AndrzejTorój Lecture8:Interpretationofparametersinspatialmodels SpatialEconometrics

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Spatial Econometrics

Lecture 8: Interpretation of parameters in spatial models

Andrzej Torój

Institute of Econometrics – Department of Applied Econometrics

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Outline

1 Interpretation of parameters Single-source models Multi-source models

2 Statistical inference for spatial multipliers and average effects Inference in models with SAR component

Inference in models without SAR component

Andrzej Torój Institute of Econometrics – Department of Applied Econometrics

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Plan prezentacji

1 Interpretation of parameters

2 Statistical inference for spatial multipliers and average effects

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Marginal effects of growth in regressors in the linear model (1)

Consider the SAR model (and multi-source models containing ρWy):

y = ρWy + Xβ + ε

If ρ = 0 then the marginal impact of the variable x_k on y is simply the parameter β_k, i.e. the respective element of β vector (hence the interpretation of parameters in a linear model):

y = Xβ + ε

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Marginal effects of growth in regressors in the linear model (1)

y = ρWy + Xβ + ε

If ρ = 0 then the marginal impact of the variable x_k on y is simply the parameter β_k, i.e. the respective element of β vector (hence the interpretation of parameters in a linear model):

y = Xβ + ε

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Marginal effects of growth in regressors in the linear model (2)

We can consider such an impact for every pair of observations in the sample by differentiating the vectory with respect to xk(each of them of length N – number of observations, hence the result is a matrix sized N × N):

S_k= _∂x^∂y

k =^∂[x_∂x^k^β^k^]

k = βkI_N=







∆xk,1= 1

↓ βk

∆xk,2= 1

↓

0 · · ·

∆xk,N= 1

↓

0 → ∆y1

0 βk · · · 0 → ∆y2

.. .

..

. . .. ... →

.. .

0 0 · · · β_k → ∆y_N







diagonal matrix: change ofxkfor i -th observation has no impact ony for any other observation

spherical matrix (equality of diagonal elements): the marginal impact is equal across observations

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Marginal effects of growth in regressors in SAR model (1)

y = ρWy + Xβ + ε

If ρ 6= 0, the computation of marginal effect requires first to solve this equation for y:

y = (I − ρW)⁻¹Xβ + (I − ρW)⁻¹ε

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Marginal effects of growth in regressors in SAR model (1)

y = ρWy + Xβ + ε

If ρ 6= 0, the computation of marginal effect requires first to solve this equation for y:

y = (I − ρW)⁻¹Xβ + (I − ρW)⁻¹ε

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Marginal effects of growth in regressors in SAR model (2)

The effect of differentiation is, again, the N × N matrix, but this time...

Sk=_∂x^∂y

k =^∂[(I−ρW)⁻¹x_kβ_k]

∂x_k = βk(I − ρW)⁻¹=







∆xk,1= 1

↓ m1,1

∆xk,2= 1

↓

m1,2 · · ·

∆xk,N= 1

↓

m1,N → ∆y₁

m2,1 m2,2 · · · m2,N → ∆y2

.. .

..

. . .. ... →

.. .

mN,1 mN,2 · · · mN,N → ∆y_N







this matrix is neigher diagonal (dependence between observations: change inxk

for i -th observation can have an impact ony for other observations)...

...nor spherical (various direct impacts across observations).

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Interactions between neighbours – complication of the multipliers (1)

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Interactions between neighbours – complication of the multipliers (2)

Cat exerts an impact on Pat (β₁). Pat carries forward this impact to Mat (β₁·ρ₂), Mat to Pat (β₁ρ₂·ρ₁), Pat to Mat again (β₁ρ2ρ1·ρ2)...

In other words (in spatial modelling terms):

S_k = β_k(I − ρW)⁻¹ = β_kI + β_kρW + β_kρ²W²+ . . . β_kI: effect without taking space into consideration (diagonal, spherical)

β_kρW: effect of first-order neighbourhood (this effect still keeps the diagonal of Sk equal, because of zero diagonal in W) βkρ²W²+ . . .: the diagonal of Sk is no longer equal

(second-order neighbourhood effects, and further-orders, operate with different strength for individual observations)

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Interactions between neighbours – complication of the multipliers (2)

Cat exerts an impact on Pat (β₁). Pat carries forward this impact to Mat (β₁·ρ₂), Mat to Pat (β₁ρ₂·ρ₁), Pat to Mat again (β₁ρ2ρ1·ρ2)...

