Basics AR process MA process ARMA process Stationarity ARIMA process
CORPFIN 2503 – Business Data Analytics: Time-series analysis
Week 9: October 5th, 2021
£ius CORPFIN 2503, Week 9 1/65
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AR process
MA process ARMA process Stationarity
ARIMA process
AR process
MA process
ARMA process
Stationarity
ARIMA process
CORPFIN 2503, Week 9
AR process MA process ARMA process Stationarity ARIMA process
Introduction
Time-series process is a series of data points recorded in time order where the time interval between any adjacent observations is the same.
For example, IBM monthly stock prices (xed interval is a month).
One should have one observation per xed interval of time for time series process.
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Basics AR process MA process ARMA process Stationarity ARIMA process
Example: IBM
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Basics AR process MA process ARMA process Stationarity ARIMA process
Example: IBM II
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Basics AR process MA process ARMA process Stationarity ARIMA process
Example: IBM III
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Basics AR process MA process ARMA process Stationarity ARIMA process
Example: IBM IV
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Basics AR process MA process ARMA process Stationarity ARIMA process
Example: IBM V
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AR process MA process ARMA process Stationarity ARIMA process
Introduction II
Suppose we are given the data for the 2003-2020 period.
Forecasting would help us to determine which scenario is the most plausible for the 2021-2023 period.
I.e., forecasting would tell us how to extend the line beyond 2020 year given the data for the 2003-2020 period and no other information.
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AR process MA process ARMA process Stationarity ARIMA process
Introduction III
The aim of time-series analysis is to model the patterns to use for forecasting.
Phases of time-series analysis:
1. Descriptive analysis
2. Modeling
3. Forecasting the future values.
CORPFIN 2503, Week 9
AR process MA process ARMA process Stationarity ARIMA process
Introduction IV
In the descriptive phase, we should understand the nature of the time series:
• about what the trend looks like, upward or downward, and whether
• there is any seasonal trend in the process.
In the modeling phase, one should model the inherent properties of
the time-series data.
In the forecasting phase, one predicts the future values of a variable of interest using its historical values:
Yt =f(Yt−1,Yt−2,Yt−3,…,Y0).
In this course, we will use ARIMA (auto-regressive integrated moving average) methods to forecast.
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AR process MA process ARMA process Stationarity ARIMA process
AR process
AR (auto-regressive) process is where the current values of the series depend on their previous values.
It is regression on itself (lagged values).
The AR process is denoted by AR(p), where p is the order of the auto-regressive process.
p determines on how many previous values the current value of the series depends.
AR process reects long-term trends.
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AR process MA process ARMA process Stationarity ARIMA process
AR process II
The general format of AR(1) process is:
Yt −μ=β(Yt−1 −μ)+εt or
Yt =μ+β(Yt−1 −μ)+εt.
where μ is mean.
If μ = 0 then AR(1) process is:
Yt = βYt−1 + εt.
For AR(1) model, SAS procedure ARIMA provides estimates for MU and AR(1):
Yt =μ+AR(1)×(Yt−1 −μ)+εt.
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AR process MA process ARMA process Stationarity ARIMA process
AR process III
Let’s regress IBM sales growth on its lagged values. First, we need to generate sales growth variable:
data work.ibm2;
set work.ibm2;
lag_sale=lag(sale);
sale_growth=sale/lag_sale-1;
lag_sale_growth=lag(sale_growth);
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AR process MA process ARMA process Stationarity ARIMA process
AR process IV
Let’s look at the rst few observations and time series plot of the sales growth:
proc print data=work.ibm2 label;
var fyear tic sale sale_growth lag_sale_growth;
symbol1 color=red interpol=join;
proc gplot data=work.ibm2;
plot sale_growth*fyear / hminor=0;
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Basics AR process MA process ARMA process Stationarity ARIMA process
AR process V
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Basics AR process MA process ARMA process Stationarity ARIMA process
AR process VI
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AR process MA process ARMA process Stationarity ARIMA process
AR process VII
Now let’s estimate AR(1) model for IBM sales growth, using SAS procedure ARIMA::
proc arima data=work.ibm2;
identify var=sale_growth;
estimate p=1 q=0;
run; quit;
CORPFIN 2503, Week 9
Basics AR process MA process ARMA process Stationarity ARIMA process
AR process VIII: SAS output
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Basics AR process MA process ARMA process Stationarity ARIMA process
AR process IX: SAS output
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Basics AR process MA process ARMA process Stationarity ARIMA process
AR process X: SAS output
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Basics AR process MA process ARMA process Stationarity ARIMA process
AR process XI: SAS output
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Basics AR process MA process ARMA process Stationarity ARIMA process
AR process XII: SAS output
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Basics AR process MA process ARMA process Stationarity ARIMA process
AR process XIII: SAS output
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AR process MA process ARMA process Stationarity ARIMA process
AR process XIV
The resulting AR(1) model is:
Yt =μ+β(Yt−1 −μ)+εt,
Yt = 0.09664 + 0.46573(Yt−1 − 0.09664) + εt.
