Heteroscedasticity Models and their Forecasting Performance

Transcription

1 U.U.D.M. Project Report 2015:32 Heteroscedasticity Models and their Forecasting Performance Sebastian Sjöholm Examensarbete i matematik, 15 hp Handledare och examinator: Rolf Larsson September 2015 Department of Mathematics Uppsala University

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3 Abstract In this paper the aim is to fit and compare six different heteroscedasticity models for which has the best forecast accuracy in two different markets: equity and exchange rates. I will compare ARCH (q), GARCH (1,1), GARCH (1,1) with student t distribution, the Integrated GARCH, the Exponential GARCH and the Threshold GARCH model. The forecast will be made for 100 days ahead and the results of the models are measured by the MSE, mean squared error, for which the least value gives the best performance for the given stock or exchange rate. It should be mentioned that the results do not differ much between the chosen models compared.

4 Table of Contents 1 Introduction Background Previous studies Aim Heteroscedasticity models Autoregressive Heteroscedasticity model GARCH (p,q) model GARCH (p,q) with student- t distribution Integrated GARCH model (IGARCH) Asymmetry in volatility Exponential GARCH model (EGARCH) TGARCH Data Building an ARCH (p) model Determining the order for an ARCH (p) model Akaike s Information Criteria (AIC) Bayesian Information Criteria (BIC) Hannan- Quinn Information Criteria (HQIC) Mean Square Error (MSE) Why use GARCH (1,1)? Why use Student- t distribution? Method Results and Analysis Discussion and conclusion References Appendix... 35

5 1 Introduction 1.1 Background Market risks associated with high volatility are catastrophic for all investors. For an average investor, the performance of a portfolio usually depends on a few trading days during which extreme returns occur. During these days the bad outcomes can completely reverse the good outcomes accumulated during the previous period. So investors are primarily concerned with the extreme risks that make the most impact on their portfolios performance. Therefore forecasting volatility plays one of the most important parts in portfolio allocation. The three main purposes of forecasting volatility are for risk management, portfolio allocation and for taking bets on future volatility. Risk management includes measuring the future potential losses of a portfolio asset, and this includes that estimates have to be done about future volatilities and correlation. In asset allocation, minimizing the risk for a given level of expected return is a standard approach. The volatility of an asset, the risk, can also be seen as the movement of a financial asset, and the more the asset moves, the more likely its value will decrease. The first model that emerged for modeling changing variance in time series is the Autoregressive Conditional Heteroscedasticity (ARCH) model, as a function of passed residuals, by Engle (1982) and its various extension. For this he later came to receive the Swedish central bank Prize in Economic Sciences in Memory of Alfred Nobel in Since the introduction of the ARCH model several more sophisticated models have appeared. One of them is the Generalized ARCH model by Bollerslev (1986), which is a generalization of the ARCH process and also allows for past conditional variances in the current conditional variance equation. Many of the different models have different features, which makes the forecast accuracy better. Both the ARCH and GARCH models capture volatility clustering, which means that periods with high volatility tend to be followed by periods of high volatility, and leptokurtosis, when the kurtosis value is largely positive. But as their distribution is symmetric, they fail to model the leverage effect. To address this problem, many nonlinear GARCH models have been proposed, such as the Exponential GARCH (EGARCH) model by Nelson (1991), where volatility can react asymmetrically to the good or bad news. It is also proposed in the analysis of empirical data that sometimes the model fits better when relaxing the Gaussian assumption in the previous models and supposing instead

6 that the distribution follows a heavier tailed zero- mean distribution such as the Student s t- distribution. So, the question is: do the more sophisticated models capture the volatility better? And which model has the best forecast accuracy? There are some studies that have been made about the forecast accuracy of the different models on different securities. 1.2 Previous studies Hansen and Lunde (2005) compared 330 different ARCH- type models in terms of their ability to describe conditional variance. They used out- of- sample data using DM- US exchange rate data and IBM return data, where IBM return data where based on a new data set of realized variance. They found no evidence that a GARCH (1,1) model is outperformed by more sophisticated models in their analysis of exchange rates. But the GARCH (1,1) model was poor in comparison to models with a leverage effect in their analysis of IBM returns. Marucci (2005) compared different standard GARCH models with more complex models like Markov Regime- Switching GARCH (MRS- GARCH) in their ability to forecast the U.S. stock market volatility at horizons that range from one day to one month. He found that MRS- GARCH was better than all standard GARCH models in forecasting volatility at horizons shorter than one week. However, at forecasts longer than one week, standard asymmetric GARCH models tend to be better. Alberg, Shalit and Yosef (2008) compared the forecast performance of several GARCH models with different distributions for two Tel Aviv stock index returns. Their results showed that, among the forecast tested, the EGARCH skewed Student- t model outperformed GARCH, TGARCH and APARCH models in forecasting the Tel Aviv stock index returns. The result also showed that the asymmetric models, such as the EGARCH, improve the forecasting performance. Zhang and Yu (2009) used the Shanghai Stock Exchange Composite Index data to construct ARCH- type models under the assumptions of normal residuals and non- normal residuals and compared the forecast performance of volatility for normal and non- normal ARCH models and the performance of VaR measure to demonstrate the effect of the distribution assumptions on GARCH model s forecasting ability. Their results showed that the forecast performances of volatility for GARCH with assumptions of a student s- t distribution for the residual terms are superior to the assumption of a normal distribution.

