FORECASTING HOMEWORK 3
1) Consider the MA (1) model xt = t + t 1, where cfw_t is zero mean white noise.
A) Use the formula 1 = Corr (xt , xt 1) =
var t
to show that 1 =
.
var xt
1 + 2
B) Using the result from Part A), determine the maximum possible valu
FORECASTING HOMEWORK 1
All data sets are on the course website, http:/www.stern.nyu.edu/churvich (click on "Forecasting"). All recent data sets are available in Minitab Portable (.MTP) format, and most are also available
in EXCEL (.XLS or .CSV). The le Re
Chapter 1: Basic Concepts of Forecasting
Types of forecasts
In time series forecasting, we seek to make statements about xn +h , the value the series will take at
the future time period n +h . The quantity h is called the lead time . Forecasts may be clas
Chapter 2: Trend-Line Fitting and Forecasting
A time series that appears to contain a smoothly increasing (or decreasing) component is said to
contain a trend term , C (t ). Examples include population, prices, global temperature (upward trends),
and the
S. Wu
Shannon Wu SSW311
Project 1 Due Nov. 11
The data set that I chose to explore is the unemployment rate of Australia. I
got the data from two different sources, both listed below, in order to get
both historical data and the most recent data that I co
2.
Housing Starts
The model for HousingStarts is
identified as ARIMA (2,0,0). Because from the above graphs, we can
see that ACF dies down while PACF has two spikes in lag 1 and 2.
LogGDEP
The above shows that LogGDP
is not stationary. Therefore, it is ne
Shannon Wu SSW311
HW 9
1.
The data seems to be stationary with no obvious signs of trend or seasonality. The correlogram
of the ACF dies down and the PACF spikes at one, making the AR(1) model seem reasonable.
2.
= ( 1)/s = (0.8577 1)/ 0.0451 = -3.155
At
Shannon Wu SSW311
Forecasting HW 8
1.
The series doesnt appear to be stationary from the ACF and PACF
graphs of the logged data.
The graphs of the differenced log values dont spike on the ACF and
PACF graphs, so this leads me to believe that this is an AR
Shannon Wu SSW311
Worked with Lucheng Li
HW 2
1. A)
1. B)
Based on these two graphs, I would say that Yesterdays Russel is a better
predictor than 0.5*Yesterdays Russel because it follows Todays Russel
more closely.
1. C)
Using the formula
(ActualForecast
Shannon Wu Ssw311
Project 2
Data set was chosen as the price of Chevrons stock from 20 years ago to today. Taken from Yahoo
Finance - http:/finance.yahoo.com/q/hp?s=CVX&a=11&b=18&c=1995&d=11&e=16&f=2015&g=d
1.
The series doesnt appear to be stationary fro
Shannon Wu SSW311
A.
Autocorrelation Function for difflogun
(with 5% significance limits for the autocorrelations)
1.0
0.8
Autocorrelation
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
1
5
10
15
20
25
30
35
40
45
50
55
60
65
70
Lag
Since the ACF of the log of
Shannon Wu SSW311
Due 9/30
Homework #1
1. A)
GDP seems to grow relatively linearly. However, it does seem like there is a
convex shape to the data, possibly making it quadratic or logarithmic.
1. B)
The log of GDP seems to grow linearly over time. This im
DIFFERENCING AND UNIT ROOT TESTS
In the Box-Jenkins approach to analyzing time series, a key question is whether to difference the
data, i.e., to replace the raw data cfw_xt by the differenced series cfw_xt xt 1. Experience indicates that
most economic t
STAT-GB.2302, STAT-UB.0018: FORECASTING TIME SERIES DATA
Clifford M. Hurvich
Ofce Hours: Wed, 12:30-1:30, 8-52 KMC. Tel: (212) 998-0449.
e-Mail: churvich@stern.nyu.edu
Course Website: http:/www.stern.nyu.edu/churvich (click on "Forecasting").
