Time Series — Algebra dan Stasioneritas

AR, MA, ARMA, Unit Root, VAR

econometrics-math

intermediate

Matematika stasioneritas, proses AR/MA/ARMA, unit root dan spurious regression, VAR systems, dan kointegrasi. Kenapa data non-stasioner berbahaya dan cara mendeteksinya.

1 Kenapa Ini Penting?

Why This Matters for Your Work

Spurious regression, unit roots, kointegrasi — ini adalah jebakan klasik time series econometrics. Kalau kamu pernah menemukan $R^2$ tinggi dan t-stats signifikan antara dua seri yang seharusnya tidak berkorelasi, itu kemungkinan besar spurious regression.

Math time series memberi kamu: - Pemahaman mengapa data non-stasioner menyebabkan masalah (distribusi OLS t-stat runtuh) - Alat formal untuk menguji stasioneritas (ADF, KPSS) - Kerangka VAR untuk multi-variable dynamics - Konsep kointegrasi — long-run equilibrium relationship

2 Stasioneritas

Definisi: Stasioneritas

Strict stationarity: distribusi joint $(y_{t_1}, y_{t_2}, \ldots, y_{t_k})$ sama dengan $(y_{t_1+h}, \ldots, y_{t_k+h})$ untuk semua $h$ — distribusi time-invariant.

Covariance/Weak stationarity: kondisi lebih lemah: 1. $E[y_t] = \mu$ (mean konstan, tidak bergantung pada $t$) 2. $\text{Var}(y_t) = \sigma^2 < \infty$ (variance finite dan konstan) 3. $\text{Cov}(y_t, y_{t-h}) = \gamma(h)$ bergantung hanya pada lag $h$, bukan $t$

Untuk proses Gaussian, weak = strict stationarity.

Autocovariance function: $\gamma(h) = \text{Cov}(y_t, y_{t-h})$

Autocorrelation function (ACF): $\rho(h) = \frac{\gamma(h)}{\gamma(0)} \in [-1, 1]$

Partial ACF (PACF): korelasi $y_t$ dan $y_{t-h}$ setelah mengontrol $y_{t-1}, \ldots, y_{t-h+1}$.

ACF dan PACF adalah tools diagnostik utama untuk identifikasi model ARMA.

3 AR(1) Process

3.1 Definisi dan Stasioneritas

\[y_t = \phi y_{t-1} + \varepsilon_t, \quad \varepsilon_t \sim WN(0, \sigma^2)\]

Kapan stasioner? Perlu $|\phi| < 1$.

Iterasi mundur: \[y_t = \varepsilon_t + \phi\varepsilon_{t-1} + \phi^2\varepsilon_{t-2} + \cdots = \sum_{j=0}^\infty \phi^j \varepsilon_{t-j}\]

Ini adalah MA($\infty$) representation — konvergen jika $|\phi| < 1$.

3.2 Momen AR(1)

Mean (dengan $E[\varepsilon_t] = 0$): \[E[y_t] = \phi E[y_{t-1}] + 0 \quad \Rightarrow \quad \mu(1-\phi) = 0 \quad \Rightarrow \quad \mu = 0\]

Variance (menggunakan stationarity $\text{Var}(y_t) = \text{Var}(y_{t-1}) = \sigma_y^2$): \[\sigma_y^2 = \phi^2 \sigma_y^2 + \sigma^2 \quad \Rightarrow \quad \sigma_y^2 = \frac{\sigma^2}{1 - \phi^2}\]

Perhatikan: variance meledak saat $\phi \to 1$.

ACF: \[\gamma(h) = \text{Cov}(y_t, y_{t-h}) = \phi^h \cdot \frac{\sigma^2}{1-\phi^2} \quad \Rightarrow \quad \rho(h) = \phi^h\]

ACF dari AR(1) adalah geometric decay. Positif/negatif bergantung pada sign $\phi$.

4 AR(p) Process dan Characteristic Equation

4.1 Setup

\[y_t = \phi_1 y_{t-1} + \phi_2 y_{t-2} + \cdots + \phi_p y_{t-p} + \varepsilon_t\]

Menggunakan lag operator $L$ (di mana $Ly_t = y_{t-1}$):

\[(1 - \phi_1 L - \phi_2 L^2 - \cdots - \phi_p L^p)y_t = \varepsilon_t\] \[\Phi(L)y_t = \varepsilon_t\]

4.2 Characteristic Polynomial

\[\Phi(z) = 1 - \phi_1 z - \phi_2 z^2 - \cdots - \phi_p z^p\]

Kondisi stasioneritas: semua akar dari $\Phi(z) = 0$ berada di luar unit circle $|z| > 1$.

Contoh: AR(2) $\Phi(z) = 1 - \phi_1 z - \phi_2 z^2 = 0$ punya dua akar. Proses stasioner jika kedua akar modulus $> 1$.

