MA7 -4.395704 -4.381284 -4.390587
MA8 -4.618436 -4.602414 -4.612751
ARMA1,1 -6.218499 -6.212089 -6.216224
ARMA1,2 -6.223872 -6.215860 -6.221029
ARMA1,3 -6.224088 -6.214472 -6.220675
ARMA1,4 -6.224339 -6.213121 -6.220358
ARMA1,5 -6.224389 -6.211568 -6.219839
ARMA1,6 -6.224472 -6.210049 -6.219354
ARMA1,7 -6.224327 -6.208302 -6.218641
ARMA1,8 -6.224919 -6.207292 -6.218664
ARMA2,1 -6.221306 -6.213292 -6.218462
ARMA2,2 -6.223877 -6.214259 -6.220464
ARMA2,3 -6.224613 -6.213393 -6.220631
ARMA2,4 -6.224118 -6.211295 -6.219567
ARMA2,5 -6.224473 -6.210047 -6.219354
ARMA2,6 -6.224263 -6.208234 -6.218575
ARMA2,7 -6.224333 -6.206701 -6.218076
ARMA2,8 -6.224608 -6.205373 -6.217782
ARMA3,1 -6.223841 -6.214222 -6.220427
ARMA3,2 -6.224281 -6.213059 -6.220299
ARMA3,3 -6.225631 -6.212805 -6.221080
ARMA3,4 -6.228891 -6.214462 -6.223770
ARMA3,5 -6.224721 -6.208689 -6.219031
ARMA3,6 -6.224220 -6.206584 -6.217961
ARMA3,7 -6.227969 -6.208731 -6.221142
ARMA3,8 -6.223896 -6.203054 -6.216500
ARMA4,1 -6.226065 -6.214840 -6.222081
ARMA4,2 -6.225748 -6.212920 -6.221196
ARMA4,3 -6.225678 -6.211246 -6.220557
ARMA4,4 -6.226041 -6.210005 -6.220350
ARMA4,5 -6.225913 -6.208274 -6.219653
ARMA4,6 -6.224840 -6.205597 -6.218011
ARMA4,7 -6.226014 -6.205168 -6.218616
ARMA4,8 -6.225512 -6.203062 -6.217545
ARMA5,1 -6.227874 -6.215043 -6.223320
ARMA5,2 -6.228751 -6.214316 -6.223628
ARMA5,3 -6.228411 -6.212372 -6.222719
ARMA5,4 -6.229221 -6.211578 -6.222960
ARMA5,5 -6.229271 -6.210024 -6.222441
ARMA5,6 -6.228676 -6.207825 -6.221276
ARMA5,7 -6.229630 -6.207176 -6.221662
ARMA5,8 -6.229176 -6.205117 -6.220638
ARMA6,1 -6.230245 -6.215807 -6.225121
ARMA6,2 -6.229981 -6.213939 -6.224288
ARMA6,3 -6.229531 -6.211885 -6.223269
ARMA6,4 -6.229407 -6.210156 -6.222575
ARMA6,5 -6.230290 -6.209434 -6.222888
ARMA6,6 -6.229504 -6.207044 -6.221533
ARMA6,7 -6.229318 -6.205254 -6.220778
ARMA6,8 -6.229167 -6.203499 -6.220058
ARMA7,1 -6.229915 -6.213868 -6.224220
ARMA7,2 -6.229944 -6.212294 -6.223680
ARMA7,3 -6.229873 -6.210618 -6.223040
ARMA7,4 -6.229641 -6.208782 -6.222238
ARMA7,5 -6.231862 -6.209397 -6.223889
ARMA7,6 -6.231862 -6.209397 -6.223889
ARMA7,7 -6.232250 -6.206577 -6.223139
ARMA7,8 -6.233330* -6.206051 -6.223649
ARMA8,1 -6.230896 -6.213241 -6.224630
ARMA8,2 -6.230880 -6.211621 -6.224045
ARMA8,3 -6.230555 -6.209691 -6.223151
ARMA8,4 -6.230092 -6.207623 -6.222118
ARMA8,5 -6.229676 -6.205602 -6.221132
ARMA8,6 -6.230799 -6.205120 -6.221686
ARMA8,7 -6.231941 -6.204657 -6.222258
ARMA8,8 -6.232627 -6.203737 -6.222374
The* codex exemplifies smallest or minimum values

Table 17- ARMA Selection (Continued)
Akaike Information Criterion
AIC=T(log.RSS)+2n Schwarz Information Criterion
BIC=T(log.RSS)+n(log.T)
AR(6) -6.230524 -6.219295
ARMA(7,8 ) -6.233330 -6.206051
ARMA(7,8) is smallest AR(6) is smallest

Table 18 – AR (6)
Dependent Variable: E
Method: Least Squares
Date: 03/26/19 Time: 21:03
Sample (adjusted): 3/23/2004 3/15/2019
Included observations: 3909 after adjustments

Variable Coefficient Std. Error t-Statistic Prob.

C 0.008019 0.003804 2.107978 0.0351
E(-1) 0.986243 0.015956 61.81023 0.0000
E(-2) -0.063604 0.022388 -2.841030 0.0045
E(-3) 0.049703 0.022394 2.219481 0.0265
E(-4) 0.054938 0.022395 2.453161 0.0142
E(-5) -0.053275 0.022388 -2.379608 0.0174
E(-6) 0.024994 0.015946 1.567420 0.1171

R-squared 0.999029 Mean dependent var 7.622632
Adjusted R-squared 0.999028 S.D. dependent var 0.343981
S.E. of regression 0.010726 Akaike info criterion -6.230524
Sum squared resid 0.448904 Schwarz criterion -6.219295
Log likelihood 12184.56 Hannan-Quinn criter. -6.226539
F-statistic 669244.3 Durbin-Watson stat 1.997115
Prob(F-statistic) 0.000000

Table 19- ARMA (7, 8)
Dependent Variable: E
Method: ARMA Maximum Likelihood (OPG – BHHH)
Date: 03/26/19 Time: 22:11
Sample: 3/24/2004 3/15/2019
Included observations: 3908
Convergence achieved after 54 iterations
Coefficient covariance computed using outer product of gradients

Variable Coefficient Std. Error t-Statistic Prob.

C 0.025365 0.013364 1.897996 0.0578
E(-1) 0.314738 0.047256 6.660298 0.0000
E(-2) 0.509666 0.051557 9.885470 0.0000
E(-3) -0.209380 0.057445 -3.644845 0.0003
E(-4) 0.248442 0.052573 4.725672 0.0000
E(-5) -0.406196 0.045504 -8.926556 0.0000
E(-6) -0.236856 0.039584 -5.983616 0.0000
E(-7) 0.776422 0.043252 17.95131 0.0000
MA(1) 0.675972 0.047891 14.11480 0.0000
MA(2) 0.089383 0.070949 1.259818 0.2078
MA(3) 0.302137 0.070482 4.286733 0.0000
MA(4) 0.121493 0.063219 1.921780 0.0547
MA(5) 0.508155 0.058125 8.742525 0.0000
MA(6) 0.736879 0.043280 17.02605 0.0000
MA(7) -0.075669 0.014711 -5.143846 0.0000
MA(8) -0.058040 0.013140 -4.417026 0.0000
SIGMASQ 0.000114 1.49E-06 76.18589 0.0000

R-squared 0.999037 Mean dependent var 7.622845
Adjusted R-squared 0.999033 S.D. dependent var 0.343765
S.E. of regression 0.010689 Akaike info criterion -6.233330
Sum squared resid 0.444541 Schwarz criterion -6.206051
Log likelihood 12196.93 Hannan-Quinn criter. -6.223649
F-statistic 252335.6 Durbin-Watson stat 2.000618
Prob(F-statistic) 0.000000

Inverted MA Roots .73-.61i .73+.61i .30 -.17+.94i
-.17-.94i -.25 -.92+.29i -.92-.29i

Equation 3- Ljung-Box Q Test
Ho: ρ_1=0,ρ_2=0,ρ_3=0,………,ρ_k=0 with (3.1)

Ha:at least one of ρ_j≠0,(j = 1,2,………,k)
Q^*=T(T+2) ∑_(j=1)^k▒ρ ̂_j^2 /(T-j) ~ χ^2 (ν=k)
Table 20 – Residual AR (6)

Date: 03/27/19 Time: 13:42
Sample: 3/15/2004 3/15/2019
Included observations: 3909
Q-statistic probabilities adjusted for 6 dynamic regressors

