```{r} #| label: setup #| include: false set.seed(2026) library(ggplot2) theme_set(theme_minimal(base_size = 18)) ``` ## ๐ฌ The Idea in 3 Minutes ::: {style="text-align:center"} ```{=html} ``` [A short film with the intuition. Watch it before the slides, or after. Captions: CC button.]{style="font-size:28px"} ::: --- ## ๐ฏ Learning Objectives ::: {style="font-size: 32px"} By the end of this lecture, you will be able to: - **Explain** why a time series is one trajectory of a process, and what that costs us - **Define** strict stationarity and the strong, semi-strong and weak versions of white noise - **Derive** the moments of MA(1), AR(1) and the random walk, and decide which are stationary - **Distinguish** stationarity from ergodicity, with a process that has one but not the other - **Choose** a transformation (log, difference, log return) that makes an economic series plausibly stationary ::: --- ## ๐บ๏ธ Where We Are ::: {style="font-size: 30px"} This is the first lecture. The course asks one question over and over: **what can we learn about a random process from a single stretch of its history?** - Lectures 1โ3: the foundations โ stationarity, ergodicity, the law of large numbers and the CLT for dependent data, and regression with time-series data - Lectures 4โ8: what to model, how to choose and evaluate models, and the linear (ARMA) workhorse - Lectures 9โ17: unit roots, nonlinear and regime models, VARs, volatility, breaks and risk ::: {.fragment} Today supplies the two assumptions everything else leans on. ::: ::: --- ## โ The Question ::: {.callout-important} ## One history, one market The S&P 500 has exactly one past. We cannot rerun 2016โ2026 a thousand times to see how else it might have gone. ::: ::: {style="font-size: 30px"} Yet we want to say "the expected daily return is about 0.05%" and attach a standard error to it. That means treating an **average over time** as if it were an **average over possible histories**. When is that legitimate? The answer has two parts: **stationarity** and **ergodicity**. ::: --- ## ๐ฅ Two Kinds of Economist ::: {style="font-size: 28px"} | | **n-economist** (cross-section) | **t-economist** (time series) | |---|---|---| | Data | $\{x_i\}_{i=1}^{n}$ | $\{x_t\}_{t=1}^{T}$ | | What it is | $n$ observations of one random variable | **one** segment of **one** realisation of $\{x_t\}_{t=-\infty}^{+\infty}$ | | Dependence | independent draws (i.i.d.) | not i.i.d.; we expect **strong** dependence | | Asymptotics | $n \to \infty$: more independent draws | $T \to \infty$: a longer stretch of the same path | ::: {.fragment} To apply large-sample theory to time series we must **impose structure** on the process $\{x_t\}_{t=-\infty}^{+\infty}$. Without it, a single path says nothing about the space of paths it was drawn from. ::: ::: --- ## ๐ The Space of Trajectories ```{r} #| label: trajectories #| echo: false #| fig-width: 11 #| fig-height: 4 set.seed(2026) T_len <- 120; n_path <- 25; rho <- 0.7 paths <- replicate(n_path, { e <- rnorm(T_len + 200); x <- numeric(T_len + 200) for (t in 2:(T_len + 200)) x[t] <- rho * x[t - 1] + e[t] tail(x, T_len) }) df <- data.frame(t = rep(1:T_len, n_path), x = as.vector(paths), path = factor(rep(1:n_path, each = T_len))) ggplot(df, aes(t, x, group = path)) + geom_line(colour = "grey75", linewidth = 0.4) + geom_line(data = subset(df, path == 1), colour = "#8b2635", linewidth = 1.1) + geom_vline(xintercept = 60, linetype = "dashed", colour = "#2f4b7c") + annotate("text", x = 62, y = 4.6, label = "ensemble