S. NAGATA 387

rameter values are the same as those for Figure 2. We

note that the shapes of the distributions are almost the

same, even if the noise ratio ρ is changed. From these

figures, the noise variance has only a small effect on the

shape of the OLS distribution.

4. Discussion

In this paper, we considered finite sample properties of

the OLS estimator for the AR(1) model with measure-

ment error. Using the formula in [9], we obtained the

Edgeworth expansion for finite sample distributions of

the OLS estimator up to O(12

T).

In the simulation work, we have compared naive OLS

estimator with the IV estimator which is a consistent es-

timator in the presence of noise. We can confirm that,

even if the measurement errors is exist, the OLS estima-

tor is more efficient than the IV estimator when the sam-

ple size is small such as T = 20 and 40. If the noise-

to-signal ratio is not so small (ρ ≥ 0.4), the IV estimator

is more efficient than the OLS estimator when T ≥ 800 (β

= 0.4) or T ≥ 100 (β = 0.8). From the graph of the nor-

malized OLS distributions, we find similar properties to

those of the distributions, which correspond to the no

noise situation examined by [6]. This result implies that

measurement error mainly distorts the OLS distributions

for mean and variance; hence we can separately deal with

the two problems of small sample size and observation

errors.

Recently, the differenced-AR(1) estimator was dis-

cussed in [12,13]. Even if the sample size T is relatively

small, this estimator has a small bias. To obtain the finite

sample distribution and to examine the robustness of

such estimators with respect to observation errors, we

can apply the technique of this paper, and this will be

dealt with in a future study.

5. Acknowledgements

I am grateful to Professor Koichi Maekawa for his guid-

ance on this topic and his valuable comments on this

paper. I am also grateful to Professor Yasuyoshi Tokutsu

for his valuable comments.

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