Stima e previsione del modello MA

Analisi delle serie temporali in R

David S. Matteson

Associate Professor at Cornell University

Tasso di inflazione USA mensile (in %, annualizzato)
Osservazioni mensili dal 1950 al 1990

data(Mishkin, package = "Ecdat")
inflation <- as.ts(Mishkin[, 1])

inflation_changes <- diff(inflation)

ts.plot(inflation) ; ts.plot(inflation_changes)

Processi MA: variazioni del tasso d’inflazione - II

Inflation_changes: variazioni del tasso d’inflazione mensile USA
Traccia la serie e la ACF campionaria:

ts.plot(inflation_changes)
acf(inflation_changes, lag.max = 24)

$Oggi = Media + Rumore + Pendenza * (Rumore di ieri)$ $$Y_t = \mu + \epsilon_t + \theta\epsilon_{t-1}$$ $$\epsilon_t ~ WhiteNoise(0, \sigma_{\epsilon}^2)$$

MA_inflation_changes <- arima(inflation_changes, 
                              order = c(0, 0, 1))

print(MA_inflation_changes)

Coefficients:
         ma1  intercept
      -0.7932    0.0010
s.e.   0.0355    0.0281
sigma^2 estimated as 8.882

ma1 = $\hat{\theta}$, intercept = $\hat{\mu}$, sigma^2 = $\hat{\sigma^2_{\epsilon}}$

Processi MA: valori adattati - I

Valori adattati MA:

$$\hat{Y_t} = \hat{\mu} +\hat{\theta}\hat{\epsilon_{t-1}}$$

Residui =

$$\hat{\epsilon_t} = Y_t - \hat{Y_t}$$

ts.plot(inflation_changes)
MA_inflation_changes_fitted <- 
    inflation_changes - residuals(MA_inflation_changes)

points(MA_inflation_changes_fitted, type = "l", 
       col = "red", lty = 2)

Previsioni

Previsioni a 1 passo:

predict(MA_inflation_changes)$pred

Jan
1991 4.831632

predict(MA_inflation_changes)$se

Jan
1991 2.980203

Previsioni (cont.)

Previsioni a h passi:

predict(MA_inflation_changes, n.ahead = 6)$pred

           Jan       Feb       Mar       Apr       May       Jun
1991  4.831632  0.001049  0.001049  0.001049  0.001049  0.001049

predict(MA_inflation_changes, n.ahead = 6)$se

           Jan       Feb       Mar       Apr       May       Jun
1991  2.980203  3.803826  3.803826  3.803826  3.803826  3.803826

Passons à la pratique !

Analisi delle serie temporali in R