Macroeconometrics - Winter 2014, Y1 (Masters students)ajripatt/opetus/metrics/slides.pdf · Macroeconometrics Winter 2014, Y1 (Masters students) Antti Ripatti1 1Economics and HECER

Macroeconometrics

MacroeconometricsWinter 2014, Y1 (Masters students)

Antti Ripatti1

1Economics and HECERUniversity of Helsinki

version: February 24, 2014

Macroeconometrics

MacroeconometricsWinter 2014, Y1 (Masters students)

Antti Ripatti1

1Economics and HECERUniversity of Helsinki

version: February 24, 2014

Macroeconometrics

OUTLINE

1 Introduction

2 Difference equations and lag operators

3 Approximating and Solving DSGE Models

4 Moments of Model and Data

5 Matching moments

6 Filtering and Likelihood approach

Macroeconometrics

Introduction

OUTLINE

1 Introduction

2 Difference equations and lagoperators

3 Approximating and Solving DSGEModels


5 Matching moments

Macroeconometrics

Introduction

THE GOAL

The goal of the course is give an overview of the econometricmethods to study how to validate macroeconomic models.If time allows, we study the use dynamic macroeconomicsmodels to address empirical issues like

Model EstimationModel ComparisonForecastingMeasurementShock IdentificationPolicy Analysis

Macroeconometrics

Introduction

EVOLUTION OF EMPIRICAL MACRO – PART 1: BIRTH

OF LARGE MODELS

The representation of theoretical models as completeprobability models dates at least to Haavelmo (1944,Econometrica)Models were loosely related to economic theory:

Production functionsNational account identitiesOld-Keynesian consumption function

And estimated using OLS or instrumental variables methods(2SLS).Phillips curve was upward-sloping even in the long-run! Policytool.

Macroeconometrics

Introduction

EVOLUTION OF EMPIRICAL MACRO – PART 2B: SCENT

OF NEO-CLASSICAL SYNTHESIS

Developments in time series econometrics: cointegrationand error correction modelsSeparation of short-run (dynamics) and long-run(cointegration)Long-run (cointegration) builds on economic theoryNot yet general equilibriumAd-hoc dynamicsNo sensible steady-stateVulnerable to Lucas critique

AWM, BOF3, BOF4

Macroeconometrics

Introduction

EVOLUTION OF EMPIRICAL MACRO – PART 3A:KYDLAND–PRESCOTT

Methodological revolutionIntertemporally optimizing agents. Budget and technologyconstraints.

Conceptual revolutionIn a frictionless markets under perfect competitionbusiness cycles are efficient: no need for stabilization;stabilization may be counter-productive.Economic fluctuations are caused by technology shocks:they are the main source of fluctuation.Monetary factors (price level) has a limited (or no) roleCalibration

Macroeconometrics

Introduction

EVOLUTION OF EMPIRICAL MACRO – PART 3A:KYDLAND–PRESCOTT. . .

Implementation1 Pose a question2 Use theory to address the question3 Construct the model economu4 Calibrate5 Run the experiment within the model

Lack of statistical formality in calibration.

Macroeconometrics

Introduction

EVOLUTION OF EMPIRICAL MACRO – PART 3B:BAYESIAN ESTIMATION

Sargent (1989 JPE) presents the mapping of linearizedDSGE models into state-space representationDeJong, Ingram and Whiteman (2000, JoE, 2000, JAE)developed methods for mapping priors over structuralparameters into priors over corresponding parameters ofthe state-space representation enabling Bayesian inference.Smets and Wouters (2003) made a real workingapplication.

Macroeconometrics

Introduction

THIS COURSE

This course builds on Part 3 (Kydland and Prescott (1982)) andSims (2011). The aim is to learn how the model meets data.

Westationarize the model: restrictions implied by thebalanced growth pathapproximate it with the first order Taylor approximationreview (some of) the solution methodsrefresh the basic statistical tools to compute momentsfilter the model and the dataintroduce the Bayesian methods to study modelparameters

Macroeconometrics

Difference equations and lag operators

OUTLINE

1 Introduction


Lag operatorsDifference equations



5 Matching moments

Macroeconometrics


Lag operators

Macroeconometrics


Lag operators

This section builds mostly on Sargent (1987).

OUTLINE

1 Introduction





5 Matching moments

Macroeconometrics


Lag operators

LAG OPERATORS

The backward shift or lag operator is defined as

Lxt = xt−1

Lnxt = xt−n, for n = . . . ,−2,−1, 0, 1, 2, . . .

Note that if n < 0 the operator shifts xt forward.This language is loose, since we start with the sequence

{xt}∞t=−∞,

where xt is a real number. We operate on {xt}∞t=−∞ by L.

This section builds mostly on Sargent (1987).

Macroeconometrics


Lag operators

OPERATIONS WITH LAG OPERATOR I

Lag operator and multiplication operator are commutative

L(βxt) = βLxt.

It is distributive over the addition operator

L(xt + wt) = Lxt + Lwt.

Hence, we are free to use the standard commutative,associative, and distributive algebraic laws for multiplicationand addition to express the compound operator in analternative form.Examples

yt = (a + bL)Lxt = (aL + bL2)xt = axt−1 + bxt−2

Macroeconometrics


Lag operators

OPERATIONS WITH LAG OPERATOR II

or

(1− λ1L)(1− λ2L)xt = (1− λ1L− λ2L + λ1λ2L2)xt

= [1− (λ1 + λ2)L + λ1λ2L2]xt

= xt − (λ1 + λ2)xt−1 + λ1λ2xt−2.

andLc = c

andL0 = 1

Macroeconometrics


Lag operators

POLYNOMIALS IN THE LAG OPERATOR IPolynomial

A(L) = a0 + a1L + a2L2 + · · · =∞

∑j=0

ajLj,

where aj’s are constants.

A(L)xt = (a0 + a1L + a2L2 + · · · )xt

= a0xt + a1xt−1 + a2xt−2 + · · · =∞

∑j=0

ajxt−j.

Rational A(L)

(L) =B(L)C(L)

,

Macroeconometrics


Lag operators

POLYNOMIALS IN THE LAG OPERATOR IIwhere

B(L) =m

∑j=0

bjLj, C(L) =n

∑j=0

cjLj,

where bj, cj, m and n are constants.Simple example

A(L) =1

1− λL= 1 + λL + λ2L2 + · · · ,

which is "useful" only if |λ| < 1. Why?Then

11− λL

xt = (1 + λL + λ2L2 + · · · )xt =∞

∑i=0

λixt−i.

Macroeconometrics


Lag operators

POLYNOMIALS IN THE LAG OPERATOR IIIConsider the case |λ| > 1, the following "trick" is useful:

11− λL

=−(λL)−1

1− (λL)−1

= − 1λL

[1 +

1λ

L−1 +

(1λ

)2

L−2 + · · ·]

= − 1λ

L−1 −(

1λ

)2

L−2 −(

1λ

)3

L−3 + · · · .

Then

11− λL

xt = −1λ

xt+1 −(

1λ

)2

xt+2 + · · · = −∞

∑i=1

λ−ixt+i,

which is geometrically declining weighted sum of future valuesof xt.

Macroeconometrics


Difference equations

OUTLINE

1 Introduction





5 Matching moments

Macroeconometrics



DIFFERENCE EQUATION IConsider the following difference equation

yt = λyt−1 + bxt + a, t = . . . ,−1, 0, 1, . . . , (1)

where yt is an endogenous variable and xt is an exogenousvariable and sequences of real number and λ 6= 1. It can bewritten as

(1− λL)yt = a + bxt.

Multiply both sides by (1− λL)−1 to obtain

yt =a

1− λL+

b1− λL

xt + cλt

=a

1− λ+ b

∞

∑i=0

λixt−i + cλt. (2)

Macroeconometrics



DIFFERENCE EQUATION IIThe reason why we have the last term is that for any constant c

(1− λL)cλt = cλt − cλλt−1 = 0.

By multiplying both sides by 1− λL, gives the originaldifference equation! This the "general solution"!To get the "particular solution" we must be able to tie down theconstant c with an additional bit of information.An example of an additional bit of information is

limn→∞

∞

∑i=n

λixt−i = 0 for all t.

It simply says that λixt−i must be "small" for large i. If xt wereconstant, say x, all time, this requires |λ| < 1. This also resultsbounded constant term in the solution.

Macroeconometrics



DIFFERENCE EQUATION III

Assume t > 0 to see the impact of an arbitrary initial condition.Solution may be written as

yt = at−1

∑i=0

λi + a∞

∑i=t

λi + bt−1

∑i=0

λixt−i + b∞

∑i=t

λixt−i + cλt

= a1− λt

1− λ+ a

λt

1− λ+ b

t−1

∑i=0

λixt−i + bλt∞

∑i=0

λix0−i + cλt,

yt = a1− λt

1− λ+ b

t−1

∑i=0

λixt−i +λt

(a

1− λ+ b

∞

∑i=t

λix0−i + c

), t ≥ 1.

Macroeconometrics



DIFFERENCE EQUATION IVThe term in braces is equals y0! Hence

yt = a1− λt

1− λ+ b

t−1

∑i=0

λixt−i + λty0

=a

1− λ+ λt

(y0 −

a1− λ

)+ b

t−1

∑i=0

λixt−i, t ≥ 1.

The original difference equation (1) may be solved "forward" byapplying the "forward inverse".

yt =−(λL)−1

1− (λL)−1 a + b−(λL)−1

1− (λL)−1 xt + dλt,

=a

1− λ− b

∞

∑i=0

(1λ

)i+1

xt+i+1 + dλt, (3)

Macroeconometrics



DIFFERENCE EQUATION V

where d is constant to be determined by some side condition.An example of such side condition is

limn→∞

∞

∑i=n

(1λ

)i

xt+i = 0.

Macroeconometrics



DIFFERENCE EQUATION VI

Equivalence of (2) and (3)If a = 0, then for any value of λ 6= 1 both (2) and (3) representsolution to the difference equation (1). They are simplyalternative representation of the solution! The equivalence willhold whenever

b1− λL

xt (4)

and

b−(λL)−1

1− (λL)−1 xt (5)

are both finite for all t.

It often happens that one of them fails to be finite.

Macroeconometrics



DIFFERENCE EQUATION VIIIf the sequence {xt} is bounded and |λ| < 1, the (4) is aconvergent sum for all t.If the sequence {xt} is bounded and |λ| > 1, the (5) is aconvergent sum for all t.

Since our desire is to impose that the sequence {yt} is bounded,and we do not have sufficient side conditions, then we must setc = 0 in (2) or d = 0 (3). To see that

If λ > 1 and c > 0, then

limt→∞

cλt = ∞.

If λ < 1 and d < 0, then

limt→∞

dλt = ∞.

All of the above stuff means that we need to solve "stable roots"|λ| < 1 backward and "unstable roots" |λ| > 1 forward.

Macroeconometrics



SECOND-ORDER DIFFERENCE EQUATIONS I

Macroeconometrics

Approximating and Solving DSGE Models

OUTLINE

1 Introduction



Examples of dynamicmacroeconomic models

Real Business Cycle ModelThe Basic New-KeynesianModel

ApproximatingSolving

Blanchard and Kahn methodKlein methodMethod of undeterminedcoefficients


5 Matching moments

Macroeconometrics


Examples of dynamic macroeconomic models

OUTLINE

1 Introduction








5 Matching moments

Macroeconometrics



HOUSEHOLDS

Households chooses consumption labour and capital stock bymaximizing their discounted present value of utility streamfrom consumption and leisure

max E0

∞

∑t=0

βt

{c1−σ

t1− σ

− φl1+γt

1 + γ

}

subject to the budget constraint

ct + kt+1 = wtlt + rtkt + (1− δ)kt

Markets are complete and Arrow-Debreu securities.

Macroeconometrics



SETTING UP LAGRANGEAN I

The Lagrangean of the above problem can be written as

L = E0

∞

∑t=0

{βt c1−σ

t1− σ

− φl1+γt

1 + γ

− λt [ct + kt+1 −wtlt − rtkt − (1− δ)kt]}

Macroeconometrics



SETTING UP LAGRANGEAN II

The first order conditions with respect to ct, ct+1, lt and kt+1:

∂L∂ct

= c−σt − λt = 0

∂L∂ct+1

= βc−σt+1 − λt = 0

∂L∂lt

= −φlγt + λtwt = 0

∂L∂kt+1

= −λt + λt+1rt+1 + λt+1(1− δ) = 0.

Macroeconometrics



THEIR OPTIMALITY CONDITIONS

c−σt = λt

β Et c−σt+1 = λt+1

φlγt = λtwt

λt = λt+1(rt+1 + 1− δ)

Macroeconometrics



FIRMSMaximize profits

max Πt = yt −wtlt − rtkt

subject to the production technology

yt = Atkαt l1−α

t .

Optimality conditions

αAtkα−1t l1−α

t = rt

(1− α)Atkαt l1−α

t l−1t = wt.

Investments it solves from the law of motion of capital:

kt+1 = (1− δ)kt + it.

Macroeconometrics



TECHNICAL DEVELOPMENT

Techological development At follows AR(1) process:

log At = ρ log At−1 + zt,

wherezt ∼ N(0, σ2

z ).

Parameter ρ determines the persistence of the process.

Macroeconometrics



A COMPETITIVE EQUILIBRIUM

c−σt = β Et c−σ

t+1(rt+1 + 1− δ)

φlγt = λtwt

rt = αAtkα−1t l1−α

t

wt = (1− α)Atkαt l1−α

t l−1t

ct + it = yt

log At = ρ log At−1 + zt.

Macroeconometrics



SOLVING THE MODEL

First, we need to specify the functional forms (or make someassumptions regarding their first and second derivatives).

Second, in this course we focus on algorithms of linear rationalexpectation models: perturbation methods. Usual steps involve

Solving deterministic steady state.Linearization: approximating the nonlinear model withthe first order Taylor approximation.

Macroeconometrics



THE BASIC NEW-KEYNESIAN MODEL I

Let

yt output gap, yt = xt − xnt , where xt is actual output

and xnt is the natural level of output.

πt inflation rate (period-by-period), ie log(Pt/Pt−1),(multiply by 400 to get annualized %-change)

σ > 0 intertemporal elasticity of substitution,it nominal interest rate,

ρ ≡ − log β discount rate (degree of time preference)Et conditional expectation operator, E(·|Ft), whereFt contais all information available at (and up-to)period t.

Macroeconometrics



THE BASIC NEW-KEYNESIAN MODEL II

The expectation-augmented IS curve is given by

yt = Et yt+1 −1σ(it − Et πt+1 − ρ) + dt,

where dt is an exogenously given stochastic process, whoseproperties we specify later. We call it demand shock

Macroeconometrics



NEW-KEYNESIAN PHILLIPS CURVE

The New-Keynesian Phillips Curve is given by the followingequation

πt = β Et πt+1 + λ(yt + mt),

where λ is a parameter and mt is an exogenously givenstochastic process, whose other properties we specify later. Wecall it mark-up shock.

Macroeconometrics



MONETARY POLICY RULE

The third unknown endogenous variable it is determined bythe Taylor rule

it = ρ + φππt + φyyt.

According to the Taylor principle, the monetary policy (hereTaylor rule) should respond to inflation (deviation from thetarget — here zero) more than one-to-one, ie φπ > 1.

