```{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: 3 · Time-Series Regression and HAC Inference](../03-time-series-regression-hac/time_series_regression_hac_lecture3.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: 34px"} By the end of this lecture, you will be able to: - **Name** the objects of non-structural modelling: conditional mean, variance, median, quantile, higher moments, probability and density - **Prove** that the conditional mean is the minimum-MSPE predictor - **Describe** the three tools of model selection: diagnostic testing, sequential testing and information criteria - **Compute** AIC, BIC and HQ for a family of autoregressions and pick an order - **Explain** why BIC is consistent and AIC is not ::: --- ## 🔁 Warm-up from last time {.quiz-question} Mishkin's three-month regression on monthly data, $\pi_{t+3} = \alpha + \beta i_{t,3} + e_{t+3}$. Why does it need HAC standard errors when the one-month regression does not? - [The three-month forecast errors overlap, so $e_{t+3}$ is MA(2) and $x_t e_{t+3}$ is serially correlated]{.correct data-explanation="✅ Consecutive three-month windows share two months of news, so e(t+3) is MA(2). OLS stays consistent, but the long-run variance of x_t e_t includes autocovariances up to lag 2: use HAC with truncation lag at least 2."} - Because OLS is inconsistent at the three-month horizon - Because inflation is not stationary at the three-month horizon - Because the interest rate $i_{t,3}$ is measured with error --- ## 🗺️ Where We Are ::: {style="font-size: 32px"} Lecture 3 ended by separating two kinds of time-series work. - **Structural**: theory with optimising agents, deep parameters with economic meaning (an Euler equation estimated by GMM). - **Non-structural**: models built to describe the data and to forecast it (AR-ARCH-M). ::: {.fragment} Today we take the non-structural side seriously and ask two questions. **What** feature of the series do we model? And among the many models that could describe it, **how do we choose one**? ::: ::: --- ## ❓ The Question ::: {.callout-important} ## Three people, one series A central-bank economist, a risk manager and a bond trader all watch the monthly change in the US unemployment rate. ::: ::: {style="font-size: 30px"} - The economist wants the **expected** change next month. - The risk manager wants to know how large a jump is **possible**: a high quantile. - The trader wants the **probability** that the rate rises at all. Same data, three different targets. Each is a feature of one object: the distribution of $x_{T+1}$ given the past. ::: --- ## 📐 The Setting ::: {style="font-size: 30px"} We observe $\{x_t\}_{t=1}^{T}$, assumed **strictly stationary and ergodic**. Non-structural modelling has two aims: 1. describe the **stylised facts** of the data; 2. **predict**: the one-step forecast $\hat x_{T+1|T}$ and perhaps multi-step forecasts $\hat x_{T+h|T}$, $h > 1$. Both require the dynamics of $x_t$ as a function of its past, $$I_{t-1} = \{x_{t-1}, x_{t-2}, \dots\}, \qquad E_{t-1}[\cdot] = E[\,\cdot \mid I_{t-1}].