Unconditional Mean, Volatility and the Fourier-GARCH Representation

Abstract

This paper proposes a new model called Fourier-GARCH that is a modification of the popular GARCH(1,1). This modification allows for time-varying first and second moments via means of Flexible Fourier transforms. A nice feature of this model is its ability to capture both short and long run dynamics in the volatility of the data, requiring only that the proper frequencies of the Fourier transform be specified. Several simulations show the ability of the Fourier series to approximate breaks of an unknown form, irrespective of the time or location of breaks. The paper shows that the main cause of the long run memory effect seen in stock returns is the result of a time varying first moment. In addition, the study suggests that allowing only the second moment to vary over time is not sufficient to capture the high persistence observed in lagged returns. [PUBLICATION ABSTRACT]

Headnote

SUBMITTED : 21 FEBRUARY 2010

ACCEPTED : 19 SEPTEMBER 2010

Abstract

This paper proposes a new model called Fourier-GARCH that is a modification of the popular GARCH(1,1). This modification allows for time-varying first and second moments via means of Flexible Fourier transforms. A nice feature of this model is its ability to capture both short and long run dynamics in the volatility of the data, requiring only that the proper frequencies of the Fourier transform be specified. Several simulations show the ability of the Fourier series to approximate breaks of an unknown form, irrespective of the time or location of breaks. The paper shows that the main cause of the long run memory effect seen in stock returns is the result of a time varying first moment. In addition, the study suggests that allowing only the second moment to vary over time is not sufficient to capture the high persistence observed in lagged returns.

Keywords:

ARCH/GARCH, Structural change, Unconditional volatility.

JEL classification:

G12, G29.

(ProQuest: ... denotes formulae omitted.)

1. Introduction

Recently there has been an upsurge interest in modeling the nonstationarities present in the volatility of financial data. The clustering and the persistence of volatility of asset returns have been well documented. The IGARCH model of Engle and Bollerslev (1986) for instance, describes in a parsimonious way the high persistence in the conditional volatility of stock returns while the underlying process remains strictly stationary. Alternatively, Granger (1980) and Granger and Joyeux (1980) model the long memory or the long range dependence of a series of log-returns as a fractionally integrated process to allow the autocorrelation functions to decay very slowly, in a fashion characteristic of stock returns. However, seminal papers from Granger and Joyeux (1986) and Lamoureux and Lastrapes (1990) and more recently from Diebold and Inoue (2001), Mikosch and Starica (2004), Starica and Granger (2005), and Perron and Qu (2007) argue that the high persistence close to unit root and long memory both in the first and the second moments may actually be caused by structural changes in the level or slope of an otherwise locally stationary process of the longrun volatility. Diebold and Inoue (2001) argue that this is due to switching regimes in the data. Mikosch and Starica (2004) provide theoretical evidence that changes in the unconditional mean or variance induce the statistical tools (e.g., sample ACF, periodogram) to behave the same way they would if used on stationary long-range dependent sequences. Starica and Granger (2005) also deliver evidence against global stationarity. Finally, Perron and Qu (2007) conclude that the S&P 500 return series is best described as a stationary short memory process contaminated by mean shifts.

These results imply that a good model for volatility should take into account the possibility of a time varying unconditional second-moment and possibly, of a time varying first moment as well.

Engle and Rangle (2008) propose the Spline-GARCH to model long-run volatility nonparametrically using an exponential quadratic spline. However, they do so only for the second moment. Further, Starica and Granger (2005) use step functions to approximate nonstationary data locally by stationary models. They apply their methodology to the S&P 500 series of returns covering a period of seventy years of market activity and find that most of the dynamics are concentrated in shifts of the unconditional variance.

