UNIT IV: TIME SERIES ANALYSIS
BASICS OF TIME SERIES
 A time series consists of a set of observations measured at specified, usually equal, time interval.
 Time series analysis attempts to identify those factors that exert influence on the values in the
series.
 Time series analysis is a basic tool for forecasting.
 Industry and government must forecast future activity to make decisions and plans to meet
projected changes.
 An analysis of the trend
of the observations is
needed to acquire an
understanding
of
the
progress of events leading
to prevailing conditions.
 The trend is defined as the long term underlying growth movement in a time series
 Accurate trend spotting can only be determined if the data are available for a sufficient length of
time
 Forecasting does not produce definitive results
 Forecasters can and do get things wrong from election results and football scores to the weather
Time series examples
• Sales data
•
• Gross national product
•
• Share prices
•
• $A Exchange rate
•
Time series components
Time series data can be broken into these four components:
1. Secular trend
2. Seasonal variation
Unemployment rates
Population
Foreign debt
Interest rates
3. Cyclical variation
4. Irregular variation
All Ords
10000
0
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



All Ords
This is the long term growth or decline of the series.
In economic terms, long term may mean >10 years
Describes the history of the time series
Uses past trends to make prediction about the future
Where the analyst can isolate the effect of a secular trend, changes due to other causes become
clearer
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The seasonal variation of a time
series is a pattern of change that
recurs regularly over time.
Seasonal variations are usually due
to the differences between seasons
and to festive occasions such as
Easter and Christmas.
Examples include:
• Air conditioner sales in Summer
• Heater sales in Winter
• Flu cases in Winter
• Airline tickets for flights during school vacations
Cyclical variations also have recurring patterns but with a longer and more erratic time scale
compared to Seasonal variations. The name
is quite misleading because these cycles can
be far from regular and it is usually
impossible to predict just how long periods
of expansion or contraction will be. There is
no guarantee of a regularly returning pattern.
Example includes:
• Floods
• Economic depressions or recessions
• Wars
• Changes in consumer spending
• Changes in interest rates
This chart represents an economic cycle, but we know it doesn’t always go like this. The timing and
length of each phase is not predictable.
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1. An Irregular (or random) variation in a time
series occurs over varying (usually short)
periods.
2. It follows no pattern and is by nature
unpredictable.
3. It usually occurs randomly and may be
linked to events that also occur randomly.
4. Irregular variation cannot be explained
mathematically.
5. If the variation cannot be accounted for by
secular trend, season or cyclical variation,
then it is usually attributed to irregular
variation. Example include:
– Sudden changes in interest rates
– Collapse of companies
– Natural disasters
– Sudden shift s in government policy
Monthly Value of Building Approvals ACT)
– Dramatic changes to the stock
market
– Effect of Middle East unrest on
petrol prices
ARIMA MODEL
• Autoregressive Integrated Moving Average models (ARIMA models) were popularized by
George Box and Gwilym Jenkins in the early 1970s.
• ARIMA models are a class of linear models that is capable of representing stationary as well
as non-stationary time series.
• ARIMA models do not involve independent variables in their construction. They make use of
the information in the series itself to generate forecasts.
• ARIMA models rely heavily on autocorrelation patterns in the data.
• ARIMA methodology of forecasting is different from most methods because it does not
assume any particular pattern in the historical data of the series to be forecast.
• It uses an interactive approach of identifying a possible model from a general class of models.
• The chosen model is then checked against the historical data to see if it accurately describes
the series.
In time series analysis, the Box-Jenkins Model is a mathematical model designed to forecast data
within a time series. Time-Series is of two types:
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If the ACF (autocorrelation factor) of the time series values either cuts off or dies down fairly
quickly, then the time series values should be considered STATIONARY.
If the ACF (auto correlation factor) of the time series values either cuts off or dies down extremely
slowly, then it should be considered NON-STATIONARY.
•
•
•
•
•
•
A stationary process is one whose statistical properties do not change over time.
A non-stationary process/time-series have properties which change over time.