In other words (in spatial modelling terms):

S_k = β_k(I − ρW)⁻¹ = β_kI + β_kρW + β_kρ²W²+ . . . β_kI: effect without taking space into consideration (diagonal, spherical)

β_kρW: effect of first-order neighbourhood (this effect still keeps the diagonal of Sk equal, because of zero diagonal in W) βkρ²W²+ . . .: the diagonal of Sk is no longer equal

(second-order neighbourhood effects, and further-orders, operate with different strength for individual observations)

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Interpreting the parameters of SEM and SLX models

Much simpler:

SEM: y = Xβ +

M

z }| {

(I −λW)⁻¹u β – like in the linear model

λ – through the lens of matrix M = [mi ,j]:

∆uj = 1 → ∆yi= mi ,j (hardly ever useful in practice...) SLX: y = Xβ +WXθ+ ε

β and θ – through the marginal effects:

Sk = _∂x^∂y

k = ^∂[x^k^(β_∂x^k^I+θ^k^W)]

k = βkI + θkW ...and further as in SAR

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Interpreting the parameters of SEM and SLX models

Much simpler:

SEM: y = Xβ +

M

z }| {

Sk = _∂x^∂y

k = ^∂[x^k^(β_∂x^k^I+θ^k^W)]

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Interpreting the parameters of SEM and SLX models

Much simpler:

SEM: y = Xβ +

M

z }| {

Sk = _∂x^∂y

k = ^∂[x^k^(β_∂x^k^I+θ^k^W)]

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Direct, indirect and total effects (1)

Presentation of the entire matrix S_k is burdensome and normally useless. There are synthetic measures instead.

Direct effectfor the variable x_k: E_k^D = ^{tr (S}_N^k⁾.

Impact of unit increase in xkin a given region on y inthe same region (averaged over regions).

Total effectfor the variable x_k: E_k^T = _N¹Σ^N_{i =1}

Σ^N_{j =1}m_{i ,j} . Average column sum over the matrix Sj.

Impact of unit increase in xk in a given region on y inall regions jointly (averaged over the regions where the impulse can potentially occur).

Indirect effect for the variable x_k: E_k^I = E_k^T− E_k^D.

Impact of unit increase in x_j in a given region on y inall other regions jointly (averaged over the regions where the impulse can potentially occur).

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Direct, indirect and total effects (1)

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Direct, indirect and total effects (1)

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Direct, indirect and total effects (1)

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Direct, indirect and total effects (2)

Usually, i.e. with ρ ∈ (0; 1), it holds that β_k < E_k^D < E_k^T. The standard interpretation of β_k as in the linear model is erroneous, a similar interpretation is valid at most for E_k^T (up to averaging over regions).

These effects are computed using post-estimation command impacts

impacts <- impacts(model, listw = W)

Effects from SLX model are only computed for one estimation method (command lmSLX) and, in such a case, the argument listw is not required.

It does not make sense to compute impacts form SEM model (why?).

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Interpretation of parameters in multi-source models

SARAR: y = (I −ρW)⁻¹Xβ +

M∗

z }| {

(I −ρW)⁻¹(I −λW)⁻¹u β and ρ – jointly as in SAR model (impacts in R) ρ and λ – jointly through the lens of M∗ (like SEM) SDM:

y = (I −ρW)⁻¹Xβ + (I −ρW)⁻¹WXθ+ (I −ρW)⁻¹ε β, θ, ρ – marginal effects: S_k =_∂x^∂y

k =

∂{xk[^βk(I−ρW)⁻¹+θ_k(I−ρW)⁻¹W]}

∂x_k = (I − ρW)⁻¹[β_kI + θkW]

...and further as in SAR model (impacts in R) SDEM: y = Xβ +WXθ+ (I −λW)⁻¹u

β and θ – like in SLX (impacts in R) λ – like in SEM

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Interpretation of parameters in multi-source models

M∗

z }| {

k =

∂x_k = (I − ρW)⁻¹[β_kI + θkW]

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Interpretation of parameters in multi-source models

M∗

z }| {

k =

∂x_k = (I − ρW)⁻¹[β_kI + θkW]

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Plan prezentacji

1 Interpretation of parameters

2 Statistical inference for spatial multipliers and average effects

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Inference in models with SAR component

SAR / SARAR: ˆS_k = (I − ˆρW)⁻¹βˆ_k SDM: ˆS_k = (I − ˆρW)⁻¹h ˆβkI + ˆθkWi

In both cases, multiplier matrices are non-linear functions of the estimated parameters (with respect to ˆρ).

