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Autoregressive Integrated Moving Average (ARIMA) models

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- Forecasting techniques based on exponential smoothing

- General assumption for the above models: times series data are represented as the sum of two distinct components (deterministc & random)
- Random noise: generated through independent shocks to the process
- In practice: successive observations show serial dependence

- ARIMA models are also known as the Box-Jenkins methodology

- very popular : suitable for almost all time series & many times generate more accurate forecasts than other methods.
- limitations:
- If there is not enough data, they may not be better at forecasting than the decomposition or exponential smoothing techniques.
- Recommended number of observations at least 30-50
- - Weak stationarity is required
- - Equal space between intervals

- - It is a process that converts the input xt, into output yt
- The conversion involves past, current and future values of the input in the form of a summation with different weights
- Time invariant do not depend on time
- Physically realizable: the output is a linear function of the current and past values of the input
- Stable if

In linear filters: stationarity of the input time series is also reflected in the output

A time series that fulfill these conditions tends to return to its mean and fluctuate around this mean with constant variance.

Note: Strict stationarity requires, in addition to the conditions of weak stationarity, that the time series has to fulfill further conditions about its distribution including skewness, kurtosis etc.

Determine stationarity

-Take snaphots of the process at different time points & observe its behavior: if similar over time then stationary time series

-A strong & slowly dying ACF suggests deviations from stationarity

Input xt stationary

Output yt

Stationary, with

&

THEN,

the linear process with white noise time series εt

Is stationary

εt independent random shocks, with E(εt)=0 &

The infinite moving average serves as a general class of models for any stationary time series

THEOREM (World 1938):

Any no deterministic weakly stationary time series yt can be represented as

where

INTERPRETATION

A stationary time series can be seen as the weighted sum of the present and past disturbances

- Impractical to estimate the infinitely weights
- Useless in practice except for special cases:
- i. Finite order moving average (MA) models : weights set to 0, except for a finite number of weights
- ii. Finite order autoregressive (AR) models: weights are generated using only a finite number of parameters
- iii. A mixture of finite order autoregressive & moving average models (ARMA)

Finite Order Moving Average (MA) process

Moving average process of order q(MA(q))

MA(q) : always stationary regardless of the values of the weights

Expected value of MA(q)

Variance of MA(q)

Autocovariance of MA(q)

Autocorelation of MA(q)

Helps identifying the MA model & its appropriate order as its cuts off after lag k

Real applications:

r(k) not always zero after lag q; becomes very small in absolute value after lag q

- Short runs where successive observations tend to follow each other
- Positive autocorrelation

- Observations oscillate successively
- negative autocorrelation

Finite Order Autoregressive Process

- World’s theorem: infinite number of weights, not helpful in modeling & forecasting
- Finite order MA process: estimate a finite number of weights, set the other equal to zero
- Oldest disturbance obsolete for the next observation; only finite number of disturbances contribute to the current value of time series
- Take into account all the disturbances of the past :
- use autoregressive models; estimate infinitely many weights that follow a distinct pattern with a small number of parameters

First Order Autoregressive Process, AR(1)

Assume : the contributions of the disturbances that are way in the past are small compared to the more recent disturbances that the process has experienced

Reflect the diminishing magnitudes of contributions of the disturbances of the past,through set of infinitely many weights in descending magnitudes , such as

Exponential decay pattern

The weights in the disturbances starting from the current disturbance and going back in the past:

Autocovariance function AR(1)

Autocorrelation function AR(1)

The ACF for a stationary AR(1) process has an exponential decay form

- The observations exhibit up/down movements

Second Order Autoregressive Process, AR(2)

This model can be represented in the infinite MA form & provide the conditions of stationarity for yt in terms of φ1& φ2

WHY?

1. Infinite MA

Apply

We need

The satisfy the second-order linear difference equation

The solution : in terms of the 2 roots m1 and m2 from

AR(2) stationary:

Condition of stationarity for complex conjugates a+ib:

AR(2) infinite MA representation:

Solutions

A. Solve the Yule-Walker equations recursively

B. General solution

Obtain it through the roots m1 & m2 associated with the polynomial

Case I: m1, m2 distinct real roots

c1, c2 constants: can be obtained from ρ(0) ,ρ(1)

stationarity:

ACF form: mixture of 2 exponentially decay terms

e.g. AR(2) model

It can be seen as an adjusted AR(1) model for which a single exponential decay expression as in the AR(1) is not enough to describe the pattern in the ACF and thus, an additional decay expression is added by introducing the second lag term yt-2

