Incorporating Uncertainties into Economic Forecasts: an Application to Forecasting

Economic Activity in Croatia

Dario RukeljMinistry of Finance of the Republic of Croatia

Barbara UlloaCentral Bank of Chile

Young Economists’ Seminar (YES)Dubrovnik Economic Conference

June 23, 2010

Uncertainty is the only certainty there is, and knowing how to live with insecurity is the only security...

John Allen Paulos

1. INTRODUCTION

2. METHODOLOGY

3. RESULTS

4. CONCLUSION

MOTIVATION

Source: European Commission

-5

-4

-3

-2

-1

0

1

2

3

4

2002 2003 2004 2005 2006 2007 2008 2009 2010 2011

%

Projection in the current year Projection one year ahead

Projection two years ahead Outturn

FORECASTS OF THE EUROZONE REAL GDP GROWTH

DEALING WITH UNCERTAINTY

Point forecasts Mode of distribution

Interval forecasts Consists of an upper and a lower limit

Density forecasts The whole probability distribution of the forecasts

1. INTRODUCTION

2. METHODOLOGY

3. RESULTS

4. CONCLUSION

STOCHASTIC SIMULATION APPROACH*

Data generating process assumed to be VAR model, estimated in the finite sample:

tit

p

iit tyy

10

1

ˆˆˆ

Forecasts incorporating future uncertainties:

)(10

)(

1

)( )(ˆˆˆˆ xhT

xihT

p

ii

xhT uhTyy

Forecasts incorporating future and parameter uncertainties: Simulate s in sample values of y

For each of these estimated models, r replications of the forecasts are calculated

)(10

)(

1

)( ˆˆˆˆ st

sit

p

ii

st utyy

),(),(

1

)(),( )(ˆˆˆˆ10

sxhT

sssxihT

p

i

ssxhT uhTyy

i

* Garrat, Pessaran and Shin (2003 and 2006)

SIMULATED SHOCKS

Parametric approach :

),()(),( ˆ srht

ssrht vPu

)'()()( ˆˆˆ sss PP

),0(),( IIINv srht

Non-parametric approach: random draws with replacements from the in sample residuals

Unbalanced risks:

xforxC

xforxC

vf srht

,2

1exp

,2

1exp

),,,'(2

22

2

21

21),(

121

kC /2k

CALCULATION

Future uncertainty Obtain the set of simulated shocks Generate the forecasts using the simulated shocks Sort the forecasted values of the variable of interest Determine probability bands by the deciles

Future and parameter uncertainty Using initial values for the number of lags determined by the order of the VAR, calculate

forecasts ahead using estimated parameters of the initial model, as well as applying a shock to each observation in each period

Re estimate the models with each set of time series obtained in this way Based on these models forecasts are made like under only future uncertainty

PRESENTATION

5.15

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5.29

2007 I IV VII X 2008 I IV VII X 2009 I IV

Pro

bab

ilit

y d

ensi

ty

90%

80%

70%

50%

60%

Probability Distribution Fanchart

EVALUATION

In Sample Fancharts

5.120

5.140

5.160

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5.200

5.220

5.240

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5.280

5.300

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Probability Integral Transform

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1.00

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KOLMOGOROV – SMIRNOV TEST

Kolmogorov – Smirnov test can be used for comparing two distributions

Comparing Probability Integral Transform of outturns with uniform distribution

Let F(a) be the cumulative distribution function of uniform distribution

Cumulative distribution function of empirical distribution is given by:

tn

baF1

,ˆ

aFbaFD i ,ˆmax

where t is the number of observations of variable b such that

If variable b comes from uniform distribution then D should be small

b a

1. INTRODUCTION

2. METHODOLOGY

3. RESULTS

4. CONCLUSION

Reduced form VECM from Rukelj (2010) considered:

ttttt uxxctbxx 661111ˆ...ˆˆ)1(ˆ'

where xt is vector of endogenous variables (m, g and y).

Rewritten in a VAR form:

titi

it utyx 10

7

1

ˆˆˆ

PORTMANTEAU TEST (H0:Rh=(r1,...,rh)=0)

