I have a hypothetical time series data which is derived from following . equation
y =0.10 + 0.75 Trend+ 1.00 𝑦𝑡−1_ + ε
When I apply standard ADF procedure to data, I see that y has a unit root while log(y) is stationary. However, when I apply ERS Point Optimal Unit Root Test, I am receiving an absurd 51,465,638 value as test statistics and I see that data is stationary even without making log transformation.
Do I skipped an important assumption for ERS test or not?
Apart from the trend component your model is autoregressive of lag 1. With a coefficient of only 0.75, this trend may not be significant in the first place. If the series is detrended, the model has a unit root since (1-B)yt=0 where B is the backshift operator defined by Byt=yt-1. It is not surprising that a powerful unit root test like the ERS Point Optimal Unit Root test is significant. This is in spite of a contrary result by the Augmented Dickey Fuller (ADF) Test.
If your data is derived from
y(t) =0.10 + 0.75 Trend+ 1.00 𝑦(𝑡−1) + ε
then y is is clearly non-stationary and I(1) because y(t) - y(t-1)) is stationary about a trend (assuming that ε is stationary). In a perfect world a DF test about a constant and trend would would not reject stationarity. The ADF-ERS test should give the same result. Depending on your sample size, the variance of ε, test size distortions and bad luck you may get other results.
I do not understand the relevance of taking logs. This is only necessary to convert a multiplicative expression to additive form and your equation is already additive.