$$ The feature of the conditional distribution we choose to describe is the **object** of modelling. ::: --- ## 📐 Object 1: The Conditional Mean ::: {style="font-size: 30px"} $$\mu_t = E_{t-1}[x_t] = E[x_t \mid x_{t-1}, x_{t-2}, \dots]$$ ::: {.callout-note} ## Why the mean Among all predictors $g$ built from $I_{t-1}$, $\mu_t$ minimises the **mean squared prediction error** (MSPE), $E[(x_t - g)^2]$. ::: The natural object for GNP, unemployment, the real exchange rate: series where the level of the next value is what matters. ::: --- ## ✏️ Proof: the Mean Minimises MSPE ::: {style="font-size: 30px"} Take any $g \in I_{t-1}$ and add and subtract $\mu_t$: $$E[(x_t - g)^2] = E[(x_t - \mu_t)^2] + 2E[(x_t - \mu_t)(\mu_t - g)] + E[(\mu_t - g)^2].$$ ::: {.fragment} The cross term vanishes by the law of iterated expectations, because $\mu_t - g$ is known at $t-1$: $$E[(x_t - \mu_t)(\mu_t - g)] = E\big[(\mu_t - g)\,E_{t-1}[x_t - \mu_t]\big] = E[(\mu_t - g)\cdot 0] = 0.$$ ::: ::: {.fragment} So $E[(x_t - g)^2] = E[(x_t - \mu_t)^2] + E[(\mu_t - g)^2] \ge E[(x_t - \mu_t)^2]$, with equality only when $g = \mu_t$. $\blacksquare$ ::: ::: --- ## 📐 Variance, Median, Quantile ::: {style="font-size: 29px"} - **Conditional variance** $\sigma_t^2 = \operatorname{Var}_{t-1}[x_t] = E_{t-1}[(x_t - \mu_t)^2]$: volatility, the object for asset returns. - **Conditional median**: optimal under **absolute** loss (minimum mean absolute prediction error). It is robust to outliers, and it exists even when thick tails leave the mean undefined. - **Conditional quantile** $q_t^{\alpha}$, defined by $\Pr_{t-1}\{x_t \le q_t^{\alpha}\} = \alpha$. A low quantile of returns is the **Value-at-Risk**; $\alpha = 0.5$ gives the median. Each is the optimal predictor under its own loss function. ::: --- ## ✏️ Why the Median, Why the Quantile ::: {style="font-size: 30px"} **Absolute loss.** For a continuous $x$ with distribution function $F$, $$\frac{d}{dm}E|x - m| = \Pr(x < m) - \Pr(x > m) = 2F(m) - 1,$$ which is zero at $F(m) = \tfrac12$: the median. ::: {.fragment} **Check loss.** Replace $|u|$ by $\rho_\alpha(u) = u\,(\alpha - \mathbf 1\{u < 0\})$, which charges $\alpha$ per unit of under-prediction and $1-\alpha$ per unit of over-prediction. Then $$\frac{d}{dm}E\,\rho_\alpha(x - m) = F(m) - \alpha = 0 \;\Rightarrow\; m = q^\alpha.$$ Conditioning on $I_{t-1}$ throughout gives $q_t^\alpha$. ::: [The check-loss argument is not in the 2016 notes: added from Koenker (2005), *Quantile Regression*, ch. 1.]{.gapnote} ::: --- ## 📐 Moments, Probability, Density ::: {style="font-size: 29px"} - **Higher moments**: $S_t = E_{t-1}[(x_t - \mu_t)^3]$ measures asymmetry and $K_t = E_{t-1}[(x_t - \mu_t)^4]$ tail thickness. Skewness and kurtosis proper are $S_t/\sigma_t^3$ and $K_t/\sigma_t^4$. - **Conditional probability** $p_t = \Pr_{t-1}\{x_t > c\}$: will the rate rise, will the firm default. - **Conditional density** $f_{t-1}(x_t)$: contains every object above, and is the hardest to specify and identify. ::: {.fragment} **Direct** modelling specifies a model for the object of interest itself (a quantile regression for $q_t^\alpha$). **Indirect** modelling specifies something richer, typically the whole conditional density, and derives the object from it. ::: ::: --- ## 💻 Same Mean, Moving Quantiles ```{r} #| label: objects-sim #| fig-width: 11 #| fig-height: 3.6 #| code-summary: "AR(1)-ARCH(1): the mean barely moves, the 5% quantile does" n <- 400; r <- e <- s2 <- numeric(n); s2[1] <- 1 for (t in 2:n) { s2[t] <- 0.2 + 0.7 * e[t - 1]^2 # conditional variance e[t] <- sqrt(s2[t]) * rnorm(1) r[t] <- 0.1 * r[t - 