Calificación de riesgos

The purpose of the AR(1) simulations presented in this section and the AR(2) and MA(1) simulations presented in Sections (3.1.5) and (3.1.6) respectively was to assess ART’s ability to correctly locate breaks in the presence of serial correlations.

There is a convention used in which causal AR models are often referred to as stationary. If an AR(p) model is non-stationary when regressed on pass values of a time series it can always be converted to a stationary model by instead regressing on future values. In practice time series analysts do not build AR models regressed on future values and the distinction between causal models, which are only regressed on past values, and stationary models is often neglected, generally with no serious consequences. Chan (2002, pp26-28) gives a short summary of the differences between causal and stationary models while Brockwell and Davis (2006, pp77-89) gives a more

elaborate description. Because it is common practice to use the term stationary when causal would be more correct, in what follows we will refer to causal AR models as stationary but the reader should be aware this terminology is somewhat loose.

0 200 400 600 800 1000 − 4 0246 (a) x y 0 200 400 600 800 1000 −5 0 5 10 (b) x y 0 200 400 600 800 1000 05 1 0 (c) x y

Figure 3.5: Three AR(1) series with φ = −0.65, break size 2 and (a) Gaussian, (b) geometric and (c) gamma noise structures.

An autoregressive process of order 1 (AR(1)),X_t, can be expressed as

X_t=φX_t−1+Z_t (3.1) whereφis the AR(1) parameter and Z_t is a random noise process.

Provided|φ|<1 the AR(1) process is stationary and invertible and we can write it as an infinite order MA process

X_t=

∞

X

i=0

0 0.5 1 1.5 2 −1 −0.5 0 0.5 1 0 5 10 15 20 Break Size AR(1) Parameter

Average Number of Breaks

Figure 3.6: Average total number of breaks reported by ART as a function of AR(1) parameter and break size with Gaussian noise and BIC tree selection.

From this it is a simple matter to show that the variance of the process is

σ2_X =σ_Z2 ∞ X i=0 φ2i = σ 2 Z 1−φ2 (3.2)

with the standard deviation obtained in the usual way. Thus for values of|φ|close to 1, the variance of the series will be much larger than the variance of the noise terms. This should be borne in mind when interpreting the performance of ART (or the BP) on series with serial correlations.

While it would have been a simple programming task to adjust the break size to correspond to the variance of the process rather than the variance of the noise terms, it was not clear that this would better reflect real time series. Instead, we chose to express all break sizes in units of the input noise terms.

All series were 1024 observations in length with a single break at the mid-point. The break was added after the full series was generated. We used three noise structures;

200 400 600 800 1000 10400 10800 11200

(a)

(b)

Figure 3.7: Residual sum of squares as a function of candidate breakpoint location for a series of length 1024 data points, break size two standard deviations, Gaussian noise and (a)φ=−0.95 and (b)φ= 0.95.

Gaussian, gamma and geometric. A plot of one realization of each of these three noise structures is presented in Figure (3.5).

Gaussian Noise

The Gaussian noise was drawn from an N(0,1) distribution. The results are presented in Figure (3.6).

When comparing Figure (3.6) with Figure (3.1) we should note the uncorrelated case is when the AR(1) parameterφ= 0 and that the regime lengths are 512 data points in Figure (3.6) and a maximum of 400 data points in Figure (3.1). For−0.5< φ <0 ART’s ability to locate a break was enhanced. When φ < −0.5 ART’s performance

−1.0 −0.5 0.0 0.5 1.0 2 4 6 8 10 12 14 AR(1) Parameter

Average Number of Breaks

Figure 3.8: Average total number of breaks reported by the BP as a function of AR(1) parameter with Gaussian noise and a break size of two standard deviations.

was worse at small break sizes. In the case φ < −0.5 the variance of the series was much larger than the variance of the noise terms. It can be shown (see Chatfield, 2004, p115) that the spectral density of an AR(1) process is given by

f(ω) = 1

π(1−2φcosω+φ2). (3.3) Thus for φ < 0 the power is concentrated in the high frequency regions. This means the series oscillates rapidly about the mean. This generates a clear minimum RSS as can be seen in panel (a) of Figure (3.7).

Figure (3.7) presents the RSS as a function of candidate break location as seen by ART when deciding where to split the root node. With φ =−0.95 the minimum was clearly at the location of the true break. It should be intuitively clear that after splitting the root node no further splits would be made in this series. This corresponds

to the large flat parameter region in the front of Figure (3.6). Part of the reduced effectiveness of ART in reporting breaks when −0.95 < φ <−0.5 could be attributed to the fact that the effect of a large value ofZ_t decayed only slowly with time.

Withφ= +0.95 (see panel (b) of Figure 3.7) the true breakpoint was not close to a local minimum. In this particular case the true break would probably not be reported. This corresponds to the parameter region at the back of Figure (3.6). It should be intuitively clear that after splitting the root node (here very close to data point 400) there would be a number of other places where ART would split the series and report candidate breaks in attempting to minimize the RSS.

Figure (3.8) shows the BP reported similar results to ART. Here the break size was two standard deviations in terms of the noise series. Once the AR(1) parameter, φ, exceeded about 0.25 the procedure began to report spurious breaks in rapidly increasing numbers. The comparable results in Figure (3.6) are the parameter region with break size two on the far right hand side of the graph.

The results for the gamma and geometrically distributed noise were nearly indis- tinguishable from the Gaussian noise and are not presented here for reasons of space.

Gamma Noise

In panel (c) of Figure (3.5) we have plotted a realization of an AR(1) process with gamma noise.

The choice of gamma parameters wasα= 2 andβ = 1. The gamma distribution is given by

f(x;α, β) = 1

βα_Γ(α)x

α−1_e−xβ _(3.4) with meanµ=αβ = 2 and varianceσ2 =αβ2 = 2.

We examined the break locations from the simulations for break size 0.4 and AR(1) parameter -0.3. This parameter combination was in the equivalent region for the gamma distribution simulations to the large flat area in the foreground of Figure (3.6). These results are presented as a histogram in Figure (3.9). ART correctly determined that the series had a single break. However, the distribution of breakpoints covered a wide

Breakpoint Frequency 300 400 500 600 700 0 100 200 300 400

Figure 3.9: The distribution of breaks reported by ART for AR(1) parameter = -0.3, break size 0.4 with gamma distributed noise whereα= 2, β= 1 and BIC tree selection.

range from less than data point 300 to higher than data point 700.

Geometric Noise

The choice of parameter for the geometric distribution was θ= 0.51. For a geometric distribution

g(x;θ) =θ(1−θ)x−1 forx= 1,2,3, . . . (3.5)

µ= 1

θ

1_{θis the standard notation for the parameter of the geometric distributions. It is also the standard} notation for the MA(q) parameters with subscripts ifq >1, and in the literature on long memoryθis often used to denote a vector of parameters to be estimated. It should be clear from the context which usage is being employed.

and

σ2 = 1−θ

θ .

Thus forθ = 0.5, µ= 2, and σ2 = 1. The results were almost indistinguishable from the gamma noise results and are omitted for reasons of space.