7 Laurent, Rombouts and Violante (2012) used 10 assets from the New York Stock Exchange and compared 125 different multivariate GARCH models on their 1-, 5-, and 20- day- ahead forecast performance of the conditional variance over a period of 10 years. The results show that in unstable markets the multivariate GARCH models perform poorly. Ramasamy and Munisamy (2012) compared three simulated exchange rates of the Malaysian Ringgit with actual exchange rates using GARCH, T- GARCH and EGARCH models. The forecasted rates are compared with the actual daily exchange rates for four different currencies, the Australian dollar, Singapore dollar, Thailand Bath and Philippine Peso in The results show that these GARCH models perform better predicting the more volatile exchange rates than predicting the exchange rates that are less volatile. The results also show that the leverage effect incorporated in the TGARCH and EGARCH models do not improve the results much. 1.3 Aim The aim of this paper is to examine six different heteroscedasticity models ARCH, GARCH, IGARCH, TGARCH and EGARCH. And for the GARCH model, a different distribution will also be considered, i.e. student- t distribution. The main goal is to examine if one of these models outperform the other models significantly in forecasting the conditional volatility for different securities. I will therefore not only focus on the stock market but also on exchange rates. I have chosen three different stocks Apple, Coca- Cola and Google and three different currencies United States Dollars (USD), Great Britain Pound (GBP) and the Swedish Krown (SEK). For all six data sets I have excluded the last 100 observations when I modeled the volatility. This is because I used the last 100 observations in purpose of comparing the models forecasting accuracy of the conditional variances to the real observations. The result was measured using the Mean Squared Error, MSE, which is the squared sum of the difference between the observed squared residuals and the forecasted conditional variance. I chose to examine the ARCH model because it is the first heteroscedasticity model, by Engle (1982), and my purpose of this choice is to compare this model with the more sophisticated models. The GARCH (1,1) model is suggested to be enough to capture the volatility, and according to Hansen and Lunde (2005) there is no evidence that the more sophisticated models outperform the GARCH (1,1) model in modeling exchange rates. The EGARCH and IGARCH model are more sophisticated models compared to the others. For example one purpose of the development of the EGARCH model is because when studying empirical data in different markets, the impact of negative price moves are bigger on future volatility that of positive price moves, which is called asymmetric volatility.

8 For the stock data I have collected 2025 observations for Apple and 2024 for Coca- Cola and Google. For the exchange rates I have 396 observations for GBP/SEK and 364 for SEK/USD and GBP/USD, so there is a quite a big difference in the sample size between the stock and exchange rates. The difference in sample size is also very interesting because the models forecast accuracy might perform worse when the sample size is smaller or bigger. For example Ng and Lam (2006) investigated how the size of the sample size affected the GARCH model and found that 1000 or more observations are recommended for the conventional GARCH model. We might therefore expect better performance in forecasting accuracy from the original GARCH model for the stocks than for the exchange rates. 2 Heteroscedasticity models 2.1 Autoregressive Heteroscedasticity model The Autoregressive Heteroscedasticity models are built to capture time varying volatility in time series. The first heteroscedasticity model introduced by Engle (1982) used this model to estimate the means and variances of inflation in the U.K. (Engle, 1982). The basic idea behind the ARCH model in modeling financial data is that (1) the change of an asset return, a!, is serially uncorrelated, but dependent because of the assumption that high volatility appears in clusters, and (2) the dependence of a!, can be described by a quadratic function of its lagged values. The model is built on the information set at time t- 1. The conditional variance function depends on p- lagged observations: σ!!!! = α! + α! a!!! + + α! a!!! (1) In equation (1) it can be seen that for larger values of previous observations of asset returns the forecast of the volatility will be bigger, and the squared values will cause them to be even bigger. This means that a shock which causes big changes in asset return, a!, the model will automatically forecast large values for the future volatility which is the same behavior as the clusters of the volatility (Tsay 2005:102). Consider the asset return function as a! = σ! ε! (2) Where [ε! ] is a sequence of independent identically distributed (iid) random variables with mean zero and variance 1.

9 σ! Is the positive ARCH (p) function, as described above, σ!!!! = α! + α! a!!! + + α! a!!! (3) Where α! > 0 and α! 0 for j > 0. One weakness of the ARCH (p) model is that the model assumes that positive and negative shocks have the same effect on volatility because it depends on the square of the previous returns. In practice it is well known that the price of a financial asset responds differently to positive and negative shocks (Tsay 2005:106). How to select the order p of an ARCH model is debatable and the required number of p might be very large, which makes this model unattractive (Brooks 2008:391). In this paper we will use the Akaike s information criteria, AIC, the Bayesian Information Criteria, BIC, and the Hannan- Quinn Information Criteria, HQIC, to decide wich order that best describes the conditional variance. 2.2 GARCH (p,q) model Although the ARCH model is simple, it often requires a lot of parameters to describe the conditional variances. The General Autoregressive Heteroscedasticity model (GARCH (p,q)) introduced by Bollerslev (1986) is a generalization of the ARCH process. To illustrate the GARCH (p,q) model, consider the log return of an asset, r!, and let a! = r! μ! be the innovations at time t. Then we define GARCH (p,q) models as: a! = σ! ε! (4) Where again [ε! ] is a sequence of iid random variables with mean zero and variance 1, σ! is the positive function: σ!!! = α! +!!!( α!!!a!!! ) +!!!( β!σ!!!! ) (5) Where α! > 0, α! 0 and β! 0 for i, j>0 and (α! + β! ) < 1!"# (!,!)!!!. We also assume that α! = 0 for j>p and β! = 0 for i>q. The constraint α! + β! implies that the unconditional variance is finite, and the conditional variance, σ!, varies over time. If in the equation (3) q=0 the parameters β! in the equation will vanish and the

