An Alternative for Time Series Models

I. INTRODUCTION

In this work a pattern for construction of new stochastic models is proposed. The models modification and generalization of the classical time series frameworks for financial analysis (Tsay, 2005). As such they are considered a possible alternative to these known ones. They can be obtained by two different methods.

One of the methods employs triangular transformations (Filus & Filus& Arnold, 2010), as the defining tool and may therefore be more useful in a further statistical analysis and possible simulation studies. This method is described in Section 2 and 3. The other, described in Section 4, relies on application of the 'parameter dependence method' (Filus & Filus, 2012), (Filus & Filus, 2013), which is more flexible than the first method in the sense that it produces more models. The models obtained by either of the two methods are stochastic processes whose terms have financial meanings, especially the meaning of log returns for a single asset.

All the stochastic processes obtained by the triangular transformations method may also be obtained by the parameter dependence (not conversely), but the possibility of a nice statistical and simulation analysis as provided by the transformations is sometimes lost. This was the reason both methods were introduced. Any of the two is very general.

The patterns employed allow us to define wide classes of conditional probability distributions_of any term $X_t$, given realizations $x_1,\dots ,x_{t-1}$ of all past terms $X_1,\dots ,X_{t-1}$ of the defined stochastic processes.

Notice, that such conditional distributions are very seldom explicitly given in efficient forms in the literature. The classical exception lies within the pattern of the multivariate normal case. The obtained conditional distributions are then used for further construction of joint probability distributions of all the random vectors $(X_1,\dots ,X_t)$, $t=2,3,\dots$ if an initial distribution of $X_1$ is given.

Perhaps the most amazing fact that follows is the easy possibility of defining non-Markovian (as well as the Markovian) stochastic processes incorporating long pasts, and still analytically tractable. Additionally, the method of parameter dependence allows us to include into the model, typically occurring in practice, 'state random variables' that describe a "stochastic environment" in which the processes evolve over time.

The generality of these new models (from a financial perspective) inclined us rather to concentrate on the formulation of fundamental ideas as beginning to possibly new theories. Therefore, in order to avoid unnecessary dissipation, the number of examples was purposefully limited. Statistical analysis problems of the new stochastic models are only mentioned. Also references are limited, somewhat, especially because the results presented are possibly at first in a financial setting. However, somewhat similar, from a pure mathematical point of view, but generally different results were published in (Filus & Filus, 2008).

II. DEFINING TRANSFORMATIONS

Consider a sequence of log returns $R_t$ of a single asset, t = 0, 1,..., T; (Tsay, 2005), as given by the following sequence T = 1, 2,... of transformations:

T = 1, 2,..., (1) where the random variables $X_1$,..., $X_T$ are assumed to be independent and identically distributed. This is then a general white noise pattern which is a source of randomness for the considered log returns $R_1$,..., $R_T$,..., $V_0$ represents a nonnegative constant initial value, while the functions $V_1$,..., $V_T$ are arbitrary positive and piecewise continuous with respect to each argument. If the variance of each random variable $X_t$ is 1, then $V_1$,..., $V_T$ will have the "conditional volatilities" interpretation conditioned on realizations of past returns $R_0$, $R_1$,... prior to a given $R_t$. Also conditioned on the same realizations of the past returns are the conditional expectations

where $B_1$,..., $B_T$ are arbitrary, piecewise continuous with respect to each argument, functions of realizations of past returns.

Example. The functions $V_t()$ and $B_t()$ ( $t=1,\dots ,T$ ) may be arbitrary continuous. However, in practical applications one could choose, for example, the following simple functions: $V_t(r_0,r_1,\dots ,r_{t-1})=1+a_0{r}_0^2+a_1{r}_1^2+\dots +a_{t-1}{r}_{t-1}^2$, $B_t(r_0,r_1,\dots ,r_{t-1})=b_0{r}_0+b_1{r}_1+\dots +b_{t-1}{r}_{t-1}$, where the coefficients $a_0$, $a_1$,..., $a_{t-1}$ are real nonnegative and $b_0$, $b_1$,..., $b_{t-1}$ are arbitrary real. These coefficients are to be statistically estimated.

