Download Autoregressive and Distributed

Agung Harjaya Buana
Budi Suhendar
Krismiyati
Yogi Sugiawan
Yusuf Wicaksono
Static Models
• A time series model where only
contemporaneous explanatory variables affect
the dependent variable
• Relate a time series variable to other time series
variables
• The effect is assumed to operate within a same
period
Dynamic Models
• A time series model where the lagged value of
explanatory variables and/or dependent variable
affect the present value of dependent variable
𝑌𝑡 = 𝛼 + 𝛽0 𝑋𝑡 + 𝛽1 𝑋𝑡−1 + 𝛽2 𝑋𝑡−2 + 𝑢𝑡
𝑌𝑡 = 𝛼 + 𝛽𝑋𝑡 + 𝛾𝑌𝑡−1 + 𝑢𝑡
What is the role of lags in economics?
 What are the reasons for the lags?
 Is there any theoretical justification for the
commonly used lagged models in empirical
econometrics?
 What is the relationship, if any, between
autoregressive and distributed lag models? Can
one be derived from the other?
 What are some of the statistical problems
involved in estimating such models?
 Does a lead-lag relationship between variables
imply causality? If so, how does one measure it?

DistributedLag Model
OLS
Almon
Aproach (PDL)
LAG
Autoregressive
Model
Koyck
Approach
Instrumental
Variable
Causality
Adaptive
Expectations
Granger
Causality Test
PAM
𝑌𝑡 = 𝛼 + 𝛽0 𝑋𝑡 + 𝛽1 𝑋𝑡−1 + 𝛽2 𝑋𝑡−2 + ⋯ + 𝛽2 𝑋𝑡−𝑘 + 𝑢𝑡
Examples
• The consumption function
• Demand deposit (creation of bank
money)
• Link between money and prices
• Lag between R&D expenditure and
productivity
• The J curve of international economic
• The accelerator model of investment
Psychological
Technological
Institutional
Lags
𝑌𝑡 = 𝛼 + 𝛽0 𝑋𝑡 + 𝛽1 𝑋𝑡−1 + 𝛽2 𝑋𝑡−2 + ⋯ + 𝑢𝑡
𝑌𝑡 = 8.37 + 0.171 𝑋𝑡
𝑌𝑡 = 8.27 + 0.111 𝑋𝑡 + 0.064𝑋𝑡−1
𝑌𝑡 = 8.27 + 0.109 𝑋𝑡 + 0.071𝑋𝑡−1 − 0.055𝑋𝑡−2
𝑌𝑡 = 8.32 + 0.108 𝑋𝑡 + 0.063𝑋𝑡−1 + 0.022𝑋𝑡−2 − 0.020𝑋𝑡−3
𝛽𝑘 = 𝛽0 𝜆𝑘
𝑘 = 0,1, …
∞
𝛽𝑘 = 𝛽0
𝑘=0
1
1−𝜆
𝑌𝑡 = 𝛼 + 𝛽0 𝑋𝑡 + 𝛽0 𝜆𝑋𝑡−1 + 𝛽0 𝜆2 𝑋𝑡−2 + ⋯ + 𝑢𝑡
𝑌𝑡−1 = 𝛼 + 𝛽0 𝑋𝑡−1 + 𝛽0 𝜆𝑋𝑡−2 + 𝛽0 𝜆2 𝑋𝑡−3 + ⋯ + 𝑢𝑡−1
𝜆𝑌𝑡−1 = 𝜆𝛼 + 𝜆𝛽0 𝑋𝑡−1 + 𝛽0 𝜆2 𝑋𝑡−2 + 𝛽0 𝜆3 𝑋𝑡−3 + ⋯ + 𝜆𝑢𝑡−1
𝑌𝑡 − 𝜆𝑌𝑡−1 = 𝛼 1 − 𝜆 + 𝛽0 𝑋𝑡 + (𝑢𝑡 − 𝜆𝑢𝑡−1 )
𝑌𝑡 = 𝛼 1 − 𝜆 + 𝛽0 𝑋𝑡 + 𝜆𝑌𝑡−1 + [𝑢𝑡 − 𝜆𝑢𝑡−1 ]
log 2
𝑀𝑒𝑑𝑖𝑎𝑛 𝐿𝑎𝑔 = −
log 𝜆
𝑀𝑒𝑎𝑛 𝐿𝑎𝑔 =
𝜆
1 −𝜆
𝑌𝑡 = 𝛽0 + 𝛽1 𝑋 ∗ 𝑡 + 𝑢𝑡
𝑋 ∗ 𝑡 − 𝑋 ∗ 𝑡−1 = 𝛾 (𝑋𝑡 − 𝑋 ∗ 𝑡−1 )
𝑌𝑡 = 𝛽0 + 𝛽1 𝛾𝑋𝑡 + 1 − 𝛾 𝑋 ∗ 𝑡−1 ) + 𝑢𝑡
𝑌𝑡 = 𝛾𝛽0 + 𝛾𝛽1 𝑋𝑡 + 1 − 𝛾 𝑌𝑡−1 + [𝑢𝑡 − 1 − 𝛾 𝑢𝑡−1 ]
Assumption (Long-Run, or Equilibrium, Demand)
𝑌 ∗ 𝑡 = 𝛽0 + 𝛽1 𝑋𝑡 + 𝑢𝑡
