2.2.1 Mean Function

Mean Function

The mean function of a time series $\{X_t\}$ is the expected value of the process at each time $t$, defined as
${\mu }_t=\mathbb{E}[X_t]={\int }_{-\infty }^{\infty }x{f}_t(x)dx,$
where $f_t(x)$ is the marginal density of $X_t$. The mean function specifies the baseline level around which the series fluctuates at each point in time.

In general, ${\mu }_t$ can vary with $t$, reflecting systematic structure such as trend or seasonality. A constant mean ${\mu }_t=\mu$ for all $t$ is a necessary (but not sufficient) condition for stationarity.

Sample vs. Population Mean sample mean $\bar{X}=\frac{1}{T}{\sum }_{t=1}^T{X}_t$ is a single number summarizing the observed series, whereas ${\mu }_t$ is a function of time describing the theoretical average at each index. If the series is stationary, $\bar{X}$ is a consistent estimator of the common $\mu$; if ${\mu }_t$ varies, $\bar{X}$ conflates different levels and may be misleading.

The

Intuition

The mean function answers the question: if you could repeat the experiment generating $\{X_t\}$ infinitely many times, what would the average value be at time $t$? It captures deterministic structure (trend, seasonal cycles, or other systematic shifts) that persists across realizations.

For a stationary process, ${\mu }_t$ is a flat line: the series has no preferred direction over time. For a process with trend, ${\mu }_t$ drifts upward or downward. For a seasonal process, ${\mu }_t$ oscillates periodically. Estimating or removing ${\mu }_t$ is often the first step in time series analysis, because the autocovariance and autocorrelation are defined in terms of deviations from the mean.

Properties

(Linearity) If $Y_t=a{X}_t+b$, then ${\mu }_Y(t)=a{\mu }_X(t)+b$.

(Sum) If $Z_t=X_t+Y_t$, then ${\mu }_Z(t)={\mu }_X(t)+{\mu }_Y(t)$.

(Stationarity Condition) A process is (weakly) stationary only if ${\mu }_t=\mu$ is constant for all $t$.

(Sample Estimator) For a stationary process, the natural estimator of $\mu$ is $\bar{X}=\frac{1}{T}{\sum }_{t=1}^T{X}_t$, which is unbiased: $\mathbb{E}[\bar{X}]=\mu$.

Examples

White Noise

Let $\{W_t\}$ be white noise with $W_t\sim (0,{\sigma }^2)$ for all $t$. Then ${\mu }_t=\mathbb{E}[W_t]=0$ for every $t$, so the mean function is identically zero.

Random Walk with Drift

Let $X_t=\delta t+{\sum }_{j=1}^t{W}_j$, where $\delta$ is the drift and $W_j\sim (0,{\sigma }^2)$ are i.i.d.

What is the mean function? ${\mu }_t=\mathbb{E}[X_t]=\delta t+{\sum }_{j=1}^t\mathbb{E}[W_j]=\delta t.$ The mean function is a straight line with slope $\delta$. This process is not stationary because ${\mu }_t$ depends on $t$.

By linearity of expectation,