Applying TimeSeries Analysis in Sport Management Research: Avoiding the Pitfalls

1
Applying Time-Series Analysis in Sport Management
Research Avoiding the Pitfalls

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Research Methodology Seminar
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Focus of seminar

• Time-series analysis the statistical analysis of
  a sample of time-ordered, periodic observations
• Example annual performance data for a sporting
  organisation
• Focus of seminar how to avoid the various
  potential pitfalls in the statistical analysis of
  time-series data that can invalidate the
  empirical results

3
Potential pitfalls in time-series analysis

• Dynamic dependence current outcome depends on
  past events (including past outcomes)
• Structural instability the non-local causal
  process (i.e. DGP) changes over time
• Residual autocorrelation estimated residuals are
  correlated over time
• Non-stationarity the statistical properties of
  the data (i.e. mean, variance and
  autocovariances) are time-dependent such that
  each observation has different distributional
  characteristics

4
Plan of workshop

• Introduction
• Dynamic dependence
• Structural instability
• The autocorrelation problem
• Non-stationarity and cointegration
• Application Man Utd average league gate
  attendances, 1946/47 2002/03
• Practical laboratory session

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Dynamic Dependence in Time-Series Models
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Dynamic dependence

• Dynamic model
• Current outcome f(past outcomes, current and
  past events)
• Formal representation (single period lag, one
  independent variable)
• Yt a0 a1Yt-1 ß1Xt ß2Xt-1 et
• Autoregressive distributed lag (ADL) model
• Mathematics stochastic linear difference equation

7
A simple dynamic model

• Yt a0 a1Yt-1 et
• 1st order difference equation
• Mathematically this can be solved to yield an
  expression for Yt in terms of the parameters,
  current and past disturbance terms and the
  initial value of Y
• Three components to solution
• long-run equilibrium position (Yt Yt-1, et
  0,)
• history of short-run disturbance effects
• current effect of initial deviation from long-run
  equilibrium

8
Dynamic dependence implies history matters

• Yt a0 a1Yt-1 et
• Solution
• Yt a0/(1- a1) ?(a1)iet-i (a1)tY0 - a0/(1
  - a1)
• a0/(1- a1) long-run equilibrium position
• ?(a1)iet-i history of short-run disturbance
  effects
• (a1)tY0 - a0/(1 - a1) current effect of
  initial deviation from long-run equilibrium
• a1 is known as the characteristic root
• Magnitude of a1 is crucial to determining the
  behaviour of Yt over time

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The nature of the adjustment path

• Yt a0 a1Yt-1 et
• Yt a0/(1- a1) ?(a1)iet-i (a1)tY0 - a0/(1
  - a1)
• a1 lt 1 gt stable, equilibrium-convergent
  adjustment path
• a1 gt 1 gt explosive, non-convergent path
• a1 1 gt unit-root process non-convergent
• a1 gt 0 gt direct adjustment
• a1 lt 0 gt oscillating (i.e. cyclical) adjustment
• Vital to determine whether or not you are dealing
  with a unit-root process

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The random walk model

• Yt a1Yt-1 et
• Random walk model a1 1
• Yt Yt-1 et
• gt ?Yt Yt Yt-1 et
• Solution
• Yt Y0 ? et-i
• Implications
• Yt follows a random, non-predictable path
• disturbances have permanent effects
• If a1lt1, disturbances have a decaying effect
• e, a1e, (a1)2 e, ...

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Short-run and long-run impacts

• Simple multivariate ADL
• Yt a0 a1Yt-1 ß1Xt et
• Example
• Dynamic win-wage benchmark model
• W/Lt a0 a1W/Lt-1 ß1WAGES t et
• Short-run impact of WAGES ß1
• Long-run impact (W/Lt W/Lt-1) ß1 /(1 - a1)
• a1Yt-1 represents learning/momentum effects

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Structural Instability
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Structural instability

• OLS estimation assumes that the DGP is
  structurally stable (i.e. parameter constancy)
  over the whole sample
• But time-series data may cover periods of
  structural change and regime shifts
• Structural stability may also be an inappropriate
  assumption for ordered cross-sectional data

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Consequences of structural instability

• Parameter estimates summarise the average effects
  across alternative regimes
• OLS estimates are biased for any individual
  regime
• Benchmarks and simulations based on the OLS
  estimates are biased

15
Methods of detecting structural instability

• Residuals
• Chow tests
• Chow test for structural instability
• Chows second test for predictive failure
• Recursive least squares
• Within-sample forecasting
• Hansen test

