```{r} #| label: setup #| include: false set.seed(2026) library(ggplot2) theme_set(theme_minimal(base_size = 18)) ``` ## π¬ The Idea in 3 Minutes ::: {style="text-align:center"} ```{=html} ``` [β¬
Previous lecture: 11 Β· Nonlinear Models and Threshold Autoregression](../11-threshold-autoregression/threshold_autoregression_lecture11.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"} ::: --- ## π― Learning Objectives ::: {style="font-size: 32px"} By the end of this lecture, you will be able to: - **State** Hansen's likelihood-ratio result for the threshold, and compute its critical values - **Construct** a confidence set for $\gamma$ by inverting $LR(\gamma)$, and read an asymmetric or disjoint one - **Explain** why linearity tests are non-standard when $\gamma$ is unidentified, and **run** a sup-F test with the fixed-regressor bootstrap - **Write down** LSTAR and ESTAR models, and describe their limits as $\delta\to0$ and $\delta\to\infty$ ::: --- ## π Warm-up from last time {.quiz-question} Lecture 11: in concentrated least squares for a TAR, what does $\hat\sigma^2(\gamma)$ look like as a function of $\gamma$? - [A step function, constant between neighbouring observed values of $q_{t-1}$]{.correct data-explanation="β
ΟΜΒ²(Ξ³) depends on Ξ³ only through the split {t : q_{tβ1} β€ Ξ³}, which changes only when Ξ³ crosses an observed q. So the grid is the set of observed values, trimmed 10β15% at each end."} - A smooth parabola with its minimum at $\hat\gamma$ - A straight line - A continuous function with a kink at $\hat\gamma$ --- ## πΊοΈ Where We Are ::: {style="font-size: 32px"} Lecture 11 ended with good news and an open question. The slopes of a TAR are $\sqrt T$-normal as if $\gamma$ were known, and $\hat\gamma$ is $T$-consistent. But the limit of $T(\hat\gamma-\gamma)$ is non-pivotal. ::: {.fragment} Today: **how to build a confidence set for $\gamma$** anyway; **how to test whether there is a threshold at all**; and what happens when the switch between regimes is **gradual** rather than abrupt. ::: ::: --- ## β The Question ::: {.callout-important} ## Three questions about unemployment Lecture 11 found that US unemployment dynamics switch when unemployment is above its level of a year earlier, with $\hat\gamma=0$. ::: ::: {style="font-size: 30px"} 1. How precisely is the threshold located? Could it be $0.3$ points instead of $0$? 2. Is the gain over a linear AR(2) real, or the product of searching over 22 candidate thresholds? 3. Do the regimes switch like a light, or fade into each other like a dimmer? ::: --- ## π Recap: TAR and Its Estimator ::: {style="font-size: 28px"} $$x_t=\begin{cases}\mu_1+\rho_1x_{t-1} & \text{if } q_{t-1}\le\gamma_0\\ \mu_2+\rho_2x_{t-1} & \text{if } q_{t-1}>\gamma_0\end{cases}\;+\;\varepsilon_t,\qquad E_{t-1}\varepsilon_t=0,\ \operatorname{var}_{t-1}\varepsilon_t=\sigma^2 .