IV. Statistical Inference for Mean Reversion of a Time Series
4.1 Estimating the Level of Mean Reversion
− −
= log +1 is the speed
of mean reversion at time t.
It is easy to see that the definition is compliant with the concept of mean reversion by the stochastic differential equations. Also an AR(1) process with a negative coefficient of correlation satisfies this definition. But processes satisfying this definition do not necessarily converge in distribution or have a mean for long time.
By this definition, level and speed of mean reversion are in a sense confounding.
That is, each target of mean reversion µt corresponds to a different speed of mean reversionκt. But such characteristics enable dealing with non-stationary processes.
On analyzing time series with this approach, a class of µt can be taken into consideration as the candidates of the target of mean reversion, for example an EWMA (Exponentially Weighted Moving Average) predictor or some other nonlinear functions of observations.
4.1 Estimating the Level of Mean Reversion
Next, we discuss how the historical data can be used to estimate of the levels of mean reversion. In many instances, authors assume that the mean is constant over all time period. As a consequence, in the forecast computations each observation carries the same weight. However, the assumption of a time invariant is restrictive, and it would be more reasonable to allow for a mean that moves slowly over time.
Heuristically, in such a case it would be reasonable to give more weight to the most recent observations and less to the observations in the distant past. The exponentially weighted moving average (EWMA) model is a particular case of the model where the weights decrease exponentially as we move back through time. In this paper, we will use the O-U stochastic process as mean reverting model. Using weekly data of VIX (Volatility Index) traded in CBOE from 1990 to 2007, interest rate of the United States Treasury Benchmark Bond 10-year from 1984 to 2007, and spread between Brent crude oil and West Texas Intermediate (WTI) crude oil traded in NYMEX from 1997 to 2007 to investigate the phenomenon of mean reversion with different time scales.
Continuous time models are useful for the theoretical properties, but in reality the trajectories of the process cannot be observed continuously, and the process must be sampled in discrete time. For estimation and testing purposes, a discrete time analogue of the continuous time model is required. First, we consider the locally constant mean model:
n j n j
X + =µ ε+ +
where µ is a constant mean level and εn+j is a sequence of uncorrelated errors. If one chooses weights that decrease geometrically with the age of the observations, the forecast of the future observation at time n can be calculated from
µˆn =
(
1−ω)
Xn +ωµˆn−1 (6) whereXt is the asset price, µˆ is the estimate of the mean price for timen n, and the weightω is a constant between zero and one. Equation (6) shows how the forecast can be updated after a new observation has become available and expresses the new forecast as a combination of the old forecast and the most recent observation. The coefficientω depends on how fast the mean level changes. If ω is small, more weight is given to the last observation and the information from previous periods is heavily discounted. If ω is close to 1, a new observation will change the old forecast only very little.Through repeated application of Equation (6), we can see that µˆ is an n
exponentially weighted average of previous observations, which can be shown that
Second, we consider the locally linear trend mean model:
t squares leads to the following estimates at time 0:
used by Holt (1957) to introduce the updating equations. We assume a linear trend for the mean µt and write it in slightly different formµt =µn +
(
t−n)
β. Then the mean considers a linear combination of these estimates,( )
n(
n n)
n ω X ω µ β
µˆ +1 = 1− 1 +1+ 1 ˆ + ˆ
where α1 =1−ω1 is a smoothing constant that determines how quickly past information is discounted.
Similarly, information about slope comes from two sources: (1) from the difference of the mean estimators µˆn+1−µˆn, and (2) from the previous estimator of the slopeβˆn. Again these estimates are linearly weighted to give
( )(
n n)
nn ω µ µ ω β
βˆ 1 ˆ ˆ ˆ
2 1
2
1 = − + − +
+
Thus, the parameter estimates are updated according to
( ) ( )
( )(
n n)
n nn n n
n X
β ω µ µ ω β
β µ ω ω
µ
ˆ ˆ 1 ˆ
ˆ
ˆ ˆ ˆ 1
2 1
2 1
1 1 1 1
+
−
−
=
+ +
−
=
+ +
+
+ (9)
We define the two different smoothing coefficients asω1 =ω2 and ω2 =2ω
(
1+ω)
where the coefficientω depends on how fast the mean level changes.
Figure 1 shows how fast the mean level changes by using the weekly data of VIX, interest rate of the United States Treasury Benchmark Bond 10-year, and spread between Brent crude oil and West Texas Intermediate crude oil traded in NYMEX. If ω is small, more weight is given to the last observation and the mean level is closer to the new observation. On the other way, if ω is close to 1, a new observation will change the old forecast only very little and the mean level changes smoothly. Figure 1 also shows that the locally constant mean model would be appropriate method for the data of VIX and spread of crude oil, and the locally downward linear trend mean would be appropriate method for the data of the United States Treasury Benchmark Bond 10-year.
VIXVIXVIX
VIXVIX w=0.9w=0.9w=0.9w=0.9 w=0.7w=0.7w=0.7w=0.7 w=0.5w=0.5w=0.5w=0.5 w=0.3w=0.3w=0.3w=0.3 w=0.1w=0.1w=0.1w=0.1
USB10Y interest reatinterest reat
interest reat w=0.9w=0.9w=0.9w=0.9 w=0.7w=0.7w=0.7w=0.7 w=0.5w=0.5w=0.5w=0.5 w=0.3w=0.3w=0.3w=0.3 w=0.1w=0.1w=0.1w=0.1
Spread of Crude Oil Spread of Crude Oil Spread of Crude Oil Spread of Crude Oil
0000
Spread w=0.9w=0.9w=0.9w=0.9 w=0.7w=0.7w=0.7w=0.7 w=0.5w=0.5w=0.5w=0.5 w=0.3w=0.3w=0.3w=0.3 w=0.1w=0.1w=0.1w=0.1
Figure 1. Estimator for Level of Mean Reversion: Index series and mean level series with different weight of the VIX (top) and interest rate of the United States Treasury Benchmark Bond 10-year
(middle), and spread between Brent crude oil and West Texas Intermediate crude oil traded in NYMEX suggested by Brown (1962).