TVP-VECM

Time-Varying Parameter Vector Error Correction Model

(Full Modeling & Estimation Procedure)

---

A. Konsep Dasar TVP-VECM

Conceptual Foundation of TVP-VECM

🇮🇩 Bahasa Indonesia

TVP-VECM merupakan pengembangan dari model VECM konvensional yang memungkinkan parameter jangka pendek, kecepatan penyesuaian, dan dinamika transmisi kebijakan berubah dari waktu ke waktu. Model ini sangat relevan untuk menganalisis kebijakan moneter non-konvensional seperti Quantitative Easing (QE) yang efektivitasnya tidak bersifat statis, terutama pada periode krisis.

TVP-VECM menggabungkan:

1. Hubungan jangka panjang (cointegration) antarvariabel,
2. Dinamika jangka pendek (error correction),
3. Perubahan struktural dan adaptasi kebijakan lintas waktu.

---

🇬🇧 English

TVP-VECM extends the conventional Vector Error Correction Model (VECM) by allowing short-run coefficients, adjustment speeds, and policy transmission mechanisms to evolve over time. This framework is particularly suitable for analyzing non-conventional monetary policies such as Quantitative Easing (QE), whose effectiveness is inherently dynamic and crisis-dependent.

TVP-VECM integrates:

1. Long-run cointegrating relationships,
2. Short-run error correction dynamics,
3. Time-varying structural and policy adaptation.

---

B. Spesifikasi Variabel dan Sistem

B. Variable Specification and System Setup

🇮🇩

Vektor variabel endogen didefinisikan sebagai:$$\mathbf{y}_t= \begin{bmatrix} BI_t\\ INF_t\\ GDP_t\\ REER_t\\ IHSG_t\\ SUN_t \end{bmatrix}$$

dengan:

• $BI_t$​: suku bunga kebijakan
• $INF_t$​: inflasi
• $GDP_t$​: pertumbuhan output
• $REER_t$​: nilai tukar efektif riil
• $IHSG_t$: indeks saham
• $SUN_t$​: instrumen obligasi pemerintah (kanal QE)

Semua variabel diasumsikan I(1) dan berpotensi memiliki hubungan keseimbangan jangka panjang.

---

🇬🇧

The endogenous variable vector is specified as:$$\mathbf{y}_t= \begin{bmatrix} BI_t\\ INF_t\\ GDP_t\\ REER_t\\ IHSG_t\\ SUN_t \end{bmatrix}$$

where:

• $BI_t$​: policy interest rate
• $INF_t$​: inflation
• $GDP_t$: output growth
• $REER_t$: real effective exchange rate
• $IHSG_t$​: stock market index
• $SUN_t$: government bonds (QE channel)

All variables are assumed to be integrated of order one (I(1)) and potentially cointegrated.

---

C. Model Dasar VECM

C. Baseline VECM Structure

🇮🇩

VECM standar dengan $k$k vektor kointegrasi ditulis sebagai:$$\Delta \mathbf{y}_t = \boldsymbol{\alpha}\boldsymbol{\beta}’\mathbf{y}_{t-1} + \sum_{i=1}^{p-1}\mathbf{\Gamma}_i\Delta \mathbf{y}_{t-i} + \boldsymbol{\varepsilon}_t$$​

di mana:

• $\boldsymbol{\beta}’\mathbf{y}_{t-1}$​: vektor kointegrasi (hubungan jangka panjang),
• $\boldsymbol{\alpha}$: matriks kecepatan penyesuaian,
• $\mathbf{\Gamma}_i$​: dinamika jangka pendek,
• $\boldsymbol{\varepsilon}_t\sim N(0,\boldsymbol{\Sigma})$

---

🇬🇧

The standard VECM with $k$k cointegrating vectors is expressed as:$$\Delta \mathbf{y}_t = \boldsymbol{\alpha}\boldsymbol{\beta}’\mathbf{y}_{t-1} + \sum_{i=1}^{p-1}\mathbf{\Gamma}_i\Delta \mathbf{y}_{t-i} + \boldsymbol{\varepsilon}_t$$

where:

• $\boldsymbol{\beta}’\mathbf{y}_{t-1}$ represents long-run equilibrium relations,
• $\boldsymbol{\alpha}$ is the speed of adjustment matrix,
• $\mathbf{\Gamma}_i$ captures short-run dynamics.

