On the alternative model of volatility studies

Jacek Leśkow, Institute of Mathematics Technical University of Wrocław WROCŁAW, POLAND. and CIMAT Gunajuato, Mexico.

1 Abstract.

Recent years has seen an outburst in the number of papers related to the study of volatility, variance, conditional variance and higher-order characteristics related to the econometric time series. Various modifications of classicial ARMA, ARIMA, ARCH and GARCH models have been proposed. Most of them, however, by-pass the basic question that one asks while analyzing econometric times series models, this question being:

’Is it true that time series is stationary ?

To provide tests for stationarity one needs to create a wider framework for analysis. In such a framework the condition of stationarity will be only a special case. One of the purposes of the lecture is to provide such framework, based on the class of nonstationary, almost periodically correlated time series. It will be also discussed how the linear structure of autoregressive models creates a false alternative for studying variance, that is a limited choice between constant and explosive variance models. The model proposed avoids that problem by not explicitely specifying the structure of the model error, focusing only on estimating its variance-covariance. We will show properties of estimators derived in the proposed model and the new spectral theory for analysis of time series. Practical applications will involve tests of stationarity and analysis of Value at Risk calculation under lack of stationarity and gaussianity.

2 Stationarization - how far should we go ?

One of the main issues in the mean-variance approach to the time series analysis and financial data modelling is the assumption of stationarity. The usual ARMA-GARCH models are based on the condition of stationarity, allowing time-dependence of the conditional variance (see Bollerslev et al (1992) or Brockwell and Davis (1991) ). But the essence of the estimation technique is based on the assumption of stationarity, which, in turn, is related to the technical condition of the absence of the roots of the characteristic polynomial inside the unit circle (see Shumway (1988)). The main line of the article is to attack that assumption, which is necessarily connected with the linear structure of the time series model under consideration. There is indeed a big danger in having too much confidence in linearization techniques while working with the time series data. For example, one of the most popular linearization technique is differencing of time series that, supposedly, leads to stationarization. Let us observe more closely, what stationary difference may really mean for the underlying time series.

Example. Assume that the time series {Y_{t}, t ∈ Z^{+}} has independent increments and that the first order differencing X_{t} = Y_{t}−Y_{t−1} leads to a stationary series. Then... V ar(Y_{t}) tends linearly to infinity!

V ar(Y_{t} − Y_{1}) = V ar(Y_{t} − Y_{t−1} + Y_{t−1} − Y_{t−2} + … + Y_{2} − Y_{1}) =

∑_{i=1}^{t−1} V ar(Y_{i+1} − Y_{i}) =

= ∑_{i=1}^{t−1} X_{i+1} = c(t − 1).

One might be doubtful about stochastic model for data, when the variance tends to infinity. We may have different scale of variances, but the steady (linear) increase of the variance does not seem to correspond to many interesting data sets. Rather, as we will see in the following pictures, financial time series data may have periods of high variance and small variance. Therefore, we should not trust too much the linear models and stationarizingby-differencing techniques as they introduce a false alternative: onstant variance vs. ever increasing variance. There is little place within RMA-GARCH models for considering a fluctuating unconditional variance, so well visible for financial data.

3 Examples of fluctuating variance 4 Model for nonstationary time series

Our starting point is the time series of returns {r_{t}; t ∈ Z} with {ϵ_{t}; t ∈ Z} denoting the innovations of the observation model. Throughout the paper we will not assume a particular form of the distributions (conditional or unconditional) of the innovations {ϵ_{t}; t ∈ Z}. In this approach the innovations ϵ(t) and ϵ(t + τ) are allowed to be correlated when τ is relatively small (for a more formal discussion see Dehay and Leśkow (1996b)). Recall, that the mean zero time series {ϵ_{t}; t ∈ Z} is stationary when for each t, τ ∈ Z the covariance function

B(t, τ) = E(ϵ(t)ϵ(t + τ))  (1)

depends only on τ, that is B(t, τ) = B(τ).

