econ 4160: econometrics-modelling and systems estimation...

Intro Dynamics is a typical feature Cobweb model Business cycle dynamics Causality and invariance References

ECON 4160: Econometrics-Modelling andSystems Estimation

Lecture 1: Introduction

Ragnar Nymoen

Department of Economics University of Oslo

20 August 2018


Lecture plan

I The lectures will follow themain route map of thetextbook, see right picture.

I Start today with Ch. 1.I Lecture 2 and 3 (joint with

ECON 5106) review thebackground in econometrics.

I Lecture 4 and 5 cover anotherbackground: the maths ofdifference equations, a mainanalytical workhorse for thiscourse.

I See posted Lecture plan formore info.

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Computer classes, seminars and obligatory

I Computer classes: They are listed as Plenary exercises on thesemester page.

I Seminars: There are 8 listed on the semester page. The exerciseset to the first seminar has already been posted. The others willfollow as the semester progresses.

I Obligatory assignment: One of the seminars is an obligatorywritten assignment. Details about dates come later

The aim of the rest of the Introductory lecture is to give a first,intuitive, introduction to several key concepts in dynamic modelling.

By the use of two simple economic models (ie bachelor level).

As noted, the literature reference to this lecture is Chapter 1 in thetextbook: Dynamic Econometrics for Empirical Macroeconomic Modelling.

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Dynamics is typical of economicsystems

I A typical trait of real-world macroeconomic systems is thatthe adjustment to a shock (ie impulse) spans several timeperiods.

I Impulses to the macroeconomic system that are propagatedby the system’s internal mechanisms, into effects that lastsfor several periods after the occurrence of the shock.

I Remark: “occurrence” is not always unambiguous.Sometimes the effects can start before the implementationof a policy change for example.

I In models of such systems, time must play an essential role.

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Let t denote time and let C denote private consumption and letI denote private income. Which of the following are dynamicmodel equations?

a) C = 1.5 + 0.75Ib) Ct = 1.5 + 0.75It

c) Ct = 1.5 + 0.75It−1

d) Ct = 1.5 + 0.25t + 0.25It−1 + 0.25Ct−1

In this Lecture we review two simple models that illustrateseveral concepts of we will use throughout the course.

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Cobweb modelLet Pt denote the market clearing price in a product market intime period t, and let Qt denote equilibrium demand andsupply in period t (i.e. QD

t = QSt = Qt) ) The model consists of

the demand equation, (1), and the supply equation (2):Qt = aPt + b, a < 0 (1)Qt = cPt−1 + d, c > 0 (2)

I The defining trait is that the short-run response of supplywith respect to a price change is zero.

I Supply is perfectly inelastic in the short run. Laggedsupply response with respect to price. Interpretations:

I Production and/or delivery lagsI Supply depends on the expected price Pe

t , but expectationformation is:

Pet = Pt−1

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Cobweb-model

I Initial stationary state is(P10,Q10).

I It is defined by the (blue)intersection point betweenthe static long-run supplyand demand schedules:

Q = aP + b D-curveQ = cP + d S-curve

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Cobweb-model dynamics

I Assume an exogenous and permanent demand increase.I Formally: a positive change in the parameter, bI In the static analysis, only the long-run effect can be

analysed.I In the dynamic analysis, the short-run, and all

“intermediate-run” effects can be analysed.I The stability (or not) of the process can be analysed.I All this, and more, means that the dynamic model offers

more content and analysis than the static model.

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Cobweb-model dynamic effectsThe system is in a stationary state in period 10, meaning thatP10 = P9 and Q10 = Q9.The Demand-shift hits the system in period 11.

I Effect in the period ofshock:P11 > P10,Q11 = Q10.

I It is custom to call this animpact effect or short runeffect.

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Cobweb-model dynamic effects

I Effect in period of shock:P11 > P10,Q11 = Q10

I Effect one period after theshock:P12 < P11,Q12 > Q11.

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Cobweb: Final form and autoregressiveparameter

I A final form equation of a model expresses an endogenousvariable as function of its own lags and of current and laggedvalues of exogenous variables.

I Final form equations are important for understanding the solutionof dynamic models, and for analysing the effects of shocks.

I For the cobweb-model, the two final form equations become: (seeExercise 1.1 in the book):

Pt = φ1Pt−1 +d− bt

a(3)

Qt = φ1Qt−1 +da− cbt−1

a, (4)

φ1 is the autoregressive parameter.