However, these models pose several problems. While spline functions may lead to over fitting, step functions may not give smooth approximations. Even major breaks, such as the stock market crash of 1929 and the oil price shocks of the 1970s did not display their full impact immediately. Structural changes may take longer to extinguish which suggests they need to be modeled as smooth or gradual changing processes. These arguments motivate the present study to propose a new approach to model the long-run first and second moments as smooth processes. The paper denotes the new process Fourier-GARCH because it uses the Flexible Fourier transform of Gallant (1981) (i.e., an expansion of a periodic function in terms of an infinite sum of sines and cosines). The basic model can be extended to incorporate the long-run volatility in the mean model. Flexible Fourier transforms have been used in the literature to approximate nonlinear structures in several ways. For instance, Becker et al. (2001) use Fourier transforms to model inflation and money demand as having smooth changes in the intercept. Also, Enders and Lee (2006) and Becker et al. (2006) propose new unit root and stationarity tests that use the Fourier approximation to model the unknown shape of the structural breaks in macro time series. The main advantage is that the issue of estimating the shape and location of the breaks reduces to selecting the proper frequency of the Fourier sine and cosine terms. A section below details how Fourier transforms can be used to approximate various types of breaks.

The study applies the new model to several of the largest stocks from S&P 500 to estimate volatility persistence in stock returns. Based on the discussion above, the paper considers several competing models. The basic Fourier-GARCH model specifies a constant first-moment, while the second-moment changes smoothly over time. A first extension to the basic model allows both the first and the second moments to vary over time, while a second extension incorporates the long run volatility in the model for the mean. The paper checks for each model the sum of the estimated coefficients in the equation for conditional volatility to assess the so called long-memory effect. The results show that allowing only the second moment to vary over time does not significantly reduce the persistence effect. In fact, the difference between this model and the simple GARCH(1,1) is negligible. However, the extended model that allows the first moment to vary over time as well, reduces the persistence effect by more than half of the value suggested by GARCH(1,1). The evidence suggests that the persistence effect seen in stock returns is mainly a result of the misspecification of the model for the mean.

The paper is structured as follows. Section 2 discusses in more detail the performance of the Fourier series to approximate various types of structural breaks. Section 3 introduces the basic Fourier-GARCH model and its extensions. Section 4 discusses the empirical estimates of the long memory effect using four different models and section 5 concludes.

2. Nonlinear Trend approximation with Fourier Transforms

The general approach to account for breaks is to approximate them using dummy variables. However, this approach has several undesirable consequences. First, one has to know the exact number and location of the breaks. These are not usually known and therefore need to be estimated. This in turn introduces an undesirable pre-selection bias (see Maddala and Kim, 1998). Second, use of dummies suggests sharp and sudden changes in the trend or level. However, for low frequency data it is more likely that structural changes take the form of large swings in the data which cannot be captured well using only dummies. Breaks should therefore be approximated as smooth processes (see Leybourne et al., 1998 and Kapetianos et al., 2003).

Flexible Fourier transforms, originally introduced by Gallant (1981), are able to capture the essential characteristics of one or more structural breaks using only a small number of low frequency components. This is true because a break tends to shiftthe spectral density function towards frequency zero. Below is illustrated the ability of Fourier transforms to capture nonlinear trends.

Using a simple form for the mean model, one can allow the intercept mt to be a deterministic function of time:

... (1)

where the driftterm is written as:

... (2)

In the above formulation et is a stationary disturbance term with variance s2e , s is the maximum number of frequencies, k is a particular frequency and T is the total number of observations. The driftterm represents the Fourier approximation written as a deterministic function of sine and cosine terms. Note that by imposing ak = b k = 0, one gets the constant mean or trend return specification. In contrast to other possible series expansions (e.g. Taylor series) the Fourier expansion has the advantage of acting as a global approximation (see Gallant, 1981). This property is obtained even if one specifies a small number of frequencies. In fact, Enders and Lee (2006) argue that a large value of s in a regression framework uses many of the degrees of freedom and leads to an over-fitting problem.

To illustrate the approximation properties of a Fourier series, the paper considers first a single frequency in the Data Generating Process (DGP):

mt=c0+cksin(2pkt/T)+dkcos(2pkt/T) (3)

where k is the single frequency selected in the approximation, and ck and dk represent the magnitudes of the sinusoidal terms.