It is easy to see that the assumption of stationarity (1) implies that innovations ϵ_{t} have constant variance. On the other hand, the usual ARMA-GARCH(p,q,r,s) model

r(t) = a_{0} + ∑_{i=1}^{p} a_{i}r(t − i) + ϵ(t) + ∑_{j=1}^{q} b_{j}ϵ(t − j)

and

ϵ(t) = σ(t)u(t),

where

σ^{2}(t) = ω + ∑_{i=1}^{r} α_{i}ϵ^{2}(t − i) + ∑_{j=1}^{s} β_{j}σ^{2}(t − j).

require stationarity both on returns {r(t)} and innovations {ϵ(t)}.

It is, therefore, essential to deal with the time variability of the covariance function and to do it beyond the context of ARMA-GARCH models. We propose to represent B(t, τ) in the form of Fourier series, i.e.

B(t, τ) = ∑_{λ∈Λ} a(λ, τ) exp(iλt).  (2)

The innovations {ϵ_{t}; t ∈ Z} with the covariance in the form (2) will be called almost periodically correlated (APC).

¿From the previous work of the author (see e.g. Dehay and Leśkow (1996)) we know that:

• The innovations {ϵ_{t}; t ∈ Z} are stationary if and only if the only nonzero element a(λ, τ) in (2) corresponds to λ = 0 for more details).

• If, for at least one λ ≠ 0, a(λ, τ) is different from zero for at least one τ, then the innovations {ϵ_{t}; t ∈ Z} are not stationary. This means that unconditional variance is NOT constant and, in particular, GARCH(p,q) model should not be applied to such innovations (see e.g. calculations of unconditional second moments for GARCH(p,q) model in Bollerslev (1992)).

• Representation (2) allows to estimate the departure from stationarity. There is a considerable literature on estimation of a(λ, τ) (see e.g. Dehay and Leśkow (1996) and the references included).

5 Inference results

Using returns data and the model we can find residuals ϵ̂(t) corresponding to innovations ϵ(t). In what follows, we will suppress the hat over ϵ understanding that the residuals calculating process was already performed.

We will first estimate a(λ, τ). For that purpose define the estimator â_{A}(λ, τ) by:

â_{A}(λ, τ) = \{ \begin{matrix}\frac{1}{A} ∑_{t=0}^{A−τ} ϵ(t + τ)ϵ(t)e^{−iλ}, & if A ≥ τ ≥ 0, \\ \frac{1}{A} ∑_{t=−τ}^{A} ϵ(t + τ)ϵ(t)e^{−iλ}, & if −A ≤ τ ≤ 0, \\ 0 & otherwise. \end{matrix}  (3)

We have the following: Theorem 5.1. (Dehay and Leśkow (1996)). Define the stochastic pro-

cess:

W_{A}(λ, τ) = A^{1/2}(â_{A}(λ, τ) − a(λ, τ)).  (4)

Then W_{A} converge in distribution uniformly in τ in the space C(R) to a mean zero gaussian time series G(λ) with a variance covariance matrix C(λ).

There are many important consequences of the Theorem 2.1 in the analysis of time series. Assume that M is the range of autocovariance B(t, τ) that is sup_{t}|B(t, τ)| is negligible for |τ| ≥ M.

As a consequence of Theorem 5.1 we get that:

Theorem 5.2. Let Λ = {λ_{1}, …, λ_{p}}. Let also

Û_{A} = [ \begin{matrix}W_{A}(λ_{1}, 0) \\ … \\ W_{A}(λ_{1}, M − 1) \\ … \\ W_{A}(λ_{p}, 0) \\ … \\ W_{A}(λ_{p}, M − 1) \end{matrix} ] ,  (5)

where W_{A} was defined in (4).

Then:

(i) The vector Û_{A} is asymptotically 2pM-dimensional centered normal with the covariance matrix C.

