# Chapter 7 - Jonathan Cryer

**Purpose**

To go over the examples in Jonathan Cryer Chan book on time series , specifically Chapter 7 on estimation

Simulate ar(1) process

> setwd("C:/Cauldron/garage/R/soulcraft/Volatility/Learn/JonathanCryer_Chan/") > x <- arima.sim(n = 200, list(ar = c(0.9))) |

Use Conditional sum of squares method.

> fit <- arima(x, order = c(1, 0, 0), method = c("CSS")) > print(fit) Call: arima(x = x, order = c(1, 0, 0), method = c("CSS")) Coefficients: ar1 intercept 0.8593 -0.5721 s.e. 0.0355 0.4637 sigma^2 estimated as 0.8486: log likelihood = -267.37, aic = NA |

Use Maximum Likelihood estimate.

> fit <- arima(x, order = c(1, 0, 0), method = c("ML")) > print(fit) Call: arima(x = x, order = c(1, 0, 0), method = c("ML")) Coefficients: ar1 intercept 0.8633 -0.4055 s.e. 0.0352 0.4637 sigma^2 estimated as 0.8527: log likelihood = -268.54, aic = 543.08 |

Use Conditional Sum of squares to find the initial values and then use MLE

> fit <- arima(x, order = c(1, 0, 0), method = c("CSS-ML")) > print(fit) Call: arima(x = x, order = c(1, 0, 0), method = c("CSS-ML")) Coefficients: ar1 intercept 0.8633 -0.4060 s.e. 0.0352 0.4636 sigma^2 estimated as 0.8527: log likelihood = -268.54, aic = 543.08 |

Using OLS

> ar.ols(x) Call: ar.ols(x = x) Coefficients: 1 0.8593 Intercept: -0.01460 (0.0653) Order selected 1 sigma^2 estimated as 0.8486 |

Using Yule Walker

> ar.yw(x) Call: ar.yw.default(x = x) Coefficients: 1 2 3 0.8583 0.1227 -0.1417 Order selected 3 sigma^2 estimated as 0.8774 |

Using MLE

> ar.mle(x) Call: ar.mle(x = x) Coefficients: 1 2 3 0.8672 0.1291 -0.1543 Order selected 3 sigma^2 estimated as 0.8324 |

Conditional MLE is far far easier to compute than Unconditional MLE Why ?

Unconditional MLE is essentially a non linear equation and hence one needs to use optimization tools to estimate the parameters

Conditional MLE on the other hand is a simple equation for which a simple differentiation would give the estimates.

Takeaway — Use conditional MLE in most of the cases…

method Fitting method: maximum likelihood or minimize conditional sum-of-squares. The default (unless there are missing values) is to use conditional-sum-of-squares to find starting values, then maximum likelihood. *'*' The above statements took me about an hour to understand. Even though I had read Hamilton just a month ago I forgot the crucial difference between Conditional Sum of Squares MLE and Unconditional Sum of Squares MLE

I had to go back to Hamilton to understand…

If you have an AR(p) process, you basically collect the first p terms, calculate the sigma, covariance matrix get the multivariate density for the first p terms, then use the multivariate density for the remaining terms and then maximize the likelihood function….. This is UNCONDITIONAL MLE

If you use the first p terms to compute the errors and then maximize MLE, then it is CONDITIONAL MLE and computationally easier…

If you have a very long time series, Conditional and Unconditional difference is hardly a matter of issue But for moderate series..you have to think abt it..