16
The limitations of OLS residuals

• OLS residuals are based on full sample
  information
• Smoothing effect that may disguise the presence
  of structural instability and outliers (i.e.
  observations far removed from rest of sample)
• Important to use other types of residuals to
  detect structural instability and outliers

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Alternative residuals to detect instability and
outliers

• Predicted residuals estimated coefficient on
  impulse dummy (i.e. predicted values obtained
  from estimated regression with observation
  omitted)
• Studentised residuals t-ratios of impulse dummy
• Time-series residuals
• Estimated innovations determined using parameter
  estimates that utilise only past sample
  information
• One-step residuals determined using parameter
  estimates that utilise only current and past
  sample information

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Chow test for structural stability

• Estimate pooled (i.e. full sample) regression
  model
• Estimate separate regression model for each
  sub-sample
• Apply F test for linear restrictions
• F(J,N-K) (RRSS URSS)/J
• URSS/(N K)
• RRSS restricted RSS
• URSS unrestricted RSS
• J no. of restrictions
• N sample size
• K no. of parameters in unrestricted model

19
Recursive least squares

• Recursive estimation involves
• estimating the regression model for an
• initial sub-sample and then re-
• estimating the model repeatedly as the
• sub-sample is expanded to include the
• next (ordered) observation

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Dealing with structural instability

• Separate sample into structurally stable
  sub-samples and estimate separate regression
  models
• Or
• Estimate single regression model for pooled
  sample with dummy variables for intercept and
  slope shifts
• Both approaches are equivalent

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TheAutocorrelation Problem
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What is the autocorrelation problem?

• Residual autocorrelation OLS residuals, et,
  serially correlated with lagged values, et-1,
  et-2,
• Common occurrence in estimated time-series models
• Autocorrelation problem multiple possible causes
  of residual autocorrelation with different
  solutions

23
Tests for detecting residual autocorrelation

• Durbin-Watson (DW) test
• Berenblut-Webb test
• Durbins h test
• Durbins alternative test
• Breusch-Godfrey test

24
The DW test for autocorrelation

• Traditional test for residual autocorrelation
• Test statistic
• d ?(et-et-1)2/?et2 ? 2(1 - ?)
• 0 lt d lt 4
• 0 lt d lt 2 ltgt positive residual autocorrelation
• 2 lt d lt 4 ltgt negative residual autocorrelation
• d 2 ltgt zero residual autocorrelation
• Appropriate only if 1st-order residual
  autocorrelation and no lagged dependent variables

25
The Breusch-Godfrey test for higher-order serial
correlation

• Regress residuals on explanatory variables and
  lagged residuals
• et f(X1t,..,Xkt, et-1,,et-j)
• Test the joint hypothesis that all the estimated
  coefficients on the lagged residuals are zero
• Test statistic, LM jF ?2(j)
• where F ? F statistic for joint significance of
  lagged residuals
• Alternatively test the overall significance of
  the auxiliary regression using LM nR2 ?2(kj)

26
One possible cause of residual autocorrelation

• One possible cause AR errors, et f(et-1, et-2,
  )
• Example AR(1) error process, et ?et-1 ut
• where
• autocorrelation parameter, ? ? corr(et, et-1)
• pure stochastic disturbance, ut IID(0, s2)
• AR error processes violate OLS assumptions
• Consequences
• OLS estimates unbiased but inefficient
• variance estimate biased rendering interval
  estimation and statistical tests invalid
• Solution error misspecification requires either
  reparameterisation to produce IID error process
  for OLS estimation (e.g. Cochrane-Orcutt
  quasi-differencing procedure) or use of
  alternative estimation methods e.g. RALS
• Conventional approach to residual autocorrelation

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But other possible causes of residual
autocorrelation

• Residuals may be autoregressive for several
  reasons
• AR disturbances (i.e. error misspecification)
• Model misspecification (e.g. excluded variables)
• Misspecified dynamics (e.g. static model
  estimated when dynamic dependence)
• Structural instability
• ARCH effects (i.e. error variance is
  autoregressive)
• Solution re-specify structural model if DGP
  misspecified
• AR disturbances are a special restricted case of
  misspecified dynamics

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The greatest non sequitur in econometrics?

