This exercise will demonstrate through a simple example how

Question

Homework 3- Econometrics III - Spring 2007

1. [Comparing linear prediction and best prediction] Let X₁ and X₂ be independent random variables with P(X_i = -1) = P(X_i = +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 min_{β_0,β_1,β_2} E (Y - [β₀ + β₁X₁ + β₂X₂])².

2. Let X_t follow an invertible MA(1) process: X_t = αX_t-1 + u_t, where |α| < 1 and u_t ∼ N(0, σ²_u). Let v_t be another white noise process such that for all s, t, v_t is uncorrelated with us, and V ar(v_t) = σ²_v.

We will now see that Z_t:= X_t + v_t 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

ε_t := Z_t - θZ_t-1 + θ²Z_t-2 - θ³Z_t-3 + · · ·                                                           (1)

is a white noise process.

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

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

(c) Assuming Z_t 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 Z_t 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 Y_t = ε_t + θε_t-1 where ε_t is iid N(0, σ²).

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

(b) Show that the conditional distribution of Y_t given Y_t-1, . . . , Y₁, ε₀ is the same as the conditional distribution of Y_t given ε_t-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 Y_t|ε_t-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 Y_t = c +?₁Y_t-1 + ?₂Y_t-2 + ε_t where ε_t 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 EY_t, and the γ_j, j ∈ {. . . , -2, -1, 0, 1, 2, . . .}.

(c) Based on the sample Y₁, . . . , Y_T , let µˆ_T and γˆ_j,T be the population moments, j = 0, 1, 2, 3. Find E (µˆ_T), V ar (µˆ_T), E γˆ_j,T , and V ar γˆ_j,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 Y_t = a+b ·t+?₁Y_t-1 +· · ·+?_pY_t-p +ε_t, b 6 ≠ 0, ε_t a white noise process.

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

(b) Define X_t = ?Y_t := Y_t - Y_t-1. What kind of process is X_t? Is it weakly stationary? If ε_t is strictly stationary, is X_t also strictly stationary?

(c) Define Z_t = ?X_t = X_t - X_t-1. What kind of process is Z_t? Is it weakly stationary? If ε_t is strictly stationary, is Z_t also strictly stationary?

7. Suppose that Y_t = a + b · t + c · t² + ?₁Y_t-1 + · · · + ?_pY_t-p + ε_t, c ≠ 0, ε_t a white noise process.

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

(b) X_t = ?Y_t := Y_t - Y_t-1. Is X_t any kind of stationary? Explain.

(c) Define Z_t = ?X_t = X_t - X_t-1. What kind of process is Z_t? Is it weakly stationary? If ε_t is strictly stationary, is Z_t also strictly stationary?

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

X˜_t = (X_t-m + ... + X_t-1 + X_t + X_t+1 + ... + X_t+m)/(2m + 1).

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

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

(c) Compare the spectral density of X_t 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.