GDP Forecasting with MIDAS Regression

Master 2 272 Quantitative Finance - Université Paris Dauphine-PSL

Academic replication and extension of Andreou, Ghysels, Kourtellos (2013):
"Should Macroeconomic Forecasters Use Daily Financial Data and How?"
Journal of Business & Economic Statistics, 31(2), 240-251.

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Authors

Théo Verdelhan --- Léo Renault --- Arthur Le Net --- Nicolas Annon

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Project Overview

Objective

This project implements and extends the ADL-MIDAS (Autoregressive Distributed Lag - Mixed Data Sampling) framework for forecasting quarterly US GDP growth using high-frequency daily financial data. The MIDAS approach addresses a fundamental challenge in macroeconomic forecasting: how to efficiently exploit the information content of variables sampled at higher frequencies than the target variable.

Academic Context

This project was developed as part of the Quantitative Management course at Paris Dauphine-PSL University. The assignment required:

1. Full replication of the original research paper using our own data sources
2. Critical analysis of the methodology and results
3. Extension of the paper with novel contributions

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Methodology

The MIDAS Model

The core ADL-MIDAS specification estimates quarterly GDP growth as:

$$y_{t+h}^Q = \alpha + \sum_{k=1}^{p} \rho_k \cdot y_{t-k+1}^Q + \beta \cdot \sum_{j=0}^{m-1} B(j;\theta) \cdot x_{t-j}^D + \varepsilon_t$$

Where:

• $y^Q$: Quarterly GDP growth rate
• $x^D$: Daily financial variables (or PCA factors)
• $B(j;\theta) = \frac{\exp(\theta_1 j + \theta_2 j^2)}{\sum_{k} \exp(\theta_1 k + \theta_2 k^2)}$: Exponential Almon polynomial weights
• $h$: Forecast horizon (quarters ahead)
• $m$: Number of daily observations used (typically 63 ≈ 1 quarter)
• $p$: Autoregressive lag order (selected via AIC)

Key Features of Our Implementation

1. Exponential Almon Weighting: Parsimonious aggregation of daily data with only 1-2 parameters
2. Recursive Out-of-Sample Forecasting: Expanding window estimation to avoid look-ahead bias
3. AIC-Based Lag Selection: Automatic selection of AR lag order
4. MSFE-Weighted Forecast Combination: Combining multiple predictors using discounted forecast errors

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Data

Sources

We use Bloomberg Terminal data covering multiple asset classes:

Asset Class	Examples	Frequency
Equity Indices	S&P 500, NASDAQ, Russell 2000	Daily
Fixed Income	2Y/10Y Treasury, Credit Spreads	Daily
Commodities	WTI Crude Oil, Gold, Copper	Daily
Foreign Exchange	EUR/USD, USD/JPY, DXY	Daily
Macro Indicators	CFNAI, ADS, ISM PMI	Monthly/Daily
GDP Growth	US Real GDP QoQ	Quarterly

Sample Periods (Following the Paper)

Sample	Data Period	Training	Out-of-Sample
Long Sample	1986-01-01 → 2008-12-31	1986Q1-2000Q4	2001Q1-2008Q4
Short Sample	1999-01-01 → 2008-12-31	1999Q1-2005Q4	2006Q1-2008Q4
Extended (Our Extension)	2020-01-01 → 2025-12-31	2020Q1-2023Q4	2024Q1-2025Q4

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Project Structure

QUANTITATIVE_MIDAS_MODEL_REPLICATION/
├── main.ipynb                 # Main analysis notebook (run this)
├── requirements.txt           # Python dependencies
├── setup.sh / setup.bat       # Environment setup scripts
├── README.md
│
├── DATAS/
│   └── bloomberg_all_tickers.csv   # Bloomberg financial data
│
├── FUNCTIONS/
│   ├── __init__.py
│   ├── data_loader.py         # Bloomberg CSV parsing with date handling
│   ├── transformations.py     # Log-returns, first differences, PCA
│   ├── midas.py               # MidasSpec, MidasModel, MidasModelWithLeads
│   ├── benchmarks.py          # AR(1), Random Walk, MSFE combination
│   ├── evaluation.py          # RMSFE computation, sub-period analysis
│   ├── analysis.py            # Sample period definitions, horizon tests
│   ├── table_replication_v2.py # Paper Table 1-5 replication
│   ├── plots.py               # Visualization functions
│   └── data_diagnostics.py    # Coverage checks, stationarity tests
│
├── PLOT ANALYSIS/             # Generated figures and tables
│
└── TOOLS/
    ├── install_requirement.py # Dependency installer
    └── convert_excel_to_csv.ipynb

