Econometric Methods 1

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Econometric Methods 1

• Richard Dickens

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Introductory remarks

• One lecture and one class per week. Classes on
  Tuesday mornings at 9am and 11am, Lecture in the
  afternoon 2-4.
• Classes cover problems and exercises, which you
  are expected to attempt before the class. I will
  take in and grade one of the exercises during the
  term.
• Computer classes to teach you to use STATA later
  this term (4-5 sessions starting in week 4 or 5).
  You can then do computer-based exercises. Very
  important for your dissertations next summer.
• Book Wooldridge or Stock and Watson are probably
  the best.
• Assessment of this part of the course by 3½-hour
  exam after Christmas.

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Aims of the course

• The aim of the course is to make you proficient
  in the practical aspects of econometrics. This
  means
• Being able to estimate an appropriate model
• To be able to interpret your results
• To be aware of the shortcomings of your results
• To be able to judge the quality of published
  econometric studies.
• This is not a theoretical course in econometrics
• No proofs!

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What is Econometrics?

• The application of mathematical and statistical
  techniques to data in order to collect evidence
  on questions of interest to economics.
• Mixture of economics (providing the theory, or,
  better, imposing a logical structure on the form
  of the question), mathematics (turning it into
  mathematical form) and statistics (estimating the
  parameters of the mathematical model).
• The core technique is estimating a linear
  regression by ordinary least squares.

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Why use econometrics?

• To measure parameters of interest. For example
  what is the impact of increased welfare payments
  on job search among the unemployed?
• To understand behavioural puzzles. Are women
  paid less than similarly experienced and
  qualified men in similar work, and if so, why?
  Why do farmers with small amounts of land work
  longer hours per hectare than farmers with larger
  plots of land?
• To make predictions about the future. Will the
  housing market pick up? What is the medium-term
  direction of oil prices?
• To help formulate policy. Welfare payment
  example, in 1 above. Impact of minimum wages,
  programme evaluation.

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The simple linear model

• We usually start off from some idea, perhaps
  suggested by economic theory, that suggests X
  influences Y.
• Let us use a concrete example and a real data
  set.
• The relationship between farm size and
  productivity
• Data are the results of a survey of 400 farms in
  Gujarat State in India in 1984

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The simple linear model

• Start with a simplest linear relationship
• V a bS
• V is value added and S is farm size. Know the
  relationship is not exact, so, add an error term
  to our model
• V a bS u
• u is a random variable. Implies we are modelling
  V as a random variable. The part of V which we
  model is a bS. Thus u is the un-modelled
  component.
• This plot doesnt fit a straight-line
  relationship but if we take natural log of both
  variables and plot the relationship between them.

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Log(Value added) 8.13 -0.15 log(Crop area)
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Potential problems - bias

• Reverse causality Assume causality runs from
  right hand side to left, i.e. farm size
  determines productivity. What if high
  productivity farmers chose to work on small
  farms?
• Omitted explanatory variables What has been left
  out that might cause productivity differences
  among farms? e.g. high quality land is found in
  small farms. The b will be biased. It would be
  zero if we controlled for land quality.
• Structural stability What if the sample contained
  observations from two populations? Coefficient
  could reflect differences between populations
  rather than within populations.
• Measurement error
• Concept error - differences between theoretical
  concept and empirical counterpart
• a persons level of human capital measured by
  years of schooling,
• Empirical error years of schooling incorrectly
  measured

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PRECISION problems

• Functional form
• We look at log-log model. Made the data plot a
  bit more linear, but many other choices I could
  have made.
• What difference to the results would these
  choices have made? Well, estimating a straight
  line through a clearly non-linear relationship
  will result in a higher residual variance than
  necessary.
• Residual variance related to an explanatory
  variable.
• Classic example in consumer spending and income
  in household data. Spending increases with
  income, but also the variance of spending
  increases with income.
• Heteroscedasticity and Autcorrelation are two
  types of problem of this type.

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The formalities!

• y Xb u
• y , X , b
  , u
• where y is an n ? 1 vector the dependent
  variable,
• X is an n ? k matrix of the independent
  variables,
• b is a k ? 1 vector of coefficients to be
  estimated and
• u is an n ? 1 vector of random disturbances.
• We take the first explanatory variable x1 to be a
  constant, so the first column of X consists of
  1s.

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Multiple Regression Model

• This is a compact way of writing
• yi b1 b2 x2i ... bk xki ui
• i 1..n, (or t 1..T for time series data)
• When k 2 we have the simple linear model (SLM)
• yi b1 b2 x2i ui

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Simple Linear Model
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Estimation of the parameters

• Can estimate b a variety of methods, we begin
  with ordinary least squares (OLS), which we
  justify shortly. It is more intuitive to
  demonstrate the method in the SLM.
• To (slightly) simplify notation, we re-write the
  equation as
• Yi a b Xi ui
• Our estimates of a and b come from the sample
  regression line
• Yi a bXi ei
• and we choose the values of a and b to minimise
  the sum of squared errors Se2. This gives us the
  ordinary least squares (OLS) estimates.

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OLS estimation

• Hence we minimise Se2 by choice of a and b.
• Minimise
• Se² S(Y-a-bX)² S(Y2 2aY a2 2abX 2bXY
  b2X2)
• 1st Order condition for b
• Hence -2 S XY 2a S X 2b S X² 0and hence
  SXY b S X² a S X 0 ? S XY b S X² a S X
• 1st Order condition for a
• Hence SY na b S X 0 ? S Y na b S X

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OLS estimation

• The equations
• S XY b S X² a S X and S Y na b S X
  are called the normal equations (2 equations, 2
  unknowns in this case). Solving them gives the
  estimates of a and b. For b S XY b S X²
  (S Y/n - b S X/n) S X (substituting for a in the
  first equation) b(S X² - (S X)²/n) S
  Y S X/n

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OLS estimates
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OLS in matrix form

• In matrix form we can write the sum of squared
  errors as the inner product
• Differentiating this with respect to the vector b
  gives
• 
  (k by 1 null vector)
• Thus the OLS estimate of the coefficient vector b
  is given by b (X'X)-1 X'Y