2. FIRST, WHAT IS TIME SERIES?
BEFORE WE ANSWER THIS, WE HAVE TO UNDERSTAND …
•WHAT IS A VARIABLE?
•WHAT IS A RANDOM VARIABLE?
•WHAT IS A STOCHASTIC PROCESS?
3. STOCHASTIC PROCESS…
•PLEASE NOTE THAT A STOCHASTIC PROCESS X = XT(Ω)
IS A FUNCTION OF TWO VARIABLES-
•ONE, T AND T T. (TIME)
•ANOTHER, Ω AND Ω Ω. (EVENT)
7. 0 1 2 3 4
All these realizations are the result of SOME
underlying stochastic process whose behaviour is
modeled through Normal Probability Distribution.
8. NOW, WE DEFINE TIME SERIES…
• REALIZATION OF A PARTICULAR STOCHASTIC
PROCESS IS CALLED A TIME SERIES.
TIME SERIES IS ONLY ONE OF THE MANY POSSIBLE
REALIZATIONS OF A STOCHASTIC PROCESS THAT
THE HISTORY MIGHT HAVE GENERATED.
TIME SERIES ANALYSIS IS TRYING TO DRAW
STATISTICAL INFERENCE FROM A SINGLE
OUTCOME (TIME SERIES) OBSERVED.
10. REVISIT THE PURPOSES FOR WHICH TIME
SERIES ARE USED …
• FORECASTING
• ESTIMATION
• ESTABLISHING RELATION
For all these, we
need
INFORMATION!
But, where is the
INFORMATION
hidden in a time
series?
11. WHERE IS THE INFORMATION HIDDEN IN A
TIME SERIES?
Is it in TIME?
Is it in the OWN PAST of a time series?
Is it in OWN RESIDUAL?
Is it in SOME OTHER TIME SERIES?
12. TOOLS TO IDENTIFY A UNDERLYING BEHAVIOR OF TIME
SERIES
•TIME SERIES PLOT
•CORRELOGRAM
13. A SIMPLE TIME SERIES…
-4.0000
-3.0000
-2.0000
-1.0000
0.0000
1.0000
2.0000
3.0000
4.0000
REALIZATION OF A VALUE WHICHIS PURELY RANDOM FOLLOWINGNORMALPROBABILITY
DISTRIBUTION
21. AUTOCORRELATION AND PARTIAL
AUTOCORRELATION…
• AUTOCORRELATION IS SIMPLE CORRELATION BETWEEN XT AND, SAY, XT+H, IT IS A
CORRELATION BETWEEN A SERIES BUT WITH A LAG.
• WHILE PARTIAL AUTOCORRELATION IS A CORRELATION BETWEEN OBSERVATIONS
XT AND XT+H AFTER REMOVING THE LINEAR RELATIONSHIP OF ALL OBSERVATIONS
THAT FALL BETWEEN XT AND XT+H.
22. POSSIBLE WAYS TO MODEL A TIME SERIES FOR
FORECASTING PURPOSES!!!
A Time Series may
be a function of
time!
• It has a trend!
A Time Series may be
a function of its own
history!
• Past determines
FUTURE!
A Time Series may be
a function of some
exogenous
variables!
• Independent
Factors
determine it!
23. APPRECIATING WAYS OF MODELING A TIME SERIES!
• A TIME SERIES MAY BE A FUNCTION OF TIME!
• IT HAS A TREND! YT = F(T)!
• A TIME SERIES MAY BE A FUNCTION OF ITS OWN HISTORY!
• PAST DETERMINES FUTURE! YT = F(YT-1)!
• A TIME SERIES MAY BE A FUNCTION OF SOME EXOGENOUS VARIABLES!
• INDEPENDENT FACTORS DETERMINE IT! YT = F(XT)!
Note the difference!
Note the difference!
Note the
difference!
24. Why
should I be
bothered
by time
series?
-8.0%
-6.0%
-4.0%
-2.0%
0.0%
2.0%
4.0%
6.0%
8.0%
10.0%
12.0%
We cannot use OLS
to get the estimates of
a regression based on
time series.
25. LET’S TRY TO FORECAST USING TIME
SERIES…
•ASSUME THAT WE HAVE TIME
SERIES DATA OF SALES OF A
COMPANY FOR THE LAST 15
YEARS; AND WE WANT TO MAKE
A FORECAST FOR THE NEXT
YEAR.
