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Time Series Econometrics:
Asst. Prof. Dr. Mete Feridun Department of Banking and Finance
Faculty of Business and Economics Eastern Mediterranean
University
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What is a time series?A time series is any series of data that
varies over time. For exampleMonthly Tourist Arrivals from
KoreaQuarterly GDP of LaosHourly price of stocks and sharesWeekly
quantity of beer sold in a pubBecause of widespread availability of
time series databases most empirical studies use time series
data.
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Caveats in Using Time Series Data in Applied Econometric
ModelingData Should be StationaryPresence of AutocorrelationGuard
Against Spurious RegressionsEstablish CointegrationReconcile SR
with LR Behavior via ECM Implications to ForecastingPossibility of
Volatility Clustering
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What is a Stationary Time Series?
A Stationary Series is a Variable with constant Mean across
time
A Stationary Series is a Variable with constant Variance across
time
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These are Examples of Non-Stationary Time Series
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These are Examples of Stationary Time Series
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What is a Unit Root?If a Non-Stationary Time Series Yt has to be
differenced d times to make it stationary, then Yt is said to
contain d Unit Roots. It is customary to denote Yt ~ I(d) which
reads Yt is integrated of order d
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Establishment of Stationarity Using Differencing of Integrated
SeriesIf Yt ~ I(1), then Zt = Yt Yt-1 is Stationary
If Yt ~ I(2), then Zt = Yt Yt-1 (Yt Yt-2 )is Stationary
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Unit Root Testing: Formal Tests to Establish Stationarity of
Time SeriesDickey-Fuller (DF) TestAugmented Dickey-Fuller (ADF)
TestPhillips-Perron (PP) Unit Root TestDickey-Pantula Unit Root
TestGLS Transformed Dickey-Fuller TestERS Point Optimal TestKPSS
Unit Root TestNg and Perron Test
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What is a Spurious Regression?A Spurious or Nonsensical
relationship may result when one Non-stationary time series is
regressed against one or more Non-stationary time series
The best way to guard against Spurious Regressions is to check
for Cointegration of the variables used in time series modeling
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Symptoms of Likely Presence of Spurious RegressionIf the R2 of
the regression is greater than the Durbin-Watson Statistic
If the residual series of the regression has a Unit Root
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CointegrationIs the existence of a long run equilibrium
relationship among time series variablesIs a property of two or
more variables moving together through time, and despite following
their own individual trends will not drift too far apart since they
are linked together in some sense
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Two Cointegrated Time Series
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Cointegration Analysis: Formal TestsCointegrating Regression
Durbin-Watson (CRDW) Test
Augmented Engle-Granger (AEG) Test
Johansen Multivariate Cointegration Tests or the Johansen
Method
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Error Correction Mechanism (ECM)Reconciles the Static LR
Equilibrium relationship of Cointegrated Time Series with its
Dynamic SR disequilibriumBased on the Granger Representation
Theorem which states that If variables are cointegrated, the
relationship among them can be expressed as ECM.
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Forecasting: Main MotivationJudicious planning requires reliable
forecasts of decision variablesHow can effective forecasting be
undertaken in the light of non-stationary nature of most economic
variables?Featured techniques: Box-Jenkins Approach and Vector Auto
regression (VAR)
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Approaches to Economic ForecastingThe Box-Jenkins ApproachOne of
most widely used methodologies for the analysis of time-series
dataForecasts based on a statistical analysis of the past data.
Differs from conventional regression methods in that the mutual
dependence of the observations is of primary interestAlso known as
the autoregressive integrated moving average (ARIMA) model
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Advantages Derived from solid mathematical statistics
foundationsARIMA models are a family of models and the BJ approach
is a strategy of choosing the best model out of this family It can
be shown that an appropriate ARIMA model can produce optimal
univariate forecastsDisadvantagesRequires large number of
observations for model identificationHard to explain and interpret
to unsophisticated usersEstimation and selection an art
formApproaches to Economic ForecastingThe Box-Jenkins Approach
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Differencing the series to achieve stationarityIdentify model to
be tentatively entertainedEstimate the parameters of the tentative
modelDiagnostic checking. Is the model adequate?NoYesUse the model
for forecasting and controlApproaches to Economic ForecastingThe
Box-Jenkins Approach
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Approaches to Economic ForecastingThe Box-Jenkins
Approach-Identification ToolsAutocorrelation function (ACF)- ratio
of sample covariance (at lag k) to sample variancePartial
autocorrelation function (PACF) measures correlation between (time
series) observations that are k time periods apart after
controlling for correlations at intermediate lags (i.e., lags less
than k). In other words, it is the correlation between Yt and Yt-k
after removing the effects of intermediate Ys. Correlogram graph
showing the ACF and the PACF at different lags.
