FE8513 Fin Time Series-Assignment

FE8513 Financial Time Series Analysis

Shanghai Stock Exchange Returns Model

Master of Science (Financial Engineering)

Academic Year 2014/15 Mini-Term 3

RAKTIM RAY

G1401584K

Ray | Shanghai Stock Exchange Returns Model

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Table Of Contents

Introduction ........................................................................................................... 3

Model Data and Elementary Analysis .................................................................. 3

Mean Model Construction and Choice ................................................................. 4

Residual Diagnostics: Modelling Volatility ......................................................... 7

Forecasting and Final Model Selection .............................................................. 10

Formal Statement of Optimal Model .................................................................. 12

Limitations .......................................................................................................... 15

Conclusion .......................................................................................................... 16

Acknowledgement .............................................................................................. 17

Bibliography ....................................................................................................... 18


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Introduction

Our main objective is to create a time series model to forecast the future

prospective returns and the volatility of these returns. The investment opportunity is in

the Shanghai Pilot Free Trade Zone (SPFTZ). Since this is a newly launched initiative for

foreign companies to invest in, we use the Shanghai Stock Exchange returns as proxy for

modelling the SPFTZ returns and volatility. Our objective is to provide investment

returns and risk insights for SPFTZ index investing based on quantitative modelling.

This report will aim to serve the Senior Management of the firm in arriving at a

quantitatively driven decision of whether or not to go ahead with this new venture

based on historical proxy returns. Theoretically, index values are themselves non-

stationary. It is difficult to model non-stationary processes. We therefore base our

model building on log returns which are stationary in nature. This removes the

subjective biases of absolute value and non-stationarity and sets an objective outlook of

the relative returns which is the end goal for all investments. The data tables used for

general analyses and model building and referred to in the report is self-contained. An

analysis support folder is also provided with all the tests and output from the software

for the technical reader.

Model Data and Elementary Analysis

The index values of the Shanghai Stock Exchange ranging from 03/Jan/2000 to

27/Nov/2014 are used which is deemed a sufficient time period for our model building

purpose. We do not account for dividends paid on the stock and assume that the price or

index level have been already adjusted to account for them. Our purpose of elementary

analysis is to understand the nature of data that we are dealing with and to check whether our

assumptions of stationarity and normality hold. Additionally, other characteristics of the data


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are also tested here namely the heteroskedasticity and autocorrelation of error terms present

in our model. Checking the distribution of price data (adjusted close price) henceforth to be

known as price, we see that the data is not normally distributed from the histogram plot.

Formal tests also reveal the same result. It is difficult to work with variables which are not

normally distributed. But this problem disappears with large number of observations as the

law of large numbers starts to prevail. A bigger problem with that of price is of non

stationarity. We conduct statistical tests to verify that this is indeed the case. We cannot deal

with non-stationary data when trying to model a time series relationship. Therefore the

variable is transferred to the log return which makes the model parsimonious since return on

investment prediction is the true focus and the price levels or the absolute value of the

investments. This data is also not normally distributed like most finance data, but as

mentioned before, the law of large numbers makes the OLS estimates unbiased and efficient

and consistent with minimum variance. The stationarity tests using the unit root method

(ADF and KPSS for the technical reader) conducted on the log return data shows that it is

indeed stationary over time and does not explode. Therefore this variable may be used to

construct a time series model. This also removes the possibility of spurious correlation in

case we try to use this variable with some others to model for some other forecasts.

Now that we have a stationary series we proceed to form linear models and test for

other desirable properties exhibited by the model which makes it robust and a good prediction

model.

Mean Model Construction and Choice

The next step in the model building process involved the determining the order of the

linear model. Since this is time series estimation, we follow the Auto Regressive/Moving

Average framework for the linear model or the ARMA framework. To determine the exact


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levels of lag terms to include, the correlogram was initially used to determine if there is any

strong trend to determine the model. This is the Box-Jenkins approach. Technical readers

may be interested in this methodology but for our purposes the description is restricted to its

use which is limited at this point. As in most instances, it was difficult to pick up the correct

lag terms for the AR as well as the MA part looking at the Auto Correlation (AC) and Partial

Auto Correlation (PAC) terms and their decays. Additionally this is an informal method and a

quantitative model demands a quantitative choice for the optimal model.

Focus was therefore shifted to forming a cohort of linear models based on the ARMA

structure and then choosing the optimal model based on some information criterion.