Autocorrelation Partial Correlation AC PAC Q-Stat Prob

| | | | 1 0.001 0.001 0.0051 0.943
| | | | 2 -0.002 -0.002 0.0182 0.991
| | | | 3 0.003 0.003 0.0484 0.997
| | | | 4 -0.003 -0.003 0.0955 0.999
| | | | 5 -0.001 -0.001 0.0984 1.000
| | | | 6 -0.020 -0.020 1.6275 0.951
| | | | 7 -0.019 -0.019 3.0326 0.882
| | | | 8 0.032 0.032 6.9521 0.542
| | | | 9
-0.008 -0.008 7.1888 0.617
| | | | 10 -0.011 -0.011 7.6782 0.660
| | | | 11 -0.007 -0.007 7.8475 0.727
| | | | 12 -0.002 -0.002 7.8639 0.796
| | | | 13 -0.009 -0.010 8.1976 0.830
| |
| | 14 -0.033 -0.033 12.597 0.558
| | | | 15 0.001 0.002 12.599 0.633
| | | | 16 -0.000 -0.002 12.600 0.702
| | | | 17 -0.009 -0.009 12.892 0.743
| | | | 18 -0.009 -0.009 13.211 0.779
| | | | 19 0.016 0.016 14.260 0.768
| | | | 20 -0.008 -0.010 14.510 0.804
| |
| | 21 -0.020 -0.021 16.053 0.767
| | | | 22 0.002 0.003 16.065 0.813
| | | | 23 -0.015 -0.017 16.989 0.810
| | | | 24 0.006 0.005 17.143 0.842
| | | | 25 -0.001 -0.001 17.151 0.876
| | | | 26 -0.001 -0.000 17.152 0.904
| |
| | 27 -0.033 -0.036 21.384 0.768
| | | | 28 -0.005 -0.007 21.497 0.804
| | | | 29 0.002 0.003 21.511 0.840
| | | | 30 0.026 0.025 24.199 0.763
| | | | 31 -0.030 -0.030 27.645 0.639
| | | | 32 -0.002 -0.003 27.664 0.686
| | | | 33 -0.005 -0.006 27.762 0.725
| | | | 34 0.007 0.004 27.937 0.759
| | | | 35 0.007 0.008 28.151 0.788
| | | |
36 0.002 0.003 28.162 0.821

8th Order of the autocorrelation – Number of estimated parameters
8- 6 = 2 degrees of freedom (d.f)
Using the Chi-squared distribution table the extracted d.f value is 5.991
6.9521 > 5.991
At a 5% percent level of significance reject Ho in favour Ha

14th Order of the autocorrelation – Number of estimated parameters
14- 6 = 8 degrees of freedom (d.f)
Using the Chi-squared distribution table the extracted d.f value is 15.51
12.597 < 15.51
At a 5% percent level of significance do no reject Ho in favour Ha

21st Order of the autocorrelation – Number of estimated parameters
21- 6 = 15 degrees of freedom (d.f)
Using the Chi-squared distribution table the extracted d.f value is 25.00
16.053 < 25.00
At a 5% percent level of significance do no reject Ho in favour Ha

27th Order of the autocorrelation – Number of estimated parameters
27- 6 = 21 degrees of freedom (d.f)
Using the Chi-squared distribution table the extracted d.f value is 32.671
21.384 < 32.671
At a 5% percent level of significance do no reject Ho in favour Ha

30th Order of the autocorrelation – Number of estimated parameters
30- 6 = 24 degrees of freedom (d.f)
Using the Chi-squared distribution table the extracted d.f value is 36.415
24.119 < 36.415
At a 5% percent level of significance do no reject Ho in favour Ha

36th Order of the autocorrelation – Number of estimated parameters
36- 6 = 30 degrees of freedom (d.f)
Using the Chi-squared distribution table the extracted d.f value is 43.773
28.162 < 43.773
At a 5% percent level of significance do no reject Ho in favour Ha

Table 21- Residuals ARMA (7,8)

Date: 03/27/19 Time: 13:50
Sample: 3/15/2004 3/15/2019
Included observations: 3908
Q-statistic probabilities adjusted for 8 ARMA terms and 7 dynamic
regressors

Autocorrelation Partial Correlation AC PAC Q-Stat Prob*

| | | | 1 -0.000 -0.000 0.0005
| | | | 2 -0.001 -0.001 0.0019
| | | | 3 0.004 0.004 0.0679
| | | | 4 0.008 0.008 0.3237
| | | | 5 -0.007 -0.007 0.5276
| | | | 6 -0.010 -0.010 0.9138
| | | | 7 -0.002 -0.002 0.9348
| | | | 8 0.000 0.000 0.9349
| | | | 9 -0.004 -0.003 0.9858 0.321
| | | | 10 -0.008 -0.008 1.2321 0.540
| | | | 11 -0.021 -0.021 3.0120 0.390
| | | | 12 -0.008 -0.009 3.2883 0.511
| | | | 13 0.000 0.000 3.2884 0.656
| | | | 14 -0.013 -0.012 3.9209 0.687
| | | | 15 -0.016 -0.016 4.9456 0.667
| | | | 16 0.009 0.008 5.2556 0.730
| | | | 17 -0.007 -0.007 5.4404 0.794
| | | | 18 -0.011 -0.011 5.8936 0.824
| | | | 19 -0.004 -0.004 5.9596 0.876
| | | | 20 -0.004 -0.004 6.0127 0.915
| | | | 21 -0.013
-0.013 6.6422 0.920
| | | | 22 -0.003 -0.004 6.6847 0.946
| | | | 23 -0.011 -0.012 7.1646 0.953
| | | | 24 0.012 0.011 7.7180 0.957
| | | | 25 0.008 0.007 7.9587 0.967
| | | | 26 -0.018 -0.019 9.2004 0.955
| | | |
27 -0.029 -0.030 12.589 0.859
| | | | 28 -0.006 -0.007 12.723 0.889
| | | | 29 0.003 0.002 12.760 0.917
| | | |
30 0.019 0.020 14.246 0.893
| | | | 31 -0.023 -0.023 16.391 0.838
| | | | 32 0.007 0.005 16.600 0.865
| | | | 33 -0.011 -0.013 17.108 0.878
| | | | 34 0.012 0.011 17.679 0.887
| | | | 35 0.009 0.009 17.973 0.904
| | | | 36 0.003 0.003 18.015 0.926

*Probabilities may not be valid for this equation specification.

21st Order of the autocorrelation – Number of estimated parameters
21- 15 = 6 degrees of freedom (d.f)
Using the Chi-squared distribution table the extracted d.f value is 12.592
6.6422 < 12.592
At a 5% percent level of significance do no reject Ho in favour Ha

27th Order of the autocorrelation – Number of estimated parameters
27- 15 = 12 degrees of freedom (d.f)
Using the Chi-squared distribution table the extracted d.f value is 21.026
12.589 < 21.026
At a 5% percent level of significance do no reject Ho in favour Ha

30th Order of the autocorrelation – Number of estimated parameters
30- 15 = 15 degrees of freedom (d.f)
Using the Chi-squared distribution table the extracted d.f value is 14.246
14.246 < 24.996
At a 5% percent level of significance do no reject Ho in favour Ha

36th Order of the autocorrelation – Number of estimated parameters
36- 15 = 21 degrees of freedom (d.f)
Using the Chi-squared distribution table the extracted d.f value is 32.671
18.015 < 32.671
At a 5% percent level of significance do no reject Ho in favour Ha

Table 22- Fixed Forecast Selection in- sample 3/15/2004- 7/30/2004
Model RMSE MAE MAPE THEIL AIC BIC
AR(6) 0.012696 0.009782 0.142836 0.000947 -5.703583* -5.514189*
ARMA (7,8) 0.011898* 0.009018* 0.131685* 0.000869* -5.683707 -5.220759
The* codex exemplifies smallest or minimum values
Figure 10 – AR (6) Forecast 3/15/2004 – 7/30/2004

Figure 11 – ARMA (7, 8) Forecast 3/15/2004 – 7/30/2004

Figure 12- Forecast 3/15/2004 – 7/30/2004

Table 23- Fixed Forecast out of sample 3/15/2004 – 10/29/2018
Model RMSE MAE MAPE THEIL AIC BIC
AR(6) 0.111411 0.101756 1.312641 0.007361 -6.223363* -6.211888*
ARMA (7,8) 0.010719* 0.007600* 0.100718* 0.000703* -6.226063 -6.198190
The* codex exemplifies smallest or minimum values

Figure 13- ARMA (7,8) Forecast 3/15/2004 – 10/29/2018

Figure 14- AR (6) Forecast 3/15/2004-10/29/2018

Figure 15- Forecast 3/15/2004 – 10/29/2018

Phase 3 – ARCH, GARCH, EGARCH & TGARCH
The ARCH effect is outlined as the relationship within heteroscedasticity where the analysis of unexplained econometric models are tested. Given that the volatility is used to represent risk, the ARCH effect in subsequent response measures risk.

Y_t= β_t+β_t X_t+u_t (3.1)
u_t ~ N(0,α_0+α_1 U_(t-1)^2 (3.2)

u_t is normally distributed with zero mean and
σ_t^2=var(u_t/(u_(t-1,) u_(t-2, …..) )) (3.3)
Where is the conditional variance of the error term. The ARCH effect is then modelled by:
u_t ~ N(0,α_0+α_1 U_(t-1)^2 (3.2)

Table 24- AR (1)
Dependent Variable: R_LOG_DAILY_RETURN
Method: Least Squares
Date: 03/25/19 Time: 11:03
Sample (adjusted): 3/17/2004 3/15/2019
Included observations: 3913 after adjustments

Variable Coefficient Std. Error t-Statistic Prob.