φ1 =ca. (5)

I Note: φ1 < 0 is implied by the assumptions of the model.

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Cobweb: Stable, unstable andexplosive dynamics

I φ1 < 0 is known as negative autocorrelationI A closer characterization of cobweb dynamics requires

additional assumptions:I Stable: −1 < φ1 < 0⇔ c < −aI Instable: φ1 = −1⇔ c = −aI Explosive: φ1 < −1⇔ c > −a

I What has been assumed in the graphs above?

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Computer simulated effects

I By using numbers for theparameters, we cancomputer-simulate thedynamic responses.

I The responses areoscillating,

I But the oscillations becomedampened over time.

I The graphs demonstratethe phenomenon of stabledynamics with negativeautocorrelation,

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Stationary state and equilibrium correction

I We can formalize the stationary state (P∗,Q∗) as the solution of thestatic model:

Q∗ = aP∗ + b D-curveQ∗ = cP∗ + d S-curve

and obtain:

Q∗ =ad− bca− c

(6)

P∗ =d− ba− c

(7)

I The stationary state, i.e. equilibrium state, is (globally asymptotically)stable if and only if:

−1 < φ1 < 1,I Note, again, only φ1 < 1 is implied by the assumption of the model.I Stability of the equilibrium (P∗,Q∗), secured by −1 < φ1 is therefore a

separate assumption.

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Stationary state and equilibrium correctionFollowing Ch. 1.4 in the book, we can re-express the two final formequations to become:

∆Pt = (φ1 − 1) [Pt−1 − P∗] , (8)∆Qt = (φ1 − 1) [Qt−1 −Q∗] , (9)

where ∆Pt and ∆Qt denote the differences from period t− 1 to t. Forexample

∆Pt = Pt − Pt−1

I In the stable case of φ1 > −1, cobweb model dynamics is equilibriumcorrecting.

I As we have seen, after the demand shock, Qt and Pt are oscillatingaround the new equilibrium

I The equilibrium correction associated with stable dynamics will play acentral role in the course.

I A much used acronym is ECM, for Error Correction Model.

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Random fluctuations (random variables)

I So far, the dynamics have been deterministic, as in dynamic economictheoretical analysis

I However, to build econometric models, by the use of data andstatistics, the endogenous economic variables need to be representedas random (stochastic) variables.

I We therefore introduce two random shocks: εdt (demand) and εst(supply):

Qt = aPt + bt + εdt, a < 0, (10)Qt = cPt−1 + d + εst, c > 0. (11)

(εdt,εst) are specified with zero means, and with fixed variances.

In econometrics, it is custom to refer to (10) and (11) as structuralequations, and to assume that the two structural disturbances areuncorrelated.

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Reduced form and final form (stochasticversions)

I A reduced form equation expresses a current endogenous variable interms of exogenous variables, and of lags of itself and of otherendogenous variables. In our case:

Pt =caPt−1 +

d− bta

+εst − εdt

a, a < 0 (12)

Qt = cPt−1 + d + εst, c > 0 (13)

I The stochastic version of the two final form equations become (usethe same steps as in Exercise 1.1 in book):

Pt = φ1Pt−1 +d− bt

a+εst − εdt

a, (14)

Qt = φ1Qt−1 +da− cbt−1

a+

aεst − cεdt−1a

. (15)

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Computer simulated effects

I By using numbers for theparameters, and thecomputer’s randomnumber generator, we cansimulate the dynamicresponses also for therandom variables case.

I Demand-shift occurring inperiod 11.

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Estimation of cobweb dynamics

I Since the plots show computer generated data, using a = −1.3 andc = 1 we know what the true value of φ1 is.

I But what if we draw “a veil of ignorance” over it all, and imaginethat we would have to rely on an estimate of φ1?

I Would OLS give a reliable estimate?I Not a trivial question to answer. For example: Does the IID

assumption hold?

We can formulate (12) as an regression model equation:

Pt = β0 + β1Pt−1 + β2S11t + ut, (16)

S11t is a step-dummy which captures the shift in the demand function:

S11t =

{1 if t ≥ 11,0 otherwise. (17)

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Estimation of cobweb dynamicsOLS estimation of (16) by the use of the 59 observations gives:

Pt = − 0.755(0.053)

Pt−1 + 4.22(0.128)

+ 0.745(0.027)

S11t (18)

I The numbers in parentheses are standard errors of the estimatedcoefficients (see Exercise 1.3 in the book)

I How close is the estimated φ1 to the true value?I Note that Pt−1 is not a strictly exogenous regressor (as defined in

introductory econometrics),I Instead Pt−1 is pre-determined: uncorrelated with current and

future disturbances, ut,ut+1, ..., but correlated with lagged (past)disturbances ut−1,ut−2, ...

