Statistics 730   Time Series Analysis

Spring 2026MWF 9-9:50, Room TBA

Instructor: Eric Slud, Statistics Program, Math. Dept.    Office:  Mth 2314, x5-5469,     email slud@umd.edu

For a set of Sample Problems for the In-Class Test, click here.

Required Course Text: 
R. Shumway & D. Stoffer, Time Series Analysis and its Applications, 2006 2nd ed. or later, Springer.
(The 2006 Second Edition is free as an e-book to UMCP students through the library: see website with datasets and errata.)

Recommended text: H. Lutkepohl, New Introduction to Multiple Time Series Analysis, 2005, Springer. (Also free as e-book.)

Overview: This course covers the concepts and tools of statistical time series analysis, both from a mathematical and a data-analytic viewpoint. Course segments on mathematical tools will be interleaved with segments emphasizing model-building, statistical analysis (in R), and simulation. The course introduces methods both in the time and frequency domains. The mathematical theorems and proofs are an essential part of the course. Students will be required to make further mathematical arguments and extensions in graded homework problems, and understanding of the conditions under which the techniques are valid will be tested.

Prerequisite: Stat 700 plus a graduate course in mathematical analysis, plus some computing familiarity.

Course requirements and Grading: there will be 6 or 7 graded homework sets (one every 1½ to 2 weeks) which together will count 1/3 of the course grade. There will also be an in-class test and a final course project (or take-home test), each of which will count as 1/3 of the course grade.

NOTE ON USE OF THEORETICAL MATERIAL.  Both in homeworks and the in-class test, there will be theoretical material involving probability theory as needed to apply the law of large numbers and central limit theorem, along with the `delta method' (Taylor linearization), linear algebra and other manipulations at advanced-calculus level, in some cases verging on measure-theoretic probability techniques. (Look at Appendix A of the Shumway-Stoffer book for the flavor and level.). There will also be some use of Hilbert space methods. The theoretical material in the Shumway and Stoffer book is concentrated in the Appendices, but that material will be supplemented in class.

Course Coverage: Chapters 1-5 and Appendixes A, B, C Shumway-Stoffer, plus some material from Chapters 6-7 and from Lutkepohl Chapters 2--3 as time permits.

NOTE ON COMPUTING.  Both in the homework-sets and the course project, you will be required to do computations on real datasets using a statistical-computing platform such as R or SAS or MATLAB. The book and various class demonstrations and scripts on this web-page will be given in R, and that is the only software platform that I will use or provide help with. If you are learning one of these packages for the first time, I strongly recommend R, and I will provide links to free online materials introducing them. In addition, there is a concise introduction to R commands in time series analysis that you should consult. R is freely available in Unix or PC form through this link. For the systematic Introduction to R and R reference manual distributed with the R software, either download from the R website or simply invoke the command

> help.start()

from within R. For a quick start, see my own Rbasics handout originally intended for a Survival Analysis class, and then read more about R objects and syntax in the Venables and Ripley text, in my Stat 705 Lecture Notes, and in the R introduction manual distributed with the R software. A really useful short summary of a lot of R commands can be found here. See also the previously mentioned concise introduction to R commands in time series analysis .

R Logs

For R practice logs that will periodically illustrate R commands related to time series data and exercises, see RLogsS730 Directory.

Homework problems are from 2017 and will be updated/modified for 2026.

Assignment 1. (First 1½ weeks of course, HW with 7 Problems due Mon., Feb. 6).
          Read Chapter 1 through Sec. 1.5, plus Section A.1 (Appendix A).
          Solve problems #1.3, 1.8, 1.9, 1.13, 1.15, 1.16(b), plus one more problem, below.
In #1.3, you may use R commands as in Example 1.10 as the
problem suggests, or you may code the generation of the time-series variables directly.
Extra Problem, not in text: (i) Prove that if   X(t)  is a stochastic process with finite second moments
for integer indices t, and for each n ≥ 1,     X_n(t)   is a strictly stationary process in t,   also with finite
second moments, such that   E(X_n(t)-X(t))² → 0   as n → ∞,   then X(t) is strictly stationary.
(ii) Prove the same assertion as in (i) with "strictly stationary" replaced by "weakly stationary".

Assignment 2. (Second 1½ weeks of course, HW with 7 Problems due Wed., Feb. 15).
          Read Chapter 2 through Example 2.7, plus Section A.2 and B.1 (AppendicesA,B).
           Solve problems #2.2, 2.4, 2.5, 2.6, 2.8 (counts as 2 problems), plus one more problem, below.
Extra Problem, not in text: Suppose that w_t for all integer t is a (0,σ²) White Noise, and that
X_t = ∑_j= -2,2 w_t-j ,    Y_t=X_t-1+X_t+X_t+1. (i) Derive   γ_X(h)   and   γ_Y(h).   (ii) Express Y_t as a Moving
Average of w_t.   (iii) Prove that w_t cannot be expressed as a finite-order moving average of X_t.

