Structural Breaks

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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.
CUSUM Tests
The Chu-Stinchcombe-White CUSUM test on levels was developed by Homm and Breitung (2012) to test departures from the log price yty_tyt​
relative to the log price yny_{n}yn​
where n>tn>tn>t
.
Sn,t=(yt−yn)(σt^t−n)−1∼N(0,1)σt^2=(t−1)−1∑i=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)^2Sn,t​=(yt​−yn​)(σt​^​t−n​)−1∼N(0,1)σt​^​2=(t−1)−1i=2∑t​(Δyi​)2
The test statistic Sn,tS_{n,t}Sn,t​
can be compared against the time dependent critical value cα[n,t]c_\alpha[n,t]cα​[n,t]
for a one sided test to test for significance
cα[n,t]=bα+log⁡(t−n)c_\alpha[n, t] = \sqrt{b_\alpha + \log(t -n)} cα​[n,t]=bα​+log(t−n)​
To prevent choosing the reference level yny_nyn​
arbitrarily, we calculate the test static over all n∈[1,t]n\in[1, t]n∈[1,t]
and take the supremum such that St=sup⁡n∈[1,t]{Sn,t}S_t = \sup_{n \in [1, t]} \{S_{n,t}\}St​=supn∈[1,t]​{Sn,t​}
​
Explosiveness Tests
The Chow-Type Dickey-Fuller (SDFC) test was developed by Gregory Chow (1960). Consider an autoregressive process
yt=ρyt−1+ϵty_t = \rho y_{t-1} + \epsilon_tyt​=ρyt−1​+ϵt​
where ϵt\epsilon_tϵt​
is white noise. Under the null hypothesis yty_tyt​
follows a random walk, i.e. ρ=1\rho=1ρ=1
, while under the alternative hypothesis yty_tyt​
starts off as a random walk, but after time τ∗T\tau^* Tτ∗T
, where τ∗∈(0,1)\tau^*\in (0, 1)τ∗∈(0,1)
the process changes into an explosive process.
H0:ρ=1H1:yt={yt−1+ϵt,t=1,...,τ∗Tρyt−1+ϵt,t=τ∗T,...,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}H0​:ρ=1H1​:yt​={​yt−1​+ϵt​,t=1,...,τ∗Tρyt−1​+ϵt​,t=τ∗T,...,T,ρ>1​
To calculate the test statistic, we fit the following specification with ordinary least squares
Δyt=δyt−1Dt[τ∗]+ϵt\Delta y_t = \delta y_{t-1} D_t[\tau^*] + \epsilon_tΔyt​=δyt−1​Dt​[τ∗]+ϵt​
 where the dummy variable Dt[τ∗]D_t[\tau^*]Dt​[τ∗]
is 0 when t<τ∗Tt < \tau^*Tt<τ∗T
and 1 when t≥τ∗Tt\ge\tau^*Tt≥τ∗T
, under the null hypothesis H0:δ=0H_0: \delta = 0H0​:δ=0
and under the alternative hypothesis H1:δ>1H_1: \delta > 1H1​:δ>1
, we define the following test statistic 
DFCτ∗=δ^σ^δDFC_{\tau^*} = \dfrac{\hat{\delta}}{\hat{\sigma}_{\delta}}DFCτ∗​=σ^δ​δ^​
In practice τ∗\tau^*τ∗
is not known, we therefore try all possible τ∗\tau^*τ∗
and take the maximum such that
​
SDFC=sup⁡τ∗∈[τ0,1−τ0]{DFCτ∗}SDFC = \sup_{\tau^*\in[\tau_0, 1-\tau_0]} \{ DFC_{\tau^*}\}SDFC=τ∗∈[τ0​,1−τ0​]sup​{DFCτ∗​}
where we take τ0\tau_0τ0​
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+βyt−1+∑l=1LγlΔyt−l+ϵt\Delta y_t = \alpha + \beta y_{t-1} + \sum_{l=1}^L \gamma_l \Delta y_{t-l} + \epsilon_tΔyt​=α+βyt−1​+l=1∑L​γl​Δyt−l​+ϵt​
where we test under the null hypothesis H0:β≤0H_0: \beta \leq 0H0​:β≤0
and under the alternative hypothesis H1:β>0.H_1: \beta > 0.H1​:β>0.
For each point t0∈[1,t−τ]t_0 \in [1, t - \tau]t0​∈[1,t−τ]
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}}}}ADFt0​,t​=σβt0​,t​​^​β^​t0​,t​​
, we then define
SADFt=sup⁡t0∈[1,t−τ]{ADFt0,t}SADF_t = \sup_{t_0 \in [1, t - \tau]}\{ ADF_{t_0, t}\}SADFt​=t0​∈[1,t−τ]sup​{ADFt0​,t​}
where τ\tauτ
is taken sufficiently large to ensure the specification at the end can be fitted with sufficient data.
References
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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