```{r} #| label: setup #| include: false set.seed(2026) library(ggplot2) theme_set(theme_minimal(base_size = 18)) ``` ## 🎬 The Idea in 3 Minutes ::: {style="text-align:center"} ```{=html} ``` [⬅ Previous lecture: 10 · Dickey–Fuller, Phillips–Perron and ADF Tests](../10-dickey-fuller-tests/dickey_fuller_tests_lecture10.html){style="font-size:28px"} [A short film with the intuition. Watch it before the slides, or after. Captions: CC button.]{style="font-size:28px"} ::: --- ## 🎯 Learning Objectives ::: {style="font-size: 32px"} By the end of this lecture, you will be able to: - **Explain** two reasons to leave linear models: persistence without a unit root, and predictable white noise - **Write down** a threshold autoregression and choose its threshold variable for a given series - **Judge** stationarity of a TAR, including one with an explosive regime - **Estimate** a SETAR by concentrated least squares over a trimmed grid - **State** the asymptotic behaviour of $\hat\gamma$ and of the slope estimates ::: --- ## 🔁 Warm-up from last time {.quiz-question} Lecture 10: $T=300$ observations from $x_t=0.999\,x_{t-1}+\varepsilon_t$. How often does a 5% DF test (with a constant) reject the unit root? - [About 5% of the time]{.correct data-explanation="✅ The simulation in Lecture 10 gave 5.3% at ρ = 0.999 against 5.2% at ρ = 1: power equals size. A unit root and ρ = 0.999 cannot be told apart in a sample of this length; they differ only in long-run forecasts."} - About 95% of the time - About 50% of the time - Never, since the series is stationary --- ## 🗺️ Where We Are ::: {style="font-size: 32px"} Lecture 10 ended on a limitation: in a sample, a unit root and a root of $0.999$ look the same. A unit-root test measures **persistence**, nothing more. ::: {.fragment} Some series are persistent, yet economic theory says they must revert: the real exchange rate under purchasing power parity is the standard case. A linear AR can only make such a series *uniformly* persistent. Today the dynamics are allowed to **depend on the state** of the series. ::: ::: --- ## ❓ The Question ::: {.callout-important} ## Does unemployment rise faster than it falls? In US recessions the unemployment rate jumps by several points within a year. In expansions it drifts down by a tenth of a point a month for years. ::: ::: {style="font-size: 30px"} A linear AR for $\Delta u_t$ has one set of coefficients, so a rise and a fall of the same size must be followed by mirror-image paths. If the dynamics differ between "unemployment rising" and "unemployment falling", a linear model averages two different economies into one that exists in neither. ::: --- ## 💡 Persistence Without a Unit Root ::: {style="font-size: 30px"} Nonlinear modelling can build a bridge between empirical properties and economic theory. - **Empirically**, real exchange rates are very persistent; unit-root tests often fail to reject. - **In theory**, PPP says the real exchange rate fluctuates around zero (its log deviation from parity). - **A nonlinear model** reconciles the two: little or no reversion near parity, strong reversion far from it. ::: {.fragment} The series is persistent, but it has **no unit root**. We return to this example with numbers below. ::: ::: --- ## 💡 Predictable White Noise ::: {style="font-size: 28px"} Let $\varepsilon_t$ be strong white noise (i.i.d., mean 0, variance 1) and $$y_t=\varepsilon_t+\theta\,\varepsilon_{t-1}\varepsilon_{t-2}.$$ ::: {.fragment} For $j\neq0$, every term of $E(y_ty_{t-j})$ contains a factor $\varepsilon_s$ that appears exactly once, so it vanishes: $\operatorname{cov}(y_t,y_{t-j})=0$. Thus $y_t$ is **white noise**, though not independent. ::: ::: {.fragment} The best **linear** forecast is therefore $\hat y^{\,\text{lin}}_{t+1}=0$. The **conditional mean** is not: $$\hat y^{\,\text{nonlin}}_{t+1}=E(y_{t+1}\mid\mathcal F_t)=\theta\,\varepsilon_t\varepsilon_{t-1}.$$ A nonlinear model can reveal information that no linear model can detect. ::: ::: --- ## 💻 White Noise You Can Forecast ```{r} #| label: bilinear-wn #| code-summary: "📦 Simulate y_t = e_t + 0.8 e_{t-1} e_{t-2}" #| results: hold n <- 5000; theta <- 0.8; e <- rnorm(n + 2) y <- e[3:(n + 2)] + theta * e[2:(n + 1)] * e[1:n] round(acf(y, lag.max = 5, plot = FALSE)$acf[2:6], 3) # autocorrelations, lags 1-5 f_nonlin <- theta * e[2:(n + 1)] * e[1:n] # E(y_t | past) c(R2_linear = 0, R2_nonlinear = round(1 - mean((y - f_nonlin)^2) / var(y), 3), theory = round(theta^2 / (1 + theta^2), 3)) ``` ::: {style="font-size: 28px"} Every autocorrelation is within $\pm0.02$ of zero, yet the conditional mean explains over a third of the variance. A close relative is the **bilinear** model: an ARMA plus cross terms $\sum_{i,j}\psi_{ij}x_{t-i}\varepsilon_{t-j}$. ::: --- ## 📐 Threshold Autoregression (TAR) ::: {style="font-size: 28px"} Models with **regimes** are the nonlinear family most used in economics. In a TAR there are most often two regimes, with linear AR dynamics in each: $$x_t=\begin{cases}\mu_1+\rho_1x_{t-1} & \text{if } q_{t-1}\le\gamma\\[2pt] \mu_2+\rho_2x_{t-1} & \text{if } q_{t-1}>\gamma\end{cases}\;+\;\varepsilon_t, \qquad E_{t-1}\varepsilon_t=0,\ \operatorname{var}_{t-1}\varepsilon_t=\sigma^2.$$ - The switch is driven by an event: the **threshold variable** $q_{t-1}$ exceeds the **threshold** $\gamma$. - $q_{t-1}$ must be realised in the previous period, so the regime is known before $x_t$ is drawn. - The switches are **endogenous** when $q_{t-1}$ depends on the series itself. ::: --- ## 📐 SETAR, and Choosing $q_{t-1}$ ::: {style="font-size: 28px"} If $q_{t-1}=x_{t-d}$ for a known or unknown delay $d$, the model is a **self-exciting** TAR (SETAR). The choice of $q_{t-1}$ depends on the frequency and nature of the data: | Series | Threshold variable | Regimes | |---|---|---| | GNP growth, quarterly | $q_{t-1}=y_{t-2}$ | contraction / expansion | | unemployment, monthly | $q_{t-1}=\Delta x_{t-d}$ or $x_{t-1}-x_{t-d}$ | rising / falling | | nominal exchange rate, weekly returns | $q_{t-1}=\frac15\sum_{j=1}^{5}\lvert x_{t-j}\rvert$ | high / low volatility | | real exchange rate, quarterly | $q_{t-1}=\lvert x_{t-1}\rvert$ | near / far from PPP | ::: --- ## 📐 The Real Exchange Rate Band ::: {style="font-size: 28px"} Let $x_t$ be the log deviation of the real exchange rate from PPP, and $q_{t-1}=|x_{t-1}|$: - **inner regime**, $|x_{t-1}|\le\gamma$: estimates give $\rho_1=1.06$, locally **explosive**; - **outer regime**, $|x_{t-1}|>\gamma$: $\rho_2\approx0.92$, **mean-reverting**. ::: {.fragment} Inside the band, transaction costs make arbitrage unprofitable, and the rate drifts away from parity. Outside it, arbitrage pulls it back. A unit-root test will typically find $I(1)$, yet the outer regime forces the process back towards the band: it is **globally stationary**. ::: ::: --- ## 💻 Simulating the Band ```{r} #| label: band #| echo: false #| fig-width: 11 #| fig-height: 3.9 sim_band <- function(n, r1 = 1.06, r2 = 0.92, g = 0.10, s = 0.03) { x <- numeric(n); for (t in 2:n) x[t] <- (if (abs(x[t - 1]) <= g) r1 else r2) * x[t - 1] + s * rnorm(1) x } set.seed(7); xb <- sim_band(400) ggplot(data.frame(t = seq_along(xb), x = xb), aes(t, x)) + annotate("rect", xmin = -Inf, xmax = Inf, ymin = -0.1, ymax = 0.1, alpha = 0.12, fill = "#2f4b7c") + geom_hline(yintercept = c(-0.1, 0.1), linetype = "dashed", colour = "#2f4b7c") + geom_line(colour = "#8b2635", linewidth = 0.9) + labs(x = "quarter", y = "log deviation from PPP") + theme_minimal(base_size = 20) ``` ::: {style="font-size: 28px"} Explosive inside the shaded band, $\rho_1=1.06$; mean-reverting outside it, $\rho_2=0.92$; $\gamma=0.10$, $\sigma=0.03$. ::: --- ## 💻 The Band Fools the DF Test ```{r} #| label: band-df #| code-summary: "📦 DF τ test (constant) on 2,000 band paths of 160 quarters" dft <- function(x) { y <- x[-1]; X <- cbind(1, x[-length(x)]); f <- lm.fit(X, y) s2 <- sum(f$residuals^2) / (length(y) - 2) (f$coefficients[2] - 1) / sqrt(s2 * solve(crossprod(X))[2, 2]) } c(rejection_rate_5pct = mean(replicate(2000, dft(sim_band(160)) < -2.86))) ``` ::: {style="font-size: 28px"} The process is globally stationary, yet in forty years of quarterly data the DF test fails to reject the unit root about four times in five. ::: --- ## 📐 Stationarity of a TAR ::: {style="font-size: 28px"} $|\rho_1|<1$ and $|\rho_2|<1$ is **sufficient but not necessary**. - One regime may have a unit root, or even $\rho>1$, and the process can still be globally stationary. - What makes this possible is an **endogenous switch**: the threshold variable depends on $x$ itself, so an explosive regime pushes the process out of that regime. ::: {.fragment} For SETAR(1) with $q_{t-1}=x_{t-1}$ and zero intercepts, the process is ergodic if and only if $$\rho_1<1,\qquad \rho_2<1,\qquad \rho_1\rho_2<1$$ (Petrucelli and Woolford, 1984). So $\rho_1=-3$, $\rho_2=0.2$ is stationary: $|\rho_1|>1$ but $\rho_1\rho_2=-0.6<1$. ::: ::: --- ## 🔧 Extensions: Lags and Regimes ::: {style="font-size: 28px"} **Different lag orders.** Regime $j$ has its own AR($p_j$): $$x_t=\mu_j+\rho_{j,1}x_{t-1}+\dots+\rho_{j,p_j}x_{t-p_j}+\varepsilon_t\quad\text{in regime } j .