(ii) The matrix C can be estimated via the matrix estimator Ĉ_{A}, whose form is presented in the Appendix.

(iii) Additionally, if the rank of the matrix C is known then the quadratic form

t_{A} = Û_{A}^{T}g(Ĉ_{A})Û_{A}  (6)

has an asymptotic χ^{2} distribution with 2pM degrees of freedom. In the formula (6) symbol g(Ĉ_{A}) denotes the generalized inverse (g-inverse) of Ĉ_{A} (see Ben-Israel and Greville (1974)).

6 Nonparametric test of stationarity

¿From the inferential results of the Section 5 we can obtain nonparametric tests of stationarity for time series data.

We have the following:

Theorem 6.1 (testing stationarity of returns). Assume that:

• The time series of returns {r(t)} has the autocovariance function B(t, τ) in the form (2).

• Assume also that the time series {r(t)} has a φ-mixing property with φ^{1/2} integrable on R.

Let us take M sufficiently large such that sup_{t}|B(t, τ)| is small for |τ| ≥ M and, for return data {r(t)} let us calculate the statistic t_{A}(r) defined in (6).

Then, at a given significance level α we should reject the assumption of stationarity of the returns if

t_{A}(r) > c_{α},  (7)

where c_{α} can be calculated from the χ^{2} distribution with 2pM degrees of freedom and t_{A} was defined in (6).

The same technique can be applied to test stationarity of innovations.

Theorem 6.2 (testing stationarity of innovations). Assume that innovations {ϵ_{t}; t ∈ Z} fulfill the assumptions of Theorem 3.1 . Let us calculate the statistic t_{A}(ϵ) defined in (6).

Then, at a given significance level α we should reject the assumption of stationarity of the innovations if

t_{A}(ϵ) > c_{α}

where c_{α} can be calculated from the χ^{2} distribution with 2pM degrees of freedom.

Proofs of Theorems 6.1 and 6.2 are standard consequences of Theorem 5.2 and will not be presented here.

Observe that the above results can be used to study the validity of ARMA-GARCH models. The usual ARMA-GARCH(p,q,r,s) model assumes the following structure of returns {r(t)} and innovations {ϵ(t)} (see Mittnik et al (1998)):

r(t) = a_{0} + ∑_{i=1}^{p} a_{i}r(t − i) + ϵ(t) + ∑_{j=1}^{q} b_{j}ϵ(t − j)

and

ϵ(t) = σ(t)u(t),

where

σ^{2}(t) = ω + ∑_{i=1}^{r} α_{i}ϵ^{2}(t − i) + ∑_{j=1}^{s} β_{j}σ^{2}(t − j).

ARMA(p,q) and GARCH(r,s) models both require stationarity both on returns {r(t)} and innovations {ϵ(t)}. Once a stationarity of time series is established, the general autocovariance can be represented via an autocovariance of a corresponding ARMA process (see Brockwell and Davis (1991), Chapter 3).

In that context, validation of ARMA-GARCH model can be performed on two levels:

(i) testing of the form of the autocovariance of the returns {r(t)},

(ii) testing of stationarity of innovations {ϵ(t)}.

The APC approach provides a relatively straightforward way of performing tests (i) and (ii). Simply, we can peform the tests of stationarity using Theorem 6.1 and 6.2.

7 How JP Morgan runs its VaR calculator

Value-at-Risk (VaR) is an important measure of risk associated with a single asset or portfolio of assets. Its definition is quite simple:

VaR is the amount of money we are willing to loose at a certain confidence level within a fixed time horizon.

