Direct, Indirect, and Total Effects

Quick Reference

Attribute: results.spillover_effects Returns: dict with keys 'direct', 'indirect', 'total' Relevant models: SAR, SDM, GNS (models with spatially lagged \(y\)) Not applicable: SEM (coefficients are directly interpretable)

Overview

In spatial models that include a spatially lagged dependent variable (\(Wy\)), the estimated coefficients \(\beta\) are not the marginal effects of the covariates. This is one of the most common mistakes in applied spatial econometrics.

The reason is the spatial multiplier: because \(y_i\) depends on \(y_j\) (through \(Wy\)), and \(y_j\) in turn depends on \(y_i\) (through \(Wy\) again), there is a feedback loop. A change in \(x_i\) affects \(y_i\) directly, which then affects neighbors' \(y_j\), which feeds back to \(y_i\), and so on. The total effect is larger than \(\beta\) because of this spatial feedback.

LeSage and Pace (2009) formalized this through the partial derivative interpretation, decomposing effects into three components:

• Direct effect: the impact of changing \(x_i\) on \(y_i\) (own-unit, including feedback)
• Indirect effect: the impact of changing \(x_i\) on \(y_j\) for \(j \neq i\) (spillover to others)
• Total effect: Direct + Indirect

The Mathematics

SAR Model

For the SAR model \(y = \rho Wy + X\beta + \varepsilon\), the reduced form is:

\[y = (I - \rho W)^{-1} X\beta + (I - \rho W)^{-1} \varepsilon\]

The partial derivative of \(y\) with respect to the \(k\)-th variable \(x_k\) across all units is:

\[\frac{\partial y}{\partial x_k'} = (I - \rho W)^{-1} I_N \beta_k = S_k(\rho)\]

This is an \(N \times N\) matrix. The \((i, j)\) element gives the effect of a change in \(x_{jk}\) on \(y_i\).

• Direct effect of \(x_k\) = average of the diagonal of \(S_k(\rho)\):

\[\text{Direct}_k = \frac{1}{N} \text{tr}\left[(I - \rho W)^{-1}\right] \beta_k\]

• Total effect of \(x_k\) = average row (or column) sum of \(S_k(\rho)\):

\[\text{Total}_k = \frac{1}{N} \mathbf{1}'(I - \rho W)^{-1}\mathbf{1} \cdot \beta_k\]

• Indirect effect of \(x_k\) = Total - Direct:

\[\text{Indirect}_k = \text{Total}_k - \text{Direct}_k\]

SDM Model

For the SDM \(y = \rho Wy + X\beta + WX\theta + \varepsilon\), the reduced form is:

\[y = (I - \rho W)^{-1}(X\beta + WX\theta) + (I - \rho W)^{-1}\varepsilon\]

The partial derivative matrix for the \(k\)-th variable is:

\[S_k(\rho) = (I - \rho W)^{-1}(I_N \beta_k + W\theta_k)\]

Now the effect depends on both \(\beta_k\) and \(\theta_k\), as well as \(\rho\). This is why reading \(\beta\) as "direct effect" and \(\theta\) as "indirect effect" is wrong.

SEM Model

For the SEM \(y = X\beta + u\), \(u = \lambda Wu + \varepsilon\):

\[\frac{\partial y}{\partial x_k'} = I_N \beta_k\]

The matrix is diagonal. There are no indirect effects:

• Direct effect = \(\beta_k\)
• Indirect effect = 0
• Total effect = \(\beta_k\)

This is a major advantage of the SEM: coefficients are directly interpretable.

Comparison Table

Model	Direct Effect	Indirect Effect	Total Effect	Notes
OLS	\(= \beta_k\)	None	\(= \beta_k\)	No spatial component
SAR	\(\neq \beta_k\)	Yes	Yes	Both depend on \(\rho\)
SEM	\(= \beta_k\)	0	\(= \beta_k\)	No spillovers
SDM	\(\neq \beta_k\)	Yes	Yes	Depends on \(\rho\), \(\beta_k\), \(\theta_k\)
SLX	\(= \beta_k\)	\(= \theta_k\)	\(= \beta_k + \theta_k\)	No feedback loop
GNS	\(\neq \beta_k\)	Yes	Yes	Most complex

Common Mistake

In the SDM, \(\beta\) is NOT the direct effect and \(\theta\) is NOT the indirect effect. The actual effects must be computed from the spatial multiplier matrix \((I - \rho W)^{-1}\).