Information criterion implies Akaike Information Criterion (AIC) and Schwarz Bayesian

Information Criterion (SBIC). These two information criterion uses the same method to rank

various models. One term is related to the residual sum of squares which may be compared

with that part of the dependent variable that is still unexplained by the model. The lower this

term is the better it is for our model. But being parsimonious is also another objective of the

model. The information criterion uses a penalty term which adds to value of the AIC or SBIC

when extra variables are added. SBIC uses a stricter penalty term. So overall, to choose the

model with the lowest Information Criterion value, one has to have a trade-off between

adding more variables to lower the first term and decreasing the number of variables to

decrease the second term. A more technical discussion regarding degrees of freedom and

residual sum of squares is beyond the scope of this paper and the technical reader may refer

to Brooks1. To sum up, models with the lowest AIC and SBIC values are to be selected as the

optimal model to explain the returns of the SSE.

For the sake of being parsimonious, the AR and the MA terms are restricted to 5

which imply that the model should not have more than 5 lagged values of the returns or the

1 Chris Brooks : Introductory Econometrics for Finance- 2

nd Edition (2008)


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error terms. Beyond 5 it gets very difficult to interpret the model. This is a minor cost

compared to the major benefit of simplicity and interpretation. What we will gain by adding

the 6th

terms and so on is small compared to the loss in the power interpretation and

implementation of the model.

Data is truncated at this point between in sample and out of sample. The most

important use of a time series model is its predictive power. If predictive power is low, the

model is useless. The most recent data is therefore reserved to be the test data or out of

sample data. This is used to test which model has the best predictive power and therefore may

be used as the optimal model. Among 3776 observations, 3210 or about 85% is kept for in

sample, truncated at the date 07/09/2012, and the rest was truncated as out of sample or the

holdout sample to be used for forecasting purposes.

The AIC and SBIC table is provided for reference. From the table the choices are

marked with circles. One model is selected based on the AIC criterion. This is the ARMA (5,

4) model. The SBIC table returns the lowest IC value for the ARMA (0, 0). This is naturally

dropped as the interpretation will be that the returns in the future will be the same for every

day which is equal to the intercept of the model. This will lead to poor predictive power and

huge errors on the estimates of future returns. Volatility will equal the variance of the error

term which is as good as not modelling at all. Therefore, this choice is dropped naturally and

we proceed to choose the next least IC which is given by ARMA (1, 0) model. But the

regression reveals that the coefficients are insignificant. Therefore we drop this model. We

move on to choose ARMA (1, 1) as it is low on the SBIC value list and gives significant

values for the intercepts and is parsimonious.

We make a third choice to broaden our search for the best model. Since the next best

model in terms of the AIC and the coefficients being significant is ARMA (4, 5) and

according to the SBIC is ARMA (2, 3) the latter is selected since it is more parsimonious.


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To sum up, the three choices are ARMA (5, 4), ARMA (1, 1) and ARMA (2, 3).

Information Criterion for ARMA models of the

log returns of the Shanghai Stock Exchange index

Residual Diagnostics: Modelling Volatility

After formulating the mean model our focus now shifts on the residuals and their

diagnostics. Our assumptions about the errors were that they are normally distributed with a

constant mean and variance. In statistical parlance, this is called white noise. Now if these

assumptions hold, then the model behaves according to the predication with accurate interval

estimation of the parameters. But if these assumptions are not true, then we may have

misleading results and our estimates may not be good. The assumption of white noise being a

stationary process will then not hold true. Residuals are used as a proxy of the errors for the

models to test the assumptions.

ARCH effect or the presence of heteroskedasticity is tested. Our assumption for the

errors was that of constant volatility. But this rarely holds true for financial data. The

AIC

SBIC


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variance of the error terms change over time (is heteroskedastic) and is required to be

modelled along with the modelling of the mean which was done above. At this point we leave

the realm of linear models and step into non-linear modelling

The three chosen models were tested for the ARCH effect. At this point, it is to be

noted that the purpose of the test is to detect the presence of heteroskedasticity and not to

model the variance. GARCH will be used to model the variance since ARCH has a problem

of determining the correct amount of lags. So long as ARCH effect exists, the test works. But

when ARCH effect is not found then the process of including more lags will go on and is thus

not effective for modelling purposes.

