Stationarity

Time Series Analysis in Python

Rob Reider

Adjunct Professor, NYU-Courant Consultant, Quantopian

What is Stationarity?

Strong stationarity: entire distribution of data is time-invariant
Weak stationarity: mean, variance and autocorrelation are time-invariant (i.e., for autocorrelation, corr$\large (X_t, X_{t-\tau})$ is only a function of $\large \tau$)

Why Do We Care?

If parameters vary with time, too many parameters to estimate
Can only estimate a parsimonious model with a few parameters

Examples of Nonstationary Series

Random Walk

Examples of Nonstationary Series

Seasonality in series

Examples of Nonstationary Series

Change in Mean or Standard Deviation over time

Transforming Nonstationary Series Into Stationary Series

Random Walk
```
plot.plot(SPY)
```

First difference
```
plot.plot(SPY.diff())
```

Transforming Nonstationary Series Into Stationary Series

Seasonality
```
plot.plot(HRB)
```

Seasonal difference
```
plot.plot(HRB.diff(4))
```

Transforming Nonstationary Series Into Stationary Series

AMZN Quarterly Revenues
```
plt.plot(AMZN)
```

# Log of AMZN Revenues
plt.plot(np.log(AMZN))

# Log, then seasonal difference
plt.plot(np.log(AMZN).diff(4))

Let's practice!

Time Series Analysis in Python

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