Structural Breaks
Structural breaks in time series occur when the underlying dynamics driving the stochastic process change significantly. For example, COVID-19 has had a significant impact on the financial markets causing big changes in ways of working, travel, shopping, et cetera. The pandemic has disrupted markets, such as oil significantly, as people were traveling much less due to restrictions. Such an event can be classified as a black swan event, where the underlying dynamics which drive the time-series changes so drastically, that it invalidates some or all history up until the event. When trading strategies are unable to detect such shifts in dynamics, it can lead to severe errors and ultimately heavy losses. It is therefore beneficial to use statistical tests to automatically detect structural breaks to preempt such events and act accordingly.
Structural break tests broadly consist of two categories:
  • CUSUM Tests: Checks whether the cumulative forecasting errors deviate significantly from white noise.
  • Explosiveness Tests: In addition to testing deviation of the errors from white noise, it also tests for exponential growth and collapse.
The Chu-Stinchcombe-White CUSUM test on levels was developed by Homm and Breitung (2012) to test departures from the log price
relative to the log price
Sn,t=(ytyn)(σt^tn)1N(0,1)σt^2=(t1)1i=2t(Δyi)2S_{n, t} = (y_t - y_n) (\hat{\sigma_t}\sqrt{t-n})^{-1} \sim \mathcal{N}(0,1) \\ \hat{\sigma_t}^2 = (t - 1)^{-1} \sum_{i=2}^{t}(\Delta y_i)^2
The test statistic
can be compared against the time dependent critical value
for a one sided test to test for significance
cα[n,t]=bα+log(tn)c_\alpha[n, t] = \sqrt{b_\alpha + \log(t -n)}
To prevent choosing the reference level
arbitrarily, we calculate the test static over all
n[1,t]n\in[1, t]
and take the supremum such that
St=supn[1,t]{Sn,t}S_t = \sup_{n \in [1, t]} \{S_{n,t}\}
Explosiveness Tests
The Chow-Type Dickey-Fuller (SDFC) test was developed by Gregory Chow (1960). Consider an autoregressive process
yt=ρyt1+ϵty_t = \rho y_{t-1} + \epsilon_t
is white noise. Under the null hypothesis
follows a random walk, i.e.
, while under the alternative hypothesis
starts off as a random walk, but after time
τT\tau^* T
, where
τ(0,1)\tau^*\in (0, 1)
the process changes into an explosive process.
H0:ρ=1H1:yt={yt1+ϵt,t=1,...,τTρyt1+ϵt,t=τT,...,T,ρ>1H_0: \rho =1\\ H_1: y_t = \begin{cases} & y_{t-1} +\epsilon_t, t=1, ... , \tau^* T \\ & \rho y_{t-1} + \epsilon_t, t = \tau^* T, ... , T, \rho > 1 \end{cases}
To calculate the test statistic, we fit the following specification with ordinary least squares
Δyt=δyt1Dt[τ]+ϵt\Delta y_t = \delta y_{t-1} D_t[\tau^*] + \epsilon_t
where the dummy variable
is 0 when
t<τTt < \tau^*T
and 1 when
, under the null hypothesis
H0:δ=0H_0: \delta = 0
and under the alternative hypothesis
H1:δ>1H_1: \delta > 1
, we define the following test statistic
DFCτ=δ^σ^δDFC_{\tau^*} = \dfrac{\hat{\delta}}{\hat{\sigma}_{\delta}}
In practice
is not known, we therefore try all possible
and take the maximum such that
SDFC=supτ[τ0,1τ0]{DFCτ}SDFC = \sup_{\tau^*\in[\tau_0, 1-\tau_0]} \{ DFC_{\tau^*}\}
where we take
sufficiently large, such that the specification always has sufficient data to fit with at the start and the end. The main drawback of this test is that a single bubble is assumed, while in reality there can be multiple.
The Supremum Augment Dickey-Fuller (SADF) test by Phillips, Wu and Yu (2011) tests for periodically collapsing bubble with the following specification
Δyt=α+βyt1+l=1LγlΔytl+ϵt\Delta y_t = \alpha + \beta y_{t-1} + \sum_{l=1}^L \gamma_l \Delta y_{t-l} + \epsilon_t
where we test under the null hypothesis
H0:β0H_0: \beta \leq 0
and under the alternative hypothesis
H1:β>0.H_1: \beta > 0.
For each point
t0[1,tτ]t_0 \in [1, t - \tau]
the above mentioned specification is fitted with backwards expanding start points and for each point we calculate
ADFt0,t=β^t0,tσβt0,t^ADF_{t_0, t} = \dfrac{\hat{\beta}_{t_0, t}}{\hat{\sigma_{\beta_{t_0, t}}}}
, we then define
SADFt=supt0[1,tτ]{ADFt0,t}SADF_t = \sup_{t_0 \in [1, t - \tau]}\{ ADF_{t_0, t}\}
is taken sufficiently large to ensure the specification at the end can be fitted with sufficient data.
  • Ulrich Homm, Jörg Breitung, Testing for Speculative Bubbles in Stock Markets: A Comparison of Alternative Methods, Journal of Financial Econometrics, Volume 10, Issue 1, Winter 2012, Pages 198–231, https://doi.org/10.1093/jjfinec/nbr009
  • Chow, Gregory C. "Tests of Equality Between Sets of Coefficients in Two Linear Regressions." Econometrica 28, no. 3 (1960): 591-605. Accessed March 15, 2021. doi:10.2307/1910133.
  • Phillips, Peter CB, Yangru Wu, and Jun Yu. "Explosive behavior in the 1990s Nasdaq: When did exuberance escalate asset values?." International economic review 52, no. 1 (2011): 201-226.
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