1] + e[t] # conditional mean 0.1 r_{t-1} } d <- data.frame(t = 1:n, r = r, mu = c(0, 0.1 * r[-n]), s = sqrt(s2)) ggplot(d, aes(t)) + geom_line(aes(y = r), colour = "grey55") + geom_line(aes(y = mu), colour = "#2f4b7c", linewidth = 1) + geom_line(aes(y = mu - 1.645 * s), colour = "#8b2635", linewidth = 0.8) + labs(x = "t", y = expression(x[t])) ``` ::: {style="font-size: 28px"} Blue: $\mu_t$. Red: the conditional 5% quantile $\mu_t - 1.645\,\sigma_t$. A mean model would call this series nearly unpredictable; a quantile model would not. ::: --- ## 🧭 Model Selection: the Principle ::: {style="font-size: 30px"} Selection runs **object → class of models → specific model**. | Object | Class | Specific models | |---|---|---| | $\mu_t$ | AR | AR(1), AR(2), … | | $\mu_t$ | TAR (threshold AR) | TAR(1,1,1), … | | $\mu_t$ | MSW (Markov switching) | two-regime AR, … | ::: {.fragment} Workflow: (1) specify the object, (2) collect the data, (3) choose the model, (4) select within and across classes. Step 4 is the hard one. The model is an **approximation** to the data-generating process (DGP), not the DGP itself. ::: ::: --- ## 🔧 Tool 1: Diagnostic Testing ::: {style="font-size: 30px"} Does the estimated model conform to its own assumptions? Test, on the residuals: - no serial correlation (Ljung–Box, Breusch–Godfrey); - no conditional heteroskedasticity; - normality, if it was assumed (Jarque–Bera); - parameter stability (Andrews' sup-type test); - a correct distributional assumption: the probability integral transforms $u_t = F_{t-1}(x_t)$ should be i.i.d. $U(0,1)$. A failed diagnostic says the model is wrong; a passed one does not say it is right. ::: --- ## 🔧 Tool 2: Sequential Testing ::: {style="font-size: 30px"} Shape the model by testing parameter restrictions. **(A) General-to-specific (GETS).** Start from a large $AR(\bar p)$ and test $H_0: \phi_{\bar p} = 0$. - $H_0$ rejected → keep $AR(\bar p)$ and stop. - $H_0$ not rejected → move to $AR(\bar p - 1)$ and test $\phi_{\bar p - 1} = 0$. Only restrictions the data accept are imposed. ::: {.fragment} **(B) Specific-to-general.** Start small and relax restrictions that are rejected. It fails, in general: every test is run inside a model that is still misspecified, so its size and power are not what the tables say. ::: ::: --- ## 🔧 Tool 3: Information Criteria ::: {style="font-size: 30px"} Choose the model that minimises $$IC(k) = \underbrace{-2\log\hat{\mathcal L}_k}_{\text{quality of fit}} + \underbrace{f(k)}_{\text{complexity}},$$ with $k$ the number of parameters and $f$ increasing. | Criterion | Penalty $f(k)$ | |---|---| | Akaike (AIC) | $2k$ | | Schwarz / Bayesian (SIC, BIC) | $k \log T$ | | Hannan–Quinn (HQ) | $2k \log\log T$ | For $T \ge 8$, $\log T > 2$: BIC punishes each parameter harder than AIC, and HQ lies between them for $T > 16$. ::: --- ## ✏️ Information Criteria for an AR(p) ::: {style="font-size: 29px"} With Gaussian errors, the maximised log-likelihood of an AR($p$) on $T$ observations is $$\log\hat{\mathcal L} = -\tfrac T2\log(2\pi) - \tfrac T2\log\hat\sigma_p^2 - \tfrac T2, \qquad \hat\sigma_p^2 = \tfrac1T\textstyle\sum_t \hat\varepsilon_t^2.