Case II: m1, m2 complex conjugates in the form

c1, c2: particular constants

ACF form: damp sinusoid; damping factor R; frequency ; period

Case III: one real root m0; m1= m2=m0

ACF form: exponential decay pattern

AR(2) process :yt=4+0.4yt-1+0.5yt-2+et

Roots of the polynomial: real

ACF form: mixture of 2 exponential decay terms

AR(2) process: yt=4+0.8yt-1-0.5yt-2+et

Roots of the polynomial: complex conjugates

ACF form: damped sinusoid behavior

If the roots of the polynomial

are less than 1 in absolute value

AR(P) absolute summable infinite MA representation

Under the previous condition

pth order linear difference equations

AR(p) :

-satisfies the Yule-Walker equations

-ACF can be found from the p roots of the associated polynomial

e.g. distinct & real roots :

- In general the roots will not be real

ACF : mixture of exponential decay and damped sinusoid

- MA(q) process: useful tool for identifying order of process
- cuts off after lag k
- AR(p) process: mixture of exponential decay & damped sinusoid expressions
- Fails to provide information about the order of AR

Partial Autocorrelation Function

- Consider :
- three random variables X, Y, Z &
- Simple regression of X on Z & Y on Z

The errors are obtained from

Partial correlation between X & Y after adjusting for Z:

The correlation between X* & Y*

Partial correlation can be seen as the correlation between two variables after being adjusted for a common factor that affects them

Partial autocorrelation function (PACF) between yt & yt-k

The autocorrelation between yt & yt-k after adjusting for yt-1, yt-2, …yt-k

AR(p) process: PACF between yt & yt-k for k>p should equal zero

- Consider
- a stationary time series yt; not necessarily an AR process
- For any fixed value k , the Yule-Walker equations for the ACF of an AR(p) process

Solutions

For any given k, k =1,2,… the last coefficient is called the partial autocorrelation coefficient of the process at lag k

AR(p) process:

Identify the order of an AR process by using the PACF

MA(2)

Decay pattern

Decay pattern

AR(1)

AR(1)

Cuts off after 1st lag

AR(2)

AR(2)

Cuts off after 2nd lag

Invertible moving average process:

The MA(q) process

is invertible if it has an absolute summable infinite AR representation

It can be shown:

The infinite AR representation for MA(q)

We need

Condition of invertibility

The roots of the associated polynomial be less than 1 in absolute value

An invertible MA(q) process can then be written as an infinite AR process

PACF of a MA(q) process is a mixture of exponential decay & damp sinusoid expressions

In model identification, use both sample ACF & sample PACF

Mixed Autoregressive –Moving Average (ARMA) Process

ARMA (p,q) model

Adjust the exponential decay pattern by adding a few terms

Stationarity of ARMA (p,q) process

Related to the AR component

ARMA(p,q) stationary if the roots of the polynomial less than one in absolute value

ARMA(p,q) has an infinite MA representation

Invertibility of ARMA(p,q) process

Invertibility of ARMA process related to the MA component

Check through the roots of the polynomial

If the roots less than 1 in absolute value then ARMA(p,q) is invertible & has an infinite representation

Coefficients:

Sample ACF & PACF: exponential decay behavior

Not constant level, exhibit homogeneous behavior over time

- yt is homogeneous, non stationary if
- It is not stationary
- Its first difference, wt=yt-yt-1=(1-B)yt or higher order differences wt=(1-B)dyt produce a stationary time series

Yt autoregressive intergrated moving average of order p, d,q –ARIMA(p,d,q)

If the d difference , wt=(1-B)dytproduces a stationary ARMA(p,q) process

ARIMA(p,d,q)

The random walk process ARIMA(0,1,0)

Simplest non-stationary model

First differencing eliminates serial dependence & yields a white noise process

- Evidence of non-stationary process
- Sample ACF : dies out slowly
- Sample PACF: significant at the first lag
- Sample PACF value at lag 1 close to 1

- First difference
- Time series plot of wt: stationary
- Sample ACF& PACF: do not show any significant value
- Use ARIMA(0,1,0)

The random walk process ARIMA(0,1,1)

Infinite AR representation, derived from:

ARIMA(0,1,1)= (IMA(1,1)): expressed as an exponential weighted moving average (EWMA) of all past values

- The mean of the process is moving upwards in time
- Sample ACF: dies relatively slow
- Sample PACF: 2 significant values at lags 1& 2

Possible model :AR(2)

Check the roots

- First difference looks stationary
- Sample ACF & PACF: an MA(1) model would be appropriate for the first difference , its ACF cuts off after the first lag & PACF decay pattern

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