TA Info: Nik
Chapter 3, Part II: Autoregressive Models
Another simple time series model is the f irst order autoregression , denoted by AR(1). The
series cfw_xt is AR(1) if it satises the iterative equation (called a dif f erence equation )
xt = xt 1 + t
,
(1)
where
Chapter 3, Part III: Mixed Autoregressive-Moving Average Models
Even though the AR(p) and MA(q) models are somewhat unrealistic by themselves, we can mix
them to form the extremely useful ARMA(p,q) models. The ARMA(p,q) series cfw_xt is generated by
xt =
Chapter 3, Part IV: The Box-Jenkins Approach to Model Building
The ARMA models have been found to be quite useful for describing stationary nonseasonal time
series. A partial explanation for this fact is provided by Wolds Theorem: "Any stationary series c
Chapter 3, Part V: More on Model Identication; Examples
Automatic Model Identication Through AIC
As mentioned earlier, there is a clear need for automatic, objective methods of identifying the
best ARMA model for the data at hand. Objective methods become
THE DURBIN-WATSON TEST
Suppose we have a time series regression model relating a "dependent" time series cfw_yt to the
"independent" time series cfw_x 1t , . . . , cfw_xpt . The model is
yt = o + 1x 1t + . . . + p xpt + t
,
t =1,2, . . . , n
,
where cfw
FORECASTING HOMEWORK 2
In Problems 1-2, we consider the mean-adjusted Russell 2000 data set, xt = Russellt .
Russell
"Todays Russell" is x 2 , . . . , xn , and "Yesterdays Russell" is x 1 , . . . , xn 1. To create xt in Minitab,
use Calc Calculator, Store
FORECASTING HOMEWORK 4
1) Consider the AR (2) process xt = xt 1 5xt 2 + t . Determine whether the process is stationary.
2) Use the ACF and PACF to identify ARIMA (p , d , q ) models for the Housing Starts series, the log of
the GDP series, the rst differ
FORECASTING HOMEWORK 5
A) For the Unemployment series, use the ACF, PACF and AICC to identify an ARIMA model (perhaps
including a constant term).
B) Estimate the parameters. Are they all statistically signicant? Do you think that a constant should be
incl
FORECASTING HOMEWORK 6
The data sets chaos1 and chaos2 (available on the course website) were generated with n = 50 by
iterating the "tent map",
x /.6 if 0 x .6
f (x ) = (1x )/.4 if .6 < x 1 .
We used x 0 = .5 for chaos1 and x 0 = .501 for chaos2.
1) Che
FORECASTING HOMEWORK 7
Recall that cfw_xt is a martingale if E [xn +h xn , xn 1 , . . . ] = xn for all n and for all lead times
h > 0. Actually, to establish that cfw_xt is a martingale, one simply needs to prove the above formula for
h = 1 since it can
FORECASTING HOMEWORK 8
In these problems, we will consider Rupee, the exchange rate for the Indian Rupee to 1 U.S. Dollar. The data is daily from July 1st, 2002 to April 8th, 2011. (n=2235). We will work with the logs of
the exchange rates. We will be tti
FORECASTING HOMEWORK 9
In these problems, we will consider the spread (long term minus short term interest rates) between
20 year UK gilts and 91 day UK Treasury Bills for the period 1952, First Quarter, to 1988 Fourth Quarter. (n = 148). You need to crea
LINEAR PREDICTION OF A RANDOM VARIABLE
Some Important Denitions
Suppose X and Y are random variables. For example, X = Todays Price, Y = Tomorrows Price.
The mean, or expectation of X is denoted by E [X ] = x . E [X ] is the average value of X over a larg
Chapter 3: Forecasting From Time Series Models
Part 1: White Noise and Moving Average Models
Stationarity
In this chapter, we study models for stationary time series. A time series is stationary if its
underlying statistical structure does not evolve with