This Paper Considers the Consistency Property of Some Test Statistics Based on a Time Series of Data. While Th Eusual Consistency Criterion Is Based on Keeping the Sampling Interval Fixed, We Let the Sampling Interval Take Any Path As the Sample Size Increases to Infinity. We Consider Tests of the Null Hypotheses of the Random Walk and Randomness Against Positive Autocorrelation We Show That Tests of the Unit Root Hypothesis Based on the First-Order Correlation Coefficient of the Original Data Are Consistent As Long As the Span of the Data Is Increasing. Tests of the Same Hypothesis Based on the First-Order Correlation Coefficient Using the First-Differenced Data Are Consistent Only If the Span Is Increasing At a Rate Greater Than Square Root of 'T'. on the Other Hand Tests of the Randomness Hypotheses Based on the First-Order Correlation Coefficient Applied to the Original Data Are Consistent As Long As the Span Is Not Increasing Too Fast. We Provide Monte Carlo Evidence on the Power, in Finite Samples, of the Tests Studied Allowing Various Combinations of Span and Sampling Frequencies. It Is Found That the Consistency Properties Summarize Well the Behavior of the Power in Finite Samples. the Power of Tests for a Unit Root Is More Influenced by the Span Than the Number O Observations While Tests of Randomness Are More Powerfull When a Small Sampling Frequency Is Available.