The assessment of the related statistical uncertainty (and the resulting uncertainty of average effects) requires to use approximations or numerical methods.

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Inference in models with SAR component

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Inference in models with SAR component

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Method 1: delta approximation (SAR model) (1)

Let ∇_k = ^{∂ ˆ}^S

∗ k

h ∂ ˆρ ∂ ˆβ_k ⁱ, where ˆS^∗_k denotes a vertical vector of length N · N composed of elements from matrix ˆS_k. The matrix of derivatives ∇_k is sized (N · N) × 2 (derivatives of N · N elements with respect to 2 parameters).

∂ ˆSk

∂ ˆβk = ^{∂(I− ˆ}^ρW)

−1βˆk

∂ ˆβk = (I − ˆρW)⁻¹ ≡ M

∂ ˆS_k

∂ ˆρ = ^{∂(I− ˆ}^ρW)

−1βˆk

∂ ˆρ = − (I − ˆρW)⁻¹(−W) (I − ˆρW)⁻¹ ≡ MWM

From matrix calculus:

∂U(x )⁻¹

∂x = −U (x )^{−1 ∂U(x)}_∂x U (x)⁻¹

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Method 1: delta approximation (SAR model) (1)

Let ∇_k = ^{∂ ˆ}^S

∗ k

∂ ˆSk

∂ ˆβk = ^{∂(I− ˆ}^ρW)

−1βˆk

∂ ˆβk = (I − ˆρW)⁻¹ ≡ M

∂ ˆS_k

∂ ˆρ = ^{∂(I− ˆ}^ρW)

−1βˆk

∂ ˆρ = − (I − ˆρW)⁻¹(−W) (I − ˆρW)⁻¹ ≡ MWM

∂U(x )⁻¹

∂x = −U (x )^{−1 ∂U(x)}_∂x U (x)⁻¹

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Method 1: delta approximation (SAR model) (1)

Let ∇_k = ^{∂ ˆ}^S

∗ k

∂ ˆSk

∂ ˆβk = ^{∂(I− ˆ}^ρW)

−1βˆk

∂ ˆβk = (I − ˆρW)⁻¹ ≡ M

∂ ˆS_k

∂ ˆρ = ^{∂(I− ˆ}^ρW)

−1βˆk

∂ ˆρ = − (I − ˆρW)⁻¹(−W) (I − ˆρW)⁻¹ ≡ MWM

∂U(x )⁻¹

∂x = −U (x )^{−1 ∂U(x)}_∂x U (x)⁻¹

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Method 1: delta approximation (SAR model) (2)

Let Σ_k =





Var ( ˆρ) Cov

ˆ ρ, ˆβ_k

. Var ˆβ_k



 be a respective sub-matrix 2 × 2 from the variance-covariance matrix of the parameter estimates.

1-st order Taylor expansion of the variance-covariance formula for ˆS^∗_k yields the following matrix sized (N · N) × (N · N):

Var ˆS^∗_k

≈ ∇^T_k · (Σ_k)⁻¹· ∇_k

Its diagonal elements can be rearranged back into the matrix sized N × N, and their square roots approximate the standard errors ˆS_k.

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Method 1: delta approximation (SAR model) (2)

Let Σ_k =





Var ( ˆρ) Cov

ˆ ρ, ˆβ_k

. Var ˆβ_k



Var ˆS^∗_k

≈ ∇^T_k · (Σ_k)⁻¹· ∇_k

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Method 1: delta approximation (SAR model) (2)

Let Σ_k =





Var ( ˆρ) Cov

ˆ ρ, ˆβ_k

. Var ˆβ_k



Var ˆS^∗_k

≈ ∇^T_k · (Σ_k)⁻¹· ∇_k

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Method 2: bootstrap (1)

Speaking of bootstrap methods, we usually mean theirnon-parametric variant:

Draw independently with replacement N regions from the population of N regions.

For each draw r = 1, ..., R compute ˆS^{(r )}_k .

From the obtained distribution ˆS^{(r )}_k we derive the desired quantiles for each element of the matrix (e.g. of order 0.025 and 0.975, yielding 95-percent confidence interval).

This will not work well in spatial models!

Observations are not independent. This sampling scheme tears apart the network of dependencies summarized by ˆρ, which will be subject to our inference later on.

One potential solution is the block bootstrap method, as in dynamic time series models (observation + lag).

Quite difficult in dynamic spatial panels (two blocking dimensions: time and space!).