---

D. Spesifikasi TVP-VECM

D. TVP-VECM Specification

🇮🇩

Dalam TVP-VECM, semua parameter utama diizinkan berubah seiring waktu:$$\Delta \mathbf{y}_t = \boldsymbol{\alpha}_t\boldsymbol{\beta}’\mathbf{y}_{t-1} + \sum_{i=1}^{p-1}\mathbf{\Gamma}_{i,t}\Delta \mathbf{y}_{t-i} + \boldsymbol{\varepsilon}_t$$

dengan:

• $\boldsymbol{\beta}$β: tetap (time-invariant) untuk menjaga interpretasi kointegrasi,
• $\boldsymbol{\alpha}_t$αt​: kecepatan penyesuaian time-varying,
• $\mathbf{\Gamma}_{i,t}$Γi,t​: koefisien jangka pendek time-varying.

---

🇬🇧

In the TVP-VECM framework, key parameters are allowed to evolve over time:$$\Delta \mathbf{y}_t = \boldsymbol{\alpha}_t\boldsymbol{\beta}’\mathbf{y}_{t-1} + \sum_{i=1}^{p-1}\mathbf{\Gamma}_{i,t}\Delta \mathbf{y}_{t-i} + \boldsymbol{\varepsilon}_t$$

where:

• $\boldsymbol{\beta}$β is time-invariant to preserve cointegration interpretation,
• $\boldsymbol{\alpha}_t$αt​ is a time-varying adjustment matrix,
• $\mathbf{\Gamma}_{i,t}$Γi,t​ represents time-varying short-run dynamics.

---

E. State-Space Representation

E. State-Space Formulation

🇮🇩

TVP-VECM dapat ditulis dalam bentuk state-space.

Measurement equation:$$\Delta \mathbf{y}_t = \mathbf{Z}_t \boldsymbol{\theta}_t + \boldsymbol{\varepsilon}_t$$

State equation:$$\boldsymbol{\theta}_t = \boldsymbol{\theta}_{t-1} + \mathbf{u}_t$$

dengan:

• $\boldsymbol{\theta}_t = (\boldsymbol{\alpha}_t,\mathbf{\Gamma}_{1,t},\dots)$
• $\mathbf{u}_t \sim N(0,\mathbf{Q})$

---

🇬🇧

TVP-VECM is formulated in a state-space framework.

Measurement equation:$$\Delta \mathbf{y}_t = \mathbf{Z}_t \boldsymbol{\theta}_t + \boldsymbol{\varepsilon}_t$$

State equation:$$\boldsymbol{\theta}_t = \boldsymbol{\theta}_{t-1} + \mathbf{u}_t$$

where:

• $\boldsymbol{\theta}_t$contains time-varying parameters,
• $\mathbf{u}_t$ governs parameter evolution.

---

---

F. Time-Varying Impulse Response Function

G. Time-Varying IRF

🇮🇩

TV-IRF pada waktu $t$t didefinisikan sebagai:$$IRF_t(h)=\frac{\partial \mathbf{y}_{t+h}}{\partial \varepsilon_{j,t}}$$

Memungkinkan analisis:

• perubahan kekuatan transmisi QE,
• perbedaan respons sebelum, saat, dan setelah krisis.

---

🇬🇧

The time-varying IRF at time $t$t is defined as:$$IRF_t(h)=\frac{\partial \mathbf{y}_{t+h}}{\partial \varepsilon_{j,t}}$$​​

This enables analysis of:

• evolving QE transmission strength,
• differential responses across crisis and non-crisis periods.

---

H. Relevansi dalam Kerangka New Monetary Trinity

H. Relevance to the New Monetary Trinity

🇮🇩

TVP-VECM memungkinkan pemetaan adaptasi kebijakan moneter terhadap:

• stabilitas inflasi,
• stabilitas nilai tukar,
• stabilitas keuangan,

secara dinamis dan empiris.

🇬🇧

TVP-VECM enables a dynamic empirical mapping of monetary policy adaptation across:

• price stability,
• exchange rate stability,
• financial stability.

---

I. Inti Metodologis

I. Methodological Insight

🇮🇩

TVP-VECM merupakan pendekatan paling konsisten untuk mengevaluasi efektivitas QE dalam konteks krisis, tekanan politik, dan perubahan rezim kebijakan.

🇬🇧

TVP-VECM provides the most coherent framework for evaluating QE effectiveness under crisis, political pressure, and regime shifts.