Also, if appropriate, one can choose as model:

with arbitrary real coefficients $a_0$, $a_1$,..., $a_{t-1}$ and $b_0$, $b_1$,..., $b_{t-1}$. Other examples of such functions can easily be given.

Returning to the main subject, notice that the sequence of the random vector transformations $(X_1,\dots ,X_T)$ $◻(R_1,\dots ,R_T)(T=1,2,\dots )$ defined by (1), is the pseudoaffine version of sequence of triangular transformations $R^T$ $◻R^T$, (Filus & Filus & Arnold, 2010). Here it is proposed to apply them as a general financial model_for values of log returns. This model can be seen as a slightly different version of time series, and is proposed to be named "triangular model". Realize that all the transformations (1) are easily invertible, and their inverses are given as follows:

T = 1, 2,... (1*) For realizations $x_1,\dots ,x_T$ and $r_0,r_1,\dots ,r_T$ of the underlying random variables, denoted by the corresponding capital letters, the jacobians, $J_{\mathrm{T}}({\mathrm{r}}_1,\dots ,{\mathrm{r}}_{\mathrm{T}})=\partial ({\mathrm{x}}_1,\dots ,{\mathrm{x}}_{\mathrm{T}})/\partial ({\mathrm{r}}_1,\dots ,{\mathrm{r}}_{\mathrm{T}})$, have the simple form of the inverse of the volatilities' products

One can see that if the sequence of probability densities (pdf) of the random vectors $({\mathrm{X}}_1,\dots ,{\mathrm{X}}_{\mathrm{T}})$ is known (which is mostly the case), then from $(1^{\ast })$ and (2) one immediately can derive the corresponding sequence of joint pdfs of the random vectors of the returns $({\mathrm{R}}_1,\dots ,{\mathrm{R}}_{\mathrm{T}})$, $T=1,2,\dots$.

In such a way, one defines a wide class of stochastic processes $\{{\mathbf{R}}_{\mathrm{T}}{\}}_{\mathrm{T}=1,2,\dots }$. (The Kolmogorov consistency theorem easily applies to this case.)

Consider these processes as "modified time series" processes for log returns $R_1$, $R_2$,.... Clearly, the model given by (1) is heteroscedastic as the underlying conditional volatilities, $V_t(r_0,r_1,\dots ,r_{t-1})$, t = 1, 2,... (conditioned on elementary events $R_0=V_0=r_0$, $R_1=r_1$,..., $R_{t-1}=r_{t-1}$ ), are, in general, distinct.

It follows from (1) that the introduced model is, in general, not Markovian but still analytically tractable.

Actually, when using model (1) one can incorporate in each conditional pdf $g_{\mathrm{T}}({\mathbf{r}}_{\mathrm{T}}\mid {\mathbf{r}}_1,\dots ,{\mathbf{r}}_{\mathrm{T}-1})$ (at present time T) all the past information on the returns, and underlying calculations are still performable.

However, this computational advantage is overshadowed by limitations of a statistical nature. As T grows, the number of parameters to be estimated also grows without bounds, so some restrictions on the past must be provided. For that one can apply the notion of k-Markovianity that limits the past to the last k observations (k = 1, 2,...). The case k = 1 means the ordinary Markovianity.

The general k-Markovian version of model (1) can be defined as the following sequence of transformations:

The k-Markovian conditional pdfs of R_{t}|R_{0}, R_{1}, \dots, R_{t-1} as derived from (3) are given by: g_{t}(r_{t}|r_{1}, \dots, r_{t-1}) if t-1 \leq k and g_{t}(r_{t}|r_{t-k}, \dots, r_{t-1}) if t-1 \geq k.

Thus, in this setting, the (conditional) distribution of the present asset log return $R_T$ only depends on the last k moments (months, years) in the past. The earlier times are considered irrelevant and are neglected.

Nevertheless, even in the case k = 2 (bi-Markovian) the amount of information incorporated in the stochastic model is significantly bigger than in the Markovian case, so one may expect more accurate predictions.

3. Examples

The following examples are based on (1) and (3).

Example 1. Assume that, for each T, the random variables $X_1,\dots ,X_T$ are independent, each having the standard normal $N(0,1)$ pdf.