4.3 Companion Matrix Form

Tulis AR(p) sebagai VAR(1) dengan state vector $\mathbf{Y}_t = (y_t, y_{t-1}, \ldots, y_{t-p+1})^T$:

\[\mathbf{Y}_t = A\mathbf{Y}_{t-1} + \mathbf{u}_t\]

\[A = \begin{pmatrix} \phi_1 & \phi_2 & \cdots & \phi_{p-1} & \phi_p \\ 1 & 0 & \cdots & 0 & 0 \\ 0 & 1 & \cdots & 0 & 0 \\ \vdots & & \ddots & & \vdots \\ 0 & 0 & \cdots & 1 & 0 \end{pmatrix}\]

Kondisi stasioneritas ekivalen: semua eigenvalue dari companion matrix $A$ memiliki modulus $< 1$.

5 Unit Root: Random Walk dan Non-Stationarity

5.1 Random Walk

Kasus spesial AR(1) dengan $\phi = 1$:

\[y_t = y_{t-1} + \varepsilon_t \quad \Rightarrow \quad y_t = y_0 + \sum_{j=1}^t \varepsilon_j\]

Properties: - $E[y_t] = y_0$ (konstan tapi bergantung initial condition) - $\text{Var}(y_t) = t\sigma^2$ — tumbuh tanpa batas! - $\text{Cov}(y_t, y_s) = \min(t,s)\sigma^2$ — bergantung pada $t,s$, bukan hanya $|t-s|$

Tidak stasioner — variance tidak finite.

5.2 Spurious Regression

Definisi: Spurious Regression

Dua proses independent I(1) $x_t$ dan $y_t$:

\[y_t = y_{t-1} + u_t, \quad x_t = x_{t-1} + v_t, \quad \text{Cov}(u_t, v_t) = 0\]

OLS dari $y_t$ pada $x_t$ memberikan: - $\hat{\beta} \xrightarrow{p}$ sesuatu selain 0 - t-statistic $\to \infty$ (diverges!) - $R^2 \to$ distribusi positif

Distribusi OLS runtuh — tidak lagi t-distributed! Ini karena asymptotics OLS bergantung pada stasioneritas.

Solusi: differencing. $\Delta y_t = y_t - y_{t-1}$ menghapus unit root.

Seri I(1): integrated order 1, perlu di-difference satu kali untuk stasioner.

5.3 Dickey-Fuller Test

Test $H_0: \phi = 1$ (unit root) vs $H_1: |\phi| < 1$ (stasioner).

Augmented Dickey-Fuller (ADF) mengakomodasi serial correlation:

\[\Delta y_t = \delta y_{t-1} + \sum_{j=1}^{p-1} \psi_j \Delta y_{t-j} + \varepsilon_t\]

Test $H_0: \delta = 0$ (unit root, $\phi = 1 \Leftrightarrow \delta = 0$).

Penting: distribusi t-stat di bawah $H_0$ bukan normal atau t — ini adalah distribusi Dickey-Fuller (tabel critical values berbeda!).

Varian: - ADF tanpa intercept: untuk zero-mean random walk - ADF dengan intercept: untuk random walk with drift - ADF dengan trend: untuk deterministic trend + unit root

6 MA(q) dan ARMA(p,q)

6.1 MA(q)

\[y_t = \varepsilon_t + \theta_1\varepsilon_{t-1} + \cdots + \theta_q\varepsilon_{t-q} = \Theta(L)\varepsilon_t\]

Selalu stasioner (finite weighted sum of WN).

ACF: $\rho(h) = 0$ untuk $h > q$ — cuts off setelah lag $q$.

6.2 ARMA(p,q)

\[\Phi(L)y_t = \Theta(L)\varepsilon_t\]

\[(1 - \phi_1 L - \cdots - \phi_p L^p)y_t = (1 + \theta_1 L + \cdots + \theta_q L^q)\varepsilon_t\]

Stasioner: roots dari $\Phi(z) = 0$ di luar unit circle. Invertible: roots dari $\Theta(z) = 0$ di luar unit circle (supaya MA$(\infty)$ converge).

6.3 Wold’s Theorem

Definisi: Wold’s Decomposition Theorem

Setiap proses covariance-stationary yang tidak deterministik dapat ditulis sebagai:

\[y_t = \sum_{j=0}^\infty c_j \varepsilon_{t-j} + \eta_t\]

di mana $\sum c_j^2 < \infty$, $\varepsilon_t \sim WN$, dan $\eta_t$ adalah purely deterministic component.

Implikasi: setiap proses stasioner punya MA($\infty$) representation — ARMA adalah parsimonious parametrization dari representasi ini.