The analysis was conducted starting with a lag of two for the ARCH

heteroskedasticity test. All the three models show ARCH effect. Both the F test and the Chi-

Squared tests were significant which shows that the volatility of the residuals is indeed not

constant and needs to be modelled along with the mean. Results are documented in the

analysis folder for the technical reader.

With the three mean models, the process of estimating the GARCH model for the

volatility was conducted. GARCH implies General Auto Regressive Conditionally

Heteroskedastic which means the volatility of the log returns will be modelled with previous

volatility terms and the square of the error terms as estimated from the mean equations.

Therefore, even if the assumptions of constant variance of error terms do not hold for the

mean equation, the variance will be constantly updated through this model. The analysis was

made thorough by modelling the volatility with EGARCH, GJR or TGARCH and GARCH in

Mean models as well. EGARCH accounts for leverage effects which means that volatility

should rise more after huge price fall than for an equivalent price rise. TGARCH accounts for

the fact that bad news affects volatility more than good news and GARCH in Mean puts the

volatility term in the mean equation to account for the fact that index level is directly


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determined by the past volatility level. For a more detailed study on these models, please

refer to Brooks2. At this point the ARCH models were also tested but it is to be noted that

most finance data volatility agrees with the GARCH model. Not all the models and their tests

are recorded. Only the optimal volatility models for the three mean models are recorded in

the analysis folder. At each stage of the GARCH/ARCH models, diagnostic checks are

conducted mainly to see if the heteroskedasticity effect is significant any more or not. In

almost all cases, after modelling for volatility, it was seen that the ARCH effect becomes

insignificant. Additionally correlogram checks are also done to ensure that the residuals are

not auto correlated. This applies to both the residuals and the squared residuals since the

volatility model has a squared residual term. The ideal model should possess no

heteroskedasticity, no auto correlation both for the residuals and the squared residuals. With

these criteria in mind, the GRACH models were tested.

The focus now is to find that volatility model which has no heteroskedasticity and the

residuals are not auto correlated. Although AIC and SBIC values were looked at but priority

was given to the residual criteria here. In order to maintain some level of parsimony, the

GARCH levels were limited to (5, 5) or five lags. The final decisions were made with low

AIC and SBIC as well as fulfilling the other essential conditions mentioned above and

maintaining parsimony and selecting a simple model. Subjectivity is important in model

where it should not always be the focus of mathematically optimising but also choosing a

model that is easy to understand and implement. Due to the large variety of options the

comparisons for AIC and SBIC have not been provided here to keep the choices

straightforward. All volatility models were checked for standard coefficient violations during

the selection process so as to make the volatility process stationary.

2 Chris Brooks : Introductory Econometrics for Finance- 2

nd Edition (2008)


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For the ARMA (5, 4) model, it was determined that the volatility model most apt to fit

was the TGARCH (1, 1) with Threshold Order=1. This conclusion was arrived at after going

through the various combinations as mentioned above and conducting residual checks to see

which model provides the best results.

For the ARMA (1, 1) model, it was determined that the TGARCH (1, 1) in Mean

model with Threshold Order = 1 produced the optimal fit. At this point it must be mentioned

that since it is a market index model, the intuition is for a GARCH term to be present in the

mean model as well. Although the GARCH term in the mean equation is insignificant, the

model produced a much reduced auto correlation for the standardized coefficient. Therefore

the above volatility model was selected for the ARMA (1, 1) model.

For the ARMA (2, 3) model the volatility prediction model is selected as the

TGARCH (1, 2) with threshold order=2. This model made most of the coefficients significant

as well as reduced the auto correlation between the error terms to a decent level so that the

assumptions behind the model may be matched with the data.

A summary of the Information criteria for the models selected is given below.

Information

Criterion

ARMA(5,4)TGARCH

(1,1)(t=1)

ARMA(1,1)TGARCH(1,1)-

in M(t=1)

ARMA(2,3)

TGARCH(1,2)(t=2)

AIC 3.5600 3.5633 3.5600

SBIC 3.5866 3.5784 3.5827

HQIC 3.5696 3.5687 3.5682

As can be seen from the table, the ICs are very close to one another and a choice for

the optimal model would involve some other criteria than the ICs and the correlogram.

Forecasting and Final Model Selection

The purpose of a good model is to be able to forecast the future values with the

greatest degree of accuracy. Some academics and statisticians are of the opinion that the as

long as forecasts are good, the model is good. The assumptions are secondary to the real


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purpose of the model which is to be able to forecast the future value with a given level of

accuracy.

