M.6 Econometrics & Time Series

Basic Prerequisites: Linear Algebra & Matrices and Probability & Measure Theory (conditional expectation, iid/white noise). (these are the folder-level prerequisites for pages 02–06; page 01 states its own, smaller, entry requirements)

1. Intuition & Practical Objective

Financial time series violate nearly every textbook-classical-statistics assumption: returns are fat-tailed and non-Gaussian, volatilities cluster in persistent regimes, and raw price series carry a unit root so that OLS on levels produces spurious regressions - statistically significant relationships between variables that share no economic link (Granger–Newbold 1974). Econometrics is the discipline that replaces “significant-looking” with “actually stationary.”

Its claim is sharp and practical: nearly every result you will use in the rest of the Atlas - a volatility forecast feeding VaR, a cointegrating pair feeding a stat-arb book, a Kalman filter tracking an alpha factor - is a statement about conditional structure (mean given the past, variance given the past) made valid only after you have settled stationarity. This folder is the topic-folder for that toolbox.

This page is a hub: it (a) gives the fast model-form and key-test lookup below (job #1), and (b) routes you to six sub-pages from raw intuition to Kalman/multivariate.

The one-sentence essence. “Every time-series question reduces to two conditional objects - $E [r_{t} ∣ F_{t - 1}]$ (the mean model) and $Var (r_{t} ∣ F_{t - 1})$ (the volatility model) - and every inference on them is only trustworthy once the series is made stationary, which for prices means differencing or finding a cointegrating combination.”

2. Mathematical Ground Truth & Derivations

Quick-Reference Lookup (job #1). All formulas below are transcribed from Tsay (2010), Ch 1–3, 8, 10–11 and cross-checked against ESL Ch 7 (model selection); the numbers in the check column were re-executed and reproduced exactly by the scripts in §3 and on the sub-pages.

Notation: $r_{t}$ (log) return, $a_{t}$ innovation/white noise ( $E [a_{t} ∣ F_{t - 1}] = 0$ , $Var = σ_{a}^{2}$ ), $B$ backshift operator ( $B x_{t} = x_{t - 1}$ ), $γ_{ℓ}, ρ_{ℓ}$ autocovariance/autocorrelation at lag $ℓ$ .

Mean (conditional-expectation) models:

Model	Form	Key facts	Verified check
White noise	$r_{t} = μ + a_{t}$	$ρ_{ℓ} = 0 \forall ℓ \geq 1$ ; Ljung–Box rejects	Ljung–Box $Q (10) = 18.14$ vs $χ_{10, 0.95}^{2} = 18.31$ on noise
AR(1)	$x_{t} = ϕ_{0} + ϕ_{1} x_{t - 1} + a_{t}$	stationary $∣ ϕ_{1} ∣ < 1$ ; mean $ϕ_{0} / (1 - ϕ_{1})$ ; var $σ_{a}^{2} / (1 - ϕ_{1}^{2})$ ; $ρ_{ℓ} = ϕ_{1}^{ℓ}$	ADF on $ϕ_{1} = 0.7$ AR(1): $- 19.1$ (rejects)
AR(2)	$x_{t} = ϕ_{1} x_{t - 1} + ϕ_{2} x_{t - 2} + a_{t}$	$ρ_{ℓ} = ϕ_{1} ρ_{ℓ - 1} + ϕ_{2} ρ_{ℓ - 2}$ ; PACF cuts at lag 2; complex roots $\Rightarrow$ damped sine	YW fit $(\hat{ϕ}_{1}, \hat{ϕ}_{2}) = (1.213, - 0.605)$ vs $(1.2, - 0.6)$
MA(1)	$x_{t} = a_{t} + θ_{1} a_{t - 1}$	always stationary; $ρ_{1} = + θ_{1} / (1 + θ_{1}^{2})$ , $ρ_{ℓ} = 0 ℓ > 1$ ; invertible $∣ θ_{1} ∣ < 1$	-
ARMA(1,1)	$x_{t} = ϕ x_{t - 1} + a_{t} + θ a_{t - 1}$	st. $∣ ϕ ∣ < 1$ , inv. $∣ θ ∣ < 1$ ; $ρ_{1} = \frac{( 1 + θϕ ) ( ϕ + θ )}{1 + 2 θϕ + θ ^{2}}$ ; decay starts at lag 2	$ρ_{1}$ theory $0.6619$ vs sample $0.6611$
ARIMA(p,1,q)	$(1 - B) x_{t}$ is stationary ARMA	unit root removed by differencing	-
Random walk	$p_{t} = p_{0} + \sum a_{i}$	var grows linearly; forecast $\overset{p}{^}_{h} (ℓ) = p_{h}$	ADF on RW: $- 0.64$ (cannot reject)
Forecast error var	via MA rep	$Var [e_{h} (ℓ)] = (1 + ψ_{1}^{2} + \dots + ψ_{ℓ - 1}^{2}) σ_{a}^{2}$	-

