Time Series Econometrics Distributed Lag Modeling

1
Time Series Econometrics- Distributed Lag
Modeling

• Main Reading Gujarati, Chapter 17,
• Griffith, Judge and Hall (2001)

2
Time and Econometrics

• Time Series elements of cross-sectional
  (Retrospective designs)
• Univariate Time Series Models
• Multivariate Static Models
• Multivariate Dynamic Models
• Stationary Variables
• Non-Stationary Variables
• Panel Econometrics
• Note Most of the methods we examine are
  single-equation methods so bear in mind potential
  extensions in to multi-equation methods

3
Some Time Series/Stochastic Processes

• Fertility in America
• Vote Share of the Democrats in the 20th Century
• Ice-cream Consumption
• Barium Chloride Imports in to the US
• Capital Expenditures and Appropriations

4
Introduction

• Economists 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.

5
Dynamic Models

• Dynamic effects.
• Policy takes time to have an effect.
• The size and nature of the effect can vary over
  time.
• Permanent vs. Temporary effects.

6

• Macroeconomics
• e.g. the effect of M on Y in short run vs. the
  long run
• this is know as impulse response function
• money supply increases by 1 in year 1
• returns to normal afterwards
• what happens to y over time

7
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
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The Distributed Lag Effect
Effect at time t1
Effect at time t2
Effect at time t
Economic action at time t
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The Distributed Lag Effect
Effect at time t
Economic action at time t-2
Economic action at time t
Economic action at time t-1
10
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

11
Unrestricted Finite DL

• Finite change in variable has an effect on
  another only for a fixed period
• e.g. Monetary policy affects GDP for 18 months
• the interval is assumed known with certainty
• Unrestricted (unstructured)
• the effect in period t1 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 i.e. will produce consistent and
unbiased estimates
13
Problems

• 1. n observations are lost with n-lag setup.
• data from 1960, 5 lags in model implies earliest
  point in regression is 1965
• use up degrees of freedom (n-k)
• 2. high degree of multicollinearity among
  xt-js
• xt is very similar to xt-1 --- little
  independent information
• imprecise estimates
• large stn errors, low t-tests
• hypothesis tests uncertain.

14

• 3. Several LHS variables
• many degrees of freedom used for large n.
• 4. Could get greater precision using structure

15
Examples

• See example in See example in Hill, Griffiths and
  Judge (Table 15.3 and 15.4).
• low t statistics
• strange pattern of coefficients
• impulse response graph
• x goes up by one unit in year 1
• what happens through time?
• Fertility and Personal Exemption Example

16
Arithmetic Lag

• Still finite the effect of X eventually goes to
  zero
• The coefficients are not independent of each
  other
• The effect of each lag will be less than previous
  one
• E.G. Monetary policy in 1995 will have less of an
  effect on GDP in 1998 than will monetary policy
  in 1996
• Note how this is different to the capital exp
  example

17
Arithmetic Lag Structure (impulse response
function)
?i
.
?0 (n1)?
.
?1 n?
.
?2 (n-1)?
linear lag
structure
.
?n ?
0 1 2 . . .
. . n n1
i
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The Arithmetic Lag Structure
yt ? ?0 xt ?1 xt-1 ?2 xt-2 . . .
??n xt-n et
only need to estimate one coefficient,
??, instead of n1 coefficients, ?0 , ... , ?n .
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• Suppose that X is (log of) money supply and Y is
  (log of) GDP, n12 and g is estimated to be 0.1
• the effect of a change in x on GDP in the current
  period is b0(n1)g1.3
• the impact of monetary policy one period later
  has declined to b1ng1.2
• n periods later, the impact is bn g0.1
• n1 periods later the impact is zero

20
Estimation

• Estimate using OLS
• only need to estimate one parameter g
• Have to do some algebra to rewrite the model in
  form that can be estimated.

