¿Cuál es la varianza a largo plazo?

Es una medida del error estándar de la media de la muestra cuando hay dependencia en serie.

Si $Y_t$ es una covarianza estacionaria con $E(Y_t)=\mu$ y $Cov(Y_t,Y_{t-j})=\gamma_j$ (en un ajuste iid, ¡esta cantidad sería cero!) Tal que $\sum_{j=0}^\infty|\gamma_j|<\infty$ . Entonces

lim_{T \to \infty} {V a r [\sqrt{T} ({\bar{Y}}_{T} - μ)]} = lim_{T \to \infty} {T E ({\bar{Y}}_{T} - μ)^{2}} = \sum_{j = - \infty}^{\infty} γ_{j} = γ_{0} + 2 \sum_{j = 1}^{\infty} γ_{j},

$\lim_{T\to\infty}\{Var[\sqrt{T}(\bar{Y}_T- \mu)]\}=\lim_{T\to\infty}\{TE(\bar{Y}_T- \mu)^2\}=\sum_{j=-\infty}^\infty\gamma_j=\gamma_0+2\sum_{j=1}^\infty\gamma_j,$ donde la primera igualdad es definitoria, lasegunda un poco más difícil de establecery el tercero una consecuencia de la estacionariedad, lo que implica que

γ_{j} = γ_{- j}

$\gamma_j=\gamma_{-j}$ .

Entonces, el problema es la falta de independencia. Para ver esto más claramente, escriba la varianza de la media muestral como

\begin{aligned} E ({\bar{Y}}_{T} - μ)^{2} & = E {[(1 / T) \sum_{t = 1}^{T} (Y_{t} - μ)]}^{2} \\ = 1 / T^{2} E [{(Y_{1} - μ) + (Y_{2} - μ) + \dots + (Y_{T} - μ)} \\ {(Y_{1} - μ) + (Y_{2} - μ) + \dots + (Y_{T} - μ)}] \\ = 1 / T^{2} {[γ_{0} + γ_{1} + \dots + γ_{T - 1}] + [γ_{1} + γ_{0} + γ_{1} + \dots + γ_{T - 2}] \\ + \dots + [γ_{T - 1} + γ_{T - 2} + \dots + γ_{1} + γ_{0}]} \end{aligned}

$\begin{align*} E(\bar{Y}_T- \mu)^2&=E\left[(1/T)\sum_{t=1}^T(Y_t- \mu)\right]^2\\ &=1/T^2E[\{(Y_1- \mu)+(Y_2- \mu)+\ldots+(Y_T- \mu)\}\\ &\quad\{(Y_1- \mu)+(Y_2- \mu)+\ldots+(Y_T- \mu)\}]\\ &=1/T^2\{[\gamma_0+\gamma_1+\ldots+\gamma_{T-1}]+[\gamma_1+\gamma_0+\gamma_1+\ldots+\gamma_{T-2}]\\ &\quad+\ldots+[\gamma_{T-1}+\gamma_{T-2}+\ldots+\gamma_1+\gamma_0]\} \end{align*}$

A problem with estimating the long-run variance is that we of course do not observe all autocovariances with finite data. Kernel (in econometrics, "Newey-West" or HAC estimators) are used to this end,

\hat{J_{T}} \equiv {\hat{γ}}_{0} + 2 \sum_{j = 1}^{T - 1} k (\frac{j}{ℓ_{T}}) {\hat{γ}}_{j}

$\hat{J_T}\equiv\hat{\gamma}_0+2\sum_{j=1}^{T-1}k\left(\frac{j}{\ell_T}\right)\hat{\gamma}_j$

k

$k$ is a kernel or weighting function, the

{\hat{γ}}_{j}

$\hat\gamma_j$ are sample autocovariances.

k

$k$ , among other things must be symmetric and have

k (0) = 1

$k(0)=1$ .

ℓ_{T}

$\ell_T$ is a bandwidth parameter.

A popular kernel is the Bartlett kernel

k (\frac{j}{ℓ_{T}}) = {\begin{cases} (1 - \frac{j}{ℓ_{T}}) & for 0 ⩽ j ⩽ ℓ_{T} - 1 \\ 0 & for j > ℓ_{T} - 1 \end{cases}

$k\left(\frac{j}{\ell_T}\right) = \begin{cases} \bigl(1 - \frac{j}{\ell_T}\bigr) \qquad &\mbox{for} \qquad 0 \leqslant j \leqslant \ell_T-1 \\ 0 &\mbox{for} \qquad j > \ell_T-1 \end{cases}$ Good textbook references are Hamilton, Time Series Analysis or Fuller. A seminal (but technical) journal article is Newey and West, Econometrica 1987.

Christoph Hanck
fuente

Thank you! I checked Time series Analysis by Hamilton. It does in fact say that a non-parametric way to estimate the spectrum is to take a weighted average of the sample covariances but It does not delve into the mathematics behind the determination of this statement. Could you suggest a reference book or paper that explains why is this a good estimator when the sample size increases?

Monolite

good point. Made some edits

Christoph Hanck

It perhaps worth mentioning that the second ("tricky") step requires dominated convergence (see stats.stackexchange.com/questions/154070/… ).

Tamas Ferenci

@TamasFerenci, thanks for the pointer, I included the link.

Christoph Hanck

@Cristoph Hanck, you're welcome, thanks for the update!

Tamas Ferenci

¿Cuál es la varianza a largo plazo?

Respuestas: