位置6的问题 A stationary time series Xt has zero autocorrelations at all lags except lags 2, 10 , 12, and 14. It also has a nonzero partial autocorrelation at lag two. Determine which of the following models is the most appropriate. a. SARIMA(2, 0, 0)(1, 0, 0)12 b. SARIMA(0, 0, 2)(0, 0, 1)12 c. SARIMA(0, 0, 10)(0, 0, 1)12 d. SARIMA(10, 0, 0)(1, 0, 0)12 e. SARIMA(0, 0, 1)(0, 0, 2)12 A stationary time series Xt has zero autocorrelations at all lags except lags 2, 10 , 12, and 14. It also has a nonzero partial autocorrelation at lag two. Determine which of the following models is the most appropriate. a. SARIMA(2, 0, 0)(1, 0, 0)12 b. SARIMA(0, 0, 2)(0, 0, 1)12 c. SARIMA(0, 0, 10)(0, 0, 1)12 d. SARIMA(10, 0, 0)(1, 0, 0)12 e. SARIMA(0, 0, 1)(0, 0, 2)12 SARIMA(10, 0, 0)(1, 0, 0)12SARIMA(2, 0, 0)(1, 0, 0)12SARIMA(0, 0, 10)(0, 0, 1)12SARIMA(0, 0, 2)(0, 0, 1)12SARIMA(0, 0, 1)(0, 0, 2)12清除选择题目解析

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BlackTom AI · Accepted Answer

We start by restating the key features of the observed stationary series: the ACF is zero at all lags except 2, 10, 12, and 14; the PACF is nonzero at lag 2. InSARIMA modeling, these patterns guide which AR and/or MA terms (both nonseasonal and seasonal) are needed to capture the dependence structure, with the seasonal period implied by the data (likely 12 months). The following analyses treat each candidate model in light of these quirks, noting how they would generate or fail to generate the observed ACF/PACF structure.

Option A: SARIMA(2, 0, 0)(1, 0, 0)12
- Nonseasonal part: AR(2). An AR(2) nonseasonal term naturally creates a PACF that may terminate after lag 2, which aligns with the nonzero PACF at lag 2. 
- Seasonal part: seasonal AR(1) with seasonal lag 12. A SAR(1,0,0) at lag 12 would produce a nonzero ACF at lag 12 and related seasonal lags, contributing to structure near 12 and possibly 24, etc. 
- Given the observed nonzero ACF at lags 12 and 14 (and 10), this model could plausibly generate a spike at 12 due to seasonality and, through the AR(2) component, generate a nonzero PACF at 2. However, the presence of a nonzero ACF at 10 and 14 is not obviously explained by a single seasonal AR(1) term at lag 12 or by an AR(2) nonseasonal term; these extra lags might require additional structure. Overall, this option is reasonable as a candidate because it matches a nonzero PACF at 2 and introduces seasonal dependence at 12, but it may not fully explain the observed ACF spikes at 10 and 14.

Option B: SARIMA(0, 0, 2)(0, 0, 1)12
- Nonseasonal MA(2) term will primarily shape the ACF with a halting pattern after lag 2, which could produce nonzero ACF at some early lags, but PACF behavior would be largely damped and not necessarily produce a distinct nonzero at lag 2. 
- Seasonal MA(1) at lag 12 would affect the ACF at lag 12 (and nearby lags) but does not naturally impose a nonzero PACF at lag 2. 
- The combination does not obviously explain nonzero ACF at 10 and 14, and it emphasizes moving-average structure rather than autoregressive structure at lag 2. Therefore this model seems less aligned with a nonzero PACF at 2 and a dominant AR-type explanation for the 2-lag behavior.

Option C: SARIMA(0, 0, 10)(0, 0, 1)12
- Nonseasonal MA(10) is a very large moving-average order; such a term would smear out many lags in the ACF rather than produce clean, sparse nonzero autocovariances at specific lags like 2, 10, and 14. The PACF at lag 2 would not be naturally nonzero due to a high-order MA term. 
- Seasonal MA(1) at lag 12 adds a seasonal moving-average effect but does not directly create a clean nonzero PACF at lag 2. Overall, this option seems inconsistent with a sharp, sparse ACF pattern and a distinct nonzero PACF at lag 2.

Option D: SARIMA(10, 0, 0)(1, 0, 0)12
- Nonseasonal AR(10) is an extensive autoregressive structure; such a high-order AR would typically produce nonzero PACF values extending beyond lag 2, and the PACF would likely decay more gradually rather than being nonzero only at lag 2. It would also induce a rich structure in ACF across many lags, not the sparse pattern observed. 
- Seasonal AR(1) at lag 12 would add seasonality in the ACF, but combining with AR(10) would not naturally yield zero ACF at lags like 3–9 and 11, which contradicts the stated sparsity. This option seems overly complex for the described pattern.

Option E: SARIMA(0, 0, 1)(0, 0, 2)12
- Nonseasonal MA(1) provides a simple moving-average effect early on, which could generate a nonzero ACF at lag 1 (not observed), but MA terms can produce nonzero ACF for several early lags depending on parameter values. The observed nonzero PACF at lag 2 is not directly explained by a nonseasonal MA(1) term.
- Seasonal MA(2) at lag 12 would create nonzero ACF at lag 12 and possibly around it, but again does not naturally target a nonzero PACF at lag 2. The combination seems unlikely to align with a lone nonzero PACF at lag 2 and a very sparse ACF pattern at other lags.

Synthesis of considerations: The central clue is the nonzero PACF at lag 2 coupled with zeros in the ACF at most other lags, except for 10, 12, and 14. This points toward a strong nonseasonal AR influence primarily at lag 2 and a seasonal component tied to lag 12 that can generate a spike near lag 12. Among the options, the model that explicitly includes a nonseasonal AR term of order 2 and a seasonal AR term at lag 12 is SARIMA(2, 0, 0)(1, 0, 0)12, which directly targets the AR behavior at lag 2 and the seasonal dependence at lag 12. The other specifications either emphasize moving-average structure (options B, C, E) or rely on a high nonseasonal AR order (option D) that would typically produce a more extensive PACF pattern than what is described.

Note: In practice, one would also examine the ACF/PACF plots of the squared series or use model selection criteria (AIC/BIC) and diagnostic checking after fitting candidate models to confirm which model reproduces the observed autocorrelation structure most faithfully. The explanation above reflects the theoretical alignment between the observed patterns and the candidate structures rather than performing a numerical fit here.

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