10 GARCH (p,q) model will be reduced to an ARCH(p) model. The parameters p and q are referred to as ARCH and GARCH parameters respectively. The GARCH (1,1) model can be written as σ!!!! = α! + α! a!!! + β! σ!!! (6) Where α! > 0 and α!, β! 0, also (α! + β! ) < 1. If we look at the equation (5) above we notice that large values for lagged variance and returns we will have a bigger variance, again the well- known behavior of volatility clustering in financial time series(tsay 2005:114) The GARCH (p,q) model also responses equally to positive and negative shocks as the ARCH (p) model (Tsay 2005:116). 2.3 GARCH (p,q) with student- t distribution When we defined the GARCH (p,q) model we assumed that ε! was iid N(0,1). So here, instead, we assume that ε! follows a student- t distribution with v degrees of freedom, t(v). In the analysis of empirical financial data it is found that sometimes it is better to assume that the distribution of a! better suits a student s- t distribution (Brockwell and Davis 2002:352). 2.4 Integrated GARCH model (IGARCH) In 1986 Engle and Bollerslev introduced a new model called the integrated GARCH model (IGARCH) that is persistent in variance because todays information remains important for forecasts on all horizons. First we consider the lag operator L! a! = a!!! and the lag polynomials α L = α! L + + α! L! and β L = β! L + + β! L!. Then the GARCH models can be written as: σ!! = α! + α L a!! + β(l)σ!! (7) and in the case when α L + β L = 1 we may represent the GARCH model as an ARMA (m,p) model where m=max (p,q),

11 1 α L β L a!! = α! + (1 β L )ν! (8) where ν! = a!! σ!!. If we look at equation (8) and consider the autoregressive polynomial 1 α L β L, Engle and Bollerslev said that when this polynomial had d > 0 unit roots, and m d roots outside the unit circle, the GARCH model is integrated in variance of order d (IGARCH (p,d,q)). With some modifications equation (6) may be written as: σ!! = α! + α L a!! + β(l)σ!! (9) which is the Integrated GARCH model by Engle and Bollerslev (1986) (Hafner 1998:103). Notice that Integrated GARCH model is the GARCH model where α L + β L = 1 (Reider 2009:11). The IGARCH (1,1) model may be written as: a!! = σ! ε! (2), σ!! = α! + (1 β! )a!! + β! σ!! (10) where ε! is iid N (0,1) and we have replaced α! by 1 β!. 2.5 Asymmetry in volatility The heteroscedasticity models that assume symmetry in volatility, some of them presented above, are models that assume that negative and positive innovations have the same impact when forecasting the conditional variance. But in many financial markets this is not the case. In fact, in many markets negative shocks are assumed to have bigger impact on volatility than positive shocks, which is known as the leverage effect. For example Ferreira, Menezes and Mendes (2007) investigated the hypothesis that the conditional variance of stock return is an asymmetric functions of past information on seven different stock market indices. Their results show that the conditional variance is in fact asymmetric, and that negative shocks have bigger impact on the variance than positive shocks. 2.6 Exponential GARCH model (EGARCH) While the GARCH (p,q) model captures the short run temporal dependencies in magnitude for a variety of speculative assets, the model does not capture the leverage

12 effect in stock returns (Bollerslev and Mikkelsen 1996:159). In the GARCH models, the conditional variances are functions of magnitude of the lagged residuals and do not consider if they are positive or negative. To capture this, inclusion of different asymmetric terms in the conditional variance equation has been suggested. One of these models is the Exponential GARCH (EGARCH) model proposed by Nelson (1991). The EGARCH model may be written as follows!! ) ln (σ!! ) = α! +!!! g! ε!!! +!!! β! ln (σ!!! (11)! The conditional variance is constrained to be non- negative by the assumption that the logarithm of σ! is a function of passed ε! s. We define g! (ε!!! ) as the weighted innovation: g! ε!!! = θ! ε!!! + γ! ε!!! E ε!, j = 1,, q (12) Where θ and γ are real constants. Both ε! and ε! E( ε! ) are zero- mean iid sequences with continuous distributions. Therefore E g! = 0 (Tsay 2005:124). In equation (10) the term of the function γ! ε! E ε! represents the magnitude effect as in the GARCH (p,q) model. Suppose for example θ = 0 and γ > 0, then ln(σ!!!! ) will be positive when the magnitude of change in ε! is larger that the expected value of ε!. Also notice that if θ < 0 and the magnitude of change in ε! is smaller than the expected value of ε!, the negative effect will be larger than if ε!!! E ε!. Therefore the conditional variances will response different in magnitude to positive and negative shocks (Nelson 1991:351). Equation (11) and (12) together form Nelsons EGARCH model (1991), and the most popular EGARCH model is the EGARCH (1,1) model with p and q equal to 1. The EGARCH (1,1) model may then be written as a combination of (11) and (12), where (12) becomes g ε!!! = θ! ε!!! + γ! [ ε!!! E ε! ] (13) which we insert to equation (11) for p, q=1, ln (σ!!! ) = α! + θ! ε!!! + γ! ε!!! E ε! + β! ln σ!!! (14) It should also be mentioned that when the EGARCH model assumes Gaussian distribution, the error term E ε! = 2/π, so we will get the equation:

13 ln σ!! = α! + θ! ε!!! + γ! ε!!!!! + β! ln (σ!!!! ) (15) Here it should be mentioned that because of the property that negative shocks of the volatility tend to have bigger impact, θ is often assumed to be negative (Tsay 2005:124). 2.7 TGARCH Another model within the asymmetric GARCH family is the Threshold GARCH (T- GARCH) model by Rebemanajara and Zakoian (1993), which is an extension of the Threshold ARCH model by including the lagged conditional standard deviations as a regressor (Miron and Tudor 2010). The TGARCH model can be written as σ!!!! = α! +!!! α! a!!!! +!!! γ! S!!! a!!!!!!!! σ!!!! + β! (16) where: S!!! = 1 if a!!! < 0 0 if a!!! 0 (17) So, by examining the equations (16) and (17) we see that negative changes will have bigger effect than the positive changes; they capture the leverage effects, which is the purpose of the asymmetric GARCH models. Notice that γ! is expected to be positive otherwise the positive effects would be greater than the negative effects.

14 3 Data The data that has been used in my study is the daily closing prices from three different stocks Apple, Coca- Cola and Google. I have also compared the result of forecasting three different exchange rates, Great Britain Pound (GBP), United States Dollars (USD) and the Swedish Krown (SEK). All the data was downloaded from Google Finance and the stock data are dated from to , and contains 2025 observations for Apple and 2024 observations for Coca- Cola and Google. The last 100 observations where left out in order to evaluate the forecasting results. The exchange rates are dated from to each containing 396 observations for GBP/SEK and SEK/USD and 398 observations GBP/USD. Also here the last 100 observations where left out to evaluate the forecast accuracy. All calculations were done in R using the rugarch package. Graph 1 (left) Daily closing price (USD) for Coca- Cola (Black) and Apple (Red) from (Right) Daily closing price (USD) for Google from to

15 Graph 2 (Top) Daily closing price (SEK) for GBP/SEK from to (Middle) Daily closing price (USD) for SEK/USD from to (Bottom) Daily closing price (USD) for GBP/SEK from to To be able to model the volatility of the financial data the trend must be removed from each data set. We do this by differentiating the time series one by one. The stock data were differentiated with the log returns: a! = ln y! ln (y!!! ), where y! is the observed closing price at time t. And the exchange rates were differentiated with the somewhat modified log returns: a! = 100 (ln y! ln (y!!! )), here y! is the exchange rate at time t. A sample Autocorrelation function, sample ACF, and the sample Partial Autocorrelation function, PACF, was plotted for all stationary data sets. The sample ACF is a tool for estimating the dependence in the data. For example, if the sample ACF is close to zero we might suggest that it is iid noise (Brockwell and Davis 2002:18). Also, the PACF could be used in order to determine the order of the ARCH (p) model (Tsay 2005:119). The Lagrange multiplier test for the Autoregressive heteroscedasticity (ARCH) effect was done for all residuals up to lag 30 and is represented in Appendix 1-6. If the LM test is not rejected, i.e. p- value exceeds 5%, then the null hypothesis that all coefficients in the ARCH model are zero, cannot be rejected and we say there is no ARCH effect (Engle 1982)

16 Graph 3 Daily log returns (USD) from Apple (top), Coca- Cola (middle) and Google (bottom). Graph 3 shows the log returns from the three equities from to The top graph represent Apple, the middle graph represents Coca- Cola and the bottom graph Google. As we can see in the graph, all series show a significant increase in volatility during the financial crisis in 2007 to It can also be seen that volatility seems to appear in clusters, which is a sign of ARCH effect, and there are periods with high volatility and periods with low volatility.

17 Graph 4 ACF function for Apple (top), Coca- Cola (middle) and Google (bottom). Graph 4 shows the sample ACF function for the three equities Apple (top), Coca- Cola (middle) and Google (bottom). Notice that lag zero is always one. By examining the graphs we see that there exist no correlations between one lag to another. It also seems to be equally likely to be positive or negative from one lag to another. But, there might still be some correlation in magnitude. If we square the residuals and plot the graphs again we can see that there still exists some correlation in magnitude from one lag to another (Graph 5). LM tests for the ARCH effect was carried out on the three stocks (see Appendix (1-3)) and all p- values are less than 0.05, so we have ARCH effect.

18 Graph 5 The ACF function of squared residuals for Apple (top), Coca- Cola (middle) and Google (bottom). Graph 6 shows the partial ACF of the squared residuals for all tree equities. The PACF shows significant lags up to lag 10 and then cuts off. The PACF cuts off at lag 3 for Coca- Cola and at lag 12 for Google.

19 Graph 6 The PACF function of squared residuals for Apple (top), Coca- Cola (middle) and Google (bottom). Graph 7 Log returns for GBP/USD (top), SEK/USD (middle) and GBP/SEK (bottom). Graph 7 shows the differentiated series for the exchange rates. Notice also here that the volatility appears in clusters and that there are periods with high volatility and periods with low volatility. If we look at the sample ACF function for the exchange rates, Graph 8, we notice that there seems to be no correlation between different lags for all three exchange rates passed lag 1. If we look back in Graph 4 we notice that there is no significance in lag 1 for the stock data, but there is for all three exchange rates. This means that the degree of dependence is higher for the exchange rates than for the stocks.

20 Although, as in the case for the three stocks we also check the squared residuals to see if there exists some correlation in the magnitude of change from one lag to another. And if we examine Graph (1.9) we notice that there might still be some correlation of the change in magnitude. Graph 8 ACF functions for GBP/USD (top), SEK/USD (middle) and GBP/SEK (bottom).

21 Graph 9 The ACF function of squared residuals for GBP/USD (top), SEK/USD (middle) and SEK/USD (bottom). In Graph 10 below, the PACF for the exchange rates shows that for all tree exchange rates the squared sample PACF cuts off at lag one. If we compare the ACF and PACF on the squared residuals for the stock and the exchange rates it can be seen that the correlation is greater for the tree stocks than for the exchange rates. And that the correlation for the tree exchange rates seems weak according to the sample ACF and PACF.

22 Graph 10 The PACF function of squared residuals for GBP/SEK (top), SEK/USD (middle) and GBP/SEK (bottom). The LM test on the residuals was done for the exchange rates, as in the case for the stocks, and it was found that for GBP/SEK the null hypothesis is rejected for all lags except at lag 27 and 30. For the exchange rate SEK/USD there is an interval between 3 and 13 where the p- value for the LM test exceeds 0.05 and we do not reject the null hypothesis for these lags, although the remaining lags have a p- value less than For GBP/USD the p- value for lags 7, 9-13 and exceeds 0.05 and we do not include them in our modeling. The remaining lags for GBP/USD are less than 0.05 and therefore included. These results can be found in Appendix 1-6. Stock/ER MEAN MAXIMUM MINIMUM Std. Dev. Skewness Kurtosis Apple Coca- Cola Google GBP/USD SEK/USD GBP/SEK Table 1 Statistics about the log return series. Table 1 shows some statistical information about the logarithmic return series. The kurtosis for normal distribution is 3 (Shumway and Stoffer 2006:282) and as we can see

23 in Table 1, the kurtosis for the three stocks exceeds 3. The kurtosis for all the exchange rates is below 3. So none of the time series seems to be normally distributed. 4 Building an ARCH (p) model In order to build an ARCH (p) model it is important to determine the correct number of lags. There are many ways of determining the order and below you will find three different information criteria that will be used in order to determine the best order for the ARCH (p) models. 4.1 Determining the order for an ARCH (p) model The methods used to compare the considered models are Akaike s, Bayesian and Hannan- Quinn information criteria. Note that these criteria only compare considered models and selects the model that best fit the given data. Also, because this selection is based on information criteria no p- values for the different variables are necessary, only those coefficients which equal zero we will consider as insignificant Akaike s Information Criteria (AIC) The first method used to compare the considered ARCH models to determine which model that best fits the given data is Akaike s Information Criteria (AIC) (Javed and Mantalos 2013). The idea is that you first fit an ARCH (p) model with lag p to the data. Maximizing the conditional maximum likelihood function with respect to the data, which also determines the estimation of the parameters. The maximum likelihood function for an ARCH model is given by:! L α!, α!.. a!,.. =!!! f!!!!..(a! a!!! ) (16) where the density function f, is the density function for the specified distribution of the model, e.g. normal or student- t distribution. When we have maximized the likelihood function we will use it for the minimization of the AIC function, which in R is given by: AIC = 2 ln L /N + 2 k /N (17)

24 where N is the number of observations in our data after differentiation, k is the number of parameters in the ARCH and GARCH models, respectively (Ghalanos 2013:26), and L is the value of the maximum Likelihood Bayesian Information Criteria (BIC) The Bayesian Information Criteria (BIC) is another way to determine the number of lags to include for the best fit of ARCH/GARCH models given a number of models (Javed and Mantalos 2013). Many studies suggest that when the sample size is large, the BIC performs better in selecting the correct order of lags in your model (McQuarrie and Tsai, 1998). The BIC function for an ARCH (p) model in R is given by: BIC = ( 2) ln L /N + k ln N /N (18) Notice that, compared to AIC, the BIC criterion penalizes more complex models, i.e. models with more parameters, relative to models with fewer parameters. Also here, the BIC function selects the model that minimizes the value of the function Hannan- Quinn Information Criteria (HQIC) Hannan and Quinn (1979) introduced another information criterion in order to compare models for model selection (Javed and Mantalos 2013). The HQIC function in R is calculated as: HQIC =!!!"!!!!"#!"! + (! ) (19) and also here N is the number of observations, ln(l) is the logarithm of the maximum likelihood function and k is the number of parameters in the model (Ghalanos 2013:26). The results from the information criteria applied on the six data sets can be seen in appendix 1-6 along with the Lagrange multiplier test results. Appendix 1 shows that for the Apple stock the criteria select ARCH (16), ARCH (8) and ARCH (11) for the AIC, BIC and HQIC respectively. For the Coca- Cola stock in Appendix 2 all three information criteria select ARCH (19). Although AIC gives the same value for ARCH (19) and ARCH (23) we will choose ARCH (19) because it is preferable to have less parameters in our models. In appendix 3 we also see that the three information criteria selects the same number of lags, namely ARCH (19).