Macroeconometrics


Approximating

OUTLINE

1 Introduction








5 Matching moments

Macroeconometrics


Approximating

(LOG)LINEARIZING

The stationarized decision rules, budget constraints andequilibrium conditions can typically be written in the followingform

Et Ψ(Zt+1, Zt) = 0, (6)

where Zt and 0 are n× 1 vectors. Ψ(·) is a matrix valuedfunction that is continuous and differentiable. The conditionalexpectation operator Et uses information up to and includingperiod t.Note, that it is not restrictive to use the first order system, i.e.having only one lead. The higher order leads/lags may beintroduced by augmenting the state vector Zt (google“companion form”).

Macroeconometrics


Approximating

The goal is to approximate (6) with a linear system, which canthen be solved using the methods that will be described in thefollowing chapters.

NotationDenote the steady state by a variable without time index, Z.Small letter denotes log of original capital letter variable,zt ≡ log(Zt).

The deterministic steady state of (6) is

Ψ(Z, Z) = 0,

Note, that Z, the deterministic steady state of the model is anonlinear function of the model’s parameters µ.

Macroeconometrics


Approximating

Compute the first-order Taylor approximation around thesteady-state Z.

0 ≈ Ψ(Z, Z) +∂Ψ∂Zt

(Z, Z)︸︷︷︸≡A(µ)

×(Zt − Z) +∂Ψ

∂Zt+1(Z, Z)︸︷︷︸

B(µ)

×(Zt+1 − Z),

where Zt − Z is n× 1, ∂Ψ(Z, Z)/∂Zt denotes the Jacobian ofΨ(Zt+1, Zt) wrt Zt+1 evaluated at (Z, Z). To shorten

A(µ)(Zt − Z) + B(µ)Et(Zt+1 − Z) = 0.

The coefficient matrices are function of deep (model’s)parameters µ.Note that this is a mechanical step that is typically done by thesoftware (e.g. Dynare, Iris). Often the Jacobian may becomputed analytically.

Macroeconometrics


Approximating

LOGARITHMIC APPROXIMATION

Logarithmic approximation is a special case of above. Note thatZt = exp(log(Zt)), and, as denoted above, Zt = exp(zt).Suppose we have

f (Xt, Yt) = g(Zt), (7)

with strictly positive X, Y, Z (ie the linearization point). Thesteady state counterpart is f (X, Y) = g(Z).This simple summarization is, for example, in the slides by Jürg Adamek.

(http://www.vwl.unibe.ch/studies/3076_e/linearisation_slides.pdf)

http://www.vwl.unibe.ch/studies/3076_e/linearisation_slides.pdf

Macroeconometrics


Approximating

Start from replacing Xt = exp(log(Xt)) in (7),

f(

elog(Xt), elog(Yt))= g

(elog(Zt)

),

i.e.f (ext , eyt) = g (ezt) ,

Taking first-order Taylor approximations from both sides:

f (X, Y) + f ′1(X, Y)X(xt − x) + f ′2(X, Y)Y(yt − y)= g(Z) + g′(Z)Z(zt − z) (8)

Often we denote xt ≡ xt − x.

Macroeconometrics


Approximating

Divide both sides of (8) by f (X, Y) = g(Z) to obtain

1 + f ′1(X, Y)X(xt − x)/f (X, Y) + f ′2(X, Y)Y(yt − y)/f (X, Y)= 1 + g′(Z)Z(zt − z)/g(Z) (9)

Note thatf ′1(X, Y)X/f (X, Y)

is the elasticity of f (Xt, Yt) with respect to Xt at the steady-statepoint.Also note, that 100× (xt − x) tells Xt’s relative deviation fromthe steady-state point.

Macroeconometrics


Approximating

WARNING!

You may only loglinearize strictly positive variable. Typicalexample of a variable that may obtain negative values is the netforeign asset. This needs to be linearized!

Macroeconometrics


Approximating

USEFUL LOG-LINEARIZATION RULESDenote xt ≡ log(Xt)− log(X)

Xt ≈ X(1 + xt)

Xρt ≈ Xρ(1 + ρxt)

aXt ≈ X(1 + xt)

XtYt ≈ XY(1 + xt + yt)

Yt(a + bXt) ≈ Y(1a + bX) + aYyt + bXY(xt + yt)

Yt(a+ bXt + cZt) ≈ Y(a+ bX+ cZ)+Y(a+ bX+ cZ)yt + bXYxt + cZYzt

Xt

aYt≈ X

aY(1 + xt − yt)

Xt

Yt + aZt≈ X

Y + aZ

[1 + xt −

YY + aZ

yt −aZ

Y + aZzt

]

Macroeconometrics


Solving

OUTLINE

1 Introduction








5 Matching moments

Macroeconometrics


Solving

BLANCHARD AND KAHN METHOD I

Blanchard and Kahn (1980) develop a solution method basedon the following setup of a model[

x1t+1Et x2t+1

]= A

[x1tx2t

]+ Eft, (10)

wherex1t is n1 × 1 vector of endogenous predetermined variables= variables for which Et x1t+1 = x1t+1.

These are typically backward-looking variables.(Do not exist with t + 1.)For example kt+1 in the standard RBC model.

Macroeconometrics


Solving

BLANCHARD AND KAHN METHOD II

x2t is n2 × 1 vector of endogenous nonpredeterminedvariables = for which x2t+1 = Et x2t+1 + ηt+1, where ηt+1represents an expectational error.

forward-looking variablesjump-variables

ft contains k× 1 vector of exogenous forcing variables: e.g.shock innovations.A is full rank.

Macroeconometrics


Solving

Use Jordan normal form1 of the matrix A as follows

A = Λ−1JΛ,

where J is a diagonal matrix consisting of eigenvalues of A thatare ordered from in increasing value. It is partitioned as follows

J =[

J1 00 J2

],

wherethe eigenvalues in J1 lie on or within the unit circle (stableeigenvalues)the eigenvalues in J2 lie outside the unit circle (unstableeigenvalues)

1See also Spectral decomposition or Eigendecomposition or canonical form

Macroeconometrics


Solving

Matrix Λ contains the corresponding eigenvectors. It ispartitioned accordingly (and E too)

Λ =

[Λ11 Λ12Λ21 Λ22

]E =

[E1E2

]

Macroeconometrics


Solving

STABILITY CONDITION

Saddle-path stabilityIf the number of unstable eigenvalues is equal to the number ofnonpredetermined variables, the system is said to besaddle-path stable and an unique solution exists.

Other cases1 If the number of unstable eigenvalues exceeds the number

of nonpredetermined variables, no solution exists.2 If the number of unstable eigenvalues is smaller than the

number of nonpredetermined variables, there are infinitesolutions

Macroeconometrics


Solving

SADDLE-PATH CASE IRewrite (10) as[

x1t+1Et x2t+1

]= Λ−1JΛ

[x1tx2t

]+

[E1E2

]ft (11)

and premultiply by Λ to obtain[x1t+1

Et x2t+1

]=

[J1 00 J2

] [x1tx2t

]+

[E1E2

]ft,

where [x1tx2t

]≡[

Λ11 Λ12Λ21 Λ22

] [x1tx2t

][

E1E2

]≡[

Λ11 Λ12Λ21 Λ22

] [E1E2

].

Macroeconometrics


Solving

SADDLE-PATH CASE IIThe system is now “de-coupled” in the sense that thenonpredetermined variables are related only to theunstable eigenvalues J2 of A.Hence, we have two “seemingly unrelated” set ofequations.As in the univariate case, we derive the solution of thenonpredetermined variables by forward iteration andpredetermined variables by backward iteration.We start by analysing the lower block, ie the system of thenonpredetermined variables by performing the forwardinteration.Denote f2t of those fts that are conformable with E2.The lower part of (11) is as follows

x2t = J−12 Et x2t+1 − J−1

2 E2f2t (12)

Macroeconometrics


Solving

SADDLE-PATH CASE III

Shift it one period and use the law-of-iterated-expectations(Et(Et+1 xt) = Et xt)

Et x2t+1 = J−12 Et x2t+2 − J−1

2 E2 Et f2t+1

and substitute it back to (12) to obtain

x2t = J−22 Et x2t+2 − J−2

2 E2 Et f2t+1 − J−12 E2f2t (13)

Because J2 contains the eigenvalues above the unit disc,(J−12

)nwill asymptotically vanish. The iteration results

x2t = −∞

∑i=0

J−(i+1)2 E2 Et f2t+i

Macroeconometrics


Solving

SADDLE-PATH CASE IV

Using the definition x2t, we may write in to the form

x2t = −Λ−122 Λ21x1t −Λ−1

22

∞

∑i=0

J−(i+1)2 E2 Et f2t+i (14)

Finally, the upper part of (11) is given by

x1t+1 = A11x1t + A22x2t + E1ft, (15)

where A11 and A22 are reshuffled A according to the aboveordering.

Macroeconometrics


Solving

AN EXAMPLE WITH AR(1) SHOCK

Supposef2t+1 = Φf2t + εt+1, εt ∼ IID(0, Σ)

and Φ is full rank and its roots are within the unit disc (iestationary VAR(1)).Then

Et f2t+i = Φif2t, i ≥ 0

and (14) becomes

x2t = −Λ−122 Λ21x1t −Λ−1

22 (I−ΦJ−12 )−1E2f2t (16)

Macroeconometrics


Solving

“VAR” FORM

Stacking sets of equations (14) and (15) gives the following[x1t+1x2t+1

]︸︷︷︸≡xt+1

=

[A11 A22

−Λ−122 Λ21 0

]︸︷︷︸

≡F1

[x1tx2t

]︸︷︷︸≡xt

+

[E1 00 −Λ−1

22 (I−ΦJ−12 )−1E2

]︸︷︷︸

≡FΓ

[f1tf2t

]︸︷︷︸≡ft

.

(17)

iext+1 = F1xt + FΓft

Macroeconometrics


Solving

ISSUES

The model variables has to be classified eitherpredetermined or nonpredetermined. −→model-specificsystem reduction may be required.A has to be a full rank matrix. Hence, identities are notallowed!

Macroeconometrics


Solving

EXAMPLE: NK MODEL IN BK FORM I

Recall the basic New-Keynesian model:

Expectation-augmented IS curve

yt = Et yt+1 −1σ(it − Et πt+1 − ρ) + dt (18)

Phillips curve

πt = β Et πt+1 + λ(yt + mt), (19)

and the monetary policy rule

it = ρ + φππt + φyyt. (20)

Macroeconometrics


Solving

EXAMPLE: NK MODEL IN BK FORM IIAnd assume that dt and mt are white noise, ie independent withzero mean, and a constant variance.We aim finding the representation (10). Substitute (20) into (18)to obtain the following system of equations. After substitution,all variables yt and πt are non-predetermined.

yt = Et yt+1 −1σ

(φππt + φyyt − Et πt+1

)+ dt

πt = β Et πt+1 + λ(yt + mt).

or (1 +

φπ

σ

)yt = Et yt+1 −

1σ(φππt − Et πt+1) + dt

πt = β Et πt+1 + λ(yt + mt).

Macroeconometrics


Solving

EXAMPLE: NK MODEL IN BK FORM III

Moving expectation-related terms to the left:

Et yt+1 +1σ

Et πt+1 =

(1 +

φπ

σ

)yt +

φπ

σπt − dt

β Et πt+1 = −λyt + πt − λmt.

Same in the matrix form[1 1/σ0 β

]︸︷︷︸

A0

Et

[yt+1πt+1

]︸︷︷︸

Et xt+1

=

[1 + φy/σ φπ/σ−λ 1

]︸︷︷︸

A1

[ytπt

]︸︷︷︸

xt

+

[−1 00 −λ

]︸︷︷︸

E?

[dtmt

]︸︷︷︸

ft

ieA0 Et xt+1 = A1xt + E?ft. (21)

Macroeconometrics


Solving

EXAMPLE: NK MODEL IN BK FORM IVTo obtain form (10), we need to pre-multiply the previousequation by A−1

0 . Naturally, the parameter values need to bedetermined such that A0 is invertible. Then

Et xt+1 = A−10 A1︸︷︷︸≡A

xt + A−10 E?︸︷︷︸≡E

ft.

The matrices A and E are the following

A =

[1 + φy/σ + λ/(βσ) φπ/σ− 1/(βσ)

−λ/β 1/β

]E =

[−1 λ/(βσ)0 −λ/β

].

Macroeconometrics


Solving

KLEIN’S METHOD I

Klein (2000) proposes a method that overcome some of thedrawbacks of Blanchard and Kahn (1980) by allowing singularA. It is also computationally fast. The system has to be in theform

A Et xt+1 = Bxt + Eft, (22)

where ft (nz × 1 vector) follows the VAR(1)2. A and B are n× nmatrices, and E n× nz matrix.

ft = Φft−1 + εt, εt ∼ IID(0, Σ)

and A may be singular (reduced rank). This mean that we mayhave static equilibrium conditions (like identities) in thesystem.

Macroeconometrics


Solving

KLEIN’S METHOD II

Decompose xt to predetermined3 x1t and nonpredetermined x2tvariables as before. Then

Et xt+1 =

[x1t+1

Et x2t+1

].

As before the system will be de-coupled according to x1t(n1 × 1) and x2t (n2 × 1). Generalized Schur decomposition, thatallows singularity is used instead of standard spectraldecomposition. Applying it to A and B gives

QAZ = S (23)QBZ = T, (24)

Macroeconometrics


Solving

KLEIN’S METHOD IIIwhere Q, Z are unitary4 and S, T upper triangular matrices withdiagonal elements containing the generalized eigenvalues of Aand B. Eigenvalues are ordered as above.Z is partitioned accordingly

Z =

[Z11 Z12Z21 Z22

].

Z11 is n1 × n1 and corresponds the stable eigenvalues of thesystem and, hence, conforms with x1, ie predeterminedvariables.Next we triangularize the system (22) to stable and unstableblocks

zt ≡[

stut

]= ZHxt,

Macroeconometrics


Solving

KLEIN’S METHOD IV

where H denotes the Hermitian transpose. and (23) and (24) canbe written as

A = Q′SZH

B = Q′TZH.

Premultiplying the partitioned system by Q we obtain

S Et zt+1 = Tzt + QEft (25)

and since S and T are upper triangular, (22) may be written as[S11 S120 S22

]Et

[st+1ut+1

]=

[T11 T120 T22

] [stut

]+

[Q1Q2

]Eft. (26)

Macroeconometrics


Solving

KLEIN’S METHOD VDue to block-diagonal (recursive) structure, the process ut isunrelated st. We iterate this forward to obtain5

ut = −T−122

∞

∑i=0

[T−122 S22]

iQ2E Et ft+i.

Since ft is VAR(1), we obtain

ut = Mftvec M = [(Φ′ ⊗ S22)− In ⊗ T22]

−1 vec[Q2E].

This solution of the unstable component is used to solve thestable block, resulting

st+1 = S−111 T11st + S−1

11 [T12M− S12MΦ + Q1E]ft − Z−111 Z12Mεt+1.

Given the definition of ut and st, we may express the solution interms of original variables.

2We drop the constant term (and other stationary deterministic stuff) tosimplify algebra.

3Klein (2000) gives more general definition to predetermidness than BK.He also allows “backward-lookingness”, which means that the predictionerror is martingale difference process.

4U′U = I5See appendix B in Klein (2000).

Macroeconometrics


Solving

METHOD OF UNDETERMINED COEFFICIENTSFollowing Anderson and Moore (1985) (AiM) and Zagaglia(2005), DSGE model may be written in the form

H−1zt−1 + H0zt + H1 Et zt+1 = Dηt, (27)

where zt is vector of endogenous variables and ηt are pureinnovations with zero mean and unit variance.The solution to (27) takes the form

zt = B1zt−1 + B0ηt,

where

B0 = S−10 D,

S0 = H0 + H1B1.