C 0.000342 0.000173 1.979794 0.0478
R_LOG_DAILY_RETURN(-1) -0.012078 0.015987 -0.755452 0.4500

R-squared 0.000146 Mean dependent var 0.000338
Adjusted R-squared -0.000110 S.D. dependent var 0.010796
S.E. of regression 0.010797 Akaike info criterion -6.218616
Sum squared resid 0.455912 Schwarz criterion -6.215411
Log likelihood 12168.72 Hannan-Quinn criter. -6.217479
F-statistic 0.570708 Durbin-Watson stat 2.002003
Prob(F-statistic) 0.450023

Table 25- ARCH (1)
Dependent Variable: R_LOG_DAILY_RETURN
Method: ML ARCH – Normal distribution (BFGS / Marquardt steps)
Date: 03/25/19 Time: 12:54
Sample (adjusted): 3/17/2004 3/15/2019
Included observations: 3913 after adjustments
Convergence achieved after 12 iterations
Coefficient covariance computed using outer product of gradients
Presample variance: backcast (parameter = 0.7)
GARCH = C(3) + C(4)*RESID(-1)^2

Variable Coefficient Std. Error z-Statistic Prob.

C 0.000561 0.000146 3.852539 0.0001
R_LOG_DAILY_RETURN(-1) 0.008558 0.013398 0.638756 0.5230

Variance Equation

C 8.05E-05 1.65E-06 48.68638 0.0000
RESID(-1)^2 0.302640 0.022573 13.40709 0.0000

R-squared -0.000721 Mean dependent var 0.000338
Adjusted R-squared -0.000977 S.D. dependent var 0.010796
S.E. of regression 0.010802 Akaike info criterion -6.328949
Sum squared resid 0.456307 Schwarz criterion -6.322538
Log likelihood 12386.59 Hannan-Quinn criter. -6.326674
Durbin-Watson stat 2.039139

Figure 16- Residuals ARCH Effect

Table 26- ARCH LM Test
Heteroskedasticity Test: ARCH

F-statistic 43.27119 Prob. F(1,3910) 0.0000
Obs*R-squared 42.81945 Prob. Chi-Square(1) 0.0000

Test Equation:
Dependent Variable: RESID^2
Method: Least Squares
Date: 03/27/19 Time: 21:12
Sample (adjusted): 3/18/2004 3/15/2019
Included observations: 3912 after adjustments

Variable Coefficient Std. Error t-Statistic Prob.

C 7.55E-08 1.73E-08 4.369493 0.0000
RESID^2(-1) 0.104622 0.015905 6.578084 0.0000

R-squared 0.010946 Mean dependent var 8.43E-08
Adjusted R-squared 0.010693 S.D. dependent var 1.08E-06
S.E. of regression 1.08E-06 Akaike info criterion -24.64366
Sum squared resid 4.54E-09 Schwarz criterion -24.64045
Log likelihood 48205.00 Hannan-Quinn criter. -24.64252
F-statistic 43.27119 Durbin-Watson stat 2.037400
Prob(F-statistic) 0.000000

Table 27 – GARCH (1,1)

Dependent Variable: R_LOG_DAILY_RETURN
Method: ML ARCH – Normal distribution (BFGS / Marquardt steps)
Date: 03/27/19 Time: 23:18
Sample (adjusted): 3/17/2004 3/15/2019
Included observations: 3913 after adjustments
Convergence achieved after 25 iterations
Coefficient covariance computed using outer product of gradients
Presample variance: backcast (parameter = 0.7)
GARCH = C(3) + C(4)*RESID(-1)^2 + C(5)*GARCH(-1)

Variable Coefficient Std. Error z-Statistic Prob.

C 0.000485 0.000136 3.562281 0.0004
R_LOG_DAILY_RETURN(-1) 0.002006 0.017327 0.115765 0.9078

Variance Equation

C 5.41E-06 6.27E-07 8.625251 0.0000
RESID(-1)^2 0.115519 0.008199 14.08868 0.0000
GARCH(-1) 0.835628 0.011556 72.31256 0.0000

R-squared -0.000241 Mean dependent var 0.000338
Adjusted R-squared -0.000497 S.D. dependent var 0.010796
S.E. of regression 0.010799 Akaike info criterion -6.429341
Sum squared resid 0.456088 Schwarz criterion -6.421326
Log likelihood 12584.01 Hannan-Quinn criter. -6.426497
Durbin-Watson stat 2.027583

Table 28 – GARCH (1,2)
Dependent Variable: R_LOG_DAILY_RETURN
Method: ML ARCH – Normal distribution (BFGS / Marquardt steps)
Date: 03/27/19 Time: 23:19
Sample (adjusted): 3/17/2004 3/15/2019
Included observations: 3913 after adjustments
Convergence achieved after 31 iterations
Coefficient covariance computed using outer product of gradients
Presample variance: backcast (parameter = 0.7)
GARCH = C(3) + C(4)*RESID(-1)^2 + C(5)*GARCH(-1) + C(6)*GARCH(-2)

Variable Coefficient Std. Error z-Statistic Prob.

C 0.000491 0.000137 3.594923 0.0003
R_LOG_DAILY_RETURN(-1) 0.001789 0.017560 0.101884 0.9188

Variance Equation

C 5.75E-06 8.64E-07 6.658223 0.0000
RESID(-1)^2 0.125660 0.016927 7.423843 0.0000
GARCH(-1) 0.699194 0.135013 5.178703 0.0000
GARCH(-2) 0.123128 0.116666 1.055395 0.2912

R-squared -0.000249 Mean dependent var 0.000338
Adjusted R-squared -0.000505 S.D. dependent var 0.010796
S.E. of regression 0.010799 Akaike info criterion -6.429014
Sum squared resid 0.456092 Schwarz criterion -6.419396
Log likelihood 12584.37 Hannan-Quinn criter. -6.425601
Durbin-Watson stat 2.027154

Table 29- GARCH (1,3)
Dependent Variable: R_LOG_DAILY_RETURN
Method: ML ARCH – Normal distribution (BFGS / Marquardt steps)
Date: 03/27/19 Time: 23:21
Sample (adjusted): 3/17/2004 3/15/2019
Included observations: 3913 after adjustments
Convergence achieved after 43 iterations
Coefficient covariance computed using outer product of gradients
Presample variance: backcast (parameter = 0.7)
GARCH = C(3) + C(4)*RESID(-1)^2 + C(5)*GARCH(-1) + C(6)*GARCH(-2) +
C(7)*GARCH(-3)

Variable Coefficient Std. Error z-Statistic Prob.

C 0.000480 0.000137 3.498058 0.0005
R_LOG_DAILY_RETURN(-1) 0.004417 0.017525 0.252045 0.8010

Variance Equation

C 5.38E-06 6.91E-07 7.785966 0.0000
RESID(-1)^2 0.123095 0.010982 11.20834 0.0000
GARCH(-1) 0.981528 0.113183 8.672050 0.0000
GARCH(-2) -0.500598 0.162495 -3.080705 0.0021
GARCH(-3) 0.347169 0.080385 4.318847 0.0000

R-squared -0.000305 Mean dependent var 0.000338
Adjusted R-squared -0.000560 S.D. dependent var 0.010796
S.E. of regression 0.010799 Akaike info criterion -6.429519
Sum squared resid 0.456117 Schwarz criterion -6.418299
Log likelihood 12586.35 Hannan-Quinn criter. -6.425538
Durbin-Watson stat 2.032046

Table 30- GARCH (2,1)
Dependent Variable: R_LOG_DAILY_RETURN
Method: ML ARCH – Normal distribution (BFGS / Marquardt steps)
Date: 03/27/19 Time: 23:20
Sample (adjusted): 3/17/2004 3/15/2019
Included observations: 3913 after adjustments
Convergence achieved after 28 iterations
Coefficient covariance computed using outer product of gradients
Presample variance: backcast (parameter = 0.7)
GARCH = C(3) + C(4)*RESID(-1)^2 + C(5)*RESID(-2)^2 + C(6)*GARCH(-1)

Variable Coefficient Std. Error z-Statistic Prob.

C 0.000495 0.000137 3.619344 0.0003
R_LOG_DAILY_RETURN(-1) 0.001945 0.017703 0.109871 0.9125

Variance Equation

C 4.67E-06 5.90E-07 7.909670 0.0000
RESID(-1)^2 0.132767 0.018052 7.354758 0.0000
RESID(-2)^2 -0.028739 0.017818 -1.612920 0.1068
GARCH(-1) 0.853695 0.011466 74.45755 0.0000

R-squared -0.000265 Mean dependent var 0.000338
Adjusted R-squared -0.000521 S.D. dependent var 0.010796
S.E. of regression 0.010799 Akaike info criterion -6.429150
Sum squared resid 0.456099 Schwarz criterion -6.419533
Log likelihood 12584.63 Hannan-Quinn criter. -6.425737
Durbin-Watson stat 2.027419

Table 31- GARCH (2,2)

Dependent Variable: R_LOG_DAILY_RETURN
Method: ML ARCH – Normal distribution (BFGS / Marquardt steps)
Date: 03/27/19 Time: 23:19
Sample (adjusted): 3/17/2004 3/15/2019
Included observations: 3913 after adjustments
Failure to improve likelihood (singular hessian) after 88 iterations
Coefficient covariance computed using outer product of gradients
Presample variance: backcast (parameter = 0.7)
GARCH = C(3) + C(4)*RESID(-1)^2 + C(5)*RESID(-2)^2 + C(6)*GARCH(-1)
+ C(7)*GARCH(-2)

Variable Coefficient Std. Error z-Statistic Prob.

