Miscellaneous Topics for Final STAT 730 Lecture
=============================================== 5/8/2026

i. Smoothing of trend and better decomposition of detrended data 
   into seasonal and irregular components

## Following up on "trend" tmp2 found by simple moving-window filter in 
##  Seasonal Adjustment RLog.

plot(tmp2)            ## already very smooth
lines(m1$data[7:317,"trend"], col="blue")   ## the seasonally adjusted trend series
       ## had sharper blips at all by 1 stationary point

According to Wikipedia, the X-13 method is given by steps 1.-8. at 

https://en.wikipedia.org/wiki/X-13ARIMA-SEATS

We already tried the first two steps, a moving-average trend estimate; subtracted from the
original data. Next smooth the seasonal.

aux2 = c(unemp[1:6], unemp[7:317]-tmp2, unemp[318:323])
for(t in 25:299) aux2[t] = mean(aux2[c(t-24,t-12,t,t+12,t+24)])
unemp2 = unemp-aux2

plot(unemp)
lines(1990+(6:316)/12, unemp2[7:317], col="blue")

## Now find new trend for this series using "Henderson weights"
## a kind of smoothing weights ("Whittaker-Henderson", from package WH)
## using in actuarial and financial math.

unemp3 = WH(y=unemp2[7:317], wt=rep(1,311), criterion="GCV")$y_hat
plot(1990+(6:316)/12, unemp3, type="b")

aux3 = c(unemp[1:6], unemp[7:317]-unemp3, unemp[318:323])
for(t in 25:299) aux3[t] = mean(aux3[c(t-24,t-12,t,t+12,t+24)])
unemp3 = unemp[7:317]-aux3[7:317]

par(mfrow=c(3,1))
plot(unemp, main="Original Data")
plot(1990+(6:316)/12, unemp2[7:317], type="b", xlab="Time",    
       ylab="unemp",col="blue", main="First Trend Estimate")
plot(1990+(6:316)/12, unemp3, xlab="Time", ylab="unemp", 
       main="Second Trend Estimate", type="b")

plot(1990+(6:316)/12, unemp[7:317]-tmp2, xlab="Time", 
       ylab="seas", type="l", main="First Seasonal Estimate")
plot(1990+(6:316)/12, unemp[7:317]-unemp3, ylab="seas", 
       xlab="Time", type="l", main="Second Seasonal Estimate")
plot(m1$data[,"seasonal"], ylab="seas", main="Package Seasonal Estimate")

### Two pictures saved as UnempPic7.pdf and UnempPic8.pdf
### So there  are more and slightly different iterative steps to do ...

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ii. Stochastic regression of interventions, point occurrences, etc.
    Combined with Sec.5.7 Transfer Function Modeling

  Stochastic Regression for output series yt as linear in covariates z_t 
    and lagged input xt with independent ARMA(p,q) autocorrelated errors

      y_t = z_t'\beta + \sum_{j=0}^\infty \alpha_j x_{t-j} + \eta_t

  Can consider first term an ordinary regression, 2nd a "distributed lag"
     where  \alpha_j = effect at time j after point event can be modeled!
 
  Stochastic regressors  z_t  could be relevant covariates including
     residuals from another model !
  x_t could either be a modelled time series as in Sec.5.7 OR
     an indicator like I[intervention event at time t]

  "Intervention event" could be point occurrence 
     like hurricane or govt shutdown OR "trading day" effect 
     like Christmas or Easter, and can superpose such effects as necessary.

   "Transfer function modeling" looks at resultant in terms of Backshift
     series and autocorrelations ... in Sec.5.7

   Final proposal in Sec.5.7 is to minimize sum of squared innovations z_t 
    (associated with eta_t) in residuals from 
            y_t - z_t'\beta - \sum_{j=0}^\infty \alpha_j x_{t-j} 

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iii. ARMAX Sec.5.8

    Here there must be exogenous inputs like z_t that are known (can be
        vectors) but not modeled distributed lags. This setup falls within
        state-space model.

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iv. "Causality" -- directions of influence may not be unique when there 
        is missing data !