•DATA IS LIKE – (PART OF THE
DATA)
SALES DATA
YEAR t SALES IN (Rs. LAKHS)
1996 1 21,777
1997 2 22,418
1998 3 22,308
1999 4 23,319
2000 5 24,180
2001 6 24,893
2002 7 25,310
26. SALES = 8860.49 + 3121.69 t
WE RUN SIMPLE REGRESSION…AND GOT
THE RESULTS…
33. OLS WORKS WHEN ITS
ASSUMPTIONS ARE TRUE!!!
•MANY TIMES, A TIME SERIES DOES NOT SATISFY THE
ASSUMPTIONS OF HOMOSCEDASTICITY AND NO
AUTOCORRELATION.
•THE CULPRIT IS NON-STATIONARITY!!!
34. STATIONARITY …
• …IS AN IDEA WHICH TALKS ABOUT THE STATIONARITY OF THE PARAMETERS OF A
TIME SERIES.
• A TIME SERIES IS SAID TO BE STATIONARY IF IT HAS –
•CONSTANT MEAN
•CONSTANT VARIANCE
•COVARIANCE FUNCTION DEPENDS ON TIME
DIFFERENCE BETWEEN THE VALUES OF THE
VARIABLE.
35. CHARACTERISTICS OF A STATIONARY
PROCESS
• IT IS MEAN REVERTING.
• IT HAS A FINITE MEMORY.
• THE IMPACT OF A SHOCK DIES OUT WITH TIME.
• IT HAS A FINITE VARIANCE.
36. YOU CAN APPRECIATE THE IMPORTANCE OF STATIONARITY BY LOOKING AT THE
FOLLOWING GRAPH
… Yt-2 Yt-1 Yt Yt+1 Yt+2 …
37. UNIT ROOT STATIONARITY TESTS…
•DICKEY-FULLER TEST
•AUGMENTED DICKEY-FULLER TEST
•THE PHILLIPS-PERRONTEST
38. LET’S NOTE THE ASSUMPTIONS ABOUT THESE TESTS…
Dickey-
Fuller Test
Error terms are
NOT correlated.
Augmented
Dickey-
Fuller Test
Error terms are
correlated.
Phillips-
Perron Test
Error terms are
correlated.
Errors terms
are
heteroscedastic
39. WHO IS THE CULPRIT…
… behind the
violation of the
stationarity
assumption in the
case of a time
series?
It is
TREND!
40. TRENDS…
•… CAN CREATE A PROBLEM IN OBTAINING THE CONSISTENT AND ASYMPTOTIC
NORMALLY DISTRIBUTED OLS ESTIMATES.
•…TRENDS MAY MAKE OUR ESTIMATES AT TIMES UNSTABLE.
•A TREND IS … A PERSISTENT UPWARD OR DOWNWARD MOVEMENTS OF VARIABLES
OVER A PERIOD OF TIME.
•…OBSERVED NOT IN SHORT-TERM BUT OVER A LONG PERIOD OF TIME.
41. TRENDS…
Trends are of TWO types:
Deterministic Trends:
• The trending variable changes by a constant
amount each period
Stochastic Trends:
• The trending variable changes by a random amount
each period (vt)
E(Yt )- E(Yt-1) vt
-1 1
( )- ( )
t t
E Y E Y
43. NON-STATIONARY TIME SERIES…
Trend Stationary: If Yt contains a deterministic trend and
{Yt – Trend} becomes stationary; then Yt is known as trend stationary.
Difference Stationary: If Yt contains a stochastic trend and {Yt – Yt-1 }
becomes stationary; then Yt is known as difference stationary.
Remember that these can be used as
a strategy to make a non-stationary
time series as STATIONARY TIME
SERIES
45. Time Series Plot of Daily Returns on Nifty
(1990-2009)
Stationary
46. BY TAKING DIFFERENCE, WE LOST THE INFORMATION OF …
… Co-Trending which explains underlying dynamics of the
behaviour of the time series.
47. GRANGER SAID THAT WE CAN WORK
WITH NON-STATIONARY TIME SERIES
PROVIDED THAT THEY ARE
COINTERGRATED!
Clive William John Granger
YES!!!
But, subject to
terms and
conditions!!!
48. COINTERGRATION…(CONTINUED)
•IF TWO TIME SERIES ARE INTEGRATED AT SAME LEVEL AND IF THEIR
LINEAR COMBINATION IS STATIONARY, THEN THE TWO SERIES ARE
COINTERGRATED.