This study considers several possible patterns for the occurrence of a break. Thus, for T=500, the paper simulates one break, two breaks, and trend breaks both in the midunconditional dle and towards the extremes. The paper illustrates the cases for temporary, permanent, and reinforcing breaks. We display the results below in Panels 1 through 9 (i.e., Figure 1). As in Enders and Lee (2006), Panels 1 and 2 illustrate approximations for breaks towards the end of a series. In Panel 3 the series has a temporary, though longlasting break. Panels 4 and 5 display permanent breaks in opposite directions while in Panel 6 the breaks are in the same direction. Finally, Panels 7-9 depict breaks in the intercept and slope of a trending series. The paper estimates the coefficients of the sinusoidal terms by performing a simple regression of yt on mt and a time trend. One can draw several conclusions based on the visual inspection of the graphs. First, a single frequency k=1 or two cumulative frequencies n=2, can approximate a large variety of breaks. Second, the Fourier transform approximates well even when the breaks are asymmetric (see Panels 1 and 2). Third, a Fourier series works best when the break is smooth over time which means it may not be suited for abrupt and sharp breaks of short duration (see Panel 5). An additional frequency of k=2 can improve the fit in this situation. Interested readers are referred to Enders and Lee (2006) and Becker et al. (2006) who have a longer discussion on the properties of the Fourier approximations. The next section introduces a new model to approximate long-run volatility.

3. a New Model for Unconditional Volatility

As the introductory part suggested, the simple GARCH(1,1) may not be appropriate because it implies a long-run level of the volatility that is constant. However previous research regarding the presence of various shifts in stock returns suggests that structural changes in the second moment induce global nonstationarity. This invalidates the use of the simple GARCH(1,1). It is known that breaks shiftthe spectral density function towards frequency zero. This indicates that the frequencies to be used are towards the low end of the spectrum (see Enders and Lee, 2006). A simple visual inspection of the autocorrelation function and periodogram of absolute returns of S&P 500 confirms this fact:

As you can note from the top graph, the most important frequencies that have an impact on the absolute returns are at the low end of the sample spectrum which is indicative of structural breaks. Both graphs confirm the presence of long memory in financial returns - slow decay with lags still significant at the 200th lag. These findings suggest the use of the following model whose aim is to capture various unknown shifts in long-run volatility. The paper denotes it the basic Fourier-GARCH:

... (4)

... (5)

... (6)

The model preserves the parsimony of the GARCH(1,1) model while it allows the unconditional expectation of the volatility to be a function of time and of cycles of different frequencies. A simple extension allows the unconditional mean to be a function of time as well - higher unconditional variance certainly requires higher unconditional mean. The time varying first moment is also approximated using a Fourier representation:

... (7)

Given its flexible setup, the Fourier-GARCH captures both short and long-run dynamics. Note that:

E(rt-m)2=E(vt2utht )=utE(ht)=ut (8)

The study uses an exponential representation of the Fourier transform to ensure its positivity. Goodness of fit measures like the BIC or AIC criterions are employed to choose the proper number of frequencies exogenously. They are computed as follows:

... (9)

... (10)

Here n denotes the number of parameters estimated by the model. The advantage of using the AIC and BIC criterions is that they include a penalty for the additional estimated parameters. Throughout the estimation the criterions employ only integer frequencies.

The advantage of using a time varying first moment for a sample of forty years of daily data of S&P 500 absolute returns is highlighted below:

Note the better fit of the second model which augments the basic Fourier-GARCH representation with a time varying intercept as in equation (7). However, given the presumption that a higher long run volatility requires a higher long-run return, the paper proposes the Fourier-M model that includes the unconditional time-varying volatility in the equation for the mean:

... (11)

In this way, both the first and the second moment change over time while the underlying model ensures a parsimonious representation.

One way to assess the persistence or long memory in stock returns is to compute the sum of the slope coefficients in conditional volatility. If the sum is close to one, then conditional volatility is said to be almost integrated and it displays very slow time decay. However the support for long-memory is weakened if one finds that a changing first and/or second moment is responsible for the persistence effect. If the sum of the coefficients is significantly less than one after one accounts for shifts in the unconditional mean or volatility, then one can conclude that the volatility process is stationary but suffers from structural shifts (see Perron and Qu, 2007).

A sample of daily returns on S&P 500 from 1963:01:02 to 2005:2:30 illustrates this discussion. The best representation is the one that specifies a single frequency both for the mean and for the unconditional volatility:

Note the slow and gradual increase of long run volatility from the 1960's until the 1980's. Also note that the estimated long run volatility of the 1990's is lower than the one for previous decades, which is consistent with market facts.

4. Model Validation and Persistence Effects

The paper uses several representative stocks of S&P 500 to assess the long memory effect of stock returns using the new models. The first 12 stocks of the index are selected according to their market percentage participation as of March 2005. Table 1 shows their ticker, sector classification and percent of total assets.

