Econometric Modelling

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Econometric Modelling
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Introduction

• To examine some econometric results from various
  financial models
• To use the results to determine levels of
  significance of the variables and whether the
  results fit the theory
• To use the results for testing specific
  restrictions.
• Suggest some potential problems when assessing
  model results

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Carrying out a regression

• Set out the model/theory, including expected
  signs and magnitudes of the coefficients
• Gather data
• Estimate the model using a relevant technique
• Interpret results, assess diagnostic tests.
• If model fails the diagnostic tests, respecify
  model

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Stock price return Model

• Given the following model, We wish to obtain
  estimates of the constant and slope coefficients

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Variables
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Estimation

• We would estimate this model using ordinary least
  squares (OLS), although as we will find out later
  other methods may be more appropriate.
• The model is estimated using monthly data from
  2000m1 to 2005 m12.
• This gives 6 years of data producing 72
  observations.

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Results
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Coefficients

• The signs on the coefficients are as we
  hypothesised with the possible exception of p.
• However as this variable is insignificant, the
  sign is of less importance.
• For y, a unit rise in y gives a 0.8 of a unit
  rise in the dependent variable s(t). For p, a
  unit rise in p gives a 0.2 of a unit rise in s(t)
  etc.

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T-statistics

• Firstly test if the 4 variables are individually
  different to 0, using the t-test (we usually
  ignore the constant)
• E.g. y 0.8-0/0.24
• Critical value is 2.000 (5) (72-5 degrees of
  freedom, 60 d of f is nearest in tables)
• As 4gt2 we reject the null hypothesis that y0,
  therefore y is said to be significantly different
  to 0.
• The t-statistics for p 1, i 10 and rp2.333, we
  conclude that y, i and rp are significant and p
  is insignificant.
• This result would suggest we might consider
  removing p from our model.

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R-squared

• The adjusted R-squared statistic is 0.58, which
  is relatively good explanatory power.
• The F-test for the significance of the goodness
  of fit is 25. The critical value for F(4,67) is
  2.53 (5).
• As 25 gt 2.53, the goodness of fit of the
  regression is significant, or the joint
  explanatory power of the variables is
  significantly different to 0.

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DW statistic

• We first need to find the dl and du values from
  the tables. As k is 4 and we have 72
  observations, the critical values are dl-1.49
  and du 1.74.
• The DW statistic is 1.84, which is between du
  (1.74) and 4-du (2.26), so we accept the null
  hypothesis of no 1st order autocorrelation.

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LM test for higher autocorrelation

• Given that we have monthly data, we test for 12th
  order autocorrelation.
• The critical value for chi-squared (12) is
  21.026.
• As the LM statistic is 12.8 lt 21.026, we accept
  the null hypothesis of no 12th order
  autocorrelation

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Whites test for heteroskedasticity

• This follows a chi-squared distribution with 14
  degrees of freedom (including cross product
  terms)
• The critical value with 14 degrees of freedom is
  23.685 using the chi-squared tables
• As 15.2 lt 23.685, we accept the null hypothesis
  that there is no heteroskedasticity.

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Market Model

• According to the market model, the return on an
  asset is determined by a constant and the return
  to the market index.

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Market Model

• As before we would wish to run an OLS regression,
  then interpret the coefficient, t-statistics and
  various diagnostic tests for autocorrelation and
  heteroskedasticity.
• In this model we would expect ß gt 0, the closer
  to 1, the closer asset i follows the market
  index.
• If we have 100 days of daily data for the
  regression, we get the following result

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Market Model
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Market Model

• The result shows that a unit rise in the market
  produces a 0.9 of a unit rise in asset i. This
  suggests this asset closely follows the market
  and would be considered safe.
• The t-statistic shows that the market index is
  significant, 0.9-0/0.19. critical value is 1.98.
  As 9 gt 1.98, we reject the null hypothesis and
  say that the market has a significant effect on
  asset i.
• The usual diagnostic tests would have been
  produced and interpreted as before.

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The Market-Adjusted-Returns-Model

• Based on the original model, termed the market
  model, we can test a restriction using the t-test
  to determine if the model we have is an
  alternative specification, termed the
  market-adjusted-returns model.

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Test

• This model implies the following
• ß1
• We can use a t-test to determine if this
  condition holds.

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Test

• A t-test can also be used to determine if ß 1.
• The critical value is the same as before, the
  test is

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Test

• As 1lt 2 (ignore the sign the t-statistic is an
  absolute value), we fail to reject the null
  hypothesis and conclude that the market adjusted
  model applies.
• In this case the hypothesis is

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F-test of a restriction

• The main use of this F-test is to determine if a
  group of explanatory variables are jointly equal
  to 0.
• However we can test alternative theories or
  restrictions
• The most common restriction is that 2 or more
  explanatory variables sum to 1.
• The best example of this is the Cobb-Douglas
  production function.

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Cobb-Douglas Production Function

• This model suggests that output is a function of
  capital and labour, in logarithmic form it can be
  expressed as

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Restriction

• We can then test if constant returns to scale
  applies by testing the restriction that the
  coefficients on the capital and labour
  coefficients sum to 1.
• Constant returns to scale are a proportionate
  increase in all inputs produces a proportionate
  increase in outputs.
• This allows us to rewrite the Cobb-Douglas
  production function in terms of output per unit
  of labour (divide through by l)

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Test for Constant Returns to Scale

• Run the regression in its unrestricted form with
  both explanatory variables (k and l)
• Collect the RSS (unrestricted)
• Run the following restricted version, with
  constant returns to scale and again collect the
  RSS (restricted), then use the formula used
  previously (see over)

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Test for constant returns to scale Cont..
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Test for constant returns to scale.

• If we get a RSS (unrestricted) of 1.2 and a RSS
  (restricted) of 1.4 and we have 60 observations.
  We would get an F statistic of

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Constant returns to scale

• We would reject the null hypothesis of constant
  returns to scale, therefore we would use the
  unrestricted model with capital and labour
  included separately. The null hypothesis is

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Conclusion

• When running an OLS regression, we need to assess
  the coefficients, t-statistics and diagnostic
  tests.
• We can also use the t-statistic to determine if a
  coefficient equals 1.
• The F-test can also be used to test a specific
  restriction in a model.