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Approaches to Economic ForecastingThe Box-Jenkins
Approach-IdentificationTheoretical Patterns of ACF and PACF
Type of ModelTypical Pattern of ACFTypical Pattern of PACFAR
(p)Decays exponentially or with damped sine wave pattern or
bothSignificant spikes through lags pMA (q)Significant spikes
through lags qDeclines exponentiallyARMA (p,q)Exponential
decayExponential decay
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Approaches to Economic ForecastingThe Box-Jenkins
Approach-Diagnostic CheckingHow do we know that the model we
estimated is a reasonable fit to the data? Check residualsRule of
thumb: None of the ACF and the PACF are individually statistically
significantFormal test: Box-Pierce Q Ljung-Box LB
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Approaches to Economic ForecastingSome issues in the Box-Jenkins
modeling Judgmental decisions on the choice of degree of order on
the choice of lags Data mining can be avoided if we confine to AR
processes only fit versus parsimony Seasonalityobservations, for
example, in any month are often affected by some seasonal
tendencies peculiar to that month. the differencing operation
considered as main limitation for a series that exhibit moving
seasonal and moving trend.
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Vector Autoregression (VAR)IntroductionVAR resembles a SEM
modeling we consider several endogenous variables together. Each
endogenous variables is explained by its lagged values and the
lagged values of all other endogenous variables in the model.In the
SEM model, some variables are treated as endogenous and some are
exogenous (predetermined). In estimating SEM, we have to make sure
that the equation in the system are identified this is achieved by
assuming that some of the predetermined variables are present only
in some equation (which is very subjective) and criticized by
Christopher Sims.If there is simultaneity among set of variables,
they should all be treated on equal footing, i.e., there should not
be any a priori distinction between endogenous and exogenous
variables.
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Vector Autoregression (VAR)Its UsesForecastingVAR forecasts
extrapolate expected values of current and future values of each of
the variables using observed lagged values of all variables,
assuming no further shocksImpulse Response Function (IRFs)IRFs
trace out the expected responses of current and future values of
each of the variables to a shock in one of the VAR equations
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Vector Autoregression (VAR)Its UsesForecast Error Decomposition
of Variance (FEDVs)FEDVs provide the percentage of the variance of
the error made in forecasting a variable at a given horizon due to
specific shock. Thus, the FEDV is like a (partial) R2 for the
forecast errorGranger Causality TestsGranger-causality requires
that lagged values of variable A are related to subsequent values
in variable B, keeping constant the lagged values of variable B and
any other explanatory variables
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Vector Autoregression (VAR)Mathematical Definition[Y]t =
[A][Y]t-1 + + [A][Y]t-k + [e]t orwhere: p = the number of variables
be considered in the systemk = the number of lags be considered in
the system[Y]t, [Y]t-1, [Y]t-k = the 1x p vector of variables[A],
and [A'] = the p x p matrices of coefficients to be estimated[e]t =
a 1 x p vector of innovations that may be contemporaneously
correlated but are uncorrelated with their own lagged values and
uncorrelated with all of the right-hand side variables.
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Vector Autoregression (VAR)ExampleConsider a case in which the
number of variables n is 2, the number of lags p is 1 and the
constant term is suppressed. For concreteness, let the two
variables be called money, mt and output, yt . The structural
equation will be:
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Vector Autoregression (VAR)Example Then, the reduced form is
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Vector Autoregression (VAR)ExampleAmong the statistics computed
from VARs are:Granger causality tests which have been interpreted
as testing, for example, the validity of the monetarist proposition
that autonomous variations in the money supply have been a cause of
output fluctuations.Variance decomposition which have been
interpreted as indicating, for example, the fraction of the
variance of output that is due to monetary versus that due to real
factors.Impulse response functions which have been interpreted as
tracing, for example, how output responds to shocks to money (is
the return fast or slow?).