7 VAR(p) System

7.1 Model Multivariate

\[\mathbf{y}_t = A_1\mathbf{y}_{t-1} + A_2\mathbf{y}_{t-2} + \cdots + A_p\mathbf{y}_{t-p} + \boldsymbol{\varepsilon}_t\]

$\mathbf{y}_t \in \mathbb{R}^K$: vektor $K$ variabel
$A_j \in \mathbb{R}^{K \times K}$: coefficient matrices
$\boldsymbol{\varepsilon}_t \sim WN(0, \Sigma)$, $\Sigma$ positive definite

Estimasi: OLS equation-by-equation (karena same regressors in each equation).

7.2 Granger Causality

$x_t$ Granger-causes $y_t$ jika past values of $x$ help predict $y$ beyond what past $y$ alone provides.

Formal test dalam VAR: test $H_0$ bahwa coefficients on lagged $x$ in the $y$-equation are all zero.

$F$-test pada blok koefisien — bukan “true” causality, hanya predictive.

7.3 Impulse Response Functions (IRF)

Bagaimana $\mathbf{y}_{t+h}$ merespons shock $\boldsymbol{\varepsilon}_t$?

VAR(1) companion form: $\mathbf{y}_t = A\mathbf{y}_{t-1} + \boldsymbol{\varepsilon}_t$

\[\frac{\partial \mathbf{y}_{t+h}}{\partial \boldsymbol{\varepsilon}_t} = A^h\]

Elemen $(i,j)$ dari $A^h$ = efek shock ke variabel $j$ pada variabel $i$ setelah $h$ periods.

7.4 Cholesky Identification

Masalah: $\boldsymbol{\varepsilon}_t$ bisa berkorelasi — butuh “structural” shocks yang orthogonal.

Cholesky decomposition: $\Sigma = PP^T$ (lower triangular $P$).

Structural shocks: $\mathbf{u}_t = P^{-1}\boldsymbol{\varepsilon}_t$, $E[\mathbf{u}_t\mathbf{u}_t^T] = I$.

Caveat: Cholesky imposes ordering — variabel yang diurut pertama tidak merespons variabel lain secara contemporaneous. Ordering harus justified by economic theory.

8 Cointegration

8.1 Definisi

Dua seri I(1) $x_t$ dan $y_t$ adalah cointegrated jika ada $\alpha$ sehingga: \[y_t - \alpha x_t \sim I(0)\]

Ini berarti ada long-run equilibrium relationship: meskipun keduanya non-stasioner, perbedaannya stasioner.

8.2 Error Correction Model (ECM)

Jika $y_t$ dan $x_t$ cointegrated:

\[\Delta y_t = \gamma_0 + \gamma_1 \Delta x_t + \pi (y_{t-1} - \alpha x_{t-1}) + \varepsilon_t\]

$(y_{t-1} - \alpha x_{t-1})$: error correction term — deviasi dari long-run equilibrium
$\pi < 0$: speed of adjustment back to equilibrium
Short-run dynamics ($\Delta x_t$) + long-run correction

8.3 Engle-Granger Test

Step 1: OLS $y_t = \hat{\alpha} x_t + \hat{e}_t$
Step 2: ADF test pada residual $\hat{e}_t$
$H_0$: residual tidak stasioner (tidak ada cointegration)

Note: critical values berbeda dari standar ADF karena residual adalah estimated.

9 Worked Example: AR(2) Stability Analysis

Worked Example: AR(2) dan Unit Root Testing di R

library(tidyverse)
library(tseries)   # for adf.test
library(vars)      # for VAR
library(forecast)  # for auto.arima

# ==========================================
# 1. AR(2) Stability Check
# ==========================================

# AR(2): y_t = 1.2*y_{t-1} - 0.5*y_{t-2} + eps_t
phi <- c(1.2, -0.5)

# Companion matrix
A <- matrix(c(phi[1], phi[2], 1, 0), nrow = 2, byrow = TRUE)
eigenvalues <- eigen(A)$values
cat("Eigenvalues of companion matrix:\n")
print(eigenvalues)
cat("Moduli:", Mod(eigenvalues), "\n")
cat("Stationarity:", all(Mod(eigenvalues) < 1), "\n")

# Characteristic polynomial roots
Phi_z <- function(z) 1 - phi[1]*z - phi[2]*z^2
roots <- polyroot(c(1, -phi[1], -phi[2]))
cat("\nRoots of characteristic polynomial:", roots, "\n")
cat("Moduli of roots:", Mod(roots), " (should be > 1 for stationarity)\n")

# ==========================================
# 2. Simulate and Test for Unit Root
# ==========================================
n <- 500

# Stationary AR(1)
y_stat <- arima.sim(model = list(ar = 0.7), n = n)

# Non-stationary (unit root)
set.seed(123)
y_nonstat <- cumsum(rnorm(n))  # random walk

# ADF tests
adf_stat   <- adf.test(y_stat)
adf_nonstat <- adf.test(y_nonstat)

cat("\nADF test - Stationary series p-value:", adf_stat$p.value)
cat("\nADF test - Random walk p-value:", adf_nonstat$p.value)

# ==========================================
# 3. Spurious Regression Demo
# ==========================================
set.seed(42)
n <- 200
x_I1 <- cumsum(rnorm(n))
y_I1 <- cumsum(rnorm(n))  # independent!

# OLS akan memberikan "significant" result
spurious <- lm(y_I1 ~ x_I1)
cat("\nSpurious regression R^2:", summary(spurious)$r.squared)
cat("\nt-stat for x:", summary(spurious)$coefficients[2, 3])
# Very high R^2 and significant t-stat, but WRONG!

# Correct: difference both series
dy <- diff(y_I1)
dx <- diff(x_I1)
correct <- lm(dy ~ dx)
cat("\n\nCorrect regression (differenced) R^2:", summary(correct)$r.squared)
cat("\nt-stat for dx:", summary(correct)$coefficients[2, 3])

# ==========================================
# 4. VAR Example
# ==========================================
data("Canada", package = "vars")
plot(Canada)

# Select VAR order
VARselect(Canada, lag.max = 8, type = "const")$selection

# Estimate VAR(2)
var_model <- VAR(Canada, p = 2, type = "const")
summary(var_model)

# Impulse Response Functions
irf_result <- irf(var_model, impulse = "prod", response = "e",
                  n.ahead = 20, boot = TRUE)
plot(irf_result)

# Granger causality
causality(var_model, cause = "prod")

10 Ringkasan: ACF/PACF Pattern untuk Identifikasi Model

Model	ACF	PACF
AR(p)	Exponential/oscillating decay	Cuts off after lag $p$
MA(q)	Cuts off after lag $q$	Exponential/oscillating decay
ARMA(p,q)	Exponential/oscillating decay	Exponential/oscillating decay
Random Walk I(1)	Very slow decay	~1 at lag 1, near 0 after

Connection: Panel Unit Roots dan Spatial-Temporal

Time series methods extend ke berbagai arah:

Panel unit root tests (Im-Pesaran-Shin, Levin-Lin-Chu): menggunakan cross-section power untuk test unit root. Lebih powerful dari individual ADF.
Long-run variance estimation: Newey-West HAC standard errors menggunakan kernel smoothing dari autocovariances — penting untuk inference dengan serial correlation.
Kalman filter: state-space models unify ARMA, unobserved components, time-varying parameters.
GARCH/Volatility models: conditional heteroskedasticity — variance juga memiliki dynamics.
Spatial-temporal: kombinasikan spatial weights matrix $W$ dengan time series dynamics → space-time AR models.

11 Practice Problems

Practice Problems

Problem 1: Tunjukkan bahwa AR(1) dengan $|\phi| < 1$ adalah covariance-stationary.

Dari MA($\infty$) representation: $y_t = \sum_{j=0}^\infty \phi^j \varepsilon_{t-j}$.

$E[y_t] = 0$ (konstan). $\text{Var}(y_t) = \sigma^2 \sum_{j=0}^\infty \phi^{2j} = \sigma^2/(1-\phi^2) < \infty$ jika $|\phi| < 1$. $\text{Cov}(y_t, y_{t-h}) = \sigma^2 \phi^h/(1-\phi^2)$ bergantung hanya pada $h$. QED.

Problem 2: Derive ACF dari AR(2).

Dari Yule-Walker equations: $\rho(1) = \phi_1 + \phi_2\rho(1) \Rightarrow \rho(1) = \phi_1/(1-\phi_2)$ $\rho(2) = \phi_1\rho(1) + \phi_2$ $\rho(h) = \phi_1\rho(h-1) + \phi_2\rho(h-2)$ untuk $h \geq 2$

ACF memenuhi recursion yang sama dengan AR process itu sendiri — solusinya: kombinasi linear dari $r_1^h$ dan $r_2^h$ di mana $r_1, r_2$ adalah inverse dari akar characteristic polynomial.

Problem 3: Mengapa t-statistic dalam spurious regression diverges?

Untuk dua independent I(1) series: OLS estimator $\hat{\beta} \xrightarrow{d}$ ratio dari dua Brownian motions (Wiener processes), bukan $N(0,\sigma^2/n)$. t-statistic $\sim T^{1/2}$ times something nonzero → diverges.

Problem 4: Apa syarat perlu dan cukup untuk cointegration dalam sistem bivariate $(y_t, x_t)$?

Perlu: keduanya I(1). Cukup: $\exists \alpha$ sehingga $y_t - \alpha x_t \sim I(0)$. Secara matrix: rank of cointegrating matrix $\Pi$ dalam VECM $\Delta\mathbf{y}_t = \Pi\mathbf{y}_{t-1} + \ldots$ harus antara 0 (no cointegration) dan $K$ (all stationary).

