TFP Decomposition

Quick Reference

Class: panelbox.frontier.utils.decomposition.TFPDecomposition Import: from panelbox.frontier.utils.decomposition import TFPDecomposition Stata equivalent: Custom post-estimation R equivalent: Custom post-estimation

Overview

Total Factor Productivity (TFP) measures the portion of output growth not explained by input growth. It captures technological progress, efficiency improvements, and scale effects. SFA provides a natural framework for TFP decomposition because it explicitly estimates technical efficiency, allowing researchers to separate frontier shifts (innovation) from catch-up effects (efficiency change).

PanelBox implements the Kumbhakar & Lovell (2000) decomposition framework, which decomposes TFP growth into three components:

\[\Delta \ln TFP = \Delta TC + \Delta TE + \Delta SE\]

where \(\Delta TC\) is technical change (frontier shift), \(\Delta TE\) is efficiency change (catch-up), and \(\Delta SE\) is scale efficiency change.

Quick Example

from panelbox.frontier import StochasticFrontier
from panelbox.frontier.utils.decomposition import TFPDecomposition

# Fit a panel SFA model
model = StochasticFrontier(
    data=panel_df,
    depvar="log_output",
    exog=["log_labor", "log_capital"],
    entity="firm_id",
    time="year",
    frontier="production",
    dist="half_normal",
    model_type="bc92",
)
result = model.fit()

# Decompose TFP growth
tfp = TFPDecomposition(result, periods=(2010, 2020))
decomp = tfp.decompose()
print(decomp[["entity", "delta_tfp", "delta_tc", "delta_te", "delta_se"]])

When to Use

• You want to understand the sources of productivity growth (or decline)
• You need to distinguish frontier innovation from catch-up effects
• You want entity-level decomposition showing which firms improved and why
• Policy analysis requires identifying whether growth comes from technology, efficiency, or scale

Key Assumptions

• Requires a fitted panel SFA model with entity and time identifiers
• The decomposition assumes a Cobb-Douglas functional form (coefficients are elasticities)
• Both comparison periods must exist in the data
• Entities must appear in both periods for decomposition

Detailed Guide

Components of TFP Growth

1. Technical Change (\(\Delta TC\)): Shift of the production frontier over time. Positive \(\Delta TC\) indicates that the best-practice technology has improved (innovation, adoption of new methods).

2. Technical Efficiency Change (\(\Delta TE\)): Movement of a firm relative to the frontier. Positive \(\Delta TE\) means the firm is catching up to best practice.

\[\Delta TE = \ln(TE_{t_2}) - \ln(TE_{t_1})\]

3. Scale Efficiency Change (\(\Delta SE\)): Gains or losses from changing the scale of operation. Depends on returns to scale (RTS):

• \(RTS > 1\) (IRS): Expansion increases scale efficiency
• \(RTS = 1\) (CRS): No scale effects (\(\Delta SE = 0\))
• \(RTS < 1\) (DRS): Expansion decreases scale efficiency

\[\Delta SE = (RTS - 1) \cdot \Delta(\text{weighted inputs})\]

TFPDecomposition API

from panelbox.frontier.utils.decomposition import TFPDecomposition

# Initialize with result and periods
tfp = TFPDecomposition(result, periods=(2010, 2020))

# Entity-level decomposition
decomp = tfp.decompose()
# Returns DataFrame with columns:
#   entity, delta_tfp, delta_tc, delta_te, delta_se, rts, verification

# Aggregate statistics
agg = tfp.aggregate_decomposition()
# Returns dict with keys:
#   mean_delta_tfp, mean_delta_tc, mean_delta_te, mean_delta_se,
#   pct_from_tc, pct_from_te, pct_from_se, std_delta_tfp, n_firms

# Text summary
print(tfp.summary())

# Visualization
fig = tfp.plot_decomposition(kind="bar", top_n=20)
fig = tfp.plot_decomposition(kind="scatter")

Complete Example

from panelbox.frontier import StochasticFrontier
from panelbox.frontier.utils.decomposition import TFPDecomposition

# Estimate panel SFA model
model = StochasticFrontier(
    data=panel_df,
    depvar="log_output",
    exog=["log_labor", "log_capital"],
    entity="firm_id",
    time="year",
    frontier="production",
    dist="half_normal",
    model_type="bc92",
)
result = model.fit()

