- B t* Ï denotes the adjusted t-statistic computed with a Bartlett kernel function and individual bandwidth parameters (Newey and West, 1994). *C pt denotes the adjusted t-statistic computed with a Quadratic Spectral kernel function and individual bandwidth parameters. Finally, * tÏ denotes the adjusted t-statistic computed with a Bartlett kernel function and a common lag truncation parameter. Corresponding p-value is in parentheses. *** indicates significant at the 0.01 level.
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- bt are the unit root test statistics based on de-factored panel data (Moon and Perron, 2004). Corresponding p-values are in parentheses. * ˆpoolÏ is the corrected pooled estimates of the auto- regressive parameter. *B at and *B bt are computed with a Bartlett kernel function in spite of a Quadratic Spectral kernel function.
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- C t* Ï -2.869 (0.002) -0.002*** (0.000) -2.866*** (0.002) -2.865*** (0.002) Im, Pesaran and Shin (2003) NTbart _ bartW , bartZ , DF NTbart _ DF bartZ , -0.824 2.477 (0.993) 2.421 (0.992) -0.770 2.668 (0.996) Maddala and Wu (1999) MWP MWZ 4.293 (1.000) -2.285 (0.989) Hadri (2000) Homo Hetero 2065.755 (1.00) 2676.339 (1.00) Panel C: Small middle-segment house prices Levin, Lin and Chu (2002) * pt ψ B t* Ï
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- C t* Ï -3.107*** (0.001) -0.003*** (0.000) -2.990 (0.001) -3.059 (0.001) Im, Pesaran and Shin (2003) NTbart _ bartW , bartZ , DF NTbart _ DF bartZ , -0.895 2.230 (0.987) 2.155 (0.984) -1.034 1.743 (0.959) Maddala and Wu (1999) MWP MWZ 6.289 (0.995) -1.952 (0.975) Hadri (2000) Homo Hetero 1374.696 (1.00) 1841.201 (1.00) Panel B: Medium middle-segment house prices Levin, Lin and Chu (2002) * pt ψ B t* Ï
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- C t* Ï -3.614*** (0.000) -0.002*** (0.000) -3.587*** (0.000) -3.629*** (0.000) Im, Pesaran and Shin (2003) NTbart _ bartW , bartZ , DF NTbart _ DF bartZ , -1.080 1.580 (0.943) 1.502 (0.933) -0.989 1.899 (0.971) Maddala and Wu (1999) MWP MWZ 8.887 (0.962) -1.519 (0.936) Hadri (2000) Homo Hetero 1552.556 (1.00) 3357.809 (1.00) Panel D: Entire middle-segment house prices Levin, Lin and Chu (2002) * pt ψ B t* Ï
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- C t* Ï 24 -2.785*** (0.003) -0.001*** (0.000) -2.775*** (0.003) -2.786*** (0.003) Im, Pesaran and Shin (2003) NTbart _ bartW , bartZ , DF NTbart _ DF bartZ , -0.866 2.331 (0.990) 2.258 (0.988) -1.080 1.581 (0.943) Maddala and Wu (1999) MWP MWZ 5.044 (0.999) -2.159 (0.985) Hadri (2000) Homo Hetero 2684.379 (1.00) 4571.854 (1.00) Notes: Levin, Lin and Chu (2002): *
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- Choi (2002): the mP test is a modified Fisher’s inverse chi-square test (Choi, 2001). The Z test is an inverse normal test. The * L test is a modified logit test. p-values are in parentheses.