C 0.000492 0.000136 3.612468 0.0003
R_LOG_DAILY_RETURN(-1) 0.004846 0.017867 0.271246 0.7862

Variance Equation

C 7.62E-08 2.56E-08 2.971416 0.0030
RESID(-1)^2 0.135801 0.011798 11.51046 0.0000
RESID(-2)^2 -0.133242 0.011548 -11.53839 0.0000
GARCH(-1) 1.743721 0.025967 67.15119 0.0000
GARCH(-2) -0.747046 0.025303 -29.52347 0.0000

R-squared -0.000348 Mean dependent var 0.000338
Adjusted R-squared -0.000604 S.D. dependent var 0.010796
S.E. of regression 0.010800 Akaike info criterion -6.436488
Sum squared resid 0.456137 Schwarz criterion -6.425268
Log likelihood 12599.99 Hannan-Quinn criter. -6.432507
Durbin-Watson stat 2.032778

Table 32- GARCH (2,3)
Dependent Variable: R_LOG_DAILY_RETURN
Method: ML ARCH – Normal distribution (BFGS / Marquardt steps)
Date: 03/27/19 Time: 23:21
Sample (adjusted): 3/17/2004 3/15/2019
Included observations: 3913 after adjustments
Convergence achieved after 46 iterations
Coefficient covariance computed using outer product of gradients
Presample variance: backcast (parameter = 0.7)
GARCH = C(3) + C(4)*RESID(-1)^2 + C(5)*RESID(-2)^2 + C(6)*GARCH(-1)
+ C(7)*GARCH(-2) + C(8)*GARCH(-3)

Variable Coefficient Std. Error z-Statistic Prob.

C 0.000484 0.000137 3.528764 0.0004
R_LOG_DAILY_RETURN(-1) 0.004376 0.017632 0.248167 0.8040

Variance Equation

C 4.84E-06 1.20E-06 4.026676 0.0001
RESID(-1)^2 0.126585 0.016781 7.543189 0.0000
RESID(-2)^2 -0.014752 0.033783 -0.436667 0.6624
GARCH(-1) 1.055794 0.204503 5.162726 0.0000
GARCH(-2) -0.544849 0.192773 -2.826384 0.0047
GARCH(-3) 0.333237 0.090986 3.662509 0.0002

R-squared -0.000313 Mean dependent var 0.000338
Adjusted R-squared -0.000569 S.D. dependent var 0.010796
S.E. of regression 0.010799 Akaike info criterion -6.429036
Sum squared resid 0.456121 Schwarz criterion -6.416213
Log likelihood 12586.41 Hannan-Quinn criter. -6.424485
Durbin-Watson stat 2.031949

Table 33- GARCH (3,1)

Dependent Variable: R_LOG_DAILY_RETURN
Method: ML ARCH – Normal distribution (BFGS / Marquardt steps)
Date: 03/27/19 Time: 23:22
Sample (adjusted): 3/17/2004 3/15/2019
Included observations: 3913 after adjustments
Convergence achieved after 38 iterations
Coefficient covariance computed using outer product of gradients
Presample variance: backcast (parameter = 0.7)
GARCH = C(3) + C(4)*RESID(-1)^2 + C(5)*RESID(-2)^2 + C(6)*RESID(-3)^2
+ C(7)*GARCH(-1)

Variable Coefficient Std. Error z-Statistic Prob.

C 0.000494 0.000137 3.604216 0.0003
R_LOG_DAILY_RETURN(-1) 0.004388 0.017757 0.247122 0.8048

Variance Equation

C 3.74E-06 5.53E-07 6.759432 0.0000
RESID(-1)^2 0.130553 0.018030 7.240862 0.0000
RESID(-2)^2 -0.003632 0.021220 -0.171153 0.8641
RESID(-3)^2 -0.039456 0.016193 -2.436643 0.0148
GARCH(-1) 0.878466 0.012273 71.57737 0.0000

R-squared -0.000339 Mean dependent var 0.000338
Adjusted R-squared -0.000595 S.D. dependent var 0.010796
S.E. of regression 0.010799 Akaike info criterion -6.429268
Sum squared resid 0.456133 Schwarz criterion -6.418048
Log likelihood 12585.86 Hannan-Quinn criter. -6.425286
Durbin-Watson stat 2.031920

Table 34- GARCH (3,2)

Dependent Variable: R_LOG_DAILY_RETURN
Method: ML ARCH – Normal distribution (BFGS / Marquardt steps)
Date: 03/27/19 Time: 23:19
Sample (adjusted): 3/17/2004 3/15/2019
Included observations: 3913 after adjustments
Failure to improve likelihood (non-zero gradients) after 93 iterations
Coefficient covariance computed using outer product of gradients
Presample variance: backcast (parameter = 0.7)
GARCH = C(3) + C(4)*RESID(-1)^2 + C(5)*RESID(-2)^2 + C(6)*RESID(-3)^2
+ C(7)*GARCH(-1) + C(8)*GARCH(-2)

Variable Coefficient Std. Error z-Statistic Prob.

C 0.000492 0.000136 3.611666 0.0003
R_LOG_DAILY_RETURN(-1) 0.004849 0.017875 0.271285 0.7862

Variance Equation

C 7.62E-08 2.63E-08 2.897060 0.0038
RESID(-1)^2 0.135852 0.018226 7.453877 0.0000
RESID(-2)^2 -0.133299 0.035087 -3.799064 0.0001
RESID(-3)^2 8.04E-06 0.019057 0.000422 0.9997
GARCH(-1) 1.743644 0.031128 56.01557 0.0000
GARCH(-2) -0.746970 0.030426 -24.55021 0.0000

R-squared -0.000348 Mean dependent var 0.000338
Adjusted R-squared -0.000604 S.D. dependent var 0.010796
S.E. of regression 0.010800 Akaike info criterion -6.435977
Sum squared resid 0.456137 Schwarz criterion -6.423154
Log likelihood 12599.99 Hannan-Quinn criter. -6.431427
Durbin-Watson stat 2.032783

Table 35 – GARCH (3,3)

Dependent Variable: R_LOG_DAILY_RETURN
Method: ML ARCH – Normal distribution (BFGS / Marquardt steps)
Date: 03/27/19 Time: 23:20
Sample (adjusted): 3/17/2004 3/15/2019
Included observations: 3913 after adjustments
Convergence not achieved after 500 iterations
Coefficient covariance computed using outer product of gradients
Presample variance: backcast (parameter = 0.7)
GARCH = C(3) + C(4)*RESID(-1)^2 + C(5)*RESID(-2)^2 + C(6)*RESID(-3)^2
+ C(7)*GARCH(-1) + C(8)*GARCH(-2) + C(9)*GARCH(-3)

Variable Coefficient Std. Error z-Statistic Prob.

C 0.000492 0.000136 3.605233 0.0003
R_LOG_DAILY_RETURN(-1) 0.004827 0.017884 0.269908 0.7872

Variance Equation

C 1.04E-07 4.90E-07 0.211920 0.8322
RESID(-1)^2 0.135876 0.017677 7.686806 0.0000
RESID(-2)^2 -0.083572 0.880353 -0.094930 0.9244
RESID(-3)^2 -0.048815 0.863696 -0.056518 0.9549
GARCH(-1) 1.374918 6.489196 0.211878 0.8322
GARCH(-2) -0.103258 11.32263 -0.009120 0.9927
GARCH(-3) -0.276195 4.854769 -0.056891 0.9546

R-squared -0.000347 Mean dependent var 0.000338
Adjusted R-squared -0.000603 S.D. dependent var 0.010796
S.E. of regression 0.010799 Akaike info criterion -6.435467
Sum squared resid 0.456137 Schwarz criterion -6.421041
Log likelihood 12599.99 Hannan-Quinn criter. -6.430348
Durbin-Watson stat 2.032742

Table 36- GARCH Selection
Model Akaiki Information Criterion (AIC) Schwarz Criterion (SIC) Insignificant Coefficient (Prob Values) MSE RMSE MAPE (%)
GARCH(1,1) -6.429341 -6.421326 0* 0.010796* 0.007599* 120.0952*
GARCH(1,2) -6.429014 -6.419396 1 0.010796 0.007599 120.3880
GARCH(1,3) -6.429519 -6.418299 0 0.010797 0.007599 120.1653
GARCH(2,1) -6.429150 -6.419533 1 0.010796 0.007599 120.6490
GARCH(2,2) -6.436488* -6.425268* 0* 0.010797 0.076000 120.8667
GARCH(2,3) -6.429036 -6.416213 1 0.010797 0.007599 120.3927
GARCH(3,1) -6.429268 -6.418048 1 0.010797 0.007600 120.9522
GARCH(3,2) -6.435977 -6.423154 1 0.010797 0.007600 120.8686
GARCH(3,3) -6.435467 -6.421041 6 0.010797 0.007600 120.8568
The* codex exemplifies smallest or minimum values
Table 37 – EGARCH
Dependent Variable: R_LOG_DAILY_RETURN
Method: ML ARCH – Normal distribution (BFGS / Marquardt steps)
Date: 03/28/19 Time: 16:57
Sample (adjusted): 3/17/2004 3/15/2019
Included observations: 3913 after adjustments
Failure to improve likelihood (singular hessian) after 312 iterations
Coefficient covariance computed using outer product of gradients
Presample variance: backcast (parameter = 0.7)
LOG(GARCH) = C(3) + C(4)*ABS(RESID(-1)/@SQRT(GARCH(-1))) + C(5)
*LOG(GARCH(-1))

Variable Coefficient Std. Error z-Statistic Prob.