# Decompose TFP between first and last period
tfp = TFPDecomposition(result, periods=(2010, 2020))

# Entity-level decomposition
decomp = tfp.decompose()
print(decomp.describe())

# Aggregate summary
agg = tfp.aggregate_decomposition()
print(f"\nAverage TFP growth: {agg['mean_delta_tfp']:.4f}")
print(f"  Technical change:   {agg['mean_delta_tc']:.4f} ({agg['pct_from_tc']:.1f}%)")
print(f"  Efficiency change:  {agg['mean_delta_te']:.4f} ({agg['pct_from_te']:.1f}%)")
print(f"  Scale effects:      {agg['mean_delta_se']:.4f} ({agg['pct_from_se']:.1f}%)")

# Print full summary
print(tfp.summary())

# Visualizations
fig_bar = tfp.plot_decomposition(kind="bar", top_n=15)
fig_scatter = tfp.plot_decomposition(kind="scatter")

Four-Component TFP

The FourComponentSFA model provides TFP decomposition with separate persistent and transient efficiency contributions:

from panelbox.frontier.advanced import FourComponentSFA

model = FourComponentSFA(
    data=panel_df,
    depvar="log_output",
    exog=["log_labor", "log_capital"],
    entity="firm_id",
    time="year",
    frontier_type="production",
)
result = model.fit()

# TFP decomposition via FourComponentResult
tfp = result.tfp_decomposition(periods=(2010, 2020))
decomp = tfp.decompose()
print(tfp.summary())

Related Analysis Methods

Returns to Scale

# Test H0: CRS (sum of elasticities = 1)
rts = result.returns_to_scale_test(
    input_vars=["log_labor", "log_capital"],
    alpha=0.05,
)
print(f"RTS = {rts['rts']:.4f} ({rts['conclusion']})")

Elasticities

# Cobb-Douglas: constant elasticities
elas = result.elasticities(input_vars=["log_labor", "log_capital"])
print(elas)

# Translog: observation-varying elasticities
elas_tl = result.elasticities(translog=True, translog_vars=["ln_K", "ln_L"])
print(elas_tl.describe())

Marginal Effects on Inefficiency

For BC95 models with inefficiency determinants:

# Effect of Z variables on mean inefficiency
me = result.marginal_effects(method="location")
print(me)

Configuration Options

TFPDecomposition Parameters

Parameter	Type	Default	Description
result	SFResult	required	Fitted SFA result object
periods	tuple[int, int]	None	Comparison periods \((t_1, t_2)\); uses first/last if None

Key Methods

Method	Returns	Description
decompose()	DataFrame	Entity-level decomposition
aggregate_decomposition()	dict	Mean decomposition with percentages
plot_decomposition(kind, top_n)	Figure	Bar chart or scatter plot
summary()	str	Formatted text summary

Aggregate Decomposition Keys

Key	Description
mean_delta_tfp	Average TFP growth across firms
mean_delta_tc	Average technical change
mean_delta_te	Average efficiency change
mean_delta_se	Average scale effects
pct_from_tc	Percentage of TFP from technical change
pct_from_te	Percentage of TFP from efficiency change
pct_from_se	Percentage of TFP from scale effects
std_delta_tfp	Standard deviation of TFP growth
n_firms	Number of firms in decomposition

Tutorials

Tutorial	Description	Link
TFP Analysis	Productivity decomposition with SFA

See Also

• Production and Cost Frontiers -- SFA fundamentals and elasticities
• Panel SFA Models -- Panel models for TFP analysis
• Four-Component SFA -- Persistent vs transient efficiency for TFP
• SFA Diagnostics -- Returns to scale tests

References

• Kumbhakar, S. C., & Lovell, C. A. K. (2000). Stochastic Frontier Analysis. Cambridge University Press. Chapter 7.
• Fare, R., Grosskopf, S., Norris, M., & Zhang, Z. (1994). Productivity growth, technical progress, and efficiency change in industrialized countries. American Economic Review, 84(1), 66-83.
• Orea, L. (2002). Parametric decomposition of a generalized Malmquist productivity index. Journal of Productivity Analysis, 18(1), 5-22.
• Kumbhakar, S. C., Lien, G., & Hardaker, J. B. (2014). Technical efficiency in competing panel data models: a study of Norwegian grain farming. Journal of Productivity Analysis, 41(2), 321-337.