Describe Model

Time Series Analysis in Python

Rob Reider

Adjunct Professor, NYU-Courant Consultant, Quantopian

Mathematical Description of MA(1) Model

$\large \quad \quad \quad \quad R_t \ = \ \ \mu \ \ + \ \ \epsilon_t \ + \ \ \theta \ \epsilon_{t-1} $

Since only one lagged error on right hand side, this is called:
- MA model of order 1, or
- MA(1) model
MA parameter is $\large \theta$
Stationary for all values of $\theta$

Interpretation of MA(1) Parameter

$\large \quad \quad \quad \quad R_t \ \ = \ \ \mu \ \ + \ \ \epsilon_t\ + \ \ \theta \ \epsilon_{t-1} $

Negative $\large \theta$: One-Period Mean Reversion
Positive $\large \theta$: One-Period Momentum
Note: One-period autocorrelation is $\large \theta/(1+\theta^2)$, not $\large \theta$

Comparison of MA(1) Autocorrelation Functions

$\large \theta=0.9$
$\large \theta=0.5$

$\large \theta=-0.9$
$\large \theta=-0.5$

Example of MA(1) Process: Intraday Stock Returns

Autocorrelation Function of Intraday Stock Returns

Higher Order MA Models

MA(1)

$\large \quad \quad R_t = \mu + \epsilon_t - \theta_1 \ \epsilon_{t-1}$

MA(2)

$\large \quad \quad R_t = \mu + \epsilon_t - \theta_1 \ \epsilon_{t-1} - \theta_2 \ \epsilon_{t-2}$

MA(3)

$\large \quad \quad R_t = \mu + \epsilon_t - \theta_1 \ \epsilon_{t-1} - \theta_2 \ \epsilon_{t-2} - \theta_3 \ \epsilon_{t-3}$

...

Simulating an MA Process

from statsmodels.tsa.arima_process import ArmaProcess
ar = np.array([1])
ma = np.array([1, 0.5])
AR_object = ArmaProcess(ar, ma)
simulated_data = AR_object.generate_sample(nsample=1000)
plt.plot(simulated_data)

Let's practice!

Time Series Analysis in Python

Preparing Video For Download...