Key stationarity / test battery:

Test	Statistic	Rule	Verified check
Ljung–Box	$Q (m) = T (T + 2) \sum_{ℓ = 1}^{m} \frac{ρ ^ _{ℓ}^{2}}{T - ℓ}$	reject autocorr. if $Q > χ_{m}^{2}$ ( $m - g$ df for AR residuals)	$18.14$ vs $18.31$
Jarque–Bera	$J B = \frac{S ^ ^{2}}{6/ T} + \frac{( K ^ - 3 ) ^{2}}{24/ T}$	reject normality if $J B > χ_{2}^{2}$	fat tails reject
ADF (unit root)	$Δ x_{t} = α + βt + γ x_{t - 1} + \sum δ_{i} Δ x_{t - i} + e_{t}$ ; test $γ = 0$	reject if $t_{γ} <$ DF critical ( $- 2.86$ @5% w/ const)	AR $- 15.6$ vs RW $- 0.64$
ARCH effect	regress $a_{t}^{2}$ on $a_{t - 1}^{2}, \dots$ ; LM $= T R^{2}$	reject if $> χ_{m}^{2}$	GARCH squares: $2817.2$ vs $11.07$
Cointegration	residual $z_{t} = y_{t} - \hat{β} x_{t}$ ; ADF on $z$	reject $\Rightarrow$ cointegrated; else spurious	coint $- 17.0$ ; indep RWs $- 0.83$

Volatility models:

Model	Form	Key facts	Verified check
ARCH(m)	$σ_{t}^{2} = α_{0} + \sum α_{i} a_{t - i}^{2}$	uncond var $α_{0} / (1 - \sum α_{i})$ ; symmetric ±	-
GARCH(1,1)	$σ_{t}^{2} = ω + α a_{t - 1}^{2} + β σ_{t - 1}^{2}$	persistence $α + β < 1$ ; uncond var $ω / (1 - α - β)$	MLE $(ω, α, β) = (0.059, 0.118, 0.826)$ vs $(0.05, 0.10, 0.85)$
IGARCH / RiskMetrics	$σ_{t}^{2} = (1 - λ) a_{t - 1}^{2} + λ σ_{t - 1}^{2}$ , $λ \approx 0.94$	uncond var undefined; multistep var forecast $\propto ℓ$	-
EGARCH	$ln σ_{t}^{2} = α_{0} + \frac{1 + β _{1} B}{1 - α _{1} B} g (ε_{t - 1})$ , $g (ε) = θε + γ [∣ ε ∣ - E ∣ ε ∣]$	leverage via $θ < 0$ (asymmetric ±); logs relax positivity	IBM −2σ shock raises vol $\approx 37%$ more than +2σ
GARCH-M	$r_{t} = μ + c σ_{t}^{2} + a_{t}$	risk premium $c$ ; induces serial corr in returns	-

Multivariate / state-space:

Model	Form	Key facts
VAR(p)	$x_{t} = ϕ_{0} + Φ_{1} x_{t - 1} + \dots + Φ_{p} x_{t - p} + a_{t}$	stationary iff eigenvalues of companion $< 1$ in modulus
Cointegration	$z_{t} = y_{t} - β x_{t} \sim I (0)$ ; Granger rep.	ECM: $Δ y_{t} = γ (y_{t - 1} - β x_{t - 1}) + \dots$ , $γ < 0$
Johansen	trace $L R_{t r} (m) = - (T - p) \sum_{i = m + 1}^{k} ln (1 - \hat{λ}_{i})$ ; max $- (T - p) ln (1 - \hat{λ}_{m + 1})$	nonstandard critical values; # coint vectors = # nonzero $λ$
Local-level SS	$y_{t} = μ_{t} + e_{t}$ , $μ_{t + 1} = μ_{t} + η_{t}$	$σ_{e} = 0$ : ARIMA(0,1,0); else ARIMA(0,1,1)
Kalman	$v_{t} = y_{t} - c_{t} - Z_{t} s_{t ∣ t - 1}$ ; $K_{t} = T_{t} Σ_{t ∣ t - 1} Z_{t}^{'} V_{t}^{- 1}$ ; $s_{t + 1 ∣ t} = d_{t} + T_{t} s_{t ∣ t - 1} + K_{t} v_{t}$	filter/predict/smooth; steady state solves Riccati

3. Computational Implementation - the test battery in one place

This runs on the standard library only (no numpy/scipy) and reproduces the key-test column above: ADF on a stationary AR(1) vs a random walk, and Ljung–Box on white noise.

4. Failure Modes & First-Principles Breakdowns

Hub signposts - the folder’s failure-mode analysis lives on the sub-pages. In one line each:

Regressing on levels is spurious. Two unrelated random walks regressed against each other show high $R^{2}$ and a residual whose ADF fails to reject - “significant” coefficients with zero economic content (Granger–Newbold). Differencing or cointegration is mandatory.
Near-unit-root blindness. ADF has low power against $ϕ_{1}$ just below 1, and near-unit-root samples destroy the clean ACF/PACF cut-off story used for ARMA identification.
The ARCH model is symmetric. ARCH/GARCH respond equally to $+$ and $-$ shocks; only EGARCH/TGARCH capture the leverage effect that real markets show.
Persistence is fragile. When $α + β$ is near 1 (IGARCH/RiskMetrics), unconditional variance is undefined and multi-step vol forecasts grow linearly - the square-root-of-time rule silently fails.

5. References

Tsay, Ruey S.: Analysis of Financial Time Series (3rd ed., 2010)
Campbell, Lo & MacKinlay: The Econometrics of Financial Markets
Hastie, Tibshirani & Friedman: Elements of Statistical Learning
Box, Jenkins & Reinsel: Time Series Analysis

6. Connected Graph Bridges

Foundational base: Linear Algebra & Matrices · Probability & Measure Theory · Stochastic Calculus & Itô
Sibling foundations: Ergodicity & Statistical Mechanics (time vs ensemble averages - the twin of stationarity)
Sub-pages (in-folder): 01 From Zero · 02 Stationarity & ARMA · 03 Forecasting & Unit Roots · 04 Volatility Modeling · 05 Cointegration & Multivariate · 06 Advanced Extensions

Beginner: start at 01 · Practitioner: start at 05

Kwant Atlas

Explorer

M.6 Econometrics & Time Series

1. Intuition & Practical Objective

2. Mathematical Ground Truth & Derivations

3. Computational Implementation - the test battery in one place

4. Failure Modes & First-Principles Breakdowns

5. References

6. Connected Graph Bridges

M.6.1 Econometrics & Time Series from Zero

M.6.2 Stationarity & ARMA Models

M.6.3 Forecasting & Unit Roots

M.6.4 Volatility Modeling

M.6.5 Cointegration & Multivariate Models

M.6.6 Advanced Extensions