25 For the exchange rate GBP/USD there is a big difference of the preferable model. Both the AIC and the HQIC function select ARCH (28), and the BIC function selects ARCH (1). This may be because as mentioned in section the BIC function penalizes models with more parameters. For the SEK/USD exchange rate in Appendix 5 both the HQIC and the BIC function select ARCH (1) and AIC selects ARCH (15). And in Appendix 6 all tree information criteria select ARCH (1) for the GBP/USD exchange rate. 4.2 Mean Square Error (MSE) When we have selected number of lags for the best- fitted model of each of the three criteria we use the MSE function to determine which model to choose. This result is then compared with the MSE result for all models, and the model with the lowest MSE value will be the best forecasting model for the given data. To evaluate the MSE function we need to forecast the future conditional variances using the selected models. These forecasts will then be compared with the future values of the squared log return for the stocks and exchange rates. The models with the least MSE values will then be the selected models. The MSE function can be described as: MSE =!!""!"" (a!! σ!! )!!!! (20) where σ! represents the estimated conditional variance and a! is the squared log returns (Hansen and Lunde 2005). Notice that this function forecasts the next 100 daily observations. This is why these observations were excluded when determined the best fit of the models so they could be used as a measurement for the best forecast. Also remember that this MSE function is the function used to compare and evaluate the different heteroscedasticity models. 4.3 Why use GARCH (1,1)? Hansen and Lunde published a study in 2005 where they compared different GARCH models to see if anything beats the GARCH (1,1) model. Their aim was to determine if the improvement of the GARCH models after Bollerslev s introduction of the first GARCH model in 1986 has led to better forecast ability. They found that the forecast ability of ARCH (1) model is outperformed by other models. This is because the ARCH (1) model does not capture the persistence in volatility as the other models. When they compared different GARCH models to the GARCH (1,1) model they found no evidence to reject the standard GARCH (1,1) model in favor of other GARCH models. They found this somewhat surprising because the GARCH (1,1) model do not capture the leverage effect like the more sophisticated models like EGARCH and TGARCH. To be able to compare the results of different GARCH models, I chose to evaluate all models using order (1,1).

26 4.4 Why use Student- t distribution? Bollerslev suggested that the GARCH model a! = σ! ε! with assumed conditionally normal distribution might not sufficiently cover the leptokurtosis in financial time series. He suggested that sometimes the model a! = σ! ε! has thicker tails and is better described by a student- t distribution. He therefore introduced the GARCH- t model which assumes a student- t distribution instead of the normal distribution, (Bollerslev 1987).

27 5 Method The steps used on the following models ARCH, GARCH, GARCH with student- t innovations, IGARCH, EGARCH and TGARCH to find the best forecast accuracy for the conditional variance are explained below. The models were estimated for the three stocks Apple, Coca- Cola and Google and the three exchange rates USD/SEK, GBP/USD and GBP/SEK. Remember that the last 100 observations where excluded in order to evaluate the forecasting accuracy of the models. All calculations have been done using the rugarch package in R. 1) First differentiate the time series using the logarithmic returns to turn the series into a stationary time series. 2) The ACF/PACF function and the Lagrange multiplier test were applied to the residuals to examine if there were any ARCH effect. 3) Then the maximum likelihood function was estimated for different lags of the ARCH (p) models, and the logarithmic value of the result was calculated. 4) With different log likelihood values the AIC, HQIC and the BIC function where evaluated to find the value that minimizes the functions. The selected value then represented the number of lags to choose for the ARCH (p) model. 5) Then the coefficients for all models where estimated by maximizing the likelihood function for the chosen number of lags. 6) Then I estimated the conditional variance. 7) Then I calculated the log returns for the remaining 100 observations for each data set. 8) And last, I evaluated the forecasting accuracy using the MSE function. The MSE function, as described in section 3.3, is: MSE =!!""!"" (a!! σ!! )!!!! (21)