Assignment 3. (Weeks 4-5 of course, HW with 7 Problems due Wed., Mar. 1).
          Read Chapter 3 through Section 3.4, plus Section B.2 and B.4 (Appendix B).
           Solve problems #3.2 (counts as 2 problems), 3.3, 3.6, 3.8, plus two more problems, below.
Extra Problems, not in text: (I). Suppose that (X_t, t=1,2,3,4)   are jointly Gaussian mean 0 with
4x4 covariance matrix Σ_j,k = r(j-k) where r(0)=2, r(1)=r(-1) = 1, r(2)=r(-2) = 0.5, r(3) = r(-3) = 0.
(a) Find the partial correlation of X₁ and X₃ (given X₂).
(b) Find the partial correlation of X₁ and X₄ (given X₂, X₃).
(II). Show that in order for the AR(2) with autoregressive polynomial φ(z) = 1 - c₁ z - c₂ z²
to be causal, the parameters (c₁, c₂) must lie in the region of pairs
such that c₁+c₂<1,   c₂-c₁ < 1,   and |c₂| < 1.
Are these conditions sufficient for causality ?

Assignment 4. (Weeks 6-7 of course, HW with 7 Problems due Fri., Mar. 17 6pm).
          Read Chapter 3, Sections 3.4 through 3.8.
           Solve problems 3.11, 3.12 (proof of P3.4 only), 3.15, 3.17, 3.23, 3.27, plus one more.
           Extra Problem Suppose that X, Y are scalar r.v.'s and Z a p-vector variable, and denote the
covariance matrix (assumed finite) of the p+2 dimensional random vector (X,Y,Z) as B. Show that
the partial correlation of X,Y given Z is 0 if and only if the (1,2) entry of B^-1 is 0.

Assignment 5. (Weeks 8-9 of course, HW with 7 Problems due Wed. April 5 in-class).
          Read Chapter 4, Sections 4.1-4.8.
           Solve problems 3.40 (Hint: project onto the space spanned by   {w_j-w₀ ,   j=1,...,n}.) ,
                plus 4.4, 4.5, 4.6, 4.10, 4.13, 4.20.

Assignment 6. (Weeks 10-11 of course, HW with 9 Problems due Mon. May 1).
          Read Chapter 4, Sections 4.5-4.8, 4.10, 4.11.
           Solve problems 4.8, 4.23, 4.25, 4.28 plus 5 more, immediately following.
(I).(Counts as 2 problems.) (a) Simulate a long (n ≥ 1000) time series with the stationary
ARMA(1,2) model   X_t - 0.3 X_t-1 = (1-0.5B)(1-0.2B)W_t ,   with   W_t   standard normal.
Verify that your estimates of the parameters   γ(0)   and   γ(1)   agree reasonably closely
with the theoretically correct values of these parameters.
       (b) Find an analytical expression for the spectral density of the   X_t   process, and plot it in
a suitably labeled graph.
       (c) Overplot on the same graph (with a different line-type or color) a smoothed periodogram
estimator (with no tapering) based on a Daniell kernel with L=21 points (each with weight 1/21).
       (d) Also overplot on the same graph (again with a different line-type or color another
smoothed periodogram estimator of the spectral density which gives greater weight to
periodogram ordinates near the center of the lag window consisting of 21 points, specifying
what kernel you used and how you implemented it in the software you used.
       (e) Make sure in your solution to part (d) that your scaling of the spectral density and
periodogram are such that the smoothed periodograms are reasonably close to the true spectral density.
(II). Simulate a long (n ≥ 1000) stationary time series with spectral density very close to
f(x) = (1 - (x/π)²)   for   -π < x ≤ π. You can find a two-sided MA process   ∑_{j: -a <j≤b} c_j W_t-j
with large positive a,b to accomplish this. Overplot a graph of this spectral density f with a
smoothed periodogram estimate of the spectral density to show that you did this correctly (and say
what lag-window smoother you used, and show the computer code that generated your picture).
(III).(Counts as 2 problems.) Simulate a pair of long, dependent, stationary time series X_t, Y_t
(t=1,...,n,   n ≥ 1000) with the model X_t - 0.9 X_t-1 = W_t   and Y_t = 0.5 X_t-3 + 0.5 V_t , where
W_t and V_t are independent white-noise sequences with Uniform[-1,1] distribution.
      (a) Find the theoretical form for the cross-covariance   γ_YX(h), and show that the form you
find is reproduced in a plot of the estimated cross-covariance from your simulated pair of
time-series.
      (b) Find the theoretical form of the cross-spectral density and coherence of X_t and Y_t.

