Fiscal Effects of Foreign Aid An Application of Vector Autoregressive VAR Techniques

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Fiscal Effects of Foreign Aid An Application of
Vector Autoregressive (VAR) Techniques

• Tim Lloyd
• Associate Professor
• CREDIT Research Fellow

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Today . . .

• VAR Methodology
• Reduced and structural forms
• Cointegrated VAR modelling
• VAR VECM
• Hypothesis testing
• Impulse response Analysis
• Groundwork for modelling effects of aid
• Morrissey

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Modelling (effectiveness of) Aid

• Emerging branch of growth literature
• Does aid work?
• Burnside and Dollar (2000)
• Aid is mediated via government behaviour
• How is aid used in recipients?
• What are mechanisms?
• Fungibility studies
• Is aid spent as intended?
• Fiscal Response Models
• Aid in fiscal balance (borrowing, taxation,
  spending)

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Modelling Aid

• Two approaches in literature
• 1 Fungibility studies
• Is aid spent as intended?
• Case studies
• descriptive,
• lack generality and theoretical rigour
• Morrissey and McGillivray (2000)

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Modelling Aid

• 2 Fiscal Response Models
• Aid as an injection in fiscal balance
• (borrowing, taxation, spending)
• Micro-economic foundations
• Governments maximise utility by attaining their
  revenue and expenditure targets, subject to
  budget constraint.

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Modelling Aid

• Serious shortcomings of FRMs
• Knowledge of economic structure
• Specification of structural (behavioural)
  equations
• Assumes government set targets
• Ignores dynamics
• Common to all structural approaches
• Estimation data intensive
• technically complex (3SLS)
• Results model dependent (sensitive)
• Heroic assumptions (incredible restrictions)

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Another Approach?

• VAR overcomes many (not all) shortcomings
• Does not impose economic structure
• Imposes few restrictions
• Exogeneous/endogenous tested not imposed
• Explicitly incorporates dynamics
• Distinguishes between long and short runs
• While data driven, theory is tested not assumed
• Structural model may be arrived at rather than
  imposed

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Vector Autoregession

• Model of choice in time series
• Originally developed for I(0) data
• Sims (1980)
• Now workhorse using I(1) data
• Johansen (1988)
• Popularity tied to concept of cointegration
• Engle and Granger (1987)

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What is a VAR?

• System of k equations with p lags VAR(p)
• zt is a (k?1) vector of variables ( equations)
  at time t
• A0 is a (k?1) vector of constants
• Ai are (k?k) matrices of coefficients
• ?t is a (k?1) vector of errors at time t
• ?t (0, ?) ? is non-diagonal
  variance-covariance matrix
• All variables potentially simultaneous, hence
  some covariances non-zero and thus ?
  non-diagonal.

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What is a VAR?

• Consider a 2 variable (z1, z2) VAR(1)
• Each et are assumed uncorrelated in time
• Ete1t,e1t-i0 for i 1 . . .T
• But may be contemporaneously linked across
  equations by non zero covariances
• Ete1t,e2t?0

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What is a VAR?

• System of equations in reduced form
• All regressors predetermined (exogenous)
• All variables potentially endogenous
• No current dated variables on RHS
• However, the VAR linked to underlying
  structural equations

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VARs Structural Models

• Consider a dynamic simultaneous equation model
• These are the structural equations because they
  describe the simultaneity explicitly
• z2t appears of RHS of z1t
• z1t appears of RHS of z2t
• Represent economic (causal) relationships

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VARs Structural Models

• In matrix form, the structural equations are,
• Pre-multiplying by the inverse of coefficient
  matrix,

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VARs Structural Models

• Pre-multiplying by the inverse of coefficient
  matrix,
• Gives a VAR(1)

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VARs Structural Models

• VARs can be derived from structural economic
  model
• VAR can be thought of as a reduced form
  representation of economic relationships defined
  by structural equations
• But its not always possible to derive one
  uniquely from the other (requires identification)
• Hence, traditionally VARs criticized
• atheoretical equations do not represent
  economic relationships
• Cannot address economic (structural) issues of
  interest

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VARs Structural Models

• However, knowing the structural Economic model is
  overly rigorous for most purposes
• Much economic information may be gleaned from a
  VAR providing data is non-stationary and
  cointegrated
• Often we dont need to know SEM to obtain what we
  want
• VARs also tractable empirically

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VAR Non-stationarity

• Most time series data in economics I(1)
• I(1) allows (potentially) cointegration
• Cointegration equilibrium relationship
• Hence VAR can be used to test for cointegration
  (equilibrium relations of economics)
• Cointegration bridges atheoretical VARs and
  economic theory