at t = 60", hjust = 0, colour = "#2f4b7c", size = 6) + labs(x = "t", y = expression(x[t])) ``` ::: {style="font-size: 28px"} Grey: 25 possible histories. Red: the one we observe. $E(x_t)$ is an average **down the dashed line**; the sample mean $\bar x_T$ is an average **along the red path**. ::: --- ## ๐ Strict Stationarity ::: {style="font-size: 30px"} Stationarity means that the statistical properties of the process are **stable in time**. ::: {.callout-note} ## Definition (strict stationarity) $\{x_t\}$ is **strictly stationary** if, for every $k \ge 0$, the joint distribution of $$(x_t,\ x_{t+1},\ \dots,\ x_{t+k})$$ does not depend on $t$. ::: So the distribution of $x_t$ is the same for every $t$; so is the joint distribution of every pair $(x_t, x_{t+j})$, every triple, and so on. In particular, if moments exist, $E x_t = \mu$ and $\operatorname{Cov}(x_t, x_{t+j}) = \Gamma_j$ depend on $j$ only. ::: --- ## ๐ White Noise: Three Strengths ::: {style="font-size: 28px"} ::: {.callout-note} ## Definitions - **Strong WN:** $\varepsilon_t \sim \text{iid}(0, \sigma^2)$ โ the point where we are closest to the n-economist. - **Semi-strong WN:** $E(\varepsilon_t \mid \varepsilon_{t-1}, \varepsilon_{t-2}, \dots) = 0$ โ the past does not help to **predict** $\varepsilon_t$. - **Weak WN:** $E\varepsilon_t = 0$, $\operatorname{Var}\varepsilon_t = \sigma^2$, $\operatorname{Cov}(\varepsilon_t, \varepsilon_{t-k}) = 0$ for all $k \ne 0$. ::: $$\text{strong} \;\Longrightarrow\; \text{semi-strong (if } \sigma^2<\infty) \;\Longrightarrow\; \text{weak}$$ ::: {.fragment} Strong WN is strictly stationary. Weak WN need not be: only its first two moments are fixed in time. For instance, $\varepsilon_t \sim N(0,1)$ for even $t$ and $\varepsilon_t \sim \pm 1$ with probability ยฝ each for odd $t$ is weak WN, but its distribution changes from one period to the next. ::: ::: --- ## ๐ Why Semi-Strong Matters: Regression ::: {style="font-size: 30px"} The middle version is the one regression with time-series data needs: $$y_t = x_t'\beta + e_t, \qquad E\left(e_t \mid x_t, x_{t-1}, \dots,\ e_{t-1}, e_{t-2}, \dots\right) = 0.$$ - The error is unpredictable from **current and past regressors** and **past errors**. - That is weaker than i.i.d. errors: the conditional **variance** of $e_t$ may still move with the past. - Lecture 3 shows that this condition is what makes OLS standard errors simple; when it fails, we need HAC standard errors. ::: --- ## ๐งฎ Example: MA(1) ::: {style="font-size: 28px"} $x_t = \varepsilon_t + \theta\varepsilon_{t-1}$, with $\{\varepsilon_t\}$ white noise, variance $\sigma^2$. ::: {.fragment} **Mean.** $E x_t = E\varepsilon_t + \theta E\varepsilon_{t-1} = 0$. ::: ::: {.fragment} **Variance.** The two shocks are uncorrelated, so $\Gamma_0 = \operatorname{Var}x_t = (1 + \theta^2)\sigma^2$. ::: ::: {.fragment} **Lag 1.** Only $\varepsilon_t$ is shared by $x_{t+1}$ and $x_t$: $$\Gamma_1 = \operatorname{Cov}(\varepsilon_{t+1} + \theta\varepsilon_t,\ \varepsilon_t + \theta\varepsilon_{t-1}) = \theta\sigma^2.$$ ::: ::: {.fragment} **Lag $k \ge 2$.** No shock in common, so $\Gamma_k = 0$. None of these depends on $t$: the MA(1) is stationary **for every** $\theta$, and its memory ends after one period. ::: ::: --- ## ๐งฎ Example: AR(1) Is a Recursion ::: {style="font-size: 28px"} $x_t = \rho x_{t-1} + \varepsilon_t$, $\ |\rho| < 1$. Substitute the recursion into itself $k$ times: $$x_t = \varepsilon_t + \rho\varepsilon_{t-1} + \dots + \rho^{k-1}\varepsilon_{t-k+1} + \rho^{k}x_{t-k}.