Short-run models and Error Correction Mechanisms

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Short-run models and Error Correction Mechanisms

Professor Bill MitchellDirector, Centre of Full Employment and Equity

Department of EconomicsUniversity of Newcastle

Australia

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yObjectives

• To introduce the concept of a short-run model in economics.

• To show how short- and long-run models interact.

• To explain the concept of an Error Correction Mechanism (ECM).

• To show how ECM and cointegration work together.

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yLong-run model review

• Economic theory is essentially static and mostly considers equilibrium relationships.

• Equilibrium (long-run) relations are normally in terms of levels.

• The problem is that with non-stationary variables we are prone to finding spurious relationships if we run regressions in levels.

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yFigure 1 Z1, Z2 and Z4

• The Z variables were simulated using random walk functions with = 1:

• Any relation between them is spurious and because they contain stochastic trends.

-30

-20

-10

0

10

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30

60 65 70 75 80 85 90 95 00

Z1 Z2 Z4

1t t ty y u

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So this equation exhibits “good” econometric results but is in fact spurious and tells us nothing at all.

The “good” is qualified b/c the DW statistic is the clue.

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-12

-8

-4

0

4

8

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60 65 70 75 80 85 90 95 00

Residual Actual Fitted

The clue is in the residuals…

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yThe long-run model quandary

• So how do we proceed?

• In the 1970s, the approach was to take differences?

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yTaking differences removes trends

-3

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-1

0

1

2

3

60 65 70 75 80 85 90 95 00

DZ1

-3

-2

-1

0

1

2

3

60 65 70 75 80 85 90 95 00

DZ2

1 1t t ty y y

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yTaking differences…

• Do we still have a relationship?

• To test it we would run

Z1t = 0 + 1Z2t + 2Z4t + t

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yLevels and differences

0 1 2

1 1 1 2 1 1

1 2

(3)

(4)

t t t t

t t t t t t t t

t t t t

t t

y a a x a z e

y y a x x a z z e e

y a x a z u

u e

Question 1: What are the problems of estimating economic relationships in difference form like Equation (4), given that it can overcome the problem of non-stationarity in the levels of the variables concerned?

Problems ?

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yError Correction Approach

• This approach attempts to use differenced data to model the short-run adjustments but also take into account and estimate long-run information.

• Consider this long-run model:

0 1t ty x

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yEquilibrium and disequilibrium

10(6)

(6 ) 0

(6 ) 0

t t

t t

t t

y x

a y x

b y x

Question 2: What are the properties of Equation (6)? Does it tell you about the path of adjustment for y if x changes?

Question 3: What are some of the reasons why equilibrium may not hold in every period? In a forecasting environment why would it be necessary to know about the nature of disequilibrium adjustment paths?

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yEquilibrium and disequilibrium

(6)

(6 ) 0

(6 ) 0

t t

t t

t t

y x

a y x

b y x

When Equation 6(b) holds we cannot observe the relationship in Equation (6).

But we can observe the short-run, dynamic relationship that would reduce to Equation (6) whenever equilibrium occurs.

So we need to learn a bit about the short-run models.

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yShort-run model

• Short-run models are also called adjustment functions or dynamic models or lagged models.

• A typical (simplified) version is the first-order model:

• The order is selected to “soak” up the serial correlation (the “missing dynamics”)

0 1 1 1 2 1t t t t ty a y x x u

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yProperties of short-run model

Question 4: What are the properties of Equation (8)? Tell a story in words about the process through which the long-run relationship is re-established if x was to change in a particular period?

0 1 1 1 2 1(8) t t t t ty a y x x u

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yProperties of short-run model

Question 4: Parameter 1 measures the immediate impact of a change in x on y. It is not the long-run impact that would occur from one equilibrium to another though.

Why not? What is the difference between 1 and 1? Can you find an expression that links 1 and 1?

Solve the steady-state properties of (8).

0 1 1 1 2 1(8)

(6)

t t t t t

t t

y a y x x u

y x

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ySolving for the steady-state…

0 1 1 1 2 1

* *1 1

* *1 0 1 2

1 2* *0

1 1

(8)

(1 )

(1 ) (1 )

(6)

t t t t t

t t t t

t t

y a y x x u

y y y x x x

y a x

ay x

y x

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yQuestions 6 to 9 …

• Question 6: Assume that 1 = 0.9, 1 = 0.6 and 2 = 0.3. Starting from an equilibrium position, how long does it take for y to return to that equilibrium, if x increases by a unit and remains at that new level?

• Question 7: What is the change in y in the first period after the shock? What is the change in y in the second period? What is the total change?

• Question 8: How is the shift in equilibrium dependent on the value taken by the AR parameter?

• Spreadsheet demonstration.

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yError Correction models

• The basic dynamic model may also suffer from non-stationarity problems.

• We have seen the differencing is unsatisfactory.

• Error Correction Mechanism (ECM) models begin with the basic short-run model.

• After re-parameterising, the ECM form has both dynamic and steady-state information in the one equation.

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yECM questions

Question 9: See if you can perform the re-parameterisation to get the ECM model which combines differences and levels. It is shown below as Equation (10).

ECM form and the steady-state

0 1 1 1 2 1

1 0 1 1 1 1 1 1 1 1 2 1

1 201 1 1 1

1 1

1 1 1 0 0 1

(8)

(1 )(1 ) (1 )

(10) (1 )

t t t t t

t t t t t t t t t

t t t t

t t t t

y a y x x u

y y a y y x x x x u

ay x y x

y x y x

Question 10: Would you say that Equation (8) and Equation (10) are equivalent? What are the advantages of Equation (10) relative to Equation (8)?

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yECM form and the steady-state

1 201 1 1 1

1 1

1 1 1 0 0 1

1 2 *0

1 1

(10) (1 )(1 ) (1 )

(10) (1 )

*(1 ) (1 )

t t t t

t t t t

ay x y x

y x y x

ay x

Question 11: Provide an interpretation of the expression in square brackets in Equation (10). Have you already encountered an expression like this earlier in this lecture?

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1 201 1 1 1

1 1

1 0 1 1 1

1 1 0 1 1

(10) (1 )(1 ) (1 )

(6 )

t t t t

t t t

t t t

ay x y x

y x e

e y x

You can see that the term in square brackets is equivalent to the expression for disequilibrium in the steady-state.

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1 201 1 1 1

1 1

1 0 1 1 1

1 1 0 1 1

(10) (1 )(1 ) (1 )

(6 )

t t t t

t t t

t t t

ay x y x

y x e

e y x

Question 12: Is the term in square brackets stationary given that it is in terms of levels? Under what conditions will it be stationary?

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yCointegration and ECM model

0 1

1 201 1 1 1

1 1

1 1

(6)

(10) (1 )(1 ) (1 )

(10 )

t t t

t t t t

t t t

y x e

ay x y x

y x e u

Two-step procedure for estimating the model:

• Test for cointegration in Equation (6).

• If null accepted then the residuals would be stationary.

• Estimate (10) with residuals from CI Equation (6) as the ECM term.

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y End of Talk

Short-run models and Error Correction Mechanisms

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