Empirical Methods in Finance

Homework 2

Problem 1: AR(p) Processes

3. Consider an AR(2) process with 1 = 1:1 and 2 = 0:25 (following the notation in

Lecture 5).

(a) Plot the autocorrelation function for this process for lags 0 through 20.

(b) Is the process stationary? Explain why or why not.

(c) Give the dynamic multiplier for a shock that occurred 6 periods ago. That is, calculate @[rt+6]

@”t

(following the notation in Lecture 5). This requires some algebra.

(d) Now, instead assume 1 = 0:9 and 2 = 0:8. Give the dynamic multiplier for a

shock that occurred 6 periods ago. Is the process stationary? Why/why not?

(e) Instead of analytically solving for dynamic multipliers, we can easily simulate a

full impulse response (that is, dynamic multipliers at all horizons). In particular,

consider a positive “t shock with magnitude one standard deviation. Assume the

standard deviation is 1 for simplicity. DeÖne xt rt as in class. Thus:

xt = 1:1xt1 0:25xt2 + “t

:

Set the initial values equal to the unconditional mean: xt1 = xt2 = 0. Set

all future shock equal to their expectations, “t+j = 0 for all j > 0. As stated

earlier, let “t = 1. Simulate xt+j for j = 0; :::; 60 given the above initial values

and sequence of shocks. Plot the resulting series from xt1 through xt+60. This is

the Impulse-Response plot for a one standard deviation positive shock to

Problem 2: Applying the Box-Jenkins methodology1

In PPIFGS.xls you will Önd quarterly data for the Producer Price Index. Our goal is

to develop a quarterly model for the PPI, so we can come up with forecasts. Our boss

needs forecasts of ináation, because she wants to hedge ináation exposure. There is not a

single ëcorrectíanswer to this problem. Well-trained econometricians can end up choosing

di§erent speciÖcations even though they are confronted with the same sample. However,

there deÖnitely are some wrong answers.

1. We look for a covariance-stationary version of this series. Using the entire sample,

make a graph with four subplots:

(a) Plot the PPI in levels.

(b) Plot P P I

(c) Plot log P P I

(d) Plot log P P I.

2. Which version of the series looks covariance-stationary to you and why? Letís call the

covariance stationary version yt = f(P P It).

3. Plot the ACF of yt for 12 quarters. What do you conclude? If the ACF converges very

slowly, re-think whether yt really is covariance stationary.

4. Plot the PACF of yt for 12 quarters. What do you conclude?

5. On the basis of the ACF and PACF, select two di§erent AR model speciÖcations that

seem the most reasonable to you. Explain why you chose these.

(a) Using the entire sample, estimate each one of these. Report the coe¢ cient estimates and standard errors. Check for stationarity of the parameter estimates.

(b) Plot the residuals. (Note: the residuals will have conditional heteroskedasticity

or ëGARCH e§ectsí. We will talk about this later. However, in well-speciÖed

models, the residuals should not be autocorrelated.)

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In Matlab, there is an Econometrics Toolbox and a series of functions : ëarima, estimate, forecast,

infer, simulate, lbqtestí that can help you solve this problem. Alternatively, you can download Kevin

Sheppardís MFE toolbox, which is freely available. You can just Google this and Önd it. In R there is a

package called íMTSí for Multivariate Time Series, by Ruey Tsay. This is a very useful package, that we

will also use when estimating time-varying volatility models.

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(c) Report the Q-statistic for the residuals for 8 and 12 quarters, as well as the AIC

and BIC. Select a preferred model on the basis of these diagnostics. Explain your

choice.

6. Re-estimate the two models using only data up to the end of 2005 and compute the

MSPE (mean squared prediction error) on the remainder of the sample for one-quarter

ahead forecasts:

1

H

X

H

t=1

v

2

t

where H is the length of the hold-out sample, and vi

is the one-step ahead prediction

error. Note: the one-step ahead prediction error means that while you Öx the parameters to those estimated up until 2005, you will use new data. So, for the and of Q2

forecast for Q3 2006 you use data up to end of Q2 2006. If you like, you can re-estimate

the model sequentially to have parameters estimated, continuing with this example, up

until end of Q2 2006. This mimicks more what you would do in a real-world situation.

Also, for comparison, report the MSPE assuming there is no predictability in yt

, i.e.

assuming yt follows a random walk with drift. What do you conclude?

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