Assignment 7. Applied Data Analysis HW set, will be due Friday, May 12.
Note that 2 problems have been deleted (because previously assigned) and one substituted:
just like number (II) from HW6 -- see HW6Notes in Rlogs for method.)
          Read Sections 2.3, 3.7-3.9, 4.10, 5.3, 5.5, 5.6, 6.1 and 6.2.
           Do problems Problems: 3.31, 3.32, plus 3 more, immediately following.
(A). Consider the SOI series, which we found to have several prominent autocorrelations at
lags k*12, filtered by the seasonal detrending operator 1-B¹².
(i) Show that this series has two spectral peaks, when the periodogram is only very slightly
smoothed. Do you think they are both real ? Try to smooth the periodogram with lag windows weighting more heavily toward the center of the window.
(ii) Follow the stepwise stochastic linear regression steps we previously used for the original SOI
series on this filtered series. Do you find that the residuals from your fitted models now pass the
Box test for model adequacy ?
(iii) If not, explain which lags in the residuals contributed most heavily to your Ljung-Box statistic.
(B). Using the smoothed bivariate periodogram    tmp = spec.pgram(SOI.Rec, kernel("daniell",4), taper=0)
as in the R Log TSAdataAnalysis.txt covered in class, find by inverse FFT the weights for the
optimal linear filter approximating Rec[t] by ∑_j b_j * SOI[t-j].
(C). Simulate a long (n ≥ 1000) stationary time series with spectral density very close to   f(x) = 1
for   -π/2 < x ≤ π/2   and   = 0   for   x ≤ -π/2 and x > π/2. You can find a one-sided MA process
 ∑_{j: 0 ≤j≤b} c_j W_t-j with large positive b to accomplish this. Overplot a graph of this spectral density f
with a smoothed periodogram estimate of the spectral density to show that you did this correctly (and
say what lag-window smoother you used, and show the computer code that generated your picture).

SYLLABUS for Stat 730

I. Definitions and Constructions of Time Series Models. (2 weeks, Ch. 1 & Appendix A)
A. White Noise AR, MA, Random Sinusoids
i. R basics and time series commands
B. Autocovariance and autocorrelation functions.
C. Strong and Weak Stationarity
D. Review: Multivariate Normal, Convergence of RVs and Distributions, & Limit Theorems (leading to Thm A.2).

II. Exploratory Data Analysis for Time Series. (2 weeks, Ch. 2)
A. Regression and ANOVA (Gaussian case)
i. Information Criteria and Model Building
ii. Differencing
B. Autocorrelation and Spectrum Estimation (Periodogram)
C. Kernel and Spline Smoothing

III. Autoregressive Integrated Moving Average (ARIMA) Models. (5 weeks, Ch. 3 & Appendices A,B)
A. Definitions, Relation to Difference Eq'ns
i. Autocorrelation and Partial Autocorrelation
ii. Prediction; Nonstationary Models
B. Estimation, Model-building
C. Decomposition into Signal, Noise, and Seasonal Components

IV. Spectral (Fourier) Analysis & Periodogram. (4 weeks, Ch. 4 & Appendix C)
A. Filtered Series, Periodogram & Discrete Fourier Transform
B. Nonparametric vs. Parametric Spectral Estimation
C. Fourier Analysis vs. Wavelets
D. Estimation, Prediction, & Filtering
E. Extensions to Multiple (Vector) Time Series

V. Miscellaneous Topics. (2-3 weeks, Ch. 5 & 6)
A. GARCH, Long-memory and ARMAX Models
B. State Space Models & Methods
C. Likelihoods in Time-Domain and Spectral Forms, Maximum Likelihood, Missing Data, Structural Models

Project Ideas -- for a list of Project paper Guidelines, click here.

Suggestions for ideas and papers which might be used as the basis for a final report or project will be added here from time to time. The Final Project will be due by 5pm Fri., May 19.

(1) Time series methods are sometimes used in connection with repeatedly collected survey data. Two technical reports that provide good exposition of how sample survey theory and time series ideas combine are Bell & Hillmer 1987 and Bell & Hillmer 1989, and there are many later references to sample-survey data with a history of using time-series methods, such as the Current Population Survey monthly employment numbers.