28 Here t represents the last 100 observations, σ! represent the estimated conditional variances and a! is the squared log returns for the last 100 observations of the data sets (Hansen and Lunde 2005). 6 Results and Analysis In section 2.1 we presented the ARCH (p) models that where chosen by the tree information criteria. The MSE results of these models are compared in Table 2 below. Stock AIC BIC HQIC Apple ARCH (16) ( e- 07) ARCH (8) ( e- 07) ARCH (11) ( e- 07) Coca- Cola ARCH (19) ( e- 08) ARCH (19) ( e- 08) ARCH (19) ( e- 08) Google ARCH (19) ( e- 07) ARCH (19) ( e- 07) ARCH (19) ( e- 07) GBP/USD ARCH (28) ( ) ARCH (1) ( ) ARCH (28) ( ) SEK/USD ARCH (15) ( ) ARCH (1) ( ) ARCH (1) ( ) GBP/SEK ARCH (1) ( ) ARCH (1) ( ) ARCH (1) ( ) Table 2 The MSE results presented in Table 2 tell us which one of these models to use. As mentioned earlier the MSE result is a measure of the performance accuracy, and here the forecast contains 100 observations. For the Apple stock all information criteria chose different lags for ARCH. And, according to the MSE result, the most accurate model in this measurement is the ARCH (11) model chosen by the Hannan- Quinn criteria. For the Coca- Cola stock all three criteria choose the same model, namely ARCH (19). This is also the case for the Google stock where all three criteria chose the same model, namely ARCH (19). For the exchange rates, AIC and HQIC chose the best model, ARCH (28), for the GBP/USD exchange rate but as we can see the constant equals zero, which contravenes the criteria for ARCH (p) models. Therefore ARCH (1) is chosen instead. For the SEK/USD exchange rate AIC chose the most accurate model, which is ARCH (15). And last, for the GBP/SEK exchange rate all three criteria chose the same model to be ARCH (1). In Appendix 7 the chosen ARCH models and coefficients are presented for all six data sets and it can be seen that ARCH (28) has a zero constant.

29 Notice that the coefficients resulting in zero coefficients are because of their given p- values. The p- values for each coefficient represent the likelihood that the given coefficient is equal to 0. So, for the zero coefficients, this means that the p- values equal 1. Resulting in 100 per cent chance to equal 0. And as was mentioned earlier in section 3.2, because information criteria are used to decide which model that best describes the given data we will oversee the remaining p- values which do not gives us zero coefficients and count them as significant. In Table 3 below the models from the GARCH family is presented with their corresponding coefficients and MSE values. The first tree models for each stock and exchange rate presents the values from the GARCH, GARCH- t and IGARCH model, i.e. assuming no leverage effect. And the remaining two are the EGARCH and TGARCH models that assumes asymmetric volatility, i.e. captures the leverage effect. Constant ARCH (α! ) GARCH (β! ) GAMMA (γ! ) Shape MSE GARCH (Apple) e - 07 GARCH- t (Apple) (0.220) e - 07 IGARCH (APPLE) (0.764) (0.317) e - 07 EGARCH (Apple) e - 07 TGARCH (Apple) (0.016) e - 07 GARCH (Coca- Cola) e - 08 GARCH- t (Coca- Cola) (0.666) (0.054) e - 08 IGARCH (Coca- Cola) (0.283) e - 08 EGARCH (Coca- Cola) e - 08 TGARCH (Coca- Cola) e - 08 GARCH (Google) e - 08 GARCH- t (Google) (0.487) e - 08 IGARCH (Google) e - 07 EGARCH (Google) e - 07 TGARCH (Google) (1.000) e - 08 GARCH

30 (GBP/USD) (0.188) (0.012) 2 GARCH- t (GBP/USD) (0.676) (0.218) IGARCH (GBP/USD) (0.133) (0.017) EGARCH (GBP/USD) TGARCH (GBP/USD) (0.069) (1.000) GARCH (SEK/USD) (0.171) (0.033) GARCH- t (SEK/USD) (0.379) (0.127) IGARCH (SEK/USD) (0.289) (0.132) EGARCH (SEK/USD) (0.362) (0.218) (0.012) TGARCH (SEK/USD) (0.0649) (0.0239) (0.049) GARCH (GBP/SEK) (0.015) (1.000) GARCH- t (GBP/SEK) (0.723) (0.301) IGARCH (GBP/SEK) (0.441) (0.679) EGARCH (GBP/SEK) (0.182) TGARCH (GBP/SEK) (0.610) (0.673) (0.724) Table 3 For the Apple stock in Table 3 both the GARCH- t and IGARCH models contain some insignificant coefficients which means that they contribute for forecasting in an negligible way. The model with best forecast accuracy in this case is the GARCH- t (1,1) model. If we look at the Coca- Cola stock in Table 3 we notice that also here the IGARCH and GARCH with student t distribution contains insignificant coefficients. The model with the best MSE value for the Coca- Cola stock is the standard GARCH (1,1) model. The TGARCH and the GARCH- t model for the Google stock contain some insignificant coefficients. Notice here that the GARCH parameter in the Threshold GARCH has a p- value of 1, resulting in a coefficient value of 0. The model with best MSE value for the Google stock is the GARCH- t (1,1) model.

