Hypothesis Testing, Model Specification and Multicollinearity

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Hypothesis Testing, Model Specification and
Multicollinearity
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Computing p-Values for t-Tests

• Selecting and committing to a significance level
  before we make the decision on a hypothesis can
  hide useful information about the tests outcome
• Example Suppose we want to test a null
  hypothesis that a coefficient is zero against a
  two-sided alternative
• Suppose that with 40 degrees of freedom in our
  model we obtain a t-statistic of 1.85
• If we select a 5 significance level, we do not
  reject the null hypothesis since the critical
  t-value is 2.021

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Computing p-Values for t-Tests

• If our goal is NOT to reject the null, we would
  report the above outcome at the 5 level of
  significance
• However, at the 10 level of significance, the
  null would be rejected (t-critical 1.648)
• Rather than selecting and testing at different
  significant levels, it is better to provide an
  answer to the following question
• What is the smallest significance level at which
  the null would be rejected?

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Computing p-Values for t-Tests

• This level is known as the p-value and gives us
  the probability of type I error if we reject the
  null hypothesis
• P-values are computed by software packages, such
  as SPSS
• Interpretation of p-value shows the probability
  of observing a t-statistic as extreme as we did
  if the null hypothesis is true
• Implication small p-values are evidence against
  the null while large p-values provide little
  evidence against the null

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The F-Test Testing the Overall Significance of
the Model

• The t-test cannot be used to test hypotheses
  about more than one coefficient in our model
• The F-test is used to test the overall
  significance of the regression equation
• In a model with K explanatory variables, the two
  hypotheses are
• H0 ?1 ?2 ?k 0
• H1 H0 not true

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The F-Test Testing the Overall Significance of
the Model

• The F-statistic is
• The decision rule is
• Reject H0 if F ? Fc
• Do not reject H0 if F
• where Fc is the critical value determined by the
  table of the F-distribution with K and n-K-1
  degrees of freedom

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Model Specification Errors Omitting Relevant
Variables and Including Irrelevant Variables

• To properly estimate a regression model, we need
  to have specified the correct model
• A typical specification error occurs when the
  estimated model does not include the correct set
  of explanatory variables
• This specification error takes two forms
• Omitting one or more relevant explanatory
  variables
• Including one or more irrelevant explanatory
  variables
• Either form of specification error results in
  problems with OLS estimates

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Model Specification Errors Omitting Relevant
Variables

• Example Two-factor model of stock returns
• Suppose that the true model that explains a
  particular stocks returns is given by a
  two-factor model with the growth of GDP and the
  inflation rate as factors
• Suppose instead that we estimated the following
  model

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Model Specification Errors Omitting Relevant
Variables

• The above model has been estimated by omitting
  the explanatory variable INF
• Thus, the error term of this model is actually
  equal to
• If there is any correlation between the omitted
  variable (INF) and the explanatory variable
  (GDP), then there is a violation of classical
  assumption III

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Model Specification Errors Omitting Relevant
Variables

• This means that the explanatory variable and the
  error term are not uncorrelated
• If that is the case, the OLS estimate of ?1 (the
  coefficient of GDP) will be biased
• As in the above example, it is highly likely that
  there will be some correlation between two
  financial (or economic) variables
• If, however, the correlation is low or the true
  coefficient of the omitted variable is zero, then
  the specification error is very small

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Model Specification Errors Omitting Relevant
Variables

• How can we correct the omitted variable bias in a
  model?
• A simple solution is to add the omitted variable
  back to the model, but the problem with this
  solution is to be able to detect which is the
  omitted variable
• Omitted variable bias is hard to detect, but
  there could be some obvious indications of this
  specification error
• For example, our estimated model has a
  significant coefficient with the opposite sign
  from that expected by our arguments

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Model Specification Errors Omitting Relevant
Variables

• The best way to detect the omitted variable
  specification bias is to rely on the theoretical
  arguments behind the model
• Which variables does the theory suggest should be
  included?
• What are the expected signs of the coefficients?
• Have we omitted a variable that most other
  similar studies include in the estimated model?
• Note, though, that a significant coefficient with
  the unexpected sign can also occur due to a small
  sample size
• However, most of the data sets used in empirical
  finance are large enough that this most likely is
  not the cause of the specification bias

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Model Specification Errors Including Irrelevant
Variables

• Example Going back to the two-factor model,
  suppose that we include a third explanatory
  variable in the model, for example, the degree of
  wage inequality (INEQ)
• So, we estimate the following model
• The inclusion of an irrelevant variable (INEQ) in
  the model increases the standard errors of the
  estimated coefficients and, thus, decreases the
  t-statistics

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Model Specification Errors Including Irrelevant
Variables

• This implies that it will be more difficult to
  reject a null hypothesis that a coefficient of
  one of the explanatory variables is equal to zero
• Also, the inclusion of an irrelevant variable
  will usually decrease the adjusted R-sq (but not
  the R-sq)
• Finally, we can show that the inclusion of an
  irrelevant variable does still allow us to obtain
  unbiased estimates of the models coefficients