I Apparently, estimation of the parameter φ1 is possible even withoutexogenous regressors, (and without the use of instrumentalvariables)?

I Or was it just a stroke of good luck?I The course will develop the econometric theory needed to give

general answers to these questions.20/28


The Fulton data set

I As first example of real world time series, we look at theFulton Fish Market Data set, Graddy (2006).

I Downloadable from the the Lecture 1 module,I where you also find the cobweb model data set used above.

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Negative and positive autocorrelation

I The cobweb model’s negative autocorrelation, with dominantshort-run oscillations for both price and quantity, can be relevant formarkets and sectors of the economy where supply is inelastic in theshort run.

I Agricultural products are examples of this (although in moderneconomies supply can be replenished)

I Housing supply is necessarily fixed in the short-run (it is a stock), butnet housing demand can change a great deal, and rapidly. Theimplication is that housing prices have a potential for volatility.

I The market for foreign exchange, under a floating exchange rate regime.Since the net supply of domestic currency is fixed, variations in the netdemand for kroner will determine the exchange rate.

I However, in markets for manufactured goods and services, the mainprinciple may be that short-run output is largely demand driven. Andpositive autocorrelation may be more typical than negativeautocorrelation

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A macro model with positive autocorrelationA Keynesian model (Appendix A shows a real Business Cycle Model,requires more symbosl and derivations):

Ct = a + bGDPt + cCt−1 + εCt, (19)GDPt = Ct + Jt, (20)

Jt = J∗ + εJt, (21)

I Ct private consumption in period t.I GDPt and Jt represent gross domestic product, and capital formation

(ie investment).The three variables are defined in real terms.

I a - c are parameters.I εCt, and εJt represent random shocks to consumption and investment,

with zero means (ie mathematical expectation). For simplicity, weassume that they are uncorrelated.

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A macro model with positiveautocorrelation

I Final form equation for private consumption (Exercise 1.4 in book):

Ct =c

1− bCt−1 +

11− b

[a + bJ∗ + εCt + bεJt

]. (22)

I We see that:φ1 =

c1− b

, (23)

is the autoregressive coefficient.I With c > 0, and 0 < b < 1, φ1 is positive, φ1 > 0, and private

consumption is therefore positively autocorrelated in this model.I Positive autocorrelation can be stable, unstable or explosive.I Stable dynamics: φ1 < 1

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A macro model with positiveautocorrelationI The stationary equilibrium

state defined by:

C∗ = a + bGDP∗ + cC∗,

GDP∗ = C∗ + J∗,J = J∗.

I is stable if 0 ≤ φ1 < 1.I Plot shows simulated

dynamics, using parametersb = 0.25 and c = 0.65, hence:φ1 = 0.713.

I ECM for Ct:

∆Ct = (φ1−1) [Ct−1 − C∗]+εCt + bεJt

1− b,

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Casuality, correlation and invariance

I Real world macroeconomicsystems are complex andchanging.

I Although we can learn aboutthe common autocorrelationfrom estimation of a singleequation, the main interest isusually relationships betweenvariables

I Therefore, methods forestimation ofmultiple-equation models formutual-depedencises will becentral.

I Models of systems can includecauses in the form of exogenousvariables, as in the picture.

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Casuality, correlation and invariance

I As we know: Correlation should not be mistaken forcausality.

I Correlation is also different from regression.I A certain form of stability of coefficient, called invariance,

can sometimes aid causality discussions in dynamiceconometrics.

I Invariance captures the idea that parts of a system (eg themutual dependence between X and Y conditional on Z)can be stable even if there is a regime change in theprocess that generate Z.

I Different concepts of econometric exogeneity will becentral in this course. Chapter 8 in the book.

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References

Graddy, K. (2006). The Fulton Fish Market. Journal of EconomicPerspectives, 20, 207–220.

Nymoen, R. (2018). Dynamic Econometrics for EmpiricalMacroeconomic Modelling. Manuscript.

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