• Perhaps the greatest non sequitur in the
  history of econometrics is the assumption that
  autocorrelated residuals entail autoregressive
  errors, as is entailed in correcting serial
  correlation using Cochrane-Orcutt.
• (Doornik and Hendry)

29
Test, test and test again

• Dont rely only on the DW test
• Use other tests for residual autocorrelation e.g.
  Breusch-Godfrey test
• Test for general model misspecification e.g.
  Ramsey RESET test
• Test for misspecified dynamics
• Test for structural instability and ARCH effects
• Test for validity of imposing restriction of AR
  errors
• Investigate robustness of parameter estimates to
  alternative model strategies

30
Non-Stationary Time Series and Cointegration
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Stationary, non-stationary and integrated
processes

• Stationary process distributional properties are
  time-independent
• Non-stationary process distributional properties
  are time-dependent
• Integrated process non-stationary process that
  can be transformed into a stationary process by
  differencing
• Yt is integrated of order d, I(d), if it requires
  to be differenced d times to become stationary
• Conventional statistical tests are invalid for
  non-stationary processes

32
ARIMA(p,d,q) models

• General representation of I(d) process
• ARMA(p,q) model of dth difference
• Autoregressive AR(p) process
• Yt a0 ?paiYt-i et
• Moving-average MA(q) error process
• Yt a0 ?qdiet-i
• ARIMA(p,1,q) model
• ?Yt ?0 ?p?i?Yt-i ?t ?q dj?t-j
• Starting point for Box-Jenkins methodology for
  modelling univariate time series

33
Dickey-Fuller (DF) tests for unit roots

• Unit-root processes are non-stationary.
• Crucial to test for unit roots
• AR(1) process
• Yt ?1Yt-1 ?t
• Subtracting Yt-1 from both sides yields
• ?Yt ?Yt-1 ?t
• where ? ?1 - 1
• Unit root gt ?1 1
• Dickey-Fuller test for unit root ? 0

34
Alternative forms of the DF test

• Pure random walk
• ?Yt ?Yt-1 ?t
• Random walk with drift
• ?Yt ?0 ?Yt-1 ?t
• Random walk with drift and time trend
• ?Yt ?0 ?Yt-1 ?2t ?t

35
The DF test statistics

• t test of null hypothesis, ? 0
• DF test statistic does not have standard t
  distribution
• Critical values depend on model specification,
  sample size and significance level
• F test of null hypothesis, ? 0, jointly with ?0
  0 and/or ?2 0
• F test has non-standard critical values

36
Augmented Dickey-Fuller (ADF) tests

• DF test assumes that the error term is a
  white-noise process
• If ARMA error process, DF regressions should be
  augmented with additional ?Yt-j terms
• Test statistics and critical values for ADF tests
  are the same as for the DF tests

37
Cointegration definition and interpretation

• Suppose Yt, Xt are both I(1) processes
• Yt and Xt are cointegrated, CI(1,1), if there
  exists ? such that Yt - ?Xt ?t I(0)
• Cointegration implies that a long-run
  relationship exists between Yt and Xt they do
  not drift apart over time

38
Spurious regression problem

• Regression of one random walk on another
  independent random walk
• Regression of integrated variables with no
  cointegrating relationship
• May have high goodness of fit
• Extremely low DW statistic indicative of spurious
  regression problem

39
Testing for cointegration

• Two principal tests for cointegration
• Engle-Granger test apply unit-root test to
  residuals from cointegrating regression (but DF
  critical values not valid)
• Cointegrating regression Durbin-Watson (CRDW)
  test no cointegrating relationship implies DW 0

40
The Granger representation theorem

• If Yt and Xt are cointegrated, there is a
  long-run relationship between the two time series
  and the short-run dynamics can be described by
  the error correction model (ECM).

41
The ECM

• Long-run relationship
• Yt ?Xt ?t
• ECM (short-run dynamics)
• ?Yt ??Xt ?Yt-1 - ?Xt-1 ?t
• Two components of the current change in Yt
• (i) the short-run response to current changes in
  Xt, ??Xt and
• (ii) the partial correction of the previous
  deviation of Yt from its optimal long-run level,
  ?Yt-1 - ?Xt-1

42
Causality tests

• Granger causality the future cannot cause the
  present or past
• Xt fails to Granger-cause Yt if, in a regression
  of Yt on lagged Ys and lagged Xs, the
  coefficients on the lagged Xs are jointly
  insignificant

43
Application
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Application football gate attendances

• Theory Gate attendance f(current and past
  playing performance, divisional demand)
• Data Man Utd, average league gate,
  1946/47-2002/03 (57 observations)
• Key results
• ManUGate is stationary process (whole sample)
• Autocorrelated residuals and possible ARCH
  effects
• RALS ? D1AvGate and PtsGap as determinants
• Dynamic model ? possible FA Cup effect
• Evidence of structural instability, 1992/93
  onwards
• FA Cup effect significant pre-1992/93
• Possible unit-root process pre-1992/93