Methodology

Financial integration through the law of one price can be measured using two models. The first consists of a traditional autoregressive (AR) model. Higher convergence speeds reflect a quicker convergence to LOOP and hence stronger financial integration. We estimate the half-life of shocks applying the Augmented Dickey-Fuller model, with one autoregressive component and other lagged differences. We use the parameter of the autoregressive factor to calculate the half-life, which also takes into account the other lags. The half-life is calculated as ln(0.5)/ln(1-beta). The model includes GARCH effects to account for the heteroskedasticity prevalent in the data. Lags are included so that no serial correlation or heteroskedasticity is present in the residuals.

The second type of model we apply is the non-linear threshold autoregressive (TAR) model. The existence of (variable) transaction costs implies that two different regimes exist, an arbitrage and a no-arbitrage regime. If the difference between the two prices is smaller than the transaction costs, arbitrage will not take place and the difference can persist. However, when a shock in either of the two markets results in a difference between the two prices that exceeds the transaction costs (that is, the premium is outside the no-arbitrage band), it will trigger profitable arbitrage trades that would elicit a strong pressure on the premium to go back inside the band.¹⁸ In other words, theoretically there will be a no-arbitrage regime, where the persistence is high, and an arbitrage regime, where there exists pressure on prices to converge. As the TAR model assumes a discrete change in the AR process once a certain threshold is crossed, this model provides a natural choice to characterize the type of regime changes that we expect to be prevalent in the DR market. To the extent that high transaction costs, and hence a broader band of no-arbitrage, are associated with a lower level of financial integration, the estimated width of the no-arbitrage bands provides a measure of effective integration.¹⁹

The TAR model was first proposed by Tong (1978) and further developed by Tong and Lim 1980) and Tong (1983). Its main premise is to describe the data-generating process by a piecewise linear autoregressive model. A TAR model works by estimating regime-switching parameters as a function of the distance of an observation from the mean.

As we expect, a reversion back to the band (and not back to the mean) once outside the noarbitrage regime, we estimate a so-called Band-TAR model first used by Obstfeld and Taylor (1997), to which we introduce two modifications. First, we correct for the presence of serial autocorrelation using a Band-TAR adaptation of the Augmented Dickey-Fuller test. Second, the residuals are corrected for GARCH effects to account for the heteroskedasticity prevalent in the data.

The resulting specification is the following:

This model is known as the TAR(k,2,d), where k is the arbitrary autoregressive length, 2 is the number of thresholds, and d is the arbitrary delay parameter (also called the threshold lag). We assume that the thresholds are symmetric and that the dynamics of the process outside the threshold are the same regardless of whether there exists a premium or a discount. Furthermore, we set d equal to one. β_in and β_out reflect the convergence speed in the no-arbitrage and arbitrage regimes, respectively. We assume that the constants in both regimes are zero. For each country, we estimate a different model, where k, p, and q are set in such a way that the residuals do not contain any serial correlation or heteroskedasticity up to lag 10 (p is the number of ARCH terms and q is the number of GARCH terms).

The model is estimated following the procedure described in Obstfeld and Taylor (1997). The estimation proceeds via a grid search on the threshold, which maximizes the log likelihood ratio LLR=2(La-Ln). This implies that, for every given threshold, the maximum likelihood estimation of the TAR model amounts to an OLS estimation on partitioned samples, i.e. sets of observations with x_t−1 either inside or outside the thresholds.

La refers to the likelihood function of the above TAR model:²⁰

The Null is an AR(1) model and Ln is its likelihood function similar to La.

the threshold is not defined under the null, standard inference is invalid and LLR does not follow the usual x² distribution. To derive the critical values of the LR test, we follow Obstfeld Taylor and use Monte Carlo simulations. First, the AR(p) null model is estimated on the actual data (x1,....., xT) . Then, 600 simulations of the model are generated. Each starts at x_-b = 0 and ends at x_T To avoid initial value bias, the first b values are discarded (we set b at 50). For each simulation, the TAR model is estimated as outlined above and the simulated LLR is calculated. The empirical distribution of the LLR can then be calculated from the 600 simulations, and this is used as the basis for the inference in judging the alternative TAR model against the AR null.

It is important to make clear that the significance test described above has the important limitation of low power. As shown by Johansson (2001), the probability that the TAR model is mistakenly rejected is high. The method introduced by Hansen (1997) and used, for example, by Imbs et al. (2003) is based on a Wald statistic and is not useful for our purpose as heteroskedasticity in our data is strong (as is common for high frequency financial data). As a result, our best approach is to use the test described above, but to take a rejection of the TAR model with caution. Nonetheless, since we also run all our estimations using a simple AR model, we can easily verify that the conclusions are not model-dependent.²¹

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