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Fitting, Diagnosing, Selecting ARIMA Models

Fitting, Diagnosing, Selecting ARIMA Models

MAS 640 – Time Series Analysis and Forecasting

2/5/2018

Schedule

I Today –
I 2/7 – Forecasting
I 2/12 – Seasonal ARIMA
I 2/14 – Seasonal ARIMA
I 2/19 – Special/Advanced Topics
I 2/21 – Review
I 2/28 – Final

Homework 1

I Submissions:
I One submission per group, with everyone’s name.
I Don’t submit the same thing 3 times.
I Don’t submit one version with your group and a different

version on your own.

Homework 1

I RMD Comments
1. Don’t leave warnings and messages in your document
2. Don’t print out several pages of residuals, fitted values, etc. . .

never type myData into your document and leave it. . .

I LOOK AT YOUR OUTPUT
I If it’s hideous, fix it
I If it’s filled with unnecessary output, remove it

ACF With Trending / Nonstationary Series

I Previously, we determined if a time series was nonstationary via
the time series plot.

I It’s generally quite obvious.
I However, if unsure, fit the ACF

I Very slow decay if non-stationary/trending
I Keep this pattern in mind, if you forget to plot the time series

and jump straight to ACF/PACF, this should help you realize
you need to difference

Example with CREF Data

CREF Time Series

Time

C
R

E
F

0 100 200 300 400 500

1
7

0
1

9
0

2
1

0

0 5 10 15 20 25

0
.0

0
.4

0
.8

Sample Autocorrelations

Lag

A
C

F

Model Specification and Diagnostics

I What do we mean by model specification?
I How do we diagnose the appropriateness of a model?

1. How do we eyeball assumptions?
2. How do we get p-values to test assumptions?

Specification

I Specification involves studying the time series plot, the ACF,
PACF, and EACF and trying to determine an appropriate
model or models for the data

I Common to come away with a couple candidate models
I Reading ACF/PACF/EACF not always straight forward

I You’re trying to find a reasonable starting point, and you want
to be able to justify that model should you ultimately use it

Residual Diagnostics

I Residual diagnostics to check appropriateness of the model
I Normality

I Histogram, Normal QQ plot, Shapiro-Wilks test

I Independence
I Time series plot, ACF of residuals, Ljung-Box p-values

Overfitting

I After going through the steps to specify a model, you fit that
model and diagnose the fit

I If it looks bad, you start iterating through the process of
overfitting

I Add an MA term, check results
I Add an AR term, check results

Overfitting

I When overfitting, you are looking at these things –
1. Are the added terms significant?
2. Do the assumptions appear to better hold (better residual

diagnostics)
3. Did the AIC/BIC improve?

Model Comparison

I There is often multiple models that will fit the data well and
look appropriate according to the residual diagnostics

I Can select between them using AIC/BIC
I Want a simple model that fits well

I If an ARMA(1,1) and an ARMA(3, 5) both fit reasonably well,
I’m going with the ARMA(1,1)