C 0.000616 0.000119 5.157953 0.0000
R_LOG_DAILY_RETURN(-1) 0.008242 0.016481 0.500100 0.6170

Variance Equation

C(3) -0.699200 0.059753 -11.70157 0.0000
C(4) 0.239185 0.012153 19.68121 0.0000
C(5) 0.943795 0.005873 160.6892 0.0000

R-squared -0.000944 Mean dependent var 0.000338
Adjusted R-squared -0.001200 S.D. dependent var 0.010796
S.E. of regression 0.010803 Akaike info criterion -6.424683
Sum squared resid 0.456409 Schwarz criterion -6.416669
Log likelihood 12574.89 Hannan-Quinn criter. -6.421839
Durbin-Watson stat 2.038076

C2 – ARCH Effect
C3 – Leverage Effect
C4 – GARCH Effect

LNϑ_t^2= α_0+ 〖α1〗_( |(ε_(t-1)^2)/ϑ_(t-1) | ) + β_1 lnϑ_(t-1)^2+ γ (ε_(t-1)^2)/ϑ_(t-1)
Table 38- TGARCH

Dependent Variable: R_LOG_DAILY_RETURN
Method: ML ARCH – Normal distribution (BFGS / Marquardt steps)
Date: 03/28/19 Time: 15:42
Sample (adjusted): 3/17/2004 3/15/2019
Included observations: 3913 after adjustments
Convergence achieved after 27 iterations
Coefficient covariance computed using outer product of gradients
Presample variance: backcast (parameter = 0.7)
GARCH = C(3) + C(4)*RESID(-1)^2 + C(5)*RESID(-1)^2*(RESID(-1)<0) +
C(6)*GARCH(-1)

Variable Coefficient Std. Error z-Statistic Prob.

C 0.000227 0.000146 1.557405 0.1194
R_LOG_DAILY_RETURN(-1) 0.005449 0.016878 0.322823 0.7468

Variance Equation

C 5.55E-06 6.37E-07 8.708256 0.0000
RESID(-1)^2 0.034300 0.006758 5.075763 0.0000
RESID(-1)^2*(RESID(-1)<0) 0.135914 0.015624 8.699323 0.0000
GARCH(-1) 0.845836 0.011460 73.80919 0.0000

R-squared -0.000264 Mean dependent var 0.000338
Adjusted R-squared -0.000519 S.D. dependent var 0.010796
S.E. of regression 0.010799 Akaike info criterion -6.444671
Sum squared resid 0.456098 Schwarz criterion -6.435054
Log likelihood 12615.00 Hannan-Quinn criter. -6.441259
Durbin-Watson stat 2.034101

C – Constant
RESID(-1)^2 – Alpha – Volatility
RESID(-1)^2*(RESID(-1)<0) – Beta – Persistence (ARCH Effect)
GARCH(-1) – Leverage Effect
ϑ_t^2= α_0+ α_1 ε_(t-1)^2 + β_1 ϑ_(t-1)^2+ γε_(t-1)^2 I_(t-1)

Table 39 – GARCH(1,1), EGARCH(1,1) and TGARCH(1,1)
Akaike Information Criterion
AIC=T(log.RSS)+2n
Schwarz Information Criterion
BIC=T(log.RSS)+n(log.T)
GARCH(1,1) -6.429341 -6.421326
EGARCH(1,1) -6.424683 -6.416669
TGARCH(1,1) -6.447423* -6.437806*
The* codex exemplifies smallest or minimum values

Phase 4 – VAR estimation

Table 40 – VAR Estimation
Vector Autoregression Estimates
Date: 03/28/19 Time: 19:53
Sample (adjusted): 4/16/2004 3/15/2019
Included observations: 3891 after adjustments
Standard errors in ( ) & t-statistics in [ ]

Z_FINANCIALS A_HEALTH_CARE E Q_INDUSTRIAL_ENG

Z_FINANCIALS(-1) 1.114045 -0.003582 -0.017094 0.014003
(0.02386) (0.01504) (0.01518) (0.02330)
[ 46.6935] [-0.23824] [-1.12604] [ 0.60094]

Z_FINANCIALS(-2) -0.189543 0.014453 0.002232 -0.019977
(0.03572) (0.02251) (0.02272) (0.03488)
[-5.30696] [ 0.64214] [ 0.09822] [-0.57270]

Z_FINANCIALS(-3) -0.036361 -0.012273 0.013875 -0.027113
(0.03596) (0.02266) (0.02288) (0.03513)
[-1.01101] [-0.54150] [ 0.60636] [-0.77188]

Z_FINANCIALS(-4) 0.120570 0.019793 0.027839 0.084427
(0.03597) (0.02267) (0.02289) (0.03513)
[ 3.35180] [ 0.87312] [ 1.21635] [ 2.40313]

Z_FINANCIALS(-5) -0.045732 -0.048138 -0.065299 -0.113268
(0.03601) (0.02270) (0.02291) (0.03517)
[-1.26986] [-2.12106] [-2.84977] [-3.22032]

Z_FINANCIALS(-6) 0.011540 0.044900 0.061517 0.082911
(0.03605) (0.02272) (0.02294) (0.03521)
[ 0.32012] [ 1.97647] [ 2.68214] [ 2.35497]

Z_FINANCIALS(-7) 0.039189 0.005377 -0.009750 -0.015275
(0.03608) (0.02274) (0.02296) (0.03524)
[ 1.08611] [ 0.23648] [-0.42472] [-0.43347]

Z_FINANCIALS(-8) -0.079551 -0.018278 -0.025160 -0.030903
(0.03605) (0.02272) (0.02294) (0.03521)
[-2.20645] [-0.80448] [-1.09681] [-0.87761]

Z_FINANCIALS(-9) 0.123049 0.001657 0.032592 0.073300
(0.03599) (0.02268) (0.02290) (0.03515)
[ 3.41935] [ 0.07308] [ 1.42346] [ 2.08560]

Z_FINANCIALS(-10) -0.056462 0.017087 -0.026142 -0.079084
(0.03605) (0.02272) (0.02294) (0.03521)
[-1.56613] [ 0.75208] [-1.13966] [-2.24607]

Z_FINANCIALS(-11) -0.060615 -0.033902 -0.009193 -0.022786
(0.03604) (0.02271) (0.02293) (0.03520)
[-1.68196] [-1.49273] [-0.40093] [-0.64738]

Z_FINANCIALS(-12) 0.071588 0.014657 0.038967 0.110381
(0.03597) (0.02267) (0.02289) (0.03513)
[ 1.99007] [ 0.64656] [ 1.70250] [ 3.14177]

Z_FINANCIALS(-13) -0.002542 0.004250 -0.004041 -0.066117
(0.03602) (0.02270) (0.02292) (0.03518)
[-0.07057] [ 0.18724] [-0.17631] [-1.87936]

Z_FINANCIALS(-14) -0.066173 -0.048794 -0.035124 -0.039941
(0.03602) (0.02270) (0.02292) (0.03518)
[-1.83715] [-2.14960] [-1.53262] [-1.13537]

Z_FINANCIALS(-15) 0.021736 -0.000189 -0.027340 0.026168
(0.03601) (0.02269) (0.02291) (0.03517)
[ 0.60365] [-0.00834] [-1.19335] [ 0.74410]

Z_FINANCIALS(-16) 0.096126 0.058694 0.076937 0.093340
(0.03591) (0.02263) (0.02285) (0.03507)
[ 2.67669] [ 2.59344] [ 3.36713] [ 2.66122]

Z_FINANCIALS(-17) -0.011549 0.010952 0.016753 -0.059540
(0.03590) (0.02262) (0.02284) (0.03506)
[-0.32174] [ 0.48414] [ 0.73353] [-1.69836]

Z_FINANCIALS(-18) -0.089943 -0.015695 -0.044875 -0.025729
(0.03588) (0.02261) (0.02283) (0.03505)
[-2.50642] [-0.69402] [-1.96544] [-0.73412]

Z_FINANCIALS(-19) 0.032739 -0.029112 -0.007342 -0.014084
(0.03590) (0.02262) (0.02284) (0.03506)
[ 0.91208] [-1.28694] [-0.32149] [-0.40175]

Z_FINANCIALS(-20) -0.006841 0.019116 0.005037 0.029921
(0.03588) (0.02261) (0.02283) (0.03505)
[-0.19065] [ 0.84533] [ 0.22061] [ 0.85373]

Z_FINANCIALS(-21) 0.040777 0.028504 -0.000769 0.004178
(0.03584) (0.02259) (0.02280) (0.03500)
[ 1.13778] [ 1.26206] [-0.03372] [ 0.11936]