The Box-Jenkins model alters the non-stationary time series to make it stationary by using the
differences between data points.
All stationary time series can be modeled as Auto Regressive (AR) or Moving Average (MA)
or ARMA models.
The BOX-JENKINS method applies Autoregressive Moving Average ARMA or ARIMA
models to find the best fit of a time-series model to past values of a time series.
This allows the model to pick out trends, typically using autoregresssion, moving averages
and seasonal differencing in the calculations.
DEFINITION of Autoregressive Integrated Moving Average – ARIMA
• A statistical analysis model that uses time series data to predict future trends.
• Box-Jenkins (ARIMA) is an important forecasting method that can yield highly accurate
forecasts for certain types of data.
• It is a form of regression analysis that seeks to predict future movements.
• It considers the random variations.
• It examining the differences between values in the series instead of using the actual data
values.
• Lags of the differenced series are referred to as “autoregressive" and lags within forecasted
data are referred to as “moving average”.
• This model type is generally referred to as ARIMA (p, d, q), model.
• p represents autoregressive, d represents integrated, and q represents the moving average parts
of the data set.
• ARIMA modeling can take into account trends, seasonality, cycles, errors and non- stationary
aspects of a data set when making forecasts.
• A seasonal Box-Jenkins model is symbolized as ARIMA(p,d,q)*(P,D,Q), where the p,d,q
indicates the model orders for the short-term components of the model and P,D,Q indicate the
model orders for the seasonal components of the model.
• Box-Jenkins is an important forecasting method that can generate more accurate forecasts than
other time series methods for certain types of data.
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•
•
•
•
•
•
As originally formulated, model identification relied upon a difficult, time consuming and
highly subjective procedure.
Today, software packages such as Forecast Pro use automatic algorithms to both decide when
to use Box-Jenkins models and to automatically identify the proper form of the model.
Box-Jenkins models are similar to exponential smoothing models.
Box-Jenkins models are adaptive, can model trends and seasonal patterns, and can be
automated.
Box-Jenkins models are based on autocorrelations (patterns in time) rather than a structural
view of level, trend and seasonality.
Box-Jenkins tends to succeed better than exponential smoothing for longer, more stable data
sets and not as well for noisier, more volatile data.
BOX-JENKINS METHOD
The Box-Jenkins method was proposed by George Box and Gwilym Jenkins
The approach starts with the assumption that the process that generated the time series can be
approximated using an ARMA model if it is stationary or an ARIMA model if it is non-stationary.
It refers to the process as a stochastic
model building and that it is an
iterative approach that consists of the
following 3 steps:
1. Identification. Use the data
and all related information to
help select a sub-class of
model
that
may
best
summarize the data.
2. Estimation. Use the data to
train the parameters of the
model (i.e. the coefficients).
3. Diagnostic
Checking.
Evaluate the fitted model in
the context of the available
data and check for areas
where the model may be
improved.
1. Identification
The identification step is further
broken down into:
1. Assess whether the time
series is stationary, and if not,
how many differences are
required to make it stationary.
2. Identify the parameters of an
ARMA model for the data.
1.1 Differencing
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Below are some tips during identification.
 Unit Root Tests. Use unit root statistical tests on the time series to determine whether or not
it is stationary. Repeat after each round of differencing.
 Avoid over differencing. Differencing the time series more than is required can result in the
addition of extra serial correlation and additional complexity.
1.2 Configuring AR and MA
Two diagnostic plots can be used to help choose the p and q parameters of the ARMA or ARIMA.
They are:
 Autocorrelation Function (ACF). The plot summarizes the correlation of an observation
with lag values. The x-axis shows the lag and the y-axis shows the correlation coefficient
between -1 and 1 for negative and positive correlation.
 Partial Autocorrelation Function (PACF). The plot summarizes the correlations for an
observation with lag values that is not accounted for by prior lagged observations.
Both plots are drawn as bar charts showing the 95% and 99% confidence intervals as horizontal lines.
Bars that cross these confidence intervals are therefore more significant and worth noting.
