先前提過兩種計算預測誤差的方式，會使 MSPE Ratio、MAPE Ratio 有所不 同，而其變動關係如下
M SP E Ratio Dif f erence = (13)
∑T
t=1be2t+1
∑T
t=1(opt+1op−opt
t )2−
∑T
t=1(opt· bet+1)2
∑T
t=1(opt+1− opt)2 ∝∼ Corr((opt+1− opt
opt )2, op2t)−Corr(be2t+1, op2t)
M AP E Ratio Dif f erence = (14)
∑T
t=1|bet+1|
∑T t=1
opt+1op−opt t −
∑T
t=1|opt· bet+1|
∑T
t=1|opt+1− opt| ∝∼ Corr(
opt+1− opt
opt
,|opt|)−Corr(|bet+1| , |opt|)
而預測 WTI 月底價變動率時發現，在 2005 至 2009 年期間 (特別是 2008:M10 到 2009:M6) 表現較佳，當時的月底價約在預測期間平均附近，各文獻中因預測評估 期間選取的不同，開始評估的時間點愈早，op2t 與|opt| 的樣本平均數就愈小，將 使 Corr((opt+1op−opt
t )2, op2t)上升的比 Corr(be2t+1, op2t)更多; 使 Corr( opt+1op−opt t , |opt|) 上升的比 Corr(|bet+1| , |opt|) 更多。9此一結果將導致兩種計算預測誤差的方式，
所得到的 MSPE Ratio 與 MAPE Ratio 差異甚大，也因此如要比較不同文獻中的 MSPE Ratio 與 MAPE Ratio，需調整至相同的預測期間且計算預測誤差的方式也 需相同才能比較。
year
19961997199819992000200120022003200420052006200720082009201020112012201320142015
Dollars per Barrel
020
40
60
80100
120
140End-of-Month WTI Oil Price End-of-Month WTI Oil Price Average of WTI Oil Price
year
19961997199819992000200120022003200420052006200720082009201020112012201320142015
Cumulative Square Prediction Error Reduction -0.02 -0.04
0
0.02
0.04
0.06
0.080.1
0.12
0.14Percentage Change of End-of-Month WTI Oil Prices simple-average MMA JMA PIA(1) PIA(2) PIA(3) AIC BIC AICc HDBIC HQ CV s-AIC s-BIC s-AICc s-HDBIC s-HQ Bates-Granger
year
19961997199819992000200120022003200420052006200720082009201020112012201320142015
Cumulative Absolute Prediction Error Reduction -0.2 -0.3
-0.1
00.1
0.2
0.3
0.4
0.5Percentage Change of End-of-Month WTI Oil Prices simple-average MMA JMA PIA(1) PIA(2) PIA(3) AIC BIC AICc HDBIC HQ CV s-AIC s-BIC s-AICc s-HDBIC s-HQ Bates-Granger
Correlation Difference ×10-3
-14 -12 -10 -8 -6 -4 -2 0 2 4 6
MSPE Ratio Difference
-0.035
simple-average
AIC
Bates-Granger
CV MSPE Ratio Difference of End-of-Month WTI Oil Prices (R2 =0.974)
Correlation Difference
-0.018 -0.016 -0.014 -0.012 -0.01 -0.008 -0.006 -0.004 -0.002 0
MAPE Ratio Difference
×10-3
simple-average
AIC
Bates-Granger
MMA CV
JMA
PIA(1)
PIA(2)
PIA(3) MAPE Ratio Difference of End-of-Month WTI Oil Prices (R2 =0.99)
year
19961997199819992000200120022003200420052006200720082009201020112012201320142015
Cumulative Square Prediction Error Reduction -0.1
00.1
0.2
0.3
0.4
0.5
0.6
0.7Percentage Change of Monthly-Averaged WTI Oil Prices simple-average MMA JMA PIA(1) PIA(2) PIA(3) AIC BIC AICc HDBIC HQ CV s-AIC s-BIC s-AICc s-HDBIC s-HQ Bates-Granger
year
19961997199819992000200120022003200420052006200720082009201020112012201320142015
Cumulative Absolute Prediction Error Reduction -0.5
00.5
11.5
22.5
33.5Percentage Change of Monthly-Averaged WTI Oil Prices simple-average MMA JMA PIA(1) PIA(2) PIA(3) AIC BIC AICc HDBIC HQ CV s-AIC s-BIC s-AICc s-HDBIC s-HQ Bates-Granger
Correlation Difference