$$ NLLS with concentration gives $\hat\theta=(\hat\mu_1,\hat\rho_1,\hat\mu_2,\hat\rho_2,\hat\gamma,\hat d)$ and $\hat\sigma^2$; the integer delay $d$ is searched jointly with $\gamma$ when unknown. - $\hat\gamma$ is superconsistent ($T$-consistent), with a non-standard, **non-pivotal** limit. - $\hat\mu_1,\hat\rho_1,\hat\mu_2,\hat\rho_2,\hat\sigma^2$ are $\sqrt T$-consistent and asymptotically normal: we can act as though $\hat\gamma=\gamma_0$. ::: --- ## π Hansen's Result ::: {style="font-size: 28px"} Define, with $\hat\sigma^2=\hat\sigma^2(\hat\gamma)=\min_\gamma\hat\sigma^2(\gamma)$, $$LR(\gamma)=T\,\frac{\hat\sigma^2(\gamma)-\hat\sigma^2}{\hat\sigma^2}.$$ ::: {.callout-important} ## Theorem (Hansen, 2000) If the threshold effect shrinks, but slowly, $(\mu_1-\mu_2,\,\rho_1-\rho_2)'\to0$ while $\sqrt T(\mu_1-\mu_2,\,\rho_1-\rho_2)'\to\infty$, then under conditional homoskedasticity $$LR(\gamma_0)\ \xrightarrow{d}\ \xi=\max_{r\in\mathbb R}\{2B(r)-|r|\}.$$ ::: $B$ is a two-sided standard Brownian motion on $\mathbb R$ with $B(0)=0$: two independent Brownian motions glued at the origin. ::: --- ## π The Limit Is Pivotal ::: {style="font-size: 28px"} $\xi$ is a random variable, and its distribution is known in closed form: $$\Pr\{\xi\le u\}=\left(1-e^{-u/2}\right)^2,\qquad u\ge0 .$$ It contains no unknown parameter: it is **pivotal**. ::: {.fragment} Inverting the CDF at $1-\alpha$: $$\left(1-e^{-c/2}\right)^2=1-\alpha\ \iff\ e^{-c/2}=1-\sqrt{1-\alpha}\ \iff\ c_{1-\alpha}=-2\ln\!\big(1-\sqrt{1-\alpha}\big).$$ ::: ::: {.fragment} Without conditional homoskedasticity the limit is $\eta^2\xi$, with a nuisance scale $\eta^2$ that must be estimated. ::: ::: --- ## βοΈ Worked Example: Critical Values ::: {style="font-size: 28px"} Compute $c_{1-\alpha}=-2\ln(1-\sqrt{1-\alpha})$ step by step. | $\alpha$ | $\sqrt{1-\alpha}$ | $1-\sqrt{1-\alpha}$ | $\ln(\cdot)$ | $c_{1-\alpha}$ | |---|---|---|---|---| | 0.10 | 0.94868 | 0.05132 | β2.9699 | **5.94** | | 0.05 | 0.97468 | 0.02532 | β3.6761 | **7.35** | | 0.01 | 0.99499 | 0.00501 | β5.2958 | **10.59** | ::: {.fragment} Compare $\chi^2(1)$: 2.71, 3.84, 6.63. The threshold critical values are roughly twice as large: $\xi$ is the maximum of a process over the whole line, not one squared normal. ::: ::: --- ## π» Simulating $\xi$ ```{r} #| label: xi-sim #| code-summary: "π¦ max of 2B(r) - |r| over a fine grid on [-20, 20]" h <- 0.001; R <- 20; m <- R / h; r <- (1:m) * h xi <- replicate(3000, { up <- cumsum(rnorm(m, sd = sqrt(h))) # B(r), r > 0 dn <- cumsum(rnorm(m, sd = sqrt(h))) # B(-r), independent side max(0, 2 * up - r, 2 * dn - r) }) # r = 0 gives 0 crit <- function(a) -2 * log(1 - sqrt(1 - a)) rbind(simulated = quantile(xi, c(0.90, 0.95, 0.99)), exact = crit(c(0.10, 0.05, 0.01))) |> round(2) ``` ::: {style="font-size: 28px"} Simulation and formula agree to about 0.1. The grid sees the path only at its nodes, which pulls the simulated values slightly down; Monte Carlo noise does the rest. ::: --- ## π The Distribution of $\xi$ ```{r} #| label: xi-fig #| echo: false #| fig-width: 11 #| fig-height: 4.1 dens <- function(u) (1 - exp(-u / 2)) * exp(-u / 2) # derivative of the CDF ggplot(data.frame(xi = xi), aes(xi)) + geom_histogram(aes(y = after_stat(density)), binwidth = 0.5, boundary = 0, fill = "#8ba3c7", colour = "white") + stat_function(fun = dens, colour = "#8b2635", linewidth = 1.3) + stat_function(fun = function(u) dchisq(u, 1), colour = "grey40", linetype = "dashed", linewidth = 1, xlim = c(0.15, 16)) + geom_vline(xintercept = crit(0.05), colour = "#8b2635", linetype = "dotted", linewidth = 1) + coord_cartesian(xlim = c(0, 16), ylim = c(0, 0.3)) + labs(x = expression(xi), y = "Density") + theme_minimal(base_size = 20) ``` ::: {style="font-size: 28px"} Bars: simulated $\xi$. Red: the density $(1-e^{-u/2})e^{-u/2}$. Dashed: $\chi^2(1)$. Dotted: $c_{0.95}=7.35$. ::: --- ## π Why Shrinking Threshold Effects? ::: {style="font-size: 30px"} The condition $(\mu_1-\mu_2,\rho_1-\rho_2)\to0$ is an example of **alternative asymptotics**. - The regime parameters are fixed in reality. We pretend that, as $T\to\infty$, they also move closer, but more slowly than $T^{-1/2}$. - Under fixed differences, the limit of $T(\hat\gamma-\gamma_0)$ depends on the whole distribution of the data, and cannot be standardised. - The shrinking device trades a little realism for a limit that is easy to use. ::: {.fragment} Compare weak instruments, where letting the first-stage coefficient shrink at rate $T^{-1/2}$ gives tractable limits. ::: ::: --- ## π A Confidence Set by Inversion ::: {style="font-size: 28px"} Collect the values of $\gamma$ that the LR statistic does not reject: $$CI_{1-\alpha}(\gamma_0)=\big\{\gamma\in\Gamma:\ LR(\gamma)\le c_{1-\alpha}\big\},\qquad c_{0.95}=7.35 .$$ On a plot of $LR(\gamma)$: draw a horizontal line at $c_{1-\alpha}$, and keep the $\gamma$ where the curve lies below it. ::: {.fragment} The set is built in a non-standard way, so - it need **not be symmetric** around $\hat\gamma$; - it may be **disjoint**: $LR(\gamma)$ is jagged, and a second dip below the line adds a separate piece. ::: ::: --- ## π» $LR(\gamma)$ by Hand ```{r} #| label: lr-sim #| code-summary: "π¦ SETAR(1) with gamma_0 = 0, its LR(gamma) and the 95% set" 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)] s2_of <- function(g) { I <- q <= g; J <- !I 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, s2_of); g_hat <- grid[which.min(s2)] LR <- n * (s2 - min(s2)) / min(s2) inside <- grid[LR <= crit(0.05)] round(c(gamma_hat = g_hat, ci_low = min(inside), ci_high = max(inside), points_in_set = length(inside), grid_points = length(grid)), 3) ``` ::: {style="font-size: 28px"} The 95% set runs from about $-0.05$ to $0.08$ around $\hat\gamma\approx0$: it is not centred on $\hat\gamma$. ::: --- ## π Reading the LR Plot ```{r} #| label: lr-fig #| echo: false #| fig-width: 11 #| fig-height: 4.1 ggplot(data.frame(g = grid, LR = LR), aes(g, LR)) + geom_step(colour = "#2f4b7c", linewidth = 0.9) + geom_hline(yintercept = crit(0.05), colour = "#8b2635", linetype = "dashed", linewidth = 1) + geom_rug(data = data.frame(g = inside), aes(x = g), inherit.aes = FALSE, colour = "#8b2635", linewidth = 1, length = unit(0.06, "npc")) + annotate("text", x = max(grid), y = crit(0.05), label = "c[0.95] == 7.35", parse = TRUE, hjust = 1, vjust = -0.5, size = 6.5, colour = "#8b2635") + labs(x = expression(gamma), y = expression(LR(gamma))) + theme_minimal(base_size = 20) ``` ::: {style="font-size: 28px"} Red ticks on the axis: the grid points with $LR(\gamma)\le7.35$, the 95% confidence set. Its two sides are not the same length. ::: --- ## π Is There a Threshold at All? ::: {style="font-size: 28px"} $$H_0:\ \mu_1=\mu_2,\ \rho_1=\rho_2\ \text{(linearity)}\qquad H_A:\ \mu_1\neq\mu_2\ \text{or}\ \rho_1\neq\rho_2 .$$ This is **testing when a nuisance parameter is not identified under the null**. If we impose $H_0$, the two regimes coincide and $\gamma_0$ loses its meaning. ::: {.fragment} Consequently $$W,\ F,\ LR,\ LM\ \ \overset{d}{\underset{H_0}{\not\to}}\ \ \chi^2,\qquad t\ \ \overset{d}{\underset{H_0}{\not\to}}\ \ N(0,1).$$ For each fixed $\gamma$ the usual theory holds; but $\gamma$ is chosen by searching, and the search inflates the statistic (Davies, 1977, 1987). ::: ::: --- ## π The sup-F Statistic ::: {style="font-size: 28px"} For each $\gamma$ in the trimmed set $\Gamma$, compare the linear AR with the TAR at $\gamma$: $$F(\gamma)=T\,\frac{\tilde\sigma^2-\hat\sigma^2(\gamma)}{\hat\sigma^2(\gamma)},\qquad \sup F=\max_{\gamma\in\Gamma}F(\gamma)=F(\hat\gamma),$$ where $\tilde\sigma^2$ is the residual variance of the linear model. The last equality holds because $\hat\gamma$ minimises $\hat\sigma^2(\gamma)$. ::: {.fragment} Under $H_0$, $\sup F$ has a non-standard distribution. Two ways out: 1. alternative asymptotics, or 2. the bootstrap. Bruce Hansen uses a special resampling algorithm, the **fixed-regressor bootstrap**. ::: ::: --- ## π§ The Fixed-Regressor Bootstrap ::: {style="font-size: 28px"} 1. **Right-hand side:** keep $x_{t-1}$ and $q_{t-1}$ from the original sample. They are never resampled. 2. **Left-hand side:** replace $x_t$ by a draw $y_t^*\sim$ i.i.d. $N(0,1)$; under heteroskedasticity, use $y_t^*=\hat\varepsilon_tz_t$ with $z_t\sim N(0,1)$. 3. Regress $y_t^*$ on the fixed regressors, under the null and under the TAR for every $\gamma\in\Gamma$; compute $\sup F^*$. 4. Repeat $B$ times. The p-value is the share of $\sup F^*$ at or above the sample $\sup F$. ::: {.fragment} It works because the null distribution of $\sup F$ depends on the data only through the regressors; the fixed-regressor bootstrap reproduces exactly that dependence (Hansen, 1996). ::: ::: --- ## π» What Searching Costs ```{r} #| label: supf-null #| code-summary: "π¦ 1,000 AR(1) samples under H0: sup F vs F at one fixed gamma" Fstats <- function(y, xl) { n <- length(y); Z <- cbind(1, xl) s2l <- mean(.lm.fit(Z, y)$residuals^2) g <- sort(xl); g <- g[g >= quantile(xl, 0.15) & g <= quantile(xl, 0.85)] s2u <- sapply(g, function(gg) { I <- xl <= gg; mean(.lm.fit(cbind(Z * I, Z * !I), y)$residuals^2) }) Fg <- n * (s2l - s2u) / s2u c(supF = max(Fg), F_at_median = Fg[ceiling(length(Fg) / 2)]) } null <- t(replicate(1000, { z <- as.numeric(stats::filter(rnorm(151), 0.5, "recursive")) Fstats(z[-1], z[-151]) })) qs <- apply(null, 2, quantile, c(0.90, 0.95, 0.99)) round(cbind(qs, chisq_2 = qchisq(c(0.90, 0.95, 0.99), df = 2)), 2) ``` ::: {style="font-size: 28px"} At a fixed $\gamma$, $F$ behaves like $\chi^2(2)$. Taking the maximum roughly doubles the 95% point. ::: --- ## π§ ThinkβPairβShare ```{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. A TAR gives $\sup F=9.5$. A colleague compares it with $\chi^2(2)$, whose 99% point is 9.21, and declares a threshold at 1%. What is wrong? 2. A 95% confidence set for $\gamma$ is $[-0.1,\,0.1]\cup[0.3,\,0.4]$. Is this a mistake? 3. Why does the fixed-regressor bootstrap not simulate a whole new AR path for $x_t^*$? ::: --- ## β
ThinkβPairβShare: Solution ::: {style="font-size: 28px"} 1. $\sup F$ is a maximum over many correlated $F(\gamma)$. Under $H_0$ its 95% point is near 12, not 6, so $9.5$ is not significant even at 10%. The $\chi^2$ table is for one $\gamma$ fixed in advance. ::: {.fragment} 2. No. The set is $\{\gamma:LR(\gamma)\le7.35\}$, and a jagged $LR(\gamma)$ can dip below the line twice: two locations fit about equally well. ::: ::: {.fragment} 3. Under $H_0$ the distribution of $\sup F$ depends on the data only through the regressors. Holding them fixed and redrawing only the error reproduces it, without needing a correct model for the whole path of $x_t$. ::: ::: --- ## π Smooth Transition (STAR) ::: {style="font-size: 28px"} Replace the indicator by a **transition function** $G$ with values in $[0,1]$: $$x_t=(\mu_1+\rho_1x_{t-1})\big(1-G(q_{t-1},\gamma_0,\delta)\big)+(\mu_2+\rho_2x_{t-1})\,G(q_{t-1},\gamma_0,\delta)+\varepsilon_t ,$$ with $E_{t-1}\varepsilon_t=0$, $\operatorname{var}_{t-1}\varepsilon_t=\sigma^2$. $\delta>0$ is the **smoothness parameter**. ::: {.fragment} Equivalently, $$x_t=(\mu_1+\rho_1x_{t-1})+\big[(\mu_2-\mu_1)+(\rho_2-\rho_1)x_{t-1}\big]\,G(q_{t-1},\gamma_0,\delta)+\varepsilon_t .$$ Usually $G$ is a CDF, or some other function with values in $[0,1]$. ::: ::: --- ## π LSTAR and ESTAR ::: {style="font-size: 28px"} **Logistic (LSTAR):** $$G(u,\gamma,\delta)=\frac{1}{1+\exp\{-\delta(u-\gamma)\}} .$$ Monotone, with $G(\gamma)=\tfrac12$. - $\delta\to0$: $G\equiv\tfrac12$, and the model is **linear**, with coefficients $\tfrac12(\mu_1+\mu_2)$ and $\tfrac12(\rho_1+\rho_2)$. - $\delta\to\infty$: $G$ becomes the indicator $\mathbb 1\{u>\gamma\}$, and the model is the **TAR**. ::: {.fragment} **Exponential (ESTAR):** bounded in $[0,1]$ but non-monotone, $$G(u,\gamma,\delta)=1-\exp\{-\delta(u-\gamma)^2\}.