B1 satisfies the identity

H−1 + H0B1 + H1B21 = 0.

Macroeconometrics


Solving

SUMMARIZING IThe general feature of the solution methods is that they result a“VAR(1)” representation of the model

ξt = F0(µ) + F1(µ)ξt−1 + FΓ(µ)vt, (28)

whereVariable vector ξt are the variables in the model.matrices Fi(µ) are complicated (and large) matrices whoseexact form depends on the solution method (See, eg,previous slide)They are also highly nonlinear function of the “deep”parameters µ of the economic model

The parameters µ include, among others, the parametersspecifying the stochastic processes of the model: shockvariances, for example

Macroeconometrics


Solving

SUMMARIZING II

We may use this representation to analytically calculatevarious model moments for given parameter values.We may use this also for simulating the model.Note that we do not know anything about the data at thisstage.

Macroeconometrics

Moments of Model and Data

OUTLINE

1 Introduction



UnivariateFrequency-Domain MethodsEstimationLinear Filters

4 Moments of Model and DataIntroductionAutocovariance functions andalikeSpectral analysisMeasuring Business CyclesMultivariate time series modelsFiltering the data and model

5 Matching moments

Macroeconometrics


Introduction

OUTLINE

1 Introduction





5 Matching moments

Macroeconometrics


Introduction

THE IDEA

The ultimate purpose of bringing model and data together is tovalidate the model. This means

comparing model and datausing data to calibrate (some) parameters of the modelfiguring out in where the model fits the data and where not

How to do it:computing model moments (statistics), andcomparing them to data moments (statistics)

More challenging task is to estimate parameters (or otherobjects) of the model.

Macroeconometrics


Introduction

HOW TO COMPUTE MODEL MOMENTS

(After all the hazzle of the previous section) the model is in thelinear form. Many statistics may be computed analyticallyusing the linear form.For some statistics this is not possible. The simulation (MonteCarlo) alternative:

N = 10000; % large numberfor iN = 1:Ndraw_shock_innovation;relying_on_the_solution_compute_endog._variables;compute_test_statistic;save_it_to_vector;

end;my_test_statistic = meanc(test_statistic)

Macroeconometrics


Introduction

COMPARING DATA AND MODEL

Note that the data moments are estimated!−→ there is uncertainty related to these estimates−→ use confidence bounds (of data moments) when

comparing data moments and model momentsVery often the comparison involves plotting. Do not forgetbounds in plots.Univariate comparisons are straightforward. Multivariatecomparison involves identification problem: impulse responses

Macroeconometrics


Introduction

WHAT MOMENTS?

Standard moments aremeans: great ratiosvariances: relative to, for example, outputcorrelation: with different lagssimilar moments in frequency domain

spectral densitycoherencegain

impulse responsesforecast error variance decompositionother VAR statistics

People are innovative in comparing, for example, forecasts!

Macroeconometrics


Introduction

STRUCTURE OF THIS SECTION

1 Introduce univariate or bivariate moments in time domain:auto and cross correlations

2 Same stuff in frequency domain3 Have a look at multivariate stuff: VARs

ExclusionsI exclude

univariate time series model (eg ARMA): you shouldknow these!nonlinear time series models (eg ARCH): the currentmodel solution methods (as we use them) do not supportnonlinearity

Macroeconometrics


Autocovariance functions and alike

OUTLINE

1 Introduction





5 Matching moments

Macroeconometrics



STOCHASTIC PROCESS

Definition (Stochastic process)A set of random variables

{Xt = (x1t, . . . , xmt)′|t ∈ T }

is called stochastic process, where t denotes time and Xt is anobservation that is related to time t.

Since this is infinite dimensional, we cannot determine its’ jointdistribution. According to Kolmogorov (1933), under certainregularity conditions we can characterize its joint distributionby its’ finite dimensional marginal distributions

FXt1 ,...,Xtp(t1, . . . , tp ∈ T ).

Large part of material of this section are based on the excellent lecture notes by Markku Rahiala

(http://stat.oulu.fi/rahiala/teachmr.html)

http://stat.oulu.fi/rahiala/teachmr.html

Macroeconometrics



STOCHASTIC PROCESS AND ITS PROPERTIES I

Definition (Stationarity)If

FXt1 ,...,Xtp= FXt1+τ ,...,Xtp+τ

for all p, τ, t1, . . . , tp, the process is strictly stationary.If it applies only to the first and second moments, ie{

E Xt ≡ µ = constant andΓ(τ) = E Xt+τX′t − µµ′ = cov (Xt+τ, Xt)

the process is weakly stationary.

Macroeconometrics



STOCHASTIC PROCESS AND ITS PROPERTIES II

Definition (Autocovariance function)

Γ(τ) = cov (Xt+τ, Xt)

is called autocovariance function.

Note that Γ(τ)′ = Γ(−τ). A weakly stationary process {Xt}that has Γ(τ) = 0 (for all τ 6= 0) is called white noise.

Macroeconometrics



STOCHASTIC PROCESS AND ITS PROPERTIES III

Definition (Autocovariance-generating function)For each covariance-stationary (weakly stationary) process Xtwe calculate the sequence of autcovariances {Γ(τ)}∞

τ=−∞. Ifthis sequence is absolutely summable, we may summarize theautocovariances through a scalar-valued function calledautocovariance-generating function

gX(z) =∞

∑τ=−∞

Γ(τ)zτ,

where z is complex scalar.

Macroeconometrics



STOCHASTIC PROCESS AND ITS PROPERTIES IV

Definition (Autocorrelation)Let {xt} be weakly stationary scalar process and γx(τ) itsautocovariance function. Then

γx(0) = var(xt) = σ2x = constant

and

ρx(τ) =γx(τ)

γx(0)=

cov(xt+τ, xt)

var(xt)

depends only on τ. The sequency ρx(τ) (for allτ = 0,±1,±2, . . . ).

Macroeconometrics



STOCHASTIC PROCESS AND ITS PROPERTIES V

Definition (Crosscorrelation)If Xt = (x1t · · · xmt)′ is multivariate, then function

ρxi,xj(τ) =cov(xi,t, xj,t+τ)√var(xi,t) var(xj,t)

is called cross correlation of components xi,t and xj,t.

Macroeconometrics



SAMPLE COUNTERPARTS AND SOME USEFULL

THEOREMS I

Definition (Sample autocorrelation)

For observations (x1, . . . , xn the sample autocorrelation is thesequence

rx(τ) =1

n−τ ∑n−τt=1 (xt − x)(xt+τ − x)1n ∑n

t=1(xt − x)2, τ = 0,±1,±2, . . . ,±(n− 1),

where x = 1n ∑n

t=1 xt.

To make inference about the size of sample autocorrelation, thefollowing theorems can be useful

Macroeconometrics



SAMPLE COUNTERPARTS AND SOME USEFULL

THEOREMS II

TheoremLet K be fixed positive integer and {εt} white noise. Then

√nR =

√n(rε(1) · · · rε(K))′ ∼asympt. NK(0, I),

when n approaches infinity.And

Qn = (n + 2)K

∑τ=1

(1− τ

n

)rε(τ)

2 ∼as. χ2K.

when n approaches infinity.

Similar results can be shown to sample crosscorrelation.

Macroeconometrics


Spectral analysis

OUTLINE

1 Introduction





5 Matching moments

Macroeconometrics


Spectral analysis

SPECTRAL ANALYSIS IThe spectral analysis section is based on the course material by Markku Lanna and Henri Nyberg, see the course“Empirical macroeconomics” http://blogs.helsinki.fi/lanne/teaching

In physics and engineering light is considered as electromagnetic waves that consists of in terms of energy,wavelength, or frequency.One may use prism to decompose light to spectrum (probably you did this in the high school).Similarly a stationary time series can be considered as constant electromagnetic wave.Economic time series can be seen as consisting of components with different periodicity.

trend, business cycle, seasonal component etc.In frequency-domain analysis, the idea is to measure the contributions of different periodic components in aseries.

Different components neet not be identified with regular(fixed) cycles, but there is only a tendency towards cyclicalmovements centered around a particular frequency.

For instance, the business cycle is typically defined asvariation with periodicity of 2–8 years.

Frequency-domain methods are complementary to time-domain methods in studying the properties of, e.g.,

detrending methods,

http://blogs.helsinki.fi/lanne/teaching

Macroeconometrics


Spectral analysis

SPECTRAL ANALYSIS II

seasonal adjustment methods,data revisions, andinterrelations between the business cycle components ofvarious economic variables.

Wave lengthwave length × frequency = constant (light speed in physics)

Macroeconometrics


Spectral analysis

PRELIMINARIESFrom high school algebra:

sin(x + y) = sin(x) · cos(y) + cos(x) · sin(y)

cos(x + y) = cos(x) · cos(y)− sin(x) · sin(y)

[sin(x)]2 + [cos(x)]2 ≡ 1

from which follows

b · cos(x) + c · sin(x) = a · cos(x + θ),

wherea =

√b2 + c2 and θ = arctan

( cb

)θ is called phase angle. Complex number

z = x + i · y ∈ C and conjugate z = x− i · y ∈ CEuler’s formula

ez = ex+i·y = exei·y = ex(cos(y) + i sin(y))

Macroeconometrics


Spectral analysis

SPECTRAL REPRESENTATION THEOREM

Any covariance-stationary process Xt can be expressed as sumof cycles

Xt = µ +∫ π

0[α(ω) cos(ωt) + δ(ω) sin(ωt)]dω.

The random processes α(·) and δ(·) have zero mean and theyare not correlated across frequencies.

Macroeconometrics


Spectral analysis

FIXED CYCLES I

Although economic time series are not characterized bycycles with fixed periodicity, let us first, for simplicity,consider fixed cycles to introduce the main concepts.For example, the cosine function is periodic, i.e., itproduces a fixed cycle.

y = cos (x) goes through its full complement of values (onefull cycle) as x (measured in radians) moves from 0 to 2π.The same pattern is repeated such that for any integer k,cos (x + 2πk) = cos (x).By defining x = ωt, where ω is measured in radians and t istime, y = cos (ωt) becomes a function of time t with fixedfrequency ω.

By setting ω equal to different values, y can be made toexpand or contract in time.

Macroeconometrics


Spectral analysis

FIXED CYCLES IIFor small ω (low frequency) it takes a long time for y to gothrough the entire cycle.For big ω (high frequency) it takes a short time for y to gothrough the entire cycle.Example: try to play with command plotcos(180*(pi/180)*(x-0)) in the sitewww.wolframalpha.com

Each frequency ω corresponds to the period of the cycle(denoted by p), which is the time taken for y to go throughits complete sequence of values.Given ω, how is the period p computed?

cos (ωt) = cos (ωt + 2π), so how many periods does it takefor ωt to increase by 2π?

2π/ω periods.

www.wolframalpha.com

Macroeconometrics


Spectral analysis

FIXED CYCLES III

A cycle with period 4 thus repeats itself every four periodsand has frequency ω = 2π/4 = π/2 ≈ 1.57 radians.

Macroeconometrics


Spectral analysis

FIXED CYCLES IAMPLITUDE AND PHASE

The range of y is controlled by multiplying cos (ωt) by theamplitude, ρ.The location of y along the time axis can be shifted byintroducing the phase, θ.Incorporating the amplitude and phase into y yields

y = ρ cos (ωt− θ) = ρ cos [ω (t− ξ)] ,

where ξ = θ/ω gives the shift in terms of time.

Macroeconometrics


Spectral analysis

FIXED CYCLES IIAMPLITUDE AND PHASE

ExampleSuppose the cosine function has a period of 4 so that it peaks att = 0, 4, 8, . . .. If we want it to have peaks at t = 2, 6, 10, . . ., i.e.ξ = 2, the phase θ = ξω = 2 (2π/4) = π. Note, however, thatthis phase shift is not unambiguous because the same effectwould have been obtained by setting ξ = −2, indicatingθ = ξω = −2 (2π/4) = −π.

Macroeconometrics


Spectral analysis

SPECTRUM I

The (power) spectrum (or spectral density) gives thedecomposition of the variance of a time series process interms of frequency.The spectral density function sy (ω) of a weakly stationaryprocess yt is obtained by evaluating the autocovariance-generating function (AGF) at e−iω and standardizing by2π:

sy (ω) =1

2πgy

(e−iω

)=

12π

∞

∑j=−∞

γje−iωj,

where i =√−1.

Macroeconometrics


Spectral analysis

SPECTRUM IIBy de Moivre’s theorem,

e−iωj = cos (ωj)− i sin (ωj) ,

we obtain

sy (ω) =1

2π

∞

∑j=−∞

γj [cos (ωj)− i sin (ωj)] ,

and making use of the fact that for a weakly stationaryprocess γ−j = γj and well-known results fromtrigonometry, sy (ω) simplifies to

sy (ω) =1

2π

[γ0 + 2

∞

∑j=1

γj cos (ωj)

].

Macroeconometrics


Spectral analysis

SPECTRUM III

Properties

Assuming the sequence of autocovariances{

γj}∞

j=−∞ isabsolutely summable, the spectrum of yt is a continuous,real-valued function of ω.If the process yt is weakly stationary, the spectrum isnonnegative for all ω.It suffices to consider sy (ω) in the range [0, π].

The spectrum is symmetric around zero, becausecos (−ωj) = cos (ωj).The spectrum is a periodic function of ω, i.e.,sy (ω + 2πk) = sy (ω), because cos [(ω + 2πk) j] = cos (ωj)for any integers k and j.

Example: White Noise

Macroeconometrics


Spectral analysis

SPECTRUM IV

To get some intuition of how the spectrum decomposes thevariance of a time series in terms frequency, let us first, forsimplicity, consider the white noise process.For a white noise process yt = εt, γ0 = σ2

ε and γj = 0 forj 6= 0.

Hence, the AGF gy (z) = σ2ε and the spectrum is flat:

sy (ω) =1

2πγ0 =

σ2ε

2π.

The area under the spectrum over the range [−π, π] equalsσ2

ε : ∫ π

−πsy (ω) dω =

σ2ε

2π(π + π) = σ2

ε .

Macroeconometrics


Spectral analysis

SPECTRUM V

FactThe area under the spectrum over the range [−π, π] always equalsthe variance of yt. Comparisons of the height of sy (ω) for differentvalues of ω indicate the relative importance of fluctuations at thechosen frequencies in influencing variations in yt.

It is useful to relate the radians ω to the associated period(the number of units of time it takes the cyclical componentwith frequency ω to complete a cycle), p = 2π/ω.

Macroeconometrics


Spectral analysis

SPECTRUM VI

ExampleThe period associated with frequency ω = π/2 is2π/ω = 2π/ (π/2) = 4. With annual data, a cyclicalcomponent with frequency π/2 thus corresponds to a cyclewith a periodicity of 4 years. With quarterly data it correspondsto a cycle with a periodicity of 4 quarters or 1 year.

Example: MA(1)

The AGF of the MA(1) process is

gy (z) = σ2ε (1 + θ1z)

(1 + θ1z−1

).

Macroeconometrics


Spectral analysis

SPECTRUM VII

Thus

sy (ω) =σ2

ε

2πgy

(e−iω

)=

σ2ε

2π

(1 + θ1e−iω

) (1 + θ1eiω

)=

σ2ε

2π

(1 + θ1eiω + θ1e−iω + θ2

1eiω−iω)

=σ2

ε

2π

(1 + θ2

1 + 2θ1 cos (ω))

.