Z_FINANCIALS(-22) 0.006851 -0.030765 0.034116 0.103364
(0.03581) (0.02257) (0.02279) (0.03498)
[ 0.19129] [-1.36309] [ 1.49713] [ 2.95504]

Z_FINANCIALS(-23) -0.043426 -0.015887 -0.063755 -0.121888
(0.03566) (0.02247) (0.02269) (0.03483)
[-1.21775] [-0.70692] [-2.80985] [-3.49964]

Z_FINANCIALS(-24) 0.004480 0.016876 0.025004 0.017846
(0.02381) (0.01500) (0.01515) (0.02325)
[ 0.18820] [ 1.12494] [ 1.65080] [ 0.76758]

A_HEALTH_CARE(-1) 0.065951 1.039133 0.031885 0.044224
(0.03402) (0.02144) (0.02165) (0.03323)
[ 1.93843] [ 48.4650] [ 1.47294] [ 1.33089]

A_HEALTH_CARE(-2) -0.103710 -0.053525 -0.066797 -0.107803
(0.04844) (0.03053) (0.03082) (0.04731)
[-2.14109] [-1.75345] [-2.16738] [-2.27877]

A_HEALTH_CARE(-3) 0.014232 -0.024755 0.002619 -0.007897
(0.04838) (0.03049) (0.03078) (0.04725)
[ 0.29417] [-0.81194] [ 0.08508] [-0.16713]

A_HEALTH_CARE(-4) -0.000868 0.049861 0.071108 0.089898
(0.04834) (0.03046) (0.03076) (0.04721)
[-0.01795] [ 1.63681] [ 2.31202] [ 1.90421]

A_HEALTH_CARE(-5) 0.053894 -0.006913 -0.001863 -0.022561
(0.04839) (0.03049) (0.03079) (0.04726)
[ 1.11375] [-0.22668] [-0.06052] [-0.47739]

A_HEALTH_CARE(-6) 0.004195 0.003078 -0.033025 -0.018053
(0.04839) (0.03050) (0.03079) (0.04726)
[ 0.08669] [ 0.10092] [-1.07259] [-0.38196]

A_HEALTH_CARE(-7) -0.047334 -0.037098 -0.007700 -0.044483
(0.04840) (0.03050) (0.03080) (0.04727)
[-0.97794] [-1.21624] [-0.25003] [-0.94102]

A_HEALTH_CARE(-8) 0.022585 0.007523 -0.014401 0.000578
(0.04839) (0.03049) (0.03079) (0.04726)
[ 0.46674] [ 0.24669] [-0.46776] [ 0.01224]

A_HEALTH_CARE(-9) -0.063353 -0.016069 0.003063 0.058341
(0.04831) (0.03044) (0.03074) (0.04718)
[-1.31147] [-0.52785] [ 0.09966] [ 1.23659]

A_HEALTH_CARE(-10) 0.124648 0.030461 0.025137 0.077921
(0.04832) (0.03045) (0.03074) (0.04719)
[ 2.57959] [ 1.00032] [ 0.81760] [ 1.65111]

A_HEALTH_CARE(-11) -0.040426 0.000638 -0.022070 -0.027820
(0.04832) (0.03045) (0.03075) (0.04720)
[-0.83659] [ 0.02095] [-0.71783] [-0.58947]

A_HEALTH_CARE(-12) 0.019046 -0.004054 0.033026 0.024584
(0.04826) (0.03042) (0.03071) (0.04714)
[ 0.39463] [-0.13330] [ 1.07549] [ 0.52154]

A_HEALTH_CARE(-13) -0.185830 -0.035667 -0.023855 -0.131894
(0.04825) (0.03040) (0.03070) (0.04712)
[-3.85172] [-1.17310] [-0.77713] [-2.79912]

A_HEALTH_CARE(-14) 0.139798 0.055428 -0.005984 0.016136
(0.04832) (0.03045) (0.03074) (0.04719)
[ 2.89329] [ 1.82033] [-0.19464] [ 0.34194]

A_HEALTH_CARE(-15) 0.010526 -0.037936 0.000943 0.015667
(0.04838) (0.03049) (0.03078) (0.04725)
[ 0.21757] [-1.24421] [ 0.03064] [ 0.33156]

A_HEALTH_CARE(-16) -0.065502 0.037206 -0.024460 0.005672
(0.04837) (0.03048) (0.03077) (0.04724)
[-1.35424] [ 1.22062] [-0.79483] [ 0.12006]

A_HEALTH_CARE(-17) 0.003815 -0.042519 0.009782 -0.019390
(0.04840) (0.03050) (0.03080) (0.04727)
[ 0.07882] [-1.39389] [ 0.31761] [-0.41017]

A_HEALTH_CARE(-18) 0.080527 0.058698 0.042682 0.094617
(0.04840) (0.03050) (0.03079) (0.04727)
[ 1.66385] [ 1.92450] [ 1.38606] [ 2.00169]

A_HEALTH_CARE(-19) 0.036767 -0.022257 -6.55E-05 0.004143
(0.04838) (0.03049) (0.03078) (0.04725)
[ 0.75995] [-0.72999] [-0.00213] [ 0.08768]

A_HEALTH_CARE(-20) -0.066583 0.004612 -0.036669 -0.040463
(0.04816) (0.03035) (0.03064) (0.04704)
[-1.38252] [ 0.15196] [-1.19666] [-0.86025]

A_HEALTH_CARE(-21) -0.055082 -0.083078 -0.047552 -0.049778
(0.04816) (0.03035) (0.03064) (0.04704)
[-1.14372] [-2.73729] [-1.55182] [-1.05828]

A_HEALTH_CARE(-22) 0.086264 0.107500 0.081428 0.002350
(0.04812) (0.03032) (0.03062) (0.04699)
[ 1.79277] [ 3.54510] [ 2.65970] [ 0.05000]

A_HEALTH_CARE(-23) 0.019328 0.023843 0.032897 0.106669
(0.04802) (0.03026) (0.03056) (0.04690)
[ 0.40248] [ 0.78784] [ 1.07664] [ 2.27425]

A_HEALTH_CARE(-24) -0.046092 -0.054456 -0.048880 -0.072694
(0.03378) (0.02129) (0.02149) (0.03299)
[-1.36443] [-2.55801] [-2.27417] [-2.20334]

E(-1) -0.171416 -0.023786 0.958072 -0.045414
(0.03520) (0.02218) (0.02240) (0.03438)
[-4.86978] [-1.07226] [ 42.7780] [-1.32101]

E(-2) 0.138988 -0.046514 -0.001271 0.031695
(0.04986) (0.03142) (0.03173) (0.04870)
[ 2.78744] [-1.48025] [-0.04006] [ 0.65084]

E(-3) 0.047234 0.096864 0.024270 0.037208
(0.04995) (0.03148) (0.03178) (0.04878)
[ 0.94568] [ 3.07733] [ 0.76369] [ 0.76275]

E(-4) 0.028895 0.008712 0.011060 -0.023555
(0.05000) (0.03151) (0.03182) (0.04884)
[ 0.57784] [ 0.27647] [ 0.34764] [-0.48231]

E(-5) -0.103447 -0.091297 -0.018761 0.003849
(0.04998) (0.03150) (0.03180) (0.04881)
[-2.06980] [-2.89863] [-0.58998] [ 0.07885]

E(-6) 0.050228 0.009235 -0.009475 0.036224
(0.05008) (0.03156) (0.03187) (0.04891)
[ 1.00292] [ 0.29259] [-0.29735] [ 0.74059]

E(-7) -0.041659 0.025052 0.025102 -0.025753
(0.05008) (0.03156) (0.03186) (0.04891)
[-0.83187] [ 0.79382] [ 0.78782] [-0.52654]

E(-8) 0.176291 0.052170 0.064426 0.110910
(0.05005) (0.03154) (0.03185) (0.04888)
[ 3.52220] [ 1.65399] [ 2.02308] [ 2.26888]

E(-9) -0.184949 -0.038131 -0.062185 -0.116053
(0.05005) (0.03154) (0.03185) (0.04889)
[-3.69500] [-1.20884] [-1.95259] [-2.37396]

E(-10) -0.011888 0.001791 -0.027563 -0.048365
(0.05006) (0.03155) (0.03185) (0.04890)
[-0.23745] [ 0.05676] [-0.86530] [-0.98915]

E(-11) 0.084449 0.013063 0.047462 0.075163
(0.05003) (0.03153) (0.03183) (0.04887)
[ 1.68785] [ 0.41431] [ 1.49092] [ 1.53817]

E(-12) 0.008487 0.008407 -0.015230 -0.045608
(0.04997) (0.03149) (0.03180) (0.04881)
[ 0.16982] [ 0.26696] [-0.47898] [-0.93445]

E(-13) 0.057381 0.027348 -0.016528 0.023982
(0.04998) (0.03150) (0.03180) (0.04881)
[ 1.14815] [ 0.86831] [-0.51976] [ 0.49133]

E(-14) -0.097448 -0.034403 -0.012316 0.032100
(0.04991) (0.03145) (0.03176) (0.04875)
[-1.95247] [-1.09380] [-0.38785] [ 0.65854]

E(-15) -0.010653 -0.002657 0.051470 -0.010998
(0.04993) (0.03146) (0.03177) (0.04876)
[-0.21336] [-0.08444] [ 1.62020] [-0.22554]