Tested order: 10

Adjusted test statistic 66.884

p-Value: 0.151

JARQUE-BERA TEST

Variable Test Statistic p-Value Skewness Kurtosis

u1 1.074 0.585 0.052 3.421

u2 14.798 0.001 0.745 3.609

u3 1.066 0.587 0.060 3.415

BENCHMARK MODEL

FUTURE UNCERTAINTY

Fanchart – Parametric Approach PIT – Parametric Approach

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2006I

IV VII X 2007I

IV VII X 2008I

IV VII X 2009I

IV0.00

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0.90

1.00

1 2 3 4 5 6 7 8 9 10 11 12

FUTURE UNCERTAINTY

Fanchart – Non Parametric Approach PIT – Non Parametric Approach

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5.30

2006I

IV VII X 2007I

IV VII X 2008I

IV VII X 2009I

IV0.00

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FUTURE UNCERTAINTY

Fanchart – Skewed Distribution PIT – Skewed Distribution

0.00

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0.90

1.00

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5.30

2006I

IV VII X 2007I

IV VII X 2008I

IV VII X 2009I

IV

FUTURE AND PARAMETER UNCERTAINTY

Fanchart – Parametric Approach PIT – Parametric Approach

4.90

5.00

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5.50

2006I

IV VII X 2007I

IV VII X 2008I

IV VII X 2009I

IV0.00

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0.90

1.00

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FUTURE AND PARAMETER UNCERTAINTY

Fanchart – Non Parametric Approach PIT – Non Parametric Approach

4.90

5.00

5.10

5.20

5.30

5.40

5.50

2006I

IV VII X 2007I

IV VII X 2008I

IV VII X 2009I

IV0.00

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0.90

1.00

1 2 3 4 5 6 7 8 9 10 11 12

KOLMOGOROV – SMIRNOV TEST RESULTS

Kolmogorov (D) 1 2 3 4 5 6 7 8 9 10 11 12

Value (D) 0.14 0.20 0.24 0.18 0.30 0.26 0.21 0.24 0.22 0.22 0.21 0.19Adjusted value 0.83 1.25 1.47 1.13 1.83 1.60 1.30 1.49 1.34 1.38 1.26 1.17Probability 0.49 0.09 0.03 0.16 0.00 0.01 0.07 0.02 0.06 0.04 0.08 0.13

Value (D) 0.17 0.22 0.26 0.22 0.29 0.28 0.24 0.28 0.25 0.24 0.22 0.20Adjusted value 1.06 1.33 1.62 1.37 1.79 1.69 1.49 1.73 1.52 1.45 1.37 1.24Probability 0.21 0.06 0.01 0.05 0.00 0.01 0.02 0.00 0.02 0.03 0.05 0.09

Value (D) 0.20 0.17 0.19 0.17 0.23 0.29 0.26 0.30 0.30 0.27 0.29 0.27Adjusted value 1.22 1.06 1.16 1.04 1.39 1.79 1.60 1.86 1.86 1.67 1.79 1.66Probability 0.10 0.21 0.14 0.23 0.04 0.00 0.01 0.00 0.00 0.00 0.00 0.01

Value (D) 0.41 0.42 0.41 0.43 0.40 0.42 0.39 0.38 0.38 0.37 0.38 0.37Adjusted value 2.53 2.61 2.52 2.65 2.45 2.59 2.42 2.35 2.33 2.30 2.35 2.28Probability 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

Value (D) 0.35 0.38 0.38 0.35 0.39 0.38 0.36 0.39 0.33 0.35 0.35 0.32Adjusted value 2.18 2.35 2.35 2.15 2.42 2.33 2.22 2.39 2.01 2.15 2.15 1.96Probability 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

Future and parameter uncertainty, non - parametric

Future uncertainty, parametric

Future uncertainty, non - parametric

Future uncertainty, skewed (σ1=1, σ2=2)

Future and parameter uncertainty, parametric

1. INTRODUCTION

2. METHODOLOGY

3. RESULTS

4. CONCLUSION

FORECASTING WITH UNCERTAINTY

Probability Forecasts for the Real GDP Growth

2010 Q1 2010 Q2 2010 Q3 2010 Q4 2011 Q1 2011 Q2 2011 Q3 2011 Q4

Central Projection -2.50 -0.50 0.50 1.00 2.50 2.00 1.50 1.00

Probability that distribution will be between specified interval, %

Pr. {<0%} 95 62 43 35 32 34 37 41Pr. {0% - 2%} <5 34 32 31 29 27 22 19Pr. {2% - 4%} <5 <5 22 21 20 20 18 16Pr. {>6%} <5 <5 <5 <5 19 19 23 24

CONCLUSION

In this paper we have shown how to calculate, present and evaluate density forecasts by stochastic simulation approach

An application of this methodological framework to the chosen benchmark model showed that: parametric and non-parametric approach yielded similar results incorporating parameter uncertainty results in a much wider probability

bands of the forecasts evaluation of the density forecasts indicate a better performance when

only future, without parameter uncertainties are considered

Future research in this topic should incorporate model uncertainty and additional goodness of fit tests

Thank you for your attention!