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Method 2: bootstrap (1)

Speaking of bootstrap methods, we usually mean theirnon-parametric variant:

Draw independently with replacement N regions from the population of N regions.

For each draw r = 1, ..., R compute ˆS^{(r )}_k .

This will not work well in spatial models!

Observations are not independent. This sampling scheme tears apart the network of dependencies summarized by ˆρ, which will be subject to our inference later on.

One potential solution is the block bootstrap method, as in dynamic time series models (observation + lag).

Quite difficult in dynamic spatial panels (two blocking dimensions: time and space!).

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Method 2: bootstrap (2)

Parametric bootstrap: having estimated the model parameters ˆρ, ˆβ...:

Generate R draws of the error vector ε^{(r )}: i.i.d., N 0, ˆσ² For every r , reconstructy^{(r )}= f

ε^{(r )},X, W, ˆβ, ˆρ . Estimate

h ˆ ρ^{(r )}, ˆβ^{(r )}

i

= f (y^{(r )},X, W) and, based on this result, compute Sˆ^{(r )}_k .

Equivalently (and more simply in practice):

Draw R timesh ˆ ρ^{(r )}, ˆβ^{(r )}

i

∼ MVNh ˆ ρ, ˆβ

i , Cov

h ˆ ρ, ˆβ

i

and, for each r , compute ˆS^{(r )}_k .

Limitation: we assume that our model is statistically correct!

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Method 2: bootstrap (2)

h ˆ ρ^{(r )}, ˆβ^{(r )}

i

i , Cov

h ˆ ρ, ˆβ

i

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Method 2: bootstrap (2)

h ˆ ρ^{(r )}, ˆβ^{(r )}

i

i , Cov

h ˆ ρ, ˆβ

i

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Method 2: bootstrap (2)

h ˆ ρ^{(r )}, ˆβ^{(r )}

i

i , Cov

h ˆ ρ, ˆβ

i

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Evaluating the uncertainty around average effects

R delivers an off-the-shelf bootstrap procecure related to the average direct, indirect and total effects.

Using the command impacts, we should supply the following additional arguments:

R = 200 (number of iterations R in the bootstrap procedure) zstats = TRUE

For models with the SAR component (SAR, SDM, SARAR, GNS), following the use of impacts, one can report the confidence intervals for the three types of effects, at a given confidence level (using the command

HPDinterval.lagImpact), and verify whether they contain zero.

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Inference in models without SAR component

Both methods can be applied to SLX and SDEM models. But senseless to use, since an an exact analytical formula for the distribution of linear functions of linear model parameters exists (derived e.g. inGreene, 2003)

In such cases, delta method provides the exact formula instead of an approximation, because Sk= βkI + θkWis a linear function with respect to βk and θk.

Bootstrap method also ensures the correct inference (though it is computationally inefficient).

For the SLX and SDEM models, summary(impacts) will test three hypotheses on the basis of the analytical distribution (p-values are printed):

H0: E_k^D = 0 H0: E_k^I = 0 H0: E_k^T = 0

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Interaction variables

Caution! Throughout this lecture (regardless of the model and inference method) we assumed that regressor matrix X does not contain interaction variables (of type x₁· x₂).

Otherwise the multipliers for variables x₁ and x₂ should be derived differently. I recommend this as an exercise.

Spatial multipliers for a given region depend on:

size of the non-spatial impact of X on y (β) intensity of the spatial processes (ρ, θ)

degree of a region’s integration into the network (W) in interaction variable models: level of other predictors in the analysed region and other regions (X)

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Interaction variables

Caution! Throughout this lecture (regardless of the model and inference method) we assumed that regressor matrix X does not contain interaction variables (of type x₁· x₂).

Otherwise the multipliers for variables x₁ and x₂ should be derived differently. I recommend this as an exercise.

Spatial multipliers for a given region depend on:

size of the non-spatial impact of X on y (β) intensity of the spatial processes (ρ, θ)

degree of a region’s integration into the network (W) in interaction variable models: level of other predictors in the analysed region and other regions (X)

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Homework 7

Select one explanatory variable from the ones discussed in HW 5 and 6.

For the best model (or, if that was the SEM model – for any other model) compute the direct, indirect and total effect for this variable.

Test the significance of this indirect effect.

Illustrate on a map the impact of a unit change of the indicated regressor in the selected region on the dependent variable across all regions. Comment on the results.

The PDF file should contain the calculation results with all interpretations and the map.