Econometric Methods 1

1
Econometric Methods 1

• Lecture 4 Specification Tests and Violations of
  the OLS assumptions

2
Tests of multiple hypotheses

• Can more than one hypothesis at time in
  regression. i.e. the constant equals -60 and the
  slope coefficients are equal but opposite in
  sign.
• By multiple hypothesis mean there is more than
  one sign in H0. e.g. the above hypothesis can
  be written
• Hypothesis is essentially a restriction on the
  model, a simplification. Here we have q 2
  restrictions in the null hypothesis. We might
  consider estimating both the unrestricted and the
  restricted model.

H0 ß1 -60 and ß2 - ß 3 vs H1 ß1 ? -60 or ß2
? - ß 3 or both.
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Tests of multiple hypotheses

• Restricted model
• y b1 b2 x2 b3 x3 u
• while the restricted version is
• y 60 b2 (x2 x3) u'

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Example

• Illustrate with farm data. Let our
  unrestricted regression be
• Source SS df MS
  Number of obs 369
• -------------------------------------------
  F( 6, 362) 29.56
• Model 67.6016953 6 11.2669492
  Prob gt F 0.0000
• Residual 137.966779 362 .381123698
  R-squared 0.3289
• -------------------------------------------
  Adj R-squared 0.3177
• Total 205.568474 368 .558609983
  Root MSE .61735
• --------------------------------------------------
  ----------------------------
• lva Coef. Std. Err. t
  Pgtt 95 Conf. Interval
• -------------------------------------------------
  ----------------------------
• lfs -.1879987 .0519786 -3.62
  0.000 -.2902165 -.0857808
• propwell 1.01175 .1179663 8.58
  0.000 .7797651 1.243736
• propcan 1.14663 .1882754 6.09
  0.000 .7763792 1.516881
• lpopd .2375268 .0527616 4.50
  0.000 .1337692 .3412845
• lplotn .2219142 .071631 3.10
  0.002 .081049 .3627793
• lfamsize .0256384 .0846921 0.30
  0.762 -.1409118 .1921886
• _cons 7.071058 .171034 41.34
  0.000 6.734713 7.407403

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Example

• Test joint restriction b(lfamsize) 0 and
  b(propwell) b(propcan). Excluding lfamsize and
  creating a variable (propirr) which is sum of
  propwell and propcan.
• Source SS df MS
  Number of obs 369
• -------------------------------------------
  F( 4, 364) 44.37
• Model 67.3785946 4 16.8446486
  Prob gt F 0.0000
• Residual 138.189879 364 .379642526
  R-squared 0.3278
• -------------------------------------------
  Adj R-squared 0.3204
• Total 205.568474 368 .558609983
  Root MSE .61615
• --------------------------------------------------
  ----------------------------
• lva Coef. Std. Err. t
  Pgtt 95 Conf. Interval
• -------------------------------------------------
  ----------------------------
• lfs -.1845138 .0516282 -3.57
  0.000 -.2860408 -.0829868
• propirr 1.034815 .11124 9.30
  0.000 .8160611 1.253569
• lpopd .2299474 .0515964 4.46
  0.000 .1284829 .3314119
• lplotn .2234388 .0682978 3.27
  0.001 .089131 .3577466
• _cons 7.110554 .1296769 54.83
  0.000 6.855544 7.365564
• --------------------------------------------------
  ----------------------------

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Example of hypothesis testing

• Can test using classical or ML methods, comparing
  the two regressions (restricted and
  unrestricted). Need information on the RSS and
  likelihoods.
• Remember that

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Example

• For the F test
• 0.2886 lt 3.0 F(2,) at 5 Cannot Reject H0
  
• q is the number of restrictions in the null.
• Sometimes tricky to work out number of
  restrictions in hypothesis tests. q (degrees of
  freedom in the numerator) can be calculated as
  the difference between the degrees of freedom in
  each of the restricted and unrestricted models.
  In the denominator we simply have the number of
  degrees of freedom in the unrestricted model.
• LR test. -2 ln (LogLr LogLu) 0.6 lt (5)
  5.99

8
Specification Problems Stability testing

• Common problem is that regression may not apply
  over whole sample.
• e.g. the oil price shocks in the 1970s may
  have altered price and wages setting and hence
  the parameters of the Phillips curve.
• We use a Chow test
• Suppose we want to test if the sample can be
  split into two sub-samples. We could
• - (a) estimate one equation for each sub-sample,
  or
• - (b) estimate one equation over the whole
  sample
• The RSS in the latter case must be larger than
  the (sum of the) former. If there is a big enough
  difference we reject the null of stability.