31 If we look at the GBP/USD exchange rate in Table 3 we notice that all models contains some insignificant coefficients except the EGARCH model. The model with the best forecast accuracy is the IGARCH (1,1) model. For the SEK/USD exchange rate in Table 3, all models contain some insignificant coefficients. For this exchange rate the EGARCH (1,1) model provides the best forecast accuracy. The last exchange rate in Table 3 is the GBP/SEK exchange rate and also here all models contain some negligible coefficients. The model with the best forecast accuracy is the GARCH (1,1) model with student t distribution. If we compare the model with the best forecast accuracy from the ARCH (p) models and the different models from the GARCH (1,1) family. It is found that the models from the GARCH (1,1) family provide better forecast accuracy for all tree stock and all tree exchange rates. Notice that this is true even though the ARCH (p) models contain a lot more parameters than the GARCH (1,1) models. 7 Discussion and conclusion The purpose of this paper was to compare the forecast accuracy of six different heteroscedasticity models. The ARCH (p) model was built with the use of tree different information criteria that determined the number of lags that best described the data. For some of the data these criteria showed different results and therefore the model with the lowest MSE value was chosen. All five models from the GARCH family were pre determined to be order (1,1) to give a fair result and because its been showed to be enough. The MSE results show that for different stocks or exchange rates, different models have the best forecast accuracy. There was no model that overall outperformed the others. Also, as can be seen in Appendix 7 and Table 3, the difference in the models forecast accuracy didn t differ much. The Apple stock shows a very large dependence on previous lags when the ACF function of the squared logarithmic values is examined. The LM test also shows strong evidence of ARCH effect for all lags. The partial ACF also shows strong ARCH effect and cuts off after 10 lags. The best forecaster evaluated with the MSE was the GARCH model with student t distribution. The Coca- Cola stock also shows very strong evidence of ARCH effect and the ACF of the squared residuals shows strong correlation between one lag to another. The partial ACF cuts off after 3 lags. The standard GARCH (1,1) model shows the best performance.

32 The Google stock shows less ARCH effect than the first two stocks in the first ten lags, even though there is still a strong ARCH effect. The ACF also shows less correlation in the first ten lags than the other two stocks, though the partial ACF cuts off at lag 12. The model with the lowest MSE value is the GARCH with student t distribution. The ARCH LM test for the GBP/USD exchange rate shows significant ARCH effect for all lags except at lag 8, and 18. The ACF and the PACF show significant result at lag 1 but then cut off. The best forecast performance was shown by the IGARCH model. The LM test for the SEK/USD exchange rate shows insignificant results at lags 3-13, and the remaining stay significant. The ACF and the PACF cut off at lag 1 and the EGARCH models did the best performance. And last, the GBP/SEK exchange rate shows significant ARCH effect for all lags except lag 27 and 30. The model with the best forecast accuracy was the GARCH model with student t distribution. The result of this study in Appendix 7 and Table 3 indicates there is a difference in forecasting performance between the different models. There is no evidence that one model outperforms the others. For the stock data the ordinary GARCH model with Gaussian and student t distribution performed better in forecasting the conditional volatility. So, here was no evidence that models that captures the leverage effect, i.e. EGARCH and TGARCH, performs better. This contradicts the results of Hansen and Lunde (2005) when they compared different GARCH models on IBM data. Their study showed that models that capture leverage effect performed better than the GARCH (1,1) model. However, in their comparison Hansen and Lunde used seven different criteria where one of them was the Mean squared Error which was also used in this paper. They also compared their models using intra- day estimate measures of realizes volatility. The forecasting performance between the stock data and the exchange rates differ in their MSE result. The forecasting performance for the stock data was better. This could obviously be an effect of the given observations used when fitting the considered models to the data. For example Ng and Lam (2006) wrote a paper examining how the sample size affects the GARCH models and one of their conclusions was that at least 1000 observations was recommended for the model estimation of the conventional GARCH model. They also concluded that the sample size requirement for the MEM- GARCH model is 700, which is less than that of the conditional GARCH model. It should also be mentioned that the ACF and the PACF of the squared residuals showed less correlation between one lag to another for the exchange rates. The PACF cuts off at lag one for all exchange rates which also is the case for ACF, indicating less correlation in magnitude between one lag to another. Also, as mentioned in section 3, the ACF function for the differentiated data sets showed significant values at lag 1 for all exchange rates. This was not the case for the stock data. So, there might still be some correlation left in the differentiated data, which might have affected the forecast performance.

33 Zhang and Yu (2009) concluded in their study that the assumption of the student t distribution was superior to the assumption of the Gaussian normal distribution. By examining Table 3 it can be seen that the GARCH (1,1) model with student t distribution outperformed the GARCH (1,1) model with normal distribution for all data sets except for one, the Coca- Cola stock. One other observation of this study shows that the long ARCH (p) models for the tree stocks with lags 11,19 and 19 were outperformed by the smaller estimated GARCH (1,1) models. Even though the ARCH models have more parameters does not indicate more accuracy in forecasting the future values of the conditional variance. For the exchange rates the result was reversed, the GBP/USD and SEK/USD favored the bigger ARCH models while the beta coefficient in the GARCH model for the GBP/SEK was 0 resulting in the same ARCH (1) model as in Table 2. Even though, the ARCH models where still outperformed by other models from the GARCH (1,1) family, indicating that the GARCH (1,1) family outperforms the standard ARCH model on every dataset and therefore are in favor. The results in this paper do not tell us whether there exists a heteroscedasticity model that outperforms all other models when forecasting the conditional volatility. On the other hand they tell us that there is a difference in the models and that they perform differently to different data sets. The in- sample data is also a good comparison of the models because the different sample sizes might affect some models more than others. To be more accurate in the comparison one might use more data sets and evaluate all models performance on different in- sample sizes and the out- sample size should also differ, determining the best forecast range for the different models.

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