Example: AR(1)

Macroeconometrics


Spectral analysis

SPECTRUM VIII

The AGF of the AR(1) process is

gy (z) = σ2ε

1φ (z) φ (z−1)

=1

(1− φ1z) (1− φ1z−1).

Macroeconometrics


Spectral analysis

SPECTRUM IXThus

sy (ω) =σ2

ε

2πgy

(e−iω

)=

σ2ε

2π

1(1− φ1e−iω) (1− φ1eiω)

=σ2

ε

2π

1(1− φ1e−iω) (1− φ1eiω)

=σ2

ε

2π

11− φ1e−iω − φ1eiω + φ2

1

=σ2

ε

2π

11 + φ2

1 − 2φ1 cos (ω).

Example: AR(2)

Macroeconometrics


Spectral analysis

SPECTRUM X

The spectrum of the AR(2) process need not bemonotonically decreasing, but its peak depends on thevalues of φ1 and φ2.A peak at ω∗ indicates a tendency towards a cycle at afrequency around ω∗.

This stochastic cycle is often called a pseudo cycle as thecyclical movements are not regular.

The peak of the spectrum can be found by setting thederivative of sy (ω) equal to zero and solving for ω.

Macroeconometrics


Spectral analysis

SPECTRUM XIThe AGF of an AR(2) process is

gy (z) = σ2ε

1φ (z) φ (z−1)

=σ2

ε

(1− φ1z− φ2z2) (1− φ1z−1 − φ2z−2).

Thus

sy (ω) =σ2

ε

2πgy

(e−iω

)=

σ2ε

2π

1(1− φ1e−iω − φ2e−i2ω) (1− φ1eiω − φ2ei2ω)

=σ2

ε

2π

11 + φ2

1 + φ22 − 2φ1 (1− φ2) cos (ω)− 2φ2 cos (2ω)

Macroeconometrics


Spectral analysis

ESTIMATION OF SPECTRUM ISAMPLE PERIODOGRAM

The spectrum of an observed time series can, in principle,be estimated by replacing the autocovariances γj in thetheoretical spectrum

sy (ω) =1

2π

[γ0 + 2

∞

∑j=1

γj cos (ωj)

]

by their estimates γj to obtain the sample periodogram

sy (ω) =1

2π

[γ0 + 2

T−1

∑j=1

γj cos (ωj)

].

Macroeconometrics


Spectral analysis

ESTIMATION OF SPECTRUM IISAMPLE PERIODOGRAM

This yields a very jagged and irregular estimate of thespectrum.

Because we are, effectively, estimating T parameters with Tobservations, this pattern persists irrespective of thesample size.

Nonparametric Estimation

It is reasonable to assume that sy (ω) will be close to sy (λ)when ω is close to λ.

Macroeconometrics


Spectral analysis

ESTIMATION OF SPECTRUM IIISAMPLE PERIODOGRAM

This suggests that sy (ω) might be estimated by a weightedaverage of the values of sy (λ) for values of λ in theneighborhood around ω, with the weights depending onthe distance between ω and λ:

sy(ωj)=

h

∑m=−h

κ(ωj+m, ωj

)sy(ωj+m

).

h is a bandwidth parameter indicating how manyfrequencies around ωj are included in estimating sy

(ωj).

Macroeconometrics


Spectral analysis

ESTIMATION OF SPECTRUM IVSAMPLE PERIODOGRAM

The kernel κ(ωj+m, ωj+m

)gives the weight of each

frequency, and these weights sum to unity,

h

∑m=−h

κ(ωj+m, ωj

)= 1.

Modified Daniell Kernel

Several kernels are available for the nonparametricestimation of the spectrum.The default kernel in R is the modified Daniell kernel,possibly used repeatedly.

The kernel puts half weights at end points.

Macroeconometrics


Spectral analysis

ESTIMATION OF SPECTRUM VSAMPLE PERIODOGRAM

For example, with bandwidth parameter h = 1, the weights

of sy

(ωj−1

), sy

(ωj

)and sy

(ωj+1

)are 1/4, 2/4 and 1/4,

respectively.

Applying the same kernel again yields weights 1/16, 4/16,6/16, 4/16, 1/16 for sy

(ωj−2

), ..., sy

(ωj+2

), respectively.

Choosing the bandwidth parameter is arbitrary. Onepossibility is to plot the estimated spectrum based onseveral different bandwidths and rely on subjectivejudgment.

Autoregressive Spectral Estimation

An estimate of the spectrum can be based on an AR(p)model fitted to the observed time series:

Macroeconometrics


Spectral analysis

ESTIMATION OF SPECTRUM VISAMPLE PERIODOGRAM

1 Estimate an adequate AR(p) model for yt.2 Plug the estimated coefficients φ1, φ2, ..., φp in the

expression of sy (ω) derived for the AR(p) model.

The order of the AR model, p, must be carefully chosen.

If p is too small, i.e., the AR model does not adequatelycapture the dynamics of yt, the ensuing spectrum may bemisleading.If p is too large, the spectrum may be inaccuratelyestimated.

Macroeconometrics


Spectral analysis

LINEAR FILTERS I

Many commonly used methods of extracting the businesscycle component of macroeconomic time series (andseasonal adjustment methods) can be expressed as linearfilters,

yt =s

∑j=−r

wjxt−j

where, say, xt is the original series and yt its business cyclecomponent.A filter changes the relative importance of the variouscyclical components. To see how different frequencies areaffected, the spectra of the unfiltered and filtered series canbe compared.

Macroeconometrics


Spectral analysis

LINEAR FILTERS II

A more direct way is to inspect the squared frequencyresponse function that gives the factor by which the filteralters the spectrum of yt.The filter may also induce a shift in the series with respectto time, and this is revealed by the phase diagram.

Squared Frequency Response Function Assuming the originalseries xt has an infinite-order MA representation xt = ψ (L) εt,the filtered series can be written asyt = W (L) xt = W (L)ψ (L) εt, whereW (L) = w−rL−r + · · ·+ w−1L−1 + w0 + w1L + · · ·wsLs.

Macroeconometrics


Spectral analysis

LINEAR FILTERS IIIThe spectrum of yt thus equals

sy (ω) =1

2πgy

(e−iω

)=

σ2ε

2πW(

e−iω)

ψ(

e−iω)

W(

eiω)

ψ(

eiω)

=σ2

ε

2πW(

e−iω)

W(

eiω)

ψ(

e−iω)

ψ(

eiω)

= W(

e−iω)

W(

eiω)

sx (ω) .

The term W(e−iω)W

(eiω) is called the squared frequency

response function and it shows how the filtering changes thespectrum of the original series.Squared Frequency Response Function:

Macroeconometrics


Spectral analysis

LINEAR FILTERS IVExampleConsider taking first differences of a variable xt,

yt = xt − xt−1 = (1− L) xt = W (L) xt.

Nowsy (ω) = W

(e−iω

)W(

eiω)

sx (ω) ,

i.e., the squared frequency response function equals

W(

e−iω)

W(

eiω)

=(

1− e−iω) (

1− eiω)

= 2− eiω − e−iω

= 2− [cos (ω)− i sin (ω) + cos (ω) + i sin (ω)]

= 2 [1− cos (ω)] .

Macroeconometrics


Spectral analysis

LINEAR FILTERS: GAIN I

In the literature, the properties of filters are often illustratedusing the gain function, G (ω), which is just the modulus of thesquared frequency response function (squared gain):

G (ω) =√

W (e−iω)W (eiω) ≡∣∣∣W (

e−iω)∣∣∣ .

Macroeconometrics


Spectral analysis

LINEAR FILTERS: GAIN II

ExampleFor the first-difference filter (1− L), the gain function is givenby

G (ω) =√(1− e−iω) (1− eiω)

=√

2 [1− cos (ω)].

This carries the same information as the squared frequencyresponse function.

Try plot sqrt(2*(1-cos(w)))

Macroeconometrics


Spectral analysis

LINEAR FILTERS: GAIN III

Given the gain function G(ω), the relationship betweensy(ω) and sx(ω) can be written

sy(ω) = G(ω)2sx(ω).

The gain function expresses how the employed filter serveto isolate cycles.

Gain function with the value 0 indicates that the filtereliminates those frequencies.

Macroeconometrics


Spectral analysis

LINEAR FILTERS: PHASE DIAGRAM I

The squared frequency response function W(e−iω)W

(eiω)

takes on real values only. However, the frequency responsefunction W

(e−iω) is, in general a complex quantity,

W(

e−iω)= W∗ (ω) + iW† (ω)

where W∗ (ω) and W† (ω) are both real.The phase diagram, i.e., the slope of the phase function

Ph (ω) = tan−1[−W† (ω) /W∗ (ω)

]gives the delay in time periods.

Macroeconometrics


Spectral analysis

LINEAR FILTERS: PHASE DIAGRAM IIThe phase diagram measures the shift that how much thelead-lag relationships in time series are altered by the filter.

ExampleConsider a filter that shifts the series back by three timeperiods,

yt = xt−3 = L3xt = W (L) xt.

The frequency reponse function equalsW(e−iω) = e−3iω = cos (3ω)− i sin (3ω) Thus

Ph (ω) = tan−1 [sin (3ω) / cos (3ω)] = tan−1 [tan (3ω)] = 3ω.In other words, the phase diagram is a straight line with slope3.

Phase Diagram: Symmetric Filter

Macroeconometrics


Spectral analysis

LINEAR FILTERS: PHASE DIAGRAM IIIMost filters used to extract the business cycle component ofmacroeconomic time series are symmetric.For a symmetric linear filter, the phase function equals zero,i.e., they exhibit no phase shift.The frequency response function equals

W(

e−iω)

= w−re−riω + · · ·+ w−1e−iω + w0

+w1eiω + · · ·+ wreriω

=r

∑j=−r

wj cos (ωj)− ir

∑j=−r

wj sin (ωj)

= w0 + 2r

∑j=1

cos (ωj) .

Macroeconometrics


Spectral analysis

LINEAR FILTERS: PHASE DIAGRAM IV

Hence, W† (ω) = 0, indicating thatPh (ω) = tan−1 [−W† (ω) /W∗ (ω)

]= 0.

Macroeconometrics


Spectral analysis

CROSS-SPECTRAL ANALYSIS I

In addition to the spectra of single time series, therelationships between pairs of variables can be examinedin the frequency domain.The spectrum can be generalized for the vector case, andthe cross spectrum between yt and xt is definedanalogously with the spectrum of a single series,

syx (ω) =1

2π

∞

∑τ=−∞

γyx (τ) e−iωτ.

Instead of the cross spectrum, functions derived from it(phase and coherence) allow for convenient interpretationof the relationship between two variables in the frequencydomain.

Macroeconometrics


Spectral analysis

CROSS-SPECTRAL ANALYSIS II

Given data on y and x, the cross spectrum and the derivedfunctions can be computed analogously to the powerspectrum. In particular, some smoothing is required toobtain consistent estimators of the phase and coherence.

The cross-spectral density function can be expressed in terms ofits real component cyx(ω) (the cospectrum) and imaginarycomponent qyx (the quadrature spectrum)

syx(ω) = cyx(ω) + i qyx(ω).

In polar form, the cross-spectral density can be written

syx(ω) = R(ω) eiθ(ω),

Macroeconometrics


Spectral analysis

CROSS-SPECTRAL ANALYSIS III

whereR(ω) =

√cyx(ω)2 + qyx(ω)2

and

θ(ω) = arctan−qyx(ω)

cyx(ω).

The function R(ω) is the gain function.θ(ω) is called the phase function.

Macroeconometrics


Spectral analysis

PHASE DIAGRAM

The phase function has the same interpretation as in thecase of a linear filter.

If in the plot of the phase function against ω (the phasediagram), the phase function is a straight line over somefrequency band, the direction of the slope tells which seriesis leading and the amount of the slope gives the extent ofthe lag.

The lag need not be an integer number.If the slope is zero there is no lead-lag relationship.If the slope is positive, the first variable is lagging.If the slope is negative, the first variable leading.

A confidence band may be computed around the estimatedphase function to evaluate statistical significance.

Macroeconometrics


Spectral analysis

COHERENCE

The coherence measures the strength of the relationshipbetween two variables, yt and xt, at different frequencies.The (squared) coherence is defined analogously tocorrelation:

C (ω) =

∣∣syx (ω)∣∣2

sx (ω) sy (ω),

and it can be interpreted as a measure of the correlationbetween the series y and x at different frequencies.0 ≤ C (ω) ≤ 1If C (ω) is near one, the ω-frequency components of y andx are highly (linearly) related, and if C (ω) is near zero,these components are only slightly related.A confidence band can be used to evaluate the statisticalsignificance.

Macroeconometrics


Measuring Business Cycles

OUTLINE

1 Introduction





5 Matching moments

Macroeconometrics



(This relies on Lanne’s and Nyberg’s material).

Macroeconometrics



METHODS FOR EXTRACTING THE CYCLICAL

COMPONENT

A number of methods are commonly used to extract thebusiness cycle component of macroeconomic time series.In this course, we concentrate on:

Linear detrending (fitting a linear trend model)DifferencingBeveridge-Nelson decompositionFilters

As discussed above, a reasonable detrending method must1 allow for variation over time in the growth rate and2 ensure that short-term fluctuations are correctly

categorized as cyclical deviations from trend.

In addition, the detrended time series must be stationary,i.e., its frequency response must be zero at frequency zero.

Macroeconometrics



LINEAR DETRENDING AND DIFFERENCING

These two methods are conducted under the implicitassumption that the growth rate of yt is constant.In linear detrending, a constant term and a linear trend isregressed to yt. In other words, yt = α0 + α1t + ct).

The estimated cyclical component is ct = yt − α0 − α1t.Parameter α1, and its estimate α1, can be interpreted as atrend coefficient.

Macroeconometrics



LINEAR DETRENDING AND DIFFERENCING

The first-order difference leads to the cyclical component

ct = yt − yt−1.

The choice between detrending and differencing dependson which one provides a more appropriate representationof yt = log(Yt).In practice, the choice of either specification is problematicas the data do not appear to follow a constant averagegrowth rate throughout the sample period.

Macroeconometrics



BEVERIDGE-NELSON DECOMPOSITION IStarting point: Any I(1) process yt consists of thestationarity component ηt and the permanent (trend)component ξt, yt = ξt + ηt.Any I(1) process yt whose first difference ∆yt satisfies

∆yt = ψ (L) εt =∞

∑j=0

ψjεt−j,

where ∑∞j=0 j

∣∣ψj∣∣ < ∞, E(∆yt) = 0, with εt a white noise

process, can be written as the sum ofa random walk ψ(1)εt or equivalently ∆ξt = ψ(1)εta stationary process ηt, andinitial conditions y0 − η0.

ηt is often interpreted as the cyclical component ct of theseries yt.

Macroeconometrics



BEVERIDGE-NELSON DECOMPOSITION IIN PRACTICE

The Beveridge-Nelson decomposition can be applied in thefollowing steps

1 Identify an appropriate ARMA model for ∆yt, estimate theparameters ψ(1) and save the residuals εt.

2 Given the initial value of ξ0, the trend component can begenerated following the presentation ∆ξt = ψ(1)εt.

3 Construct the cyclical component ct = yt − ξt.

Macroeconometrics



BEVERIDGE-NELSON DECOMPOSITION IIIN PRACTICE

ExampleAssume that ARMA(1,1) model is an appropriate model for∆yt. This means that ψ(1) = θ(1)

φ(1) =1+θ11−φ1

. Given the initial value

ξ0, the estimated parameters θ1 and φ1 and residuals εt, thetrend component can be constructed as ∆ξt =

1+θ11−φ1

εt.