Therefore the criterion for the final model selection will be based on the degree of

forecasting accuracy possessed by the three models. Both dynamic and static forecasts are

used to determine model forecasting strength since both long run and short run predictions

are of importance in time series modelling. Forecasting both ways gives information

regarding whether the model is a good predictor in the short or the long run. The summary of

the results for the forecasting errors are represented below with the forecasting of the chosen

model being represented in the graphical form.

Static Forecast

Forecast

Statistics

ARMA(5,4)TGARCH

(1,1)(t=1)


in M(t=1)

ARMA(2,3)

TGARCH(1,2)(t=2)

RMSE 1.0212 1.0231 1.0294

MAPE 107.4325 105.3958 117.7223

THEIL U 0.9469 0.9724 0.9157

Dynamic Forecast

Forecast

Statistics

ARMA(5,4)TGARCH

(1,1)(t=1)


in M(t=1)

ARMA(2,3)

TGARCH(1,2)(t=2)

RMSE 1.0238 1.0229 1.0235

MAPE 103.6614 102.6099 99.9349

THEIL U 0.9839 0.9780 0.9954

Static and Dynamic Forecast of ARMA(1,1) TGARCH(1,1)-in M(t=1)


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-4

-2

0

2

4

III IV I II III IV I II III IV

2012 2013 2014

DPRICEF ± 2 S.E.

Forecast: DPRICEF

Actual: DPRICE

Forecast sample: 7/10/2012 11/27/2014

Included observations: 566

Root Mean Squared Error 1.022977

Mean Absolute Error 0.746364

Mean Abs. Percent Error 102.6099

Theil Inequality Coefficient 0.978045

Bias Proportion 0.000154

Variance Proportion 0.987612

Covariance Proportion 0.012234

1.4

1.6

1.8

2.0

2.2

2.4

2.6


2012 2013 2014

Forecast of Variance

-6

-4

-2

0

2

4

6


2012 2013 2014

DPRICEF ± 2 S.E.

Forecast: DPRICEF

Actual: DPRICE

Forecast sample: 7/10/2012 11/27/2014

Included observations: 566

Root Mean Squared Error 1.023112

Mean Absolute Error 0.746558

Mean Abs. Percent Error 105.3958

Theil Inequality Coefficient 0.972364

Bias Proportion 0.001734

Variance Proportion 0.947801

Covariance Proportion 0.050465

0

1

2

3

4

5


2012 2013 2014

Forecast of Variance

Since they objective is to predict returns in the long run the dynamic forecast statistics

are used to select the optimal model. Static forecasts work well more one or two step ahead

time periods. Dynamic forecasts predict into the future. Considering the fact that index

returns will be viewed as a long term investment as opposed to daily investment, the model is

chosen based on the dynamic forecasting power.

The optimal model should have the minimum RMSE and THEIL U. Lower values for

these variables indicate lower deviation from the actual values.

The chosen model is therefore the ARMA (1, 1) TGARCH (1, 1) in Mean with

threshold order of one.

Formal Statement of Optimal Model

The general statement of the model is as follows:

yt = µ + a1yt-1 + a2ut-1 + bσt-1 + ut , ut ~ N(0,σt2)

σt2 = α0 + α1 u

2t-1 + βσ

2t-1 + γ u

2t-1It-1

where yt is the value of the returns to be predicted.

µ is the mean equation intercept


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a1yt-1 is the previous period return and its coefficient

a2ut-1 is the previous period error term and its coefficient

bσt-1 is the previous period standard deviation and its coefficient

ut is the error term for the mean equation estimation

α0 is the volatility model intercept

α1 u2

t-1 is the previous period error term squared and its coefficient

βσ2

t-1 is the previous period variance and its coefficient

γ u2

t-1It-1 is the dummy variable and its coefficient where

It-1 = 1 if ut-1 < 0

= 0 otherwise

The TGARCH or GJR model has some pre conditions on the coefficients for the

model to be valid and stationary.

α0 ≥ 0 and α1 + γ ≥ 0

The TGARCH model measures the leverage effect which states that the volatility

increases more with a price fall or bad news than it does for a price rise or good news. For

leverage effect to exist in the model, the value of γ should be ≥ 0.

