Econometrics Lecture 1

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Econometrics Lecture 1

• A Review of OLS, Model Specification and
  Mis-specification Testing

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OUR OBJECTIVES

• To establish whether the hypothesised economic
  relationship is supported by empirical evidence
• To obtain 'good' estimates of the unknown
  parameters ?1 and ?2
• To be able to test hypotheses about the unknown
  parameters or to construct confidence intervals
  surrounding our estimates.
• To use our estimated regression model for other
  purposes, including particularly
• forecasting
• policy analysis/simulation modelling.

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THE REQUIREMENTS OF A GOOD ECONOMETRIC MODEL

• THE MODEL MUST BE DATA ADMISSIBLE.
• It must be logically possible for the data to
  have been generated by the model that is being
  assumed to generate the data. This implies that
  the model should impose any constraints that the
  data are required to satisfy (e.g. they must lie
  within the 0-1 interval in a model explaining
  proportions).
• THE MODEL MUST BE CONSISTENT WITH SOME ECONOMIC
  THEORY.
• REGRESSORS SHOULD BE (AT LEAST) WEAKLY EXOGENOUS
  WITH RESPECT TO THE PARAMETERS OF INTEREST
• This is a requirement for valid conditioning in a
  single equation modelling context, to permit
  efficient inference on a set of parameters of
  interest.
• THE MODEL SHOULD EXHIBIT PARAMETER CONSTANCY.
• This is essential if parameter estimates are to
  be meaningful, and if forecasting or policy
  analysis is to be valid.
• THE MODEL SHOULD BE DATA COHERENT.
• The residuals should be unpredictable from their
  past history. This implies the need for
  exhaustive misspecification testing.
• THE MODEL SHOULD BE ABLE TO ENCOMPASS A RANGE OF
  ALTERNATIVE MODELS.

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Model specification

• Step 1 Specify a statistical model that is
  consistent with the relevant prior theory
• (i) The choice of the set of variables to include
  in the model.
• (ii)The choice of functional form of the
  relationship (is it linear in the variables,
  linear in the logarithms of the variables, etc.?)
• Step 2 Select an estimator (OLS, GLS, GMM, IV,
  etc etc)
• Step 3 Estimate the regression model using the
  chosen estimator
• Step 4 Test whether the assumptions made are
  valid (in which case the regression model is
  statistically well-specified) and the estimator
  will have the desired properties.
• Step 5a
• If these tests show no evidence of
  misspecification in any relevant form, go on to
  conduct statistical inference about the
  parameters.
• Step 5b
• If these tests show evidence of misspecification
  in one or more relevant forms, then two possible
  courses of action seem to be implied
• If you are able to establish the precise form in
  which the model is misspecified, then it may be
  possible to find an alternative estimator which
  will is optimal or will have other desirable
  qualities when the regression model is
  statistically misspecified in a particular way.
• Regard statistical misspecification as a symptom
  of a flawed model. In this case, one should
  search for an alternative, well-specified
  regression model, and so return to Step 1.

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CLRM Assumption

• The assumptions of the CLRM are
• The dependent variable is a linear function of
  the set of possibly stochastic, covariance
  stationary regressor variables and a random
  disturbance term. The model specification is
  correct.
• The set of regressors is not perfectly collinear.
  This means that no regressor variable can be
  obtained as an exact linear combination of any
  subset of the other regressor variables.
• The error process has zero mean. That is, E(ut)
  0 for all t.
• The errors terms, ut, t1,..,T, are serially
  uncorrelated. That is, Cov(ut,us) 0 for all s
  not equal to t.
• The errors have a constant variance. That is,
  Var(ut) s2 for all t.
• Each regressor is asymptotically correlated with
  the equation disturbance, ut.
• We sometimes wish to make the following
  assumption
• The equation disturbances are normally
  distributed, for all t.

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ASSUMPTIONS ABOUT THE SPECIFICATION OF THE
REGRESSION MODEL
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Testing Restrictions

• F Test

NOTE the sample must be the same for the two
regressions
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Functional Form

• We know yf(x) but not the correct functional form

LINEAR
LOGARITHMIC (LOG-LINEAR)
LIN-LOG (SEMI-LOG)
LOG-LIN (SEMI-LOG)
RECIPROCAL/RATIO FORM

POLYNOMIAL
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Ramseys RESET test

• First Step run regression by OLS and save
  residuals and fitted values
• Second Step run auxiliary regression

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Assumptions about Equation Disturbance Term

• ABSENCE OF DISTURBANCE TERM SERIAL CORRELATION
• CONSTANCY OF DISTURBANCE TERM VARIANCE
  (HOMOSCEDASTICITY)
• NORMALITY OF DISTURBANCE TERM

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ABSENCE OF DISTURBANCE TERM SERIAL CORRELATION

• THE DURBIN-WATSON (DW) TEST\
• Only first order autocorrelation
• Only for models without lagged variables

• DURBIN'S h STATISTIC

• GODFREY'S LAGRANGE MULTIPLIER (LM) TESTS

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Responses to presence of Autocorrelation

• a Type 1 error has occurred (you have incorrectly
  rejected a true null)
• your chosen regression model is misspecified in
  some way, perhaps because a variable has been
  incorrectly omitted, there are insufficient lags
  in the model, or the functional form is
  incorrect.
• the disturbance term is actually serially
  correlated GLS

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CONSTANCY OF DISTURBANCE TERM VARIANCE
(HOMOSCEDASTICITY)

• Types of heteroscedasticity

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Assumptions about the Parameters of the Model

• PARAMETER CONSTANCY OVER THE WHOLE SAMPLE PERIOD

Chow 1 If enough observation in the two
subsample
Chow 2 predictive failure test not enough
observations in the subsample