Macroeconometrics



FILTERSUSING FILTERS TO ISOLATE CYCLES

Filters are tools designed to eliminate the influence ofcyclical variation at various frequencies.Low frequencies are associated with cycles of long periodsof oscillation (infrequent shifts from a peak to trough) andvice versa with high frequencies.

Slowly evolving trends (i.e. cycles with an infiniteperiodicity) are associated with very low frequencies.Constant trend is associated with zero frequency ω = 0.

Macroeconometrics



HIGH-PASS AND BAND-PASS FILTERS I

Two kinds of filters are used to extract the business cyclecomponent (with period of 2–8 years) of quarterly timeseries.

A high-pass filter passes only components with periodicityless than or equal to 32 quarters (or with frequency ωgreater than or equal to 2π/32 = π/16 ≈ 0.20).

The frequency response function equals 0 for ω < π/16 and1 for ω ≥ π/16.A high-pass filter passes also variation associated withseasonal frequencies, so it can only be applied to seasonallyadjusted data.

A band-pass filter passes only components with periodicitybetween 8 and 32 quarters (or with frequency between2π/8 = π/4 ≈ 0.79 and 2π/32 = π/16 ≈ 0.20).

Macroeconometrics



HIGH-PASS AND BAND-PASS FILTERS II

The frequency response function equals 1 forπ/16 ≤ ω ≤ π/4 and zero otherwise.

With a finite number of observations it is not possible tofilter out exactly the desired frequencies.

Macroeconometrics



HODRICK-PRESCOTT FILTER I

The Hodrick-Prescott filtering is probably the mostcommonly used method of extracting business cyclecomponents in macroeconomics.The general idea is to compute the growth (trend)component gt and cyclical component ct of yt byminimizing the magnitude

T

∑t=1

(yt − gt)︸︷︷︸ct

2 + λT−1

∑t=1

[(gt+1 − gt)− (gt − gt−1)]2 .

The second term minimizes the changes in the growth rateover time while the first term bringing gt as close aspossible to the observed series yt.

Macroeconometrics



HODRICK-PRESCOTT FILTER IIThe smoothing parameter λ tells how much weight is givento the first objective.

If λ = 0, gt = yt (no smoothing).The greater λ is, the smoother the growth component. Whenλ→ ∞, gt is a straight line.

Using the lag operator L, the cyclical component producedby the Hodrick-Prescott filter can be written as (see, Baxterand King, 1999)

ct =

[λ (1− L)2 (1− L−1)2

1 + λ (1− L)2 (1− L−1)2

]yt.

The filter removes unit root components.

Macroeconometrics



HODRICK-PRESCOTT FILTER III

The filter is a symmetric infinite-order two-sided movingaverage.

ct depends on past and future values of yt, t = 1, 2, ..., T.Therefore, the first and last values of ct are inaccurate andmay change considerably as new observations becomeavailable.The filter introduces no phase shift.

Macroeconometrics



HODRICK-PRESCOTT FILTERGAIN

With quarterly data λ is most often set to 1600, whichproduces a filter reasonably close to the ideal high-passfilter.With annual data three values of λ, 400, 100 and 10 are themost common. In this case the filter is not so close to theideal filter.

Macroeconometrics



BAXTER-KING FILTER

The cyclical component extracted by all high-pass andband-pass filters are infinite two-sided symmetric movingaverages of yt,

ct =∞

∑j=−∞

wjyt−j.

Baxter and King suggest approximating these filters bytruncating,

ct =K

∑j=−K

wjyt−j,

and selecting the weights wj based on the frequencies thatare filtered out.

K observations at the beginning and end of the series arelost.

Macroeconometrics



BAXTER-KING FILTERFREQUENCY RESPONSES: HIGH-PASS FILTER (BAXTER& KING (1999)

Macroeconometrics



BAXTER-KING FILTERFREQUENCY RESPONSES: BAND-PASS FILTER

Macroeconometrics


Multivariate time series models

OUTLINE

1 Introduction





5 Matching moments

Macroeconometrics



This material is based on professor Lanne’s“Macroeconometrics” course.

Macroeconometrics



VAR MODEL I

The VAR(p) model is a generalization of the univariate AR(p)model. For example, a two-variable VAR(2) model can bewritten as

y1t = φ1,11y1,t−1 + φ1,12y2,t−1 + φ2,11y1,t−2 + φ2,12y2,t−2 + u1t

y2t = φ1,21y1,t−1 + φ1,22y2,t−1 + φ2,21y1,t−2 + φ2,22y2,t−2 + u2t

or in matrix form,[y1ty2t

]=

[φ1,11 φ1,12φ1,21 φ1,22

] [y1,t−1y2,t−1

]+[

φ2,11 φ2,12φ2,21 φ2,22

] [y1,t−2y2,t−2

]+

[u1tu2t

]

Macroeconometrics



VAR MODEL IIor

yt = Φ1yt−1 + Φ2yt−2 + ut.

Let yt = (y1t · · · yKt)′ be a K× 1 vector.

Definition (VAR(p) model)VAR(p) model is

yt = Φ1yt−1 + · · ·+ Φpyt−p + ut ut ∼ NIDm(0, Σ).

Denote matrix polynomial Φ(L) ≡ I−Φ1L− · · · −ΦpLp.

ut = (u1t, ..., uKt)′ is a K-dimensional white noise process:

E (ut) = 0,E (utu′t) = Σu, andE (utu′s) = 0 for s 6= t.

Macroeconometrics



VAR MODEL IIIIn subsequent derivations, it is often convenient to write theVAR(p) model using the lag operator L,

yt = Φ1Lyt+Φ2L2yt + · · ·+ ΦpLpyt + ut

or (IK −Φ1L−Φ2L2 − · · · −ΦpLp

)yt = ut

orΦ (L) yt = ut.

The VAR(p) model may also contain deterministic termssuch as an intercept, linear trend or seasonal dummies.The VAR(p) model is a simple linear model to describe thejoint probability distribution of the data.It is hard to interprete and, as such, rarely tells nothingabout the cause and effect.

Macroeconometrics



COMPANION FORM I

Sometimes it is more convenient to use the so-calledcompanion form of the VAR(p) model. The VAR(p) process

yt = Φ1yt−1 + · · ·+ Φpyt−p + ut

can be written as the following VAR(1) process

ξt = Fξt−1 + vt,

where

ξt ≡

yt

yt−1...

yt−p+1

,

Macroeconometrics



COMPANION FORM II

F ≡

Φ1 Φ2 · · · Φp−1 ΦpIK 0 · · · 0 00 IK · · · 0 0...

... · · ·...

...0 0 · · · IK 0

,

vt ≡

ut0...0

and

E(vtv′s

)=

{Q t = s0 otherwise

Macroeconometrics



COMPANION FORM III

with

Q ≡

Σu 0 · · · 00 0 · · · 0...

... · · ·...

0 0 · · · 0

.

Macroeconometrics



STATIONARITY I

The VAR(p) process yt is covariance-stationary (weaklystationary) if E (yt) and E

(yty′t−j

)do not depend on the date t.

In this case, the consequences of any given shock ut musteventually die out. Weak stationarity imposes the followingstability condition: a VAR(p) process yt is weakly stationary ifthe roots of the equation∣∣∣IK −Φ1z−Φ2z2 − · · · −Φpzp

∣∣∣ = 0

are greater than unity in absolute value. To see how thiscondition arises, consider for simplicity the VAR(1) process

yt = Φ1yt−1 + ut.

Macroeconometrics



STATIONARITY IIBy recursive substitution, this can be written as

yt = Φj+11 yt−j−1 +

j

∑i=0

Φi1ut−i,

and the effect of any shock will die out only if Φj1 → 0 as

j→ ∞. From linear algebra we know that this, in turn, is thecase only if all the eigenvalues of Φ1 are less than unity inabsolute value, which is equivalent to the equation

|IK −Φ1z| = 0

having all roots greater than unity in absolute value. Because aVAR(p) process can be written in the companion form, thiscondition holds for a general VAR process. It can generally be

Macroeconometrics



STATIONARITY III

expressed as the matrix F of the VAR(1) form having alleigenvalues less than unity in absolute value or the equation

|IK − Fz| =∣∣∣IK −Φ1z−Φ2z2 − · · · −Φpzp

∣∣∣ = 0

having all roots greater than unity in absolute value.

Macroeconometrics



ESTIMATION AND TESTING

The unrestricted stationary VAR can be estimatedconsistently by OLS, equation by equation.If the distribution of ut is known, it can be estimated bymaximum likelihood.Under normality of ut, ML estimator is numerically equalto OLS. (Otherwise quasi-ML)Under regularity conditions (incl. stationarity), the OLS(ML) is asymptotically normal−→ standard t-tests can be applied

Macroeconometrics



MODEL CHECKING

The error term ut is assumed to be normal white noise. Theproperties of the estimated error term (residuals) may be studiedby

autocorrelation, and cross-correlation of individualresidual seriesPormanteau test and LM-test for (vector) autocorrrelation.Autoregressive conditional heteroscedasticity may bestudied by squared residuals.and normality by Jarque-Bera test.

Macroeconometrics



MOVING AVERAGE REPRESENTATION IThe roots of the VAR representation corresponds to the mpfactors of equation

det[Φ(s−1)

]= 0.

If they lie inside of the unit circle, and all elements in Xt arestationary the process Xt is stationary and we have the Wolddecomposition

yt = Φ(L)−1ut =∞

∑j=0

Ψjut−j,

where m×m coefficient matrices Ψj (j = 0, 1, 2, . . . ) are calledimpulse responses. They can be computed recursively from

Ψi = Φ1Ψi−1 + Φ2Ψi−2 + · · ·+ ΦpΨi−p i = 1, 2, . . . ,

Macroeconometrics



MOVING AVERAGE REPRESENTATION II

with Ψ0 = Im and Ψi = 0 for i < 0. They tell how earlierinnovations ut−j are reflected by yt observations.The MA(∞) representation allows for tracing out the dynamiceffects of shocks to the variables. i.e.,

Ψs =∂yt+s

∂u′t.

The row i, column j element of Ψs gives the effect of a one-unitincrease in the error of the jth variable at time t on the ithvariable at time t + s, holding all other errors constant. Forinstance, the first column of Ψ1 gives the effect in period 1 of aunit increase in the error of the first variable in period 0 on eachvariable in the system.

Macroeconometrics



STRUCTURAL VAR I

The problem is that each component in ut may correlate withanother component. Identification problem arises. Theinnovations ut may be orthogonalized by Choleskydecomposition:Let εt be orthogonalized innovation so that

E εtε′t = I.

Define a matrix B such that

B−1ΣB′−1= I.

which impliesΣ = BB′. (29)

Macroeconometrics



STRUCTURAL VAR II

εt may be constructed using

εt = B−1ut

Note, that this is not the only way to make the innovationsorthogonal!The Cholesky decomposition of Σ is a lower-triangular matrixthat satisfies (29), with diagonal elements containing the squareroot of the diagonal elements of Σ . Note, that the ordering ofthe variables in yt affects B. It is not unique in that sense.The whole structural VAR litterature aims finding (economic)theoretically consistent ways to orthogonalize innovations.

Macroeconometrics



STRUCTURAL VAR: B-MODEL IThe recursive structure implied by the Choleski decompositionmay not always be reasonable.Let us keep assuming that the errors of the VAR model arelinear combinations of the economic shocks

ut = Bεt

and let us, for simplicity assume that "t has covariance matrixIK. As seen above, this implies that

Σu = BB′.

Because Σu is symmetric, this involves K (K + 1) /2different equations.

Macroeconometrics



STRUCTURAL VAR: B-MODEL IIHowever, there are K2 elements in the matrix B, so that weneed K2 − K (K + 1) /2 = K (K− 1) /2 further restrictionsto identify all the K2 elements.The K (K− 1) /2 restrictions must be inferred fromeconomic theory or institutional knowledge. This happensin Choleski decomposition.Once a sufficient number of restrictions have been found,the structural VAR model imposing these restrictions canbe estimated by ML and its dynamic properties examinedthrough impulse response analysis.The VAR model based on the assumption that ut = B"t iscalled the B-model (Lütkepohl 2005, Chapter 9) and it isprobably the most commonly employed structural VARmodel.

Macroeconometrics



STRUCTURAL VAR: AB-MODEL IA natural extension of the B-model is the so-called AB-model,where

Aut = Bεt.

In this model, the structural shocks are identified by modellingthe instantaneous relations between the observable variablesdirectly.Here

ut = A−1Bεt,

and, hence,Σu = A−1BB′A−1.

Thus, again, we have K (K + 1) /2 restrictions, but now thematrices A and B have together 2K2 parameters, and, therefore,we need 2K2 − K (K + 1) /2 additional restrictions.

Macroeconometrics



STRUCTURAL VAR: AB-MODEL II

With enough restrictions, the model can be estimated byrestricted ML.Note that the B-model is a special case of the AB-model, whereA is the identity matrix. By restricting B to the identity matrix,we obtain the so-called A-model.

Macroeconometrics



FORECAST ERROR VARIANCE DECOMPOSITION I

In addition to impulse response functions, the effects of theidentified structural shocks can be studied by means of theirproportional contribution to the forecast error variance of eachof the variables at different forecast horizons. Let us firstconsider the VAR(1) model

yt = Φ1yt−1 + ut.

Previously it was shown that in this case the forecast error ofyt+h is

yt+h − Et (yt+h) = yt+h −Φh1yt =

h−1

∑j=0

Φj1ut+h−i

Macroeconometrics



FORECAST ERROR VARIANCE DECOMPOSITION II

and, thus, the forecast error covariance matrix is

Σy (h) = E

[(h−1

∑j=0

Φj1ut+h−i

)(h−1

∑j=0

Φj1ut+h−i

)′]=

h−1

∑j=0

Φj1Σu

(Φj

1

)′= Σu + Φ1ΣuΦ′1 + Φ2

1Σu

(Φ2

1

)′+ · · ·+ Φh−1

1 Σu

(Φh−1

1

)′Because any VAR(p) model can be written as a VAR(1) model incompanion form, for a general VAR(p) model, this is

Σy (h) = Σu + Ψ1ΣuΨ′1 + Ψ2ΣuΨ′2 + · · ·+ Ψh−1ΣuΨ′h−1,

where Ψjs are the coefficient matrices of the MA(∞)representation.

Macroeconometrics



FORECAST ERROR VARIANCE DECOMPOSITION III

In terms of the structural shocks "t with diagonal covariancematrix Σε such that ut = B"t, this can be written as

Σy (h) = Θ0ΣεΘ′0 + Θ1ΣεΘ′1 + Θ2ΣεΘ′2 + · · ·+ Θh−1ΣεΘh−1,

where Θj = ΨjB.The forecast error variance of the kth variable is thus the kthdiagonal element of Σy (h),

σ2k (h) =

h−1

∑n=0

(θ2

n,k1σ21 + θ2

n,k2σ22 + · · ·+ θ2

n,kKσ2K

)=

K

∑j=1

σ2j

(θ2

0,kj + θ21,kj + · · ·+ θ2

h−1,kj

),

Macroeconometrics



FORECAST ERROR VARIANCE DECOMPOSITION IV

and the contribution of the jth shock εkt at horizon h to this isσ2

j

(θ2

0,kj + θ21,kj + · · ·+ θ2

h−1,kj

). Dividing the latter by σ2

k (h)hence gives the proportional contribution of the jth shock tovariable k.By plotting the proportions of the forecast error varianceaccounted for by each shock at diffent forecast horizons, theirimportance can be assessed. For instance, it may be concludedthat a shock is important only for certain variables or that itscontribution for all variables is minor etc.

