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🇮🇩 VERSI BAHASA INDONESIA

1. Kerangka Metodologis Umum

Penelitian ini memandang asset pricing empiris sebagai persoalan estimasi ekspektasi kondisional return, yang secara formal dapat dinyatakan sebagai:$$\mathbb{E}\left(R_{i,t+1} \mid \mathcal{F}_t\right)$$

dengan:

• $R_{i,t+1}$​ adalah excess return saham $i$i pada periode $t+1$,
• $\mathcal{F}_t = \{X_{i,t}, Z_t\}$ adalah himpunan informasi yang tersedia pada waktu $t$,
• $X_{i,t}$ mencerminkan karakteristik spesifik perusahaan,
• $Z_t$​ merepresentasikan kondisi makroekonomi dan pasar agregat.

Berbeda dari pendekatan klasik yang mengasumsikan bentuk linier dan faktor risiko terstruktur, studi ini mengadopsi kerangka supervised learning, di mana fungsi prediksi $f(\cdot)$ dipelajari langsung dari data:$$R_{i,t+1} = f(X_{i,t}, Z_t) + \varepsilon_{i,t+1}$$

tanpa pembatasan awal terhadap linearitas maupun additivitas.

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2. Struktur Data dan Ruang Prediktor

2.1 Vektor Prediktor

Vektor prediktor berdimensi tinggi dibangun sebagai:$$W_{i,t} = \begin{bmatrix} X_{i,t} \\ X_{i,t} \otimes Z_t \\ D_i \end{bmatrix}$$

di mana:

• $X_{i,t} \in \mathbb{R}^{94}$ adalah karakteristik saham,
• $X_{i,t} \otimes Z_t$​ adalah interaksi karakteristik dengan variabel makro,
• $D_i$ adalah dummy industri.

Total dimensi prediktor:$$\dim(W_{i,t}) > 900$$

yang menempatkan studi ini dalam konteks high-dimensional regression.

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3. Spesifikasi Model Matematis

3.1 Model Linear Dasar (Benchmark)

Ordinary Least Squares (OLS)

$$R_{i,t+1} = \alpha + W_{i,t}’\beta + \varepsilon_{i,t+1}$$

OLS berfungsi sebagai benchmark, namun tidak konsisten secara out-of-sample ketika:$$\dim(W_{i,t}) \approx N \quad \text{atau} \quad \dim(W_{i,t}) > N$$

akibat varians estimator yang meningkat tajam.

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3.2 Penalized Linear Regression

Elastic Net

Model diestimasi dengan meminimalkan:$$\min_{\beta} \sum_{i,t} \left(R_{i,t+1} – W_{i,t}’\beta\right)^2 + \lambda_1 \|\beta\|_1 + \lambda_2 \|\beta\|_2^2$$

dengan:

• penalti $\ell_1$mendorong sparsity,
• penalti $\ell_2$ menstabilkan estimasi pada prediktor berkorelasi tinggi.

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3.3 Reduksi Dimensi

Principal Component Regression (PCR)

1. Dekomposisi:

$$W = U \Sigma V’$$

2. Pemilihan $K$K komponen utama:

$$\tilde{W} = U_K \Sigma_K$$

3. Regresi:

$$R_{t+1} = \tilde{W}_t \gamma + \varepsilon_{t+1}$$​

Partial Least Squares (PLS)

PLS memilih vektor bobot $\omega_k$ωk​ yang memaksimalkan:$$\text{Cov}(W\omega_k, R)^2$$

sehingga lebih fokus pada prediktivitas return dibanding variansi prediktor.

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3.4 Model Nonlinier Terbatas

Generalized Linear Model dengan Spline

$$R_{i,t+1} = \alpha + \sum_{j=1}^p g_j(W_{i,t,j}) + \varepsilon_{i,t+1}$$

dengan:

• $g_j(\cdot)$gj​(⋅) adalah fungsi spline nonlinier,
• penalti group LASSO diterapkan untuk seleksi kelompok fungsi.

Keterbatasan utama:$$\frac{\partial^2 R}{\partial W_j \partial W_k} = 0 \quad \forall j \neq k$$

(interaksi tidak dimodelkan).