VaR, therefore, corresponds to the low α quantile of a distribution of returns and can be formalized as:

P {r_{t} < −V aR_{t}} = α.  (8)

We would like to discuss here a very popular methodology of estimating VaR used by JP Morgan. It is described in the RiskMetrics^{T M} Technical Document (see JP Morgan (1996)) on the page 66. First, authors assume that the returns on a security for which the VaR is calculated are normally distributed, however the variance may be fluctuating over time. The assumption of normality is particularly appealing to the authors of RiskMetrics^{T M} since it allows a direct calculation of the upper α- quantile. Given N securities, the RiskMetrics^{T M} assumes that the returns are generated following the model:

r_{i,t} = σ_{i,t}ϵ_{i,t},  (9)

where ϵ_{t} = (ϵ_{1,t}, …, ϵ_{N,t}) is a normal vector with the variance-covariance matrix dependent on time t and the variance σ_{i,t} depends both on i and t. Then, the value at risk is calculated from the standard normal distribution after standardizing r_{i,t} by σ_{i,t}.

Our main point of criticism here is that returns are seldom normally distributed and the appealing feature of using the number 1.65 for the upper 5 percent quantile of the distribution can be easily a source of large modelling errors. Moreover, the model (9) is hard to validate when the data on returns cannot be replicated.

In order to improve the JP Morgan VaR calculator we propose the modified historical method (see e.g. Gamrowski and Rachev (1995)) based on the following

Procedure

:

Step 1 Check, whether the returns {r_{t}, r_{t−1}, …, r_{t−n}} are normally distributed and stationary. If so, use the JP Morgan VaR calculator.

Step 2 If the returns are not normally distributed, check whether they are stationary using the test statistic t_{A} presented in Theorem 6.1.

Step 3 If no, reduce the sample size n and repeat Step 2. If it results in not rejecting the hypotheses of stationarity of returns, calculate VaR using the historical method (Gamrowski and Rachev (1995)).

The procedure presented above has the following advantages over the JP Morgan’s VaR calculator:

(i) It allows the returns to be nonstationary, (ii) It does not require normality of returns, (iii) It includes the JP Morgan VaR calculator as a special case .

8 Appendix

Form of the variance estimator

Let us have a look at the form of the estimator Ĉ_{A}:

Ĉ_{A}(λ_{i}, λ_{i′}, τ_{j}, τ_{j′}) = Ĉ_{A}^{1}(λ_{i}, λ_{i′}, τ_{j}, τ_{j′}) + Ĉ_{A}^{2}(λ_{i}, λ_{i′}, τ_{j}, τ_{j′})

where

Ĉ_{A}^{1}(λ_{i}, λ_{i′}, τ_{j}, τ_{j′})  (10)
= lim_{A→∞} \frac{1}{A} ∑_{u=−M−τ_{j′}}^{−1} ∑_{t= −u}^{A} X(t)X(t + τ_{j})X(t + u)X(t + u + τ_{j′})S(λ_{i}, λ_{i′}, t, t + u)
+ lim_{A→∞} \frac{1}{A} ∑_{u=0}^{M+τ_{j}} ∑_{t=0}^{A−u} X(t)X(t + τ_{j})X(t + u)X(t + u + τ_{j′})S(λ_{i}, λ_{i′}, t, t + u).

(11)

and Ĉ_{A}^{2}(λ_{i}, λ_{i′}, τ_{j}, τ_{j′})

= ∑_{u=−M−τ_{j′}}^{M+τ_{j}} Re ( ∑_{α=λ−λ′} â_{A}(λ_{i} + λ_{i′} − α, τ_{j})â_{A}(α, τ_{j′}) exp(−iαu)) S_{1}(λ_{i′}u)

+ ∑_{u=−M−τ_{j′}}^{M+τ_{j}} Im ( ∑_{α=λ−λ′} â_{A}(λ_{i} + λ_{i′} − α, τ_{j})â_{A}(α, τ_{j′}) exp(−iαu)) S_{2}(λ_{i′}u)

+ ∑_{u=−M−τ_{j′}}^{M+τ_{j}} Re ( ∑_{α=λ−λ′} â_{A}(λ_{i} − λ_{i′} − α, τ_{j})â_{A}(α, τ_{j′}) exp(−iαu)) S_{3}(λ_{i′}u)