Partial (Stock) Adjustment Hypothesis
𝑌𝑡 − 𝑌𝑡−1 = 𝛿 (𝑌 ∗ 𝑡 − 𝑌𝑡−1 )
Adjustment Mechanism
𝑌𝑡 = 𝛿𝑌 ∗ 𝑡 + (1 − 𝛿)𝑌𝑡−1
Partial Adjustment Model (Short-Run Demand)
𝑌𝑡 = 𝛿𝛽0 + 𝛿𝛽1 𝑋𝑡 + 1 − 𝛿 𝑌𝑡−1 + 𝛿𝑢𝑡
Comments
• The PAM model resembles
both the Koyck and the AE
model in that it is
autoregressive
• Its disturbance term is
simpler
PAM vs AE Model
The gradual adjustment due to rigidities
• AE model is based on
uncertainty of the future
(interest rate, prices, etc)
• PAM model is based on
technical or institutional
rigidities
Koyck
𝑌𝑡 = 𝛼 1 − 𝜆 + 𝛽0 𝑋𝑡 + 𝜆𝑌𝑡−1 + [𝑢𝑡 − 𝜆𝑢𝑡−1 ]
Adaptive Expectations
𝑌𝑡 = 𝛾𝛽0 + 𝛾𝛽1 𝑋𝑡 + 1 − 𝛾 𝑌𝑡−1 + [𝑢𝑡 − 1 − 𝛾 𝑢𝑡−1 ]
Partial Adjustment
𝑌𝑡 = 𝛿𝛽0 + 𝛿𝛽1 𝑋𝑡 + 1 − 𝛿 𝑌𝑡−1 + 𝛿𝑢𝑡
Common Form
𝑌𝑡 = 𝛼0 + 𝛼1 𝑋𝑡 + 𝛼2 𝑌𝑡−1 + 𝑣𝑡
Koyck and AEM
• Presence of stochastic explanatory
variables and possibility of serial
correlation.
• Cannot be estimated by using OLS
Partial Adjustment
• OLS estimation of PAM yield consistent
estimates although tend to bias.
Comments
• Although PAM provides consstent
estimation, one should not assume that it
applies rather than the Koyck or adaptive
expectations model.
• A model should be chosen on the basis of
strong theoretical considerations.
OLS for Koyck and AEM
•Cannot be used because the explanatory variable tends to be
correlated with error term
•Can be used if the correlation can be removed
How to Remove It?
•Use a proxy called Instrumental Variable (IV)
•Based on Liviatan suggestion, Xt-1 is the instrumental variable for Yt-1
Does It Solve the Problem?
•No, a problem with multicollinearity occurs
•Thus, although Liviatan procedure yields consistent estimates, the
estimator are likely to be inefficient
How to Find a Good Proxy?
•Refer to the previous presentation
Can we use d stat?
• No, we cannot, because the computed d value in
such model generally tends toward 2
How to detect it?
• Durbin propose the h statistic test
𝑛
ℎ= 𝜌
1 − 𝑛[𝑣𝑎𝑟 𝛼2 ]
• where
𝑑
𝜌≈1−
2
Features of the h statistic
• To compute h, we need consider only the
variance of the coefficient of the lagged
Yt-1
• The test is not applicable if [nvar(a2)]
exceeds 1
• Since the test is a large-sample test, its
application in small samples is not strictly
justified (Inder and Kiviet). It has been
suggested to use the more powerful test,
the BG test.
