Parameter Estimation

Question 1

When we aim to estimate the parameters of our model (including sigma squared) what are we asusming

Answer 1

We assume we observe a trajectory from a causal and invertible gaussian arma p,q process where p and q are known

Question 2

Explain the idea of the method of moments

Answer 2

Construct a system of equations by equating population moments to sample moments and then solving for the parameters in terms of the sample moments

Question 3

What’s the significant of the yule walker equations

Answer 3

For a causal AR(p) process the yule walker estimators are asymptotically normal distributed and estimate for sigma squared is consistent for sigma squared. This means we can construct confidence intervals for the AR parameters

Question 4

How do we obtain the yule walker estimators?

Answer 4

By method of moments we replace gamma h with gamma h hat

Question 5

How do you obtain the standard errors for the parameter estimates

Answer 5

Find the diagronals of the variance covariance matrix of the estimates and get the square root.

Question 6

How is a confidence interval constructed?

Answer 6

Estimate+- quantile X Standard error

Question 7

What is the r command for finding the YWalker estimates?

Answer 7

ar.yw(x, order=)

Question 8

Describe the method to propose estimators for parameters and sigma squared in case of a MA(1) process

Answer 8

We have two parameters to estimate - two equations to solve so we use equations for gamma 1 and gamma 0
We get gamma 1 and gamma 0 in terms of the parameters to be estimated and then we use the method of moments: We replace gamma 0 and gamma 1 with estimates of gamma 0 and gamma 1 and solve for the estimated parameters in terms of gamma’s

Question 9

What is the liklihood function of a time series

Answer 9

The joint density function of X1, …, Xn

Question 10

How do we know AR(1) is markovian

Answer 10

By definition - defined on only the previous step

Question 11

Why is white noise being normal essential in defining the distribution of Xt given Xt-1

Answer 11

A linear combination of normal random variables is also a normal distribution

Question 12

How do we find the MLE estimates of mew phi and sigma for an AR(1) model

Answer 12

MLEs are found by maximising the log likelihood function

Question 13

How do we find the CMLE estimates of mew phi and sigma for an AR(1) model

Answer 13

The CMLEs are found by maximising the conditional log likelihood function

Question 14

Learn the CMLES for mew phi and sigma squared

Answer 14

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