E(-16) 0.047246 0.003624 -0.017750 -0.000578
(0.04996) (0.03149) (0.03179) (0.04880)
[ 0.94563] [ 0.11510] [-0.55836] [-0.01185]

E(-17) -0.056727 -0.006833 -0.042734 -0.063294
(0.04997) (0.03149) (0.03179) (0.04880)
[-1.13527] [-0.21699] [-1.34415] [-1.29696]

E(-18) 0.010779 -0.009680 0.029011 -0.030615
(0.04993) (0.03147) (0.03177) (0.04876)
[ 0.21588] [-0.30764] [ 0.91322] [-0.62782]

E(-19) -0.006772 0.035351 0.024276 0.050073
(0.04979) (0.03138) (0.03168) (0.04863)
[-0.13600] [ 1.12658] [ 0.76625] [ 1.02967]

E(-20) 0.004005 -0.043210 -0.038673 -0.021806
(0.04971) (0.03133) (0.03163) (0.04855)
[ 0.08057] [-1.37939] [-1.22276] [-0.44917]

E(-21) 0.033756 0.006162 0.023198 0.034921
(0.04970) (0.03132) (0.03162) (0.04854)
[ 0.67915] [ 0.19671] [ 0.73355] [ 0.71937]

E(-22) 0.045460 0.016925 -0.007351 0.053665
(0.04966) (0.03129) (0.03159) (0.04850)
[ 0.91549] [ 0.54086] [-0.23267] [ 1.10656]

E(-23) -0.067824 -0.032648 -0.027403 -0.076683
(0.04961) (0.03126) (0.03156) (0.04845)
[-1.36727] [-1.04437] [-0.86823] [-1.58281]

E(-24) 0.012227 0.024731 0.036967 0.024232
(0.03514) (0.02214) (0.02236) (0.03432)
[ 0.34800] [ 1.11690] [ 1.65361] [ 0.70614]

Q_INDUSTRIAL_ENG(-1) 0.034073 -0.006869 0.029953 1.078250
(0.02294) (0.01445) (0.01459) (0.02240)
[ 1.48553] [-0.47523] [ 2.05246] [ 48.1330]

Q_INDUSTRIAL_ENG(-2) 0.030044 0.038430 -0.016275 -0.073302
(0.03330) (0.02099) (0.02119) (0.03252)
[ 0.90222] [ 1.83127] [-0.76815] [-2.25384]

Q_INDUSTRIAL_ENG(-3) 0.002120 -0.023446 0.002725 -0.000365
(0.03331) (0.02099) (0.02119) (0.03253)
[ 0.06364] [-1.11688] [ 0.12856] [-0.01120]

Q_INDUSTRIAL_ENG(-4) -0.096397 -0.027779 -0.046337 -0.088748
(0.03332) (0.02100) (0.02120) (0.03254)
[-2.89312] [-1.32295] [-2.18573] [-2.72723]

Q_INDUSTRIAL_ENG(-5) 0.014775 0.012648 0.038675 0.066126
(0.03335) (0.02102) (0.02122) (0.03257)
[ 0.44297] [ 0.60174] [ 1.82246] [ 2.02998]

Q_INDUSTRIAL_ENG(-6) 0.005489 -0.003198 -0.015055 -0.018200
(0.03335) (0.02102) (0.02122) (0.03257)
[ 0.16458] [-0.15214] [-0.70949] [-0.55873]

Q_INDUSTRIAL_ENG(-7) -0.004172 0.014425 -0.021837 0.026956
(0.03331) (0.02099) (0.02120) (0.03254)
[-0.12522] [ 0.68711] [-1.03019] [ 0.82850]

Q_INDUSTRIAL_ENG(-8) 0.032632 -0.011693 0.021618 -0.005699
(0.03329) (0.02098) (0.02118) (0.03251)
[ 0.98019] [-0.55732] [ 1.02056] [-0.17528]

Q_INDUSTRIAL_ENG(-9) -0.006828 0.015223 -0.010221 -0.012838
(0.03331) (0.02099) (0.02119) (0.03253)
[-0.20500] [ 0.72525] [-0.48228] [-0.39465]

Q_INDUSTRIAL_ENG(-10) -0.022614 -0.005345 0.044542 0.023119
(0.03331) (0.02099) (0.02119) (0.03253)
[-0.67895] [-0.25465] [ 2.10184] [ 0.71070]

Q_INDUSTRIAL_ENG(-11) 0.038444 0.006282 -0.024929 0.046256
(0.03333) (0.02100) (0.02121) (0.03255)
[ 1.15350] [ 0.29909] [-1.17560] [ 1.42103]

Q_INDUSTRIAL_ENG(-12) -0.092494 -0.044600 -0.048489 -0.095980
(0.03331) (0.02099) (0.02119) (0.03253)
[-2.77691] [-2.12473] [-2.28802] [-2.95043]

Q_INDUSTRIAL_ENG(-13) 0.095344 0.027200 0.038526 0.102410
(0.03333) (0.02100) (0.02121) (0.03255)
[ 2.86063] [ 1.29498] [ 1.81674] [ 3.14607]

Q_INDUSTRIAL_ENG(-14) 0.013279 0.040283 0.020609 -0.015554
(0.03332) (0.02100) (0.02120) (0.03254)
[ 0.39851] [ 1.91836] [ 0.97210] [-0.47795]

Q_INDUSTRIAL_ENG(-15) -0.037294 -0.002101 0.016748 -0.037104
(0.03332) (0.02100) (0.02120) (0.03254)
[-1.11919] [-0.10006] [ 0.78995] [-1.14010]

Q_INDUSTRIAL_ENG(-16) 0.024306 -0.063068 -0.049951 -0.019096
(0.03330) (0.02098) (0.02119) (0.03252)
[ 0.72995] [-3.00553] [-2.35772] [-0.58719]

Q_INDUSTRIAL_ENG(-17) -0.024079 0.019724 0.018836 0.058152
(0.03337) (0.02103) (0.02123) (0.03259)
[-0.72162] [ 0.93799] [ 0.88719] [ 1.78439]

Q_INDUSTRIAL_ENG(-18) 0.001888 -0.014972 -0.019208 -0.074714
(0.03338) (0.02104) (0.02124) (0.03260)
[ 0.05655] [-0.71170] [-0.90434] [-2.29169]

Q_INDUSTRIAL_ENG(-19) 0.004807 0.042001 0.011772 0.057743
(0.03338) (0.02103) (0.02124) (0.03260)
[ 0.14403] [ 1.99673] [ 0.55431] [ 1.77129]

Q_INDUSTRIAL_ENG(-20) 0.042344 0.002619 0.042567 0.005143
(0.03337) (0.02103) (0.02123) (0.03260)
[ 1.26877] [ 0.12452] [ 2.00459] [ 0.15779]

Q_INDUSTRIAL_ENG(-21) -0.048940 -0.002451 -0.010220 -0.020430
(0.03338) (0.02104) (0.02124) (0.03260)
[-1.46600] [-0.11651] [-0.48114] [-0.62659]

Q_INDUSTRIAL_ENG(-22) -0.092532 -0.050967 -0.066434 -0.144966
(0.03334) (0.02101) (0.02121) (0.03256)
[-2.77539] [-2.42577] [-3.13176] [-4.45201]

Q_INDUSTRIAL_ENG(-23) 0.072327 0.051358 0.062824 0.105634
(0.03333) (0.02100) (0.02120) (0.03255)
[ 2.17036] [ 2.44549] [ 2.96291] [ 3.24555]

Q_INDUSTRIAL_ENG(-24) 0.017916 -0.014655 -0.020028 0.026415
(0.02286) (0.01441) (0.01455) (0.02233)
[ 0.78362] [-1.01712] [-1.37678] [ 1.18295]

C 0.020581 0.006405 0.010078 0.004747
(0.01011) (0.00637) (0.00643) (0.00988)
[ 2.03533] [ 1.00517] [ 1.56635] [ 0.48070]

R-squared 0.997439 0.997843 0.999054 0.992905
Adj. R-squared 0.997374 0.997788 0.999030 0.992725
Sum sq. resids 1.055549 0.419205 0.427315 1.006851
S.E. equation 0.016680 0.010511 0.010613 0.016290
F-statistic 15393.73 18279.35 41730.93 5530.436
Log likelihood 10456.06 12252.64 12215.36 10547.95
Akaike AIC -5.324625 -6.248082 -6.228919 -5.371858
Schwarz SC -5.168407 -6.091864 -6.072701 -5.215640
Mean dependent 4.929787 6.438681 7.626196 4.849842
S.D. dependent 0.325522 0.223498 0.340738 0.190994

Determinant resid covariance (dof adj.) 1.87E-16
Determinant resid covariance 1.69E-16
Log likelihood 48565.65
Akaike information criterion -24.76363
Schwarz criterion -24.13876

Table 41 – VAR Selection Criteria
VAR Lag Order Selection Criteria
Endogenous variables: Z_FINANCIALS A_HEALTH_CARE E Q_INDUSTRIAL_ENG
Exogenous variables: C
Date: 03/28/19 Time: 19:44
Sample: 3/15/2004 3/15/2019
Included observations: 3891

