Spatial Econometrics - web.sgh.waw.pl - prace trwaja › ... › 9_panels.pdf · 2019-02-25 · Not...

Basics of panel econometrics Basic spatial panel models Extensions to speci�cation

Spatial EconometricsLecture 9: Spatial panel models

Andrzej Torój

Institute of Econometrics � Department of Applied Econometrics

Andrzej Torój Institute of Econometrics � Department of Applied Econometrics

(9) Spatial Econometrics 1 / 37

Outline

1 Basics of panel econometricsNon-spatial panel modelsBasic diagnostics of panel models

2 Basic spatial panel modelsPanel SARAR modelML estimation

3 Extensions to the speci�cation of spatial panelsStatic panelsAccounting for dynamics

Plan prezentacji

1 Basics of panel econometrics

2 Basic spatial panel models

3 Extensions to the speci�cation of spatial panels

Non-spatial panel models

Panel data

yt,i : data indexed in two dimensions � time and observationunits

t: unidimensional �space� with a prede�ned orientation (past→ future)i : units � in non-spatial panels mutually independent

Network of units summarized by W can be referred to the'spatial' panel dimension, i .

Two sorts of panels:

balanced: T × N observations available for all variablesunbalanced: missing data (more serious consequences inspatial econometrics)

Panel data

Unbalanced panels (1)

obs = 7 · 5− 2

obs = 7 · 5− 3

obs = 7 · 5− 4− 3

obs = 7 · 5− 4− 3; ρ1?, ρ2 ↓, in consequence β2 ↑

obs = 7 · 5− 2 · 5

Pooled model (classical linear regression)

Pooled model:

yt,i = c + xt,iβ + εi ,t

εi ,t ∼ N(0, σ2

)i .i .d .

Matrix notation:

yNT×1

= 1NT×1

· c1×1

+ XNT×K

βK×1

+ εNT×1

ε ∼ MVN

(0, σ2 I

NT×NT

Pooled model (classical linear regression)

Pooled model:

yt,i = c + xt,iβ + εi ,t

εi ,t ∼ N(0, σ2

)i .i .d .

Matrix notation:

yNT×1

= 1NT×1

· c1×1

+ XNT×K

βK×1

+ εNT×1

ε ∼ MVN

(0, σ2 I

NT×NT

Fixed e�ects model (FE)

FE model:

yt,i = αi + xt,iβ + εi ,t

εi ,t ∼ N(0, σ2

)i .i .d .

Matrix notation:

yNT×1

(1T×1 ⊗ IN

N×1+ X

NT×Kβ

K×1+ ε

ε ∼ MVN

(0, σ2 I

NT×NT

Fixed e�ects model (FE)

FE model:

yt,i = αi + xt,iβ + εi ,t

εi ,t ∼ N(0, σ2

)i .i .d .

Matrix notation:

yNT×1

(1T×1 ⊗ IN

N×1+ X

NT×Kβ

K×1+ ε

ε ∼ MVN

(0, σ2 I

NT×NT

Kronecker product

12×1⊗ α

α1α2α3

1·α11·α21·α31·α11·α21·α3

Random e�ects model (RE)

RE model:

yt,i = α+ xt,iβ + ui , ui = αi + εi ,t ,

αi ∼ N(0, σ2α

)i .i .d ., ui ,t ∼ N

(0, σ2u

)i .i .d .

Matrix notation:

yNT×1

= 1NT×1

· c1×1

+ XNT×K

βK×1

(1T×1 ⊗ IN

N×1+ u

µ ∼ MVN

(0, σ2µ I

), u ∼ MVN

(0, σ2u I

NT×NT

Random e�ects model (RE)

RE model:

yt,i = α+ xt,iβ + ui , ui = αi + εi ,t ,

αi ∼ N(0, σ2α

)i .i .d ., ui ,t ∼ N

(0, σ2u

)i .i .d .

