Aurlie Lemmens, Erasmus University Rotterdam

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Measuring and Testing Granger Causality over the
Spectrum

• Aurélie Lemmens, Erasmus University Rotterdam
• Christophe Croux, K.U. Leuven
• Marnik G. Dekimpe, Tilburg University and K.U.
  Leuven

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The Concept of Granger Causality

• The 2003 Nobel Prize in Economics
• Granger (1969)
• A process Xt is said to Granger cause another
  process Yt
• if future values of Yt can be predicted better
  using past values of Xt and Yt
• than using the past of Yt alone.
• Incremental predictability the extent to which a
  process leads another.
• In practice, several bivariate tests (e.g., the
  Granger-Sargent test, the Granger-Wald test, the
  Haugh-Pierce test, or the Sims test), or
  multivariate tests (El Himdi and Roy test) exist.

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Previous Marketing Literature

• More than 20 articles using Granger causality in
  the major marketing journals over the last 3
  decades.
• Issues covered using Granger causality
• Advertising effect on sales (Leone, 1983
  Hanssens, 1980 Holak and Tang 1990) or aggregate
  consumption (Ashley, et al. 1980 Jacobson and
  Nicosia, 1981)
• Competitive (price) reactions and price
  leadership (Hanssens, 1980 Kuiper and
  Meulenberg, 2004 Leeflang and Wittink, 1992
  Roy, et al. 2006)
• Private labels shares during economic expansions
  and contractions (Lamey, et al. 2007)
• Objective vs. perceived quality (Mitra and
  Golder, 2006).
• Movies theater distribution box office demand
  relation (Krider, et al. 2005).
• Cross-category purchases interactions
  (Chintagunta and Haldar, 1998)
• Market shares price relationships (Bass and
  Pilon, 1980)
• Variables selection procedure (Horvath, Leeflang
  and Otter, 2002).
• Or to rule out reverse causality (Luo and Homburg
  2007 McAlister, et al. 2007).

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Importance of the Periodicity

• Increasing evidence of different (in direction or
  importance) relations over different planning
  horizons
• Short and long-run effectiveness of the
  marketing-mix (see e.g. Nijs, Dekimpe, Steenkamp,
  Hanssens, 2001 Pauwels, Hanssens, Siddarth,
  2002).
• Cooperative versus competitive interactions
  (Bronnenberg, Mela and Boulding 2004)

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4 weeks
25 weeks
Bronnenberg, Mela and Boulding (2005)
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Importance of the Periodicity

• Increasing evidence of different (in direction or
  importance) relations over different planning
  horizons
• Short and long-run effectiveness of the
  marketing-mix (see e.g. Nijs, Dekimpe, Steenkamp,
  Hanssens, 2001 Pauwels, Hanssens, Siddarth,
  2002).
• Cooperative versus competitive interactions
  (Bronnenberg, Mela and Boulding 2004)
• Aggregate advertising and GNP relation at the
  business cycle (Deleersnyder, Dekimpe and
  Leeflang, 2004)
• ..

Decomposing Granger causality over different
periodicities or frequencies
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Methodology

• Two approaches exist (and can be extended)
• Pierce (1979)
• Based on the decomposition of the R-squared over
  each frequency of the spectrum through the
  coherence function.
• Non-parametric approach
• No testing procedure yet
• Geweke (1982)
• Based on the spectral decomposition of the vector
  autoregressive polynomial in a bivariate vector
  autoregressive model containing Xt and Yt.
• Parametric approach
• Available testing procedures proposed by Hosoya
  (1991) Yao Hosoya (2000) Breitung and
  Candelon (2006).
• Although these two approaches are different in
  their operationalizations, they are similar in
  spirit (Pierce, 1982).

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Methodology Spectral Analysis

• The basic concept of spectral analysis
• Any time series Yt can be decomposed into an
  infinite sum
• of components, each having a different frequency ?

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Spectral Analysis
Some Intuition
time
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Spectral Analysis
High Frequency - Short-run variations
Granger causality
. . .
. . .
Medium Frequency - Middle-run variations
Granger causality
. . .
. . .
Low Frequency - Long-run variations
Granger causality
Time
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Dynamic Granger Causality (Example)
LR
MR
Granger causality
SR
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The Gewekes Framework

• Let Xt and Yt be 2 stationary series (possibly
  after transformation)
• We have a bivariate VAR model with (Yt , Xt).
• Take the first equation
• Xt does not Granger cause Yt at frequency ? if
• Use standard F-test (Breitung and Candelon 2006)

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The Pierces Framework

• We model Xt and Yt stationary (possibly after
  transformation) as 2 univariate ARMA processes
• We can define the coefficient of coherence
• Between zero and one (R-squared)
• Measure of the strength of linear association
  between two time series

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The Pierces Framework (2)

• It is possible to decompose the cross-spectrum in
  3 parts
• Hence, we define a Granger coefficient of
  coherence as
• And the test statistics is given by
  with

0 when no Granger causality
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Monte Carlo Simulations

• We simulate 10,000 series of length T 216 from
  the following model
• ?

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Monte Carlo Simulations (2)
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Monte Carlo Simulations (3)
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Application
Production expectations

• 68,000 companies and 27,000 consumers
• Ex-ante judgments about the future industrial
  production levels,
• i.e. Production Accounts (OECD)

What it their predictive content ?

• Significant predictive content in Finland and
  Sweden (Bergström 1995, Lindström 2000, Öller and
  Tallbom 1996, Teräsvirta 1986)
• No systematic predictive content in Belgium,
  France, Germany, The Netherlands and Italy
  (Hanssens and Vanden Abeele 1987)
• Partial within-country and cross-country
  causality in Europe (Lemmens, et al. 2005)

Is the timeliness of the surveys a major
advantage?
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Segment 1 Long-Run Causality

• Consistent pattern across 7 countries
• i.e. Austria, Belgium, Finland, France, Italy,
  The Netherlands, and the United Kingdom.
• Longer-run GC gtgt Shorter-run GC

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Segment 2 No Causality

• Consistent pattern across 4 countries
• i.e. Greece, Ireland, and, to a lesser extent,
  Denmark and Luxembourg
• No predictive content at any frequency

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Segment 3 Short- and Long-run Causality

• Single-country segment with short- and long-run
  predictive content for Germany

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Thanks