$$ $G(\gamma)=0$, and $G\to1$ as $|u-\gamma|\to\infty$: the regimes are "near $\gamma$" and "far from $\gamma$". ::: ::: --- ## ποΈ Explore: Light Switch or Dimmer? ```{ojs} //| echo: false viewof delta = Inputs.range([0.1, 50], {value: 2, transform: Math.log, label: "Ξ΄ (log scale)"}) viewof shape = Inputs.radio(["logistic (LSTAR)", "exponential (ESTAR)"], {value: "logistic (LSTAR)", label: "G"}) ``` ```{ojs} //| echo: false curve = d3.range(-3, 3.0001, 0.01).map(u => ({ u, G: shape.startsWith("logistic") ? 1 / (1 + Math.exp(-delta * u)) : 1 - Math.exp(-delta * u * u) })) tarStep = shape.startsWith("logistic") ? [{u: -3, G: 0}, {u: 0, G: 0}, {u: 0, G: 1}, {u: 3, G: 1}] : [{u: -3, G: 1}, {u: 3, G: 1}] Plot.plot({ width: 1100, height: 300, marginLeft: 70, marginBottom: 50, style: {fontSize: "18px"}, x: {label: "u β Ξ³", domain: [-3, 3]}, y: {label: "G", domain: [0, 1.05]}, marks: [ Plot.ruleY([0.5], {stroke: "#bbb", strokeDasharray: "3,3"}), Plot.line(tarStep, {x: "u", y: "G", stroke: "#999", strokeDasharray: "6,4", strokeWidth: 2}), Plot.line(curve, {x: "u", y: "G", stroke: "#8b2635", strokeWidth: 3}) ] }) ``` ::: {style="font-size: 28px"} Small $\delta$: a dimmer, nearly one linear model. Large $\delta$: a light switch. Grey dashes: the limit $\delta\to\infty$, which for LSTAR is the TAR. ::: --- ## π§ Extensions of STAR ::: {style="font-size: 28px"} 1. More lags; heteroskedasticity. 2. **Three regimes, same transition variable**, $\gamma_1<\gamma_2$: $$\begin{aligned}x_t={}&(\mu_1+\rho_1x_{t-1})+\big[(\mu_2-\mu_1)+(\rho_2-\rho_1)x_{t-1}\big]G_1(q_{t-1},\gamma_1,\delta_1)\\&+\big[(\mu_3-\mu_2)+(\rho_3-\rho_2)x_{t-1}\big]G_2(q_{t-1},\gamma_2,\delta_2)+\varepsilon_t .\end{aligned}$$ 3. **Four regimes, different transition variables** $y_{t-1}$, $y_{t-2}$, with $G_1=G_1(y_{t-1},\gamma_1,\delta_1)$, $G_2=G_2(y_{t-2},\gamma_2,\delta_2)$: $$\begin{aligned}x_t={}&\big[(\mu_1+\rho_1x_{t-1})(1-G_1)+(\mu_2+\rho_2x_{t-1})G_1\big](1-G_2)\\&+\big[(\mu_3+\rho_3x_{t-1})(1-G_1)+(\mu_4+\rho_4x_{t-1})G_1\big]G_2+\varepsilon_t .\end{aligned}$$ ::: --- ## π° Case Study: Testing the Regimes :::: columns ::: {.column width="50%"} ::: {.callout-note} ## π The question 1. Which thresholds does the 95% LR set keep? 2. Does sup-F reject linearity, with bootstrap critical values? 3. Do the data prefer a sharp or a smooth switch? ::: ::: ::: {.column width="50%"} ::: {.callout-tip} ## π The data **Series**: FRED `UNRATE`, monthly, Jan 1948 β Dec 2019 Same model as Lecture 11: $x_t=\Delta u_t$, two lags per regime, $q_{t-1}=u_{t-1}-u_{t-13}$, grid trimmed 15% Fixed-regressor bootstrap, $B=1000$; LSTAR fitted by concentration over $(\gamma,\delta)$ ::: ::: :::: ```{.r .display-only} download.file("https://fred.stlouisfed.org/graph/fredgraph.csv?id=UNRATE", "data/UNRATE.csv") # then keep Jan 1948 - Dec 2019 ``` --- ## π° Case Study: The Computation ```{r} #| label: case-inf #| code-summary: "π¦ LR set, sup-F with fixed-regressor bootstrap, LSTAR profile" #| results: hold u <- read.csv("data/UNRATE.csv")$UNRATE; N <- length(u); du <- c(NA, diff(u)); tt <- 14:N yy <- du[tt]; Z <- cbind(1, du[tt - 1], du[tt - 2]); qq <- round(u[tt - 1] - u[tt - 13], 1) TT <- length(yy); G <- sort(unique(qq)); G <- G[G >= quantile(qq, 0.15) & G <= quantile(qq, 0.85)] s2_tar <- function(y) sapply(G, function(g) { I <- qq <= g mean(.lm.fit(cbind(Z * I, Z * !I), y)$residuals^2) }) s2 <- s2_tar(yy); LRu <- TT * (s2 - min(s2)) / min(s2) supF_of <- function(y) { s2l <- mean(.lm.fit(Z, y)$residuals^2); s2u <- min(s2_tar(y)) TT * (s2l - s2u) / s2u } supF <- supF_of(yy); boot <- replicate(1000, supF_of(rnorm(TT))) # fixed regressors s2_lstar <- function(g, d) { Gq <- 1 / (1 + exp(-d * (qq - g))) mean(.lm.fit(cbind(Z, Z * Gq), yy)$residuals^2) } dgrid <- exp(seq(log(0.5), log(300), length.out = 40)); ggrid <- seq(-0.8, 1.2, by = 0.02) prof <- sapply(dgrid, function(d) min(sapply(ggrid, s2_lstar, d = d))) cat("gamma_hat:", G[which.min(s2)], " 95% LR set:", G[LRu <= crit(0.05)], "\n") cat("sup F:", round(supF, 1), " bootstrap 95% point:", round(quantile(boot, 0.95), 1), " p-value:", mean(boot >= supF), "\n") cat("LSTAR best delta:", round(dgrid[which.min(prof)], 1), " s2 LSTAR:", signif(min(prof), 4), " s2 TAR:", signif(min(s2), 4), "\n") ``` ::: {style="font-size: 28px"} Three answers: the LR set, the sup-F p-value, and the $\delta$ the data choose. ::: --- ## π° Case Study: What the Data Say ```{r} #| label: case-fig #| echo: false #| fig-width: 11 #| fig-height: 3.6 pd <- rbind( data.frame(x = G, y = LRu, panel = "LR(Ξ³) against Ξ³; dashed: 7.35"), data.frame(x = log10(dgrid), y = TT * (prof - min(s2)) / min(s2), panel = "LSTAR fit minus TAR fit, against log10 Ξ΄")) hl <- data.frame(panel = unique(pd$panel), yint = c(crit(0.05), 0)) ggplot(pd, aes(x, y)) + geom_step(data = subset(pd, panel == unique(pd$panel)[1]), colour = "#2f4b7c", linewidth = 1) + geom_line(data = subset(pd, panel == unique(pd$panel)[2]), colour = "#2f4b7c", linewidth = 1) + geom_point(colour = "#2f4b7c", size = 1.6) + geom_hline(data = hl, aes(yintercept = yint), colour = "#8b2635", linetype = "dashed") + facet_wrap(~ panel, scales = "free") + labs(x = NULL, y = NULL) + theme_minimal(base_size = 18) + theme(strip.text = element_text(size = 15)) ``` ::: {style="font-size: 28px"} **What the data say:** the 95% set is **disjoint**, $\{-0.1,0,0.1\}\cup\{0.3,0.4\}$. Linearity is rejected far beyond the bootstrap critical value. Right: $T(\hat\sigma^2_{\text{LSTAR}}-\hat\sigma^2_{\text{TAR}})/\hat\sigma^2_{\text{TAR}}$ falls to zero only for large $\delta$; with $q$ recorded to one decimal, a steep dimmer and a switch fit alike. ::: --- ## π Quiz #1: The LR Set {.quiz-question} $LR(\gamma)$ takes the values 9.1, 6.2, 0, 3.4, 8.0, 5.1 on six consecutive grid points. Which points are in the 95% confidence set? - [The 2nd, 3rd, 4th and 6th]{.correct data-explanation="β
Keep every Ξ³ with LR(Ξ³) β€ c_0.95 = 7.35: 6.2, 0, 3.4 and 5.1. The 5th point (8.0) is excluded, so the set is disjoint: two pieces, as Hansen's construction allows."} - The 2nd, 3rd and 4th only, since a confidence set must be an interval - All six, since $\hat\gamma$ is superconsistent - The 3rd only --- ## π Quiz #2: Critical Value {.quiz-question} What is the 90% critical value $c_{0.90}$ of $\xi$, from $\Pr\{\xi\le u\}=(1-e^{-u/2})^2$? - [$-2\ln(1-\sqrt{0.9})\approx5.94$]{.correct data-explanation="β