Matrix notation:

yNT×1

= 1NT×1

· c1×1

+ XNT×K

βK×1

(1T×1 ⊗ IN

N×1+ u

µ ∼ MVN

(0, σ2µ I

), u ∼ MVN

(0, σ2u I

NT×NT

Basic diagnostics of panel models

Some testing procedures

Testing individual e�ects:

poolability FE (Wald)variance RE (Breusch-Pagan)

Hausman test:

H0: RE estimator consistent (and then preferred as moree�cient)H1: RE estimator inconsistent (and then FE preferred in spiteof lower e�ciency)

Testing of spatial e�ects in non-spatial models

Basic diagnostics of panel models

Special models

Dynamic: Arellano-Bond, Blundell-Bond, etc.

Limited dependent variable

Cointegration

Booming literature over the last decadeNot covered by this lecture, but: Elhorst P., 2014, SpatialEconometrics: From Cross-Sectional Data to SpatialPanels (ch. 4)Software: mostly Stata (e.g. handling unbalanced panels)

Plan prezentacji

Panel SARAR model

SARAR model with individual e�ects

yNT×1

T×T⊗ W

NT×1+ X

NT×Kβ

K×1+ (1T×1 ⊗ IN)µ+ [INT − λ (IT ⊗W)]−1 ε︸︷︷︸

ε ∼ MVN(0, σ2εI

)Option spatial.error in R command spml:

�none�: λ = 0 (SAR)�b�: λ 6= 0�kkp�: spatial autocorrelation also related to individual e�ects(Kapoor et al., 2007 � not discussed below, but can be easilyderived as an exercise)

[INT − λ (IT ⊗W)]−1 [(1T×1 ⊗ IN)µ+ ε]

The case above easily collapses to SEM (ρ = 0, lag = FALSE).

Panel SARAR model

yNT×1

T×T⊗ W

NT×1+ X

NT×Kβ

K×1+ (1T×1 ⊗ IN)µ+ [INT − λ (IT ⊗W)]−1 ε︸︷︷︸

[INT − λ (IT ⊗W)]−1 [(1T×1 ⊗ IN)µ+ ε]

Panel SARAR model

yNT×1

T×T⊗ W

NT×1+ X

NT×Kβ

K×1+ (1T×1 ⊗ IN)µ+ [INT − λ (IT ⊗W)]−1 ε︸︷︷︸

[INT − λ (IT ⊗W)]−1 [(1T×1 ⊗ IN)µ+ ε]

Panel SARAR model

Spatial weight matrix

Spatial weight matrix (IT ⊗W) is now block diagonal:W

W. . .

W − spatial weights in period 1W − spatial weights in period 2W − spatial weights in period 3

...W − spatial weights in period T

Caution! We assume here that observations in matrices y and X areordered:

region 1 period 1, region 2 period 1, region 1 period 2, region2 period 2...In spml in R it almost does not matter how the data istechnically imported (details in the code), but the derivationsbelow are only true for such a case.

Panel SARAR model

Spatial weight matrix

Spatial weight matrix (IT ⊗W) is now block diagonal:W

W. . .

W − spatial weights in period 1W − spatial weights in period 2W − spatial weights in period 3

...W − spatial weights in period T

Caution! We assume here that observations in matrices y and X areordered:

region 1 period 1, region 2 period 1, region 1 period 2, region2 period 2...In spml in R it almost does not matter how the data istechnically imported (details in the code), but the derivationsbelow are only true for such a case.

Panel SARAR model

Types of individual e�ects

Individual e�ects can also occur for periods, not only forregions (effect: �individual�, �time�, �twoways�).

Statistically, a spatial model can also be handled in two ways,just like a non-spatial one:

FE: L(y|X,β, ρ, λ, σ2ε ,µ

)RE: L

(y|X,β, ρ, λ, σ2ε , σ2µ

), where µ ∼ MVN

(0;σ2µI

ML estimation

Composite error

Moving from the structural form to the reduced form:

y = [INT − ρ (IT ⊗W)]−1Xβ + [INT − ρ (IT ⊗W)]−1 ν

ν = ε (FE, SAR) � µ is treated as part of X

ν = (1T×1 ⊗ IN)µ+ ε (RE, SAR)

ν = [INT − λ (IT ⊗W)]−1 ε (FE, SARAR) � µ is treated aspart of X

ν = (1T×1 ⊗ IN)µ+ [INT − λ (IT ⊗W)]−1 ε (RE, SARAR)

Regardless of the above variant:

∂ν∂y = [INT − ρ (IT ⊗W)] = IT ⊗ (IN − ρW) ≡ IT ⊗Mρ

ML estimation

Likelihood function

In spatial panel models, we always use the general likelihood function ofobservations, whose logarithm takes the form:

ln L(y|X,β, ρ, λ, σ2ε , ...