$$ ::: {.fragment} Dropping the constants, every criterion becomes $$IC(p) = T\log\hat\sigma_p^2 + f(p+1),$$ where $k = p+1$ counts the slopes and the intercept. Dividing by $T$ gives the per-observation form $\log\hat\sigma^2_p + 2k/T$ for AIC. ::: ::: {.fragment} Adding a lag never raises $\hat\sigma_p^2$. The penalty is the only thing that stops the search at a finite $p$. Compare all orders on the **same** sample: lose the first $\bar p$ observations for every $p$. ::: ::: --- ## 🧮 Worked Example: Pick an Order ::: {style="font-size: 28px"} $T = 100$ observations; residual variances from AR($p$), $p = 0, \dots, 3$. Penalty per parameter: AIC $2$, BIC $\log 100 = 4.61$. | $p$ | $\hat\sigma_p^2$ | $100\log\hat\sigma_p^2$ | AIC | BIC | |---|---|---|---|---| | 0 | 1.500 | 40.55 | 42.55 | 45.15 | | 1 | 1.200 | 18.23 | 22.23 | 27.44 | | 2 | 1.150 | 13.98 | **19.98** | **27.79** | | 3 | 1.130 | 12.22 | 20.22 | 30.64 | ::: {.fragment} AIC chooses $p = 2$ (19.98). BIC chooses $p = 1$ (27.44 < 27.79): the second lag lowers $T\log\hat\sigma^2$ by 4.25, more than AIC's price of 2 but less than BIC's 4.61. ::: ::: --- ## ✏️ Why AIC Overfits ::: {style="font-size: 30px"} Let the true order be $p_0$, and let it be in the candidate set. Compare $p_0$ with $p_0 + 1$. The gain in fit from the redundant lag is the likelihood-ratio statistic, $$T\log\big(\hat\sigma^2_{p_0}/\hat\sigma^2_{p_0+1}\big) \xrightarrow{d} \chi^2_1 .$$ The criterion adds the lag when that gain exceeds the extra penalty $c_T = f(k+1) - f(k)$: - AIC: $c_T = 2$, so $\Pr(\chi^2_1 > 2) = 0.157$ for **every** $T$; - BIC: $c_T = \log T \to \infty$, so the probability tends to $0$. ::: --- ## ✏️ Why BIC Is Consistent ::: {style="font-size: 30px"} **Underfitting.** Drop a true lag and $\hat\sigma^2$ converges to a strictly larger number, so $T\log\hat\sigma^2$ rises by an amount of order $T$. Any penalty with $c_T/T \to 0$ is outweighed, and the short model loses. ::: {.callout-note} ## Consistency A criterion selects $p_0$ with probability tending to one if its per-parameter penalty satisfies $c_T \to \infty$ and $c_T/T \to 0$, and the true model is among finitely many candidates. ::: BIC ($\log T$) and HQ ($2\log\log T$) satisfy both; HQ's is the slowest-growing penalty that still does. AIC fails the first. ::: --- ## 💻 AIC vs BIC as $T$ Grows ```{r} #| label: ic-sim #| echo: false #| fig-width: 11 #| fig-height: 4.1 sel <- function(x, pmax = 8) { X <- embed(x, pmax + 1); y <- X[, 1]; n <- length(y) s2 <- sapply(0:pmax, function(p) mean(lm.fit(cbind(1, X[, -1][, seq_len(p), drop = FALSE]), y)$residuals^2)) k <- 1:(pmax + 1) c(AIC = which.min(n * log(s2) + 2 * k), BIC = which.min(n * log(s2) + k * log(n)), HQ = which.min(n * log(s2) + 2 * k * log(log(n)))) - 1 } Ts <- c(50, 100, 200, 500, 1000, 2000) sim <- do.call(rbind, lapply(Ts, function(T) { r <- replicate(300, sel(arima.sim(list(ar = c(0.5, -0.3)), n = T))) rbind(data.frame(T = T, crit = rownames(r), what = "chooses p = 2 (true)", share = rowMeans(r == 2)), data.frame(T = T, crit = rownames(r), what = "chooses p > 2 (overfit)", share = rowMeans(r > 2))) })) ggplot(sim, aes(T, share, colour = crit)) + geom_line(linewidth = 1.2) + geom_point(size = 2.5) + facet_wrap(~ what) + scale_x_log10(breaks = Ts) + scale_colour_manual(values = c(AIC = "#8b2635", BIC = "#2f4b7c", HQ = "#d39c2f"), name = NULL) + labs(x = "sample size T (log scale)", y = "share of 300 samples") + theme_minimal(base_size = 18) + theme(legend.position = "top") ``` ::: {style="font-size: 28px"} DGP: AR(2) with $\phi = (0.5, -0.3)$; candidates $p = 0, \dots, 8$. BIC's hit rate climbs to 1; AIC's overfit share stays near 0.27 for every $T$. ::: --- ## 🎛️ Explore: the IC Curves ```{ojs} //| echo: false viewof T = Inputs.range([30, 2000], {step: 10, value: 100, label: "Sample size T"}) viewof seed = Inputs.range([1, 30], {step: 1, value: 1, label: "Sample number"}) ``` ```{ojs} //| echo: false icdata = { // AR(2) sample from a seeded generator, then Durbin-Levinson for sigma^2_p let s = seed * 9301 + 49297; const u = () => { s = (s * 16807) % 2147483647; return s / 2147483647; }; const z = () => Math.sqrt(-2 * Math.log(u())) * Math.cos(2 * Math.PI * u()); const N = T + 200, x = new Array(N).fill(0); for (let t = 2; t < N; t++) x[t] = 0.5 * x[t-1] - 0.3 * x[t-2] + z(); const y = x.slice(200), m = d3.mean(y), P = 10; const g = d3.range(P + 1).map(k => d3.sum(d3.range(T - k), t => (y[t]-m)*(y[t+k]-m)) / T); let phi = [], s2 = [g[0]]; for (let k = 1; k <= P; k++) { const a = (g[k] - d3.sum(d3.range(k - 1), j => phi[j] * g[k-1-j])) / s2[k-1]; phi = phi.map((p, j) => p - a * phi[k-2-j]).concat([a]); s2.push(s2[k-1] * (1 - a * a)); } const pen = {AIC: 2, BIC: Math.log(T), HQ: 2 * Math.log(Math.log(T))}; const out = []; for (const c of ["AIC", "BIC", "HQ"]) { const v = s2.map((v2, p) => T * Math.log(v2) + pen[c] * (p + 1)); const mn = d3.min(v); v.forEach((vv, p) => out.push({p, crit: c, ic: Math.min(vv - mn, 30), best: vv === mn})); } return out; } ``` ```{ojs} //| echo: false Plot.plot({ width: 1100, height: 300, marginLeft: 78, marginBottom: 58, style: {fontSize: "18px"}, x: {label: "AR order p", domain: [0, 10], ticks: 11}, y: {label: "IC(p) − min IC (capped at 30)", domain: [0, 30]}, marginTop: 40, marginRight: 70, color: {domain: ["AIC", "BIC", "HQ"], range: ["#8b2635", "#2f4b7c", "#d39c2f"]}, marks: [ Plot.line(icdata, {x: "p", y: "ic", stroke: "crit", strokeWidth: 3}), Plot.dot(icdata.filter(d => d.best), {x: "p", y: "ic", fill: "crit", r: 9}), Plot.text(icdata.filter(d => d.p === 10), {x: "p", y: "ic", text: "crit", fill: "crit", dx: 34, fontSize: 20, fontWeight: "bold"}), Plot.ruleY([0]) ] }) ``` --- ## 🎛️ What the Curves Show ::: {style="font-size: 30px"} The true order is $p_0 = 2$. Large dots mark each criterion's choice. - To the **left** of $p_0$ all three curves are steep, and the steepness grows with $T$: underfitting costs order $T$. - To the **right** of $p_0$ AIC's curve is nearly flat. A lucky lag can tip it, at any $T$. - BIC's right-hand side climbs by about $\log T$ per lag, so at $T = 2000$ it rises by 7.6 per lag and noise cannot overturn that. - At $T = 30$, try a few samples: BIC often stops at $p = 1$. Consistency is a large-sample property. ::: --- ## 🧠 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: 28px"} An analyst fits AR($p$), $p = 0, \dots, 4$, to $T = 200$ monthly observations and reports $200\log\hat\sigma_p^2$: | $p$ | 0 | 1 | 2 | 3 | 4 | |---|---|---|---|---|---| | $200\log\hat\sigma^2_p$ | 60.0 | 31.0 | 24.0 | 20.0 | 18.5 | 1. Which order does AIC choose? Which does BIC ($\log 200 = 5.30$)? 