Macroeconometrics



SOME VAR BASED STATISTICS I

Definition (Contemporaneous variance-covariancematrix)Let

Γ(0) = E ξtξ′t = E(Fξt−1 + vt)(Fξt−1 + vt)

′ = FΓ(0)F′ + Σ,

where Γ(0) denotes the contemporaneous variance-covariancematrix of ξt. The solution of above is given by

vec (Γ(0)) = (I− F⊗ F)−1 vec (Σ) .

Definition (τth order covariance matrix)

Γ(τ) = E ztz′t−τ

Macroeconometrics



SOME VAR BASED STATISTICS II

Note that

Γ(1) = E(ξtξ′t−1 = E(Fξt−1 + vt)ξ

′t−1 = FΓ(0),

and, in general,Γ(τ) = FτΓ(0)

Macroeconometrics


Filtering the data and model

OUTLINE

1 Introduction





5 Matching moments

Macroeconometrics



TO PRE-FILTER

Often the theoretical model show no growth. The balancedgrowth path issues are then ignored.

Ironically, the RBC models that build on the neo-classicalgrowth model do not — as a prototype — model the economicgrowth.

Data portrays growth−→ in order to compare data and model moments, one needsto get rid of the growth in data.−→ detrend the data!

Macroeconometrics



TO PRE-FILTER THE DATA

The idea is to remove certain frequencies of the dataSeasonal filters remove seasonalityHodrick-Prescott filter removes the trendBand-pass filter removes chosen frequencies

Univariate filtersHodrick-Prescott filterBand-pass filterBaxter-King filterBeverage-Nelson filterSeasonal filters

Multivariate filters

Macroeconometrics



TO PRE-FILTER OR NOT TO PRE-FILTER

Problems in pre-filtering the dataDo not take into account the restriction by the theoreticalmodel: number of trends, style of trends.

Filtering is subsitute of poor modeling of trends/balancedgrowth path.

Filtering is never perfect. Leakage.

Macroeconometrics

Matching moments

OUTLINE

1 Introduction




5 Matching momentsCalibrating steady-stateComparing model and datamomentsGMM

Macroeconometrics

Matching moments

MATCHING MOMENTS

Moment is a fancy word to describe the statistics we use tocompare model and data.

There is no clear guide which moments to match. It maydepend on

purpose of the modeling: policy vs. forecast, . . .style of the model: for example growth vs. dynamics orboth!frequency: short-run vs long-run. . .

In the previous chapter we had large list of moments. It is easyto invent more!

Macroeconometrics

Matching moments

BENEFITS OF CALIBRATION/MOMENT MATCHING

Get aquainted with your model!

Macroeconometrics

Matching moments

Calibrating steady-state

OUTLINE

1 Introduction





Macroeconometrics

Matching moments


THREE SETS OF PARAMETERS

We may typically partition the parameters of the theoreticalmodel µ into three sets

1 Parameters that affect only the steady-state2 Parameters that does not affect steady state but affect

dynamics (temporal dependency) of the system3 Parameters that affect both

We may utilize the separation of the sets in calibration.

Macroeconometrics

Matching moments


HOW TO CALIBRATE STEADY-STATE: MOMENTS OF

THE STEADY STATE

Consider the first set of parameters, ie parameters that affectonly the steady-state.They can be chosen such as they exactly match

great ratios in the data: I/K, C/Y, I/Y, (X−M)/Y, . . .average growth rates in the data: ∆y, πt.

given the chosen parameters of the set 3.

Macroeconometrics

Matching moments


HOW TO DO ITYou may follow the procedure

1 Choose the data and the sample!2 Compute the great ratios that can be found from the model

too. Plot them to check that they really are stationary3 Compute sample average.4 Solve the steady-state model

1 Compute the same great ratios using steady-state model2 Try another set of parameter values

5 Repeat the above until the distance between great ratioscomputed from the data and from the model areminimized

Often conflict: Matching certain moment results unmatchingother one.−→ try GMM and do it formally (statistically consistent way)

Macroeconometrics

Matching moments

Comparing model and data moments

OUTLINE

1 Introduction





Macroeconometrics

Matching moments

Comparing model and data moments

CALIBRATING DYNAMICS: MOMENTS OF TEMPORAL

DEPENDENCE

These are difficult to calibrate!

You may try the following momentsCross-correlationsCoherence could be useful.Impulse response function

look the response at impact,the shape,the persistenceIdentification problem: how to compute the impulseresponses we observe in the data. Structural VAR!

Macroeconometrics

Matching moments

GMM

OUTLINE

1 Introduction





Macroeconometrics

Matching moments

GMM

THE GENERALIZED METHOD OF MOMENTS ILet wt denote the variables we observe at time t and let Θ bethe unknown parameters of interest. They are related by thefunction h(Θ, wt), where the dimension of the function vector isl.

The orthogonality conditions lies at the hearth of thegeneralized methods of moments.The orthogonality condition says that when evaluated atΘ∗ (the true value), the unconditional expections satisfy

E[h(Θ∗, wt)] = 0, (30)

The sample mean of these conditions are

g(Θ; w1, . . . , wT) ≡1T

T

∑t=1

h(Θ, wt),

Macroeconometrics

Matching moments

GMM

THE GENERALIZED METHOD OF MOMENTS IIThe GMM objective function to be minimized is

Q(Θ) = g(Θ; w1, . . . , wT)′Wg(Θ; w1, . . . , wT), (31)

where W is the weighting matrix of the orthogonalityconditions. Any positive definite matrix is ok, but optimalweighting is obtained by using the covariance matrix ofthe orthogonality conditions ST.The estimator of the covariance matrix is the following

S = limT→∞

(1/T)T

∑t=1

∞

∑v=−∞

E[h(Θ∗, wt)h(Θ∗, wt−v)

′] ,

where Θ∗ denotes the true value of Θ. I have usedthe VARHAC estimator by den Haan and Levin (1996) and

Macroeconometrics

Matching moments

GMM

THE GENERALIZED METHOD OF MOMENTS IIIquadratic the spectral kernel estimator by Newey and West(1994) that allow autocorrelation (and heteroscedasticity) inthe sample orthogonality conditions.

If the number of orthogonality conditions, l, exceeds thenumber of parameters, j, the model is overidentified.Hansen (1982) shows that it is possible to test theoveridentification restrictions (J-test), since[√

Tg(Θ, w1, . . . , wT)]′

S−1T

[√Tg(Θ, w1, . . . , wT)

]d−→ χ2(l− j),

where d−→ denotes convergence in distribution.Problems and solutions

In the GMM estimation, we encounter the problem ofdefining the orthogonality conditions (instruments in thetypical case).

Macroeconometrics

Matching moments

GMM

THE GENERALIZED METHOD OF MOMENTS IVThe GMM estimator is different for different set oforthogonality conditions.According to the simulation experiments of Tauchen (1986)and Kocherlakota (1990), increasing the number oforthogonality conditions reduces the estimators’ variancebut increases the bias in small samples.In the iteration of the GMM objective (31), follow theguidence of Hansen et al. (1996): Since the weightingmatrix in the objective function is also a function of theparameters, it is useful to iterate that as well. This, ofcourse, increases the computational burden.

Extensions (that follow GMM spirit)Simulated method of moments: extends the generalizedmethod of moments to cases where theoretical momentfunctions cannot be evaluated directly, such as whenmoment functions involve high-dimensional integrals.

Macroeconometrics

Matching moments

GMM

THE GENERALIZED METHOD OF MOMENTS V

Indirect inference: introduce auxiliary model and match thesimulated and data moments of the auxialiary model. (Use,eg, VAR as the auxiliary model) If the auxiliary model iscorrectly specified this is maximum likelihood.Empirical likelihood. . .

Macroeconometrics

Filtering and Likelihood approach

OUTLINE

1 Introduction




5 Matching moments

Macroeconometrics


Kalman filter and the maximum likelihood

OUTLINE

1 Introduction




5 Matching moments

Macroeconometrics



THE STATE SPACE FORM I

In following I follow the notation of Hamilton (1994) and Yadamanual but the logic of Harvey (1989, Structural Time SeriesModels). Harvey’s setup is more general than that of Hamilton.

Macroeconometrics



THE STATE SPACE FORM II

Definition (Measurement equation)Let yt be multivariate time series with n elements. They areobserved and related to an r× 1 state vector ξt viameasurement equation

yt = Htξt + xt + wt, t = 1, . . . , T (32)

where Ht is an n× r matrix, xt is k× 1 vector and wt is ab n× 1vector of serially uncorrelated disturbances with mean zeroand covariance matrix Rt:

E wt = 0 and var(wt) = Rt.

In general the elements of ξt are not observable.

Macroeconometrics



THE STATE SPACE FORM III

Definition (Transition equation, state equation)The state variables ξt are generated by a first-order Markovprocess

ξt = Ftξt−1 + ct + Btvt, (33)

where Ft is an r× r matrix, ct is an r× 1 vector, Bt is an m× gmatrix and vt is a g× 1 vector of serially uncorrelateddisturbances with mean zero and covariance matrix Qt, that is

E vt = 0 and var(vt) = Qt.

Assumption

E ξ0 = ξ0 and var(ξ0) = P0

Macroeconometrics



THE STATE SPACE FORM IV

Assumption

E(wtv′k) = 0, for all k, t = 1, . . . , T

E(wtξ′0) = 0, E(vtξ

′0) = 0 for t = 1, . . . , T

The first line in above assumption may be relaxed.Matrices Ft, xt, Rt, Ht, ct, Bt and Qt are called system matrices. Ifthey do not change over time, the model is said to betime-invariant or time-homogenous. Stationary models are specialcase.

Macroeconometrics



STATE SPACE FORM AND THE ECONOMIC MODEL

Equation (28) on slide 13 looks familiar:

ξt = F0(µ) + F1(µ)ξt−1 + FΓ(µ)vt, (28)

Redefining ct = F0(µ) and Bt = FΓ(µ), the solution of theeconomic model is the transition (state) equation of the statespace representation of (32) and (33). This makes the statespace representation attractive!

With the measurement equation, we are able to map the(theoretical) model variables to actual data!

Macroeconometrics



THE KALMAN FILTER I

Definition (The Kalman filter)The Kalman filter is a recursive procedure for computing theoptimal estimator of the state vector ξt|t at time t, based on theinformation available at time t.

Combined with an assumption that the disturbances and theinitial state vector are normally distributed, it enables thelikelihood function to be calculated via what is known as theprediction error decomposition. The derivation of Kalman filterbelow rests on that assumption.

Macroeconometrics



CONDITIONAL MULTIVARIATE NORMALThis a sidestep to statistics. We need this result in constructingthe Kalman filter!If X ∼ N(µ, Σ) where µ and Σ are mean and covariance ofmultivariate normal distribution and are partitioned as follows

X =

[X1X2

]

µ =

[µ1µ2

], Σ =

[Σ11 Σ12Σ21 Σ22

]then the distribution of X1 conditional on X2 = a is multivariatenormal, i.e. (X1|X2 = a) ∼ N(µ, Σ), where

µ = µ1 + Σ12Σ−122 (a− µ2)

and covariance matrix

Σ = Σ11 − Σ12Σ−122 Σ21.

Macroeconometrics



DERIVATION OF THE KALMAN FILTER I

Consider the state space representation of (32) and(33).Under normality, ξ0 is multivariate normal withmean ξ0 and covariance P0 = E(ξ0 − ξ0)(ξ0 − ξ0)

′.t = 1

ξ1 = F1ξ0 + c1 + B1v1

−→ Due to normality, the conditional expectation of ξ1is

ξ1|0 = F1ξ0 + c1

Macroeconometrics



DERIVATION OF THE KALMAN FILTER IIand its conditional covariance matrix

P1|0 = E(ξ1 − ξ1|0)(ξ1 − ξ1|0)′

= E(F1ξ0 + c1 + B1v1 − F1ξ0 − c1)(F1ξ0

+ c1 + B1v1 − F1ξ0 − c1)′

= E[F1(ξ0 − ξ0) + B1v1][F1(ξ0 − ξ0) + B1v1]′

= E[F1(ξ0 − ξ0) + B1v1][(ξ0 − ξ0)′F′1 + v′1B′1]

= E[F1(ξ0 − ξ0)(ξ0 − ξ0)′F′1 + F1(ξ0 − ξ0)v′1B′1

+ B1v1(ξ0 − ξ0)′F′1 + B1v1v′1B′1]= F1P0F′1 + B1Q1B′1, (34)

since — by assumption — E(ξ0 − ξ0)v′1 = 0.

Macroeconometrics



DERIVATION OF THE KALMAN FILTER IIIFor the distribution of ξ1 conditional on y1 write

ξ1 = ξ1|0+ (ξ1 − ξ1|0)

y1 = H1 ξ1|0 + x1 + H1 (ξ1 − ξ1|0) + w1.

The second equation is rearranged measurementequation (32).

−→ vector [ξ ′1 y′1]′ is multivariate normal with mean

[ξ ′1|0 (H1ξ1|0 + x1)′]′

and a covariance[P1|0 P1|0H′1

H1P1|0 H1P1|0H′1 + R1

].

Macroeconometrics



DERIVATION OF THE KALMAN FILTER IV

Applying the conditional distribution ofmultivariate normal to the case: ξ1 conditional onparticular value of y1 results mean

ξ1 = ξ1|0 + P1|0H′1S−11 (y1 −H1ξ1|0 − x1)

and covariance

P1 = P1|0 − P1|0H′1S−11 H1P1|0,

whereS1 = H1P1|0H′1 + R1

Iterating this over t = 2, 3, . . . , T gives us the Kalman filter!

Macroeconometrics



THE KALMAN FILTER IConsider, again, the state space representation of (32) and (33).

Define ξt−1 denotes the optimal estimator of ξt−1 basedon the observations up to and including yt−1.Pt−1 denotes the r× r covariance matrix of theestimations error

Pt−1 = E[(ξt−1 − ξt−1)(ξt−1 − ξt−1)

′]Prediction equations Given ξt−1 and Pt−1, the optimal

estimator of ξt is given by

ξt|t−1 = Ftξt−1 + ct

and the covariance matrix of estimation error is

Pt|t−1 = FtPt−1F′t + BtQtB′t, t = 1, . . . , T

Macroeconometrics



THE KALMAN FILTER II

Updating equations Once the new observation yt becomesavailable, the estimator ξt|t−1 of ξt can be updated

ξt = ξt|t−1 + Pt|t−1H′1S−1t (yt −Htξt|t−1 − xt)

andPt = Pt|t−1 − Pt|t−1H′tS

−1t HtPt|t−1,

whereSt = HtPt|t−1H′t + Rt

Macroeconometrics



THE KALMAN FILTER IIIThe equations in these two slides make up the Kalman filter.They can be written in the single set of recursion from ξt−1 to ξtor from ξt|t−1 as follows

ξt+1|t = (Ft+1 − KtHt)ξt|t−1 + Ktyt + (ct+1 − Ktxt),

where the Kalman gain is given by

Kt = Ft+1Pt|t−1H′tS−1t , t = 1, . . . , T.

The recursion for the error covariance matrix is

Pt+1|t = Ft+1(Pt|t−1−Pt|t−1H′tS−1t HtPt|t−1)F

′t+1 +Bt+1Qt+1B′t+1, t = 1, . . . , T

It is know as a Riccati equation.

