Post on 06-Apr-2018
8/2/2019 Lags and Lag Structure
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Lags and Lag Structure
Distributed Lag Modeling
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Introduction
• We are often interested in variables that change
across time rather than across individuals.
• Simple Static models relate a time series variable
to other time series variables.
• The effect is assumed to „operate‟ within the
period.
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Dynamic Models
• Dynamic effects
– Policy takes time to have an effect.
– Size and nature of effect can vary over time.
– Permanent vs. Temporary effects.
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The „Principle‟
• Macroeconomics
– e.g. the effect of X on Y in short run vs. the long run
• impulse response function
– X increases by 1 in year 1
– returns to normal afterwards
– what happens to y over time?
time
Y
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Distributed Lag
• Effect is distributed through time
– consumption function: effect of income through time
– effect of income taxes on GDP happens with a lag
– effect of monetary policy on output through time
yt = + 0 xt + 1 xt-1 + 2 xt-2 + et
it
t
i x
y E
)(
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The Distributed Lag Effect
Economic action
at time t
Effect
at time t
Effect
at time t+1
Effect
at time t+2
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The Distributed Lag EffectEffect at time t
Economic action
at time t
Economic action
at time t-1
Economic action
at time t-2
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Two Questions
1. How far back?
- What is the length of the lag?
- finite or infinite
2. Should the coefficients be restricted?
- e.g. smooth adjustment
- let the data decide
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Unrestricted Finite DL
• Finite: change in variable has an effect on another only for
a fixed period
– Fare Price Changes affect Ridership for 6 months
– the interval is assumed known with certainty
• Unrestricted (unstructured)
– the effect in period t+1 is not related to the effect in period t
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yt = + 0 xt + 1 xt-1 + 2 xt-2 + . . . +n xt-n + et
n unstructured lags
no systematic structure imposed on the ‟s
the ‟s are unrestricted
OLS will work
produce consistent and unbiased estimates
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Problems
1. n observations are lost with n-lag setup.
• data from 1990, 5 lags in model implies earliest point in regression is 1995
• use up degrees of freedom (n-k)
2. high degree of multicollinearity among xt-j‟s
• xt is very similar to xt-1 --- little independent information
• imprecise estimates
• large std errors, low t-tests
• hypothesis tests uncertain.
3. Several LHS variables• many degrees of freedom used for large n.
4. Could get greater precision using structure
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Arithmetic Lag Structure
i
i
0 = (n+1)
1 = n
2 = (n-1)
n =
.
.
.
0 1 2 . . . . . n n+1
..
.
.
linear
lag
structure
1. effect of X eventually goes to zero
2. Effect of each lag will be less than previous of
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The Arithmetic Lag Structure
Imposing the relationship:
ii = (n - i+ 1)
0 = (n+1)
1 = n
2
= (n-1)
3 = (n-2)
n-2 = 3
n-1 = 2 n =
only need to estimate one coefficient, ,
instead of n+1 coefficients, 0 , ... , n .
yt = + 0 xt + 1 xt-1 + 2 xt-2 + . . . +n xt-n + et
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• Suppose that X is (log of) fare price and Y is (log of)
Ridership, n=12 and is estimated to be 0.1
• the effect of a change in x on Ridership in the current
period is 0(n+1)=1.3
• the impact of fare price changes one period later has
declined to 1n=1.2
• n periods later, the impact is n 0.1
• n+1 periods later the impact is zero
it
t i
x
y E
)(
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Estimation
• Estimate using OLS
• only need to estimate one parameter:
• Have to do some algebra to rewrite the model inform that can be estimated.
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Advantages/disadvantages
• Fewer parameters to be estimated (only one) thanin the unrestricted lag structure
– Lower standard errors
– Higher t-statistics
– More reliable hypothesis tests
• What if the restriction is untrue?
– Biased and inconsistent
– A bit like wrong exclusion restrictions in 2SLS• Is the linear restriction likely to be true?
– Look at unrestricted model
– Do f-test
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2
1
/
/ )(
df SSE
df SSE SSE
F U
U R
F-test
• estimate the unrestricted model
• estimate the restricted (arithmetic lag) model
• calculate the test statistic
• compare with critical value F(df1,df2)
– df1=n the number of restrictions
• number of betas less number of gammas = (n+1)-1
– df2=number of observations-number of variables in the
unrestricted model (incl. constant)
– df2=(T-n)-(n+2)
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Polynomial Distributed Lag
• Linear shape of lags maybe restrictive
• Want “hump” shape
• Polynomial --- quadratic or higher
i = 0 + 1i + 2i2
i
it
t
x
y E
)(
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Polynomial Lag Structure
.. . .
.
0 1 2 3 4 i
i
0
12 3
4
• Similar idea to Arithmetic DL model; different shape to lags• Still Finite: the effect of X eventually goes to zero
• The coefficients are related to each other
• the effect of each lag will not necessarily be less than previous one i.e.
not uniform decline
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Estimation
• Estimate using OLS
• only need to estimate p parameters:
– number of parameters is equal to degree of polynomial
• Have to do some algebra to rewrite the model inform that can be estimated.
– model reduces to arithmetic model if polynomial is of
degree 1
• Do OLS on transformed model
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Advantages/Disadvantages
• Fewer parameters to be estimated (only the degree of
polynomial) than in the unrestricted lag structure
– more precise
• What if the restriction is untrue?
– biased and inconsistent
• Is the polynomial restriction likely to be true?
– more flexible than arithmetic DL
• what if approximately true?
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Lag Length
• For all three finite models we need to choose the lag length(DL,ADL,PDL)
• Think of this as choosing the cut-off point
– The time beyond which a variable will cease to have an impact
• No satisfactory objective criterion for deciding this• Choose n to be infinite?
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Lag-Length Criteria
• Akaike‟s AIC criterion
• Schwarz‟s SC criterion
• For each of these measures we seek that lag length that minimizes thecriterion used. Since adding more lagged variables reduces SSE , thesecond part of each of the criteria is a penalty function for addingadditional lags.
2( 2)ln n
SSE n AIC
T N T N
2 ln( )( ) ln n
n T N SSE SC n
T N T N
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Key Points1. How far back?
- What is the length of the lag?
- No conclusive answer
2. Should the coefficients be restricted?
- Let the data decide: unrestricted
- Arithmetic or polynomial- What degree of polynomial
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Geometric Lag Model
• DL is infinite --- infinite lag length
• But cannot estimate an infinite number of parameters
• restrict the lag coefficients to follow a pattern – estimate the parameters of this pattern
• For the geometric lag the pattern is one of continuous
decline at decreasing rate
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Geometric Lag Structure
i
.
..
. .0 1 2 3 4 i
1 =
2 = 2
3 =
3
4 = 4
0 = geometrically
declining
weights
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Estimation
• Cannot Estimate using OLS
– OLS will be inconsistent (just as with simultaneous equations)
• Only need to estimate two parameters: ,
• Have to do some algebra to rewrite the model in form thatcan be estimated.
• Then apply Koyck transformation
• Then use 2SLS