You might wonder how these results are dierent from OLS. Let’s have a look:
proc reg data=work.ibm2;
model sale_growth=lag_sale_growth;
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Basics AR process MA process ARMA process Stationarity ARIMA process
AR process XV: SAS output
The resulting model:
Yt = 0.04764 + 0.46574Yt−1 + εt.
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AR process MA process ARMA process Stationarity
AR process XVI
Let’s compare the models:
AR(1): Yt = 0.09664 + 0.46573(Yt−1 − 0.09664) + εt.
OLS: Yt = 0.04764 + 0.46574Yt−1 + εt. Intercepts are dierent but slopes are very similar.
We should not use OLS regressions to estimate AR models.
ARIMA process
£ius CORPFIN 2503, Week 9
AR process MA process ARMA process Stationarity
AR process XVII
AR(2) process is:
Yt =μ+β1(Yt−1 −μ)+β2(Yt−2 −μ)+εt
Let’s estimate AR(2) model for IBM sales growth, using SAS
procedure ARIMA:
proc arima data=work.ibm2;
identify var=sale_growth;
estimate p=2 q=0;
run; quit;
ARIMA process
CORPFIN 2503, Week 9
Basics AR process MA process ARMA process Stationarity ARIMA process
AR process XVIII: SAS output
The resulting model:
Yt = 0.10187+0.34549(Yt−1−0.10187)+0.27025(Yt−2−0.10187)+εt.
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AR process MA process ARMA process Stationarity ARIMA process
AR process XIX
So which model is the best for IBM sales growth: AR(1), AR(2), or perhaps AR(3)?
It is not easy to determine the order of an AR process by simply looking at the shape of the time-series curve alone.
Statistical tests are used to determine the order of AR process.
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AR process MA process ARMA process Stationarity ARIMA process
MA process
A moving average (MA) process is a time-series process where the current value and the previous values in the series are almost the same.
But the current deviation in the series depends upon the previous error or residual or shock (εt, εt−1. . . ).
εt is considered to be an unobserved term.
E.g., if we consider monthly stock returns, the εt includes shocks a
stock price is subject to:
• unexpected changes to macroeconomic environment • investor sentiment
• rm operating performance
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AR process MA process ARMA process Stationarity ARIMA process
MA process II
In an AR process, Yt depends on the previous values of Y (e.g.,Yt−1), whereas in the MA process, the previous values of Y are irrelevant.
MA process is about the short-term shocks in the series.
MA is not related to the long-term trends, like in an AR process.
In the MA process, the current value of the series is a factor of the previous errors.
The MA process is denoted by MA(q), where q determines how many previous error values have an eect on the current value of an MA series.
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AR process MA process ARMA process Stationarity ARIMA process
MA process: notation
The rst order MA (MA(1)) process:
Yt = μ + εt − MA(1)εt−1, where
MA(1) is the factor or the quantied impact of εt−1 on the
current deviation
εt is the residual at time t.
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AR process MA process ARMA process Stationarity ARIMA process
MA process III
Let’s estimate the MA(1) model for monthly returns on IBM stock:
proc arima data=work.ibm;
identify var=ret;
estimate p=0 q=1;
run; quit;
CORPFIN 2503, Week 9
AR process MA process ARMA process Stationarity
ARIMA process
SAS output:
MA process IV
The resulting model:
Yt = 0.006 + εt + 0.028εt−1.
£ius CORPFIN 2503, Week 9
AR process MA process ARMA process Stationarity ARIMA process
MA process V
Let’s estimate the MA(2) model for monthly returns on IBM stock:
proc arima data=work.ibm;
identify var=ret;
estimate p=0 q=2;
run; quit;
CORPFIN 2503, Week 9
AR process MA process ARMA process Stationarity
ARIMA process
SAS output:
MA process VI
The resulting model:
Yt = 0.006 + εt + 0.033εt−1 + 0.053εt−2.
£ius CORPFIN 2503, Week 9
AR process MA process ARMA process Stationarity ARIMA process
MA process VII
The shape of any AR(p) and MA(q) series curves depend upon the actual parameters.