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yt ? ?0 xt ?1 xt-1 ?2 xt-2 . . .
??n xt-n et
Step 1 impose the restriction ?i (n - i
1) ?
yt ? (n1) ?xt n ?xt-1 (n-1) ?xt-2 .
. . ?xt-n et
Step 2 factor out the unknown coefficient, ? .
yt ? ? (n1)xt nxt-1 (n-1)xt-2 . .
. xt-n et
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Step 3 Define zt .
zt (n1)xt nxt-1 (n-1)xt-2 . . . xt-n
Step 4 Decide number of lags, n.
For n 4 zt 5xt 4xt-1 3xt-2
2xt-3 xt-4
Step 5 Run least squares regression on
yt ? ? zt et
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Advantages/disadvantages

• Fewer parameters to be estimated (only one) than
  in 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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F-test

• estimate the unrestricted model
• estimate the restricted (arithmetic lag) model
• calculate the test statistic

25

• compare with critical value F(df1,df2)
• df1n the number of restrictions
• number of betas less number of gammas (n1)-1
• df2number of observations-number of variables in
  the unrestricted model (incl. constant)
• df2(T-n)-(n2)

26
Polynomial Distributed Lag

• Linear shape to impulse response function usually
  thought to be two restrictive -- see monetary
  policy
• Want hump shape
• Polynomial --- quadratic or higher

27
Polynomial Lag Structure
?i
?2
?1
?3
?0
?4
0 1 2 3 4
i
28

• Similar idea to Arithmetic DL model
• just a different shape to the impulse response
  function
• 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

29
Estimation

• Estimate using OLS
• only need to estimate p parameters g
• number of parameters is equal to degree of
  polynomial
• Have to do some algebra to rewrite the model in
  form that can be estimated.
• model reduces to arithmetic model if polynomial
  is of degree 1
• Do OLS on transformed model

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n the length of the lag p degree of polynomial
where i 1, . . . , n
For example, a quadratic polynomial
?0 ?0 ?1 ?0 ?1 ?2 ?2 ?0
2?1 4?2 ?3 ?0 3?1 9?2 ?4
?0 4?1 16?2
where i 1, . . . , n p 2 and n 4
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yt ? ?0 xt ?1 xt-1 ?2 xt-2 ?3 xt-3
??4 xt-4 et
yt ? ?0?xt ??0 ?1 ?2?xt-1 (?0
2?1 4?2)xt-2 (?0
3?1 9?2)xt-3 (?0 4?1 16?2)xt-4 et
Step 2 factor out the unknown coefficients
?0, ?1, ?2.
yt ? ?0 xt xt-1 xt-2 xt-3 xt-4
?1 xt-1 2xt-2 3xt-3 4xt-4 ?2
xt-1 4xt-2 9xt-3 16xt-4 et
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yt ? ?0 xt xt-1 xt-2 xt-3 xt-4
?1 xt-1 2xt-2 3xt-3 4xt-4 ?2
xt-1 4xt-2 9xt-3 16xt-4 et
Step 3 Define zt0 , zt1 and zt2 for ?0 , ?1
, and ?2.
z t0 xt xt-1 xt-2 xt-3 xt-4
z t1 xt-1 2xt-2 3xt-3 4xt- 4
z t2 xt-1 4xt-2 9xt-3 16xt- 4
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Do OLS on
yt ? ?0 z t0 ?1 z t1 ?2 z t2 et
34
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?

35
F-test

• estimate the unrestricted model
• estimate the restricted (polynomial lag) model
• calculate the test statistic as before
• compare with critical value F(df1,df2)
• df1number of restrictionsnumber of b less the
  number of g(n1)-(p1)
• df2number of observations-number of variables in
  the unrestricted model (incl. constant)
• df2(T-n)-(n2)

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Example Capital Expenditure

• See example in Hill, Griffiths and Judge (Table
  15.3 and 15.4).
• high t statistics
• reasonable pattern of coefficients
• impulse response graph (figure 15.4)
• x goes up by one unit in year 1
• what happens through time?

37
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
• E.G. Monetary policy does not affect GDP after
  two years
• No satisfactory objective criterion for deciding
  this
• Book gives brief discussion of two
• Choose n to be infinite?

38
Lag-Length Criteria

• Akaikes AIC criterion
• Schwarzs SC criterion
• For each of these measures we seek that lag
  length that minimizes the criterion used. Since
  adding more lagged variables reduces SSE, the
  second part of each of the criteria is a penalty
  function for adding additional lags.

39
Summary

• 1. How far back?
• - What is the length of the lag?
• - No good answer
• 2. Should the coefficients be restricted?
• - Let the data decide unrestricted
• - Arithmetic or polynomial
• - What degree of polynomial

40
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

41
Geometric Lag Structure(impulse response
function)
?i
geometrically declining weights
42
Estimation

• Cannot Estimate using OLS
• Only need to estimate two parameters f,b
• Have to do some algebra to rewrite the model in
  form that can be estimated.
• Then apply Koyck transformation
• Then use 2SLS

