Fdtxjr

Multi tool use

Friend wants my recommendation but I don't want to

What is 管理しきれず?

Would mining huge amounts of resources on the Moon change its orbit?

Was World War I a war of liberals against authoritarians?

Writing in a Christian voice

How to test the sharpness of a knife?

Exposing a company lying about themselves in a tightly knit industry: Is my career at risk on the long run?

Why is participating in the European Parliamentary elections used as a threat?

Print a physical multiplication table

Recursively updating the MLE as new observations stream in

Hot air balloons as primitive bombers

Unfrosted light bulb

Why didn’t Eve recognize the little cockroach as a living organism?

What is the reasoning behind standardization (dividing by standard deviation)?

PTIJ: At the Passover Seder, is one allowed to speak more than once during Maggid?

What is the difference between something being completely legal and being completely decriminalized?

Why do I have a large white artefact on the rendered image?

PTIJ: Why do we make a Lulav holder?

Why are there no stars visible in cislunar space?

What is the tangent at a sharp point on a curve?

Determine voltage drop over 10G resistors with cheap multimeter

What is it called when someone votes for an option that's not their first choice?

Asserting that Atheism and Theism are both faith based positions

How to find the largest number(s) in a list of elements, possibly non-unique?

OLS estimator of AR($1$) is biased

Unbiased Estimator for a Uniform Variable SupportOLS standard error that corrects for autocorrelation but not heteroskedasticityRates of convergence of an OLS estimatorIs this estimator biased?AR(1) process: Finding the distribution of the prediction errorHow to show $Y = sum_i=1^n fracY_in-2$ is a biased estimator of the mean?show $bar x$ is biased estimator $f(x)=e^(delta-x)$A question about nonstationary test in time series.Determine if an Estimator is Biased (Unusual Expectation Expression)Conditional expectation $operatorname E[varepsilon_svarepsilon_tmid X_1,ldots,X_n-1]$

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