$$ Choose $(p_1,p_2)$ by minimising $IC_1+IC_2$, for example $$BIC_j=T_j\log\hat\sigma_j^2+p_j\log T_j,$$ where $T_j$ is the number of observations in regime $j$. ::: {.fragment} **Multiple regimes, same threshold variable**, with $\gamma_0<\gamma_1<\dots<\gamma_m$: $$x_t=\mu_j+\rho_jx_{t-1}+\varepsilon_t\quad\text{if }\gamma_{j-1}\le q_{t-1}<\gamma_j,\qquad j=1,\dots,m.$$ ::: ::: --- ## 🔧 Extensions: Two Thresholds ::: {style="font-size: 28px"} Four regimes, governed by $x_{t-1}$ and $x_{t-2}$ with thresholds $\gamma_1,\gamma_2$: $$x_t=\begin{cases}\mu_1+\rho_1x_{t-1} & \text{if }x_{t-2}\le\gamma_2,\ x_{t-1}\le\gamma_1\\ \mu_2+\rho_2x_{t-1} & \text{if }x_{t-2}\le\gamma_2,\ x_{t-1}>\gamma_1\\ \mu_3+\rho_3x_{t-1} & \text{if }x_{t-2}>\gamma_2,\ x_{t-1}\le\gamma_1\\ \mu_4+\rho_4x_{t-1} & \text{if }x_{t-2}>\gamma_2,\ x_{t-1}>\gamma_1\end{cases}\;+\;\varepsilon_t .$$ ::: {.fragment} Each added threshold multiplies the number of regimes, and divides the observations available to each. ::: ::: --- ## 🔧 Extensions: Rules and Variances ::: {style="font-size: 28px"} **A general switching rule.** Replace $q_{t-1}\lessgtr\gamma$ by a condition on $$f(x_{t-1},\dots,x_{t-d},\vartheta),$$ where the form of $f$ is known a priori, the delay $d$ may be unknown, and $\vartheta$ is an unknown finite-dimensional parameter. ::: {.fragment} **Regime-wise heteroskedasticity:** $$\operatorname{var}_{t-1}\varepsilon_t=\begin{cases}\sigma_1^2 & \text{if }q_{t-1}\le\gamma\\ \sigma_2^2 & \text{if }q_{t-1}>\gamma\end{cases}$$ **Conditional heteroskedasticity:** $\operatorname{var}_{t-1}\varepsilon_t$ not constant, for example GARCH-type errors (Lecture 15). ::: ::: --- ## 📐 Estimation: The Problem ::: {style="font-size: 28px"} Back to the basic model, written with indicators: $$x_t=(\mu_1+\rho_1x_{t-1})\,\mathbb 1\{q_{t-1}\le\gamma\}+(\mu_2+\rho_2x_{t-1})\,\mathbb 1\{q_{t-1}>\gamma\}+\varepsilon_t .$$ Parameters: $\theta=(\mu_1,\rho_1,\mu_2,\rho_2,\gamma,d)$, with $d$ possibly, and $\sigma^2$. Because $\gamma$ is unknown, the model is **nonlinear**. ::: {.fragment} Gaussian QML is equivalent to NLLS: $$\min_\theta\sum_t\Big(x_t-(\mu_1+\rho_1x_{t-1})\mathbb 1\{q_{t-1}\le\gamma\}-(\mu_2+\rho_2x_{t-1})\mathbb 1\{q_{t-1}>\gamma\}\Big)^2 .$$ With respect to $\gamma$ the objective is not only non-differentiable but **not even continuous**. Gradient methods are useless. ::: ::: --- ## 🔧 The Concentration Method ::: {style="font-size: 28px"} Split the parameters into $\gamma$ and the rest. For fixed $\gamma$ the model is **linear**. 1. **Grid.** Make a grid $\Gamma$ for $\gamma$. If $q$ is a lag of $x$, put the grid points at the observed values of $q_{t-1}$. **Trim** the lowest and highest 10–15%: too few observations in a regime do not allow its parameters to be estimated. 2. **OLS for each $\gamma\in\Gamma$.** Regress $x_t$ on the regime-split regressors and record $\hat\sigma^2(\gamma)$. 3. **Minimise.