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Finally, the Fisher (ADF) statistic proposed by Maddala and Wu (1999), and Choi (2001), is based on the idea of Fisher (1932) of combining p-values from individual unit root tests. Again, for a Fisher test based on ADF specifications for each cross-section unit, information criteria are required to choose the required lag order to have white noise residuals. The combined test statistic is asymptotically distributed as a Chi-square with 2N degrees of freedom.

All the three test statistics are constructed using the Modified Akaike Information Criterion (MAIC), proposed by Ng and Perron (2001), which reduces the well known size distortions commonly present in unit root tests when the autocorrelation structure is characterized by high negative MA components. When required, the long-run variance is estimated through a Quadratic Spectral Kernel with automatic bandwidth selection method proposed by Andrews (1991). As deterministic components, both fixed effects and fixed effects plus a time trend are considered for each test statistic.

As shown in Table 4, the null hypothesis of a unit root is rejected for the two annuity rates, SOFI, LEV, AP(OLD), MIS, and MSHARE. The non-rejection of unit roots in the other variables seems to be related to the presence of a structural change in the intercept of their deterministic trend, which occurs at the end of the sample. As stated by Perron (1989), unit root tests are not consistent against the alternative hypothesis of trend stationary processes when the trend contains a shift in the slope. Moreover, although the tests are consistent when the shift is in the level of the trend (intercept), their power is remarkably lower. Therefore, breaks in the deterministic function of a trend stationary series are likely to produce the non-rejection of the null of a unit root.

**-** **<u>Table 4:  Panel Unit Root Tests (p values)</u>**

||**L**<br>|**LC**<br>|**IP**<br>|**S**<br>|**FISHE**<br>|**R(ADF)**<br>|
|---|---|---|---|---|---|---|
||**Const.**|**+Trend**|**Const.**|**+Trend**|**Const.**|**+Trend**|
|AR(EARLY)|0.9894|1.0000|0.5757|1.0000|0.0240|0.9997|
|AR(OLD)|0.0048|0.0004|0.2611|0.7633|0.0000|0.7318|
|RF|0.9994|0.9947|0.9998|0.9778|0.9996|0.9775|
|SOFI|0.9527|0.8834|0.9997|0.1185|0.9953|0.0008|
|SOA|0.9231|1.0000|0.9260|0.8663|0.9703|0.2582|
|LEV|0.0061|0.9875|0.0000|0.0412|0.0000|0.0064|
|MIS|0.7051|0.0000|0.0487|0.0000|0.0000|0.0000|
|MSHARE|0.0000|0.0001|0.0013|0.9666|0.0000|0.0000|
|AP(EARLY)|1.0000|1.0000|0.4185|0.8368|0.4600|0.5527|
|AP(OLD)|1.0000|1.0000|0.5273|0.0203|0.1218|0.0000|
|CR|0.9990|0.9999|0.9997|0.9997|0.9998|0.7720|
|HER(EARLY)|1.0000|1.0000|0.9988|1.0000|0.9983|0.9122|
|HER(OLD)|0.0018|0.9918|0.0000|0.0000|0.0000|0.0000|

If the series are considered as trend-stationary and some of them affected by a discrete change in the intercept of their deterministic trend, this deterministic non-stationarity can be addressed by including in the regressions a time trend plus an additive dummy variable. According to the Frisch-Waugh-Lovell theorem (Frisch and Waugh (1933), and Lovell (1963)), the numeric value of the coefficients of interest are the same, either if the regression includes detrended variables (the residual from the regression of the original