Boundary Crossing Counts and Autocorrelation

Let us discuss the case when there is autocorrelation in the error term in Equation (2.1). Typically, such as in Bai and Ng (2002), Choi (2006), Im et al. (2003), Chang (2002), Levin et al. (2002), Maddala and Wu (1999) and Pesaran (2007), panel unit root tests are based on individual ADF tests and the number of lags are estimated from the data. This can safely be done as (Said and Dickey (1984)) the variable of interest in the ADF test and in the DF test have the same limiting distribution even in the case when the number of lags, m, is unknown, if m³/T → ∞ as T → ∞. This result holds under more general conditions as well, as discussed in Chang and Park (2002). Alternatively, as in Moon and Perron (2004), instead of using the ADF test, one may incorporate the lag structure into the factor model as well.

In case of BCC test, based on Equation (2.16), autocorrelation effects the convergence probability in finite samples. In particular, in the case of positive autocorrelation, the convergence probability is less than 0.5 while in

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the case of negative autocorrelation, the convergence probability is greater than 0.5 under the null hypothesis in finite samples. Thus, autocorrelation may induce size distortion. Similarly, in the case of stationary process, autocorrelation may induce loss of power.

We continue by outlining two potential solutions to the problem of autocorrelation in finite samples. Note that we do not provide detailed resolution due to space constrains. The first option is to filter out the autocorrelation of the error term in Equation (2.1) and continue by carrying out the BCC test on the autocorrelation-adjusted data.

In order to implement this option, the first step is to assume that the autocorrelation structure can be captured by the following model.

∆X_it =µ_i+(α_i−1)Xit−1+ρ_i1∆Xit−1+ρ_i2∆Xi,t−2+...ρ_imi∆Xi,t−m_i+_it (2.20) where _it is free of autocorrelation and m_i is the number of lags. The convergence probability can be restored to 0.5 by first estimating the autocorrelation structure and second by defining the counting process over the autocorrelation-adjusted differences as shown in Equation (2.21).

Xf⁰_it =







0 ift ≤mi

Pt

j=mi+1∆Xij −(ρ^E_i1∆Xit−1+...+ρ^E_imi∆xi,t−mi) for t > mi. (2.21) where ρ^E_i1, ...ρ^E_im

i are the estimated autocorrelation coefficients. This option may be problematic because, besides the usual issues such as selection of lags as discussed in Harris (1992), it requires a parametric estimation which is somewhat alien to the original nonparametric philosophy of the test.

Alternatively, we can account for potential autocorrelation via the contingency table. The idea is similar to what has been discussed in section

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two, but instead of analyzing the relationship between the full history of the process,Y_ik−1^D , and the next boundary crossing event,Z_ik, as in Table 2.1, here, we analyze the relationship between the “immediate history”, represented by the last, Y_ik−1^U , or last few, boundary crossing events, and the next boundary crossing event.

More precisely, the condition Y_ik−1^U = 1 implies that the DGP’s immediate history was characterized by positive shocks. If there is no autocorrelation, this information should not affect the next boundary crossing event.

p(Z_ik = 1|Y_ik−1^U = 0) =p(Z_ik = 1|Y_ik−1^U = 1); (2.22) Likewise, the lower crossing probabilities below should also not depend on the previous boundary crossing events:

p(Z_ik =−1|Y_ik−1^U = 0) =p(Z_ik =−1|Y_ik−1^U = 1); (2.23) By combining these two equations, we obtain the following equality under the null:

p(Z_ik = 1|Y_ik−1^U = 0) +p(Z_ik =−1|Y_ik−1^U = 1) =

=p(Zik =−1|Y_ik−1^U = 0) +p(Zik = 1|Y_ik−1^U = 1)

(2.24) For ease of notation, let us introduce an additional variable which describes the effect of the process’s immediate history.

A_ik =−1 if







Z_kt = 1 andY_ik−1^U = 0 or Z_kt =−1 and Y_ik−1^U = 1.

(2.25) Also, the events of the right hand side of Equation (2.24) are denoted as follows:

Aik = 1 if







Z_kt= 1 and Y_ik−1^U = 1 or Z_kt=−1 and Y_ik−1^U = 0.

(2.26)

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Using these notations, the null hypothesis, which is somewhat analogue with the null hypothesis of no autocorrelation in the parametric case, stating that the stochastic process’s immediate history does not affect its next realization, can be described as follows:

H₀ :p(A_ik = 1) =p(A_ik =−1) = 1

2 (2.27)

From now on, we refer to p(Aik = 1), as autocorrelation probability. Under the alternative hypothesis of autocorrelation, these probabilities are no longer equal, the stochastic process’s immediate history has an effect on the next event.

H₁ :p(A_ik = 1)6=p(A_ik = 0) (2.28)

More precisely, in case of positive correlation, p(A_ik = 1) > p(A_ik = 0), while in case of negative autocorrelation, p(A_ik = 1) < p(A_ik = 0). All this is summarized in Table 2.8.

Last Boundary Crossing Event Y_ik−1^U = 0 Y_ik−1^U = 1

Next BC Event

Z_ik =−1 A_ik = 1 A_ik =−1 Z_ik = 1 A_ik =−1 A_ik = 1

Table 2.8:Contingency table describing the effect of autocorrelation.

In case of the parametric approach, one can include additional lags into the autocorrelation structure. In this nonparametric, state-based approach, it is also possible to include additional states. Assuming we find significant autocorrelation probability in the above-described first step, we can check for additional autocorrelation by adding an additional states. For example, we may ask if

p(Z_ik = 1|Y_ik−1^U = 0, Y_ik−2^U = 0)=^? p(Z_ik = 1|Y_ik−1^U = 0, Y_ik−2^U = 1)? (2.29)

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The null hypothesis in this case would state that once we have controlled for the immediate history of the process by controlling for Y_ik−1^U , the additional history represented by Y_ik−2^U = 0 does not affect the next boundary crossing probabilities significantly, while the alternative hypothesis would state that the additional history is of importance.

Finally, let us discuss how to combine the effect of the process’s immediate and full history. More precisely, let us assume that we have found significant autocorrelation probability in the first state but did not find a significant relationship in the second state. Table 2.9 summarizes the potential states for this case.

Cumulative Upper minus Lower Crossing Y_ik−1^D <0 Y_ik−1^D = 0 0< Y_ik−1^D Z_ik=−1 C_ik¹ =−1 C_ik¹ = 1

Zik= 1 C_ik¹ = 1 C_ik¹ =−1 Y_ik−1^U = 0 Zik=−1 C_ik² =−1 C_ik² = 1

Next Boundary

Crossing

Event Zik= 1 C_ik² = 1

Non Informative

C_ik² = 1

Y_ik−1^U = 1

Previous Boundary Crossing Event

Table 2.9: Contingency table describing the logic of the BCC unit root test in case of autocorrelation.

The remaining steps are almost identical to what has been described above for the case of no autocorrelation. The null hypothesis states that:

H₀ :p(C_ik¹ = 1) +p(C_ik² = 1) =p(C_ik¹ =−1) +p(C_ik² =−1), (2.30) while alternative hypothesis states that

H₁ :p(C_ik¹ = 1) +p(C_ik² = 1)> p(C_ik¹ =−1) +p(C_ik² =−1). (2.31) In case of cross-sectionally independent error terms, the distribution of the test statistics can be calculated by using the fact that the sum of two

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binomial distributions is also binomial. In case of cross-sectional dependence, the simulation-based methods can be used. Further elaborating on the method outlined above is well beyond the scope of our paper. At this stage, we can conclude that potentially, we can adjust for autocorrelation in a nonparametric manner as well.