You have been asked a rather strange and indeed nonsensical question, as the motivations for a "log transformation" and a "unit root test" are unrelated to each other. The log transformation is relevant when one thinks a variable grows (or shrinks) at a percentage rate rather than an absolute rate, or when one thinks the (untransformed) variables in a model are related to each other in a multiplicative way, that is, when percentage changes in one are related to percentage changes in another. On the other hand, unit root tests examine whether individual variables are stationary around some constant mean or deterministic trend and will shed light on the appropriate technique to estimate the model. Your interrogator seems to imply that the log transformation might obviate the need for a unit root test, which is wrong.

Simple example, the quantity theory of money: MV = Py or P = MV/y. If M (or alternatively y or V) changes by, say, 10% then P changes by 10%, holding the other variables constant. To further simplify the example, assume V is constant. Next, the equation can be linearized by taking its log and an error term included: ln(P) = ln(M) + c - ln(y) + e. Empirically, one might wonder if the coefficients on the right-hand variables ln(M) and ln(y) are indeed 1.0. If so, one can estimate the equation with a simple linear technique. However, if some or all of the variables are nonstationary, OLS (for example) will give erroneous estimates of standard errors. This provides the motivation for unit root tests and the results will suggest whether OLS is appropriate or whether instead either converting the model to first differences or investigating a cointegrating relationship is appropriate.

As for your request for a reference, any econometrics book will likely explain these issues (although pinning it down to a specific page might be difficult) and there are countless empirical papers in top quality journals that apply log transformations and then test for unit roots.

Running the ADF (Augmented Dickey-Fuller) and PP (Phillips-Perron) tests on log-transformed variables is valid and often done in time series analysis to stabilize variance and meet the assumptions of stationarity.

1. Why Use Log-Transformed Variables? Log transformation is used in time series to stabilize variance (i.e., remove heteroscedasticity) and normalize data that may exhibit exponential trends. Many economic and financial variables, such as GDP, stock prices, or consumption, exhibit multiplicative trends, making the log transformation more appropriate for meaningful interpretations. 2. Running ADF and PP Tests on Log-Transformed Variables ADF Test: The Augmented Dickey-Fuller test checks for unit roots in a time series. When applied to log-transformed data, it determines whether the series is stationary in its log-transformed form. If a log-transformed variable is non-stationary, it implies that shocks to the variable have a persistent effect even in its transformed state. PP Test: The Phillips-Perron test complements the ADF test and is less sensitive to autocorrelation and heteroscedasticity. It also handles log-transformed variables and provides robust results. Note: The underlying principles of stationarity and unit root testing remain the same for both original and log-transformed variables.3. Interpretation of Results If the log-transformed series is found to be stationary (I(0)), it means the variance-stabilized variable does not have a unit root and fluctuates around a constant mean. If the log-transformed series is non-stationary (I(1)), differencing may still be required even after log transformation. 4. Key References for Log-Transformed Variables in ADF and PP Tests Books: Hamilton, J. D. (1994). Time Series Analysis. Princeton University Press.Discusses log transformations and their relevance in time series analysis. Enders, W. (2014). Applied Econometric Time Series (4th Edition). Wiley.Explains the use of log transformations and unit root tests, including ADF and PP tests. Papers: Phillips, P. C. B., & Perron, P. (1988). "Testing for a Unit Root in Time Series Regression." Biometrika, 75(2), 335–346.Foundational paper on the PP test, applicable to log-transformed data. Dickey, D. A., & Fuller, W. A. (1979). "Distribution of the Estimators for Autoregressive Time Series with a Unit Root." Journal of the American Statistical Association, 74(366), 427–431.Original ADF test formulation, applicable regardless of transformation. Practical Resources: Gujarati, D. N., & Porter, D. C. (2009). Basic Econometrics (5th Edition). McGraw-Hill.Covers practical aspects of transformations and unit root testing. Greene, W. H. (2018). Econometric Analysis (8th Edition). Pearson.Discusses transformations and diagnostic tests in time series models. 5. How to Explain as an Expert When asked why you performed the tests on log-transformed variables, you can explain: Theoretical Justification: Log transformation is standard in econometrics to stabilize variance and improve interpretability, especially for variables with exponential growth or multiplicative trends. Practical Relevance: The ADF and PP tests evaluate stationarity in the transformed domain, which ensures that the series adheres to the assumptions of further econometric analysis, such as ARDL, VAR, or cointegration testing. References Support: Use the aforementioned references to substantiate your approach and explain that these tests are agnostic to the transformation used as long as the assumptions are met.

As both answers above suggest, many economic variables have exponential trends such that they grow with constant percentage increases. Taking the difference of a variable with an exponential trend will leave the data trended and likely to be I(1). However, taking the natural log of an exponential trended variable gives a variable with a linear trend. Taking the first difference of a linear trend will leave a process that varies around a constant mean, which will likely be I(0). Hence, one reason for taking the logs of a variable that is likely to have an exponential trend is that the log of the variable will likely be I(1) whereas the variable without taking the log will likely be I(2). That is, if X(t) has an exponential trend it is likely that X(t)~I(2) whereas lnX(t)~I(1). The most widely used econometric methods are most easily applied to variables that are I(1) and I(0) rather than I(2). This is one reason econometricians take logs of variables that are believed to have exponential trends. Macroeconomic variables like money supply, GDP, consumption are typically used after taking logs. In contrast, variables already measured in proportions, such as interest rates and unemployment are typically used without taking logs.

Log transformation is commonly used to stabilize variance and make data more linear, but it does not inherently ensure stationarity. The ADF and PP tests are still necessary to confirm if the log-transformed variables are stationary. A reference that justifies this approach is:

• Gujarati, D. N., & Porter, D. C. (2009). "Basic Econometrics" (5th ed.), which explains the need for unit root tests (like ADF and PP) even after transformation to ensure stationarity.

Thus, the ADF and PP tests on log-transformed data are appropriate for verifying stationarity after transformation.

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