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Chow Test

• Formally, we wish to test for a break after n1
  periods.
• estimate y Xb e over the first n1
  observations. Obtain RSS1.
• then estimate y Xb' e over the remaining n2
  observations. Obtain RSS2.
• calculate RSSU RSS1  RSS2
• estimate the equation over the whole sample.
  Obtain RSSR.
• use the F-test F
• If greater than the critical value, reject H0
  coefficients are stable.

10
Specification Problems Omitted explanatory
variables

• The true model is Y b1 b2 X2 b3 X3 u
• we estimate Y b1 b2 X2 u by mistake,
  i.e. leave out X3.
• Consequences OLS estimate of b2 is in general
  biased.
• b2
• and E(b2)
• This equals b2 only if (a) b3 0 or (b) there is
  zero correlation between X2 and X3.

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Omitted explanatory variables other
consequences

• The estimates are also inconsistent, unless all
  correlations between included and excluded
  variables disappear as n ? ?.
• The estimated variance of b2 in the mis-specified
  model could be larger or smaller than in the
  correct model. In practice, it is often 'too
  large'.
• Omission could lead to symptoms such as
  autocorrelation or heteroscedasticity in the
  error term (see later)
• Tests of functional form, stability and normality
  of the errors may be failed.
• Exercise had example of omission of house
  characteristics in regression of price on
  distance to incinerator

12
Specification Problems Inclusion of an
irrelevant regressor

• This is generally a less severe problem.
• The true model is Y b1 b2 X2 u
• We estimate Y b1 b2 X2 b3 X3 u
• The coefficient b3 should be insignificantly
  different from zero (its true value). The
  estimate of b2 will be unbiased but, in general,
  there is some loss in efficiency,
• i.e. the sampling variance of b2 is larger than
  it need be. Hence, should omit X2 .

13
Specification ProblemsMulticollinearity

• If two variables move together (over time or in
  cross-section) then it is difficult to tell which
  is influencing Y.
• If move exactly together, then it is impossible.
  Multicollinearity is problem of degree, ranging
  from perfect to zero.
• Perfect multicollinearity
• Linear relationship variables, e.g. X2 a bX3.
  
• - X has less than full column rank.
• - (X'X)-1 doesn't exist, so no estimates.
• In practice, perfect multicollinearity often
  occurs by mistake - entering the same X variable
  twice or too many dummy variables.

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Near Perfect Multicollinearity

• Approximate linear dependence X still has full
  column rank, so OLS estimates exist.
• (X'X) is nearly singular (i.e. determinant is
  near to 0) and (X'X)-1 is large.
• OLS is still BLUE, but V(b) may be large
• Hence inference may be affected - t-ratios are
  small and you will think variables are not
  significant when in fact they are.
• All models contain some multicollinearity,
  question of when it is problem.

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Near Perfect Multicollinearity

• Symptoms
• There is no one test, since multicollinearity is
  a question of degree.
• High R2 and F, but high variances and low
  t-statistics
• High correlations between X variables (neither
  necessary nor sufficient).
• Small changes in sample data lead to big changes
  in estimates (of coefficients and variances)
• Cures
• respecify after thinking hard about the
  possible causalities and correlations .

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Specification ProblemsFunctional form

• Have assumed the equation is linear. Ramsey's
  RESET tests whether this is appropriate
• Add powers of the X variables to the estimating
  equation.
• Instead of using powers of X, we use powers of
  
• The procedure is
• 1. estimate Y b1 b2 X2 ... bk Xk u
• 2. obtain fitted values
• 3. estimate Y b'1 b'2 X2 ... b'k Xk c
  2 u'
• 4. test for the significance of c. If it is
  significantly different from 0, reject the linear
  specification.

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Example RESET Test on Farm data

• Take fitted values and square them ? fits2.
• reg lva lfs propirr lpopd frag fits2
• Source SS df MS
  Number of obs 369
• -------------------------------------------
  F( 5, 363) 35.79
• Model 67.8727105 5 13.5745421
  Prob gt F 0.0000
• Residual 137.695763 363 .379327172
  R-squared 0.3302
• -------------------------------------------
  Adj R-squared 0.3209
• Total 205.568474 368 .558609983
  Root MSE .6159
• --------------------------------------------------
  ----------------------------
• lva Coef. Std. Err. t
  Pgtt 95 Conf. Interval
• -------------------------------------------------
  ----------------------------
• lfs -.6413701 .4036006 -1.59
  0.113 -1.435059 .1523188
• propirr 3.653044 2.296729 1.59
  0.113 -.8635211 8.169609
• lpopd .7999175 .5020516 1.59
  0.112 -.1873774 1.787212
• frag .7609251 .4758566 1.60
  0.111 -.1748567 1.696707
• fits2 -.1606089 .1407221 -1.14
  0.254 -.4373418 .116124
• _cons 15.21461 7.101781 2.14
  0.033 1.248809 29.18041

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Specification Problems Normality

• Errors assumed to be Normal for inference. Are
  they?
• Tests based on the skewness and kurtosis
• - skewness related to the 3rd moment kurtosis to
  the 4th
• - these should be 0 and 3 respectively in Normal
• Jarque-Bera Test is based on this, H0 Normal
• JB
  Under the null
• Where sum of residuals each
  raised to the
• power j, divided by the sample size

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Heteroscedasticity

• Relax some of the classical assumptions. We look
  now at assumption A2 and relax this.
• Recall
• A2. E(uu) s2 I.
• This is a matrix with s 2 on the diagonals,
  zeroes everywhere else. We can relax this in two
  ways
• 1. have non-constant values on the diagonal
• 2. have non-zero elements off the diagonal
• 1. leads to the problem of heteroscedasticity,
• 2. leads to autocorrelation (later).

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Heteroscedasticity

• Variance of error term now varies across
  observations.
• It may increase with one X variable, with time,
  or in some more complex way.
• (NB we still assume independence of the errors).
• Often occurs in cross-section data when wide
  range to the X variables,
• e.g. holiday expenditure and income probably
  rises with income but also becomes more variable.
  
• It can also arise in grouped data, when each
  observation is an average for a group (village,
  industry, etc) and groups different sizes.

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Heteroscedasticity

• We now assume
• V(b) s2O
• and in general s12 ? s 22 etc.

22
Heteroscedasticity
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Consequences of heteroscedasticity

• OLS estimates of ß unbiased and consistent
• - but inefficient (not BLUE) and not
  asymptotically efficient.
• - E(b) b E(S(xu)/Sx2) ) still no
  correlation between x and u
• (E(xu) 0). Hence OLS still unbiased.
• estimate of the error variance is biased,
  invalidating inference.
• Inefficiency is harder to demonstrate,
• - OLS gives equal weight to all observations.
  
• - If know which error terms have high variance,
  better to give less weight to them.

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Goldfeld-Quandt test

• Applicable if heteroscedasticity is related to
  only one of the x variables, say x1. The
  procedure is (where s2 increases with x1)
• order your observations in increasing order of x1
• omit the central r ? n/3 observations
• run a regression on the first n1 (n-r)/2 obs,
  obtaining RSS1
• run a regression on the last n2 (n-r)/2 obs,
  obtaining RSS2.
• the test statistic for H0 s12 s22 ... sn2
  is
• F RSS2/RSS1 Fc1,c2 where ci (ni - r -
  2k)/2.
• bit restrictive since you have to know which is
  x1.

