Accreditation Commission for Acupuncture and Oriental Medicine
Stat
MATH 522

Fall 2017
101 Formulaic Alphas
Zura Kakushadze1, Geoffrey Lauprete2 and Igor Tulchinsky3
Quantigic Solutions LLC,4 1127 High Ridge Road, #135, Stamford, CT 06905
Free University of Tbilisi, Business School & Sc
Accreditation Commission for Acupuncture and Oriental Medicine
DKFJAL
MATH 200

Spring 2009
Solutions to HW 4
Exercise 5.1
X t +1 = Z t +1 + Z t ,
X t + h = Z t + h + Z t + h 1
^ X t +1 = E ( X t +1  X t , X t 1 ,.) = Z t ^ X t + h = E ( X t + h  X t , X t 1 ,.) = 0, h 2
^ et +1 = X t +
Accreditation Commission for Acupuncture and Oriental Medicine
DKFJAL
MATH 200

Spring 2009
Solutions to HW 6
Exercise 6.1 (a)
X t = X t 1 + Z t , or (1  B ) X t = Z t ,
Then
2 2 ( B) = Z ( B ) ( B 1 ) = Z
i.e.,
( B) =
1 1 B
1 , (1  B)(1  B 1 )
f ( ) =
2 2 Z 1 1 = . e i = Z 2 2 (1 
Accreditation Commission for Acupuncture and Oriental Medicine
DKFJAL
MATH 200

Spring 2009
Solutions to HW 7
Exercise 12.1 VAR(1) model, X t = X t 1 + Z t , where = i.e.,
1 0.5 , 0.2 0.7
X t , 1 1 0.5 X t 1, 1 Z t , 1 . = + X t , 2 0.2 0.7 X t 1, 2 Z t , 2
The model is nonstationar
Accreditation Commission for Acupuncture and Oriental Medicine
DKFJAL
MATH 200

Spring 2009
Solutions to HW 8
Exercise 12.4
All pure MA processes, whether univariate or multivariate, are stationary. The model is invertible if all the roots of  ( B )  = I + B  = 0 are outside of the unit
Accreditation Commission for Acupuncture and Oriental Medicine
DJAOI
MATH 200

Spring 2009
Matlab Solutions Time Series (2DD23) Exercise 2.5
a.
12 X t
Week 2
= X t  X t12 = (a + bt + St + t )  (a + b(t  12) + St12 + t12 ) = a  a + bt  bt + 12b + St  St12 + t  t12 = 0 + 0 + 12b +
Accreditation Commission for Acupuncture and Oriental Medicine
DJAOI
MATH 200

Spring 2009
Matlab Solutions Time Series (2DD23) Exercise 3.1
The formula of the autocorrelationfunction is: (k) =
(k) . (0)
Week 3
(k) = Cov(X t , X t+k ) = Cov(Z t + 0.7Z t1  0.2Z t2 , Z t+k + 0.7Z t+k1 
Accreditation Commission for Acupuncture and Oriental Medicine
DJAOI
MATH 200

Spring 2009
Matlab Solutions Time Series (2DD23) Exercise 3.4
X t  = 0.7(X t1  ) + Z t = 0.7(0.7(X t2  ) + Z t1 ) + Z t = 0.7(0.7(0.7(X t3  ) + Z t2 ) + Z t1 ) + Z t = 0.73 (X t3  ) + 0.72 Z t2 + 0.7
Accreditation Commission for Acupuncture and Oriental Medicine
DJAOI
MATH 200

Spring 2009
Matlab Solutions Time Series (2DD23) Exercise 4.4
Consider the AR(2) process 1 2 X t1 + X t2 + Z t 3 9
Week 5
Xt =
In exercise 3.6 it is shown that the autocorrelationfunction is: 16 21 2 3
k
(k)
Accreditation Commission for Acupuncture and Oriental Medicine
DJAOI
MATH 200

Spring 2009
Matlab Solutions Time Series (2DD23) Solution to the Matlab Exercise
(D + C sin(2 t + 2 ) sin(1 t + 1 ) Use: 2 cos A cos B = cos(A + B) + cos(A  B) (see exercise 2.3, page 26. So
Week 8
(1)
1 1 1 1
Accreditation Commission for Acupuncture and Oriental Medicine
DJAOI
MATH 200

Spring 2009
Matlab Solutions Time Series (2DD23) Exercise 6.2b
X t = Z t + 0.5Z t1  0.3Z t2 The power spectral density function is: 1
Week 9
f () =
2
k=1
(k) cos k + (0) =
1
2
k=0
(k) cos k  (0)
The autoc
Accreditation Commission for Acupuncture and Oriental Medicine
DJAOI
MATH 200

Spring 2009
Matlab Solutions Time Series (2DD23) Exercise 8.1
X t = Z 1,t + 11 Z 1,t1 + 12 Z 2,t1 Yt = Z 2,t + 21 Z 1,t1 + 22 Z 2,t1
Week 10
X Y (0) = Cov(X t , Yt ) = Cov(Z 1,t + 11 Z 1,t1 + 12 Z 2,t1 , Z
Accreditation Commission for Acupuncture and Oriental Medicine
DJAOI
MATH 200