-0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015
MSPE Ratio Difference
-0.025
simple-average
AIC
Bates-Granger
CV
MSPE Ratio Difference of Monthly-Averaged WTI Oil Prices (R2 =0.992)
Correlation Difference
-0.03 -0.025 -0.02 -0.015 -0.01 -0.005 0
MAPE Ratio Difference
-0.012
simple-average
AIC
Bates-Granger
CV
MAPE Ratio Difference of Monthly-Averaged WTI Oil Prices (R2 =0.99)
圖 7: MSPE、MAPE Ratio Difference and Correlation Difference
5 結論
本研究探討 1986:M2 到 2015:M10 的月資料，以過去文獻使用的多個相關變數 配合模型選取或組合預測方法，預測原油價格。近年研究發現若與 no-change 預測 比較，有若干變數可能具有預測油價月資料的能力，像是消費者物價指數、工業 原料價格指數、原油相關性股票價格與美元澳幣匯率等，受限於電腦計算速度本 研究中僅挑選各類別較具有代表性的變數，以預測組合方法進行研究。而美國能 源局公佈的歷史資料包含油價的日資料與月資料，其中月資料則是以該月份的每 日油價簡單平均計算，相關文獻中預測油價的月資料可能是月底價 (end-of-month price) 或是月均價 (monthly-averaged price)，本研究對兩種月資料分別進行探討。
挑選預測變數時，增加一個新變數用於預測月均價，藉由探討 Working effect 及其 衍伸的性質說明此一新變數的必要性。另外，研究過程中發現預測月底價時，其 預測效果有集中於特定期間的情況，此現象會造成兩種算法所得的 MSPE Ratio 有 所差異，並分析評估期間選取如何影響 MSPE Ratio 的差異大小，進而影響評估預 測能力的結果。
樣本內估計的相關結果顯示，預測原油價格月均價變動率時，由於 y†t 與預測 變數的相關性與 yt†特別高的 t-statistic，而使運用這些預測變數所做的樣本內與樣 本外結果皆會受到 Working effect 的影響。利用各種模型選取 (Model Selection) 與 模型組合 (Model Averaging) 對原油價格做出預測，觀察樣本外的預測表現隨時間 的變化，大致上僅能在特定期間預測月底價的變動率，而預測月均價的變動率方 面，與 no-change 預測比較，整個預測期間 1996:M1 到 2015:M10 皆能具有一定的 預測能力。若是去除 Working effect 的影響，預測月均價的效果與預測月底價相 似，僅能於特定期間內保有預測能力。不同 P/R ratio 的選擇可能影響到樣本外檢 定是否顯著，但不影響僅能於特定期間內保有預測能力的結論。
探討的過程中發現，計算預測誤差的方式不同，選取預測期間的不同皆可能導 致預測結果有相當大的差異。藉由驗證 (13) 與 (14) 式的關係，我們了解到在預測 效果於評估期間內並非均勻分布的情況下，會導致 MSPE Ratio 有所差異，而評估
注意到這個因素的影響。
綜 合 性 的 比 較 各 種 模 型 選 取 與 模 型 組 合 方 法 結 果 上 的 差 異， 大 致 上 使 用 Information Criterion 來選取單一模型的方法其結果較其它方法差，在預測原油價 格上，應盡量避免單獨使用此方法。而各模型組合方法中，PIA(2) 不論在預測月 底價或是月均價，相對於其它模型組合方法，均能取得不錯的效果。另外，於預 測月均價變動率時，CV 表現相當不錯，表現較差的反而是 simple average，原因 是因為不包含 yt†的模型表現遜於包含 yt†的模型，而 simple average 因為無法依照 各模型表現改變權重，有一半的模型都不包含 yt†，導致其表現遠遜於其他方式。
過往預測原油價格的相關文獻數量甚鉅，因為原油價格定義、資料處理、取樣 期間、評估方式等等的不同，而可能產生看似矛盾的結論，希望能藉由本研究釐 清部分造成矛盾的原因。於實用預測的角度，尋找至今仍能持續保持預測效果的 變數與方式是相當重要的課題。不同變數的選取、P/R ratio 的選擇皆可能對預測 結果產生影響。另外，此一預測油價的方式僅能於特定時期保有預測能力的原因，
還有待後續研究者作更進一步的探討。
附錄一
Working Effect
由 Working(1960) Note on the correlation of first differences of averages in a random chain 一文中提到，若某一價格的日資料 Pt走勢為 random-walk，則其月均價的變 動量，會有前後期的相慣性產生 [36]。推論過程如下
Pt= Pt−1+ et t = 1, 2, . . . et∼ (0, 1), corr(ei, ej) = 0 if i̸= j (15) 設定每 k 期平均一次，此均價變動量 Dt,k可做以下轉換
Dt,k = 1 k
t+k∑−1 n=t
Pn− 1 k
t−1
∑
n=t−k
Pn (16)
= 1
k[Pt+ (Pt+ et+1) +· · · + (Pt+ et+1+ et+2+· · · + et+k−1)− (Pt− et)
− (Pt− et− et−1)− · · · − (Pt− et− et−1− · · · − et−k+1)]
= 1
k[et+k−1+ 2et+k−2+· · · + (k − 1) et+1+ ket+ (k− 1) et−1+· · · + et−k+1] 同樣的可以對前 k 期的均價變動量 Dt−k,k 一樣的轉換
Dt−k,k = 1
k [et−1+ 2et−2+· · · + (k − 1) et−k+1+ ket−k+ (k− 1) et−k−1+· · · + et−2k+1] (17) 則前後期的均價變動量的共變異數與相關係數
cov(Dt,k, Dt−k,k) = 1
k2[1(k−1)+2(k−2)+· · ·+(k−1)1] = 2k2− 1
