lnYt

lnYt refers to the natural logarithm of a time-indexed variable Y_t, where ln denotes the logarithm with base e. The expression is defined only when Y_t is positive. In practice, lnYt is used to transform data in statistics, econometrics, and related fields to stabilize variance, linearize multiplicative relationships, and facilitate interpretation of growth.

In a common modeling context, taking logs converts a multiplicative model into an additive one. For example, if Y_t = A X_t^β, then lnY_t = lnA + β lnX_t, which simplifies estimation and interpretation. The difference lnY_t − lnY_{t−1} represents the continuously compounded growth rate of Y_t and, for small changes, approximates the proportional change Y_t / Y_{t−1} − 1.

Key considerations include the positivity requirement and interpretation. When the dependent variable is lnY_t, coefficients in regression models are interpreted as elasticities: a 1% change in a regressor is associated with a β% change in Y_t. If Y_t contains zeros or negative values, practitioners may add a constant, use a different transformation (such as log(1+Y_t) or alternative links), or apply non-log transformations.

In time series analysis, applying the natural log affects variance stabilization and linearization but does not by itself ensure stationarity. Researchers often combine log transformations with differencing or other techniques to address non-stationarity in Y_t.

Common uses include transforming macroeconomic series (e.g., GDP, price levels, consumption), financial data (e.g., asset prices in log-returns form), and various regression models where a log specification is appropriate for interpretation and statistical properties.