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柚柚_柚 · 2020年05月31日

问一道题：NO.PZ2020011101000020 [ FRM I ]

  1. FRM I
  2. Quantitative

问题如下：

Suppose an hourly time series has a calendar effect where the hour of the day matters. How would the dummy variable approach be implemented to capture this calendar effect? How could differencing be used instead to remove the seasonality?

解释：

Let s = 24 represent the hour of the day in military time (e.g. 13 = 1 p.m.). Then Yt=g(t)+γ1I1t+...+γ23I23t+ϵtY_t = g(t) + \gamma_1I_{1t} + ... + \gamma_{23}I_{23t} + \epsilon_t.

Differencing this series can be achieved by looking at observation 24 periods (hours) apart from each other (the following presumes that the error terms are iid and normal):

Yt+24−Yt=g(t+24)−g(t)+ϵt+24−ϵtY_{t + 24} - Y_t = g(t + 24) - g(t) + \epsilon_{t + 24} - \epsilon_t

Once the deterministic time trend is removed the remaining is a covariance-stationary MA(1) process.

Yt+24那条式子哪来的, 为什么remove后等于MA1 ？ 谢谢
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袁园_品职助教 · 2020年06月01日

同学你好！

Yt+24那个式子就是把 Yt+24 和 Yt 分别代入 Yt 那个式子，由于 24 是一个循环，所以中间那些都是一样的，可以全部相减抵消。

所以 Yt+24−Yt 相当于是 remove 了 calendar effect，得到的表达式是一个只跟 当期扰动项  ϵt+24​ 和 前期扰动项 ϵt 有关的式子，即 MA（1）模型。

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