A classical ordinary least squares (OLS) cannot be applied if the dependent variable exhibits autocovariance

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Question restatement: The prompt asks whether a classical ordinary least squares (OLS) regression cannot be applied when the dependent variable shows autocovariance, with True/False options provided.
Option 1: True. The claim here is that OLS cannot be applied if the dependent variable exhibits autocovariance. In practice, this is not correct. OLS can still be estimated on time series data even when autocovariance is present, but the usual OLS assumptions about the error terms (no autocorrelation) are violated. This violation means that standard error estimates may be biased and inference (t-stats, p-values) may be invalid unless adjustments are made. So saying OLS cannot be applied at all is too strong and inaccurate.
Option 2: False. This option correctly challenges the statement by noting that OLS can be applied; the issue is not applicability but inference. When autocovariance is present, the OLS estimator of the coefficients remains unbiased and consistent under certain conditions, but the standard errors are biased, leading to unreliable hypothesis tests. To address this, one might use robust (heteroskedasticity- and autocorrelation-consistent) standard errors (e.g., Newey-West), or switch to generalized least squares (GLS) or feasible GLS if the form of autocorrelation is known. The key nuance is that the method itself is still usable, with caveats for inference.
Additional nuance: If the model is truly dynamic with autoregressive structure, researchers often explicitly model the autocorrelation (e.g., using ARIMA errors or GLS with a specified correlation structure) to obtain efficient estimates. However, merely having autocovariance in the dependent variable does not automatically render OLS invalid for estimation of coefficients; it mainly affects standard errors and the efficiency of the estimates. 
In sum, the statement that OLS cannot be applied is incorrect because estimation can proceed, while the consequences for inference and standard errors must be addressed with appropriate adjustments or alternative methods.

AS.440.617.80.SP25 Quiz 4. Time Series regression

A classical ordinary least squares (OLS) cannot be applied if the dependent variable exhibits autocovariance

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