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Replication Results

Table 1: RMSFE Comparisons — No Leads (Ratios to RW)

Model	Long h=1	Long h=4	Short h=1	Short h=4
RW (absolute RMSFE)	2.69	3.18	3.46	4.66
AR	1.01	0.91	1.13	1.00
FAR (CFNAI)	0.91	0.90	0.94	0.98
ADL (5 DF)	1.09	1.12	1.20	1.14
ADL-MIDAS (5 DF)	1.11	1.11	1.24	1.13
FADL (CFNAI, 5 DF)	0.96	1.12	1.00	1.14
FADL-MIDAS (CFNAI, 5 DF)	1.07	0.86	1.02	1.00

Values < 1 indicate improvement over Random Walk

Table 3: RMSFE Comparisons — With Leads (Ratios to RW)

Model	Long h=1	Long h=4	Short h=1	Short h=4
ADL-MIDAS (J_D=2)	0.97	0.87	0.94	0.89
FADL-MIDAS (J_D=2)	0.77	0.73	0.70	0.62
FADL-MIDAS (J_M=1, J_D=2)	0.93	0.81	0.86	0.82
FAR (J_M=1)	0.87	0.73	0.84	0.72
FADL (J_M=1)	0.90	0.88	0.92	0.86

Table 5: ADS Index Comparisons (Ratios to RW)

Model	Long h=1	Long h=4	Short h=1	Short h=4
ADL-MIDAS (J_D,ADS=2)	0.57	0.48	0.56	0.42
FADL-MIDAS (J_M=1, J_D,ADS=2)	0.60	0.52	0.60	0.46

Key Findings from Replication

1. Leads are crucial: Models with daily leads (J_D=2) substantially outperform no-lead specifications
2. ADS dominates: The Aruoba-Diebold-Scotti daily macro index achieves the best performance (40-50% improvement vs RW)
3. CFNAI adds value: Factor AR with CFNAI beats pure AR benchmark
4. FADL-MIDAS with leads is the best financial-factor model (0.70-0.77 vs RW)
5. Short sample challenges: Pure financial factors underperform on shorter estimation windows without leads

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Extension: Two-β MIDAS Model (Lags vs Leads)

Motivation

In the standard MIDAS specification (Andreou et al., 2013), a single β parameter determines the weighting applied to the entire daily block—both lagged data and nowcast leads. This imposes the same weighting dynamics on historical information and intra-quarter data.

Our contribution: We propose a Two-β MIDAS extension that separates this parameter into:

• β_lag: Weights applied to the lag block (past data, m = 63 days ≈ 1 quarter)
• β_lead: Weights applied to the lead block (nowcast data, m_L ≈ 42 days ≈ 2 months)

This relaxes a potentially restrictive constraint while remaining parsimonious (only one additional parameter).

Model Specification

The extended model at horizon h = 1 is:

$$y_{t+1} = \alpha + \rho y_t + \beta_{lag} \sum_{k=1}^{m} B(k;\theta_{lag}) x_{t-k} + \beta_{lead} \sum_{j=1}^{m_L} B(j;\theta_{lead}) x_{t+j} + \varepsilon_{t+1}$$

Where $B(k;\theta) = \frac{\exp(\theta k)}{\sum_\ell \exp(\theta \ell)}$ is the normalized exponential Almon weighting function.

Economic Intuition

The leads correspond to the beginning of the target quarter—the most recent data available at forecast time. Under a single-β constraint, this valuable nowcast information may be under-weighted if the weighting shape is primarily driven by the lag block. Allowing β_lag ≠ β_lead lets the model treat historical and nowcast information differently, which is more economically coherent for nowcasting exercises.