$$ ::: {.fragment} As $k \to \infty$, $\rho^k x_{t-k} \to 0$ for any path with bounded variance, which leaves the **stationary solution** $$x_t = \sum_{j=0}^{\infty}\rho^j\varepsilon_{t-j}, \qquad \Gamma_0 = \frac{\sigma^2}{1-\rho^2}, \qquad \Gamma_k = \rho^{|k|}\,\Gamma_0.$$ ::: ::: {.fragment} The recursion has **other** solutions. If $x_t$ solves it, so does $y_t = x_t + \rho^t$, since $\rho^t = \rho\cdot\rho^{t-1}$. But $E y_t = \rho^t$ changes with $t$: $y_t$ is **not** stationary. Only one solution is. ::: ::: --- ## ๐ป Many Solutions, One Stationary ```{r} #| label: ar1-solutions #| fig-width: 11 #| fig-height: 3.3 set.seed(2026) rho <- 0.85; T_len <- 60; e <- rnorm(T_len) solve_ar <- function(x0) { x <- numeric(T_len); prev <- x0 for (t in 1:T_len) { x[t] <- rho * prev + e[t]; prev <- x[t] }; x } df <- data.frame(t = rep(1:T_len, 3), x = c(solve_ar(10), solve_ar(0), solve_ar(-10)), start = rep(c("x0 = 10", "x0 = 0", "x0 = -10"), each = T_len)) ggplot(df, aes(t, x, colour = start)) + geom_line(linewidth = 1) + scale_colour_manual(values = c("#8b2635", "#2f4b7c", "#e09f3e"), name = NULL) + labs(x = "t", y = expression(x[t])) + theme(legend.position = "top") ``` ::: {style="font-size: 28px"} Same shocks, three starting values. The gaps are $\pm 10\rho^t$: they die out, and every solution merges into the stationary one. ::: --- ## ๐งฎ Example: The Random Walk ::: {style="font-size: 28px"} Set $\rho = 1$: $\ x_t = x_{t-1} + \varepsilon_t$, started at $x_0 = 0$. Then $x_t = \varepsilon_1 + \varepsilon_2 + \dots + \varepsilon_t$. ::: {.fragment} $$E x_t = 0, \qquad \operatorname{Var}(x_t) = t\sigma^2, \qquad \operatorname{Cov}(x_t, x_{t+k}) = t\sigma^2.$$ ::: ::: {.fragment} - The **variance is not stable**: it grows linearly with $t$. The random walk is not stationary. - The correlation $\sqrt{t/(t+k)}$ between $x_t$ and $x_{t+k}$ tends to 1 as $t$ grows: a shock is never forgotten. It is not ergodic either. - Back-substitution fails for $\rho = 1$: $\rho^k x_{t-k} = x_{t-k}$ does not vanish, so there is no stationary solution to fall back on. ::: ::: --- ## ๐ฆ๏ธ What Breaks Stationarity? ```{r} #| label: seasonal-arch #| echo: false #| fig-width: 11 #| fig-height: 3.5 set.seed(2026) n <- 144 # twelve years of monthly data season <- 1.5 + 1.5 * cos(2 * pi * (1:n) / 12) # winter peak 3, summer trough 0 seas <- season + rnorm(n, sd = 0.5) m <- 240 eps <- numeric(m); s2 <- numeric(m); s2[1] <- 1; eps[1] <- rnorm(1) for (t in 2:m) { s2[t] <- 0.2 + 0.8 * eps[t - 1]^2; eps[t] <- sqrt(s2[t]) * rnorm(1) } df <- rbind(data.frame(t = 1:n, x = seas, panel = "Seasonality: mean depends on t"), data.frame(t = 1:m, x = eps, panel = "ARCH: variance clusters, yet stationary")) df$panel <- factor(df$panel, levels = unique(df$panel)) ggplot(df, aes(t, x)) + geom_line(colour = "#2f4b7c", linewidth = 0.5) + facet_wrap(~ panel, scales = "free") + labs(x = "t", y = NULL) + theme(strip.text = element_text(size = 18, face = "bold")) ``` ::: {style="font-size: 28px"} **Left:** monthly data whose winter and summer means differ, so the distribution of $x_t$ depends on the season of $t$ โ not stationary. **Right:** $\sigma_t^2 = 0.2 + 0.8\,\varepsilon_{t-1}^2$ moves with the past, but the **unconditional** variance $0.2/(1-0.8) = 1$ is constant โ stationary. ::: --- ## ๐๏ธ Explore: Persistence ```{ojs} //| echo: false viewof rho = Inputs.range([-0.9, 1], {step: 0.01, value: 0.5, label: "AR coefficient ฯ"}) ``` ```{ojs} //| echo: false ar_data = { const T = 500, rng = d3.randomNormal.source(d3.randomLcg(0.2026))(0, 1); const out = []; let x = 0, s = 0; for (let t = 1; t <= T; t++) { x = rho * x + rng(); s += x; out.push({t, value: x, series: "path x_t"}); out.push({t, value: s / t, series: "running mean"}); } return out; } ``` ```{ojs} //| echo: false Plot.plot({ width: 1100, height: 300, marginLeft: 78, marginBottom: 50, style: {fontSize: "18px"}, x: {label: "t"}, y: {label: null, grid: true}, color: {domain: ["path x_t", "running mean"], range: ["#8ba3c7", "#8b2635"]}, marks: [ Plot.ruleY([0]), Plot.line(ar_data, {x: "t", y: "value", stroke: "series", strokeWidth: d => d.series === "running mean" ? 3 : 1.2}) ] }) ``` ::: {style="font-size: 28px"} Blue: the path $x_t$; red: its running mean $\bar x_t$. Same shocks for every $\rho$. Near 0 the running mean settles at $E x_t = 0$ quickly; near 1 it wanders for a long time; at $\rho = 1$ it never settles. ::: --- ## ๐ Ergodicity: Loss of Memory ::: {style="font-size: 30px"} ::: {.callout-note} ## Definition (heuristic) $\{x_t\}$ is **ergodic** if $x_t$ and $x_{t+k}$ become independent as $k \to \infty$: the process **loses its memory**. ::: - Strictly, asymptotic independence is a *mixing* condition. It is sufficient for ergodicity, and it is the version to keep in mind. - For asymptotic theory we require **both** stationarity and ergodicity (S&E). - The random walk is neither stationary nor ergodic. - Stationarity alone is **not** enough: the next slide has a stationary process with a perfect memory. ::: --- ## ๐ง Stationary but Not Ergodic ::: {style="font-size: 28px"} Draw $z \sim N(0,1)$ **once**, and set $x_t = z$ for every $t$. ::: {.fragment} - **Stationary:** every vector $(x_t, \dots, x_{t+k}) = (z, \dots, z)$ has the same distribution for every $t$. - **Not ergodic:** $\operatorname{Cov}(x_t, x_{t-k}) = \operatorname{Var}(z) = 1$ for every $k$. The dependence never dies out. ::: ::: {.fragment} The sample mean is $\bar x_T = z$ for **every** $T$. It does not converge to $E x_t = 0$. One path shows one value of $z$, and no amount of waiting reveals the others. A slightly richer version, $x_t = \varepsilon_t + z$ with $\varepsilon_t$ white noise, looks like noise on every path, yet each path hovers around its own level $z$. ::: ::: --- ## ๐ The Law of Large Numbers ::: {style="font-size: 30px"} ::: {.callout-important} ## Law of large numbers for S&E series If $\{x_t\}$ is strictly stationary and ergodic and $E|x_t| < \infty$, then $$\operatorname*{plim}_{T\to\infty}\ \frac1T\sum_{t=1}^{T}x_t = E(x_t).$$ ::: This is the licence to replace an average over histories (the dashed line) by an average over time (the red path). - For **non-ergodic** series the LLN does not hold. - For **non-stationary** series $E(x_t)$ is not even a single number to converge to. ::: --- ## ๐ป The LLN Working, and Failing ```{r} #| label: lln-sim #| fig-width: 11 #| fig-height: 3.6 set.seed(2026) T_len <- 2000; n_path <- 15; rho <- 0.5 run_mean <- function(x) cumsum(x) / seq_along(x) erg <- replicate(n_path, run_mean(as.numeric(filter(rnorm(T_len), rho, "recursive")))) non <- replicate(n_path, run_mean(rnorm(T_len) + rnorm(1))) # x_t = eps_t + z df <- rbind( data.frame(t = 1:T_len, m = as.vector(erg), p = rep(1:n_path, each = T_len), panel = "AR(1), rho = 0.5: S&E"), data.frame(t = 1:T_len, m = as.vector(non), p = rep(1:n_path, each = T_len), panel = "eps_t + z: stationary, not ergodic")) ggplot(df, aes(t, m, group = p)) + geom_line(colour = "#2f4b7c", alpha = 0.6) + geom_hline(yintercept = 0, colour = "#8b2635", linewidth = 1) + facet_wrap(~ panel) + scale_x_log10() + labs(x = "T (log scale)", y = "running mean") + theme(strip.text = element_text(size = 17, face = "bold")) ``` ::: {style="font-size: 28px"} Left: fifteen paths, one limit, $E x_t = 0$. Right: fifteen paths, fifteen limits โ each converges to its own $z$, not to $E x_t = 0$. ::: --- ## ๐ Functions of S&E Series ::: {style="font-size: 28px"} ::: {.callout-important} ## Proposition If $\{x_t\}$ is S&E and $f$ is a measurable function that does **not** depend on $t$, then $$y_t = f(x_t, x_{t-1}, \dots, x_{t-n})$$ is S&E. The same holds with infinitely many lags, provided the function is well defined. ::: - So $x_t^2$, $x_t x_{t-1}$, $\log|x_t|$ and $\mathbf 1\{x_t > 0\}$ inherit S&E from $x_t$. - **MA($\infty$).** If $\varepsilon_t$ is i.i.d. and $\sum_j|\psi_j| < \infty$, then $x_t = \sum_{j\ge0}\psi_j\varepsilon_{t-j}$ is S&E. With $\psi_j = 0.9^j$ the sum converges: S&E. - With $\psi_j = 1$ for all $j$ the sum does not converge, so $f$ is not well defined; the partial sums form a random walk, neither S nor E. ::: --- ## ๐ง Making a Series Stationary ::: {style="font-size: 28px"} Most economic levels are not stationary. We **transform** them first. - **Prices** wander: use **log returns** $r_t = \log P_t - \log P_{t-1}$. - **Output** (GNP) grows exponentially, so its mean is growing. Take logs: $\log \text{GNP}_t$ grows roughly linearly. Then difference: $$\Delta\log\text{GNP}_t = \log\frac{\text{GNP}_t}{\text{GNP}_{t-1}} \approx \frac{\Delta\text{GNP}_t}{\text{GNP}_{t-1}},$$ the **growth rate**, which fluctuates around a constant mean. - The trend in $\log\text{GNP}$ may be **deterministic** or **stochastic**; telling them apart is the subject of Lectures 9โ10. - **Seasonality** and **structural breaks** also violate stationarity, and must be modelled rather than ignored. ::: --- ## ๐ง ThinkโPairโShare ```{r} #| label: tps-timer #| echo: false countdown::countdown(minutes = 3, top = 0, right = 0, font_size = "2em", warn_when = 30) ``` ::: {style="font-size: 28px"} $\varepsilon_t$ is i.i.d. $N(0,1)$; $z \sim N(0,1)$ is drawn once, independent of $\{\varepsilon_t\}$. For each process: is it strictly stationary? Is it ergodic? 1. $x_t = \varepsilon_t + 3\varepsilon_{t-1}$ 2. $x_t = z\,\varepsilon_t$ 3. $x_t = \mu_{q(t)} + \varepsilon_t$, where $q(t)$ is the quarter of month $t$ and $\mu_1, \dots, \mu_4$ are not all equal *Hint for 2:* does $\bar x_T$ converge to $E x_t$? Does the sample mean of $x_t^2$? ::: --- ## โ
ThinkโPairโShare: Solution ::: {style="font-size: 28px"} 1. **S&E.** An MA(1) is stationary for every $\theta$, including $\theta = 3$, and $x_t$, $x_{t+k}$ are independent once $k \ge 2$. ::: {.fragment} 2. **Stationary, not ergodic.** $(x_t, \dots, x_{t+k}) = z\,(\varepsilon_t, \dots, \varepsilon_{t+k})$, whose law does not depend on $t$. The mean is fine: $\bar x_T = z\,\bar\varepsilon_T \to 0 = E x_t$. But $$\frac1T\sum_t x_t^2 = z^2\,\frac1T\sum_t\varepsilon_t^2 \xrightarrow{p} z^2 \ne E x_t^2 = 1.$$ $|x_t|$ and $|x_{t+k}|$ share $|z|$ for every $k$: the memory is in the **scale**. ::: ::: {.fragment} 3. **Not stationary.** $E x_t = \mu_{q(t)}$ depends on $t$. Removing the quarterly means (seasonal adjustment) gives back $\varepsilon_t$, which is S&E. ::: ::: --- ## ๐ฐ Case Study: Levels or Returns? :::: columns ::: {.column width="50%"} ::: {.callout-note} ## ๐ The question Is the S&P 500 a stationary series? If not, are its daily log returns? 