# Why is long-run variance a positive function of the spectral density at frequency zero?

Müller (2014) provides the following definition of the long-run variance $\omega^2$:

$\omega^2=2\pi f(0)$

where $f(0)$ is the spectral density of a time series process, evaluated at frequency zero.

As I understand it, $f(0)$ is roughly a measure of how well our process can be described with a constant (i.e. with its mean). And the long-run variance is the infinite sum of autocovariances.

So my question is, why is the long-run variance a positive function of $f(0)$?

Also, could you please suggest some introductory literature on the spectral density representation of a time series process?

EDIT: I'm looking for the intution of this relationship, since the mathematical reasoning is already available in the paper:

$\omega^2=\sum_{j=-\infty}^{\infty}\gamma(j)$

$f(\lambda)=\frac{1}{2\pi}\sum_{j=-\infty}^{\infty}cos(j\lambda)\gamma(j)$

So...

$f(0)=\frac{1}{2\pi}\sum_{j=-\infty}^{\infty}\gamma(j)$

And therefore...

$\omega^2=2\pi f(0)$

• It's hard to see anything to answer here, since the relationship you describe is immediately implied by $\cos(0)=1.$ What kind of "intuition" do you seek about that?
– whuber
Apr 30, 2018 at 13:54
• @whuber yes, the math is clear. Let me try to say it another way.The spectral density plot of a time series has different heights for different frequencies. Why is it that a plot with a lot of mass at λ=0 is indicative of a process with a large long-run variance? Conversely, why is a large sum of autocovariances indicative of a process that is (roughly speaking) mostly a constant? Apr 30, 2018 at 16:04
• $\omega$ is the standard error of the sample mean $\hat{\mu}$ under serial correlation. Why is our confidence in the estimate $\hat{\mu}$ lower when $f(0)$ is larger? Apr 30, 2018 at 19:59

The flipside of your question would be "What does the value of the spectral density at $$0$$ represent?"

More generally, the spectral measure $$\mu$$ of a (mean-zero) covariance-stationary time series is a positive measure on $$[-\frac12, \frac12]$$. In cases where it assigns positive mass to $${0}$$, the same question can be asked about $$\mu({0})$$ instead of $$f(0)$$.

Any covariance-stationary time series $$x_t$$ has frequency domain representation $$x_t = \int_{-\frac12}^{\frac12} e^{-2 \pi i \, \lambda\, t} d \nu (\lambda) \quad \quad (*)$$ as a stochastic integral in frequency domain. Here $$\nu$$ is a vector-valued measure on $$[-\frac12, \frac12]$$ that assigns mean-zero $$L^2$$-random variables to Borel sets, such that $$Var(\nu(B)) = \mu(B), \; \forall \mbox{ Borel } B \subset [-\frac12, \frac12].$$ Moreover, if $$B_1, B_2 \subset [-\frac12, \frac12]$$ are disjoint, then $$Cov(\nu(B_1), \nu(B_2)) = 0$$.

For simplicity, suppose the spectral measure $$\mu$$ consists of point masses at frequency $$\lambda = 0$$ and $$\lambda = \frac13$$. Then the frequency domain representation (*) immediately implies that $$\mu(0) = Var(\nu(0))$$ is the long run variance of $$x_t$$.

According to the stochastic integral representation, $$x_t = \nu(0) + ( e^{-2 \pi i \, \frac13 \, t} + e^{2 \pi i \, \frac13 \, t} ) \cdot \nu(\frac13) = \nu(0) + 2 \cos \frac{2 \pi}{3} t \cdot \nu(\frac13),$$ where $$\nu(0)$$ and $$\nu(\frac13)$$ are uncorrelated random variables. Therefore \begin{align*} Var(\frac{1}{n} \sum_{t=1}^n x_t) &= Var(\nu (0) \cdot \frac{1}{n} \sum_{t=1}^n 1 ) + Var (\nu(\frac13) \cdot \frac{2}{n} \sum_{t=1}^n \cos \frac{2 \pi}{3} t ) \\ &\rightarrow Var( \nu (0) ), \, \mbox{ as } n \rightarrow \infty. \end{align*}

Empirically, $$Var( \nu (0) ) = \mu(0)$$ is the intensity, as measured by variance, of the constant part of the series (at frequency $$\lambda = 0$$). In the long-run, it dominates contributions from higher frequencies, making it the long-run variance.

(When the spectral measure is not absolutely continuous, the ACF is not summable and the normalization factor for long-run variance is $$\frac{1}{n}$$, rather than $$\frac{1}{\sqrt{n}}$$.)

The situation where there is a spectral density $$f$$ is similar. Take a discrete approximation $$\mu(0, \epsilon) \approx f(0)\cdot \epsilon$$, with $$\epsilon = \frac{1}{\sqrt{n}}$$. Then the dominating term in $$Var(\frac{1}{\sqrt{n}} \sum_{t=1}^n x_t )$$ would be $$Var( \frac{1}{\sqrt{n}} \cdot n \nu (0, \epsilon)) = f(0),$$ i.e. $$f(0)$$ is equal to long-run variance.

This is not surprising. From the frequency domain perspective, the long run variance is the variance of the series obtained by applying the two-sided filter $$(\cdots,1,1,1,\cdots)$$, but the Fourier transform of this filter is precisely point-mass at $$\lambda = 0$$.

Suppose $$x_t$$ has causal MA representation $$x_t = \Psi (L) \epsilon_t,$$ where $$\Psi(L) = \sum_{h \geq 0} \psi_h L^h$$ with $$\sum_{h \geq 0} |\psi_h| < \infty.$$ Then $$f(0) = \Psi(1)^2$$.
On the other hand, we can write the lag polynomial $$\Psi(L)$$ as $$\Psi(L) = \Psi(1) + \Psi(L) - \Psi(1) = \Psi(1) + (1 - L)\alpha(L)$$ where $$\Psi(L) - \Psi(1) = (1 - L)\alpha(L)$$. This is the so-called "Beveridge-Nelson decomposition". Therefore $$x_t = \Psi (1) \epsilon_t + \eta_t$$ where $$\eta_t = (1 - L)\alpha(L) \epsilon_t$$ is an over-differenced $$I(-1)$$ series. Any over-differenced series has long-run variance $$0$$. Therefore the long-run variance of $$x_t$$ is same as that of $$\Psi (1) \epsilon_t$$, which is $$\Psi (1)^2 = f(0)$$.
The frequency domain answer says that the long run variance is equal to the intensity of the zero-frequency (constant) part of the series, which is $$f(0)$$.
The time domain answer is obtained via a different decomposition. The series is decomposed into a white noise part and an over-differenced part. Only the white noise part contributes to long run variance, and it has variance $$\sum_h \gamma_h$$, which is $$\Psi(1)^2 = f(0)$$.