Are More Data Always Better for Factor Analysis?

Methodological Notes on Factor Modeling, Estimation, and Forecasting

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1. Conceptual Framework: Approximate Factor Model (AFM)

The paper is built on the approximate factor model, which is widely used in macroeconomics to summarize large datasets using a small number of latent factors.

Model specification

For each variable $i = 1, \dots, N$ and time period $t = 1, \dots, T$ $$X_{it} = \lambda_i^{0\prime} F_t^0 + e_{it}$$

or equivalently,$$X_{it} = \chi_{it} + e_{it}$$

Interpretation of symbols

• $X_{it}$ ​: observed macroeconomic variable $i$ at time $t$
• $F_t^0$ ​: $r \times 1$ vector of unobserved common factors
• $\lambda_i^0$ ​: $r \times 1$ vector of factor loadings for variable $i$
• $\chi_{it} = \lambda_i^{0\prime} F_t^0$ ​: common component shared across variables
• $e_{it}$ ​: idiosyncratic component, specific to variable $i$i

Key modeling assumption

Unlike strict factor models, the idiosyncratic errors $e_{it}$ ​:

• may be heteroskedastic,
• may exhibit weak cross-sectional correlation,
• and may be serially correlated.

This flexibility motivates the term approximate factor model.

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2. Population Covariance Structure and Identification

Let:

• $X_t = (X_{1t}, \dots, X_{Nt})’$
• $\Sigma_X = \text{Cov}(X_t)$

Then:$$\Sigma_X = \Sigma_\chi + \Omega$$

where:

• $\Sigma_\chi = \Lambda \Sigma_F \Lambda’$ is the covariance of the common component
• $\Omega = \text{Cov}(e_t)$ is the idiosyncratic covariance matrix

Identification logic

• The matrix $\Sigma_\chi$ has rank r
• Its eigenvalues diverge with N
• Therefore, the first $r$ eigenvectors of $\Sigma_X$ span the factor space

This property underpins estimation via Principal Components (PC).

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3. Factor Estimation via Principal Components

Sample covariance

Let:$$\widehat{\Sigma}_X = \frac{1}{T} \sum_{t=1}^T X_t X_t’$$

Denote:

• $v_j$ eigenvector associated with the $j$ -th largest eigenvalue of $\widehat{\Sigma}_X$

Estimated factors

$$\widehat{F}_{t,N}^{(j)} = \sqrt{\frac{1}{N}} \sum_{i=1}^N X_{it} v_{ij}$$

Stacking the first $r$r components:$$\widehat{F}_{t,N} = (\widehat{F}_{t,N}^{(1)}, \dots, \widehat{F}_{t,N}^{(r)})’$$

Estimated loadings

$$\widehat{\lambda}_{ij} = \sqrt{N} \, v_{ij}$$

Important implication

PC implicitly assumes that:

• idiosyncratic errors are weakly correlated
• their covariance does not dominate the signal from common factors

When these assumptions fail, adding more variables may reduce factor quality.

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4. Forecasting Framework: Diffusion Index Model

The paper evaluates factor quality using out-of-sample forecasting.

Baseline AR model

$$\widehat{y}_{t+1|t} = \widehat{\alpha}_0 + \sum_{j=1}^p \widehat{\gamma}_j y_{t-j+1}$$

Factor-augmented forecast

If true factors were observable:$$\widehat{y}_{t+1|t} = \widehat{\beta}_0 + \widehat{\beta}_1′ F_t^0 + \sum_{j=1}^p \widehat{\gamma}_j y_{t-j+1}$$

In practice, factors are replaced by estimates:$$\widehat{y}_{t+1|t} = \widehat{\beta}_0 + \widehat{\beta}_1′ \widehat{F}_{t,N} + \sum_{j=1}^p \widehat{\gamma}_j y_{t-j+1}$$

Interpretation

Forecast accuracy now depends on:

• the number of variables $N$ ,
• the quality of factor estimation,
• the structure of idiosyncratic errors.

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5. Monte Carlo Design: When More Data Hurt

Data-generating process

$$X_{it} = \sum_{m=1}^r \lambda_{im} F_{mt} + e_{it}$$

Target variable:$$y_{t+1} = \sum_{m=1}^r \beta_m F_{mt} + \varepsilon_{t+1}$$

Error structure (key innovation)

Variables are divided into three groups:

1. Low-noise variables

$$e_{it} = \sigma_1 u_{it}$$

2. High-noise variables

$$e_{it} = \sigma_2 u_{it}, \quad \sigma_2^2 \gg \sigma_1^2$$

3. Cross-correlated variables

$$e_{it} = \sigma_3 \left(u_{it} + \sum_{j=1}^C \rho_{ij} u_{jt} \right)$$

Key insight

Adding variables from groups (2) or (3):

• inflates idiosyncratic covariance,
• distorts eigenstructure,
• and reduces forecasting performance.

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6. Oversampling and Factor Dominance

When one factor is over-represented in the dataset:

• it dominates the principal components,
• other relevant factors become poorly estimated,
• forecasts targeting those factors deteriorate.

This phenomenon is labeled oversampling bias.

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7. Weighted Principal Components

To address these issues, the paper proposes weighted PC estimation.

Standard PC objective

$$V(k) = \frac{1}{NT} \sum_{i=1}^N \sum_{t=1}^T e_{it}^2$$

Weighted objective

$$W(k) = \frac{1}{NT} \sum_{i=1}^N w_{iT} \sum_{t=1}^T e_{it}^2$$

where:

• $w_{iT}$ ​: weight reflecting informativeness or contamination of series $i$

Practical rules

• down-weight variables with high residual variance,
• drop variables with extreme residual correlations,
• group variables by economic blocks (real, nominal, volatile).

Empirically, smaller and cleaner datasets outperform large noisy panels.

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8. Methodological Takeaways

This paper establishes that:

1. Factor estimation quality depends on data structure, not data size.
2. Adding variables may:
   • weaken common components,
   • amplify correlated noise,
   • bias factor space estimation.
3. Intelligent selection and weighting dominate brute-force data expansion.

More data are not always better data.