Navigasi: ← Panel Data Math | Spatial Econometrics →

--- title: "Time Series — Algebra dan Stasioneritas" subtitle: "AR, MA, ARMA, Unit Root, VAR" description: "Matematika stasioneritas, proses AR/MA/ARMA, unit root dan spurious regression, VAR systems, dan kointegrasi. Kenapa data non-stasioner berbahaya dan cara mendeteksinya." categories: [econometrics-math, intermediate] --- ## Kenapa Ini Penting? ::: {.callout-note title="Why This Matters for Your Work"} Spurious regression, unit roots, kointegrasi — ini adalah jebakan klasik time series econometrics. Kalau kamu pernah menemukan $R^2$ tinggi dan t-stats signifikan antara dua seri yang seharusnya tidak berkorelasi, itu kemungkinan besar spurious regression. Math time series memberi kamu: - Pemahaman **mengapa data non-stasioner menyebabkan masalah** (distribusi OLS t-stat runtuh) - Alat formal untuk **menguji stasioneritas** (ADF, KPSS) - Kerangka **VAR** untuk multi-variable dynamics - Konsep **kointegrasi** — long-run equilibrium relationship ::: --- ## Stasioneritas ::: {.callout-important title="Definisi: Stasioneritas"} **Strict stationarity**: distribusi joint $(y_{t_1}, y_{t_2}, \ldots, y_{t_k})$ sama dengan $(y_{t_1+h}, \ldots, y_{t_k+h})$ untuk semua $h$ — distribusi time-invariant. **Covariance/Weak stationarity**: kondisi lebih lemah: 1. $E[y_t] = \mu$ (mean konstan, tidak bergantung pada $t$) 2. $\text{Var}(y_t) = \sigma^2 < \infty$ (variance finite dan konstan) 3. $\text{Cov}(y_t, y_{t-h}) = \gamma(h)$ bergantung **hanya pada lag** $h$, bukan $t$ Untuk proses Gaussian, weak = strict stationarity. ::: **Autocovariance function**: $\gamma(h) = \text{Cov}(y_t, y_{t-h})$ **Autocorrelation function (ACF)**: $\rho(h) = \frac{\gamma(h)}{\gamma(0)} \in [-1, 1]$ **Partial ACF (PACF)**: korelasi $y_t$ dan $y_{t-h}$ setelah mengontrol $y_{t-1}, \ldots, y_{t-h+1}$. ACF dan PACF adalah tools diagnostik utama untuk identifikasi model ARMA. --- ## AR(1) Process ### Definisi dan Stasioneritas $$y_t = \phi y_{t-1} + \varepsilon_t, \quad \varepsilon_t \sim WN(0, \sigma^2)$$ **Kapan stasioner?** Perlu $|\phi| < 1$. Iterasi mundur: $$y_t = \varepsilon_t + \phi\varepsilon_{t-1} + \phi^2\varepsilon_{t-2} + \cdots = \sum_{j=0}^\infty \phi^j \varepsilon_{t-j}$$ Ini adalah **MA($\infty$) representation** — konvergen jika $|\phi| < 1$. ### Momen AR(1) **Mean** (dengan $E[\varepsilon_t] = 0$): $$E[y_t] = \phi E[y_{t-1}] + 0 \quad \Rightarrow \quad \mu(1-\phi) = 0 \quad \Rightarrow \quad \mu = 0$$ **Variance** (menggunakan stationarity $\text{Var}(y_t) = \text{Var}(y_{t-1}) = \sigma_y^2$): $$\sigma_y^2 = \phi^2 \sigma_y^2 + \sigma^2 \quad \Rightarrow \quad \sigma_y^2 = \frac{\sigma^2}{1 - \phi^2}$$ Perhatikan: variance meledak saat $\phi \to 1$. **ACF**: $$\gamma(h) = \text{Cov}(y_t, y_{t-h}) = \phi^h \cdot \frac{\sigma^2}{1-\phi^2} \quad \Rightarrow \quad \rho(h) = \phi^h$$ ACF dari AR(1) adalah **geometric decay**. Positif/negatif bergantung pada sign $\phi$. --- ## AR(p) Process dan Characteristic Equation ### Setup $$y_t = \phi_1 y_{t-1} + \phi_2 y_{t-2} + \cdots + \phi_p y_{t-p} + \varepsilon_t$$ Menggunakan lag operator $L$ (di mana $Ly_t = y_{t-1}$): $$(1 - \phi_1 L - \phi_2 L^2 - \cdots - \phi_p L^p)y_t = \varepsilon_t$$ $$\Phi(L)y_t = \varepsilon_t$$ ### Characteristic Polynomial $$\Phi(z) = 1 - \phi_1 z - \phi_2 z^2 - \cdots - \phi_p z^p$$ **Kondisi stasioneritas**: semua akar dari $\Phi(z) = 0$ **berada di luar unit circle** $|z| > 1$. Contoh: AR(2) $\Phi(z) = 1 - \phi_1 z - \phi_2 z^2 = 0$ punya dua akar. Proses stasioner jika kedua akar modulus $> 1$. ### Companion Matrix Form Tulis AR(p) sebagai VAR(1) dengan state vector $\mathbf{Y}_t = (y_t, y_{t-1}, \ldots, y_{t-p+1})^T$: $$\mathbf{Y}_t = A\mathbf{Y}_{t-1} + \mathbf{u}_t$$ $$A = \begin{pmatrix} \phi_1 & \phi_2 & \cdots & \phi_{p-1} & \phi_p \\ 1 & 0 & \cdots & 0 & 0 \\ 0 & 1 & \cdots & 0 & 0 \\ \vdots & & \ddots & & \vdots \\ 0 & 0 & \cdots & 1 & 0 \end{pmatrix}$$ **Kondisi stasioneritas ekivalen**: semua eigenvalue dari companion matrix $A$ memiliki modulus $< 1$. --- ## Unit Root: Random Walk dan Non-Stationarity ### Random Walk Kasus spesial AR(1) dengan $\phi = 1$: $$y_t = y_{t-1} + \varepsilon_t \quad \Rightarrow \quad y_t = y_0 + \sum_{j=1}^t \varepsilon_j$$ **Properties**: - $E[y_t] = y_0$ (konstan tapi bergantung initial condition) - $\text{Var}(y_t) = t\sigma^2$ — **tumbuh tanpa batas!** - $\text{Cov}(y_t, y_s) = \min(t,s)\sigma^2$ — bergantung pada $t,s$, bukan hanya $|t-s|$ Tidak stasioner — variance tidak finite. ### Spurious Regression ::: {.callout-important title="Definisi: Spurious Regression"} Dua proses **independent** I(1) $x_t$ dan $y_t$: $$y_t = y_{t-1} + u_t, \quad x_t = x_{t-1} + v_t, \quad \text{Cov}(u_t, v_t) = 0$$ OLS dari $y_t$ pada $x_t$ memberikan: - $\hat{\beta} \xrightarrow{p}$ sesuatu selain 0 - t-statistic $\to \infty$ (diverges!) - $R^2 \to$ distribusi positif Distribusi OLS runtuh — tidak lagi t-distributed! Ini karena asymptotics OLS bergantung pada stasioneritas. ::: Solusi: differencing. $\Delta y_t = y_t - y_{t-1}$ menghapus unit root. Seri I(1): integrated order 1, perlu di-difference satu kali untuk stasioner. ### Dickey-Fuller Test Test $H_0: \phi = 1$ (unit root) vs $H_1: |\phi| < 1$ (stasioner). **Augmented Dickey-Fuller (ADF)** mengakomodasi serial correlation: $$\Delta y_t = \delta y_{t-1} + \sum_{j=1}^{p-1} \psi_j \Delta y_{t-j} + \varepsilon_t$$ Test $H_0: \delta = 0$ (unit root, $\phi = 1 \Leftrightarrow \delta = 0$). **Penting**: distribusi t-stat di bawah $H_0$ **bukan** normal atau t — ini adalah distribusi Dickey-Fuller (tabel critical values berbeda!). Varian: - ADF tanpa intercept: untuk zero-mean random walk - ADF dengan intercept: untuk random walk with drift - ADF dengan trend: untuk deterministic trend + unit root --- ## MA(q) dan ARMA(p,q) ### MA(q) $$y_t = \varepsilon_t + \theta_1\varepsilon_{t-1} + \cdots + \theta_q\varepsilon_{t-q} = \Theta(L)\varepsilon_t$$ Selalu stasioner (finite weighted sum of WN). ACF: $\rho(h) = 0$ untuk $h > q$ — **cuts off** setelah lag $q$. ### ARMA(p,q) $$\Phi(L)y_t = \Theta(L)\varepsilon_t$$ $$(1 - \phi_1 L - \cdots - \phi_p L^p)y_t = (1 + \theta_1 L + \cdots + \theta_q L^q)\varepsilon_t$$ Stasioner: roots dari $\Phi(z) = 0$ di luar unit circle. Invertible: roots dari $\Theta(z) = 0$ di luar unit circle (supaya MA$(\infty)$ converge). ### Wold's Theorem ::: {.callout-important title="Definisi: Wold's Decomposition Theorem"} Setiap proses covariance-stationary yang tidak deterministik dapat ditulis sebagai: $$y_t = \sum_{j=0}^\infty c_j \varepsilon_{t-j} + \eta_t$$ di mana $\sum c_j^2 < \infty$, $\varepsilon_t \sim WN$, dan $\eta_t$ adalah purely deterministic component. Implikasi: setiap proses stasioner punya **MA($\infty$) representation** — ARMA adalah parsimonious parametrization dari representasi ini. ::: --- ## VAR(p) System ### Model Multivariate $$\mathbf{y}_t = A_1\mathbf{y}_{t-1} + A_2\mathbf{y}_{t-2} + \cdots + A_p\mathbf{y}_{t-p} + \boldsymbol{\varepsilon}_t$$ - $\mathbf{y}_t \in \mathbb{R}^K$: vektor $K$ variabel - $A_j \in \mathbb{R}^{K \times K}$: coefficient matrices - $\boldsymbol{\varepsilon}_t \sim WN(0, \Sigma)$, $\Sigma$ positive definite Estimasi: OLS equation-by-equation (karena same regressors in each equation). ### Granger Causality $x_t$ **Granger-causes** $y_t$ jika past values of $x$ help predict $y$ beyond what past $y$ alone