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VAR VECM

• VARs can be re-written as Vector Error Correction
  Models (VECM)
• Consider the VAR(p)
• This can be re-written as
• where

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Deriving the VECM

• Consider the VAR(2)
• Subtract to give
• From RHS add and subtract

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The VECM

• VECM rewriting of VAR in I(0) space
• Dependent variable now DztI(0)
• Under cointegration all terms are I(0)
  facilitates hypothesis testing
• Observationally equivalent to VAR
• Contains identical information
• Unlike VAR, VECM is economically interpretable
• short run dynamics (Gi) and
• long run (cointegrating) relationships (P)
• But can only have VECM if VAR is cointegrated

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Interpreting the VECM

• Consider a VAR(2) where
• which has an VECM representation
• Noting that where are both
• Assume one cointegrating relation (r1)

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Interpreting the VECM

• Ignoring G1 for moment, focus on
• bji are the cointegrating or long run
  parameters defining the equilibrium relation.
• aij error correction coefficients quantifying
  speed at which deviation from each equilibrium
  is corrected

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Interpreting the VECM

• In equilibrium, (b11z1t-2 b21z2t-2)0
• Hence (b11z1t-2 b21z2t-2) ? 0 measures
  disequilbrium
• Error correction coefficients measure how fast
  each variable adjusts to disequilibrium (if at
  all)
• Where aij 0 no adjustment and variable i is
  weakly exogenous
• Does not adjust to disequilibrium
• It enters long run relationship but does not
  adjust to restore the equilibrium (Long run
  forcing)

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Interpreting the VECM

• We can also examine a related concept of Granger
  Causality in the VECM which relates to the short
  run parameters
• If then z2t is not a Granger cause
  of z1t
• If a variable z1t is weakly exogenous and z2t
  does not Granger-cause it then it is also
  strongly exogenous.

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Impulse Response Analysis

• VECM distinguishes long from short run
• However VAR (VECM) readily facilitate dynamic
  simulation (IR analysis)
• Simulate the effect of hypothetical shock
• Account for knock on and feedback effects

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Impulse Response Analysis

• VAR and VECM are reduced forms
• Economically empty
• Although readily facilitate IR analysis
• We seek to investigate structural hypotheses
• Results not valid unless the VECM is identified
• IR Analysis prone to mis-interpretation
• Structural VARs and VECMs allow valid impulse
  response analysis
• Whole system need not be identified
• But part of it must be.

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Impulse Response Analysis

• A VAR(2) model,
• With VECM representation
• Can be re-written in VMA form,

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Impulse Response Analysis

• IR function defines the effect of a shock of
  known size on and element of on
• Represents a dynamic simulation
• Traces the effect of the shock through the entire
  system
• Allows for interdependencies in the system
• Akin to what if experiment rather than more
  usual ceteris paribus

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Summary

• VAR is very versatile tool in time series
  econometrics
• Reduced form of wide class of simultaneous
  equation systems
• Utilizing VAR/VECM embody cointegration
• Test for equilibrium relations of theory
• Allow investigation of both equilibrium
  tendencies and dynamic adjustment

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References

• Burnside C. and D. Dollar (2000) Aid, Policies
  and Growth American Economic Review 904,
  847-868.
• Engle R. and C.W. Granger (1987) Cointegration
  and Error Correction Representation and
  Testing, Econometrica 48, 1-48.
• Hendry, D.F. and K. Juselius (2000) Explaining
  Cointegration Part I Energy Journal.
• Hendry, D.F. and K. Juselius K (2001) Explaining
  Cointegration Part II Energy Journal.
• Harris R. and R. Sollis (2003) Applied Time
  Series Modelling and Forecasting, Wiley
  Chicester.
• Hamilton, J.D. (1994) Time Series Analysis,
  Princeton University Press, Princeton. Chapter
  11.
• Johansen, S.(1988) Statistical Analysis of
  Cointegration Vectors, Journal of Economic
  Dynamics and Control. 12 231-254.
• McGillivray M. and Morrissey (2000) Aid
  Fungibility in Assessing Aid Red Herring or True
  Concern? Journal of International Development,
  123 413-428.
• Osei, R., Morrissey, W.O. and Lloyd T. (2003)
  Modelling the Fiscal Effects of Aid An Impulse
  Response Analysis for Ghana CREDIT Research
  Paper 03/10. University of Nottingham.
• Patterson, K. (2000) An Introduction to Applied
  Econometrics A Time Series Approach, Macmillan.
  Chapter 14.