Macroeconometrics



ISSUES I

The state space representation and Kalman filter tellsnothing how to interprete the state variables ξt. Economicmodel does tell! Its elements may or may not beidentifiable.Same economic model may have several state spacerepresentations. The one that has smallest state vector iscalled minimal realisation.Sometimes transition equation is written in the form wherestate variables are leaded with one period.The system matrices Ft, Rt, Ht, Bt and Qt may depend on aset of unknown parameters, and one of the main statisticaltasks will often be the estimation of these parameters.Given initial conditions, the Kalman filter delivers theoptimal estimator of the state vector.

Macroeconometrics



ISSUES IIInformation filter is a version of the Kalman filter, where therecursion involves P−1. This is handy when P0 is infinite orwhen system matrices are time-varying.Square root filter is the one where P is obtained via Choleskydecomposition. It is known to be numerically stable.

ξ = Et(ξt) = E(ξ|Yt)

andPt = Et

{[ξt − Et(ξt)] [ξt − Et(ξt)]

′}

Hence, the conditional mean is the minimum mean squareestimate of ξt.The conditional mean can also be regarded as an estimatorof ξt. Hence it applies to any set of observations. It is alsominimum mean square estimator of ξt.

Macroeconometrics



ISSUES III

In the above case Pt can be considered as the the meansquare error (MSE) matrix of the estimator.For time invariant model the necessary and sufficientcondition for stability is that the characteristic roots of thetransition matrix F should have modulus less than one.

Macroeconometrics



MAXIMUM LIKELIHOOD ESTIMATION AND THE

PREDICTION ERROR DECOMPOSITION I

Classical theory of maximun likelihood estimation is based ona situation in which the T sets of observations, y1, . . . , yT areindependently and identically distributed. The joint densityfunction is given by

L(y; φ) =T

∏t=1

p(yt),

where p(yt) is the joint probability density function (p.d.f.) ofthe t-th set of observations. Once the observations have beenmade, L(y; φ) is reinterpreted as a likelihood function and theML estimator is found by maximizing this function wrt φ.

Macroeconometrics




PREDICTION ERROR DECOMPOSITION IIIn time series or economic models observations are typicallynot independent. We write instead the conditional probabilitydensity function

L(y; φ) =T

∏t=1

p(yt|Yt−1),

where Yt−1 = {y1, y2, . . . , yt−1}.

If the disturbances vt and wt and initial state vector ξ0 aremultivariate normal, p(yt|Yt−1) is itself normal with the meanand covariance given by Kalman filter.

Conditionally Yt−1, ξt ∼ N(ξt|t−1, Pt|t−1).

Macroeconometrics




PREDICTION ERROR DECOMPOSITION III

From the measurement equation

yt|t−1 ≡ Et−1(yt) = Htξt|t−1 + xt

with covariance matrix St.Define vt = yt − Et−1 yt = yt − yt|t−1

Then the likelihood can be written as

log L = −nT2

log 2π − 12

T

∑t=1

log |St| −12

T

∑t=1

v′tS−1t vt.

Macroeconometrics




PREDICTION ERROR DECOMPOSITION IV

Since yt|t−1 is also the MMSE of yt, vt can be interpreted asvector of prediction errors. The above likelihood is,therefore, called prediction error decomposition form of thelikelihood.Consequently, the likelihood function may easilycomputed with Kalman filter.And numerically maximized wrt unknown parameter φ.

Macroeconometrics



ML ISSUES IThe limiting distribution of the ML estimator of φ isnormal with covariance matrix obtained as the inverse ofthe information matrix

φ has to be interior pointderivatives of log L exists up to order three, and they arecontinuous in the neighbourhood of true φφ is identifiable

In nonstationary models, the influence of the initial pointdoes not vanish. No steady-state point exists. Then, the KFhas to be initialized using first observations and “large“matrix P0. This is called diffuse prior.Forecasting is easy. Multistep forecast may be obtained byapplying KF prediction equations repeteadly. Note: PT+j|Tdo not take into account the uncertainty related toestimating unknown parameters.

Macroeconometrics



SMOOTHING

Filtering is prediction/forecasting. In smoothing we want toknow the distribution of ξt conditional on all the smaple, ieE(ξt|YT).

it is known as smoothed estimateand the corresponding estimator as smootherseveral algorithms: fixed point, fixed lag, fixed interval

Covariance matrix of ξt|T conditional on all T observations

Pt|T = ET[(ξt − ξt|T)(ξt − ξt|T)

′]when ξt|T is viewed as an estimator, Pt|T ≤ Pt (t = 1, . . . , T) is itsMSE matrix.

Macroeconometrics



FIXED-INTERVAL SMOOTHING

Starts with final quantities ξT and PT of the Kalman filter andwork backwards

ξt|T = ξt + P?t (ξt+1|T − Ft+1ξt)

andPt|T = Pt + P?

t (Pt+1|T − Pt+1|t)P?t′

whereP?

t = PtF′t+1P−1t+1|t t = T− 1, . . . , 1

with ξT|T = ξT and PT|T = PT.

Macroeconometrics



IDENTIFICATION IBased on Andrle (2010)

Fundamental issue in economic and statistical modeling.Easily acute in state space models.Y set of observationsModel: distribution for the variable in question.S structure, ie parameters of that distribution, ie completeprobability specification of Y of the form

S = F(Y, θ),

where θ ∈ Θ ⊂ Rn is the vector of parameters that belongsto parameters space Θ.Observational equivalence: two structures, S0 = F(Y, θ0) andS? = F(Y, θ?), having the same joint density function, ie ifF(Y, θ0) = F(Y, θ?) for almost all Y.

Macroeconometrics



IDENTIFICATION II

A structure is identifiable if the exists no otherobservationally equivalent structure, ie θ0 = θ?.A model is identifiable if all its possible structures areidentifiable. If no structure is identifiable, the model is saidto be underidentified.Local identification: the structure is locally identified if thereexists an open neighbourhood of θ0 containing no otherθ ∈ Θ, which produces observationally equivalentstructure.

Macroeconometrics



IDENTIFICATION IIITheorem (Rotemberg, 1971)

Subject to some regularity conditions, θ0 is locally identified for agiven structure if and only if the Information matrix evaluated at θ0

is not singular.

EquationT(θ, τ) = 0

defines the mapping between the reduced form parametersτ ∈ T ⊂ Rm and structural parameters θ.If reduced form parameters are (locally) identified, thenecessary and sufficient condition for identification is that

T ≡ ∂T∂θ′

Macroeconometrics



IDENTIFICATION IV

is of full rank.

Bayesian perspective“Identification is a property of likelihood” and, hence, it theBayesian approach makes no difference. Unidentification affectlarge sample inference and frequency properties of Bayesianinference since likelihood does not dominate prior as thesample size grows.Data may be marginally informative even for conditionallyinidentified parameter and marginal posterior and priordensities may differ.

Macroeconometrics


Filtering and decompositions

OUTLINE

1 Introduction




5 Matching moments

Macroeconometrics


Filtering and decompositions

APPLICATIONS

The KF opens up many applications even within the context ofDSGE models

Missing observationsDifferent frequencies for different variables, eg quarterly,annually, live happily togetherDifferent vintages of data

Macroeconometrics


Bayesian methods

OUTLINE

1 Introduction




5 Matching moments

Macroeconometrics


Bayesian methods

FEATURES OF THE MAXIMUM LIKELIHOOD

ESTIMATION

Takes probability distribution generated by the DSGEmodel very seriouslyMaximum likelihood works well if the misspecification ofthe DSGE model is small. However, likelihood-basedanalysis potentially very sensitive to misspecification.State space models are not the best setup to recovermisspecification.As stressed above, it there is a lack of identification, thelikelihood function will be flat in certain directions.In practice, likelihood function tends to be multi-modal,difficult to maximize, and often peaks in regions of theparameters space in which the model is hard to interpret.

−→ fix subset of parameters.

Macroeconometrics


Bayesian methods

BAYESIAN INFERENCE IN A NUTSHELL

Parameter is a random variable!Combine prior distribution of the structural parameterswith likelihood function

This prior density can contain information from othersources than the current data.

Use Bayesian theorem to update the prior density bycalculating conditional distribution of the parametersgiven the data (posterior distribution)Inference and decisions are then based on this posteriordistribution

Macroeconometrics


Bayesian methods

NUTSHELL OF A NUTSHELL

P(A|B) = P(A∩ B)P(B)

Macroeconometrics


Bayesian methods

INTRODUCTION I

Structural models have interest parameterization in thesense that we may have a priori information from

microeconometric studies;studies from other fieldsother countries’ data...

Bayesians love this since prior information has an essentialrole in Bayesian inference.Bayesians interprete parameters as random variables.Objective is to make conditional probabilistic statementsregarding the parameterization of the model:

1 structure of the model (model specification)2 the observed data3 prior distribution of the parameters.

Macroeconometrics


Bayesian methods

INTRODUCTION IILikelihood is 1. plus 2.Combining likelihood and 3. using Bayes’ rule yields anassociated posterior distribution.Data speaks in likelihood and the researcher speaks inprior distribution.

The incorporation of prior information is not whatdistinguishes classical (“frequentist”) from Bayesiananalysis; the distinguishing feature is the probabilisticinterpretation assigned to parameters under the Bayesianperspective.

We may interprete calibration as a Bayesian analysisinvolving very, very strict prior (probability massconcentrated on a single point).

Macroeconometrics


Bayesian methods

INTRODUCTION III

By choosing a diffuse prior we may let data to speak asmuch as possible.

Objectives in using Bayesian procedures in DSGEs1 Implement DSGEs as a source of prior information

regarding the parameterization of reduced form models(like VAR).

2 Facilitate direct estimation of the parameters of DSGEs andto implement estimated models to pursuit different tasks.

3 To facilitate model comparisons. Posterior odds analysis.Works also for false and non-nested models.

Macroeconometrics


Bayesian methods

PRELIMINARIES INotation:

A structural modelµ vector of parameters, primary focus of analysis

Λ(µ) parameters of state space representation, no(extra) identification problem here. This is themodel’s solution.

X sample of observationsL(X|µ, A) likelihood function

L(X|µ) likelihood function if A is clear/grantedπ(µ) prior distribution of µ

p(X, µ) joint probability of X and µp(X) probability of X.

Classical (frequentist) viewparameters are fixed, but unknown objects

Macroeconometrics


Bayesian methods

PRELIMINARIES IIlikelihood function is a sampling distribution for the dataX is one of many possible realizations from L(X|µ) thatcould have obtained.Inference regarding µ center on statements regardingprobabilities associated with the particular observation ofX for given values of µ.

Bayesian viewX taken as given.inference interested in alternative specification of µconditional on X.This probabilistic interpretation of µ gives rise toincorporate prior information on µ.This is facilitated by the prior distribution for µ, denotedby π(µ).

Macroeconometrics


Bayesian methods

PRELIMINARIES III

How to incorporate prior information

Calculate joint probability of (X, µ) with the help ofconditional and unconditional distribution

p(X, µ) = L(X|µ)π(µ)

or reversing the role

p(X, µ) = p(µ|X)p(X)

In the first equation the likelihood function L(X|µ) has therole of conditional probability and conditioning is madewrt µ.In the second equation conditioning is wrt X.

Macroeconometrics


Bayesian methods

PRELIMINARIES IV

Get rid of the joint probability by equating those two:

p(µ|X)p(X) = L(X|µ)π(µ).

Solving for p(µ|X) gives Bayes’ rule:

p(µ|X) =L(X|µ)π(µ)

p(X)∝ L(X|µ)π(µ).

The ∝ is because p(X) is a constant from the point of viewof the distribution of µ.p(µ|X) is now the posterior distribution.Conditional on X and the prior π(µ) it assigns probabilitiesto alternative values of µ.

Macroeconometrics


Bayesian methods

PRELIMINARIES VBayesian analysis typically involves calcualting theconditional expected value of a function of the parametersg(µ):

E[g(µ)] =∫

g(µ)P(µ|X)dµ∫P(µ|X)

,

where the denominator∫

P(µ|X) exists to handle themissing p(X) in the Bayes’ rule.Examples of g(µ)

Identitity function would result posterior mean.g(µ) = 1 for µj ∈ [µj, µj), and 0 other wise: If repeated toeach element in µ would enable to construction of marginalpredictive density functions (p.d.f.:s) for each µi. HereMarginal means that the p.d.f. is unconditional on the otherelements of µ.marginal p.d.f. of spectra

Macroeconometrics


Bayesian methods

PRELIMINARIES VImarginal p.d.f. of impulse response functionsmarginal p.d.f. of predictive densitiesmarginal p.d.f. of unobserved shock processesmarginal p.d.f. of forecasts!

Hence, E[g(µ)] is the weighted average of g(µ) whereweights are given by posterior distribution P(µ|X), ie bydata (likelihood function) and the prior.Rarely E[g(µ)] can be calculated analytically −→ numericalintegration.

Numerical integrationSuppose we may draw directly from P(µ|X)we, naturally, know g(µ)Let’s do Monte Carlo integration:for i=1:N % where N is a large number like 10000

1 draw µi (one realization) from P(µ|X)

Macroeconometrics


Bayesian methods

PRELIMINARIES VII2 compute g(µi)3 store it to a vector

endcompute the average.

Typically you may do this without loops by usingvector/matrix operations, eggN = mean(g(randmu(N)))By law of large numbers

gN =1N

N

∑i

g(µi)p−→ E(g(µ)],

wherep−→means convergence in probability.

Macroeconometrics


Bayesian methods

PRELIMINARIES VIII

Std(gN) is given by

std(gN) =σ(g(µ))√

N.

and its sample (simulated) counterpart

σN[g(µ)] =

[1N

N

∑i=1

g(µi)2 − g2

N

]2

.

Marginal p.d.f. of g(µ) may obtained from the draws ofg(µ) by kernel methods.

Macroeconometrics


Simulating posterior

OUTLINE

1 Introduction




5 Matching moments

Macroeconometrics



IMPLEMENTING DSGES

Typically posterior distribution p(µ|X) is not directly(analytically) available.

likelihood is available for Λ(µ), whereaspriors are specified in terms µ.

We can calculate p(µ?|X) for given value of µ? with thehelp of the likelihood function L(X|µ?, A) (ie with Kalmanfilter).The last problem is to define from where to draw µi suchthat the procedure will be

accurateefficient

DeJong and Dave (2007) studiesImportance samplingMCMC: GibbsMCMC: Metropolis-Hastings

Macroeconometrics



MARKOV CHAIN SIMULATIONS IA Markov chain is a sequence of random variables, such thatthe probability distribution of any one, given all precedingrealizations, depend on the immediately preceding realization.Hence, {xi} has the property

Pr(xi+1|xi, xi−1, xi−2, . . . ) = Pr(xi+1|xi)

Here i refers to Monte Carlo replication.xi can take one of n possible values, which collectivelydefines the state space over xMovements of xi over i is characterized via P, an n× ntransition matrix. Hence, a Markov chain can be fullydescribed by its initial state and a rule describing how thechain moves from its state in i to a state in i + 1.pqr is the (q, r)th element of P:

Macroeconometrics



MARKOV CHAIN SIMULATIONS II

it is the probability that xi will transition to state r inreplication i + 1 given its current state q.qth row of P is the conditional probability distribution forxi+1 given the current state q in replication i

The posterior distribution is not standard. The idea of MarkovChain Monte Carlo is to construct a stochastic process (MarkovChain), such that:

it has stationary distributionit converges to that stationary distributionthe stationary distribution is the target distribution, ie theposterior distribution p(µ|X). (ergodic distribution)

Macroeconometrics



MARKOV CHAIN SIMULATIONS III

−→ construct a Markov chain in µ, ie the transition matrix P,such that the process converges to the posterior distribution.The ‘process’ here is the sequence of draws, ie the simulations.Metropolis algorithm creates a sequence of random points(µ1, µ2, . . . ) whose distribution converges to p(µ|X).