The model also has a GARCH in Mean term which indicates a volatility term in the

mean equation. This is expected of financial data since higher risk should be compensated

with higher return. The volatility term models risk and the dependent variable is the return

and the expectation is that the value of b is > 0.

Stating the model with the coefficients:

yt = -0.0931 + -0.7546yt-1 + 0.7702ut-1 + 0.0753σt-1 + ut3 ,

σt2 = 0.0304 + 0.0441u

2t-1 + 0.9264σ

2t-1 + 0.0341u

2t-1It-1

Forecast Band: Returns ± 3.208

3 Model is executed with Bollerslev-Wooldridge robust standard errors & covariance due to non-normality of

residuals for SE adjustments


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The model satisfies all the constraints. The mean equation has a negative intercept and

is negatively attached with the previous period return which indicates oscillation about

positive and negative values which is common in financial time series returns variables. It is

positively linked with the error term of the previous period which also reverses the effect of

the prediction shortfall from the previous period. Both these values are below absolute value

1 which means that these are individually stationary effects on the return. As mentioned

above, the volatility term has a positive coefficient which models the fact that high volatility

in the previous period must be compensated by high return in this period.

The volatility model has a positive intercept indicating positive values of volatility

always. It is a stationary volatility process. It is positively linked with the error and variance

terms of the previous period which is to be expected since finance data exhibits clustering.

The volatility coefficient is very high indicating momentum effect in volatility. The leverage

coefficient is positive indicating that this model captures the fact that bad news affects

volatility more than good news. Standard forecast values and errors (twice standard error) are

also indicated which is in general agreement with financial time series models. Technical

readers may browse the model specifics attached below.


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Limitations

Every optimal exercise has drawbacks. Since there are constraints to be satisfied

especially model assumptions and parsimony, some criterion for good models may not be met

or may be at a sub optimal level. Listed below are some limitations faced in building the

model.

Choosing the perfect model involves a lot of permutations and combination. To test

all of these and arrive at a metric that makes one model better than the rest is not an

easy or efficient task. Although the model chosen is optimal with respect to some

criterion, it is sub optimal with respect to others.

The choice of the optimal model finally comes down to subjectivity and is not

completely quantitative. Although the framework is purely mathematical and

statistical, the final choice depends a lot on judgement and experience in model

picking since the number of choices is very large.

The forecasting period is short due to the lack of availability of data. Ideally greater

forecasting window gives more room for forecasting and testing the validity of the

model in the long run. But with the number of data points being constrained, the

forecasting power of the model cannot be completely determined

The assumption of residual autocorrelation being absent is not completely achieved

along with normality of the residuals. These are difficult criteria to be fulfilled and

very rarely are they satisfied. Although autocorrelation has been removed for ten lags,

beyond that auto correlation exists. The consequence is a loss of efficiency of the

coefficient estimates.


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Due to the adjustments to fit model assumptions on all sides, the regressed sum of

squares value may be quite low, which is a limitation of time series analysis.

Conclusion

Although there are plenty of drawbacks, the model for estimating future earnings or

index or return levels of the Shanghai Stock Exchange index is a robust one and may be used

to successfully predict a range of possible outcomes of the index in the future and thus base

business and investment decisions on these forecasts. The model is driven with business logic

and is designed quantitatively. It updates the volatility level at every time period to reflect

new changes in risk parameters and environment. This in turn provides better predictions for

the returns. The risk also directly influences the returns which follows business logic.

Therefore, this model provides a good framework to estimate the returns on investments and

ventures in the Shanghai Pilot Free Trade Zone (SPFTZ) with the margin for error also

clearly defined. Therefore overall risk exposures may also be assessed when relying on

the model predictions to take investment decisions at the firm level.


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Acknowledgement

I would like to thank Dr. Wu Yuan for providing me the opportunity to work on the

data provided by him to design the time series. My thanks extends to Nanyang Technological

University for providing me with the facilities needed to complete the assignment. In

particular, the Business Library and Financial Trading Room resources including modelling

software managed by the Nanyang Business School were of utmost importance.


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Bibliography

1. Brooks, Chris: Introductory Econometrics for Finance- 2nd

Edition: Cambridge

University Press 2008.

2. Kutner Michael H., Nachsheim Christopher J., Neter John, Li William: Applied

Linear Statistical Models: McGraw-Hill/Irwin 2005

3. Shanghai Stock Exchange Index level Data: NTULearn Web Portal

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FE8513 Fin Time Series-Assignment