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3.5 Tree-Based Ensemble Models

Random Forest (RF)

$$\hat{f}(W) = \frac{1}{B} \sum_{b=1}^B T_b(W)$$

dengan:

• $T_b(\cdot)$ adalah decision tree ke-$b$,
• setiap tree dibangun dari bootstrap sample.

RF secara implisit menangkap interaksi:$$W_j \times W_k$$

melalui struktur percabangan.

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Gradient Boosted Trees (GBRT)

Model aditif:$$f_M(W) = \sum_{m=1}^M \nu h_m(W)$$

di mana setiap tree $h_m$hm​ mengaproksimasi negative gradient dari fungsi loss.

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3.6 Neural Networks

Feedforward Neural Network

$$\begin{aligned} h^{(1)} &= \sigma(W^{(1)} W + b^{(1)}) \\ h^{(l)} &= \sigma(W^{(l)} h^{(l-1)} + b^{(l)}) \\ \hat{R}_{i,t+1} &= W^{(L)} h^{(L-1)} + b^{(L)} \end{aligned}$$

dengan:

• $L = 1,\dots,5$ hidden layers,
• $\sigma(\cdot)$ fungsi aktivasi nonlinier.

Estimasi dilakukan melalui:$$\min_{\theta} \sum (R – \hat{R})^2 + \lambda \|\theta\|^2$$

disertai early stopping untuk mencegah overfitting.

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4. Evaluasi dan Implementasi Ekonomi

Model dievaluasi berdasarkan:

• Out-of-sample R2,
• Sharpe ratio portofolio hasil prediksi,
• Strategi long–short decile dan market timing.

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🇬🇧 ENGLISH VERSION

Methodology and Model Specification

Empirical Asset Pricing via Machine Learning: A High-Dimensional Predictive Framework

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1. General Methodological Framework

This study formulates empirical asset pricing as a conditional expectation problem, expressed as:$$\mathbb{E}(R_{i,t+1} \mid \mathcal{F}_t)$$

where:

• $R_{i,t+1}$​ denotes excess stock returns,
• $\mathcal{F}_t = \{X_{i,t}, Z_t\}$ represents the information set,
• $X_{i,t}$ are firm-level characteristics,
• $Z_t$ captures aggregate macroeconomic conditions.

The predictive model is written as:$$R_{i,t+1} = f(X_{i,t}, Z_t) + \varepsilon_{i,t+1}$$

without imposing linearity or additivity ex ante.

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2. High-Dimensional Feature Space

The predictor vector is constructed as:$$W_{i,t} = \begin{bmatrix} X_{i,t} \\ X_{i,t} \otimes Z_t \\ D_i \end{bmatrix}$$

resulting in more than 900 predictive signals, placing the analysis in a high-dimensional regression environment.

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3. Mathematical Model Specifications

3.1 Linear Benchmark Model

Ordinary Least Squares

$$R_{i,t+1} = \alpha + W_{i,t}’\beta + \varepsilon_{i,t+1}$$

OLS serves as a baseline but suffers from poor out-of-sample performance under dimensionality.

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3.2 Penalized Linear Models

Elastic Net

$$\min_{\beta} \sum (R – W\beta)^2 + \lambda_1 \|\beta\|_1 + \lambda_2 \|\beta\|_2^2$$

balancing sparsity and stability.

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3.3 Dimension Reduction

Principal Component Regression

$$W = U\Sigma V’ \quad \Rightarrow \quad R = U_K \Sigma_K \gamma + \varepsilon$$

Partial Least Squares

PLS maximizes:$$\text{Cov}(W\omega, R)^2$$

to extract predictive components.

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3.4 Restricted Nonlinear Models

Spline-Based GLM

$$R_{i,t+1} = \alpha + \sum_j g_j(W_{i,t,j}) + \varepsilon$$

Nonlinear but additive, hence interaction-free.

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3.5 Tree-Based Ensembles

Random Forest

$$\hat{f}(W) = \frac{1}{B} \sum_{b=1}^B T_b(W)$$

Gradient Boosting

$$f_M(W) = \sum_{m=1}^M \nu h_m(W)$$

capturing nonlinear interactions.

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3.6 Neural Networks

$$\hat{R} = f(W;\theta)$$

with multi-layer nonlinear transformations estimated via regularized least squares and early stopping.

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5. Methodological Contribution

The methodology demonstrates that predictive gains in asset pricing arise primarily from nonlinear interactions, not merely from expanding predictor sets.