Stock Markets - A Unique Intensity Based Information Share
C.3 VECM and Hasbrouck Shares
According to the law of one price, prices in different trading venues that refer to the same underlying asset are cointegrated, meaning that they can only deviate from each other in the short run. Assume that TSX and NYSE price dynamics can be described by a bivariate vector autoregression of orderq, we model price changes, ∆pt=pt−pt−1, as a bivariate vector error correction model (VECM),
∆pt=αβ′pt−1+ Γ1∆pt−1+. . .+ Γq−1∆pt−q+1+ut , (C.3) wherept= (p1t, p2t)′, Γ1 to Γq−1 are 2×2 parameter matrices. ut= (u1t, u2t)′ is a white noise vector with zero means and covariance matrix Σu. The vector α = (α1, α2)′ contains the coefficients associated with the speed of adjustment of each price series to deviations from the equilibrium. β denotes the 2×1 cointegration vector, which implies that there exists one common stochastic trend, which can be considered as the stock’s underlying efficient price.
Hasbrouck’s information shares are then derived as the contribution of an innovation in one market’s price series to the underlying efficient price innovations variance. Since the VECM innovations (ut) tend to be contemporaneously correlated, the shares cannot be uniquely identified. To solve this dilemma, Hasbrouck applies the Cholesky decomposition to the covariance matrix of innovations (Σu =CC′).
With the home market ordered first, Hasbrouck information shares of TSX (HIS1) and NYSE (HIS2) can be computed as,
HIS1 = [ξ′C[1]]2
ξ′CC′ξ and HIS2 = [ξ′C[2]]2
ξ′CC′ξ , (C.4)
where ξ′C[j]denotes the jth element of the vector ξ′C andξ gives the common row vector in the matrix of long run impacts (Ξ) of timet idiosyncratic innovations on the efficient price.
It is derived as,
Ξ=β⊥[α′⊥(In−
q−1
X
i=1
Γi)β⊥]−1α′⊥ . (C.5) The Cholesky decomposition implies that the contribution of the market ordered first is maximized and that of the market ordered second is minimized. Since there is no theoretical justification for such a hierarchy, the common solution is to permutate the ordering
of the markets. This yields upper and lower bounds of information shares. The main drawback of Hasbrouck’s methodology is that these bounds can diverge considerably, as the contemporaneous correlation between the composite innovations u1t and u2t tends to increase with decreasing sampling frequency.
In our application to Canadian stocks we choose a one minute sampling frequency using transaction prices. Thereby the US stock prices are converted to Canadian Dollars. After testing for cointegration using the maximum eigenvalue and trace statistic, we confirm the existence of one cointegration relation. The number of lags in the VECM is determined by the Schwarz information criterion (see Schwarz 1978). Standard errors for Hasbrouck’s information shares are derived by a nonparametric bootstrap as proposed by Grammig et al.
(2004).
Conclusion
Modeling data on their lowest observation level is an extremely active research field in the empirical econometrics literature. Since irregular spacing of many financial and economic data is a key characteristic, new methodologies are introduced that account for this data property. These approaches are based on point processes which describe the history of events that occur consecutively in time. A process consisting of points at which we observe simultaneously variables that “mark” the points is called a marked point process. In this thesis we present new univariate and multivariate empirical (marked) point process studies that use irregular observed monetary and financial data.
Since precise predictions of short term interest rates are of key interest to investors and financial institutions, we investigate a marked point process model for the federal funds rate target and its forecast performance in Chapter 2. By modeling the target as a marked point process, Hamilton and Jord`a (2002) considerably reduce the forecast mean squared errors compared to a standard time series method that uses equidistant data.
In Chapter 2 of this thesis we present a new marked point process for the target and show that the suggested specifications deliver improved results in terms of goodness of fit and in-sample forecast performance. The proposed methodology to evaluate probability function forecasts reveals useful target probability forecasts up to a six months horizon. Out-of-sample results are promising as well and Bayesian type model averaging robustifies the point forecast performance. We conclude from our findings that the model seems to capture very well the target characteristics: target changes occur in discrete time with discrete increments and
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have an autoregressive nature.
As Chapter 2, Chapter 3 of this thesis investigates the forecast ability of a marked point process model. We present a time series model to predict total return variation. Due to the importance of accurate volatility forecasts in the valuation of derivatives, portfolio management and risk management, the prediction of volatility plays a central role in financial econometrics literature.
Chapter 3 is based on the work of Andersen et al. (2003) and Andersen and Bollerslev (1998) who use high frequency intraday data to introduce the concept of a nonparametric realized volatility measure. As shown by Barndorff-Nielsen and Shephard (2004) realized volatility can be decomposed into a continuous and a jump variation part. Since continuous variation is serially correlated, we model it by an autoregressive conditional time series model. Daily variation jumps that occur irregularly in time are conceived as marked point process. Continuous and jump variation models are combined to assess the accuracy of point and density forecasts of total return variation. The main findings of the empirical section can be summarized as follows. The estimation of the models yields sensible results in terms of diagnostics and parameter estimates. Density forecast evaluations confirm the suitability of this approach with respect to modeling the evolution of realized, continuous and jump variation. A point forecast analysis shows that the suggested model yields at least as accurate and in some cases even more accurate volatility forecasts than standard models that use equally spaced data.
In Chapter 4 we extend the univariate point process of the previous chapters to a multivariate point process and propose a new information share that measures the home and foreign market share in price discovery. We apply a bivariate autoregressive conditional intensity approach that accounts for the irregularity of the data, the informational content of time between consecutive trades and the timing interdependencies between two markets’
transaction processes. In contrast to the commonly applied Hasbrouck (1995) methodology that requires equidistant data we deliver a unique information share rather than lower and upper bounds.
Since national stock exchanges fear to lose their attractiveness for investors and are
ambitious to remain the leading market with regard to price discovery, the identification of an information share is of paramount concern for a trading venue. We apply Hasbrouck’s (1995) and our intensity based information share to analyze the price discovery process of Canadian stocks, which are traded on the Toronto Stock Exchange (TSX) and cross-listed on the New York Stock Exchange (NYSE). We find that despite the concern of the TSX to lose its share in price discovery to the NYSE, trading on the TSX still plays the most important role. Further, we show that the leadership of the TSX is even more pronounced than indicated by previous studies. The average TSX information share amounts to 71%.
We also compare our results to the Hasbrouck (1995) information shares. On average we find a larger home market contribution than indicated by the Hasbrouck midpoints.
Summarizing the results of Chapter 2 through 4 we find evidence that using (marked) point processes to model non-aggregated data helps to explore the timing related information.
Further, we conclude that accounting for the irregular spacing of economic and financial variables can improve a model’s fit and forecast or solve problems arising using equidistant data. Overall, the presented findings are promising and in favor for (marked) point processes compared to standard econometric methods that use equally spaced data.
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