Some useful patterns you may observe on these plots are:
 The model is AR if the ACF trails off after a lag and has a hard cut-off in the PACF after a
lag. This lag is taken as the value for p.
 The model is MA if the PACF trails off after a lag and has a hard cut-off in the ACF after the
lag. This lag value is taken as the value for q.
 The model is a mix of AR and MA if both the ACF and PACF trail off.
2. Estimation
Estimation involves using numerical methods to minimize a loss or error term.
We will not go into the details of estimating model parameters as these details are handled by the
chosen library or tool.
3. Diagnostic Checking
The idea of diagnostic checking is to look for evidence that the model is not a good fit for the data.
Two useful areas to investigate diagnostics are:
1. Over fitting
2. Residual Errors.
3.1 Over fitting
The first check is to check whether the model over fits the data. Generally, this means that the model
is more complex than it needs to be and captures random noise in the training data. This is a problem
for time series forecasting because it negatively impacts the ability of the model to generalize,
resulting in poor forecast performance on out of sample data. Careful attention must be paid to both
in-sample and out-of-sample performance and this requires the careful design of a robust test harness
for evaluating models.
3.2 Residual Errors
Forecast residuals provide a great opportunity for diagnostics. A review of the distribution of errors
can help tease out bias in the model. The errors from an ideal model would resemble white noise,
which is a Gaussian distribution with a mean of zero and a symmetrical variance.
For this, you may use density plots, histograms, and Q-Q plots that compare the distribution
of errors to the expected distribution. A non-Gaussian distribution may suggest an opportunity for
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data pre-processing. A skew in the distribution or a non-zero mean may suggest a bias in forecasts
that may be correct.
Additionally, an ideal model would leave no temporal structure in the time series of forecast
residuals. These can be checked by creating ACF and PACF plots of the residual error time series.
The presence of serial correlation in the residual errors suggests further opportunity for using this
information in the model.
4. Forecasting
• One of the most important tests of how well a model performs is how well it forecasts.
• One of the most useful models for forecasting is the ARIMA model.
• To produce dynamic forecasts the model needs to include lags of either the variables or error
terms.
AR, MA, ARMA, and ARIMA MODELS
AR, MA, ARMA, and ARIMA models are used to forecast the observation at (t+1) based on
the historical data of previous time spots recorded for the same observation. However, it is necessary
to make sure that the time series is stationary over the historical data of observation overtime period.
If the time series is not stationary then we could apply the differencing factor on the records and see if
the graph of the time series is a stationary overtime period.
ACF (Auto Correlation Function)
Auto Correlation function takes into consideration of all the past observations irrespective of
its effect on the future or present time period. It calculates the correlation between the t and (t-k) time
period. It includes all the lags or intervals between t and (t-k) time periods. Correlation is always
calculated using the Pearson Correlation formula.
PACF (Partial Correlation Function)
The PACF determines the partial correlation between time period t and t-k. It doesn’t take into
consideration all the time lags between t and t-k. For e.g. let's assume that today's stock price may be
dependent on 3 days prior stock price but it might not take into consideration yesterday's stock price
closure. Hence we consider only the time lags having a direct impact on future time period by
neglecting the insignificant time lags in between the two-time slots t and t-k.
How to differentiate when to use ACF and PACF?
Let's take an example of sweets sale and income generated in a village over a year. Under the
assumption that every 2 months there is a festival in the village, we take out the historical data of
sweets sale and income generated for 12 months. If we plot the time as month then we can observe
that when it comes to calculating the sweets sale we are interested in only alternate months as the sale
of sweets increases every two months. But if we are to consider the income generated next month
then we have to take into consideration all the 12 months of last year.
So in the above situation, we will use ACF to find out the income generated in the future but
we will be using PACF to find out the sweets sold in the next month.
AR (Auto-Regressive) Model
The time period at t is impacted by the observation at various slots t-1, t-2, t-3…., t-k. The
impact of previous time spots is decided by the coefficient factor at that particular period of time.
