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🔹 BAB 1 — Mathematical Foundations (Fondasi Matematis)

(A) Struktur Matematis

Semua model ekonometrika ditulis dalam bentuk optimisasi:$$\min_{\theta} Q(\theta)$$

Contoh (OLS):$$Q(\beta) = (y – X\beta)'(y – X\beta)$$

(B) Aljabar Matriks

• Vektor observasi:

$$y = (y_1, y_2, \dots, y_T)’$$

• Matriks regresor:

$$X = \begin{bmatrix} 1 & x_{11} & \dots & x_{k1} \\ 1 & x_{12} & \dots & x_{k2} \\ \vdots & \vdots & & \vdots \\ 1 & x_{1T} & \dots & x_{kT} \end{bmatrix}$$

(C) Diferensiasi

Digunakan untuk:

• OLS
• Maximum Likelihood
• Kalman Filter
• GMM

➡️ Bab ini = bahasa matematika minimum untuk seluruh buku

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🔹 BAB 2 — Statistical Foundations & Financial Data

(A) Proses Stokastik

Data keuangan diperlakukan sebagai:$$\{y_t\}_{t=1}^T \sim \text{stochastic process}$$

(B) Return Keuangan

Harga aset tidak stasioner, maka digunakan return:$$r_t = \ln(P_t) – \ln(P_{t-1})$$

Sifat:

• mean ≈ 0
• volatility clustering
• fat tails

(C) Varians & Kovarians

$$Var(y) = E[(y – \mu)^2]$$

➡️ Landasan untuk ARCH/GARCH & VAR covariance

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🔹 BAB 3 — Classical Linear Regression Model (CLRM)

(A) Model Dasar

$$y_t = X_t \beta + u_t$$

(B) Asumsi Gauss–Markov

1. Linear in parameters
2. $E(u|X)=0$
3. $Var(u|X)=\sigma^2 I$
4. No perfect multicollinearity

(C) Estimator

$$\hat{\beta} = (X’X)^{-1}X’y$$

(D) Distribusi Estimator

$$\hat{\beta} \sim N(\beta, \sigma^2 (X’X)^{-1})$$

➡️ OLS = baseline semua model lanjutan

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🔹 BAB 4 — Hypothesis Testing & Specification

(A) Uji Parameter

$$H_0: \beta_j = 0$$

Statistik:$$t = \frac{\hat{\beta}_j}{se(\hat{\beta}_j)}$$

(B) Restriksi Bersama

$$H_0: R\beta = r$$$$F = \frac{(R\hat{\beta}-r)'[R(X’X)^{-1}R’]^{-1}(R\hat{\beta}-r)}{q}$$

(C) Dummy & Regime

$$y_t = \beta_0 + \beta_1 D_t + \beta_2 x_t + u_t$$

➡️ Digunakan untuk policy change, krisis, QE episode

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🔹 BAB 5 — Diagnostic Testing (KRUSIAL)

(A) Heteroskedastisitas

$$Var(u_t|X) = \sigma_t^2$$

→ OLS tidak efisien

(B) Autokorelasi

$$u_t = \rho u_{t-1} + \varepsilon_t$$

→ standard error salah

(C) Structural Break

$$\beta_t = \begin{cases} \beta_1 & t \le T_b \\ \beta_2 & t > T_b \end{cases}$$​​

➡️ Jantung riset krisis & kebijakan

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🔹 BAB 6 — Univariate Time Series

(A) AR(p)

$$y_t = c + \sum_{i=1}^p \phi_i y_{t-i} + u_t$$

Syarat stasioner:$$|\phi_i| < 1$$

(B) MA(q)

$$y_t = u_t + \sum_{j=1}^q \theta_j u_{t-j}$$

(C) ARMA(p,q)

Digunakan untuk:

• inflasi
• suku bunga
• return

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🔹 BAB 7 — VAR (Multivariate Dynamics)

(A) VAR(p)

$$Y_t = c + \sum_{i=1}^p A_i Y_{t-i} + u_t$$

(B) Impulse Response

$$IRF(h) = \frac{\partial Y_{t+h}}{\partial u_t}$$

(C) FEVD

$$FEVD_{ij}(h)$$

➡️ Standar analisis transmisi moneter

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🔹 BAB 8 — Cointegration & VECM

(A) Unit Root

$$\Delta y_t = \gamma y_{t-1} + \varepsilon_t$$

(B) Cointegration

$$y_t – \beta’x_t \sim I(0)$$

(C) VECM

$$\Delta Y_t = \Pi Y_{t-1} + \sum \Gamma_i \Delta Y_{t-i} + u_t$$$$\Pi = \alpha \beta’$$

➡️ Keseimbangan jangka panjang + dinamika jangka pendek

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🔹 BAB 9 — ARCH & GARCH (Volatilitas)

(A) Conditional Variance

$$u_t = \sigma_t \varepsilon_t$$

(B) GARCH(1,1)

$$\sigma_t^2 = \omega + \alpha u_{t-1}^2 + \beta \sigma_{t-1}^2$$

(C) Asymmetric Volatility

• EGARCH
• GJR

➡️ Risk, uncertainty, crisis modeling

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🔹 BAB 10 — Regime Switching & State Space