** $\hat\gamma=\arg\min_{\gamma\in\Gamma}\hat\sigma^2(\gamma)$, and $\hat\theta=\hat\theta(\hat\gamma)$, the OLS estimates at $\hat\gamma$. If $d$ is unknown, minimise over the pairs $(d,\gamma)$. ::: --- ## ✍️ Worked Example: The Data ::: {style="font-size: 28px"} Six observations, and a TAR with regime means only: $x_t=\mu_1\mathbb 1\{q_{t-1}\le\gamma\}+\mu_2\mathbb 1\{q_{t-1}>\gamma\}+\varepsilon_t$. | $q_{t-1}$ | −1.0 | −0.6 | −0.2 | 0.1 | 0.4 | 0.9 | |---|---|---|---|---|---|---| | $x_t$ | −0.5 | −0.7 | −0.3 | 0.6 | 0.4 | 0.8 | ::: {.fragment} **Grid.** Put candidate thresholds at the observed $q$. Require at least two observations in each regime: this trims $\gamma=-1.0$, $0.4$ and $0.9$, leaving $$\Gamma=\{-0.6,\ -0.2,\ 0.1\}.$$ For each $\gamma$ the OLS estimates are the two regime means, and $SSR(\gamma)$ is the sum of squared deviations within regimes. ::: ::: --- ## ✍️ Worked Example: The Grid ::: {style="font-size: 28px"} | $\gamma$ | regime 1 | $\hat\mu_1$ | regime 2 | $\hat\mu_2$ | $SSR(\gamma)$ | |----|----------------------|--------|-----------------------|-------|-----------------------| | −0.6 | −0.5, −0.7 | −0.600 | −0.3, 0.6, 0.4, 0.8 | 0.375 | 0.020 + 0.688 = 0.708 | | −0.2 | −0.5, −0.7, −0.3 | −0.500 | 0.6, 0.4, 0.8 | 0.600 | 0.080 + 0.080 = **0.160** | | 0.1 | −0.5, −0.7, −0.3, 0.6 | −0.225 | 0.4, 0.8 | 0.600 | 0.988 + 0.080 = 1.068 | ::: {.fragment} So $\hat\gamma=-0.2$, $\hat\mu_1=-0.5$, $\hat\mu_2=0.6$, $\hat\sigma^2=0.160/6=0.027$. Any $\gamma\in[-0.2,\,0.1)$ splits the sample identically, so $\hat\sigma^2(\gamma)$ is flat there: the objective is a **step function**, and only its values at the observed $q$ matter. ::: ::: --- ## 💻 SETAR by Hand ```{r} #| label: setar-sim #| code-summary: "📦 Simulate a SETAR(1), then estimate it by concentration" set.seed(2028); n <- 300; x <- numeric(n + 1) for (t in 2:(n + 1)) x[t] <- if (x[t - 1] <= 0) 0.4 + 0.6 * x[t - 1] + 0.5 * rnorm(1) else -0.2 + 0.3 * x[t - 1] + 0.5 * rnorm(1) y <- x[-1]; q <- x[-(n + 1)] # q_{t-1} = x_{t-1}: self-exciting setar_s2 <- function(g) { I <- q <= g; J <- !I # regime indicators mean(lm.fit(cbind(I, I * q, J, J * q), y)$residuals^2) } grid <- sort(q); grid <- grid[grid >= quantile(q, 0.15) & grid <= quantile(q, 0.85)] s2 <- sapply(grid, setar_s2) g_hat <- grid[which.min(s2)]; I <- q <= g_hat; J <- !I b <- lm.fit(cbind(I, I * q, J, J * q), y)$coefficients round(c(gamma_hat = g_hat, mu1 = b[[1]], rho1 = b[[2]], mu2 = b[[3]], rho2 = b[[4]], grid_points = length(grid)), 3) ``` ::: {style="font-size: 28px"} True values: $\gamma_0=0$, $(\mu_1,\rho_1)=(0.4,0.6)$, $(\mu_2,\rho_2)=(-0.2,0.3)$. The threshold is found almost exactly; the slopes carry ordinary sampling error of about $\pm0.1$. ::: --- ## 📊 The Objective Is a Step Function ```{r} #| label: s2-step #| echo: false #| fig-width: 11 #| fig-height: 3.9 ggplot(data.frame(g = grid, s2 = s2), aes(g, s2)) + geom_step(colour = "#2f4b7c", linewidth = 0.9) + geom_point(colour = "#2f4b7c", size = 1.3) + geom_vline(xintercept = 0, linetype = "dashed", colour = "grey40") + geom_point(data = data.frame(g = g_hat, s2 = min(s2)), colour = "#8b2635", size = 4) + annotate("text", x = 0.02, y = min(s2) + 0.15 * diff(range(s2)), label = "true~gamma[0] == 