2. Starting from $\bar p = 4$, a GETS search finds $|t| = 1.1$ for $\hat\phi_4$ and $|t| = 2.4$ for $\hat\phi_3$. Where does it stop? ::: --- ## ✅ Think-Pair-Share: Solution ::: {style="font-size: 28px"} 1. Add $2(p+1)$ for AIC and $5.30(p+1)$ for BIC: | $p$ | 0 | 1 | 2 | 3 | 4 | |---|---|---|---|---|---| | AIC | 62.0 | 35.0 | 30.0 | **28.0** | 28.5 | | BIC | 65.3 | 41.6 | **39.9** | 41.2 | 45.0 | AIC picks $p = 3$; BIC picks $p = 2$. The third lag gains 4.0 in fit: above 2, below 5.30. ::: {.fragment} 2. $\phi_4 = 0$ is not rejected ($1.1 < 1.96$), so drop to AR(3). $\phi_3 = 0$ is rejected ($2.4 > 1.96$), so GETS stops at **AR(3)**, as AIC did. At the 5% level, GETS behaves like a criterion with a per-lag price of $1.96^2 \approx 3.84$. ::: ::: --- ## 💰 Case Study: Unemployment Lags :::: columns ::: {.column width="50%"} ::: {.callout-note} ## 📈 The question How many lags of its own past does the monthly change in US unemployment need? 1. Which orders do AIC, BIC and HQ choose? 2. Where does GETS from $\bar p = 12$ stop? 3. Are the chosen model's residuals white noise? ::: ::: ::: {.column width="50%"} ::: {.callout-tip} ## 📊 The data **Series**: civilian unemployment rate (FRED `UNRATE`), seasonally adjusted **Period**: Jan 1960 – Dec 2019, 720 months (the 2020 pandemic jump left out) $x_t = U_t - U_{t-1}$, percentage points ::: ::: :::: ```{.r .display-only} u <- read.csv("https://fred.stlouisfed.org/graph/fredgraph.csv?id=UNRATE") ``` --- ## 💰 Case Study: Three Criteria and GETS ```{r} #| label: case-ic #| code-summary: "📦 AR(0)–AR(12): criteria, GETS, Ljung–Box" u <- read.csv("data/UNRATE.csv"); x <- diff(u$UNRATE) pmax <- 12; X <- embed(x, pmax + 1); y <- X[, 1]; n <- length(y) fits <- lapply(0:pmax, function(p) if (p == 0) lm(y ~ 1) else lm(y ~ X[, 1 + seq_len(p)])) s2 <- sapply(fits, function(f) mean(resid(f)^2)); k <- 1:(pmax + 1) ic <- data.frame(p = 0:pmax, AIC = n*log(s2) + 2*k, BIC = n*log(s2) + k*log(n), HQ = n*log(s2) + 2*k*log(log(n))) p_ic <- sapply(ic[, -1], which.min) - 1 p <- pmax # GETS: drop the last lag while |t| < 1.96 while (p > 0 && abs(coef(summary(fits[[p + 1]]))[p + 1, 3]) < 1.96) p <- p - 1 lb <- sapply(c(p_ic, GETS = p), function(q) Box.test(resid(fits[[q + 1]]), lag = 24, type = "Ljung", fitdf = q)$p.value) data.frame(rule = names(lb), order = c(p_ic, p), LjungBox24_pval = signif(lb, 2), row.names = NULL) ``` ::: {style="font-size: 28px"} The criteria disagree by eight lags; every residual series fails the Ljung–Box test at lag 24. ::: --- ## 💰 Case Study: What the Data Say ```{r} #| label: case-ic-fig #| echo: false #| fig-width: 10 #| fig-height: 3.9 icl <- rbind(data.frame(p = ic$p, crit = "AIC", v = ic$AIC - min(ic$AIC)), data.frame(p = ic$p, crit = "BIC", v = ic$BIC - min(ic$BIC)), data.frame(p = ic$p, crit = "HQ", v = ic$HQ - min(ic$HQ))) ggplot(icl, aes(p, pmin(v, 60), colour = crit)) + geom_line(linewidth = 1.2) + geom_point(data = subset(icl, v == 0), size = 5) + scale_colour_manual(values = c(AIC = "#8b2635", BIC = "#2f4b7c", HQ = "#d39c2f"), name = NULL) + scale_x_continuous(breaks = 0:12) + labs(x = "AR order p", y = "IC(p) − min IC (capped at 60)") + theme_minimal(base_size = 20) + theme(legend.position = "top") ``` ::: {style="font-size: 28px"} **What the data say:** BIC stops at 4 lags, HQ at 5; AIC and GETS run to the boundary at 12. Even AR(12) fails Ljung–Box: its residuals keep a correlation of −0.18 at lag 24, a trace of seasonal adjustment. And unemployment rises fast and falls slowly, an asymmetry no linear AR captures (Lecture 11). ::: --- ## 📝 Quiz #1: The Right Object {.quiz-question} A bank must hold capital against the loss it will exceed on only 1% of days. Which object of the daily return distribution should it model? - [The conditional 1% quantile $q_t^{0.01}$]{.correct data-explanation="✅ The loss exceeded with probability 1% is the conditional 1% quantile of returns: the Value-at-Risk. It can be modelled directly (quantile regression) or derived from a model of the conditional density."