MATH/STAT 461
MIDTERM EXAM I
Fall, 2013
[Due by 11:59 PM, 23rd October, 2013, Wednesday]
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Applied Time Series Analysis
Soutir Bandyopadhyay
e-mail: sob210@lehigh.edu
Fall 2013
1
Time Domain Models
Box & Jenkins popularized an approach to time series analysis
based on
Auto-Regressive
Integrated
Moving Average
(ARIMA) models.
1
Autoregressiv
Seasonal ARIMA Models
Many time series collected on a monthly or quarterly basis
have seasonal behavior.
Similarly hourly data and daily behavior.
E.g. Johnson & Johnson quarterly earnings; discussion typically focuses on comparison with:
previous qua
The Integrated ARMA model: ARIMA(p, d, q )
Some series are nonstationary, but their dierences are stationary; e.g. the random walk.
Recall: the rst dierences of xt are
xt xt1 = (1 B )xt =
xt.
The second dierences are
xt
If
xt1 = (1 B ) xt =
2x .
t
d
The Frequency Domain
Time domain methods:
regress present on past;
capture dynamics in terms of velocity (rst order), acceleration (second order), inertia, etc.
Frequency domain methods:
regress present on periodic sines and cosines;
capture dynamic
The Periodogram
Recall: the discrete Fourier transform
n
d j = n1/2
xte2ij t,
j = 0, 1, . . . , n 1,
t=1
and the periodogram
I j = d j
2
, j = 0, 1, . . . , n 1,
where j is one of the Fourier frequencies
j
j = .
n
1
Sine and Cosine Transforms
For j =
The Spectral Density
The periodogram shows which frequencies are strong in a
nite sample x1; x2; : : : ; xn.
What about a population model for xt, such as a stationary
time series?
The spectral density plays the corresponding role.
1
The Discrete Fourier
Nonparametric Spectrum Estimates
Recall:
2
I !j
dc !j + ds !j
1 f !j =
1 f !j
2
2
2
approximately 2;
2
and I (!1), I (!2), . . . , I (!m) are approximately independent.
Problem: I !j is an approximately unbiased estimator of
f !j , but with too few degre
Tapering
The periodogram works well with data containing only Fourier
frequencies:
w = rnorm(128, sd = 0.01);
x5 = cos(2*pi*(5/128)*(1:128) + w;
x6 = cos(2*pi*(6/128)*(1:128) + w;
par(mfcol = c(3, 1), mar = c(2, 2, 1, 1);
spectrum(x5, taper = 0, ylim = c
Multiple Series
Jointly stationary series xt and yt have cross covariances
xy (h) = E xt+h x (yt y ) :
h
i
The cross spectral density is
fxy (!) =
and
xy (h) =
I
X
h=I
Z
1=2
1=2
xy (h)e2i!h;
fxy (!)e2i!hd!:
1
The cross spectral density is complex:
fxy
Linear Filters
A linear lter takes input
yt =
xt
into output yt:
I
X
r=I
ar xtr
for coecients ar ; I < r < I.
If the input is a unit impulse at t = 0,
8
<
1 t=0
0 t T= 0;
the output is yt = at, so the coecients are called the impulse
response function o
Comments on Choice of ARMA model
Keep it simple! Use small p and q .
Some systems have autoregressive-like structure.
E.g. rst order dynamics:
dx(t)
= x(t)
dt
or in stochastic form,
dx(t) = x(t)dt + dW (t)
where W (t) is a Wiener process, the continuou
Applied Time Series Analysis
Soutir Bandyopadhyay
e-mail: sob210@lehigh.edu
Fall 2013
1
Forecasting
General problem: predict xn+m given xn, xn1, . . . , x1.
General solution: the (conditional) distribution of xn+m given
xn, xn1, . . . , x1.
In particul
MATH/STAT 461
MIDTERM EXAM I (SOLUTION)
Fall, 2013
1. From the plots blow we see that the variation of the square root series is smaller and independent of time (See Figure 1).
2. (a) See Figure 2.
(b) The linear model t gives the following output.
Call:
Applied Time Series Analysis
Soutir Bandyopadhyay
e-mail: sob210@lehigh.edu
Fall 2013
1
Characteristics of Time Series
A time series is a collection of observations made at dierent
times on a given system.
For example:
Earnings per share of Johnson and
Applied Time Series Analysis
Soutir Bandyopadhyay
e-mail: sob210@lehigh.edu
Fall 2013
1
Estimating Means and Covariances
In other statistical applications, means, variances, and covariances are estimated by averaging across samples.
In time series, we o
Applied Time Series Analysis
Soutir Bandyopadhyay
e-mail: sob210@lehigh.edu
Fall 2013
1
Means
Recall: We model a time series as a collection of random
variables: x1, x2, x3, . . . , or more generally cfw_xt, t T .
The mean function is
x,t = E(xt) =
xft(
Applied Time Series Analysis
Soutir Bandyopadhyay
e-mail: sob210@lehigh.edu
Fall 2013
1
Time Series Regression
A regression model relates a response xt to inputs zt,1, zt,2, . . . , zt,q :
xt = 1zt,1 + 2zt,2 + + q zt,q + error.
Time domain modeling: the
Applied Time Series Analysis
Soutir Bandyopadhyay
e-mail: sob210@lehigh.edu
Fall 2013
1
Statistical Models
The primary objective of time series analysis is to develop mathematical models that provide plausible descriptions for sample data. . .
We model
Applied Time Series Analysis
Soutir Bandyopadhyay
e-mail: sob210@lehigh.edu
Fall 2013
1
Vector-Valued SeriesNotation
Studies of time series data often involve p > 1 series.
E.g. Southern Oscillation Index and recruitment in a sh
population (p = 2).
Tre
Applied Time Series Analysis
Soutir Bandyopadhyay
e-mail: sob210@lehigh.edu
Fall 2013
1
Exploratory Data Analysis (or Searching for Stationarity)
When an observed time series appears stationary, we can
calculate its sample autocorrelations, and use them
Applied Time Series Analysis
Soutir Bandyopadhyay
e-mail: sob210@lehigh.edu
Fall 2013
1
Moving Average Model
Moving average model of order q (MA(q ):
xt = wt + 1wt1 + 2wt2 + + q wtq
where:
1, 2, . . . , q are constants with q = 0;
2
wt is Gaussian whit
Lagged regression
The sheries recruitment series (yt) and the Southern Oscillation Index (xt) are cross-correlated with lags of several
months.
Perhaps we can model them as
yt =
r xtr + vt
r=
where vt is uncorrelated with xtr at all lags r. That is,
th