Macroeconometrics



PROPOSAL DISTRIBUTION

Let ι(µ|µi−1, θ) be a stand-in density (proposal distribution orjumping distribution), where θ represents the parameterizationof this density.

Ideally ι(µ|µi−1, θ) should have fat tails relative toposterior: algorithm should visit all parts of the posteriordistribution.center µi−1 and “rotation” θ should different fromposterior.

Macroeconometrics



METROPOLIS ALGORITHM I

Let µ?i denote a draw from ι(µ|µi−1, θ). It is a candidate to

become next successful µi.Calculate the ratio of the densities

r =p(µ?

i |X)

p(µi−1|X).

Set

µi =

{µ?

i with probability min(r, 1)µi−1 otherwise.

p(µ?i |X) ≥ p(µi−1|X)

−→ always include candidate as a new point in thesequence.

Macroeconometrics



METROPOLIS ALGORITHM II

p(µ?i |X) < p(µi−1|X)

−→ µ?i not always included; the lower p(µ?

i |X), the lowerthe chance it is included.

Given the current value µi−1, the Markov chain transitiondistribution pµi−1,µi is a mixture of the jumping distributionι(µ|µi−1, θ) and a point mass at µi = µi−1.Note! If µi = µi−1, ie the jump is not accepted, this counts as aniteration in the algorithm.The algorithm can be viewed as a stochastic version of astepwise mode-finding algorithm: always stepping to increasethe density but only sometimes stepping to decrease. This isthe idea in simulated annealing algorithm.

Macroeconometrics



WHY DOES THE METROPOLIS ALGORITHM WORK? I

The proof of the convergence to the target distribution has twosteps:

1 Show that the simulated sequence (µ1, µ2, . . . ) is a Markovchain with a unique stationary distribution: holds if theMarkov chain is irreducible, aperiodic, and not transient−→ hold if the jumping distribution is chosen such that itcovers all possible states with positive probability→ hasfat tails.

2 Show that the stationary distribution is the targetdistribution.

To proof the second part:Suppose µi−1 a draw from the target distribution p(µ|X).

Macroeconometrics



WHY DOES THE METROPOLIS ALGORITHM WORK? IIConsider two draws, µa and µb, from the targetdistribution such that p(µb|X) ≥ p(µa|X).The unconditional transitional probability from µa to µb is

pµaµb = p(µa|X)ι(µb|µa, θ)× 1,

where the acceptance probability is 1 (becausep(µb|X) ≥ p(µa|X)).The unconditional transitional probability from µb to µa

pµbµa = p(µb|X)ι(µa|µb, θ)p(µa|X)

p(µb|X)

= p(µa|X)ι(µa|µb, θ)

They are the same, because we have assumed that ι(·|·, θ) issymmetric.

Macroeconometrics



WHY DOES THE METROPOLIS ALGORITHM WORK? III

Since the joint distribution is symmetric, µi and µi−1 havethe same marginal distributions, and sop(µ|X) is the stationary distribution of the Markov chain.

Macroeconometrics



METROPOLIS-HASTING ALGORITHM ILet µ?

i denote a draw from ι(µ|µi−1, θ). It is a candidate tobecome next successful µi.It has the following probability to become successful:

q(µ?i |µi−1) = min

[1,

p(µ?i |X)/ι(µ?

i |µi−1, θ)

p(µi−1|X)/ι(µi−1|µi−1, θ)

]This probability can be simulated as follows

draw f from uniform distribution over [0, 1].if q(µ?

i |µi−1) > f then µi = µ?i else redraw new

candidate µ?i from ι(µ|µi−1, θ).

Note that if ι(µ|θ) = p(µ|X) q(µ?i |µi−1) = 1.

Finally, E[g(µ)] can be calculated usual way using sequence ofaccepted draws.Two variants:

Macroeconometrics



METROPOLIS-HASTING ALGORITHM II1 independence chain: ι(µ|µi−1, θ) = ι(µ|θ).2 random walk MH: candidate draws

µ?i = µi−1 + εi

then stand-in density evolves

ι(µ|µi−1, θ) = ι(µ− µi−1|θ),

so that the center (mean) of ι(µ|µi−1, θ) evolves over MonteCarlo replications following a random walk.

Choosing the stand-in distribution:The “quality” of stand-in distribution helps inconvergence. It should be as close as possible to posteriordistribution.

Macroeconometrics



METROPOLIS-HASTING ALGORITHM III

Natural candidate: ML estimates of the parameters µfollow asymptotically multivariate Normal distribtionN(µ, Σ).Since we want to be sure to have fatter tails, lets scale thecovariance matrix:

N(µ, c2Σ)

This would be our first draw from stand-in density, ieι(µ?

0 |θ).Random walk Metropolis-Hastings would involve

ι(µ− µi−1|θ) = N(µi−1, c2Σ).

Macroeconometrics



METROPOLIS-HASTING ALGORITHM IV

Acceptance rate =accepted draws

all drawsOptimal acceptance rate is 0.44 for a model with one parameterand 0.23 if there more than five parameters.This guides us in choosing c (the scale factor of the asymptoticcovariance matrix):

1 start with c = 2.4/√

number of parameters.2 Increase (decrease) c if the acceptance rate is too high (low).

Macroeconometrics



MARKOV CHAIN DIAGNOSTICS I

In the ML estimation the problems related to likelihood (egidentification, or the model specification in general) showtypically up as numerical problem in maximizing thelikelihood.In Bayesian estimation they show up in the MCMC.Fundamental questions is the convergence of the chain(s):

Is the target distribution achieved?What is the impact of the starting values of the chain?Are blocks of draws correlated?

Partial solutions to possible problem

Remember that this is not standard Monte Carlo where 10000 replication is a big number. Here it is million (Markovchain)

Macroeconometrics



MARKOV CHAIN DIAGNOSTICS II

Throw away early iterations of each MCMC chain. Even 50%Asses the convergence

Plot the raw draws, ie parameter value against draw: ifthey are trending, there is a trouble.Plot multiple chains to the same graph: are they in the sameregion.Plot:

1ns

ns

∑s=1

g(µ(s))

as a function of ns.start Markov chain at extreme values of µ and checkwhether different runs of the chain settle to the samedistribution

Macroeconometrics



MARKOV CHAIN DIAGNOSTICS III

Simulate multiple chains: Look, among other things,whether the moments of a single parameter µ(j) aresimilar across the chainsCompute test statistics that measure variation within a chainand between the chains.These test statistic builds on the classical test set-ups thattry to detect structural breaks.

Testing the structural change, ie convergence.

Cusum-type of tests of structural change (see Yu andMykland (1998) or Yada manual)Partial means test by Geweke (2005) splits the chain intogroups:

Macroeconometrics



MARKOV CHAIN DIAGNOSTICS IVN is number of draws, Np = N/(2p), where p is positiveinteger. Define p separated partial means

S(N)j,p =

1Np

Np

∑m=1

S(µ(m+Np(2j−1))), j = 1, . . . , p

where S is some summary statistic of the parameters µ (egoriginal parameters). Let τj,p be the Newey-West (1987)numerical standard error for j = 1, . . . , p. Define the (p− 1)vector S(N)

p with typical element S(N)j+1,p − S(N)

j,p and the

(p− 1)× (p− 1) tridiagonal matrix V(N)p where

V(N)j,j = τ2

j,p + τ2j+1,p, j = 1, . . . , p− 1

andV(N)

j,j+1 = V(N)j+1,j = τ2

j+1,p, j = 1, . . . , p− 1.

Macroeconometrics



MARKOV CHAIN DIAGNOSTICS VThe statistic

G(N)p =

(S(N)

p

)′[V(N)

p ]−1S(N)p

d−→ χ2(p− 1),

as N → ∞ under the null that the MCMC chain hasconverged with the separated partial means being equal.

Multiple chain diagnosticsi = 1, . . . , N chain length; j = 1, . . . , M number of chains.Between chain variance is

B =N

M− 1

M

∑j=1

(Sj − S)2,

where

Sj =1N

N

∑i=1

Sij S =1M

M

∑j=1

Sj.

Macroeconometrics



MARKOV CHAIN DIAGNOSTICS VIand within chain variance

W =1

M(N− 1)

M

∑j=1

N

∑i=1

(Sij − Sj)2.

Define variance of S

ΣS =N− 1

NW +

1N

B,

andV = ΣS +

BMN

.

The potential scale reduction factor R should decline to 1 ifsimulation converges

R =

√VW

=

√N− 1

N+

(M + 1)BMNW

.

Macroeconometrics



CHOOSING PRIOR DISTRIBUTION IThink carefully:

What is the domain?Is it bounded?Shape of distribution: symmetric, skewed, which side?

How to choose prior:The choice of a prior distribution for a deep parameteroften follows directly from assumptions made in the DSGEmodel.Results from other studies (Microstudies or DSGEestimates) are already available.

Pre-sample.Other countries and data setsNOT the data you are currently using the estimation.

Should not be too restrictive — allow for a wide support.

Macroeconometrics



CHOOSING PRIOR DISTRIBUTION II

Most common distributions employed are:

Beta distribution; range is ∈ (p1, p2); autoregressiveparameters ∈ (0, 1)Gamma distribution; range is ∈ (p1, ∞);Normal distributionUniform distribution; range is ∈ (p1, p2); diffuse priorsInverted-gamma distribution; range is R+; variances

Yada manual has nice plots of the distributions.

Look at the model moments implied by priors!

Sensitivity analysis: how robust are the results to choice ofprior?

Macroeconometrics


Model comparison

OUTLINE

1 Introduction




5 Matching moments

Macroeconometrics


Model comparison

MODEL COMPARISON I

Posterior distribution can be used to assess conditionalprobabilities associated with alternative model specifications.This lies at the heart of the Bayesian approach to modelcomparison.

Relative conditional probabilities are calculated using oddsratio.

Two models: A and B with corresponding parametervectors that may differ µA and µB.Study model A (and change the symbols to study themodel B).

Macroeconometrics


Model comparison

MODEL COMPARISON II

Posterior (conditional on the model A)

p(µA|X, A) =L(X|µA, A)π(µA|A)

p(X|A)

Integrate both sides by µA (ie calculate the marginalprobabilities; ie average over the all possible parametervalues) to have

p(X|A) =∫

L(X|µA, A)π(µA|A)dµA.

This is the marginal likelihood of model A.

Macroeconometrics


Model comparison

MODEL COMPARISON III

Use Bayes’ rule to calculate conditional probabilityassociated with model A:

p(A|X) =p(X|A)π(A)

p(X)

(use the similar logic as in the start of the section, ie definejoint probability of data and model (X, A).Substitute the marginal likelihood into the above equation

p(A|X) =

[∫L(X|µA, A)π(µA|A)dµA

]π(A)

p(X)

Macroeconometrics


Model comparison

MODEL COMPARISON IVLook at the ratio of the above condtional density to that of themodel B. This is called posterior odds ratio:

POA,B =

[∫L(X|µA, A)π(µA|A)dµA

][∫L(X|µB, B)π(µB|B)dµB

]︸︷︷︸Bayes factor

π(A)

π(B)︸︷︷︸prior odds ratio

Beauty of this approachall models are treated symmetrically; no “null hypothesis”all models can be falseworks equally well for non-nested models

Hence, favour the model whose marginal likelihood is larger:Odds ∈ (1− 3) “very slight evidence” in favour of AOdds ∈ (3− 10) “slight evidence” in favour of A

Macroeconometrics


Model comparison

MODEL COMPARISON VOdds ∈ (10− 100) “strong to very strong evidence” infavour of AOdds > 100 “decisive evidence” in favour of A

Implementation is tough, because we need to integrate theparameters out.Easy solution would be to assume functional form to marginallikelihood. Laplace approximation utilizes Gaussian large sampleproperties

p(X|A) = (2π)T2 |Σµm

A| 12 p(µm

A |X, A)p(µmA |A),

where µmA is the posterior mode, ie the initial (ML estimated)

value of the parameters in stand-in distribution.

Macroeconometrics


Model comparison

MODEL COMPARISON VIHarmonic mean estimator is based on the following idea:

E[

f (µA)

p(µA|A)p(X|µA, A)|µA, A

]=

∫f (µA)dµA∫

p(µA|A)p(X|µA, A)dµA= p(X|A)−1,

where f is a probability density function. This suggest thefollowing estimator of the marginal density

p(X|A) =

[1B

B

∑b=1

f (µbA)

p(µbA|A)p(X|µb

A, A)

]−1

,

where µbA is a draw of µA in Metropolis-Hastings iteration.

Typically f is replaced by a truncated normal, this estimator isthen called by modified harmonic mean estimator.

Macroeconometrics


Model comparison

REFERENCESAnderson, Gary, and George Moore (1985) ‘A linear algebraic

procedure for solving linear perfect foresight models.’Economics Letters 17(3), 247–252

Andrle, Michal (2010) ‘A note on identification patterns inDSGE models.’ Working Paper Series 1235, European CentralBank, August

Blanchard, Olivier J., and C. Kahn (1980) ‘Solution of lineardifference models under rational expectations.’ Econometrica38, 1305–1311

DeJong, David N., and Chetan Dave (2007) StructuralMacroeconometrics (Princeton and Oxford: PrincetonUniversity Press)

den Haan, Wouter J., and Andrew Levin (1996) ‘A practioner’sguide to robust covariance matrix estimation.’ DiscussionPaper 96-17, University of California, San Diego. to beappear in Handbook of Statistics 15 (Chapter 12, 291–341)

Hansen, Lars Peter (1982) ‘Large sample properties ofgeneralized method of moments estimator.’ Econometrica50(4), 1029–1054

Hansen, Lars Peter, John Heaton, and Amir Yaron (1996)‘Finite-sample properties of some alternative GMMestimators.’ Journal of Business and Economic Statistics14(3), 262–280

Klein, Paul (2000) ‘Using the generalized schur form to solve amultivariate linear rational expectations model.’ Journal ofEconomic Dynamics and Control 24(10), 1405–1423

Kocherlakota, Narayana R. (1990) ‘On tests of representativeconsumer asset pricing models.’ Journal of MonetaryEconomics 26, 385–304

Kydland, Finn E., and Edward C. Prescott (1982) ‘Time to buildand aggregate fluctuations.’ Econometrica 50(6), 1345–1370

Newey, Whitney K, and Kenneth West (1994) ‘Automatic lagselection in covariance matrix estimation.’ Review of EconomicStudies 61, 631–653

Sargent, Thomas J. (1987) Macroeconomic Theory, second editioned. (Emerald)

Sims, Christopher A. (2011) ‘Nobel lecture.’ Technical ReportTauchen, George (1986) ‘Statistical properties of generalized

method of moments estimators of structural parametersobtained from financial market data.’ Journal of BusinessEconom. Statistics 4, 397–425

Zagaglia, Paolo (2005) ‘Solving rational-expectations modelsthrough the Anderson-Moore algorithm: An introduction tothe Matlab implementation.’ Computational Economics26(1), 91–106

Documents

Macroeconometrics - Winter 2014, Y1 (Masters students)ajripatt/opetus/metrics/slides.pdf · Macroeconometrics Winter 2014, Y1 (Masters students) Antti Ripatti1 1Economics and HECER