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The price of a share of any particular company X may depend on all the previous share prices
in the time series. This kind of model calculates the regression of past time series and calculates the
present or future values in the series in know as Auto Regression (AR) model.
Consider an example of a milk
distribution company that produces milk every
month in the country. We want to calculate the
amount of milk to be produced current month
considering the milk generated in the last year.
We begin by calculating the PACF values of all
the 12 lags with respect to the current month. If
the value of the PACF of any particular month
is more than a significant value only those values will be considered for the model analysis.
For e.g in the above figure the values 1,2, 3 up to 12 displays the direct effect(PACF) of the
milk production in the current month w.r.t the given the lag t. If we consider two significant values
above the threshold then the model will be termed as AR(2).
MA (Moving Average) Model
The time period at t is impacted by the unexpected external factors at various slots t-1, t-2, t-3,
….., t-k. These unexpected impacts are known as Errors or Residuals. The impact of previous time
spots is decided by the coefficient
factor α at that particular period of
time. The price of a share of any
particular company X may depend
on some company merger that
happened overnight or maybe the
company resulted in shutdown due
to bankruptcy. This kind of model
calculates the residuals or errors of
past time series and calculates the
present or future values in the series in know as Moving Average (MA) model.
Consider an example of Cake distribution during my birthday. Let's assume that your mom
asks you to bring pastries to the party. Every year you miss judging the no of invites to the party and
end upbringing more or less no of cakes as per requirement. The difference in the actual and expected
results in the error. So you want to avoid the error for this year hence we apply the moving average
model on the time series and calculate the no of pastries needed this year based on past collective
errors. Next, calculate the ACF values of all the lags in the time series. If the value of the ACF of any
particular month is more than a significant value only those values will be considered for the model
analysis.
For e.g in the above figure the values 1,2, 3 up to 12 displays the total error(ACF) of count in
pastries current month w.r.t the given the lag t by considering all the in-between lags between time t
and current month. If we consider two significant values above the threshold then the model will be
termed as MA (2).
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ARMA (Auto Regressive Moving Average) Model
This is a model that is
combined from the AR and MA
models. In this model, the impact of
previous lags along with the
residuals
is
considered
for
forecasting the future values of the
time series. Here β represents the
coefficients of the AR model and α
represents the coefficients of the
MA model.
Consider the above graphs
where the MA and AR values are
plotted with their respective significant values. Let's assume that we consider only 1 significant value
from the AR model and likewise 1 significant value from the MA model. So the ARMA model will
be obtained from the combined values of the other two models will be of the order of ARMA (1, 1).
ARIMA (Auto-Regressive Integrated Moving Average) Model
We know that in order to apply the various models we must in the beginning convert the
series into Stationary Time Series. In order to achieve the same, we apply the differencing or
integrated method where we subtract the t-1 value from t values of time series. After applying the
first differencing if we are still unable
to get the Stationary time series then we
again
apply
the
second-order
differencing.
The ARIMA model is quite
similar to the ARMA model other than
the fact that it includes one more factor
known as Integrated( I ) i.e.
differencing which stands for I in the
ARIMA model. So in short ARIMA
model is a combination of a number of
differences already applied on the
model in order to make it stationary, the
number of previous lags along with
residuals errors in order to forecast future values.
Consider the above graphs where the MA and AR values are plotted with their respective
significant values. Let's assume that we consider only 1 significant value from the AR model and
likewise 1 significant value from the MA model. Also, the graph was initially non-stationary and we
had to perform differencing operation once in order to convert into a stationary set. Hence the
ARIMA model which will be obtained from the combined values of the other two models along with
the Integral operator can be displayed as ARIMA (1,1,1).
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ERROR MEASUREMENTS
Measurement error is the difference between the observed value of a Variable and the true, but
unobserved, value of that Variable.
Measurement error refers to a circumstance in which the true empirical value of a variable
cannot be observed or measured precisely. The error is thus the difference between the actual value of
that variable and what can be observed or measured.