The data has been obtained from the Center of Research in Security Prices made available through the WRDS database. The longest sample period available is 1926:01:02 - 2005:12:30 and corresponds to Exxon, IBM, Chevron, Phillip-Morris and General Electric. Other stock returns have shorter sample periods (i.e. Procter & Gamble from 1929:01:02 onwards, Pfizer and Johnson & Johnson start in 1944; Intel from 1972, while the rest start in 1986). For each stock return, the study chooses exogenously an integer or cumulative frequencies according to the AIC and BIC criterions. According to Enders and Lee (2006), a frequency greater than 5 uses many of the degrees of freedom and leads to an over-fitting problem.

Table 2 displays the results from applying the AIC and BIC criterions to identify the best in sample fitting model. The above mentioned criterions indicate that in most cases the best representation is the basic Fourier-GARCH(1,1) model. The coefficients of the sine and cosine terms with up to 5 frequencies are significant at the 5% level both for the basic and for the extended models. However, given that in the model for the mean each additional frequency requires the estimation of two more coefficients, the additional penalty increases the values of the AIC and BIC criterions relative to the ones for the basic model. This is not surprising given that the BIC criterion favors more parsimonious representations. Several exceptions to the finding above are noteworthy. In the case of Microsoftfor instance, both criterions select the Fourier-M model to be the optimal representation. Also, the Fourier-M model gives the best fit for Chevron as well. Note that the basic Fourier-GARCH(1,1) and the Fourier-M models have very close values for the BIC and SBC criterions. This is true because they estimate the same number of parameters (i.e. six coefficients). In rest, the increased penalty due to the additional coefficients that are estimated in the models with two or more cumulative frequencies is greater than the better fit that is obtained. Therefore, the single frequency representation fits the data best for all models. Figures 5 through 7 show several graphs of the conditional and long-run volatilities obtained using both a constant and a time varying first moment. Note that for all series the long run volatility changes smoothly over time.

Next the paper investigates whether the selected returns display the long memory property that is usually observed in financial data. To this end, the study estimates four competing models:

* the common GARCH(1,1) developed by Bollerslev (1987) denoted M0 ;

* the basic Fourier-GARCH(1,1) with constant first moment, denoted M1 ;

* the Fourier-GARCH(1,1) with a time varying first moment, denoted M2 ;

* the Fourier-M(1,1) with long-run volatility in the mean, denoted M3 .

Table 3 shows the results. Clearly, model M2 provides the best reduction of the persistence effect for most series. For 10 of the 12 stock returns considered, the long memory effect is dramatically reduced in many instances by half or even more (i.e. GE, Pfizer, IBM, Phillips-Morris, Chevron, Intel, Procter & Gamble, Exxon, Johnson & Johnson, and Citigroup).

Note that the basic representation (i.e. the M1 model above) has only little impact on overall persistence in the short-run volatility. In most cases, its persistence is only slightly lower than the one of the GARCH(1,1) representation.

This is surprising given that this model gives the best fit according the AIC and BIC criterions in 10 out of the 12 stocks considered. Note that model M3 clearly outperforms model M1 in terms of reduced long memory effect as well. The main conclusion is that allowing only for the second moment to vary over time is not enough to account for the strong persistence effect observed in financial returns. However, in contrast to the basic model, a time-varying first moment in the equation for the mean reduces significantly the persistence in short run volatility.

5. conclusion

The paper proposes a new model to estimate the short and long run dynamics in financial data that takes into account the possibility of a time varying first and second moment. The Flexible Fourier transform of Gallant (1981) approximates the unknown date and shape of any structural break in the first and second moment as smooth processes. The study shows that Fourier series are able to approximate a wide variety of breaks of an unknown form. The basic Fourier-GARCH representation modifies the popular GARCH(1,1) to include a time varying unconditional variance. The paper proposes two extensions to the basic model. The first extension specifies a time varying first moment, while the second extension includes the longrun volatility in the equation for the mean. The results suggest that persistence still remains significant in the short run volatility for the basic model. However, the so called long memory effect disappears if one includes a time varying first moment in the model for the mean. This suggests that conditional volatility persistence is an artifact of the misspecification of the model for the mean.

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