Results

Comparison: Two-β vs Single-β (OOS 2024-2025)

Model	RMSFE	Rel. to RW	vs RW
Two-β MIDAS (J_D = 2)	2.727	1.288	-28.8%
Single-β MIDAS (J_D = 2)	3.240	1.530	-53.0%

→ Two-β improves upon Single-β by 15.8%

Application to Paper Samples (Long & Short)

Sample	Model	RMSFE	Rel. to RW	Improvement vs Single-β
Long	Two-β MIDAS	2.397	1.132	+26.0%
Short	Two-β MIDAS	2.827	1.335	+12.8%

Estimated Parameters (Average across OOS forecasts)

Parameter	Value	Interpretation
θ_lag	0.0266	Decay rate for historical weights
θ_lead	0.0176	Decay rate for nowcast weights
β_lag	2.5252	Scale for lag contribution
β_lead	3.6708	Scale for lead contribution

The distinct estimated dynamics confirm the value of separating lag and lead effects.

Complete Model Ranking (OOS 2024-2025)

Rank	Model	RMSFE	vs RW
1	ADL(flat)	0.951	+55.1%
2	FAR(CFNAI)	1.802	+14.9%
3	AR	1.884	+11.0%
4	FADL(J_M = 1)	2.114	+0.2%
5	RW (baseline)	2.117	—
6	FAR(J_M = 1)	2.261	-6.8%
7	⭐ Two-β MIDAS (J_D = 2)	2.727	-28.8%
8	ADL-MIDAS(J_D = 2)	3.240	-53.0%
9	FADL-MIDAS(J_M=1, J_D=2)	3.653	-72.5%
10	FADL-MIDAS	3.732	-76.3%

⭐ indicates our novel contribution

Discussion

The Two-β extension consistently improves over the standard single-β MIDAS across all samples. The separate weighting profiles allow the model to:

1. Apply a regular decay on the lag block (classical fading memory)
2. Treat nowcast information with its own dynamics

Limitations: The extension does not beat RW on the recent 2024-2025 period. This is likely due to:

• Short OOS window (8 quarters), sensitive to individual large errors
• Post-COVID regime instability in the factor-GDP relationship

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Visualizations

The notebook generates comprehensive visualizations saved to PLOT ANALYSIS/:

• Data Exploration: Asset class coverage, stationarity tests, volatility analysis
• Factor Analysis: PCA loadings, variance explained, GDP correlations
• Forecast Comparison: MIDAS vs AR(1) vs Random Walk time series
• MIDAS Weights: Exponential Almon weight decay patterns
• Sub-Period Performance: Crisis vs normal periods breakdown
• Recent Period: 2024-2025 nowcasting results

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Installation

Option 1: Automatic Setup (Recommended)

macOS/Linux:

chmod +x setup.sh
./setup.sh

Option 2: Manual Setup

# Clone the repository
git clone https://github.com/[your-username]/MIDAS-GDP-Forecasting.git
cd MIDAS-GDP-Forecasting

# Create virtual environment
python -m venv .venv

# Activate (macOS/Linux)
source .venv/bin/activate

# Install dependencies
pip install -r requirements.txt

Dependencies

• numpy, pandas: Data manipulation
• scipy, statsmodels: Econometric estimation
• scikit-learn: PCA, preprocessing
• matplotlib: Visualization
• jupyter: Notebook execution

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Usage

1. Activate environment:

   source .venv/bin/activate  # macOS/Linux

2. Run the notebook:

   • VS Code: Open main.ipynb, select .venv as kernel
   • Jupyter: jupyter notebook main.ipynb

3. Execute all cells — outputs are saved to PLOT ANALYSIS/

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References

Primary Reference

Andreou, E., Ghysels, E., & Kourtellos, A. (2013). Should Macroeconomic Forecasters Use Daily Financial Data and How?

Additional References

• Ghysels, E., Santa-Clara, P., & Valkanov, R. (2004). The MIDAS Touch: Mixed Data Sampling Regression Models
• Ghysels, E., Sinko, A., & Valkanov, R. (2007). MIDAS Regressions: Further Results and New Directions
• Stock, J.H., & Watson, M.W. (2002). Macroeconomic Forecasting Using Diffusion Indexes

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🎓 Academic Information

Course: M2 272 - Quantitative Finance and Financial Engineering
Institution: Université Paris Dauphine-PSL
Academic Year: 2025-2026

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📄 License

This project is for academic purposes. The code is provided as-is for educational use.

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📧 Contact

For questions about this project:

• Théo Verdelhan - theo.verdelhan@dauphine.eu

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Developed at Université Paris Dauphine-PSL