For instance, household consumption/expenditures over some interval are often of great
empirical interest (in many applications because of the theoretical role they play under the forwardlooking theories of consumption). These are usually observed or measured via household surveys in
which respondents are asked to catalog their consumption/expenditures over some recall window.
However, these respondents often cannot recall precisely how much they spent on the various items
over that window. Their reported consumption/expenditures are thus unlikely to reflect precisely what
they or their households actually spent over the recall interval.
Unfortunately measurement error is not without consequence in many empirical applications.
Perhaps the most widely recognized difficulty associated with measurement error is bias to estimates
of regression parameters.
Measurement error causes the recorded values of Variables to be different from the true ones.
In general the Measurement error is defined as the sum of Sampling error and Non-sampling error.
Measurement errors can be systematic or random, and they may generate both Bias and extra
variability in statistical outputs. Usually the term measurement refers to measuring the values of a
Variable at the unit level, for example, measuring a Household’s consumption or measuring the
wages paid by a business. The term Measurement error is also applicable for aggregates at the
population level.
UNIVARIATE TIME SERIES MODELLING
The term "univariate time series" refers to a time series that consists of single (scalar)
observations recorded sequentially over equal time increments. Although a univariate time series data
set is usually given as a single column of numbers, time is in fact an implicit variable in the time
series. If the data are equi-spaced, the time variable, or index, does not need to be explicitly given.
The time variable may sometimes be explicitly used for plotting the series. However, it is not used in
the time series model itself.
UNIT ROOT
Stationarity: A stationary time series is one whose statistical properties such as mean, variance,
autocorrelation, etc. are all constant over time’. The consistency of these variables makes predictions
easier to do. On average, a stationary time series occurs less frequently than one with a stochastic
trend.
Seasonality: Seasonality is a pretty straight forward concept. It suggests that factors outside of the
control of the business model are going to still have a strong pull on how the company is performing
financially. With a Broadway show we can see the dips happen around September due to people
returning to school and summer vacations being over. The large spikes happen at times like
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Christmas and New Years when wallets are open and people are gift giving. An important thing to
keep in mind with seasonality is that it can exist even with an unpredictable outcome. In those
situations, the time series is even harder to follow and seasonality must be removed from it in order to
gain a better perspective of the trends.
Stochastic: Stochasticity (Also a Random Walk with a Drift) can be defined as a variable or process
that has uncertainty in it. The features lack dependence between one another. Stochasticity would be
used over the word randomness when the probability of a feature is important. Whereas a term such
as ‘random sampling’ just refers to having a lack of bias but does not inference an outcome.
A unit root test tests whether a time series variable is non-stationary and possesses a unit
root. The null hypothesis is generally defined as the presence of a unit root and the alternative
hypothesis is either stationarity, trend stationarity or explosive root depending on the test used.
In general, the approach to unit root testing implicitly assumes that the time series to be tested
can be written as,
Yt= Dt+Zt+€t
Where,
 Dt is the deterministic component (trend, seasonal component, etc.)
 Zt is the stochastic component.
€t is the stationary error process.
The task of the test is to determine whether the stochastic component contains a unit root or is
stationary.
Unit root (also called a unit root process or a difference stationary process) is a stochastic
trend in a time series, sometimes called a “random walk with drift”. If a time series has a unit root, it
shows a systematic pattern that is unpredictable.
A possible unit root. The red line shows the drop in output and
path of recovery. Blue shows the recovery if there is no unit root and the
series is trend-stationary.These are tests for stationarity in a time series.
A time series has stationarity if a shift in time doesn’t cause a change in
the shape of the distribution; unit roots are one cause for non-stationarity.
Methods of Unit Root Test





The Dickey Fuller Test (sometimes called a Dickey Pantula test), which is based on linear
regression. Serial correlation can be an issue, in which case the Augmented Dickey-Fuller
(ADF) test can be used. The ADF handles bigger, more complex models. It does have the
downside of a fairly high Type I error rate.