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Vector Autoregression (VAR)Granger CausalityIn a regression
analysis, we deal with the dependence of one variable on other
variables, but it does not necessarily imply causation. In other
words, the existence of a relationship between variables does not
prove causality or direction of influence.In our GDP and M example,
the often asked question is whether GDP M or M GDP. Since we have
two variables, we are dealing with bilateral causality. Given the
previous GDP and M VAR equations:
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Vector Autoregression (VAR)Granger CausalityWe can distinguish
four cases: Unidirectional causality from M to GDP Unidirectional
causality from GDP to M Feedback or bilateral causality
Independence Assumptions:Stationary variables for GDP and MNumber
of lag termsError terms are uncorrelated if it is, appropriate
transformation is necessary
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Vector Autoregression (VAR)Granger Causality Estimation
(t-test)A variable, say mt is said to fail to Granger cause another
variable, say yt, relative to an information set consisting of past
ms and ys if: E[ yt | yt-1, mt-1, yt-2, mt-2, ] = E [yt | yt-1,
yt-2, ].mt does not Granger cause yt relative to an information set
consisting of past ms and ys iff 21 = 0.yt does not Granger cause
mt relative to an information set consisting of past ms and ys iff
12 = 0.In a bivariate case, as in our example, a t-test can be used
to test the null hypothesis that one variable does not Granger
cause another variable. In higher order systems, an F-test is
used.
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1. Regress current GDP on all lagged GDP terms but do not
include the lagged M variable (restricted regression). From this,
obtain the restricted residual sum of squares, RSSR.2. Run the
regression including the lagged M terms (unrestricted regression).
Also get the residual sum of squares, RSSUR.3. The null hypothesis
is Ho: i = 0, that is, the lagged M terms do not belong in the
regression. 5. If the computed F > critical F value at a chosen
level of significance, we reject the null, in which case the lagged
m belong in the regression. This is another way of saying that m
causes y.Vector Autoregression (VAR)Granger Causality Estimation
(F-test)
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Vector Autoregression (VAR)Variance DecompositionOur aim here is
to decompose the variance of each element of [Yt] into components
due to each of the elements of the error term and to do so for
various horizon. We wish to see how much of the variance of each
element of [Yt] is due to the first error term, the second error
term and so on. Again, in our example:The conditional variance of,
say mt+j, can be broken down into a fraction due to monetary shock,
mt and a fraction due to the output shock, yt .
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Vector Autoregression (VAR)Impulse Response FunctionsHere, our
aim is to trace out the dynamic response of each element of the
[Yt] to a shock to each of the elements of the error term. Since
there are n elements of the [Yt], there are n2 responses in all.
From our GDP and money supply example:We have four impulse response
functions:
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Vector Autoregression (VAR)Pros and ConsAdvantagesThe method is
simple; one does not have to worry about determining which
variables are endogenous and which ones exogenous. All variables in
VAR are endogenousEstimation is simple; the usual OLS method can be
applied to each equation separatelyThe forecasts obtained by this
method are in many cases better than those obtained from the more
complex simultaneous-equation models.
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Vector Autoregression (VAR)Pros and ConsSome Problems with VAR
modelingA VAR model is a-theoretic because it uses less prior
information. Recall that in simultaneous equation models exclusion
or inclusion of certain variables plays a crucial role in the
identification of the model.Because of its emphasis on forecasting,
VAR models are less suited for policy analysis.Suppose you have a
three-variable VAR model and you decide to include eight lags of
each variable in each equation. You will have 24 lagged parameters
in each equation plus the constant term, for a total of 25
parameters. Unless the sample size is large, estimating that many
parameters will consume a lot of degree of freedom with all the
problems associated with that.
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Vector Autoregression (VAR)Pros and ConsStrictly speaking, in an
m-variable VAR model, all the m variables should be (joint)
stationary. If they are not stationary, we have to transform (e.g.,
by first-differencing) the data appropriately. If some of the
variables are non-stationary, and the model contains a mix of I(0)
and I(1), then the transforming of data will not be easy.Since the
individual coefficients in the estimated VAR models are often
difficult to interpret, the practitioners of this technique often
estimate the so-called impulse response function. The impulse
response function traces out the response of the dependent variable
in the VAR system to shocks in the error terms, and traces out the
impact of such shocks for several periods in the future.