IPS test using the plm R package gives different T-rho values from individually pefromed ADF test statistics

Question

I have been throught the Im, Pesaran and Shin paper many times to solve this issue but have failed so far. When I perform the IPS test in R using the plm package [purtest() function] I get exactly the same rho values for every panel as when I do the ADF test individually for it, but the problem is that the T-rho individual statistic differs. Shouldn't it be the same with ADF test?

Since I do have the same rho coefficient value I understand that the difference should be coming from the rho coefficient's standard error.

I use same lags and intercept as an exogenous variable at both tests.

Has anyone encountered this before?

As you see to a simple example with one only panel below (same happens with multiple panels), IPS rho value is equal to the ADF z.lag.1 Estimate ( -0.376026893)

However, IPS trho = -3.6235638, while ADF t-statistic is -3.5532

Why is this difference?

Im-Pesaran-Shin Unit-Root Test

Exogenous variables: Individual Intercepts
Automatic selection of lags using AIC: 0 - 0 lags (max: 5)

statistic (Wtbar): -2.41   
p-value: 0.008

          lags obs          rho         trho         p.trho         mean    var

Section1    0  52      -0.376026893  -3.6235638   0.005353902223  -1.5254  0.7578


###############################################
# Augmented Dickey-Fuller Test Unit Root Test #
###############################################
    
Test regression drift

Call:    
lm(formula = z.diff ~ z.lag.1 + 1)

Residuals:
        Min          1Q      Median          3Q         Max

-0.51672098 -0.13117631 -0.02416759  0.09858407  0.47207489

Coefficients:

               Estimate  Std. Error  t value   Pr(>|t|)   

(Intercept) -0.00638350  0.02452175 -0.26032 0.79568610       
z.lag.1     -0.37602689  0.10582777 -3.55320 0.00084152 ***
---

Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.1768252 on 50 degrees of freedom

Multiple R-squared:  0.2015994, Adjusted R-squared:  0.1856314

F-statistic: 12.62521 on 1 and 50 DF,  p-value: 0.0008415186

Value of test-statistic is: -3.5532 6.3408

Critical values for test statistics:
      1pct  5pct 10pct
tau2 -3.51 -2.89 -2.58    
phi1  6.70  4.71  3.86

Answer 1

Statistical tests come with a lot of parameters to specify. Your observation seems to relate to the parameter dfcor to purtest.

See this example where rho is the same no matter the value for dfcor but trho is slightly different:

library(plm)
data(Grunfeld)
pG <- pdata.frame(Grunfeld)
b  <- purtest(pG$value, test = "ips", exo = "intercept", lags = 0, dfcor = TRUE)
b2 <- purtest(pG$value, test = "ips", exo = "intercept", lags = 0, dfcor = FALSE)
summary(b)
#> Im-Pesaran-Shin Unit-Root Test 
#> Exogenous variables: Individual Intercepts 
#> User-provided lags
#> statistic (Wtbar): -1.419 
#> p-value: 0.078 
#> 
#>    lags obs        rho       trho       p.trho    mean    var
#> 1     0  19 -0.7221173 -3.0980241 0.0267255342 -1.5204 0.8654
#> 2     0  19 -0.8376784 -3.9708351 0.0015756584 -1.5204 0.8654
#> 3     0  19 -0.5503275 -2.4918122 0.1174272537 -1.5204 0.8654
#> 4     0  19 -0.9812049 -4.4232340 0.0002648235 -1.5204 0.8654
#> 5     0  19 -0.0246934 -0.2247631 0.9329996429 -1.5204 0.8654
#> 6     0  19  0.1313902  2.0376353 0.9999127394 -1.5204 0.8654
#> 7     0  19 -0.2767321 -1.5074566 0.5300598000 -1.5204 0.8654
#> 8     0  19 -0.2343526 -1.4013933 0.5833071748 -1.5204 0.8654
#> 9     0  19 -0.3064189 -1.4852375 0.5413593097 -1.5204 0.8654
#> 10    0  19 -0.6898608 -2.8137835 0.0562945264 -1.5204 0.8654

summary(b2)
#> Im-Pesaran-Shin Unit-Root Test 
#> Exogenous variables: Individual Intercepts 
#> User-provided lags
#> statistic (Wtbar): -1.796 
#> p-value: 0.036 
#> 
#>    lags obs        rho       trho       p.trho    mean    var
#> 1     0  19 -0.7221173 -3.2751947 1.605723e-02 -1.5204 0.8654
#> 2     0  19 -0.8376784 -4.1979203 6.603715e-04 -1.5204 0.8654
#> 3     0  19 -0.5503275 -2.6343147 8.602350e-02 -1.5204 0.8654
#> 4     0  19 -0.9812049 -4.6761911 8.947014e-05 -1.5204 0.8654
#> 5     0  19 -0.0246934 -0.2376169 9.313144e-01 -1.5204 0.8654
#> 6     0  19  0.1313902  2.1541642 9.999458e-01 -1.5204 0.8654
#> 7     0  19 -0.2767321 -1.5936655 4.858266e-01 -1.5204 0.8654
#> 8     0  19 -0.2343526 -1.4815366 5.432370e-01 -1.5204 0.8654
#> 9     0  19 -0.3064189 -1.5701756 4.979070e-01 -1.5204 0.8654
#> 10    0  19 -0.6898608 -2.9746990 3.733484e-02 -1.5204 0.8654