The Elliott–Rothenberg–Stock Test, which has two subtypes:
o The P-test takes the error term’s serial correlation into account,
o The DF-GLS test can be applied to detrended data without intercept.
The Schmidt–Phillips Test includes the coefficients of the deterministic variables in the null
and alternate hypotheses. Subtypes are the rho-test and the tau-test.
The Phillips–Perron (PP) Test is a modification of the Dickey Fuller test, and corrects for
autocorrelation and heteroscedasticity in the errors.
The Zivot-Andrews test allows a break at an unknown point in the intercept or linear trend.
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COINTEGRATION TEST
Cointegration is a statistical method used to test the correlation between two or more nonstationary time series in the long run or for a specified period. The method helps identify long-run
parameters or equilibrium for
two or more variables. In
addition, it helps determine the
scenarios wherein two or more
stationary time series are
cointegrated so that they cannot
depart
much
from
the
equilibrium in the long run.The
method
determines
the
sensitivity of two or more
variables to the same set of
conditions or parameters.
Let us understand the method with the help of a graph. The prices of two commodities, A and
B, are shown on the graph. We can infer that these are perfectly cointegrated commodities in terms of
price, as the difference between the prices of both commodities has remained the same for decades.
Though this is a hypothetical example, it perfectly explains the cointegration of two non-stationary
time series.
Examples of Cointegration
 Cointegration as correlation does not measure whether two or more time-series data or
variables move together in the long run. In contrast, it measures whether the difference
between their means remains constant or not.
 So, that means that two random variables completely different from each other can have one
common trend that combines them in the long run. If this happens, variables are said to be
cointegrated.
 Now let’s take the example of Cointegration in pair trading. In pair trading, a trader purchases
two cointegrated stocks, stock A at the long position and stock B in the short. The trader was
unsure about the price direction for both the stocks but was sure that stock A’s position would
be better than that of stock B.
 Now, let us say that if the prices of both the stocks go down, the trader will still make a profit
as long as stock A’s position is better than stock B’s if both stocks weigh equally at the
purchase time.
Condition of Cointegration
The cointegration test is based on the logic that more than two-time series variables have
similar deterministic trends that one can combine over time. Therefore, it is necessary for all
cointegration testing for non-stationary time series variables. One should integrate them in the same
order, or they should have a similar identifiable trend that can define a correlation between them. So,
they should not deviate much from the average parameter in the short run. In the long run, they
should be reverting to the trend.
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Methods for testing cointegration
You can test for cointegration using four different methods. These methods have some similarities,
but each one has a particular strength and may give you different results. You can use the four
methods below to test for cointegration:
Engle-Granger test
The Engle-Granger test was the first method to test for cointegration during its early
development. You can use the Engle-Granger test to errors based on doing a regression of two
variables. It uses the two hypotheses below:
H0: No cointegration exists between the two variables
H1: Cointegration exists between the two variables
Where: H0 is the null hypothesis for the test
H1 is the alternative hypothesis for the test
While this form of cointegration testing is quicker than the others, it has two drawbacks. The
first is that it can only work with two time-related series. The second is that if one of the two timerelated series depends on the other, then the test can sometimes output an incorrect conclusion that
the two variables have no cointegration. You can address both of these by using the other methods to
test for cointegration.
Phillips-Ouliaris test
The Phillips-Ouliaris test is an improvement on the original Engle-Granger test. Its major
benefit is that it handles the issue related to independent variables because it can recognize if one of
the time-related series depends on the other and factors this into the outcome it presents, meaning that
if one series depends on the other, it still considers them cointegrated. Like the Engle-Granger test,
the two hypotheses for this test are:
H0: No cointegration exists between the two variables
H1: Cointegration exists between the two variables
Also like the Engle-Granger test, this method cannot work with over two variables, which is
an issue that you can resolve by using the Gregory and Hansen test of cointegration.