0", parse = TRUE, hjust = 0, size = 6, colour = "grey30") + labs(x = expression(gamma ~ "(grid = observed values of " * q[t-1] * ", 15% trimmed)"), y = expression(hat(sigma)^2 * (gamma))) + theme_minimal(base_size = 20) ``` ::: {style="font-size: 28px"} Flat between neighbouring observations, jumping at each one; rough overall, with a sharp dip whose lowest step (red) is $\hat\gamma$. ::: --- ## 📐 Asymptotic Theory ::: {style="font-size: 28px"} Because of the discontinuity, the usual asymptotic theory does not apply. Yet the outcome is good news. - $\hat\mu_1,\hat\rho_1,\hat\mu_2,\hat\rho_2,\hat\sigma^2$ are standard: consistent, $\sqrt T$-rate, asymptotically normal, with the **same** limit as if $\gamma$ were known. - $\hat\gamma$ is $T$-consistent: $\hat\gamma-\gamma=O_p(T^{-1})$. Because it converges so fast, we may act as if $\hat\gamma$ were the true value when doing inference on the slopes. - The discontinuity looks like a curse; it is in fact a **blessing**. ::: {.fragment} The catch: the limit of $T(\hat\gamma-\gamma)$ is **non-pivotal**. It depends on unknown parameters and is very hard to standardise. How, then, do we build a confidence interval for $\gamma$? That question opens the next lecture. ::: ::: --- ## 🧠 Think–Pair–Share ```{r} #| label: tps-timer #| echo: false countdown::countdown(minutes = 3, seconds = 0, top = 0, right = 0, font_size = "2em", warn_when = 30) ``` ::: {style="font-size: 30px"} 1. In the step plot, why is $\hat\sigma^2(\gamma)$ constant between two neighbouring observed values of $q_{t-1}$? 2. What goes wrong if we search $\gamma$ all the way to the smallest observed $q_{t-1}$? 3. A colleague estimates $\gamma$ with a Newton–Raphson routine started at the median of $q_{t-1}$. What will happen? ::: --- ## ✅ Think–Pair–Share: Solution ::: {style="font-size: 28px"} 1. $\hat\sigma^2(\gamma)$ depends on $\gamma$ only through the split $\{t: q_{t-1}\le\gamma\}$. Between two neighbouring observations the split does not change, so neither does the OLS fit. ::: {.fragment} 2. Regime 1 would hold one or two observations: its two parameters are barely or not at all estimable, and $\hat\sigma^2(\gamma)$ can be driven down by fitting a handful of points exactly. Trimming 10–15% at each end prevents this. ::: ::: {.fragment} 3. The derivative of a step function is zero wherever it exists. Newton–Raphson sees a flat objective and stops where it started. Grid search over observed values is exact, and cheap: one OLS per grid point. ::: ::: --- ## 💰 Case Study: Unemployment Regimes :::: columns ::: {.column width="50%"} ::: {.callout-note} ## 📈 The question 1. Do the dynamics of $\Delta u_t$ change when unemployment is above its level a year earlier? 2. Where is the threshold? 