+ ∑_{u=−M−τ_{j′}}^{M+τ_{j}} Im ( ∑_{α=λ−λ′} â_{A}(λ_{i} − λ_{i′} − α, τ_{j})â_{A}(α, τ_{j′}) exp(−iαu)) S_{4}(λ_{i′}u)

where the matrices S_{1}, …, S_{4} come from the decomposition:

S(λ_{i}, λ_{i′}, t, t + u) = \frac{1}{2} cos[(λ_{i} + λ_{i′})t]S_{1}(λ_{2}u)

\frac{1}{2} sin[(λ_{i} + λ_{i′})t]S_{2}(λ_{2}u) + \frac{1}{2} cos[(λ_{i} − λ_{i′})t]S_{3}(λ_{2}u)

\frac{1}{2} sin[(λ_{i} − λ_{i′})t]S_{4}(λ_{2}u)

and have the form:

S_{1}(λu) = [ \begin{matrix}cos(λu) & sin(λu) \\ sin(λu) & − cos(λu) \end{matrix} ] ,

S_{2}(λu) = [ \begin{matrix}− sin(λu) & cos(λu) \\ cos(λu) & sin(λu) \end{matrix} ] ,

S_{3}(λu) = [ \begin{matrix}cos(λu) & sin(λu) \\ sin(λu) & cos(λu) \end{matrix} ] ,

and

S_{4}(λu) = [ \begin{matrix}sin(λu) & − cos(λu) \\ cos(λu) & sin(λu) \end{matrix} ] .

In spite of the complexity of the above representation, one can succintly establish convergence of Ĉ_{A} to C, when we realize that the limit in (??) can be split into two parts according to the formula:

E(Z(t, τ_{j})Z(t + u, τ_{j′})) =  (12)
EX(t)X(t + τ_{j})X(t + u)X(t + u + τ_{j′})
− EX(t)X(t + τ_{j}) · EX(t + u)X(t + u + τ_{j′})

The convergence of the first term follows from the APC property of the fourth order moments (see Section 2). The ’mixed moments’ term can be decomposed via:

E(Z(t, τ_{j}) · EZ(t + u, τ_{j′})) =  (13)

∑_{λ} ∑_{λ′} a(λ, τ_{j})a(λ′, τ_{j′}) exp(jλt) exp(jλ′(t + u))

Now, applying the rules (12) and (13) to the variance/covariance form and doing some technical calculations we can easily see that the estimator Ĉ_{A}^{1} consistently estimates the ’fourth-order’ part of C and the estimator Ĉ_{A}^{2} estimates the ’mixed-moment’ part. □

9 References

Bollerslev,T., Chou, R.Y and Kroner, K.F., (1992), ARCH modeling in finance, a review of theory and empirical evidence, Journal of Econometrics 52, pp 5-59. Brockwell,P.J. and Davis, R.A. (1991), Time Series: Theory and Methods, 2nd edition, Springer-Verlag, New York. Dehay, D., Leśkow, J. (1996), ’Functional Limit Theory for a Covariance Estimator’, Journal of Applied Probability, Vol 33, pp 1077 - 1092. Gamrowski, B. and Rachev, S. (1995), Testing the validity of Value-at-Risk Measures, Athens Conference on Applied Probability and Time Series Analysis, Lecture Notes in Statistics 114, Springer Verlag. JP Morgan and Co. (1996), RiskMetrics^{T M} Technical Document, Morgan Guaranty Trust Company, New York, December 17, 1996, available at http://www.riskmetrics.com. Mittnik, S., Paolella, M. and Rachev, S. (1998), Unconditional and Conditional Distributional Models for the Nikkei Index, Asia-Pacific Financial Markets 5,99-128,1998 preprint. Shumway, R (1988), Applied Statistical Time Series Analysis, rentice-Hall, New Jersey.