w5d: Fitting AR(p)

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Modeling Procedure:

( Stationarity Check ) Does \(X_t\) look stationary?

• perform stationarity transformation if it look stationary

( De-mean ) Is the mean of this process 0?

( Model Selection ) What model should we use? AR(p)? or something else?

( Order Selection ) How did we decide on the value of \(p\) to fit?

( Paramete Estimation ) What estimator was used for \(\phi_i\)? Can you reject he null hypothesis that \(\phi_i=0\)?

• Use \(\hat \phi \pm 2(S.E.)\) CI to check significance.

( Residual Analysis ) How was the residual calculated? What does it say about how the model fits the data?

1. De-mean or Not

Theoretically, AR(\(1\)) has mean of 0. \[ X_t = \phi X_{t-1} + \epsilon_t \hspace10mm \epsilon_t \sim WN(0,\sigma^2). \] If the observed series looks like AR(\(1\)) with constant mean, our model is \[ Y_t = \mu + X_t, \hspace10mm \mbox{ where } \mu \mbox{ is a constant and $X_t$ is AR(p).} \] That means, \(Y_t\) has a mean of \(\mu\), and we need to de-mean (de-trend) by letting \[ \hat X_t = Y_t - \hat \mu = Y_t - \bar Y. \] before modeling \(\hat X_t\) with zero-mean AR(\(1\)).

include.mean=TRUE is default in Arima() of forecast package.

If you use include.mean=FALSE option, that means our model is \[ Y_t = X_t. \] and \(Y_t\) is believed to have mean of \(0\). \(Y_t\) is directly being modeled with zero-mean AR(p) model.

2. Ex: Simulated Data

  set.seed(3563)
  X = arima.sim(list(ar = c(.6, .3) ), 100 )  + 8.5
  plot(X, type="o")

  mean(X)

## [1] 8.11923

  library(forecast)              # install.packages("forecast") if you haven't

  Fit01 = Arima(X, order=c(2,0,0))                             # de-mean

  Fit02 = Arima(X, order=c(2,0,0), include.mean=TRUE)          # de-mean

  Fit03 = Arima(X, order=c(2,0,0), include.mean=FALSE)         # don't de-mean

  Fit04 = Arima(X-mean(X), order=c(2,0,0), include.mean=FALSE) # de-mean by hand

Fit01 is fitting AR(2) with mean using MLE.

Fit03 is fitting AR(2) with zero-mean using MLE.

Fit04 is fitting AR(2) with zero-mean after you demean by hand.

- We usually de-mean, unless we have a reason to believe the true mean is equal to 0.

3. Ex: Dow Jones 1972

  # read in daily Dow Jones data from my website
  D  = read.csv("https://nmimoto.github.io/datasets/dowj.csv")
  head(D)                             # see only first 6 lines

##     dowj
## 1 110.94
## 2 110.69
## 3 110.43
## 4 110.56
## 5 110.75
## 6 110.84

  plot(D$dowj)

  is.ts(D)                            # is it TS object  -> no

## [1] FALSE

  Y = ts(D$dowj, start=c(1,1), freq=1)   # turn the data into TS object
  head(Y)

## Time Series:
## Start = 1 
## End = 6 
## Frequency = 1 
## [1] 110.94 110.69 110.43 110.56 110.75 110.84

  is.ts(Y)                           # D1 is TS object

## [1] TRUE

  plot(Y)

  X = diff(log(Y))                  # Take log-difference log(X_{t}) - log(X_{t-1})
  head(X)                             # see first 6 lines

## Time Series:
## Start = 2 
## End = 7 
## Frequency = 1 
## [1] -0.0022560132 -0.0023516653  0.0011765240  0.0017170489  0.0008123111
## [6] -0.0034342555

  plot(X, type='o', main="Log-diff of DOWJ")

  #--- Fit AR(3) ---
  library(forecast)                   # install.packages("forecast")
  Fit1 = Arima(X, order=c(3,0,0), include.mean=FALSE)     # fit AR(3) to Y
  Fit1                                # see the result of fit

## Series: X 
## ARIMA(3,0,0) with zero mean 
## 
## Coefficients:
##          ar1     ar2     ar3
##       0.4201  0.1127  0.0760
## s.e.  0.1147  0.1250  0.1156
## 
## sigma^2 estimated as 1.093e-05:  log likelihood=331.92
## AIC=-655.85   AICc=-655.29   BIC=-646.47

- When differencing is used, (true mean = 0) has a special meaning.