Using standard calculations based on the knowledge of (1*) and (2) one first obtains the (unconditional) normal pdf $g_1(r_1)=N[B_1(R_0),V_1(R_0)]$ for $R_1$ and then for each t = 2, 3,..., T one obtains the conditional pdf: $g_t(r_t|r_1,ext...,r_{t-1})$

Realize that the latter conditional pdf is normal with respect to the single variable r_{t}. The joint probability density g_{T}($\mathbf{r}${1}, \ldots, $\mathbf{r}${T}) for each random vector (R_{1}, \ldots, R_{T}), T = 2, 3, \ldots, is given by the common formula: g_{T}($\mathbf{r}${1}, \ldots, $\mathbf{r}${T}) = g_{1}($\mathbf{r}${1}) \prod{t=1}^{T} g_{t}($\mathbf{r}${t} | $\mathbf{r}${1}, \ldots, $\mathbf{r}${t-1}), (5) where g{t}($\mathbf{r}${t} | $\mathbf{r}${1}, \ldots, ${\mathbf{r}}_{t-1}$) is given by (4).

The so obtained T-dimensional pdf is the FF-normal (former name "pseudonormal"), (Kotz & Balakrishnan & Johnson, 2000).

Example 2.

Consider the following "pseudolinear" part of the pseudoaffine transformation (1) which one obtains by setting in (1) all the "pseudotranslation" coefficients $B_t(R_0,R_1,\dots ,R_{t-1})$ to zero. One then has the pseudolinear transformations:

Investigate how the transformations (6) act on set of independent Pareto distributed random variables $X_t$ ( $t=1,2,\dots ,T;T=1,2,\dots$ ) so, in this case, the expected values of $X_t$ 's are positive.

Recall that the Pareto density is given by

$f_t(x_t)=1/\beta (1+x_t/\beta \gamma {)}^{1+\gamma },(7)$ where $\beta$ and $\gamma$ are positive real parameters.

Using (6), for every $t=1,\dots ,T$, express $x_t$ as

Also realize, that the jacobian of inverse to (6) equals to the inverse product: ${\mathrm{J}}_{\mathrm{T}}({\mathrm{r}}_0,{\mathrm{r}}_1,\dots ,{\mathrm{r}}_{\mathrm{T}-1})=[{\mathrm{V}}_1({\mathrm{r}}_0){\mathrm{V}}_2({\mathrm{r}}_0,{\mathrm{r}}_1)\dots {\mathrm{V}}_{\mathrm{T}}({\mathrm{r}}_0,{\mathrm{r}}_1,\dots ,{\mathrm{r}}_{\mathrm{T}-1}){]}^{-1}$.

As the next step, one obtains (for each t = 1, 2, \ldots, T) the conditional pdfs g_{t}(r_{t} | r_{0}, r_{1}, \ldots, r_{t-1}) of each rv R_{t}, given the past realizations r_{0}, r_{1}, \ldots, r_{t-1} of the rvs R_{0}, R_{1}, \ldots, R_{t-1}, as follows:

So, the effect of each $t^{th}$ line in transformation (6) on the rv $X_t$ is to change its Pareto density (7) for the (conditional) Pareto density (8) of $R_t$.

The two Pareto densities (7) and (8) only differ by the scale parameters, namely: $\beta$ in (7) was transformed into the product $\beta \mid V_{t-1}(r_1,r_2,\dots ,r_{t-1})\mid$ in (8).

Given the conditional densities (8) one obtains the joint density of each random vector $({\mathbf{R}}_0,{\mathbf{R}}_1,\dots ,{\mathbf{R}}_{\mathrm{T}})$, T = 1, 2,... using formula (5). In such a way the "Pareto stochastic process" $\{R_T{\}}_{T=1,2,\dots }$ is well defined.

Example 3.

In the same way as for the independently Pareto distributed random variables ${\mathrm{X}}_1,\dots ,{\mathrm{X}}_{\mathrm{T}},(\mathrm{T}=1,2,\dots )$, one can apply transformation (6) to any sequence of independent identically, and exponentially distributed random variables that will be denoted by the same symbols ${\mathrm{X}}_t$ 's.