3. How different are the two regimes? ::: ::: ::: {.column width="50%"} ::: {.callout-tip} ## 📊 The data **Series**: US unemployment rate, FRED `UNRATE`, monthly, Jan 1948 – Dec 2019 $x_t=\Delta u_t$, two lags in each regime; $q_{t-1}=u_{t-1}-u_{t-13}$, the change over the past year; grid trimmed 15% 2020 is left out: one month's jump of 10 points would dominate least squares ::: ::: :::: ```{.r .display-only} download.file("https://fred.stlouisfed.org/graph/fredgraph.csv?id=UNRATE", "data/UNRATE.csv") # then keep Jan 1948 - Dec 2019 ``` --- ## 💰 Case Study: The Estimates ```{r} #| label: case-setar #| code-summary: "📦 SETAR(2) for monthly unemployment changes, by hand" #| results: hold un <- read.csv("data/UNRATE.csv"); u <- un$UNRATE; N <- length(u) du <- c(NA, diff(u)); tt <- 14:N # d = 12 needs u_{t-13} yy <- du[tt]; Z <- cbind(1, du[tt - 1], du[tt - 2]) qq <- round(u[tt - 1] - u[tt - 13], 1) # one-decimal data: avoid float noise G <- sort(unique(qq)); G <- G[G >= quantile(qq, 0.15) & G <= quantile(qq, 0.85)] s2_u <- sapply(G, function(g) { I <- qq <= g mean(lm.fit(cbind(Z * I, Z * !I), yy)$residuals^2) }) gh <- G[which.min(s2_u)]; I <- qq <= gh fit <- lm(yy ~ 0 + cbind(Z * I, Z * !I)); b <- coef(fit); se <- sqrt(diag(vcov(fit))) print(c(T = length(yy), grid_from = min(G), grid_to = max(G), gamma_hat = gh, share_falling = round(mean(I), 2))) round(cbind(falling = b[1:3], se = se[1:3], rising = b[4:6], se = se[4:6]), 3) |> `rownames<-`(c("const", "du(t-1)", "du(t-2)")) ``` ::: {style="font-size: 28px"} Read the columns side by side: a steady fall with no momentum against a rise that feeds on itself. ::: --- ## 💰 Case Study: What the Data Say ```{r} #| label: case-fig #| echo: false #| fig-width: 10 #| fig-height: 3.5 ggplot(data.frame(g = G, s2 = s2_u), aes(g, s2)) + geom_step(colour = "#2f4b7c", linewidth = 1) + geom_point(colour = "#2f4b7c", size = 2) + geom_point(data = data.frame(g = gh, s2 = min(s2_u)), colour = "#8b2635", size = 4.5) + geom_hline(yintercept = mean(lm.fit(Z, yy)$residuals^2), linetype = "dashed", colour = "grey40") + annotate("text", x = 0.6, y = mean(lm.fit(Z, yy)$residuals^2), vjust = 1.6, hjust = 0, label = "linear AR(2)", size = 6, colour = "grey30") + labs(x = expression(gamma ~ "(change in unemployment over the past year, points)"), y = expression(hat(sigma)^2 * (gamma))) + theme_minimal(base_size = 20) ``` ::: {style="font-size: 28px"} **What the data say:** $\hat\gamma=0$. When unemployment is no higher than a year ago it drifts down by about 0.04 points a month; when it is higher, it rises with momentum (coefficient 0.37 on $\Delta u_{t-2}$). The model is silent on whether the gain over the linear AR(2) is more than chance: next lecture. ::: --- ## 📝 Quiz #1: The Threshold Variable {.quiz-question} Which threshold variable separates high- and low-volatility regimes for weekly exchange-rate returns $x_t$? - [$q_{t-1}=\frac15\sum_{j=1}^5|x_{t-j}|$]{.correct data-explanation="✅ The average absolute return over the past five weeks measures recent volatility, and it is known at t − 1. The sign of x_{t−1} carries no information about volatility."