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OLS estimator of AR($1$) is biased

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0

$begingroup$

Suppose that we have a sample $X_0,X_1,ldots,X_n$ from the AR($1$) model given by
$$
X_t=phi X_t-1+varepsilon_t
$$
for $tinmathbb Z$, where $|phi|<1$ and $varepsilon_t_tinmathbb Z$ are iid random variables such that $operatorname Evarepsilon_0=0$ and $operatorname Evarepsilon_0^2=sigma^2<infty$. The OLS estimator of $phi$ is given by
$$
hatphi
=phi+sum_t=1^nBigl(fracX_t-1sum_t=1^nX_t-1^2Bigr)varepsilon_t.
$$

How can I show that
$$
sum_t=1^n-1operatorname EBigl[Bigl(fracX_t-1sum_t=1^nX_t-1^2Bigr)varepsilon_tBigr]ne0,
$$
i.e. that the estimator $hatphi$ is biased?

Intuitively, if we increase $varepsilon_t$ and $phi>0$, then we also increase $X_t+1,X_t+2,ldots$ and it seems that $operatorname Ehat phi<phi$ if $phi>0$ and $operatorname Ehat phi>phi$ if $phi<0$. Is this correct? This is only an heuristic argument. How can I rigorously show that the expected value is not equal to $0$?

Any help is much appreciated!

statistics least-squares time-series

share|cite|improve this question

edited Mar 13 at 11:29

Cettt

1,888622

asked Mar 13 at 10:11

Cm7F7BbCm7F7Bb

12.6k32243

$endgroup$

add a comment | 

0

$begingroup$

Suppose that we have a sample $X_0,X_1,ldots,X_n$ from the AR($1$) model given by
$$
X_t=phi X_t-1+varepsilon_t
$$
for $tinmathbb Z$, where $|phi|<1$ and $varepsilon_t_tinmathbb Z$ are iid random variables such that $operatorname Evarepsilon_0=0$ and $operatorname Evarepsilon_0^2=sigma^2<infty$. The OLS estimator of $phi$ is given by
$$
hatphi
=phi+sum_t=1^nBigl(fracX_t-1sum_t=1^nX_t-1^2Bigr)varepsilon_t.
$$

How can I show that
$$
sum_t=1^n-1operatorname EBigl[Bigl(fracX_t-1sum_t=1^nX_t-1^2Bigr)varepsilon_tBigr]ne0,
$$
i.e. that the estimator $hatphi$ is biased?

Intuitively, if we increase $varepsilon_t$ and $phi>0$, then we also increase $X_t+1,X_t+2,ldots$ and it seems that $operatorname Ehat phi<phi$ if $phi>0$ and $operatorname Ehat phi>phi$ if $phi<0$. Is this correct? This is only an heuristic argument. How can I rigorously show that the expected value is not equal to $0$?

Any help is much appreciated!

statistics least-squares time-series

share|cite|improve this question

edited Mar 13 at 11:29

Cettt

1,888622

asked Mar 13 at 10:11

Cm7F7BbCm7F7Bb

12.6k32243

$endgroup$

add a comment | 

$begingroup$

Suppose that we have a sample $X_0,X_1,ldots,X_n$ from the AR($1$) model given by
$$
X_t=phi X_t-1+varepsilon_t
$$
for $tinmathbb Z$, where $|phi|<1$ and $varepsilon_t_tinmathbb Z$ are iid random variables such that $operatorname Evarepsilon_0=0$ and $operatorname Evarepsilon_0^2=sigma^2<infty$. The OLS estimator of $phi$ is given by
$$
hatphi
=phi+sum_t=1^nBigl(fracX_t-1sum_t=1^nX_t-1^2Bigr)varepsilon_t.
$$

How can I show that
$$
sum_t=1^n-1operatorname EBigl[Bigl(fracX_t-1sum_t=1^nX_t-1^2Bigr)varepsilon_tBigr]ne0,
$$
i.e. that the estimator $hatphi$ is biased?

Intuitively, if we increase $varepsilon_t$ and $phi>0$, then we also increase $X_t+1,X_t+2,ldots$ and it seems that $operatorname Ehat phi<phi$ if $phi>0$ and $operatorname Ehat phi>phi$ if $phi<0$. Is this correct? This is only an heuristic argument. How can I rigorously show that the expected value is not equal to $0$?

Any help is much appreciated!

statistics least-squares time-series

share|cite|improve this question

edited Mar 13 at 11:29

Cettt

1,888622

asked Mar 13 at 10:11

Cm7F7BbCm7F7Bb

12.6k32243

$endgroup$

share|cite|improve this question

edited Mar 13 at 11:29

Cettt

1,888622

asked Mar 13 at 10:11

Cm7F7BbCm7F7Bb

12.6k32243

share|cite|improve this question

edited Mar 13 at 11:29

Cettt

1,888622

asked Mar 13 at 10:11

Cm7F7BbCm7F7Bb

12.6k32243

edited Mar 13 at 11:29

Cettt

1,888622

edited Mar 13 at 11:29

Cettt

1,888622

asked Mar 13 at 10:11

Cm7F7BbCm7F7Bb

12.6k32243

asked Mar 13 at 10:11

Cm7F7BbCm7F7Bb

12.6k32243