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L13 Cyclicity and Test for Stationarity

  • Cyclical variations
    • gradual, long-term, up-and-down irregular repetitive movements
    • rise and fall are not of fixed frequency
    • period usually extends beyond a single year
  • 6 phases (Business Cycle)
    • Expansion
    • Peak
    • Recession
    • Depression
    • Trough
    • Recovery

L13__Cyclicity and Test for Stationarity-1768972967531.webp

  • Examples
    • Business cycle
    • Price cycle : production decision
    • Solar cycle: Sun's magnetic field (every 11 years, Sun's north and south poles shift entirely)
Seasonality Cyclicality
Calendar Efffects Fluctuations not of fixed frequency
Average length is smaller
Magnitude of cycles are higher

Unit Roots

\[ Y_{t} - 1.9 Y_{t-1} + 0.9 Y_{t-2} = e_{t} - 0.5 e_{t-1} \]

can be REwritten as

\[ (1- 1.9B + 0.9B^{2}) Y_{t} \]

The roots of this equation are \(1\) and \(\dfrac{10}{9}\). Thus, this model has a unit (1) root.

Unit roots make the process non-stationary.

But why?

Difference once: \(W_{t} = \nabla Y_{t}\)

\[ w_{t} - 0.9 w_{t-1} = e_{t} - 0.5e_{t-1} \]

which is a stationary ARMA(1,1). Thus the original model is ARIMA(1,1,1) which is non-stationary.

Why is Unit Root a problem?

Unit roots imply that the model structure is \((1-B)Y_{t} = Y_{t} - Y_{t-1}= f(e_{t})\), implying that the current value is same as the past value plus some function of random errors. Thus, our model will never converge to a mean.

  • Causes of Non-stationary
    • Unit roots present
    • Deterministic polynomial trend present (\(+b\) exact structure can be determined)
    • Stochastic trend present (\(+bt\) exact structure cannot be determined, time dependent.)

Tests of Stationarity

Augmented Dicky Fuller (ADF) Test

Is a unit root present?

\[ \begin{matrix} H_{0}: & \text{series is non-stationary} \\ H_{\alpha}: & \text{series is stationary} \end{matrix} \]

Kwiatkowski-Phillips-Schmidt-Shin (KPSS) Test

Is there a deterministic trend (or mean)?

\[ \begin{matrix} H_{0}: & \text{series is stationary} \\ H_{\alpha}: & \text{series is non-stationary} \end{matrix} \]

Phillips-Perron (PP) Test

Goal and Hypothesis setup same as #Augmented Dicky Fuller (ADF) Test

  • Different assumptions about error terms
  • More robust in the presence of
    • autocorrelation
    • heteroskedasticity

Variance Ratio Test

Test for random walk.

\[ \begin{matrix} H_{0}: & \text{series is non-stationary} \\ H_{\alpha}: & \text{series is stationary} \end{matrix} \]
  • Ratio significantly different from \(1\) \(\implies\) Not a random walk