Consider the following time-series process given by

where denotes the lag operator and . What is the lag 3 autocorrelation of , ?

Question

Consider the following time-series process given by

where  denotes the lag operator and . What is the lag 3 autocorrelation of , ?

BlackTom AI · Accepted Answer

The question as provided lacks the actual time-series equation and the definition of the process, so a precise computation of the lag-3 autocorrelation cannot be carried out from the information shown. In general, the lag-k autocorrelation rho(k) is defined as Cov(y_t, y_{t-k}) / Var(y_t).

If the underlying process y_t is white noise (i.i.d. with zero mean and finite variance), then Cov(y_t, y_{t-k}) = 0 for all k ≠ 0, which implies rho(k)=0 for all k≠0. Therefore, in the common white-noise case, the lag-3 autocorrelation would be 0.

If instead the process has temporal dependence (for example AR, MA, or ARMA components), the value of rho(3) would generally be nonzero unless the model structure enforces a zero cross-covariance at lag 3 (which can happen in certain models, but not universally).

Because the prompt didn't include the explicit equation, we cannot verify whether the 0 result comes from a white-noise assumption, a model-specific property, or an algebraic cancellation within a particular process definition. If you intended a white-noise or orthogonal error structure, the zero value is consistent with standard results for lag-3 in such contexts. Otherwise, please provide the full model equation so we can compute rho(3) explicitly.

econ_475_120251_244434 Problem Set 3

Consider the following time-series process given by where denotes the lag operator and . What is the lag 3 autocorrelation of , ?

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