provides. Formal test dalam VAR: test $H_0$ bahwa coefficients on lagged $x$ in the $y$-equation are all zero. $F$-test pada blok koefisien — bukan "true" causality, hanya predictive. ### Impulse Response Functions (IRF) Bagaimana $\mathbf{y}_{t+h}$ merespons shock $\boldsymbol{\varepsilon}_t$? VAR(1) companion form: $\mathbf{y}_t = A\mathbf{y}_{t-1} + \boldsymbol{\varepsilon}_t$ $$\frac{\partial \mathbf{y}_{t+h}}{\partial \boldsymbol{\varepsilon}_t} = A^h$$ Elemen $(i,j)$ dari $A^h$ = efek shock ke variabel $j$ pada variabel $i$ setelah $h$ periods. ### Cholesky Identification Masalah: $\boldsymbol{\varepsilon}_t$ bisa berkorelasi — butuh "structural" shocks yang orthogonal. **Cholesky decomposition**: $\Sigma = PP^T$ (lower triangular $P$). Structural shocks: $\mathbf{u}_t = P^{-1}\boldsymbol{\varepsilon}_t$, $E[\mathbf{u}_t\mathbf{u}_t^T] = I$. **Caveat**: Cholesky imposes ordering — variabel yang diurut pertama tidak merespons variabel lain secara contemporaneous. Ordering harus justified by economic theory. --- ## Cointegration ### Definisi Dua seri I(1) $x_t$ dan $y_t$ adalah **cointegrated** jika ada $\alpha$ sehingga: $$y_t - \alpha x_t \sim I(0)$$ Ini berarti ada **long-run equilibrium relationship**: meskipun keduanya non-stasioner, perbedaannya stasioner. ### Error Correction Model (ECM) Jika $y_t$ dan $x_t$ cointegrated: $$\Delta y_t = \gamma_0 + \gamma_1 \Delta x_t + \pi (y_{t-1} - \alpha x_{t-1}) + \varepsilon_t$$ - $(y_{t-1} - \alpha x_{t-1})$: **error correction term** — deviasi dari long-run equilibrium - $\pi < 0$: speed of adjustment back to equilibrium - Short-run dynamics ($\Delta x_t$) + long-run correction ### Engle-Granger Test 1. Step 1: OLS $y_t = \hat{\alpha} x_t + \hat{e}_t$ 2. Step 2: ADF test pada residual $\hat{e}_t$ 3. $H_0$: residual tidak stasioner (tidak ada cointegration) Note: critical values berbeda dari standar ADF karena residual adalah estimated. --- ## Worked Example: AR(2) Stability Analysis ::: {.callout-tip title="Worked Example: AR(2) dan Unit Root Testing di R" collapse="true"} ```r library(tidyverse) library(tseries) # for adf.test library(vars) # for VAR library(forecast) # for auto.arima # ========================================== # 1. AR(2) Stability Check # ========================================== # AR(2): y_t = 1.2*y_{t-1} - 0.5*y_{t-2} + eps_t phi <- c(1.2, -0.5) # Companion matrix A <- matrix(c(phi[1], phi[2], 1, 0), nrow = 2, byrow = TRUE) eigenvalues <- eigen(A)$values cat("Eigenvalues of companion matrix:\n") print(eigenvalues) cat("Moduli:", Mod(eigenvalues), "\n") cat("Stationarity:", all(Mod(eigenvalues) < 1), "\n") # Characteristic polynomial roots Phi_z <- function(z) 1 - phi[1]*z - phi[2]*z^2 roots <- polyroot(c(1, -phi[1], -phi[2])) cat("\nRoots of characteristic polynomial:", roots, "\n") cat("Moduli of roots:", Mod(roots), " (should be > 1 for stationarity)\n") # ========================================== # 2. Simulate and Test for Unit Root # ========================================== n <- 500 # Stationary AR(1) y_stat <- arima.sim(model = list(ar = 0.7), n = n) # Non-stationary (unit root) set.seed(123) y_nonstat <- cumsum(rnorm(n)) # random walk # ADF tests adf_stat <- adf.test(y_stat) adf_nonstat <- adf.test(y_nonstat) cat("\nADF test - Stationary series p-value:", adf_stat$p.value) cat("\nADF test - Random walk p-value:", adf_nonstat$p.value) # ========================================== # 3. Spurious Regression Demo # ========================================== set.seed(42) n <- 200 x_I1 <- cumsum(rnorm(n)) y_I1 <- cumsum(rnorm(n)) # independent! # OLS akan memberikan "significant" result spurious <- lm(y_I1 ~ x_I1) cat("\nSpurious regression