IV. Statistical Inference for Mean Reversion of a Time Series
4.2 Hypothesis Testing about Mean Reversion
4.2 Hypothesis Testing about Mean Reversion
According to EWMA model with locally constant mean, we can calculate the estimate of the mean price for timen(µˆ ) by historical data (n X1,X2,L,Xn), and l-step-ahead forecast of mean from time original n by µˆn+l =µˆn. The forecasts are the same for all l. Therefore, the distribution of the priceXt which follows the O-U process with constant mean is
( ) ( )
and the log likelihood function is given by
( ) ∑ [ ( ) ( ) ]
2 . We need to maximize the equation (10) to obtain estimators
and test hypotheses. The Maximum Likelihood Estimators (MLE) of κand σ2 for the O-U process with constant mean are
( )
If the priceXt follows the O-U process with linear trend mean, the distribution is
( ) ( ) ( )
and the log likelihood function is given by linear trend mean are
( )
Figure 2 shows how the updated estimators for speeds of mean reversion change 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. We discuss three different time scales which are 6-month, 1- year, and 2-year respectively to estimate the speeds of mean reversion. The fact is that the less recent observations are considered which means the shorter time scale, the faster speed of mean reversion are reflected.
VIX_k_MLE
2-year 1-year1-year1-year1-year 6-month6-month6-month6-month
USB10Y_k_MLE
2-year 1-year1-year1-year1-year 6-month6-month6-month6-month
Figure 2. MLE for Speed of Mean Reversion: Estimators of the updated estimators for speeds of the
mean reversion with different time scales by using the VIX series (top), interest rate series of the United States Treasury Benchmark Bond 10 years (middle), and spread between Brent crude oil and
Spread of Crude Oil_k_MLE Spread of Crude Oil_k_MLE Spread of Crude Oil_k_MLE Spread of Crude Oil_k_MLE
0.0
2-year 1-year1-year1-year1-year 6-month6-month6-month6-month
West Texas Intermediate crude oil traded in NYMEX (bottom). The fact is that the speeds of mean reversion with shorter time scale are almost faster and than longer time scale. And the speed of mean reversion of VIX series and the spread of crude oil series are obviously faster than the interest rate series of the United States Treasury Benchmark Bond 10 years. In particular, the speeds are all zeroes which mean that there is no evidence for mean reversion since 1986 with shortest time scale.
We utilize a Chi-Square test for the presence/absence of a mean reversion effect.
The O-U process is no longer mean reverting whenκ =0. We wish to test precisely the null hypothesis of no mean reversion (Η0 :κ =0) against the alternative of mean reversion (Η1:κ >0).The LR statistic Λ
( )
X can be shown that freedom equal to the difference in dimensionality of Θ andΘ . Therefore, 0The log likelihood function for the O-U process with constant mean under the null hypothesis is
( ) ∑ ( )
The log likelihood function for the O-U process with linear trend mean under the null hypothesis is
(
log log)
~( )
12 L0 − L χ2
−
( ) ∑ ( )
VIX_Chi-Square Value
2-year 1-year1-year1-year1-year 6-month6-month6-month6-month 95% significant level95% significant level95% significant level95% significant level
VIX_speed
2-year 1-year1-year1-year1-year 6-month6-month6-month6-month
Figure 3. Chi-Square Test for VIX: Set the Chi-Square test for the presence of mean reversion effect
with different time scales by using VIX weekly data from 1990 to 2007. The results show that mean reversion phenomenon is significant but the speeds of mean reversion are slower with longer time scale.
And the mean reversion is not always significant, but recurring, and the speeds of mean reversion are also clearly faster with shorter time scale.
USB10Y_Chi-Square Value
2-year 1-year1-year1-year1-year 6-month6-month6-month6-month 95% significant level95% significant level95% significant level95% significant level
USB10Y_speed
2-year 1-year1-year1-year1-year 6-month6-month6-month6-month
Figure 4. Chi-Square Test for Interest Rate of U.S. Treasury Bond 10-Year: Set the Chi-Square test
for the presence of mean reversion effect with different time scales by using United States Treasury Benchmark Bond 10-year weekly data from 1984 to 2007. The results show that mean reversion is mostly higher significant but the speeds of mean reversion are slow, not exceeding 0.1, with longer time scale. On the contrast, the mean reversion phenomenon is mostly insignificant, but the speeds of mean reversion in 1986 is nearly eight times faster, about 0.8, with short time scale than which with longer time scale.
Spread of Crude Oil_Chi-Square_Value Spread of Crude Oil_Chi-Square_Value Spread of Crude Oil_Chi-Square_Value Spread of Crude Oil_Chi-Square_Value
0 00 0 3 33 3 6 66 6 9 99 9 12 12 12 12 1515 1515
May-99May-99May-99May-99 May-00May-00May-00May-00 May-01May-01May-01May-01 May-02May-02May-02May-02 May-03May-03May-03May-03 May-04May-04May-04May-04 May-05May-05May-05May-05 May-06May-06May-06May-06 May-07May-07May-07May-07
2-year 2-year 2-year
2-year 1-year1-year1-year1-year 6-month6-month6-month6-month 95% Significant Level95% Significant Level95% Significant Level95% Significant Level
Spread of Crude Oil_speed Spread of Crude Oil_speedSpread of Crude Oil_speed Spread of Crude Oil_speed
0 00 0 0.2 0.20.2 0.2 0.40.40.4 0.4 0.6 0.60.6 0.6 0.8 0.80.8 0.8 1 11 1
May-99May-99May-99May-99 May-00May-00May-00May-00 May-01May-01May-01May-01 May-02May-02May-02May-02 May-03May-03May-03May-03 May-04May-04May-04May-04 May-05May-05May-05May-05 May-06May-06May-06May-06 May-07May-07May-07May-07
2-year 2-year2-year
2-year 1-year1-year1-year1-year 6-month6-month6-month6-month
Figure 5. Chi-Square Test for Spread of Crude Oil between Brent and WTI: Set the Chi-Square
Test for the presence of mean reversion effect with different time scales by using weekly data of spread between Brent crude oil and West Texas Intermediate crude oil from 1997 to 2007. The results are similar to the series of VIX, which also show that mean reversion phenomenon is mostly significant but the speeds of mean reversion are slower and smoother with longer time scale. And the mean reversion phenomenon is not always significant, but recurring and the speeds of mean reversion are also faster with shorter time scale.