The case of tourism-led growth hypothesis: global question, global evidence.

"There is no mixed evidence, only poorly synthesized"" (D. Campbell)

• Use a linear representation of the series:

\[ y_{i,t} = \alpha_{i} + \sum\limits_{k=1}^{K} \gamma_{k} y_{i,t-k} + \sum\limits_{k=1}^{K} \beta_{k} x_{i,t-k} + \epsilon_{i,t}, \]

• Granger = Test signifficance of \(\beta_{k}\): IV ( Holtz-Eakin, Newey & Rosen, 1988 ), GMM ( Arellano & Bond, 1991)

• Tourism: Sequeira & Nunes (2008); Debt: Panizza & Presbitero (2014)

• Coefficients differ in cross-sections

\[ y_{i,t}=\alpha_{i} + \sum\limits_{k=1}^{K} \gamma_{ik} y_{i,t-k} + \sum\limits_{k=1}^{K} \beta_{ik} x_{i,t-k} + \epsilon_{i,t}, \]

• Use cross-section weighted average of Wald statistics: Dumitrescu & Hurlin, (2012); López & Weber, (2017).

• Test is asymptotically normal for large N, small T. Bootstrap as an alternative.

• structural breaks,
• non-linear function form,
• outliers,
• higher-moment causality

• Hiemstra & Jones (1994) propose a bivariate kernel for time-series. Bai et al. (2016) reformulate and extend.

• Our proposal:
  • A model-free casuality test for panel data that is
  • robust and simple,
  • based on symbolic transfer entropy (STE).

• (Shannon) entropy: measures amount of information or uncertainty in stochastic data (Shannon, 1984)

\[ H_y = - E_0(\log p(y)) \]

• TE quantifies reduction in uncertainty (information flow) in \(Y\) from the state of \(X\) (conditional on past Y).

\[ TE_{x \rightarrow y} = H_{y_t|y_{t-1}} - H_{y_t|x_{t-1},y_{t-1}} \]

• TE and Geweke's GC measures are equivalent under Gaussian linear processes.
• Under weak stationarity, all GC will be detected by TE (Serés et al. 2016).

• An extension of TE defined on rank points (Schreiber, 2000)
• Window (embedding dimension) \(m\). Sort data and compute ranks. Ranked vectors are symbols.
• Example: \(x_t\) = { 3,9,7,6,5,10,4 }

• Symbolic Transfer Entropy (STE): TE with symbolized data cross-section pool.
• Assess significance using Markov block bootstrap (keep time-dim structure).
• Warnings: Cross-Section Dependence and weak stationarity is assumed.

• Consider five DGPs.
• Sensititity of
  • shock persistence, \(\alpha=\{ 0,0.3,0.9 \}\)
  • and sample size, T={ 15,30,60 } , N={ 30,60,120 }.
• Estimate Surface Responses of test accept/reject outcomes, 1000 sims.
• Compute Granger-OLS, Dumitrescu-Hurlin and STE tests and compare power/size.

\[ \begin{eqnarray*} y_{it} &=& \alpha y_{i(t-1)} + \beta x_{i(t-1)} + \varepsilon_{it} \\ x_{it} &\sim& N(0,1) \\ \varepsilon_{it} &\sim& N(0,1) \\ \beta &\sim & U(0,2) \\ \end{eqnarray*} \]

\[ \begin{eqnarray*} y_{it} &=& \alpha y_{i(t-1)} + \varepsilon_{it}\\ \varepsilon_{it} &\sim& N(0,|x_{i(t-1)}|) \\ x_{it} &\sim& N(0,1) \\ \end{eqnarray*} \]

\[ \begin{eqnarray*} y_{it} &=& \alpha y_{i(t-1)} + \beta x_{i(t-1)} + \varepsilon_{it} \\ x_{it} &\sim& N(0,1) \\ \varepsilon_{it} &\sim& N(0,1) \\ \beta &=& 0 \\ \end{eqnarray*} \]

with extreme obs. at endpoints \(y_{2,1}= x_{1,1}\)=-10, \(y_{T,N} = x_{(T-1),N}\)=10.

\[ \begin{eqnarray*} y_{it} &=& y_{i(t-1)} x_{i(t-1)} + \varepsilon_{it} \\ x_{it} &\sim& N(0,1) \\ e_{it} &\sim& N(0,1) \\ \end{eqnarray*} \]

\[ \begin{eqnarray*} y_{it} &=& c_1 + \alpha y_{i(t-1)} + \beta_1 x_{i(t-1)} + % \varepsilon_{it} ~~\forall t=1,\ldots,T_1\\ y_{it} &=& c_2 + \alpha y_{i(t-1)} + \beta_2 x_{i(t-1)} + % \varepsilon_{it} ~~\forall t=T_1,\ldots,T\\ x_{it} &\sim& N(0,1) \\ e_{it} &\sim& N(0,1) \\ \alpha &=& \{ 0 , 0.3, 0.9 \} \\ c_1 &=& -c_2 = 1 \\ \beta_1 &\sim & U(0,2) \\ \beta_2 &=& -\beta_1 \\ \end{eqnarray*} \]

• The STE test of causality is
  • robust to many relevant data problems.
  • simple to run (package in Matlab/R available from authors)
• In panel data, robust and simple global answer to a global question.
• Shannon entropy can be extended to non-stationarity processes.
• Work in progress:
  • Missing observations/unbalanced panel.
  • The effect of cross-section dependence.