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- eP cMQ fMQ 2 0.554 (0.290) 21.325 (0.263) 1 2 Moon and Perron (2004) * at * bt * ˆpoolÏ *B at *B bt -8.579*** (0.000) -4.582*** (0.000) 0.969 -8.499*** (0.000) -4.538*** (0.000) Choi (2002) mP Z * L -0.760 (0.776) -0.111 (0.456) -0.099 (0.461) Pesaran (2003) *
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- eP cMQ fMQ 2 0.717 (0.237) 22.301 (0.219) 1 2 Moon and Perron (2004) * at * bt * ˆpoolÏ *B at *B bt -8.586*** (0.000) -4.966*** (0.000) 0.968 -8.742*** (0.000) -4.980*** (0.000) Choi (2002) mP Z * L -0.932 (0.824) 0.419 (0.662) 0.373 (0.645) Pesaran (2003) *
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- eP cMQ fMQ 3 -0.297 (0.617) 16.220 (0.577) 2 3 Moon and Perron (2004) * at * bt * ˆpoolÏ *B at *B bt -4.451*** (0.000) -3.158*** (0.001) 0.979 -4.738*** (0.000) -3.080*** (0.001) Choi (2002) mP Z * L -0.263 (0.604) -0.304 (0.380) -0.251 (0.401) 26 Pesaran (2003) *
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- eP cMQ fMQ 3 1.313* (0.095) 25.881 (0.103) 1 2 Moon and Perron (2004) * at * bt * ˆpoolÏ *B at *B bt -9.083*** (0.000) -4.761*** (0.000) 0.964 -10.035*** (0.000) -4.934*** (0.000) Choi (2002) mP Z * L -0.877 (0.810) 0.115 (0.546) 0.110 (0.544) Pesaran (2003) *
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- Hardi (2000): Hadri (homo) and Hadri (Hetero) denote the Hadri (2000) KPSS test assuming homogeneity and heterogeneity, respectively, in the estimation of the long-run variance. 25 Table 7. Second generation panel unit root test Panel A: Large house prices Bai and Ng (2004) ˆr ˆ c eZ ˆ c
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Im, K. S., Pesaran, M.H. and Shin, Y. 2003. “Testing for Unit Roots in Heterogeneous Panels.†Journal of Econometrics, 115:53-74.
- Im, Pesaran and Shin (2003): _ DF NTt bar (respectively _ NTt bar ) denotes the mean of Dickey Fuller (respectively Augmented Dickey Fuller) individual statistics. , DF t barZ is the standardized _ DF NTt bar statistic and associated p-values are in parentheses. ,t barZ is the standardized _ NTt bar statistic based on the moments of the Dickey Fuller distribution. ,t barW denotes the standardized _ NTt bar statistic based on simulated approximated moments (Im, Pesaran and Shin, 2003, table 3). The corresponding p-values are in parentheses. * indicates significant at the 5% level.
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- Maddala and Wu (1999): MWP denotes the Fisher’s test statistic defined as ( )2 logMW iP p= − ∑ ; where ip are the p-values from ADF unit root tests for each cross-section. Under 0H ; MWP has 2 χ distribution with 2N of freedom when T tends to infinity and N is fixed. MWZ is the standardized statistic used for large N samples: under H0; MWZ has a N (0, 1) distribution when T and N tend to infinity.
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- P CIPS * CIPS 2 -3.082** (0.010) -3.082** (0.010) Chang (2002) IV* t 0.483 (0.685) Notes: Bai and Ng (2004): ˆr is the estimated number of common factors, based on IC criteria functions. ˆ c eP be is a Fisher’s type statistic based on p-values of the individual ADF tests. ˆ c eZ be is a standardized Choi’s type statistic for large N samples. P-values are in parentheses. The firrst estimated value 1 ˆr is derived from the filtered test fMQ and the second one is derived from the corrected test cMQ . ** and *** indicate significant at the 5% and 1% levels, respectively.
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- P CIPS * CIPS 2-2.804** (0.010) -2.804** (0.010) Chang (2002) IV* t 2.392 (0.992) Panel C: Small house prices Bai and Ng (2004) ˆr ˆ c eZ ˆ c
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- P CIPS * CIPS 2-3.114** (0.010) -3.114** (0.010) Chang (2002) IV* t 5.378 (1.000) Panel B: Medium house prices Bai and Ng (2004) ˆr ˆ c eZ ˆ c
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- P CIPS * CIPS 3 -3.511** (0.010) -3.511** (0.010) Chang (2002) IV* t 4.881 (1.000) Panel D: All house prices Bai and Ng (2004) ˆr ˆ c eZ ˆ c
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- Pesaran (2003): CIPS is the mean of individual Cross sectionally augmented ADF statistics (CADF). * CIPS denotes the mean of truncated individual CADF statistics. Corresponding p-values are in parentheses. * P denotes the nearest integer of the mean of the individual lag lengths in ADF tests.
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Pesaran, M.H. 2004. “General Diagnostic Tests for Cross Section Dependence in Panels.†IZA Discussion Papers 1240, Institute for the Study of Labor (IZA).
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- pt denotes the adjusted t-statistic computed with a Bartlett kernel function and a common lag truncation parameter given by 1 3 3.21K T= (Levin and Lin, 2002). Corresponding p-value is in parentheses. ψ is the pooled least squares estimator. Corresponding standard error is in parentheses.
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