Koyck Scheme
Almon Polynomial-Lag Scheme
1st Step
• Rewrite the finite distributed-lag model into a more compact
form:
𝑘
𝑌𝑡 = 𝛼 +
𝛽𝑖 𝑋𝑡−𝑖 + 𝑢𝑡
𝑖=0
2nd Step
• By following the Weierstrass’ theorem, Almon assumes that βi
can be approximated by:
𝛽𝑖 = 𝑎0 + 𝑎1 𝑖 + 𝑎2 𝑖 2 + ⋯ + 𝑎𝑚 𝑖 𝑚
3rd Step
• Substitutes the equation from the second step to the first step
we get:
𝑘
(𝑎0 + 𝑎1 𝑖 + 𝑎2 𝑖 2 + ⋯ + 𝑎𝑚 𝑖 𝑚 )𝑋𝑡−𝑖 + 𝑢𝑡
𝑌𝑡 = 𝛼 +
𝑖=0
4th Step
• Defines new variables Z’s:
𝑘
𝑘
𝑍0𝑡 =
𝑋𝑡−𝑖 ;
𝑖=0
𝑍1𝑡 =
𝑘
𝑖𝑋𝑡−𝑖 ; … … ;
𝑖=0
𝑖 𝑚 𝑋𝑡−𝑖
𝑍𝑚𝑡 =
𝑖=0
5th Step
• Rewrite the equation in the third step by using the
new variables
𝑌𝑡 = 𝛼 + 𝑎0 𝑍0𝑡 + 𝑎1 𝑍1𝑡 + ⋯ + 𝑎𝑚 𝑍𝑚𝑡 + 𝑢𝑡
6th Step
• Estimates the equation on the fifth
step by using OLS
• Satisfies the CLRM assumption
• Estimates β from a’s:
𝛽0 = 𝑎0
𝛽1 = 𝑎0 + 𝑎1 + 𝑎2
𝛽2 = 𝑎0 + 2𝑎1 + 4𝑎2
𝛽3 = 𝑎0 + 3𝑎1 + 9𝑎2
𝜷𝒌 = 𝒂𝟎 + 𝒌𝒂𝟏 + 𝒌𝟐 𝒂𝟐
Determine the Length of Lag (AIC or SIC)
Omission of relevant variable
bias
Inclusion of irrelevant variable
bias
Degree of Polynomial
Number of turning points
the choose of m is subjective
Constructing Z’s
𝑘
𝑖 𝑚 𝑋𝑡−𝑖
𝑍𝑚𝑡 =
𝑖=0
Imply Causality
• . . . time does not run backward. That is, if event A
happens before event B, then it is possible that A is
causing B. However, it is not possible that B is causing
A. In other words, events in the past can cause events
to happen today. Future events cannot …
Predictive Causality
• . . . the statement “yi causes yj” is just shorthand for
the more precise, but longwinded, statement, “yi
contains useful information for predicting yj (in the
linear least squares sense), over and above the past
histories of the other variables in the system.” To save
space, we simply say that yi causes yj
GDP and Money Supply
𝑛
𝐺𝐷𝑃𝑡 =
𝑛
𝛼𝑖 𝑀𝑡−1 +
𝑖=1
𝑛
𝑀𝑡 =
𝛽𝑗 𝐺𝐷𝑃𝑡−1 + 𝑢1𝑡
𝑖=1
𝑛
𝜆𝑖 𝑀𝑡−1 +
𝑖=1
𝛿𝑗 𝐺𝐷𝑃𝑡−1 + 𝑢2𝑡
𝑖=1
• Assumes that u1t and u2t are uncorrelated
Four Possible Cases from F-test
• Unidirectional causality from GDP to M
• Unidirectional causality from M to GDP
• Feedback, or bilateral causality
• Independence
1st Step
• Regress current GDP on all lagged GDP terms and
other variables (if any)
• Calculate the restricted residual sum of squares, RSSR
2nd Step
• Regress all variables including the lagged M terms
• Calculate the unrestricted residual sum of squares,
RSSUR
3rd Step
• The null hypothesis is H0: αi = 0, i = 1,2,…, n, that is
lagged M terms do not belong in the regression
4th Step
5th Step
6th Step
•.
(𝑅𝑆𝑆𝑅 − 𝑅𝑆𝑆𝑈𝑅 )/𝑚
𝐹=
𝑅𝑆𝑆𝑈𝑅 /(𝑛 − 𝑘)
• If the computed F value exceeds the critical F value at
the chosen level of significance, we reject the null
hypothesis
• M causes GDP
• Repeat steps 1 to 5 to test whether GDP causes M
Stationarity
• We have to be sure that all variables are stationary, if
they are not, make them stationary
• Beware of spurious causality
Number of Lag
• AIC or SIC
• The direction of causality may depend critically on
the number of lagged terms included
No Autocorrelation
• Make sure that the error terms entering the causality
test are uncorrelated
Exogeneity
Weak
Estimating
and Testing
Strong
Forecasting
Super
Policy
Analysis
Koyck, AEM and PAM
• The Koyck is developed based on a purely algebraic
approach, meanwhile the AEM and PAM are built based on
economic principles
• A unique feature of all of those model is that they all are
autoregressive in nature
• Bias and inconsistency are the case with the Koyck and
AEM. To overcome them, we can use the instrument
variable
Almon PDL
• Avoid the estimation problems associated with
autoregressive model
• Problems in determining the lag length and the degree of
the polynomial