Estimate moving average models

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Reference no: EM131120008

Homework 3- Econometrics III - Spring 2007

1. [Comparing linear prediction and best prediction] Let X1 and X2 be independent random variables with P(Xi = -1) = P(Xi = +1) = ½, i = 1, 2 and define Y = X1 · X2.

(a) Give the distribution of Y.

(b) Give E (Y|X1), E (Y|X2), and E (Y|X1, X2).

(c) Solve the problem minβ_0,β_1,β_2 E (Y - [β0 + β1X1 + β2X2])2.

2. Let Xt follow an invertible MA(1) process: Xt = αXt-1 + ut, where |α| < 1 and ut ∼ N(0, σ2u). Let vt be another white noise process such that for all s, t, vt is uncorrelated with us, and V ar(vt) = σ2v.

We will now see that Zt:= Xt + vt can also be represented as an invertible MA(1) process. To do this, it is sufficient to show that there exists |θ| < 1 such that the process

εt := Zt - θZt-1 + θ2Zt-2 - θ3Zt-3 + · · ·                                                           (1)

is a white noise process.

(a) Why does the claim follow from eqn. (1)?

(b) Find the variance and autocovariances of Zt. Are the autocovariances consistent with the MA(1) pattern in general? Give the spectral density of Zt.

(c) Assuming Zt indeed has an MA(1) representation, set up a system of equations that θ would have to satisfy by equating variances and autocovariances from this representation with those under (b).

(d) Show that this system of equations has a solution 0 < |θ| < 1.

(e) Using the θ just described under (d), define Zt as in eqn. (1). Show that this is a white noise process.

3. This exercise will demonstrate through a simple example how to use maximum likelihood or method of moments to estimate moving average models. Consider the MA(1) model Yt = εt + θεt-1 where εt is iid N(0, σ2).

(a) Show that the conditional distribution of Yt given Yt-1, . . . , Y1, ε0 is the same as the conditional distribution of Yt given εt-1, . . . , ε0.

(b) Show that the conditional distribution of Yt given Yt-1, . . . , Y1, ε0 is the same as the conditional distribution of Yt given εt-1.

(c) Suppose we have three observations Y3, Y2 and Y1. Using the previous, Express the conditional density of (Y3, Y2, Y1) given ε0 in terms of the conditional densities Ytt-1, t = 3, 2, 1. Using the normality assumption, write down an explicit expression for this conditional density.

(d) Suppose ε0 = 0. Substitute out ε2 and ε1 in terms of Y2, Y1. Find the conditional MLE of θ if Y1 = -0.5, Y2 = 0, Y3 = -0.5.

(e) Using the same sample, calculate the method of moments estimator of θ by equating the population autocorrelations of the process with their sample counterparts.

4. [For both this problem and the next one, you can also simulate to get your answers. If you do this, be explicit about your simulation process.] Suppose that Yt = c +?1Yt-1 + ?2Yt-2 + εt where εt is a white noise process, (?1, ?2) = (0.5, 0.24), and t ∈ {. . . , -2, -1, 0, 1, 2, . . .}.

(a) Find the eigen-values of the associated F matrix and show that they are inside the unit circle.

 (b) Give EYt, and the γj, j ∈ {. . . , -2, -1, 0, 1, 2, . . .}.

(c) Based on the sample Y1, . . . , YT , let µˆT and γˆj,T be the population moments, j = 0, 1, 2, 3. Find E (µˆT), V ar (µˆT), E γˆj,T , and V ar γˆj,T.

5. One expects problems when the eigen-values get close to the edge of the unit circle. Repeat the previous problem for (?1, ?2) = (1.85, -0.9) and explain the source(s) of the difference(s).

6. Suppose that Yt = a+b ·t+?1Yt-1 +· · ·+?pYt-pt, b 6 ≠ 0, εt a white noise process.

(a) Is Yt any kind of stationary? Explain.

(b) Define Xt = ?Yt := Yt - Yt-1. What kind of process is Xt? Is it weakly stationary? If εt is strictly stationary, is Xt also strictly stationary?

(c) Define Zt = ?Xt = Xt - Xt-1. What kind of process is Zt? Is it weakly stationary? If εt is strictly stationary, is Zt also strictly stationary?

7. Suppose that Yt = a + b · t + c · t2 + ?1Yt-1 + · · · + ?pYt-p + εt, c ≠ 0, εt a white noise process.

(a) Is Yt any kind of stationary? Explain.

(b) Xt = ?Yt := Yt - Yt-1. Is Xt any kind of stationary? Explain.

(c) Define Zt = ?Xt = Xt - Xt-1. What kind of process is Zt? Is it weakly stationary? If εt is strictly stationary, is Zt also strictly stationary?

8. Volatile time series are often "smoothed out" by some sort of averaging. In particular, let Xt be a covariance stationary time series and let X˜t denote its smoothed version defined by an m-period centered moving average

t = (Xt-m + ... + Xt-1 + Xt + Xt+1 + ... + Xt+m)/(2m + 1).

(a) Let m = 1. Using lag operator notation, write down the linear filter that transforms Xt into X˜t.

(b) Find the filter function (i.e. the function by which the spectral density of Xt has to be multiplied to obtain the spectral density of X˜t).

(c) Compare the spectral density of Xt with the spectral density of X˜t. Which frequencies are missing from the spectrum of X˜t? Which frequencies are dampened down? Which are amplified?

(d) Let m = 2. Write down and graph the filter function (most easily done with a computer program). By averaging over more observations in the time domain, we should get a smoother series than before. Justify this claim in the frequency domain by comparing the graphs of the filter functions obtained for m = 1 and m = 2.

Reference no: EM131120008

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