−N·T2

ln (2π)+ln∣∣∣ ∂ν∂y ∣∣∣− 1

2ln |Σν |− 1

2ν (y|X,β, ρ,W, ...)′Σν

−1ν (y|X,β, ρ,W, ...)

N � spatial dimension

T � time dimension

y, X � data

W � known spatial weight matrix

ν � vector of error terms

Σν � variance-covariance matrix of ν∂ν∂y

� Jacobian of the relationship between structural and reduced form(derivative of ν with respect to the dependent variable)

Caution! Rede�nition of ν should be accordingly re�ected in Σν and ∂ν∂y

ML estimation

ML estimation of FE-SARAR model (1)

Error term:ν = [INT − λ (IT ⊗W)]−1 ε = IT⊗(IN − λW)−1ε ≡ IT⊗Mλε

Its variance-covariance matrix (henceforth: T � number ofperiods or a related upper/lower index, ′ � transposition):

Συ = (IT ⊗Mλ)σ2ε I (IT ⊗Mλ)

′= σ2ε (IT ⊗Mλ)

(I′T ⊗Mλ

= σ2ε

(IT I′T

)⊗(MλMλ

′)= σ2ε IT ⊗

(MλMλ

Συ−1 = 1

σ2εIT ⊗

(MλMλ

′)−1

ln |Συ| = NT · lnσ2ε + T · ln∣∣∣MλMλ

′∣∣∣

ML estimation

Error term:ν = [INT − λ (IT ⊗W)]−1 ε = IT⊗(IN − λW)−1ε ≡ IT⊗Mλε

Its variance-covariance matrix (henceforth: T � number ofperiods or a related upper/lower index, ′ � transposition):

Συ = (IT ⊗Mλ)σ2ε I (IT ⊗Mλ)

′= σ2ε (IT ⊗Mλ)

(I′T ⊗Mλ

= σ2ε

(IT I′T

)⊗(MλMλ

′)= σ2ε IT ⊗

(MλMλ

Συ−1 = 1

σ2εIT ⊗

(MλMλ

′)−1

ln |Συ| = NT · lnσ2ε + T · ln∣∣∣MλMλ

′∣∣∣

ML estimation

Substituting the relevant elements into the likelihood function:

ln LFE = −N·T2

ln (2π) + ln∣∣∣ ∂ν∂y ∣∣∣− 1

2ln |Σν | − 1

2υ′Σν

−1υ =

= −N·T2

ln (2π)− T · ln |Mρ| − N·T2

ln(σ2ε)− T

∣∣∣MλMλ′∣∣∣+

− 12σ2ευ′[IT ⊗

(MλMλ

′)−1]

υ = [INT − ρ (IT ⊗W)] y − Xβ − (1T×1 ⊗ IN)µ

Maximisation algorithm:

1 Within transformation: from every yi,t subtract the respectiveyi = 1

t=1yi,t (and likewise for X).2 Maximisation of ln LFE with respect to ρ, λ, β, σ2ε skipping the terms

(1T×1 ⊗ IN)µ.3 Computing µ as vi = 1

t=1vi,t after the estimation in step 2.