1. Do the mean and standard deviation of the **level** stay put from one half of the sample to the other? 2. Do those of the **returns**? 3. What is left over that stationarity does not rule out? ::: ::: ::: {.column width="50%"} ::: {.callout-tip} ## ๐ The data **Series**: S&P 500 index, daily close (FRED `SP500`) **Period**: 17 Oct 2016 โ 14 Oct 2026, about 2,500 trading days **Transformation**: $r_t = 100\,(\log P_t - \log P_{t-1})$, in percent ::: ::: :::: ```{.r .display-only} # How the vendored file was obtained (not run here) download.file("https://fred.stlouisfed.org/graph/fredgraph.csv?id=SP500", "data/SP500.csv") ``` --- ## ๐ฐ Case Study: Two Halves in R ```{r} #| label: case-sp #| code-summary: "๐ฆ Compare the two halves of the sample, level and returns" sp <- read.csv("data/SP500.csv") sp$P <- suppressWarnings(as.numeric(sp$SP500)) # FRED marks holidays "." sp <- sp[!is.na(sp$P), ] sp$date <- as.Date(sp$observation_date) r <- 100 * diff(log(sp$P)) # daily log returns, % half <- rep(c("first half", "second half"), each = ceiling(length(r) / 2))[seq_along(r)] lvl <- sp$P[-1] acf1 <- function(x) acf(x, plot = FALSE)$acf[2] # lag-1 autocorrelation round(rbind(level_mean = tapply(lvl, half, mean), level_sd = tapply(lvl, half, sd), return_mean = tapply(r, half, mean), return_sd = tapply(r, half, sd), acf1_return = tapply(r, half, acf1), acf1_return_sq = tapply(r^2, half, acf1)), 3) ``` ::: {style="font-size: 28px"} The level's mean nearly doubles between halves; the return's mean and sd barely move. Note the last row: $r_t^2$ is clearly autocorrelated in both halves. ::: --- ## ๐ฐ Case Study: Rolling Windows ```{r} #| label: case-sp-fig #| echo: false #| fig-width: 10 #| fig-height: 3.9 w <- 250 roll <- function(x, f) zoo::rollapply(x, w, f, align = "right", fill = NA) d <- sp$date[-1] df <- rbind( data.frame(date = d, value = lvl / 1000, what = "data", panel = "Level (thousands)"), data.frame(date = d, value = roll(lvl, mean) / 1000, what = "250-day mean", panel = "Level (thousands)"), data.frame(date = d, value = r, what = "data", panel = "Daily log return, %"), data.frame(date = d, value = roll(r, mean), what = "250-day mean", panel = "Daily log return, %"), data.frame(date = d, value = roll(r, sd), what = "250-day sd", panel = "Daily log return, %")) df$panel <- factor(df$panel, levels = c("Level (thousands)", "Daily log return, %")) df$what <- factor(df$what, levels = c("data", "250-day mean", "250-day sd")) ggplot(df, aes(date, value, colour = what, linewidth = what)) + geom_line() + facet_wrap(~ panel, scales = "free_y") + scale_colour_manual(values = c("#b9c7dc", "#8b2635", "#e09f3e"), name = NULL) + scale_linewidth_manual(values = c(0.4, 1.3, 1.3), name = NULL) + labs(x = NULL, y = NULL) + theme_minimal(base_size = 20) + theme(legend.position = "top", strip.text = element_text(face = "bold")) ``` ::: {style="font-size: 28px"} **What the data say:** the level trends upward, so its rolling mean has no fixed value to estimate; the returns fluctuate around a stable mean near zero, which is why we model returns. Stationarity is not the whole story: the rolling sd jumps in 2020 and stays high through 2022, and $r_t^2$ is autocorrelated โ volatility clustering, which a stationary but conditionally heteroskedastic model (Lecture 15) can capture. ::: --- ## ๐ Quiz #1: Which White Noise? {.quiz-question} $\varepsilon_t = \sigma_t\eta_t$ with $\eta_t$ i.i.d. $N(0,1)$ and $\sigma_t^2 = 0.2 + 0.5\,\varepsilon_{t-1}^2$. Which is the strongest version of white noise that $\varepsilon_t$ satisfies? - Strong: $\varepsilon_t$ is i.i.d. - [Semi-strong: $E(\varepsilon_t \mid \varepsilon_{t-1}, \dots) = 0$, but not i.i.d.]{.correct data-explanation="โ