Johansen test
The Johansen test is an improvement on both the Engle-Granger test and the Phillips-Ouliaris
test. You can use this test to work with time-related series in which one series depends on the other,
such as supply and demand, and eliminates errors from the previous two methods, which allows this
method to work with two or more variables. There are also two subtests to the Johansen test. You can
use the subtests below to help find if two or more time-related series are cointegrated:
CAUSALITY TEST
Prof. Clive W.J. Granger, recipient of the 2003 Nobel Prize in Economics developed the
concept of causality to improve the performance of forecasting.
It is basically an econometric hypothetical test for verifying the usage of one variable in forecasting
another in multivariate time series data with a particular lag.
A prerequisite for performing the Granger Causality test is that the data need to be stationary
i.e it should have a constant mean, constant variance, and no seasonal component. Transform the nonstationary data to stationary data by differencing it, either first-order or second-order differencing. Do
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not proceed with the Granger causality test if the data is not stationary after second-order
differencing.
Let us consider three variables Xt , Yt , and Wt preset in time series data.
Case 1: Forecast Xt+1 based on past values Xt .
Case 2: Forecast Xt+1 based on past values Xt and Yt.
Case3 : Forecast Xt+1 based on past values Xt , Yt , and Wt, where variable Yt has direct dependency
on variable Wt.
Here Case 1 is univariate time series also known as the autoregressive model in which there is
a single variable and forecasting is done based on the same variable lagged by say order p.
The equation for the Auto-regressive model of order p (RESTRICTED MODEL, RM)
Xt = α + 𝛾1 X𝑡−1 + 𝛾2X𝑡−2 + ⋯ + 𝛾𝑝X𝑡−𝑝
where p parameters (degrees of freedom) to be estimated.
In Case 2 the past values of Y contain information for forecasting Xt+1. Yt is said to “Granger cause”
Xt+1 provided Yt occurs before Xt+1 and it contains data for forecasting Xt+1.
Equation using a predictor Yt (UNRESTRICTED MODEL, UM)
Xt = α + 𝛾1 X𝑡−1 + 𝛾2X𝑡−2 + ⋯ + 𝛾𝑝X𝑡−𝑝 + α1Yt-1+ ⋯ + α𝑝 Yt-p
2p parameters (degrees of freedom) to be estimated.
If Yt causes Xt, then Y must precede X which implies:
 Lagged values of Y should be significantly related to X.
 Lagged values of X should not be significantly related to Y.
Case 3 cannot be used to find Granger causality since variable Yt is influenced by variable Wt.
Hypothesis test
Null Hypothesis (H0): Yt does not “Granger cause” Xt+1 i.e., 𝛼1 = 𝛼2 = ⋯ = 𝛼𝑝 = 0
Alternate Hypothesis (HA): Yt does “Granger cause” Xt+1, i.e., at least one of the lags of Y is
significant.
Calculate the f-statistic
Fp,n-2𝑝−1 = (𝐸𝑠𝑡𝑖𝑚𝑎𝑡𝑒 𝑜𝑓 𝐸𝑥𝑝𝑙𝑎𝑖𝑛𝑒𝑑 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒) / (𝐸𝑠𝑡𝑖𝑚𝑎𝑡𝑒 𝑜𝑓 𝑈𝑛𝑒𝑥𝑝𝑙𝑎𝑖𝑛𝑒𝑑 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒)
Fp,n-2𝑝−1 = ( (𝑆𝑆𝐸𝑅𝑀−𝑆𝑆𝐸𝑈𝑀) /𝑝) /(𝑆𝑆𝐸𝑈𝑀 /𝑛−2𝑝−1)
Where n is the number of observations and SSE is Sum of Squared Errors.
If the p-values are less than a significance level (0.05) for at least one of the lags then reject the null
hypothesis.
Perform test for both the direction Xt->Yt and Yt->Xt.
Try different lags (p). The optimal lag can be determined using AIC.
Limitation
 Granger causality does not provide any insight on the relationship between the variable hence
it is not true causality unlike ’cause and effect’ analysis.
 Granger causality fails to forecast when there is an interdependency between two or more
variables (as stated in Case 3).
 Granger causality test can’t be performed on non-stationary data.
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