Loading [MathJax]/jax/output/HTML-CSS/jax.js
  • Modeling Procedure:
  • 1. Ex: Dow Jones 1972 with AR(3)
  • 2. Parameter Significance


Class Web Page



Modeling Procedure:

( Stationarity Check ) Does Xt 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 ϕ1,ϕ2,ϕ3? What is the accuracy of the estimation, and what does it say about the choice of p? Test for the parameter significance using the Standrad Error of the estimators. Can you reject he null hypothesis that ϕi=0?

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

1. Ex: Dow Jones 1972 with AR(3)

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

## [1] FALSE
## [1] 110.94 110.69 110.43 110.56 110.75 110.84
## [1] TRUE

## [1] -0.0022560132 -0.0023516653  0.0011765240  0.0017170489  0.0008123111
## [6] -0.0034342555

## Registered S3 method overwritten by 'quantmod':
##   method            from
##   as.zoo.data.frame zoo
## Series: X 
## ARIMA(1,0,0) with zero mean 
## 
## Coefficients:
##          ar1
##       0.5042
## s.e.  0.0996
## 
## sigma^2 estimated as 1.094e-05:  log likelihood=330.89
## AIC=-657.78   AICc=-657.62   BIC=-653.09


Our Model:

In this example, Yt was the original data: Yt=daily colse price of Dow Jones from 1972 Then the series was transformed to log-difference, Xt=ln(Yt)ln(Yt1=ln(Yt/Yt1) which represents apprximate percentage of daily gain/loss of any financial data.

To Xt, AR(1) was fit. That means Xt was modeled by the model: Xt=ϕ1Xt1+ϵt where ϵtWN(0,σ2). All parameter estimates, ˆϕ1,ˆσ, were calculated.

2. Parameter Significance

## Series: X 
## ARIMA(1,0,0) with zero mean 
## 
## Coefficients:
##          ar1
##       0.5042
## s.e.  0.0996
## 
## sigma^2 estimated as 1.094e-05:  log likelihood=330.89
## AIC=-657.78   AICc=-657.62   BIC=-653.09

- For large-sample MLE ˆϕ, parameter significance can be tested by calculating CI ˆϕ±2(Standard Error) and checking if 0 is inside the CI or not. (0 is inside = not significant).