Set (1 β e^(βc/2))Β² = 0.9, so e^(βc/2) = 1 β β0.9 = 0.0513 and c = β2 ln 0.0513 = 5.94."} - $-2\ln(0.1)\approx4.61$ - $2.71$, the $\chi^2(1)$ value - $\sqrt{0.9}\approx0.95$ --- ## π Quiz #3: Limits of LSTAR {.quiz-question} In an LSTAR model with logistic $G(u,\gamma,\delta)$, what happens as $\delta\to0$? - [$G\to\tfrac12$ and the model becomes a linear AR]{.correct data-explanation="β
With Ξ΄ = 0 the logistic equals 1/2 for every u, so x_t = Β½(ΞΌ1+ΞΌ2) + Β½(Ο1+Ο2)x_{tβ1} + Ξ΅_t. As Ξ΄ β β instead, G becomes the indicator and the model is the TAR."} - $G$ becomes the indicator and the model becomes a TAR - $G\to0$ and only regime 1 remains - The model becomes an ESTAR --- ## π Key Formulas ::: {style="font-size: 28px"} | | | |---|---| | LR statistic | $LR(\gamma)=T\{\hat\sigma^2(\gamma)-\hat\sigma^2\}/\hat\sigma^2$ | | Hansen's limit | $LR(\gamma_0)\xrightarrow{d}\xi=\max_r\{2B(r)-\lvert r\rvert\}$, $\Pr\{\xi\le u\}=(1-e^{-u/2})^2$ | | critical values | $c_{1-\alpha}=-2\ln(1-\sqrt{1-\alpha})$: 5.94, 7.35, 10.59 | | confidence set | $\{\gamma:LR(\gamma)\le c_{1-\alpha}\}$ | | sup-F | $\max_\gamma T\{\tilde\sigma^2-\hat\sigma^2(\gamma)\}/\hat\sigma^2(\gamma)$, fixed-regressor bootstrap | | STAR | $x_t=(\mu_1+\rho_1x_{t-1})(1-G)+(\mu_2+\rho_2x_{t-1})G+\varepsilon_t$ | | LSTAR / ESTAR | $[1+e^{-\delta(u-\gamma)}]^{-1}$ / $1-e^{-\delta(u-\gamma)^2}$ | ::: --- ## π Summary ::: {style="font-size: 30px"} - Under shrinking threshold effects, $LR(\gamma_0)$ has a pivotal limit with a closed-form CDF - Inverting $LR(\gamma)$ gives a confidence set that may be asymmetric or disjoint - Under linearity $\gamma$ is unidentified: use sup-F with bootstrap critical values, never $\chi^2$ - The fixed-regressor bootstrap keeps the regressors and redraws only the dependent variable - STAR models blend the regimes with $G\in[0,1]$; $\delta$ runs from linear ($0$) to TAR ($\infty$) ::: --- ## π Further Reading ::: {style="font-size: 28px"} - Hansen, B. E. (2000), "Sample splitting and threshold estimation", *Econometrica* - Hansen, B. E. (1996), "Inference when a nuisance parameter is not identified under the null hypothesis", *Econometrica* - Davies, R. B. (1977, 1987), *Biometrika*; Andrews, D. W. K. and Ploberger, W. (1994), *Econometrica* - TerΓ€svirta, T. (1994), "Specification, estimation, and evaluation of smooth transition autoregressive models", *JASA* - Franses, P. H. and van Dijk, D. (2000), *Non-linear Time Series Models in Empirical Finance*, ch. 3 **Next lecture: Markov Switching and State-Space Models.** Regimes driven not by an observed threshold variable but by a hidden state. [β¬
Previous lecture: 11 Β· Nonlinear Models and Threshold Autoregression](../11-threshold-autoregression/threshold_autoregression_lecture11.html) :::