Even if the order of two MA(q) series is the same, their plot shapes can dier based upon the actual parameter values in their respective equations.
It is not easy to determine the order of an MA process by simply looking at the shape of the time-series curve alone.
Statistical tests are used to determine the order of MA process.
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AR process MA process ARMA process Stationarity ARIMA process
ARMA process
An ARMA process is a process that shows the properties of an auto-regressive process and a moving average process.
In an ARMA time-series process, the current value of the series depends on its previous values.
The small deviations from the mean value in an ARMA process are a factor of the previous errors.
An ARMA process is a series with both: • long-term trend (AR process) and • short-term shocks (MA process).
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AR process MA process ARMA process Stationarity ARIMA process
ARMA process: notation
ARMA(p,q) is the general notation for an ARMA process, where: p is the order of the AR process and
q is the order of the MA process.
E.g., ARMA(1,1):
Yt =μ+a1(Yt−1 −μ)+εt −b1εt−1,where
a1 is the quantied impact of (Yt−1 − μ) on Yt b1 is the quantied impact of εt−1 on Yt.
Similarly, ARMA(2,2):
Yt =μ+a1(Yt−1 −μ)+a2(Yt−2 −μ)+εt −b1εt−1 −b2εt−2.
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AR process MA process ARMA process Stationarity
ARIMA process
ARMA process II
Let’s estimate ARMA(1,1) model for IBM sales growth:
proc arima data=work.ibm2;
identify var=sale_growth;
estimate p=1 q=1;
run; quit;
CORPFIN 2503, Week 9
Basics AR process MA process ARMA process Stationarity ARIMA process
ARMA process III
The resulting model:
Yt = 0.16419 + 1(Yt−1 − 0.16419) + εt − 0.84763εt−1.
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AR process MA process ARMA process Stationarity
ARMA process IV
Let’s estimate ARMA(2,2) model for IBM sales growth:
proc arima data=work.ibm2;
identify var=sale_growth;
estimate p=2 q=2;
run; quit;
ARIMA process
CORPFIN 2503, Week 9
Basics AR process MA process ARMA process Stationarity ARIMA process
ARMA process V
The resulting model:
Yt = 0.183+1.999(Yt−1−0.183)−1(Yt−2−0.183)+εt−1.871εt−1+0.878εt−2. £ius CORPFIN 2503, Week 9 44/65
AR process MA process ARMA process Stationarity ARIMA process
ARMA process VI
At this stage, we cannot tell which modelARMA(1,1) or ARMA(2,2)ts the data better.
This will be determined by statistical tests.
It is also possible that AR(p) model or MA(q) model are superior to ARMA(p,q) model.
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AR process MA process ARMA process Stationarity ARIMA process
Stationarity
If we want to use ARIMA models for forecasting, our time series needs to be stationary.
Stationary time series has no trend and no systematic change in variance.
In other words, in the stationary time-series process, the mean and variance hover around a single value.
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Basics AR process MA process ARMA process Stationarity ARIMA process
Stationarity II
Source: Konasani, V. R. and Kadre, S. (2015). Practical Business Analytics Using SAS: A Hands-on Guide, p. 454.
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Basics AR process MA process ARMA process Stationarity ARIMA process
Stationarity III
Source: Konasani, V. R. and Kadre, S. (2015). Practical Business Analytics Using SAS: A Hands-on Guide, p. 455.
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AR process MA process ARMA process Stationarity ARIMA process
Stationarity IV
If time series’ mean or variance is increasing over time (exploding), the time series is likely to be non-stationary.
One needs to transform it in order to achieve stationarity.
The easiest transformation is the rst dierence of the time series:
∆Yt = Yt − Yt−1. Then use ∆Yt instead of Yt in the analysis.
Some series may not be stationary even after the rst dierentiation.
Then one should try to use the second dierentiation (∆Yt − ∆Yt−1).
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AR process MA process ARMA process Stationarity ARIMA process
Stationarity V
Let’s test whether IBM sales data is stationary time series using augmented Dickey-Fuller (DF) unit root test:
proc arima data=work.ibm2;
identify var= sale stationarity=(DICKEY);
run; quit;
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Basics AR process MA process ARMA process Stationarity ARIMA process
SAS output:
Stationarity VI
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Stationarity VII
Figure as part of SAS output:
`Spikes’ outside 2 standard error bands are statistically signicant.
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AR process MA process ARMA process Stationarity ARIMA process
Stationarity VIII
From the table, we focus on the Single Mean portion.
The DF test has two sets of P-values:
1. The rst set of P-values (Pr