ML estimation

ln LFE = −N·T2

ln (2π) + ln∣∣∣ ∂ν∂y ∣∣∣− 1

2ln |Σν | − 1

2υ′Σν

−1υ =

= −N·T2

ln (2π)− T · ln |Mρ| − N·T2

ln(σ2ε)− T

∣∣∣MλMλ′∣∣∣+

− 12σ2ευ′[IT ⊗

(MλMλ

′)−1]

ML estimation

ln LFE = −N·T2

ln (2π) + ln∣∣∣ ∂ν∂y ∣∣∣− 1

2ln |Σν | − 1

2υ′Σν

−1υ =

= −N·T2

ln (2π)− T · ln |Mρ| − N·T2

ln(σ2ε)− T

∣∣∣MλMλ′∣∣∣+

− 12σ2ευ′[IT ⊗

(MλMλ

′)−1]

ML estimation

ln LFE = −N·T2

ln (2π) + ln∣∣∣ ∂ν∂y ∣∣∣− 1

2ln |Σν | − 1

2υ′Σν

−1υ =

= −N·T2

ln (2π)− T · ln |Mρ| − N·T2

ln(σ2ε)− T

∣∣∣MλMλ′∣∣∣+

− 12σ2ευ′[IT ⊗

(MλMλ

′)−1]

ML estimation

ML estimation of RE-SARAR model (1)

Composite error term:ν = (1T×1 ⊗ IN)µ+ [INT − λ (IT ⊗W)]−1 ε =

(1T×1 ⊗ IN)µ+ IT ⊗ (IN − λW)−1 ε ≡ (1T×1 ⊗ IN)µ+ (IT ⊗Mλ) ε

Its variance-covariance matrix (assuming the independence of µ and ε):Συ = Var [(1T×1 ⊗ IN)µ] + Var [(IT ⊗Mλ) ε] =

= σ2µ

(1T×11

′T×1

)⊗(IN I

)+ σ2ε

)⊗(MλMλ

= σ2ε

σ2ε︸︷︷︸φ

(1T×T ⊗ IN) + IT ⊗(MλMλ

︸︷︷︸Συ

Συ−1 = 1

σ2εΣυ−1

ln |Συ | = NT · lnσ2ε + ln∣∣∣Συ

∣∣∣Andrzej Torój Institute of Econometrics � Department of Applied Econometrics

ML estimation

Composite error term:ν = (1T×1 ⊗ IN)µ+ [INT − λ (IT ⊗W)]−1 ε =

(1T×1 ⊗ IN)µ+ IT ⊗ (IN − λW)−1 ε ≡ (1T×1 ⊗ IN)µ+ (IT ⊗Mλ) ε

Its variance-covariance matrix (assuming the independence of µ and ε):Συ = Var [(1T×1 ⊗ IN)µ] + Var [(IT ⊗Mλ) ε] =

= σ2µ

(1T×11

′T×1

)⊗(IN I

)+ σ2ε

)⊗(MλMλ

= σ2ε

σ2ε︸︷︷︸φ

(1T×T ⊗ IN) + IT ⊗(MλMλ

︸︷︷︸Συ

Συ−1 = 1

σ2εΣυ−1

ln |Συ | = NT · lnσ2ε + ln∣∣∣Συ

∣∣∣Andrzej Torój Institute of Econometrics � Department of Applied Econometrics

ML estimation

ln LRE = −N·T2

ln (2π) + ln∣∣∣ ∂ν∂y ∣∣∣− 1

2ln |Σν |+

− 12υ′Σν

−1υ =

= −N·T2

ln (2π)− T · ln |Mρ| − N·T2

ln(σ2ε)− 1

2ln∣∣∣Συ (λ, φ)

∣∣∣+− 1

2σ2ευ′Συ (λ, φ)−1 υ

υ = {[INT − ρ (IT ⊗W)] y − Xβ}

1 Use some starting values λ(0), ρ(0), φ(0).2 Based on �rst-order conditions for ln LRE and values from step 1, set β

and σ2v .3 Conditionally upon β and σ2v from step 2, �nd new values of λ, ρ, φ by

maximising ln LRE .4 Iterate steps (2)-(3) until convergence.