Spring 2009
Matlab Solutions Time Series (2DD23) Exercise 10.2
The special case of the linear growth model is described by the following equations: X t = t + n t t = t1 + t1 t = t1 + wt
Week 12
(1) (2) (3)
Ple
Accreditation Commission for Acupuncture and Oriental Medicine
DJAOI
MATH 200

Spring 2009
H1 A time series is a collection of observations cfw_xt made sequentially through time. When the variation of some quantity over a region of space is studied, t is a spatial variable.
Examples of tim
Accreditation Commission for Acupuncture and Oriental Medicine
DJAOI
MATH 200

Spring 2009
H2 HW1 (due Jan. 17). Exercise 3.1 from the textbook and
Problem 1
For the autocovariance function,
(t1 , t2 ) = E X t1 X t2  (t1 ) (t2 ) .
Problem 2 Problem 3
cfw_
(t1 , t2 ) = E X t1  (t1 ) X t2
Accreditation Commission for Acupuncture and Oriental Medicine
DJAOI
MATH 200

Spring 2009
H3
For MA(q) processes,
2 q h j =0 j j + h , 0 h q, (h) = h > q. 0,
qh j =0 j j + h , 0 h q, (h) = 1 + 2 + + 2 q 1 h > q. 0,
The ACF of a MA(q ) process "cuts off" after the point q. This is a be
Accreditation Commission for Acupuncture and Oriental Medicine
DJAOI
MATH 200

Spring 2009
H4 HW2 (due Jan. 24). Exercise 3.9 from the textbook and
Problem 1 Let
cfw_Zt ~ IID(0, 2 ) , and let c
be a constant. Consider the process
X t = Z1 cos(ct ) + Z 2 sin(ct ) .
Find the mean and autocov
Accreditation Commission for Acupuncture and Oriental Medicine
DJAOI
MATH 200

Spring 2009
H6 HW3 (due Jan. 31). Exercise 4.4 from the textbook and
Problem 1 Explore the correlation structure of the following models,
X t = .7 X t 1  .5 X t 2 + Zt , X t = Zt + .7 Zt 1  .3Zt 2 ,
by simu
Accreditation Commission for Acupuncture and Oriental Medicine
DJAOI
MATH 200

Spring 2009
H7
Given a set of observations cfw_ X 1 ,., X n from a stationary time series, the ACVF is estimated by the sample autocovariance function defined as
^ (h) =
1 nh ( X t +h X n )( X t  X n ) , n t
Accreditation Commission for Acupuncture and Oriental Medicine
DJAOI
MATH 200

Spring 2009
H8 HW4 (due Feb. 7). Exercises 5.1, 5.2 from the textbook and
Problem 1
^ ^ Suppose that in a sample of size 100, you obtain (1) = 0.432 and (2) = 0.145 . Assuming that the data were generated from an
Accreditation Commission for Acupuncture and Oriental Medicine
DJAOI
MATH 200

Spring 2009
H10
Fitting an appropriate
ARMA(p, q) model,
X t = 1 X t 1 + + p X t  p + Z t + 1Z t 1 + + q Z t q , cfw_Z t ~ WN (0, 2 ) ,
to an observed time series data set ( x1 , . , xn ) involves
determin
Accreditation Commission for Acupuncture and Oriental Medicine
DJAOI
MATH 200

Spring 2009
H9
cfw_ X t is called an autoregressive integrated moving average (ARIMA) process of order ( p, d , q ) , denoted as cfw_ X t ~ ARIMA( p, d , q ) , where d 1 is an integer,
if its dorder difference
Accreditation Commission for Acupuncture and Oriental Medicine
DJAOI
MATH 200

Spring 2009
H11 HW5 (due Feb. 28)
Problem 1
Consider the general ARIMA(1, 2, 1) model. (a) (b) Convert the model to the equivalent ARMA(3, 1) form.
^ Find the forecasts, X t + h , h 1 , for the model.
Problem 2
A
Accreditation Commission for Acupuncture and Oriental Medicine
DJAOI
MATH 200

Spring 2009
H12 Spectral Analysis 1. A function that satisfies the equation,
f ( x) = f ( x + kp) ,
is called periodic with period Virtually any periodic function
all x ,
k = 0, 1 , 2, . ,
p , if p is the smalles
Accreditation Commission for Acupuncture and Oriental Medicine
DJAOI
MATH 200

Spring 2009
H13 HW6 (due Mar. 6). Exercises 6.1, 6.2 from the textbook.
The spectral density function or simply the spectrum of a stationary time series,
f ( ) =
1 2
h =
h e i h ,
in frequency domain.
(1)
is t
Accreditation Commission for Acupuncture and Oriental Medicine
DJAOI
MATH 200

Spring 2009
H14
The secondorder properties of a time series are completely described by its ACVF or equivalently, under mild conditions (a sufficient condition is
h ,
h =
 h  < ), by its Fourier transform,
w
Accreditation Commission for Acupuncture and Oriental Medicine
DJAOI
MATH 200

Spring 2009
H15 HW7 (due Mar. 20). Exercises 12.1, 12.2 (a, d) from the textbook
In many cases, at each time t, several related quantities are observed and, therefore, we want to study these quantities simultaneo