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White's test

• We assume that V(ui) s 2 g(E(Yi))2
• i.e. the variance of u depends upon the expected
  value of Y (and hence on at least one of the x
  variables). Hence
• V(ui) s2 g(b1 b2 x2 ... bk xk)2
• If H0 g 0 then we have homoscedasticity,
  otherwise heteroscedasticity.
• Hence we regress e2 on x22, x32, etc and x2 x3,
  x3 x4, etc (ie all the cross-products).
• From this auxiliary regression we obtain nR2
  c2(k-1) as an LM statistic.

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Example Farm data

• After running our standard regression, regress
  squared residuals on original Xs, plus their
  squares and cross-product
• nR2 369.0803 29.6 ?2(14) Here 5 is 26.1
  and 1 is 29.1, so it is significant at the 1
  level.
• Reject the null of homoscedasticity
• Simpler version uses instead of all the Xs
• In other words the auxiliary regression is
• e2 a b 2 v
• Test is H0 b 0.
• This gives similar results. t -2.1, again
  between the 5 and 1 levels.

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A now-standard cure

• Whites heteroscedasticity consistent standard
  errors (HCSEs)
• Since it is only the standard errors rather than
  the coefficients themselves which are biased (and
  inconsistent), another approach is to find
  alternative estimates of the standard errors.
• White suggests replacing si2 s2 with si2 ei2,
  the appropriate OLS residual. This is
  implemented in STATA using the robust option.

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Estimation when heteroscedasticity is present

• We illustrate the use of generalised least
  squares (GLS).
• OLS assumes E(uu) s2 I, a very restricted form
  of the error structure.
• GLS allows the assumption that E(uu) s2 W,
  where W is any n ? n matrix.
• Taking account of W GLS estimates give us BLUEs.
• There are two (equivalent) ways of obtaining GLS
  estimates
• use a different formula from OLS
• transform the data and use OLS on the transformed
  data.

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Generalised least squares (GLS)

• Need to use the GLS estimator for efficient
  estimates b (X' W-1 X)-1 X' W-1 y
• Let y Xb u with E(uu) s2 W. e.g. of W li
  on diagonal, zeroes elsewhere.
• Let PP W. (P always exists if W is
  symmetric and positive-definite).
• Hence P-1 W P-1 I (easy to show)
• and P-1 P-1 W-1 (NOTE prime)
• Now premultiply the original model by P-1
• P-1 y P-1 Xb P-1 u ? y Xb u
• where y P-1 y, etc.

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Generalised least squares (GLS)

• Note now that E(uu) E(P-1 u u P-1 )
  P-1 E(uu) P-1 P-1 s2 W P-1 s2 I
• So the starred model satisfies the classical
  assumptions about u and we could apply OLS to
  obtain BLUEs.
• We have b (XX)-1 Xy (X P-1 P-1
  X)-1 X P-1 ' P-1 y (X' W-1 X)-1 X' W-1 y
• Note that W-1 matrix with 1/li on diagonal.
• V(b) s2 (X X)-1 s2 (X W-1 X)-1
• and our estimate of s2 is e W-1 e/(n-k)

31
Generalised least squares (GLS)

• If know W can use the GLS formulae to find BLUEs.
  Alternatively, from W find P-1 and transform the
  data and use OLS
• How do we find P-1? Have to assume something
  giving us feasible GLS. Suppose a graph suggests
  that s2 ? x22, i.e. E(ui2) s2 x2i2.
• Hence W

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Generalised least squares (GLS)

• Hence P-1
• and so P-1 y y1/x21 y2 /x22 ... yn /x2n
• Similarly, all the explanatory variables are
  divided by x2. Hence instead of estimating
• y b1 b2 x2 ... bkxk u. We estimate
  (by OLS)
• y/x2 b1 /x2 b2 b3 x3 /x2 ... bk xk /x2
  u/x2
• i.e. we deflate all variables by x2.
• Note that V(u/x2) V(u)/x22 s2 x2/x2 s2.