ML estimation

ln LRE = −N·T2

ln (2π) + ln∣∣∣ ∂ν∂y ∣∣∣− 1

2ln |Σν |+

− 12υ′Σν

−1υ =

= −N·T2

ln (2π)− T · ln |Mρ| − N·T2

ln(σ2ε)− 1

∣∣∣+− 1

υ = {[INT − ρ (IT ⊗W)] y − Xβ}

ML estimation

ln LRE = −N·T2

ln (2π) + ln∣∣∣ ∂ν∂y ∣∣∣− 1

2ln |Σν |+

− 12υ′Σν

−1υ =

= −N·T2

ln (2π)− T · ln |Mρ| − N·T2

ln(σ2ε)− 1

∣∣∣+− 1

υ = {[INT − ρ (IT ⊗W)] y − Xβ}

ML estimation

ln LRE = −N·T2

ln (2π) + ln∣∣∣ ∂ν∂y ∣∣∣− 1

2ln |Σν |+

− 12υ′Σν

−1υ =

= −N·T2

ln (2π)− T · ln |Mρ| − N·T2

ln(σ2ε)− 1

∣∣∣+− 1

υ = {[INT − ρ (IT ⊗W)] y − Xβ}

ML estimation

ln LRE = −N·T2

ln (2π) + ln∣∣∣ ∂ν∂y ∣∣∣− 1

2ln |Σν |+

− 12υ′Σν

−1υ =

= −N·T2

ln (2π)− T · ln |Mρ| − N·T2

ln(σ2ε)− 1

∣∣∣+− 1

υ = {[INT − ρ (IT ⊗W)] y − Xβ}

Plan prezentacji

Static panels

Panel versions of spatial cross-section models

SAR, SEM, SARAR: maximum likelihood estimation asdiscussed above

SLX: estimation procedure the same as for non-spatial FE/REmodels

SDM, SDEM: extension of SAR or SEM by an additionalregressor (IT ⊗W)X

Estimation

model <- spml(formula = ..., model = ..., effect = ,

lag = ..., spatial.error = ...)

Static panels

Speci�cation tests in spatial panel models (1)

Baltagi et al., 2003:

Test H0 additional assumptions bsktest(test = ...)

LM1 σ2µ = 0 ρ = 0 �LM1�

LM2 ρ = 0 σ2µ = 0 �LM2�

LMH ρ = σ2µ = 0 � �LMJOINT�

LMλ ρ = 0 σ2µ ≥ 0 �CLMlambda�

LMµ σ2µ = 0 ρ ∈ (−1; 1) �CLMmu�

Static panels

Spatial version of the Hausman test (Mutl, Pfa�ermayr, 2011): sphtest

Regardless of the result, there are controversies arout using RE test tospatial data (Elhorst vs LeSage).

e�ciency of RE and the problematic N-asymptotics of µconsistency (if con�rmed by Hausman test);

spatial data frequently describes the entire population of regions, not asample drawn from a population or a process with variance σ2µwe do not de�ne any sampling scheme, but it would surely a�ect thenetwork Winference on µ is not necessary, while the problem of N-asymptotics doesnot a�ect β � Lancaster, 2000

Static panels

Accounting for dynamics

Dynamics in spatial panel models

Taking account of the time dimension, we increase the number of potentialmodel speci�cations by a multiple.The possibility of time lags gets things even more complicated... (Elhorst,2001).

The scheme below does not even take lags of X into consideration.

Main problem with speci�cation of dynamic panels

Combination of di�erent dimensions of impact:

in time: yt−1 → ytthrough space: Wyt → ytvia variables: xt → ytbottom line: yt ← xt, yt−1,Wyt,Wyt−1, xt−1,Wxt,Wxt−1, ...

Time and space dimensions are not independent, spatial panelsshould be jointly regarded as a spatio-temporal process (Cooket al., 2017: �Right Place, Right Time� � Monte Carlo study).

Consequence: frequent bias of estimates when the process ismisspeci�ed (Elhorst, 2010).

Basic rules

Omitting the temporal autoregression usually leads tooverestimating the spatial autoregression (or the dependenceupon WX) � Achen (2000).

This � in turn � leads to underestimating β (Hays, 2003).

One should be very cautious with any from-speci�c-to-generalinference scheme (though, one cannot discard it in general, asthe problem of weak spatial identi�cation becomes evenstronger under a rich spatio-temporal dynamics):

At least: estimate a dynamic non-spatial model and test itsresiduals for spatial autocorrelation (Beck and Katz, 2011).Better: consider one spatial and one temporal lag jointly (andtest via LR).