Lag LogL LR FPE AIC SC HQ

0 5208.188 NA 8.10e-07 -2.674987 -2.668545 -2.672701
1 48083.88 85641.20 2.19e-16 -24.70516 -24.67295* -24.69372
2 48147.74 127.4242 2.14e-16 -24.72976 -24.67178 -24.70918
3 48186.11 76.47304 2.11e-16 -24.74125 -24.65751 -24.71153*
4 48215.90 59.31820 2.10e-16 -24.74834 -24.63883 -24.70947
5 48237.45 42.87172 2.09e-16 -24.75119 -24.61591 -24.70317
6 48265.73 56.19361 2.08e-16 -24.75751 -24.59646 -24.70034
7 48279.52 27.39065 2.08e-16 -24.75637 -24.56956 -24.69006
8 48294.36 29.41202 2.08e-16 -24.75577 -24.54319 -24.68031
9 48310.12 31.22168 2.08e-16 -24.75565 -24.51730 -24.67104
10 48326.22 31.85769 2.08e-16 -24.75570 -24.49158 -24.66195
11 48342.48 32.16107 2.08e-16 -24.75584 -24.46595 -24.65294
12 48353.85 22.44553 2.09e-16 -24.75346 -24.43780 -24.64141
13 48379.09 49.78247 2.08e-16 -24.75820 -24.41678 -24.63701
14 48405.12 51.31478 2.07e-16 -24.76336 -24.39617 -24.63302
15 48422.51 34.22211 2.07e-16 -24.76407 -24.37111 -24.62459
16 48435.99 26.50758 2.07e-16 -24.76278 -24.34405 -24.61414
17 48467.97 62.83635 2.05e-16 -24.77100 -24.32650 -24.61321
18 48484.06 31.58159 2.05e-16* -24.77104* -24.30078 -24.60411
19 48494.85 21.14195 2.06e-16 -24.76836 -24.27233 -24.59229
20 48504.37 18.63724 2.06e-16 -24.76503 -24.24323 -24.57981
21 48511.13 13.22831 2.07e-16 -24.76028 -24.21271 -24.56591
22 48523.64 24.45109 2.08e-16 -24.75849 -24.18515 -24.55497
23 48554.54 60.32001* 2.06e-16 -24.76615 -24.16704 -24.55348
24 48565.65 21.66133 2.07e-16 -24.76363 -24.13876 -24.54182

* indicates lag order selected by the criterion
LR: sequential modified LR test statistic (each test at 5% level)
FPE: Final prediction error
AIC: Akaike information criterion
SC: Schwarz information criterion
HQ: Hannan-Quinn information criterion

Table 42- Granger Causality
VAR Granger Causality/Block Exogeneity Wald Tests
Date: 03/28/19 Time: 19:57
Sample: 3/15/2004 3/15/2019
Included observations: 3897

Dependent variable: Z_FINANCIALS

Excluded Chi-sq df Prob.

A_HEALTH_CARE 46.82684 18 0.0002
E 64.96374 18 0.0000
Q_INDUSTRIAL_ENG 44.24601 18 0.0005

All 154.9367 54 0.0000

Dependent variable: A_HEALTH_CARE

Excluded Chi-sq df Prob.

Z_FINANCIALS 39.04058 18 0.0028
E 34.26967 18 0.0117
Q_INDUSTRIAL_ENG 33.66410 18 0.0139

All 115.5254 54 0.0000

Dependent variable: E

Excluded Chi-sq df Prob.

Z_FINANCIALS 56.01788 18 0.0000
A_HEALTH_CARE 20.88721 18 0.2851
Q_INDUSTRIAL_ENG 37.63999 18 0.0043

All 103.6529 54 0.0001

Dependent variable: Q_INDUSTRIAL_ENG

Excluded Chi-sq df Prob.

Z_FINANCIALS 51.41136 18 0.0000
A_HEALTH_CARE 42.54832 18 0.0009
E 29.94544 18 0.0380

All 144.0022 54 0.0000

Table 43- Johansen cointegration test
Date: 03/28/19 Time: 21:15
Sample (adjusted): 4/09/2004 3/15/2019
Included observations: 3896 after adjustments
Trend assumption: Linear deterministic trend
Series: Z_FINANCIALS Q_INDUSTRIAL_ENG E A_HEALTH_CARE
Lags interval (in first differences): 1 to 18

Unrestricted Cointegration Rank Test (Trace)

Hypothesized Trace 0.05
No. of CE(s) Eigenvalue Statistic Critical Value Prob.**

None 0.007414 43.99713 47.85613 0.1100
At most 1 0.002373 15.00588 29.79707 0.7797
At most 2 0.001182 5.750728 15.49471 0.7247
At most 3 0.000293 1.141231 3.841466 0.2854

Trace test indicates no cointegration at the 0.05 level
* denotes rejection of the hypothesis at the 0.05 level
**MacKinnon-Haug-Michelis (1999) p-values

Unrestricted Cointegration Rank Test (Maximum Eigenvalue)

Hypothesized Max-Eigen 0.05
No. of CE(s) Eigenvalue Statistic Critical Value Prob.**

None * 0.007414 28.99124 27.58434 0.0328
At most 1 0.002373 9.255153 21.13162 0.8113
At most 2 0.001182 4.609498 14.26460 0.7901
At most 3 0.000293 1.141231 3.841466 0.2854

Max-eigenvalue test indicates 1 cointegrating eqn(s) at the 0.05 level
* denotes rejection of the hypothesis at the 0.05 level
**MacKinnon-Haug-Michelis (1999) p-values

Unrestricted Cointegrating Coefficients (normalized by b’*S11*b=I):

Z_FINANCIALS Q_INDUSTRIAL_ENG E A_HEALTH_CARE
-9.025268 14.49300 -11.49986 7.682560
3.167097 1.926900 3.703854 -5.646590
-0.434318 -1.246732 3.533224 -1.520167
1.491149 -1.605058 0.404530 4.242465

Unrestricted Adjustment Coefficients (alpha):

D(Z_FINANCIALS) 0.000389 -0.000750 6.25E-06 -7.02E-05
D(Q_INDUSTRIAL_ENG) -0.000731 -0.000651 -5.31E-05 -5.53E-05
D(E) 5.97E-05 -0.000218 -0.000222 -0.000120
D(A_HEALTH_CARE) -3.62E-05 -0.000166 7.58E-05 -0.000164

1 Cointegrating Equation(s): Log likelihood 48545.07

Normalized cointegrating coefficients (standard error in parentheses)
Z_FINANCIALS Q_INDUSTRIAL_ENG E A_HEALTH_CARE
1.000000 -1.605825 1.274184 -0.851228
(0.14267) (0.08986) (0.13544)

Adjustment coefficients (standard error in parentheses)
D(Z_FINANCIALS) -0.003513
(0.00242)
D(Q_INDUSTRIAL_ENG) 0.006594
(0.00237)
D(E) -0.000539
(0.00154)
D(A_HEALTH_CARE) 0.000327
(0.00152)

2 Cointegrating Equation(s): Log likelihood 48549.70

Normalized cointegrating coefficients (standard error in parentheses)
Z_FINANCIALS Q_INDUSTRIAL_ENG E A_HEALTH_CARE
1.000000 0.000000 1.198249 -1.526896
(0.31031) (0.47752)
0.000000 1.000000 -0.047287 -0.420761
(0.19644) (0.30229)

Adjustment coefficients (standard error in parentheses)
D(Z_FINANCIALS) -0.005888 0.004196
(0.00256) (0.00391)
D(Q_INDUSTRIAL_ENG) 0.004532 -0.011845
(0.00251) (0.00383)
D(E) -0.001228 0.000446
(0.00163) (0.00249)
D(A_HEALTH_CARE) -0.000200 -0.000845
(0.00161) (0.00247)

3 Cointegrating Equation(s): Log likelihood 48552.01

Normalized cointegrating coefficients (standard error in parentheses)
Z_FINANCIALS Q_INDUSTRIAL_ENG E A_HEALTH_CARE
1.000000 0.000000 0.000000 -0.714632
(0.67492)
0.000000 1.000000 0.000000 -0.452815
(0.22426)
0.000000 0.000000 1.000000 -0.677875
(0.54211)

Adjustment coefficients (standard error in parentheses)
D(Z_FINANCIALS) -0.005891 0.004188 -0.007232
(0.00256) (0.00393) (0.00337)
D(Q_INDUSTRIAL_ENG) 0.004555 -0.011778 0.005803
(0.00251) (0.00385) (0.00330)
D(E) -0.001132 0.000723 -0.002276
(0.00163) (0.00250) (0.00215)
D(A_HEALTH_CARE) -0.000233 -0.000939 6.85E-05
(0.00162) (0.00248) (0.00212)

Bibliography
Bera, A. a. (1993). ARCH Models: Properties, Estimation and Testing.
Bollerslev. (1984). Generalized autoregressive conditional heteroscedasticity.
Brooks, & Burke. (2003). Information Criteria for GARCH model selection.
Davidson, & Mackinnon. (2004). Econometric Theory and Methods.
Lim, C. M., & Sek, S. K. (2013). Comparing the performances of GARCH-type models in.
Olbryś, J. (2013). Asymmetric Impact of Innovations on Volatility.

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