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A Solution Manual to

The Econometrics of Financial Markets

Petr Adamek John Y Campbell Andrew W Lo

A Craig MacKinlay Luis M ViceiraAuthor address:

MIT Sloan School, 50 Memorial Drive, Cambridge, MA 02142{1347

Department of Economics, Harvard University, Littauer Center,

Cam-bridge, MA 02138

MIT Sloan School, 50 Memorial Drive, Cambridge, MA 02142{1347

Wharton School, University of Pennsylvania, 3620 Locust Walk,

Philadel-phia, PA 19104{6367

Department of Economics, Harvard University, Littauer Center,

Cam-bridge, MA 02138

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List of Figures

3.3 Histogram of IBM Price Changes Falling on Odd or Even Eighth 15

12.1 Kernel Regression of IBM Returns on S&P 500 Returns 63

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List of Tables

2.3 Statistics for Daily and Monthly Simple and Continuously Compounded

3.4 Unconditional and Conditional Distributions of Bid/Ask Spreads 20

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PrefaceThe problems inThe Econometrics of Financial Marketshave been tested in PhD courses

at Harvard, MIT, Princeton, and Wharton over a number of years We are grateful to the

students in these courses who served as guinea pigs for early versions of these problems,

and to our teaching assistants who helped to prepare versions of the solutions We also

thank Leonid Kogan for assistance with some of the more challenging problems in Chapter

9

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Problems in Chapter 2

Solution 2.1 2.1.1Recall the martingale property given by (2.1.2) and observe that the mean-squared

error of the time-tforecastXt of pricePt +1is

(S2.1.2)

= E[0(Pt ; k ; l ;Pt ;2 k ; l)] = 0:

Solution 2.2Denote the martingale property (2.1.2) by M Then

nand2 n  jn j ;1=2 satis es RW3 but not M; (ii)n nn ;1 satis es M

but not RW2; (iii)n nnsatis es RW2 but not RW1

Solution 2.3

A necessary condition for the log-price processptin (2.2.9) to satisfy RW1 is+ = 1 Let

c+ and consider the set of all non-RW1 Markov processes (2.2.9), i.e.,c6= 1 The

restriction CJ = 1 is equivalent to =c=4 The constraints 0; 1 are satis ed

exactly forc2[1;4=3] and therefore the set of all two-state Markov chains represented by

the pair (; ) that cannot support any RW1 process but still yields CJ = 1 is simply

f(1 p

Thus, we have

Var[Zt(q)] = q;1

X

k =0Var[Zt ; k] + 2q;1

X

k =1(q;k)Cov[Zt;Zt ; k](S2.4.1)

which yields (2.4.19) The coecients of Cov[Zt;Zt ; k] are simply the number of k-th

order autocovariance terms in the variance of the multiperiod returnZt(q) (recall that this

multiperiod return is the sum ofq one-period returns) The coecients decline linearly

Ten individual stocksused for problem 2.5, identi aop=

1; ap Sincer =a, combining (S5.1.4) with (S5.1.8), (S5.1.15), and (S5.1.16) gives

0= 0 which completes the solution

Solution 5.2Begin with the excess return market model from (5.3.1) forNassets Taking unconditional

expectations of both sides and rearranging gives

= ; m:

(S5.2.1)

Given that the market portfolio is the tangency portfolio, from (5.2.28) we have the (N1)

weight vector of the market portfolio

Using! m we can calculate the (N1) vector of covariances of theN asset returns with

the market portfolio return, the expected excess return of the market, and the variance of

the market return,

SOLUTION 5.4 27

Solution 5.3The solution draws on the statistical analysis of Section 5.3 The calculations for three

selected stocks are left to the reader

Solution 5.4LetZ



t be a (N+11) vector of excess asset returns with mean

and covariance matrix

 Designate assetN+1 as the market portfoliom is full rank (If the

market portfolio is a combination of theN included assets, this assumption can be met

by eliminating one asset.)

From (5.2.28) the tangency portfolioqof theseN+ 1 assets has weight vector

can be partitioned in the rstNassets and the market portfolio,



 2 4

(S5.4.4)

 2 4

0;1

1

 2 m +

0;1

(S5.4.6)

;1 :

(S5.4.9)

From (S5.4.2) and (S5.4.9) we have

0;1 =2

the estimation greatly: no additional sampling theory is needed) Use the asymptotic

normality of your estimators to perform the. .. statistically signicantly, hence we cannot

reject the hypothesis of independence on these grounds On the other hand, there is a

statistically signicant discrepancy between these sample... us to test the proposed hypotheses

temporallyeach other That being the case, testing existence of unilateral causality may be

next to meaningless Therefore, let us test \causality"