•TWO POINTS TO BE NOTED IN THE CONTEXT OF COINTERGRATION:
1. BOTH THE SERIES SHOULD BE INTEGRATED AT SAME LEVEL; AND
2. THEIR LINEAR COMBINATION SHOULD BE STATIONARY!
49. ARE THE FOLLOWING SERIES COINTERGRATED?
0.00
2000.00
4000.00
6000.00
8000.00
10000.00
12000.00
0
5000
10000
15000
20000
25000
30000
35000
4-Jan-10
4-Mar-10
4-May-10
4-Jul-10
4-Sep-10
4-Nov-10
4-Jan-11
4-Mar-11
4-May-11
4-Jul-11
4-Sep-11
4-Nov-11
4-Jan-12
4-Mar-12
4-May-12
4-Jul-12
4-Sep-12
4-Nov-12
4-Jan-13
4-Mar-13
4-May-13
4-Jul-13
4-Sep-13
4-Nov-13
4-Jan-14
4-Mar-14
4-May-14
4-Jul-14
4-Sep-14
4-Nov-14
4-Jan-15
4-Mar-15
May be or may not be … it depends!!!
51. COINTERGRATION IS NOT CORRELATION!
• CORRELATION DETERMINES THE DEGREE, NATURE AND STRENGTH RELATIONSHIP
BETWEEN TWO VARIABLES WHILE COINTEGRATION MEASURES ONLY WHETHER OR
NOT THE DISTANCE BETWEEN THEM REMAINS STABLE OVER TIME.
• IF TWO TIME SERIES ARE NEGATIVELY CORRELATED THEY CANNOT BE
COINTERGRATED!
• PRICES ARE NORMALLY COINTERGRATED WHILE RETURNS MAY HAVE CORRELATION!
52. VECM…
• IT TELLS US HOW THE LONG-TERM EQUILIBRIUM RELATIONSHIP IS RESTORED THROUGH THE DYNAMICS OF
ERROR TERM.
• ONCE THE LONG-TERM EQUILIBRIUM RELATIONSHIP IS DISTURBED…… THEN, HOW THE MECHANISM AND THE
SPEED OF ADJUSTMENT!!!!
54. BUT, TIME SERIES HAS ITS OWN
LOGIC OF CAUSE AND EFFECT!!!
And, it takes us to the
discussion on…
55. CAUSALITY…
•A VARIABLE X WOULD BE CAUSAL TO A VARIABLE Y IF X
COULD BE INTERPRETED AS THE CAUSE OF Y AND/OR Y
AS THE EFFECT OF X.
CAUSE EFFECT
WHY? WHAT?
56. GRANGER CAUSALITY…
•FOR TIME SERIES, WE TAKE INFORMATION FROM
THE CONCEPT OF GRANGER CAUSALITY FOR
DETERMINING CAUSALITY!
But, what does
Granger Causality
mean?
57. GRANGER CAUSALITY…
•THE CONCEPT IS BASED ON THE PREMISE:
CAUSE ALWAYS PRECEDES EFFECT
•IT MEANS THAT THE CONCEPT OF GRANGER
CAUSALITY IS RELATED TO THE IDEA OF
SUCCESSION IN TIME.
•IT MEANS THAT ON A TIME-LINE, IF AN EVENT X
HAPPENS BEFORE THE OTHER, Y, THEN WE CAN SAY
THAT X GRANGER CAUSES Y!
58. ANOTHER WAY TO UNDERSTAND GRANGER
CAUSALITY IS…
•IF XT IMPROVES THE FORECASTING
PERFORMANCE OF YT, THEN XT GRANGER
CAUSES YT.
•IF XT DOES NOT IMPROVE THE
FORECASTING PERFORMANCE OF YT,
THEN XT DOES NOT GRANGER CAUSE YT.
59. PRECISELY, GRANGER
CAUSALITY MEANS …
•IF PAST VALUES OF X HELP TO EXPLAIN Y,
THEN X GRANGER CAUSES Y
•IT IS A STATISTICAL CONCEPT
•A LACK OF GRANGER CAUSALITY DOES NOT
IMPLY THAT THERE IS NO CAUSAL
RELATIONSHIP
60. TRUST THAT YOU MUST BE FEELING
COMFORTABLE WITH TIME SERIES !!!!!!!!
Editor's Notes
(Ω, F, P)
Ω is the Sample Space; also called space of elementary events.
F is a class of events in a random experiment.
P is a probability measure defined over F.
