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Advances in Econometrics

Essays in Honor of Cheng Hsiao

Edited by

Emerald Publishing Limited

Volume

DOI:

https://doi.org/10.1108/S0731-9053202041

ISBN electronic:

978-1-78973-957-2

ISBN print:

978-1-78973-958-9

Series ISSN:

0731-9053

Publication date:

2020

The proof is the same as the proof for Theorem 1 in Phillips (2008), and hence omitted.

When the true DGP is the LTUE model, we have 0 < c < ∞ and

\begin{array}{l} I C_{0} = \log {\hat{σ}}_{0}^{2} = \log {\frac{1}{n} \sum_{t = 1}^{n} {(X_{t} - X_{t - 1})}^{2}} \\ = \log {\frac{1}{n} \sum_{t = 1}^{n} {[(ρ_{n} - 1) X_{t - 1} + u_{t}]}^{2}} \\ = \log {\frac{1}{n} {(ρ_{n} - 1)}^{2} \sum_{t = 1}^{n} X_{t - 1}^{2} + \frac{2}{n} (ρ_{n} - 1) \sum_{t = 1}^{n} X_{t - 1} u_{t} + \frac{1}{n} \sum_{t = 1}^{n} u_{t}^{2} .} \end{array}

By Lemma 1 in Phillips (1987b), when the process is initialized at X₀, we know

\frac{1}{n^{2}} \sum_{t = 1}^{n} X_{t - 1}^{2} \Rightarrow σ^{2} \int_{0}^{1} J_{c}^{2},

and

\frac{1}{n} \sum_{t = 1}^{n} X_{t - 1}^{2} u_{t} \Rightarrow σ^{2} \int_{0}^{1} J_{c} d B,

where

J_{c} (r) = \int_{0}^{1} e^{c (r - s)} d B (s) .

Therefore, by Equations (A.1) and (A.2), we have

\begin{array}{l} I C_{0} = \log {\frac{σ^{2} c^{2}}{n} \int_{0}^{1} J_{c}^{2} + \frac{2 c σ^{2}}{n} \int_{0}^{1} J_{c} d B + σ^{2} + o_{p} (n^{- 1})} \\ = \log σ^{2} + \log {1 + \frac{2 c}{n} \int_{0}^{1} J_{c} (r) d B + \frac{c^{2}}{n} \int_{0}^{1} J_{c}^{2} + o_{p} (n^{- 1})} . \end{array}

We also know from Phillips (1987b) that

n ({\hat{ρ}}_{n} - ρ_{n}) \Rightarrow \frac{\int_{0}^{1} J_{c} d B}{\int_{0}^{1} J_{c}^{2}},

Hence,

\begin{array}{l} I C_{1} = \log {\hat{σ}}_{1}^{2} + \frac{p_{n}}{n} \\ = \log {n^{- 1} \sum_{t = 1}^{n} {(X_{t} - {\hat{ρ}}_{n} X_{t - 1})}^{2}} + \frac{p_{n}}{n} \\ = \log {\frac{1}{n} \sum_{t = 1}^{n} {[(ρ_{n} - {\hat{ρ}}_{n}) X_{t - 1} + u_{t}]}^{2}} + \frac{p_{n}}{n} \\ = \log {\frac{1}{n} {(ρ_{n} - {\hat{ρ}}_{n})}^{2} \sum_{t = 1}^{n} X_{t - 1}^{2} + \frac{2}{n} (ρ_{n} - {\hat{ρ}}_{n}) \sum_{t = 1}^{n} X_{t - 1} u_{t} + \frac{1}{n} \sum_{t = 1}^{n} u_{t}^{2}} + \frac{p_{n}}{n} \\ \Rightarrow \log {- \frac{σ^{2}}{n} \frac{{(\int_{0}^{1} J_{c} d B)}^{2}}{\int_{0}^{1} J_{c}^{2}} + σ^{2}} + \frac{p_{n}}{n} \\ = \log σ^{2} + \log {1 - \frac{1}{n} {\frac{{(\int_{0}^{1} J_{c} d B)}^{2}}{\int_{0}^{1} J_{c}^{2}}}^{2}} + \frac{p_{n}}{n} . \end{array}

Therefore, by Equations (A.3) and (A.5), we have

I C_{1} - I C_{0} \Rightarrow \log {1 - \frac{1}{n} \frac{(\int_{0}^{1} J_{c} d B)}{\int_{0}^{1} J_{c}^{2}}} - \log {1 + \frac{2 c}{n} \int_{0}^{1} J_{c} d B + \frac{c^{2}}{n} \int_{0}^{1} J_{c}^{2}} + \frac{p_{n}}{n} .

Hence, if p_n = 2 (as in AIC), as n → ∞, we have

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