1. Autoregressive Integrated Moving Average (ARIMA) models

2. - 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

3. ARIMA Models
- 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

6. Linear Models for Time series

7. Linear Filter
- 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

8. Stationarity

9. 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

12. Infinite Moving Average
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 &

13. autocovariance function
Linear Process
Infinite moving average

14. 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

15. Infinite moving average:
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)

16. 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

17. εt white noise
Expected value of MA(q)
Variance of MA(q)
Autocovariance of MA(q)
Autocorelation of MA(q)

18. ACF function:
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

19. First Order Moving Average Process MA(1)
q=1
Autocovariance of MA(q)
Autocorelation of MA(q)

20. Mean & variance : stable
Short runs where successive observations tend to follow each other
Positive autocorrelation
Observations oscillate successively
negative autocorrelation

21. Second Order Moving Average MA(2) process
Autocovariance of MA(q)
Autocorelation of MA(q)

22. The sample ACF cuts off after lag 2

23. 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

24. 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:

25. First order autoregressive process AR(1)
where
WHY AUTOREGRESSIVE ?
AR(1) stationary if

26. Mean AR(1)
Autocovariance function AR(1)
Autocorrelation function AR(1)
The ACF for a stationary AR(1) process has an exponential decay form

27. Observe:
The observations exhibit up/down movements

28. 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

29. where

30. Calculate the weights
We need

31. Solutions
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:

32. Mean
Autocovariance function
For k=0:
For k>0:
Yule-Walker equations

33. Autocorrelation function
Solutions
A. Solve the Yule-Walker equations recursively
B. General solution
Obtain it through the roots m1 & m2 associated with the polynomial

34. 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

35. Case II: m1, m2 complex conjugates in the form
c1, c2: particular constants
ACF form: damp sinusoid; damping factor R; frequency ; period

36. Case III: one real root m0; m1= m2=m0
ACF form: exponential decay pattern

37. 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

38. AR(2) process: yt=4+0.8yt-1-0.5yt-2+et
Roots of the polynomial: complex conjugates
ACF form: damped sinusoid behavior

39. General Autoregressive Process, AR(p)
Consider a pth order AR model or

40. AR(P) stationary
If the roots of the polynomial
are less than 1 in absolute value
AR(P) absolute summable infinite MA representation
Under the previous condition

41. Weights of the random shocksas

42. For stationary AR(p)

43. ACF
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

44. ACF
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

45. Partial Autocorrelation Function
Consider :
three random variables X, Y, Z &
Simple regression of X on Z & Y on Z
The errors are obtained from

46. 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

47. 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

48. Matrix notation
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

49. MA(1) MA(2) AR(1) AR(1) Cuts off after 1st lag AR(2) AR(2)
Decay pattern
Decay pattern
AR(1)
AR(1)
Cuts off after 1st lag
AR(2)
AR(2)
Cuts off after 2nd lag

50. Invertibility of MA models
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)

51. Obtain
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

52. PACF possibly never cuts off
PACF of a MA(q) process is a mixture of exponential decay & damp sinusoid expressions
In model identification, use both sample ACF & sample PACF

53. Mixed Autoregressive –Moving Average (ARMA) Process
ARMA (p,q) model
Adjust the exponential decay pattern by adding a few terms

54. 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

55. 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:

56. ARMA(1,1)
Sample ACF & PACF: exponential decay behavior

60. Non Stationary Process
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)dyt produces a stationary ARMA(p,q) process
ARIMA(p,d,q)

61. The random walk process ARIMA(0,1,0)
Simplest non-stationary model
First differencing eliminates serial dependence & yields a white noise process

62. yt=20+yt-1+et 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)

63. 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

64. ARIMA(0,1,1) Possible model :AR(2) Check the roots
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