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Dummy Variables Incorporating Qualitative
Information in the Model

• In several cases, it may be necessary to include
  explanatory information in the model in the form
  of a qualitative variable
• Example Suppose we want to estimate a model of
  the relationship between firm performance (ROE)
  and board independence
• We are interested in empirically testing the
  argument that greater board independence leads to
  better firm performance
• In our estimated model, the null hypothesis would
  be that greater board independence will not
  affect or result in worse firm performance

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Dummy Variables Incorporating Qualitative
Information in the Model

• We can measure board independence by, for
  example, the proportion of independent directors
• However, how can we measure the impact of the
  fact that in some firms the CEO is also the
  Chairman of the Board?
• We assume that performance will be different in
  firms with this attribute (given everything else)
  compared to those where this is not true
• We can capture this effect through the inclusion
  of a dummy variable as an explanatory variable in
  our model

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Dummy Variables Incorporating Qualitative
Information in the Model

• In this example, the dummy variable will take the
  value of
• 1 for firms where the CEO is also the Chairman
• 0 otherwise
• Therefore, we estimate the following model (in
  general form)
• where Di 1 if the ith observation satisfies
  our condition, and 0 otherwise

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Dummy Variables Incorporating Qualitative
Information in the Model
Y
Di 0
?k1
Di 1
?0
X
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Dummy Variables Incorporating Qualitative
Information in the Model

• Example Suppose we empirically test the
  relationship between a firms size and its
  monthly stock returns with a sample of time
  series data
• In our model, we should include a dummy variable
  that accounts for the well-known phenomenon of
  January effect
• The dummy variable will take the following values
• 1 for the observation of returns in the month of
  January
• 0 for all other observations of monthly returns

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Dummy Variables Incorporating Qualitative
Information in the Model

• The above cases are examples of intercept dummy
  variables meaning that inclusion of the dummy
  variable shifts the regression line, but does not
  change its slope
• Another form of a dummy variable is a slope dummy
  variable that allows the slope of a regression to
  change depending on whether a condition is
  satisfied
• Example Suppose we want to test the argument
  that the relationship between credit card lending
  and loan losses for a particular bank has changed
  in the last five years

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Dummy Variables Incorporating Qualitative
Information in the Model

• I.e., suppose that credit card lending
  contributes less to this banks loan losses due
  to the implementation of better risk evaluation
  methods (credit scoring)
• We estimate the following model
• where the variable T is a time dummy variable
  that takes the value
• 1 for observations in the past five years
• 0 otherwise

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Dummy Variables Incorporating Qualitative
Information in the Model

• In this case, the coefficient of the CARDLN
  variable is
• ?1 - ?2 for observations from the past five years
• ?1 otherwise
• Dummy variable trap Always include one less
  dummy variable than the possible qualitative
  states in the data
• Example If there are 3 qualitative states, for
  example, small, medium, large firms, we should
  include 2 dummy variables

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Multicollinearity

• Multicollinearity occurs when some or all of the
  explanatory variables in the regression model are
  highly correlated
• In this case, assumption VI of the classical
  model does not hold and OLS estimates lose some
  of their nice properties
• It is common, particularly in the case of time
  series data, that two or more explanatory
  variables are correlated
• When multicollinearity is present, the estimated
  coefficients are unstable in the degree of
  statistical significance, magnitude and sign

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The Impact of Multicollinearity on the OLS
Estimates

• Multicollinearity has the following consequences
  on the OLS estimated model
• The OLS estimates remain unbiased
• The standard errors of the estimated coefficients
  are higher and, thus, the t-statistics fall
• OLS estimates become very sensitive to the
  addition or removal of explanatory variables or
  to changes in the data sample
• The overall fit of the regression (and the
  significance of non-multicollinear coefficients)
  is to a large extent unaffected
• This implies that a sign of multicollinearity is
  a high adjusted R-sq and no statistically
  significant coefficients

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Detecting Multicollinearity

• One approach to detect multicollinearity is to
  examine the simple correlation coefficients
  between explanatory variables
• This will be shown in the correlation matrix
  between the models variables
• Some researchers consider a correlation
  coefficient with an absolute value above .80 to
  be an indication of concern for multicollinearity
• A second detection approach is to use the
  Variance Inflation Factor (VIF)

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Detecting Multicollinearity

• The VIF method tries to detect multicollinearity
  by examining the degree to which a given
  explanatory variable is explained by the others
• The method involves the following steps
• Run an OLS regression of the explanatory variable
  Xi on all other explanatory variables
• Calculate the VIF for the coefficient of variable
  Xi given by 1/(1 R2i) where the R-sq
  is that given by the regression
• Evaluate the size of the VIF
• Rule of thumb if the VIF of the coefficient of
  explanatory variable Xi is greater than 5 then
  the higher is the impact of multicollinearity on
  the estimated coefficient of this variable