Financial Time Series Analysis
Jiajing Sun
University of Chinese Academy of Sciences

22 September 2013

J. Sun (UCAS)

ARMA(1)

22 September 2013

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Review

As pointed out in the last week, the objective of studying …nancial time series analysis is to predict future values of some …nancial variable. This objective is achieved by following methodology based on:
1 2 3 4

Model Speci…cation; Model Estimation; Model Validation; Forecasting (or prediction)

Note: The model to be used for forecasting is in fact estimated from history, i.e. data from the past is used to make probabilistic statements about future values.

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Stationarity

Historical values are used to identify the dynamic properties of the time series (relationship between current and past values) and these are used to forecast the future. A critical assumption behind the methodology that we develop …rst in this course is that such dynamics do not change over time. In a heuristic sense, we require the future to be like the past in a probabilistic sense. This requirement is formalized by the concept of stationarity.

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Stationarity
Strict Stationarity

A time series Y is strictly stationary if its probabilistic distribution does not change over time. That is, for any k, the joint distribution of (Yt , Yt 1 , ..., Yt k +1 ) does not depend on t. In other words, the joint distribution of any adjacent set of k observations (k 1) is not a¤ected by an arbitrary shift along the time axis. In practice, such a property is di¢ cult to check empirically, so we use a (generally) weaker condition.

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Stationarity
Weak Stationarity

1

Usually, it is su¢ cient to require the mean, variance and covariances of a time series to be independent of time, rather than the entire distribution. This is referred to as weak stationarity or covariance stationarity. Formally, a process fYt g is said to be weakly stationary if, for all t, it holds that E (Yt ) = µ < ∞ Var (Yt ) = E (Yt µ )2 = γ0 < ∞ (1) (2)

2

Cov (Yt , Yt

k)

= E (Yt

µ) (Yt

k

µ) = γk < ∞, k = 1, 2, 3, ...
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(3)
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J. Sun (UCAS)

ARMA(1)

Stationarity

Equation (3) de…nes the autocovariance of process Y at lag k. Henceforth term ’ stationary’will be taken to mean ’ weakly stationary’ . Equation (1) and (2) require the process to have a constant, …nite mean and variance. Equation (3) states that autocovariance depend only on the distance in time between the two observations.

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Autocorrelation Function (ACF)

Autocovariances are not independent of the units in which the variables are measured, it is common to standardize by de…ning autocorrelations ρk as ρk = Evidently, ρ0 = 1, which γ Cov (Yt , Yt k ) = k Var (Yt ) γ0 1 ρk 1. (4)

The autocorrelations considered as a function of k are referred to as the autocorrelation function acf (or ACF).

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Autocorrelation Function (ACF)

The ACF is also called the correlogram of the series Yt and plays a major role in modelling the dependencies among observations. This is because it characterizes the process describing the evolution of Yt over time. From the ACF we can infer how one value of the process is correlated with previous values and it indicates for how long and how strongly a shock εt a¤ects the value of Yt . For weakly stationary series the ACF is even, i.e. ρ course, γ k = γk ). k = ρk (and, of

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Example Sample Paths and Correlograms

Figure: First order autoregressive processes: data series and autocorrelation functions

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Example Sample Paths and Correlograms

Figure: First order moving average processes: data series and autocorrelation functions

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General ARMA processes
Formulating ARMA Processes

ARMA stands for AutoRegressive Moving Average. We shall deal with demeaned processes yt = Yt simple. µ to keep algebra

The AR(1) and MA(1) models encountered are special cases of the more general ARMA(1,1) model yt = θ 1 yt
1

+ α1 εt

1

+ εt .

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General ARMA processes
Formulating ARMA Processes

An autoregressive (AR) process of order p,an AR(p) process, is given by yt = θ 1 yt 1 + θ 2 yt 2 + + θ p yt p + εt . It is an AR(1) model generalized to include higher order lags. An moving average (AR) process of order q,an MA(1) process, is de…ned as yt = εt + α1 εt 1 + + αq εt q . It is a weighted combination of q + 1 white noise terms.

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General ARMA processes
Formulating ARMA Processes

Obviously, it is possible to combine the autoregressive and moving average speci…cation into an ARMA(p, q) model, which consists of an AR part of order p and an MA part of order q. yt = θ 1 yt
1

+ θ 2 yt

2

+

+ θ p yt

p

+ εt + α1 εt

1

+

+ αq εt

q.

In fact, there is no fundamental di¤erence between moving average and autoregressive processes. An AR model can be written as an MA model and vice versa, under suitable conditions, as we shall see from following slides.

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The Lag Operator
The lag operator, denoted by L (some authors use B, backshift operator). It is de…ned by Lyt = yt
1.

The operator obeys many of the standard rules of algebra, e.g. L2 yt = L (Lyt ) = yt 2 , or more generally, Lp yt = yt p with L0 1 and L 1 yt = yt +1 .Operating L on a constant leaves the constant una¤ected, e.g. Lµ = µ. Most of the time, the lag operator can be manipulated just as if it were a constant. For instance, L2 yt = L (Lyt ) = yt
2.

It is a linear operator and is well suited for use with the linear time series model introduced.
J. Sun (UCAS) ARMA(1) 22 September 2013 14 / 23

The Lag Operator
Using this lag operator allows us to write ARMA models in a concise way, for instance an AR(1) model can be written as yt = (1 + θL)εt , or

(1

θL)yt =

j =0

∑ θ j Lj .