} - The conditional mean $\mu_t$ - The conditional skewness $S_t/\sigma_t^3$ - The unconditional variance of returns --- ## 📝 Quiz #2: Overfitting Rate {.quiz-question} The true model is AR(1) and the candidates are AR(1) and AR(2) only. As $T \to \infty$, how often does AIC choose AR(2)? - [About 15.7% of the time, for every large $T$]{.correct data-explanation="✅ The fit gain from the redundant lag is asymptotically χ²(1); AIC adds it when that exceeds 2, and P(χ²(1) > 2) = 0.157. The probability does not shrink with T, so AIC is not consistent."} - With probability tending to zero - About 5% of the time - Never, because the second coefficient is zero --- ## 📝 Quiz #3: GETS {.quiz-question} In a general-to-specific search, the test of $H_0: \phi_{\bar p} = 0$ in $AR(\bar p)$ is **not rejected**. What is the next step? - [Move to $AR(\bar p - 1)$ and test its last coefficient]{.correct data-explanation="✅ GETS imposes only restrictions the data accept. Not rejecting φ_p̄ = 0 means the restriction is accepted, so the model shrinks by one lag and the procedure tests again."} - Keep $AR(\bar p)$ and stop - Move to $AR(\bar p + 1)$ - Switch to an information criterion --- ## 📋 Key Formulas ::: {style="font-size: 28px"} | | | |---|---| | Conditional mean | $\mu_t = E_{t-1}[x_t]$, minimises $E[(x_t - g)^2]$ over $g \in I_{t-1}$ | | Conditional variance | $\sigma_t^2 = E_{t-1}[(x_t - \mu_t)^2]$ | | Conditional quantile | $\Pr_{t-1}\{x_t \le q_t^\alpha\} = \alpha$; median at $\alpha = 0.5$ | | Information criterion | $IC(k) = -2\log\hat{\mathcal L}_k + f(k)$ → min | | AR($p$), Gaussian | $IC(p) = T\log\hat\sigma_p^2 + f(p+1)$ | | Penalties | AIC $2k$; BIC $k\log T$; HQ $2k\log\log T$ | | Consistency | penalty per parameter $c_T \to \infty$, $c_T/T \to 0$ | ::: --- ## 📋 Summary ::: {style="font-size: 30px"} - Non-structural modelling begins by choosing an **object**: a feature of the distribution of $x_t$ given its past. - The conditional mean is optimal under squared loss, the median under absolute loss, a quantile under check loss. - Selection runs object → class → model; diagnostics, GETS and information criteria are the tools. - BIC and HQ are consistent; AIC overfits with positive probability at every $T$, which is not always a defect when the goal is prediction. - All of these criteria judge **in-sample** fit. ::: --- ## 📚 Further Reading and Next Lecture ::: {style="font-size: 29px"} - Hamilton (1994), *Time Series Analysis*, ch. 4–5. - Tsay (2010), *Analysis of Financial Time Series*, ch. 2.4–2.5 (order selection). - Franses and van Dijk (2000), *Non-Linear Time Series Models in Empirical Finance*, ch. 2. - Akaike (1974); Schwarz (1978); Hannan and Quinn (1979), the three criteria. - Hendry (1995), *Dynamic Econometrics*, on general-to-specific modelling. **Next lecture: In-Sample Fit, Out-of-Sample Forecasts, Diebold–Mariano.** A model that fits well in sample often forecasts badly. Why, and how do we test it? :::
