Definition Moving Average model order 1

An MA(1) is defined as: \[ y_t = \psi_0 +\psi_1 \ \epsilon_{t-1} +\epsilon_t \] with the following assumptions \(\forall t\):

  • \(\mathbb{E}[\epsilon_t]= 0\), \(\mathbb{V}[\epsilon_t]= \sigma_{\epsilon}^2\) and the errors \(\epsilon_{t_1}\) and \(\epsilon_{t_2}\) are independent \(\forall t_1\neq t_2\).
  • \(\mathbb{E}[y_t]= \mu_y=\psi_0\) (stationary in mean).
  • \(\mathbb{V}[y_t]= \sigma_y^2=\gamma_0=\mathbb{E}[ (\psi_1 \ \epsilon_{t-1} +\epsilon_t) (\psi_1 \ \epsilon_{t-1} +\epsilon_t) ]=(\psi_1^2+1)\ \sigma_{\epsilon}^2\) (stationary in variance).
  • The covariance between \(y_t\) and \(y_{t-k}\) is: \[ \mathbb{C}\text{ov}[y_t,y_{t-k}]=\gamma_k=\mathbb{E}[(y_t-\mu_y)(y_{t-k}-\mu_y)]=\mathbb{E}[(\psi_1 \ \epsilon_{t-1} +\epsilon_t) (\psi_1 \ \epsilon_{t-1-k} +\epsilon_{t-k})] \] hence \(\gamma_k=0\) for \(k>1\), and \(\gamma_1=\psi_1 \ \sigma_{\epsilon}^2\).

consequently the autocorrelation between \(y_t\) and \(y_{t-k}\) is \(ACF(0)=1\), \(ACF(1)=\frac{\psi_1}{\psi_1^2+1}\) and \(ACF(k)=0\) for \(k>1\).

ACF and PACF of MA(1)

In * arima.sim* the model simulated below corresponds to \[ y_{t}= \psi_1 \ \epsilon_{t-1} + \epsilon_{t} \] with \(\psi_1=0.8\).

require(fma)
myts<-arima.sim(n=100000,list(order = c(0,0,1),ma = c(0.8)), sd = sqrt(0.1796))

# amplitude of ACF(1) usin
acf1=0.8/(1+0.8^2)
acf1
[1] 0.4878049
tsdisplay(myts)

# fitting the model MA(1) imposing $\mu_y=0$ since arima.sim uses  $\mu_y=0$
Arima(myts,order=c(0,0,1),include.mean = FALSE)
Series: myts 
ARIMA(0,0,1) with zero mean 

Coefficients:
         ma1
      0.7971
s.e.  0.0019

sigma^2 estimated as 0.179:  log likelihood=-55875.16
AIC=111754.3   AICc=111754.3   BIC=111773.4
  • the ACF has one major peak at lag \(k=1\) and then all ACF for \(k>1\) are 0s.
  • the PACF shows damped sine wave pattern which can be explained when rewritting the MA(1) as an AR(\(\infty\)).

Read https://otexts.com/fpp2/MA.html about \(AR(p)\equiv MA(\infty)\) and \(MA(q)\equiv AR(\infty)\).