} - $q_{t-1}=x_{t-1}$ - $q_{t-1}=x_t$ - $q_{t-1}=\frac15\sum_{j=1}^5x_{t-j}$ --- ## 📝 Quiz #2: Stationarity {.quiz-question} A SETAR(1) with $q_{t-1}=x_{t-1}$ and zero intercepts has $\rho_1=1.2$ in the lower regime and $\rho_2=0.5$ in the upper. Is it stationary? - [No: the condition $\rho_1<1$ fails]{.correct data-explanation="✅ The conditions are ρ1 < 1, ρ2 < 1, ρ1ρ2 < 1. With ρ1 = 1.2 > 0 and zero intercept, a negative x stays negative and grows, never reaching the upper regime. Contrast ρ1 = −3: it flips the sign and sends x into the stable regime."} - Yes: $\rho_1\rho_2=0.6<1$ - Yes: the upper regime pulls it back - Only if $\sigma^2$ is small --- ## 📝 Quiz #3: Superconsistency {.quiz-question} $\hat\gamma$ is $T$-consistent. What does this buy us? - [Inference on $\mu_j,\rho_j$ can proceed as if $\gamma$ were known]{.correct data-explanation="✅ Because γ̂ − γ = O_p(1/T), faster than the √T rate of the slopes, the slope estimates have the same normal limit as with known γ, and the usual OLS standard errors at γ̂ apply."} - A normal confidence interval $\hat\gamma\pm1.96\,SE(\hat\gamma)$ - The objective becomes differentiable in large samples - No trimming is needed --- ## 📋 Key Formulas ::: {style="font-size: 28px"} | | | |---|---| | predictable white noise | $y_t=\varepsilon_t+\theta\varepsilon_{t-1}\varepsilon_{t-2}$, $E(y_{t+1}\mid\mathcal F_t)=\theta\varepsilon_t\varepsilon_{t-1}$ | | TAR | $x_t=(\mu_1+\rho_1x_{t-1})\mathbb 1\{q_{t-1}\le\gamma\}+(\mu_2+\rho_2x_{t-1})\mathbb 1\{q_{t-1}>\gamma\}+\varepsilon_t$ | | SETAR | $q_{t-1}=x_{t-d}$ | | SETAR(1) ergodic | $\rho_1<1,\ \rho_2<1,\ \rho_1\rho_2<1$ | | lag orders | $\min_{p_1,p_2}(IC_1+IC_2)$, $BIC_j=T_j\log\hat\sigma_j^2+p_j\log T_j$ | | concentration | $\hat\gamma=\arg\min_{\gamma\in\Gamma}\hat\sigma^2(\gamma)$, $\hat\theta=\hat\theta(\hat\gamma)$ | | rates | $\hat\gamma-\gamma=O_p(T^{-1})$; slopes $\sqrt T$-normal | ::: --- ## 📋 Summary ::: {style="font-size: 30px"} - Nonlinear models give persistence without a unit root, and forecasts where linear ones see white noise - A TAR switches between linear regimes when an observed $q_{t-1}$ crosses $\gamma$ - An endogenous switch can make a TAR stationary despite an explosive regime - Estimation: OLS for each $\gamma$ on a trimmed grid of observed values; take the minimum - Slopes are $\sqrt T$-normal as if $\gamma$ were known; $\hat\gamma$ is $T$-consistent with a non-pivotal limit ::: --- ## 📚 Further Reading ::: {style="font-size: 28px"} - Franses, P. H. and van Dijk, D. (2000), *Non-linear Time Series Models in Empirical Finance*, ch. 3 - Tsay, R. S. (2010), *Analysis of Financial Time Series*, ch. 4 - Tong, H. (1990), *Non-linear Time Series: A Dynamical System Approach* - Granger, C. W. J. and Andersen, A. P. (1978), *An Introduction to Bilinear Time Series Models* - Chan, K. S. (1993), *Annals of Statistics* (consistency of $\hat\gamma$) - Petrucelli, J. and Woolford, S. (1984), *J. Applied Probability*; Hansen, B. E. (1997), *Studies in Nonlinear Dynamics and Econometrics* **Next lecture: Threshold Inference and Smooth Transition.** A confidence set for $\gamma$, a test of linearity, and regimes that blend. [⬅ Previous lecture: 10 · Dickey–Fuller, Phillips–Perron and ADF Tests](../10-dickey-fuller-tests/dickey_fuller_tests_lecture10.html) :::