If, for any $t=1,2,\dots$, $g_t(x_t)$ is the exponential density of $X_t$ given by the expression $(1/\theta )\exp [-x_t/\theta ]$ then it can easily be verified that the corresponding conditional density of $R_t|R_0,R_1,\dots ,R_{t-1}$ will be given as follows: $h_t(r_t|r_0,r_1,\dots ,r_{t-1})=(1/\theta |V_{t-1}(r_0,r_1,\dots ,r_{t-1})|)\exp [-r_t/\theta |V_{t-1}(r_0,r_1,\dots ,r_{t-1})|]$. It is then clear that as in Example 2, the parameters $\theta$ is multiplied by the "coefficient" $|V_{t-1}(r_0,r_1,\dots ,r_{t-1})|$.

The same actually will happen with the parameter $\sigma$ in Example 1, if one would assume all the random variables $X_t$ in (1) are normal $N(0,\sigma )$. Also in this case, the parameter $\sigma$ will be turned to the conditional volatility of $R_t$: $\sigma \mid V_{t-1}(r_0,r_1,\dots ,r_{t-1})\mid$.

This regularity for the parameter transformations will be applied in the next section.

III. PARAMETER DEPENDENCE MODELS

4.1 In all three examples in the previous section transformation (1) or (6) were used in order to obtain the conditional densities, say, \varphi_{t}(r_{t}|r_{0},r_{1},\ldots,r_{t-1}) describing the stochastic dependence of the return R_{t} on the past.

Realize that in this derivation the underlying operations only result in changing the value of a parameter of the given density of $X_t$, into other value that depends on the past return values $r_0,r_1,\dots ,r_{t-1}$.

This observation opens the way for the method of conditioning (on values $r_0$, $r_1$,..., $r_{t-1}$ ), which is significantly more efficient than the method of triangular transformations (1) or (6). This method, called the "parameter dependence", is presented in (Filus & Filus, 2012) and (Filus & Filus, 2013).

In the considered framework one can describe this method as follows.

Suppose there is given a sequence of independent random variables (now, instead of $X_t$, denoted by $R_t$ t = 1, 2,...) all having the same arbitrary probability density $f_t(r_t;\alpha )$, $\alpha \in A$. In this situation any past in this artificial "no memory process" has no influence on the current density $f_t(r_t;\alpha )$ of $R_{ft}$.

The density depends on a constant (original) scalar or vector parameter \alpha. Instead of applying transformation (1) or (6) to the random vectors (R_{f_{1}}, \ldots, R_{f_{T}}) one can $\text{directlytransform}$ each density f_{t}(r_{t}; \alpha) into a conditional density \varphi_{t}(r_{t} | r_{0}, r_{1}, \ldots, r_{t-1}) of R_{t} | r_{0}, r_{1}, \ldots, r_{t-1} just by setting the parameter \alpha of f_{t}(r_{t}; \alpha) to \text{\textquotedbl{}become\textquotedbl{}} a function of the values r_{0}, r_{1}, \ldots, r_{t-1}.

In such a way one defines the sequence of conditional pdfs by the formula: ${\phi }_t(r_t|r_0,r_1,\dots ,r_{t-1})=f_t(r_t;{\alpha }_t(r_0,r_1,\dots ,r_{t-1}))$, t=1,2,... (9) which, for an arbitrary function ${\alpha }_t(r_0,r_1,\dots ,r_{t-1})$, defines a legitimate density with respect to $r_t$ if all the values ${\alpha }_t(r_0,r_1,\dots ,r_{t-1})$ still belong to the set A of the parameters $\alpha$ of $f_t(r_t;\alpha )$. Each sequence of the so obtained conditional densities $\{{\phi }_t(r_t|r_0,r_1,\dots ,r_{t-1}){\}}_{t=1,2,\dots }$ defines a corresponding stochastic process $\{R_t{\}}_{t=1,2,\dots }$.

The parameter dependence method allows for relatively free choice for the functions ${\alpha }_t(r_0,r_1,\dots ,r_{t-1})$ and therefore the class of the so obtained stochastic processes is much wider than that obtained by the triangular transformation from the same sequence of independent random variables $X_t$ or $R_{ft}$. On the other hand, the factor that, in applications, often may limit the range of choices of the functions ${\alpha }_t(r_0,r_1,\dots ,r_{t-1})$ is reality.