(A) Markov Switching

$$y_t = \beta_{s_t} x_t + u_t$$$$P(s_t=j|s_{t-1}=i)=p_{ij}$$

(B) State Space

$$\begin{aligned} y_t &= Z_t \alpha_t + \varepsilon_t \\ \alpha_t &= T_t \alpha_{t-1} + \eta_t \end{aligned}$$

➡️ Dasar TVP-VAR & Kalman Filter

🔹 BAB 11 — Panel Data Models

(A) Model Dasar Panel

Panel menggabungkan dimensi waktu (t) dan individu (i):$$y_{it} = \alpha + \beta x_{it} + u_{it}, \quad i=1,\dots,N;\; t=1,\dots,T$$

Dengan error:$$u_{it} = \mu_i + \varepsilon_{it}$$

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(B) Fixed Effects (FE)

Model

$$y_{it} = \alpha_i + \beta x_{it} + \varepsilon_{it}$$

• $\alpha_i$αi​: efek individu tak teramati (negara, bank, perusahaan)
• boleh berkorelasi dengan $x_{it}$

Transformasi Within

$$(y_{it}-\bar{y}_i) = \beta(x_{it}-\bar{x}_i) + (\varepsilon_{it}-\bar{\varepsilon}_i)$$

➡️ Menghilangkan $\alpha_i$αi​

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(C) Random Effects (RE)

Model

$$y_{it} = \alpha + \beta x_{it} + \mu_i + \varepsilon_{it}$$

Asumsi:$$Cov(\mu_i, x_{it}) = 0$$

Estimator: GLS

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(D) Uji Pemilihan Model

• Hausman Test

$$H = (\hat{\beta}_{FE}-\hat{\beta}_{RE})’ [Var(\hat{\beta}_{FE})-Var(\hat{\beta}_{RE})]^{-1} (\hat{\beta}_{FE}-\hat{\beta}_{RE})$$

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(E) Relevansi

• Panel QE lintas negara
• Bank-level lending
• Firm-level stock response

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🔹 BAB 12 — Limited Dependent Variable Models

(A) Binary Choice Framework

Variabel laten:$$y_i^* = x_i’\beta + u_i$$

Observed:$$y_i = \begin{cases} 1 & y_i^* > 0 \\ 0 & y_i^* \le 0 \end{cases}$$

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(B) Logit Model

Asumsi:$$u_i \sim \text{Logistic}$$

Probabilitas:$$P(y_i=1|x_i) = \frac{e^{x_i’\beta}}{1+e^{x_i’\beta}}$$

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(C) Probit Model

Asumsi:$$u_i \sim N(0,1)$$

Probabilitas:$$P(y_i=1|x_i) = \Phi(x_i’\beta)$$

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(D) Estimasi (MLE)

Log-likelihood:$$\ln L(\beta)=\sum_i [y_i\ln P_i+(1-y_i)\ln(1-P_i)]$$

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(E) Interpretasi

• Koefisien ≠ marginal effect langsung
• Marginal effect:

$$\frac{\partial P}{\partial x_j} = f(x’\beta)\beta_j$$

➡️ Digunakan untuk:

• probabilitas krisis
• default risk
• kebijakan on/off (QE dummy)

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🔹 BAB 13 — Simulation Methods (Monte Carlo & Bootstrap)

(A) Monte Carlo Simulation

Tujuan:

• menguji sifat estimator (bias, variance, consistency)

Data Generating Process (DGP)

$$y_t = \beta_0 + \beta_1 x_t + u_t, \quad u_t \sim N(0,\sigma^2)$$

Langkah:

1. generate $x_t, u_t$
2. hitung $\hat{\beta}$​
3. ulangi $M$ kali

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(B) Bootstrap

Digunakan saat:

• sampel kecil
• distribusi asimtotik lemah

Bootstrap estimator

$$\hat{\theta}^* = f(X^*)$$

Distribusi empiris:$$\hat{\theta}^* \Rightarrow \text{sampling distribution}$$

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(C) Relevansi

• validasi VAR/VECM
• IRF confidence bands
• robust inference krisis

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🔹 BAB 14 — Financial Econometrics Techniques

(A) Event Study

Model Return Abnormal

$$AR_{it} = R_{it} – E(R_{it}|X_t)$$

Cumulative:$$CAR_{i}(\tau_1,\tau_2)=\sum_{t=\tau_1}^{\tau_2} AR_{it}$$

➡️ Digunakan untuk:

• pengumuman QE
• policy shock
• crisis announcement

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(B) Fama–MacBeth Regression

Tahap 1 (Time-series)

$$R_{it} = \alpha_i + \beta_i F_t + \varepsilon_{it}$$

Tahap 2 (Cross-section)

$$\bar{R}_i = \gamma_0 + \gamma_1 \hat{\beta}_i + u_i$$

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(C) Extreme Value Theory (EVT)

Distribusi tail:$$P(X>x) \sim x^{-\alpha}$$

➡️ Risiko ekstrim & krisis keuangan

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(D) Generalized Method of Moments (GMM)

Kondisi momen:$$E[g(Z_t,\theta)] = 0$$

Estimator:$$\hat{\theta}_{GMM} = \arg\min_\theta \left(\bar{g}(\theta)’ W \bar{g}(\theta)\right)$$