Skattning och prognos med MA-modell

Tidsserieanalys i R

David S. Matteson

Associate Professor at Cornell University

En månads amerikansk inflationstakt (i procent, årstakt)
Månadsdata från 1950 till 1990

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

inflation_changes <- diff(inflation)

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

MA-processer: förändringar i inflationstakten – II

Inflation_changes: förändringar i en månads amerikansk inflationstakt
Plotta serien och dess stickprovs-ACF:

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

$Today = Mean + Noise + Slope * (Yesterday's Noise)$ $$Y_t = \mu + \epsilon_t + \theta\epsilon_{t-1}$$ $$\epsilon_t ~ WhiteNoise(0, \sigma_{\epsilon}^2)$$

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 = $\hat{\sigma^2_{\epsilon}}$

MA-processer: anpassade värden – I

MA-anpassade värden:

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

Residualer =

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

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)

Prognoser

Prognoser ett steg framåt:

predict(MA_inflation_changes)$pred

Jan
1991 4.831632

predict(MA_inflation_changes)$se

Jan
1991 2.980203

Prognoser (forts.)

Prognoser h steg framåt:

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

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Tidsserieanalys i R