Introduction to Econometrics

1
Introduction to Econometrics

• Lecture 9
• More on Dynamic Models
• Modelling Strategies

2
Dynamic model formulations

• Simple delayed effects models
• Distributed lag models and the Koyck
  transformation
• A review of the Partial Adjustment Mechanism
• Autoregressive models
• Autoregressive Distributed Lag (ADL) models
• Error Correction models (ECM)

3
Simple delayed effect models

• Yt a b Xt-1 ut
• Changes in X affect Y but with a known lag (in
  this case one period).
• Provided the length of the lag is known, or is
  easily established, this raises no new problems.
  Indeed it can be helpful from a forecasting point
  of view because the value of the independent
  variable will be known with certainty at the time
  when the next forecast of Y is to be made.
• EXAMPLE Forecasting employment in Orange County
  (California)
• EMPt a b RGNPt ut
• where EMPt denotes total employment in the
  county in quarter t,
• RGNPt-1denotes real GNP for the whole of the US
  in the previous quarter.
• Source Doti and Adibi (1998)
• Here we can actually exploit the lags in the
  relationship for forecasting purposes

4
Distributed lag models

• Yt a b0Xt b1 Xt-1 . bsXt-s ut
• where s is the maximum lag allowed for.
• Rather than assume that the whole of the affect
  is delayed, this model has the effect distributed
  over a number of periods.
• Problems establishing the maximum lag s
• loss of degrees of freedom
• possible multicollinearity
• Example. Accidents and safety training Koop
  (2000)
• Yt a b0Xt b1 Xt-1 .. b4Xt-4 ut
• where Yt losses due to accidents for a company
  (/month)
• Xt hours of safety training
  provided to each worker in month t
• A simple regression of Y on X appeared to show
  no relationship between these variables -
  although the DW stat suggested misspecification.

5
The Koyck transformation

• Suppose that we anticipate a gradual decline in
  the affect of X on Y as the number of periods
  increase. For example Y might be sales and X
  advertising expenditure. If we can assume a
  geometric rate of decline and an infinite lag
  structure we can use the Koyck transformation to
  produce a simple model with just Xt and Yt-1 as
  regressors
• Writing Yt a b0Xt b1 Xt-1 . bsXt-s
  .. ut 1
• If bj1/bj ? for all j (with b0 just b)
• 1 becomes
• Yt a bXt ?bXt-1 . ?sbXt-s ....
  ut 2
• Lag 2 by one period and multiply by ?
• ? Yt-1 ? a ? bXt-1 ? 2bXt-2 . ? sbXt-s
  . ? ut 3
• Subtract 3 from 2 and rearrange
• Yt a(1- ?) bXt ? Yt-1 ut - ? ut-1 4

6
A review of the partial adjustment model
7
A review of the partial adjustment model (2)
8
The underlying rationale of the partial
adjustment model
9
The underlying rationale of the partial
adjustment model ( 2)
10
The Error Correction Mechanism (ECM)
11
The simple ECM specification a consumption
function example
12
Autoregressive-distributed lag models(ARDL
models)
13
An example of an AR model in economics

• Robert E Hall (JPE 1978) suggested that
  consumption would follow a simple first-order
• autoregressive process if
• (1) consumption depends only upon permanent
  income (YP)
• (2) agents expectations are formed rationally
• The second assumption means that YPt YPt-1 ?t
  where E(?t) 0
• ?t represents the revision made to agents
  perceived permanent income in period t.
• Individuals out not to expect their permanent
  income to change if they did this knowledge
• should already have been used to reassess
  permanent income so Halls consumption
• function is sometimes known as the surprise
  consumption function ?t is the surprise.
• (1) requires Ct K YPt
• Substituting we find that
• Ct Ct-1 K ?t
• or Ct Ct-1 et
• Consumption should follow a random walk.

14
Modelling strategies
The three golden rules of econometrics are test,
test and test. David F. Hendry (1980)
15
General to specific modelling

• Begin with a general model which nests the
  restricted model and so allows any restrictions
  to be tested
• These restrictions may be suggested either by
  theory or by empirical results

16
General to specific modelling (2) diagnostic
testing of the general model

• TEST 1
• First ensure that the general model does not
  suffer from any diagnostic problems. Examine the
  residuals in the general model to ensure that
  they possess acceptable properties.
• (Test for problems of autocorrelation,
  heteroskedasticity, non-normality, incorrect
  functional form etc.)

17
General to specific modelling
General to specific modelling (3) testing
restrictions on the general model

• TEST 2
• Now test the restrictions implied by the
  specific model against the general model either
  by exclusion tests or other tests of linear
  restrictions.

18
General to specific modelling
General to specific modelling (4) diagnostic
testing of the simple model

• TEST 3
• If the restricted model is accepted, test its
  residuals to ensure that this more specific model
  is still acceptable on diagnostic grounds

19
General to specific modelling (5) the two
blades of the scissors
Test parameter restrictions on the more general
model
Then check diagnostics for the restricted model
20
Frequently (and recently) asked questions!

• Should I include all the variables in the
  database in my model?
• How many explanatory variables do I need in my
  model?
• How many models do I need to estimate?
• What functional form should I be using?
• Do I need to include lagged variables?
• What are interactive dummies do I need them?
• Which regression model will work best and how do
  I arrive at it?

21
Typical cross-section model

• Maybe several hundred observations
• Maybe 10-12 potential explanatory variables, some
  of which will be dummy variables.
• So plenty of degrees of freedom but still lots of
  potential models to try, especially if you
  consider alternative functional forms,
  interactive dummies
• Maybe problems of multicollinearity,
  heteroskedasticity and non-normality
• Model selection is not just a matter of
  maximizing Rbar-squared over all possible models
  (or some other criterion)
• Use economic theory and past studies to identify
  core variables
• Test exclusion restrictions from a general model
  but balanced against misspecification tests.
  Informed searches.

22
Typical time series model

• Maybe only around a hundred observations
• Maybe four or five potential explanatory
  variables, some of which may be dummy variables.
• Relatively few degrees of freedom but still lots
  of potential models to try, especially if you
  consider alternative functional forms, lagged
  variables and interactive dummies
• As well as problems of multicollinearity,
  heteroskedasticity and non-normality there may be
  issues of autocorrelation and non-stationarity
• Model selection is not just a matter of
  maximizing Rbar-squared over all possible models
• Use economic theory and past studies to identify
  core variables and if possible functional form
• Test exclusion and other restrictions from a
  general model but balanced against
  misspecification tests. Informed searches.