Differencing

$$ \text{Differenced series}:\Delta Y_t=Y_t-Y_{t-1} $$

• When fitting ARMA model, we want the data to be close to stationary
• Stationary means variance, auto-correlation,… will be constant over time
• Differencing will often make the time series stationary
• Log return in stock price series is level one differencing

ARIMA model

1. ARIMA(p,d,q) is just a model where we’ve differenced d times before applying ARMA(p,q)
2. ARIMA(0,1,0) is I(1) and this is a random walk

$$ \Delta Y_t = \varepsilon_t \\ Y_t-Y_{t-1}=\varepsilon_t \\ Y_t=Y_{t-1}+\varepsilon_t $$

1. Log return is level one difference and according to random walk, return can’t be predicted from previous values

   $$ R_t=P_t-P_{t-1}=\varepsilon_t \sim \N(\mu,\sigma^2) $$

2. Equation

   $$ Y_t=\beta_0+\beta_1Y_{t-1}+...+\beta_pY_{y-p}+\varepsilon_t+\theta_1\varepsilon_{t-1}+...+\theta_q\varepsilon_{t-q} $$

ARIMA - ADF (Augmented Dickey–Fuller) Test for stationary

1. First apply differencing (order d)
2. Then fit ARMA(p,q)

Code & Test

Data