R^2:", summary(spurious)$r.squared) cat("\nt-stat for x:", summary(spurious)$coefficients[2, 3]) # Very high R^2 and significant t-stat, but WRONG! # Correct: difference both series dy <- diff(y_I1) dx <- diff(x_I1) correct <- lm(dy ~ dx) cat("\n\nCorrect regression (differenced) R^2:", summary(correct)$r.squared) cat("\nt-stat for dx:", summary(correct)$coefficients[2, 3]) # ========================================== # 4. VAR Example # ========================================== data("Canada", package = "vars") plot(Canada) # Select VAR order VARselect(Canada, lag.max = 8, type = "const")$selection # Estimate VAR(2) var_model <- VAR(Canada, p = 2, type = "const") summary(var_model) # Impulse Response Functions irf_result <- irf(var_model, impulse = "prod", response = "e", n.ahead = 20, boot = TRUE) plot(irf_result) # Granger causality causality(var_model, cause = "prod") ``` ::: --- ## Ringkasan: ACF/PACF Pattern untuk Identifikasi Model | Model | ACF | PACF | |-------|-----|------| | AR(p) | Exponential/oscillating decay | Cuts off after lag $p$ | | MA(q) | Cuts off after lag $q$ | Exponential/oscillating decay | | ARMA(p,q) | Exponential/oscillating decay | Exponential/oscillating decay | | Random Walk I(1) | Very slow decay | ~1 at lag 1, near 0 after | --- ::: {.callout-caution title="Connection: Panel Unit Roots dan Spatial-Temporal"} Time series methods extend ke berbagai arah: - **Panel unit root tests** (Im-Pesaran-Shin, Levin-Lin-Chu): menggunakan cross-section power untuk test unit root. Lebih powerful dari individual ADF. - **Long-run variance** estimation: Newey-West HAC standard errors menggunakan kernel smoothing dari autocovariances — penting untuk inference dengan serial correlation. - **Kalman filter**: state-space models unify ARMA, unobserved components, time-varying parameters. - **GARCH/Volatility models**: conditional heteroskedasticity — variance juga memiliki dynamics. - **Spatial-temporal**: kombinasikan spatial weights matrix $W$ dengan time series dynamics → space-time AR models. ::: --- ## Practice Problems ::: {.callout-warning title="Practice Problems" collapse="true"} **Problem 1**: Tunjukkan bahwa AR(1) dengan $|\phi| < 1$ adalah covariance-stationary. Dari MA($\infty$) representation: $y_t = \sum_{j=0}^\infty \phi^j \varepsilon_{t-j}$. $E[y_t] = 0$ (konstan). $\text{Var}(y_t) = \sigma^2 \sum_{j=0}^\infty \phi^{2j} = \sigma^2/(1-\phi^2) < \infty$ jika $|\phi| < 1$. $\text{Cov}(y_t, y_{t-h}) = \sigma^2 \phi^h/(1-\phi^2)$ bergantung hanya pada $h$. QED. **Problem 2**: Derive ACF dari AR(2). Dari Yule-Walker equations: $\rho(1) = \phi_1 + \phi_2\rho(1) \Rightarrow \rho(1) = \phi_1/(1-\phi_2)$ $\rho(2) = \phi_1\rho(1) + \phi_2$ $\rho(h) = \phi_1\rho(h-1) + \phi_2\rho(h-2)$ untuk $h \geq 2$ ACF memenuhi recursion yang sama dengan AR process itu sendiri — solusinya: kombinasi linear dari $r_1^h$ dan $r_2^h$ di mana $r_1, r_2$ adalah inverse dari akar characteristic polynomial. **Problem 3**: Mengapa t-statistic dalam spurious regression diverges? Untuk dua independent I(1) series: OLS estimator $\hat{\beta} \xrightarrow{d}$ ratio dari dua Brownian motions (Wiener processes), bukan $N(0,\sigma^2/n)$. t-statistic $\sim T^{1/2}$ times something nonzero → diverges. **Problem 4**: Apa syarat perlu dan cukup untuk cointegration dalam sistem bivariate $(y_t, x_t)$? Perlu: keduanya I(1). Cukup: $\exists \alpha$ sehingga $y_t - \alpha x_t \sim I(0)$. Secara matrix: rank of cointegrating matrix $\Pi$ dalam VECM $\Delta\mathbf{y}_t = \Pi\mathbf{y}_{t-1} + \ldots$ harus antara 0 (no cointegration) dan $K$ (all stationary). ::: --- **Navigasi**: [← Panel Data Math](05-panel-data-math.qmd) | [Spatial Econometrics →](07-spatial-econometrics.qmd)