Given the past, ฯ_t is known and E(ฮท_t)=0, so the conditional mean is zero. But the conditional variance depends on ฮตยฒ_{tโ1}, so the ฮต_t are dependent and not i.i.d."} - Only weak: uncorrelated, but predictable in mean - None: the variance changes over time --- ## ๐ Quiz #2: The Random Walk {.quiz-question} $x_t = x_{t-1} + \varepsilon_t$, $x_0 = 0$, $\operatorname{Var}\varepsilon_t = 4$. What is $\operatorname{Var}(x_{25})$? - $4$ - $\infty$ - [$100$]{.correct data-explanation="โ
x_25 is the sum of 25 uncorrelated shocks, so Var(x_25) = 25 ร 4 = 100. The variance grows like tฯยฒ, which is why the random walk is not stationary."} - $4/(1-1^2)$, which is undefined --- ## ๐ Quiz #3: Stationary, Not Ergodic {.quiz-question} $x_t = \varepsilon_t + z$, with $\varepsilon_t$ white noise and $z \sim N(0,1)$ drawn once. As $T \to \infty$, the sample mean $\bar x_T$ converges in probability to โฆ - $0 = E x_t$ - [$z$, a random variable that differs from path to path]{.correct data-explanation="โ
xฬ_T = ฮตฬ_T + z, and ฮตฬ_T โ 0. The limit is z, not E x_t = 0: the process is stationary but not ergodic, so the LLN fails."} - Nothing: $\bar x_T$ diverges - $1 = \operatorname{Var}(z)$ --- ## ๐ Key Formulas ::: {style="font-size: 28px"} | | | |---|---| | Strict stationarity | law of $(x_t, \dots, x_{t+k})$ free of $t$, every $k$ | | Strong / semi-strong / weak WN | iid$(0,\sigma^2)$ / $E(\varepsilon_t\mid\text{past})=0$ / uncorrelated | | MA(1) | $\Gamma_0 = (1+\theta^2)\sigma^2$, $\Gamma_1 = \theta\sigma^2$, $\Gamma_k = 0$ for $k\ge2$ | | AR(1), $\lvert\rho\rvert<1$ | $x_t = \sum_{j\ge0}\rho^j\varepsilon_{t-j}$, $\Gamma_k = \rho^{\lvert k\rvert}\sigma^2/(1-\rho^2)$ | | Random walk, $x_0 = 0$ | $\operatorname{Var}x_t = t\sigma^2$ | | LLN for S&E, $E\lvert x_t\rvert<\infty$ | $\frac1T\sum_{t=1}^T x_t \xrightarrow{p} E x_t$ | | Log return / growth rate | $\log P_t - \log P_{t-1} \approx \Delta P_t / P_{t-1}$ | ::: --- ## ๐ Summary ::: {style="font-size: 28px"} - A time series is one segment of one trajectory; to learn from it we **impose** stationarity and ergodicity - Strict stationarity: every finite-dimensional distribution is invariant to shifts in time - White noise comes in three strengths; the semi-strong (MDS) version is what time-series regression needs - MA(1) is always stationary; AR(1) has a unique stationary solution when $|\rho|<1$; the random walk has none - Ergodicity is loss of memory; $x_t = z$ is stationary with a perfect memory, and its sample mean never learns $E x_t$ - Under S&E the time average converges to the expectation; we transform levels (logs, differences, returns) to get there ::: --- ## ๐ Further Reading ::: {style="font-size: 28px"} - Hamilton, J. D. (1994). *Time Series Analysis*. Princeton UP, ch. 3.1โ3.2 (stationarity, ergodicity, white noise) and ch. 7.2 (the LLN for dependent data). - Tsay, R. S. (2010). *Analysis of Financial Time Series*, 3rd ed. Wiley, ch. 1 and 2.1โ2.4 (returns, stationarity, AR and MA basics). - Franses, P. H. and van Dijk, D. (2000). *Non-Linear Time Series Models in Empirical Finance*. Cambridge UP, ch. 2 (stylised facts of returns). - Hayashi, F. (2000). *Econometrics*. Princeton UP, ยง2.2 (ergodic stationarity, MDS). **Next lecture:** Ergodic Theorem, CLT and the Long-Run Variance โ how fast $\bar x_T$ converges, and how to put a standard error on it when the data have memory. :::