∞

This can be extended to AR(p) and MA(q) models, i.e.

(1 and θ1 L

θ 1 L)yt = θ (L) yt = εt

yt = ( 1 + α 1 L +

+ αq Lq ) εt = α (L) εt ,

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Invertibility of MA processes
Consider the MA(1) case and ask if the polynomial α (L) can be inverted, i.e. does a polynomial α 1 (L) exist such that α 1 (L) α (L) = 1?If it does, the MA(1) model is said to be invertible. We can investigate this by matching coe¢ cients. Let the coe¢ cients of the inverse polynomial α 1 (L) (if it exists) be labelled fαi g. Then we require 1 + α1 L + α2 L2 + (1 + αL) = 1. Thus, matching coe¢ cients of L1 we need α1 = α, matching coe¢ cients of L2 requires that α2 + α1 α = 0, or α2 = α. This implied that in…nite order lag polynomial in brackets is convergent i¤ jαj < 1; this is the invertibility condition for an MA(1) model.

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Invertibility of AR processes
Similarly, for an AR(1) case, i.e. (1 θ 1 (L) θ (L) = 1. We …nd that θL) yt = εt , it is invertible if
∞

(1

θL)

1

=

j =0

∑ θ j Lj

(5)

provided that jθ j < 1. This is similar to the result that the in…nite sum ∑j∞ 0 θ j equals = 1 (1 θL) if jθ j < 1, while it does not converge for jθ j 1. With (5) we can write the AR(1) model as yt =

j =0

∑ θ j Lj εt =

∞

j =0

∑ θ j εt

∞

j,

i.e. an AR(1) process can be written as an in…nite-order moving average process, provided jθ j < 1. There is a connection between AR and MA process!
J. Sun (UCAS) ARMA(1) 22 September 2013 17 / 23

Invertibility of Lag Polynomial

As seen from the previous slide, the …rst order lag polynomial (1 θL) is invertible, if jθ j < 1. What happens if the order is higher? Here we shall generalize this condition to higher order lag polynomials. Consider a second order polynomial, given by 1 θ 1 L θ 2 L2 , which can be factored into (1 φ1 L) (1 φ2 L) . φ1 and φ2 can be solved for from φ1 + φ2 = θ 1 and φ1 φ2 = θ 2 . Thus the conditions for invertibility of the second order polynomial are just the conditions that both the …rst order polynomials (1 φ1 L) and (1 φ2 L) are invertible, which translate into jφ1 j < 1 and jφ2 j < 1.

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Invertibility of Lag Polynomial

These requirements can also be formulated in terms of the so-called characteristic equation (1 φ1 z ) (1 φ2 z ) = 0. The two solutions of this equation are referred to as the characteristic roots. The requirement jφi j < 1 corresponds to jzi j > 1.If jzi j 1 the corresponding polynomial is non-invertible. A solution that equals unity is referred to as a unit root. The presence of a unit root in the lag polynomial θ (L) can be detected relatively easily, without solving the characteristic equation, speci…cally, the presence of a …rst unit root can be veri…ed by checking whether the sum of the polynomial coe¢ cients equals one, p ∑j =0 θ j = 1.

J. Sun (UCAS)

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Invertibility of Lag Polynomial

Example: yt = 1.2yt This can be written into (1
1

0.32yt

2

+ εt .

(6)

0.8L)(1

0.4L)yt = εt .

The characteristic equation is 1 1.2z + 0.32z 2 = (1 0.8z ) (1

0.4z ) .

The solutions (characteristic roots) are 1/0.8 and 1/0.4, which are both larger than one. Consequently, the AR polynomial in (6) is invertible.

J. Sun (UCAS)

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Common Roots

Decomposing the moving average and autoregressive polynomials into products of linear functions in L also shows the problem of common roots or cancelling roots. This means that the AR and the MA part of the model have a root that is identical and the corresponding linear functions in L cancel out. In the case of one cancelling root, an ARMA(p, q ) model can be written equivalently as an ARMA(p 1, q 1) model.

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ARMA(1)

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Common Roots
Consider the model yt = yt which can be written as
1

0.25yt

2

+ εt

0.5εt

1,

(1

0.5L) (1

0.5L) yt = (1

0.5L) εt .

Clearly, this can be reduced to an AR(1) model as

(1

0.5L) yt = εt .

The problem of common roots illustrates why it may be problematic, in practice, to estimate an ARMA model with an AR and an MA part of a high order. The reason is that identi…cation and estimation are hard if roots of the MA and AR polynomial are almost identical. In this case, a simpli…ed ARMA(p 1, q 1) model will yield an almost equivalent representation.
J. Sun (UCAS) ARMA(1) 22 September 2013 22 / 23

Generalizations
1

It can be shown that there is no fundamental di¤erence between moving average and autoregressive process. As we have seen, under suitable conditions an MA model can be inverted into an AR model, and vice versa. While the constant in the MA representation corresponds to the mean of the process, the constant in the AR representation is a function of the mean (and of the dependence parameter θ). The order of these inverse polynomial in 1. or 2. is in…nite in theory and usually quite long for satisfactory modeling of data in practice. The choice of an MA or AR or a combined ARMA representation is sometimes a matter of convenience. We shall see later that a MA representation is convenient to determine the mean, variances and covariances, whereas an AR process is often preferable for estimation and prediction.
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