Computing Effects in PanelBox

From SAR Results

from panelbox.models.spatial import SpatialLag

model = SpatialLag("y ~ x1 + x2", data, "region", "year", W=W)
results = model.fit(effects='fixed', method='qml')

# Access pre-computed spillover effects
effects = results.spillover_effects

# Direct effects (including spatial feedback)
print("Direct effects:")
for var, val in effects['direct'].items():
    print(f"  {var}: {val:.4f}")

# Indirect effects (spillovers)
print("\nIndirect effects:")
for var, val in effects['indirect'].items():
    print(f"  {var}: {val:.4f}")

# Total effects
print("\nTotal effects:")
for var, val in effects['total'].items():
    print(f"  {var}: {val:.4f}")

From SDM Results

from panelbox.models.spatial import SpatialDurbin

model = SpatialDurbin("y ~ x1 + x2", data, "region", "year", W=W)
results = model.fit(method='qml', effects='fixed')

# Effects decomposition
effects = results.spillover_effects

# Compare raw coefficients vs proper effects
print("Raw coefficients (NOT marginal effects):")
print(f"  beta_x1  = {results.params['x1']:.4f}")
print(f"  theta_x1 = {results.params['W_x1']:.4f}")
print(f"  rho      = {results.rho:.4f}")

print("\nProper marginal effects:")
print(f"  Direct(x1)   = {effects['direct']['x1']:.4f}")
print(f"  Indirect(x1) = {effects['indirect']['x1']:.4f}")
print(f"  Total(x1)    = {effects['total']['x1']:.4f}")

Interpreting the Results

Consider a regional economics application where \(y\) is GDP growth and \(x_1\) is infrastructure investment:

Direct effect of x1:   0.45
Indirect effect of x1: 0.18
Total effect of x1:    0.63

Interpretation:

• A 1-unit increase in infrastructure investment in region \(i\) increases region \(i\)'s GDP growth by 0.45 (direct effect)
• This investment also raises GDP growth across all other regions by a total of 0.18 on average (indirect effect / spillover)
• The total impact on the economy is 0.63 (0.45 direct + 0.18 spillover)

The Spatial Multiplier

For SAR

The spatial multiplier for the mean effect is:

\[\text{Multiplier} = \frac{1}{1 - \rho}\]

For \(\rho = 0.3\): multiplier \(= 1/(1 - 0.3) = 1.43\)

This means the total effect is 43% larger than the direct coefficient \(\beta\) due to spatial feedback.

Multiplier Decomposition

\(\rho\)	Multiplier	Direct/Total Ratio	Indirect Share
0.0	1.00	100%	0%
0.1	1.11	~95%	~5%
0.3	1.43	~80%	~20%
0.5	2.00	~65%	~35%
0.7	3.33	~50%	~50%
0.9	10.00	~30%	~70%

As \(\rho\) increases, a larger share of the total effect comes from spatial spillovers.

Standard Errors for Effects

Standard errors for direct, indirect, and total effects are typically computed using one of two methods:

1. Delta method: analytical approximation using the gradient of the effect with respect to parameters
2. Simulation: draw from the asymptotic distribution of \((\hat{\rho}, \hat{\beta}, \hat{\theta})\) and compute effects for each draw

Both methods yield confidence intervals and p-values for each effect component.

Practical Guidance

Reporting Effects

When publishing results from SAR or SDM models, always report the effect decomposition, not just the raw coefficients. A standard table should include:

Variable    | Direct  | Indirect | Total
------------|---------|----------|-------
x1          |  0.450  |  0.182   | 0.632
            | (0.052) | (0.041)  | (0.067)
x2          |  0.123  |  0.050   | 0.173
            | (0.031) | (0.019)  | (0.038)

Common Pitfalls

1. Interpreting SDM \(\beta\) as direct effect: The direct effect in SDM depends on \(\rho\), \(\beta\), and \(\theta\) through the spatial multiplier. Always use results.spillover_effects.

2. Ignoring feedback in SAR: Even in the SAR model, the direct effect is not \(\beta\) but \(\beta\) times the average diagonal of \((I - \rho W)^{-1}\). The feedback through \(W\) makes the direct effect slightly larger than \(\beta\).

3. Comparing effects across models: Direct effects from different model specifications (SAR vs SDM) are not directly comparable because they are computed differently. Compare total effects if the goal is to assess overall impact.

4. Applying to SEM: The SEM has no indirect effects — do not compute or report them. The coefficients are directly interpretable.

Tutorials

Tutorial	Description	Links
Spatial Econometrics	Effect decomposition examples

See Also

• Spatial Lag (SAR) — Model with \(\rho Wy\)
• Spatial Error (SEM) — Model with no indirect effects
• Spatial Durbin (SDM) — Most common model needing effect decomposition
• General Nesting Spatial (GNS) — Full model with all effect channels
• Choosing a Spatial Model — Selecting the appropriate specification

References

1. LeSage, J. and Pace, R.K. (2009). Introduction to Spatial Econometrics. Chapman & Hall/CRC. (Chapter 2: Spatial Effects)
2. Elhorst, J.P. (2010). Applied spatial econometrics: raising the bar. Spatial Economic Analysis, 5(1), 9-28.
3. LeSage, J. and Pace, R.K. (2014). The biggest myth in spatial econometrics. Econometrics, 2(4), 217-249.
4. Pace, R.K. and LeSage, J. (2010). Omitted variable biases of OLS and spatial lag models. In Progress in Spatial Analysis, 17-28.