A.     What is method of moments estimate of ö, the autoregressive parameter?

B.     What is method of moments estimate of è, the moving average parameter?

 

You can solve for ö easily. Solve for è with a quadratic equation, and choose the root whose absolute value is less than or equal to one.

 

These are method of moments estimates. We don’t know the actual values of ö and è. The method of moments is not the best estimator. We use maximum likelihood estimation if we have statistical software that gives the estimates. The method of moments shows the logic of the time series parameters and the sample autocorrelation function. It uses only pencil and paper, so it is good for homework assignments and final exam problems.

 

The final exam derives the parameters of AR(1), AR(2), MA(1), and ARMA(1,1) models by the method of moments. The ARMA(1,1) model takes a few minutes to solve the quadratic equation; the other models are easier. Review this homework assignment and the practice problems as preparation for the final exam.

 

 

are we going to solve the estimates of theta and phi or the actual values?
regards to the equation (7.1.6) on the book, is the theta missing "^"? as the next line said it is to solve for the theta ^

[NEAS: Correct, we are solving for the estimates of the time series parameters. The actual values are not known.]

ben, we are estimating.  Equation 7.1.6 says we use the phi-hats to solve 7.1.6 for theta-hat.  In the homework, we are given the lag 1 and 2 sample autocorrelations, r₁ and r₂, which, as you can see in 7.1.5 and 7.1.6 can easily be used to find phi-hat and theta-hat.  Best of luck!

RDH

thanks rdh
so the theta in the Equation 7.1.6 is theta-hat, and the book has a typo on this?
r-one = (1 minus theta-hat times phi-hat) (phi-hat minus theta-hat) divided by 1 minus 2 times theta-hat times phi-hat plus theta-hat square

ben, I wouldn't call it a typo, we're simply using that relation to estimate theta.  We use the values we know (or estimate) to find an estimation of theta.

RDH

NEAS:  How do we know which solution is invertible for an ARMA(1,1) model?  Page 151 states that a quadratic equation must be solved and only the invertible solution retained.  Unless I calculated the homework wrong, it only has one solution.  That said, I'm not sure how to proceed in cases with two solutions.

[NEAS: Generally, the solution with |theta| < 1.]

In general, quadratic equations have two solutions (think of the +/- before the square root term in the numerator).

Page 79 describes how to test for invertability, but I didn't fully understand it.  Since equation 7.1.4 tells you to use the positive sign before the square root term for the MA(1) case, I guessed that we should use that for the ARMA(1,1) case as well.

By the way, the 2 roots I got were 1.558 (using the positive sign before the sqrt) and 0.642 (using the negative sign before the sqrt).

[NEAS: Use the solution whose absolute value is less than one. The textbook gives the rules; this homework assignment is an example. The textbook says to solve the quadratic equation and choose the root that is less than one in absolute value.]

I agree with Michelle, and without giving my answer away, I got something for theta that wasn't less than 1, but at the same time, wasn't greater than 1.  And the part I'm agreeing with is that the two roots are equal.  Soooooooo.........

   -    Dave

Dave and Michelle, you are absolutely correct!  I got your answer also.

Did anyone actually find an invertible sol'n for theta?