43
infinite distributed lag model
yt ? ?0 xt ?1 xt-1 ?2 xt-2 . . .
et
geometric lag structure
?i ???i?? where 0lt?lt 1 and ??i??????
44
infinite unstructured lag
yt ? ?0 xt ?1 xt-1 ?2 xt-2 ?3 xt-3
. . . et
Substitute ?i ???i
infinite geometric lag
yt ? ??xt ? xt-1 ?? xt-2 ?? xt-3
. . .) et
45
Dynamic Response
yt ? ??xt ? xt-1 ?? xt-2 ?? xt-3
. . .) et
impact multiplier
?
interim multiplier (3-period)
?? ? ? ? ??
long-run multiplier
46
Koyck Transformation
Lag everything once, multiply by ??and subtract
from original
yt ? ??xt ? xt-1 ?? xt-2 ?? xt-3
. . .) et
? yt-1 ??? ??? xt-1 ?? xt-2 ?? xt-3
. . .) ? et-1
yt ? ? yt-1 ??????? ?xt (et ??? et-1)
yt ??????? ? yt-1 ?xt (et ??? et-1)
yt ??? ?? yt-1 ??xt ?t
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Need to Use 2SLS

• yt-1 is dependent on et-1 (see original eqn.)
• This implies that yt-1 is correlated with vt-1
• So OLS will be inconsistent (just as with
  simultaneous equations
• OLS cannot distinguish between change in yt
  caused by yt-1 that caused by vt
• OLS will treat changes in vt as being changes in
  yt-1

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ct ?1 ?2 yt et
yt ct it
49

• Use 2SLS
• 1. Regress yt-1 on xt-1 and calculate the
  fitted value
• 2. Use the fitted value in place of yt-1 in
  the Koyck regression

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• Why does this work?
• from the first stage the fitted value is not
  correlated with et-1 whereas yt-1 is
• so the fitted value is uncorrelated with
• vt (et -et-1 )
• 2SLS will produce consistent estimates of the
  Geometric Lag Model

51
Adaptive Expectations Model

• A version of the geometric lag model
• If we assume that individuals have adaptive
  expectation then the geometric lag model will
  emerge
• Assume expectations
• Formed on the basis of past experience
• Expectations are updated in the light of errors
• AE is not always consistent with rational
  expectations

52
Example Money Demand
yt ? ? xt et
yt demand for money
xt expected (anticipated) interest rate
(xt is not observable)
adjust expectations based on past error
xt - xt-1 ? (xt-1 - xt-1)
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Need some Algebra to get a form that can be
estimated
xt - xt-1 ? (xt-1 - xt-1)
rearrange to get xt on one side
xt ? xt-1 (1- ?) xt-1
or
? xt-1 xt - (1- ?) xt-1
54
Take the original model Lag this model once and
multiply by (1???)
yt ? ? xt et
(1???)yt-1 (1???)? (1???)? xt-1 (1???)et-1
subtract bottom equation from the top to get
yt ?? - (1???)yt-1 ? xt - (1???)xt-1
et -
(1???)et-1
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substitute in ? xt-1 xt - (1- ?)
xt-1
yt ?? - (1???)yt-1 ??xt-1 ut

• This is identical to the equation for the
  geometric lag where f(1-l)
• We can estimate consistently using 2SLS
• Question why does AE result in DL

where ut et - (1???)et-1
56
ExampleConsumption Function

• C is consumption and Y is expected income
• in order to decide on level of consumption the
  individual must make some guess about future
  income
• Assume that individual adjusts his expectations
  according to the AE hypothesis

57

• substitute in to the a form we can estimate
• Use 2SLS
• estimate by OLS
• use in place of

58
Partial Adjustment Model

• Another version of the geometric lag model
• Assume individuals adjust to the ideal gradually
• Cost of adjusting, so dont adjust quickly
• Example firms inventories

yt ? ? xt et
59

• Inventories partially adjust towards the optimal
  value.
• The parameter l is the fraction of difference
  between actual and desired that is adjusted.
• implicit there are costs that prevent instant
  adjustment.
• Note the equations are similar to but not exactly
  the same as the AE model (note position of the
  star).

yt - yt-1 ? (yt - yt-1)
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yt - yt-1 ? (yt - yt-1) ? (? ?xt
et - yt-1) ?? ??xt - ?yt-1 ?et
Solving for yt
yt ?? (1 - ??yt-1 ??xt ?et
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Conclusions

• This lecture has examined distributed lag models.
• An advance from the static models that we have
  previously examined.
• But generally assumes that we are working with
  stationary processes.
• The implications of non-stationarity is the topic
  for the next set of lectures.