Basic rules

Omitting the temporal autoregression usually leads tooverestimating the spatial autoregression (or the dependenceupon WX) � Achen (2000).

This � in turn � leads to underestimating β (Hays, 2003).

One should be very cautious with any from-speci�c-to-generalinference scheme (though, one cannot discard it in general, asthe problem of weak spatial identi�cation becomes evenstronger under a rich spatio-temporal dynamics):

At least: estimate a dynamic non-spatial model and test itsresiduals for spatial autocorrelation (Beck and Katz, 2011).Better: consider one spatial and one temporal lag jointly (andtest via LR).

STADL model

Generalisation: STADL(p, q, r ,P,Q,R) model �Spatio-Temporal Autoregressive Distributed Lag:

Fyt = X′tG+Hεt

I−︸︷︷︸(φ1L+ . . .+ φpLp)︸︷︷︸

−ρ1W − . . .− ρPWP︸︷︷︸FS

G =(β0 + L′β1 + . . .+ (Lq)′ βq −W′θ1 − . . .−

)′θQ

(I− δ1L− . . .− δrLr − λ1W − . . .− λRWR

)−1Spatio-temporal stationarity of a process: this is ageneralisation of notions typical to spatial econometrics andtime series econometrics. Roots of the characteristicpolynomial F should be outside the unit circle.

STADL model

Generalisation: STADL(p, q, r ,P,Q,R) model �Spatio-Temporal Autoregressive Distributed Lag:

Fyt = X′tG+Hεt

I−︸︷︷︸(φ1L+ . . .+ φpLp)︸︷︷︸

−ρ1W − . . .− ρPWP︸︷︷︸FS

G =(β0 + L′β1 + . . .+ (Lq)′ βq −W′θ1 − . . .−

)′θQ

(I− δ1L− . . .− δrLr − λ1W − . . .− λRWR

)−1Spatio-temporal stationarity of a process: this is ageneralisation of notions typical to spatial econometrics andtime series econometrics. Roots of the characteristicpolynomial F should be outside the unit circle.

Spatio-temporal stationarity

Consider an example STADL(1, 0, 0, 1, 0, 0):

(I− φ1L− ρ1W) yt = Xt′β0 + εt

Temporal stationarity (for ρ1 = 0): |φ1| < 1

Spatial stationarity (for φ1 = 0): λMIN < ρ1 < λMAX , whereλMAX , λMIN � eigenvalues of W with highest and lowestabsolute value

In the standard case: row-wise normalisation of W and ρ1 > 0,the condition simpli�es to: ρ1 < 1.

Spatio-temporal stationarity (Franzese, Hays, 2008):∣∣∣FT (FS)−1∣∣∣ < 0

In the abovementioned standard case, and additionally underφ1 > 0, the condition simpli�es to: φ1 + ρ1 < 1.

Spatio-temporal multipliers

Non-spatial multipliers:

short-term: ∂yi∂xi′

long-term: ∂yi∂xi′

1−φ1

Spatial multipliers:

static (φ1 = 0) / short-term: ∂y∂xk′

= (I− ρ1W)−1β0,k

long-term: ∂y∂xk′

= (I− φ1L− ρ1W)−1β0,k

1−φ1

= (I− ρ1W)−1β0,k

= (I− φ1L− ρ1W)−1β0,k

1−φ1

= (I− ρ1W)−1β0,k

= (I− φ1L− ρ1W)−1β0,k

What if W is non-constant?

Most likely when we use a �gurative, non-geographical notionof distance.

In principle: not a problem, because one can de�ne thefollowing instead of (IT ⊗W):

The problem appears when the development in W isendogenous!

Network co-evolution models

Solution: network co-evolution models � Franzese, Hays,Kachi, 2011.

1 Consider the fact of smoking (y) among N = 500 employees ofa corporation over the period of T = 24 months.

2 Whether one is a smoker or not depends on a number ofindividual characteristics (X), but also on the smokingbehaviour in the network of one's colleagues (Wy).

3 Our network evolves over time. Some contacts die out, someintensify (Wt).

4 It might be the case that employees build their personalnetwork by i.a. socialising in the smoking area(Wt = f (yt−1)).

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