6k corr(Dt,k, Dt−k,k) = k2− 1 4k2+ 2 (18)
每月份交易日約為 20 日，因此這邊設定 k=20 帶入 (18) 式 corr(Dt,k, Dt−k,k) = 0.249
訊並沒有被充分利用，由於原始價格日資料走勢為 random-walk, 在第 t-1 期時，
對次月份均價 k1∑t+k−1
n=t Pn的最佳預測理應為 Pt−1，也就是說對月均價的變動量 最佳預測應為
Dt†−1,k = Pt−1− 1 k
t−1
∑
n=t−k
Pn = 1
k[(k− 1)et−1+ (k− 2)et−2+· · · + et−k+1] (19)
cov(Dt,k, Dt†−1,k) = (k− 1)(2k − 1)
6k corr(Dt,k, D†t−1,k) =
√2(k− 1)(2k − 1)(2k2+ 1) 2(2k2+ 1)
(20) 同樣因為每月份交易日約為 20 日，因此這邊設定 k=20 帶入 (20) 式
corr(Dt,k, D†t−1,k) = 0.680
由於在第 t-1 期時，對 Dt,k = 1k∑t+k−1
n=t Pn− 1k∑t−1
n=t−kPn 的最佳預測是 Dt†−1,k = Pt−1− 1k∑t−1
n=t−kPn，且在 t-1 期時 1k∑t−1
n=t−kPn是已知的資訊，所以對 1 Dt,k k
∑t−1 n=t−kPn
的最佳預測就是 1 D†t−1,k k
∑t−1
n=t−kPn。這邊可以發現，1 Dt,k k
∑t−1
n=t−kPn 就是以月均價計算的變動 率，而 1 D†t−1,k
k
∑t−1
n=t−kPn 就是本文中所提到的預測變數 yt†。
附錄二
預測月均價的預測變數選擇
當預測油價的目標是月均價時，選擇被預測變數為 yt = (opt− opt−1)/opt−1， 如附錄一中所述，將會有 Working Effect。而 Working Effect 的影響太過劇烈，以 致於除了 y†t 其餘的預測變數的預測能力不易評估，為解決這個問題可以選擇被預
MSPE Ratio =
∑T MSPE Ratio =
∑T 累計預測誤差平方縮減 (CSPER,Cummlative Squared Prediction Error Reduction)
CSP ER(p) =
累計預測絕對誤差縮減 (CAPER,Cummlative Absolute Prediction Error Reduction)
CAP ER(p) =
估各方法的預測能力在 1% 的顯著水準下皆為不顯著。
表 5: In-Sample Predictability of WTI Oil Prices
Monthly-Averaged Prices bγ t-stat p value NYSE ARCA OIL & GAS INDEX 0.165 1.551 0.121 PPI: Durable Raw Goods 0.409 2.086 0.038
AUD/USD 0.085 0.451 0.652
Consumer Price Index -0.277 -0.789 0.431 yt†= (opt− opt)/opt 0.076 0.754 0.452
表中結果樣本期間為 1986:M1 至 2015:M10。t-statistic 使用 Newey-West HAC standard errors 計算，粗體代表檢定量在 5% 的顯著水準下為顯著。
year
19961997199819992000200120022003200420052006200720082009201020112012201320142015
Cumulative Square Prediction Error Reduction -0.02 -0.03
-0.01
0
0.01
0.02
0.03
0.04
0.05Forecast Monthly-Averaged WTI Oil Prices simple-average MMA JMA PIA(1) PIA(2) PIA(3) AIC BIC AICc HDBIC HQ CV s-AIC s-BIC s-AICc s-HDBIC s-HQ Bates-Granger
year
19961997199819992000200120022003200420052006200720082009201020112012201320142015
Cumulative Absolute Prediction Error Reduction -0.25
-0.2-0.15
-0.1-0.05
0
0.050.1
0.150.2
0.25Forecast Monthly-Averaged WTI Oil Prices simple-average MMA JMA PIA(1) PIA(2) PIA(3) AIC BIC AICc HDBIC HQ CV s-AIC s-BIC s-AICc s-HDBIC s-HQ Bates-Granger
Correlation Difference ×10-3
-20 -15 -10 -5 0 5
MSPE Ratio Difference
-0.04
simple-average
AIC
Bates-Granger
CV
MSPE Ratio Difference of End-of-Month WTI Oil Prices (R2 =0.941)
Correlation Difference
-0.025 -0.02 -0.015 -0.01 -0.005 0
MAPE Ratio Difference
-0.012
simple-average
AIC
Bates-Granger
CV
MMA
JMA
PIA(1)
PIA(2) PIA(3)
MAPE Ratio Difference of End-of-Month WTI Oil Prices (R2 =0.982)
表 6: Forecast Monthly-Averaged WTI Oil Prices
Percentage Error (22) Price Error (23)
MSPE Ratio MAPE Ratio MSPE Ratio MAPE Ratio Success Ratio simple average 0.977
(0.016)
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