Economics 105: Statistics

1
Economics 105 Statistics

• Any questions?
• Please hand in Review 3

2
Time Series Multiple Regression

• Assumptions
• (1)
• Linear function in the parameters, plus error
• Variation in Y is caused by ?, the error (as
  well as X)
• (2)
• Sources of error
• Idiosyncratic, white noise
• Measurement error on Y
• Omitted relevant explanatory variables
• If (2) holds, we have exogenous explanatory vars
• If some Xj is correlated with error term for
  some reason, then that Xj is an endogenous
  explanatory var

3
Time SeriesMultiple Regression

• Assumptions
• (3)
• Homoskedasticity
• (4)
• No autocorrelation
• (5)
• Errors and the explanatory variables are
  uncorrelated
• (6)
• Errors are i.i.d. normal

4
Time Series Multiple Regression

• Assumption (7) No perfect multicollinearity
• no explanatory variable is an exact linear
  function of other Xs
• Venn diagram
• Other implicit assumptions
• data are a random sample of n observations from
  proper population
• n gt K
• the little xijs are fixed numbers (the same in
  repeated samples) or they are realizations of
  random variables, Xij, that are independent of
  error term then inference is done CONDITIONAL
  on observed values of xijs

5
Nature of Serial Correlation

• Violation of (4)
• Error in period t is a function of error in
  prior period alone first-order autocorrelation,
  denoted AR(1) for autoregressive process
• Usual assumptions apply to new error term
• is positive serial correlation
• is negative serial correlation

6
Nature of Serial Correlation

• Error in period t can be a function of error in
  more than one prior period
• Second-order serial correlation
• Higher orders generated analogously
• Seasonally-based serial correlation

7
Causes of Serial Correlation

• The error term in the regression captures
• Measurement error
• Omitted variables, that are uncorrelated with the
  included explanatory variables (hopefully)
• Frequently factors omitted from the model are
  correlated over time
• Persistence of shocks
• Effects of random shocks (e.g., earthquake, war,
  labor strike) often carry over through more than
  one time period
• Inertia
• times series for GNP, (un)employment, output,
  prices, interest rates, etc. follow cycles, so
  that successive observations are related

8
Causes of Serial Correlation

• 3. Lags
• Past actions have a strong effect on current ones
• Consumption last period predicts consumption this
  period
• 4. Misspecified model, incorrect functional form
• 5. Spatial serial correlation
• In cross-sectional data on regions, a random
  shock in one region can cause the outcome of
  interest to change in adjacent regions
• Keeping up with the Joneses

9
Consequences for OLS Estimates

• Using an OLS estimator when the errors are
  autocorrelated results in unbiased estimators
• However, the standard errors are estimated
  incorrectly
• Whether the standard errors are overstated or
  understated depends on the nature of the
  autocorrelation
• For positive AR(1), standard errors are too
  small!
• Any hypothesis tests conducted could yield
  erroneous results
• For positive AR(1), may conclude estimated
  coefficients ARE significantly different from 0
  when we shouldnt !
• OLS is no longer BLUE
• A pattern exists in the errors
• Suggesting an estimator that exploited this would
  be more efficient

10
Detection of Serial Correlation

• Graphical

11
Detection of Serial Correlation

• Graphical

no obvious patternthe errors seem random.
Sometimes, however, the errors follow a
patternthey are correlated across observations,
creating a situation in which the observations
are not independent with one another.
12
Detection of Serial Correlation
Here the residuals do not seem random, but rather
seem to follow a pattern.
13
Detection The Durbin-Watson Test

• Provides a way to test H0 ? 0
• It is a test for the presence of first-order
  serial correlation
• The alternative hypothesis can be
• ? ? 0
• ? gt 0 positive serial correlation
• Most likely alternative in economics
• ? lt 0 negative serial correlation
• DW Test statistic is d

14
Detection The Durbin-Watson Test

• To test for positive serial correlation with the
  Durbin-Watson statistic, under the null we expect
  d to be near 2
• The smaller d, the more likely the alternative
  hypothesis

The sampling distribution of d depends on the
values of the explanatory variables. Since every
problem has a different set of explanatory
variables, Durbin and Watson derived upper and
lower limits for the critical value of the test.
15
Detection The Durbin-Watson Test

• Durbin and Watson derived upper and lower limits
  such that d1 ? d ? du
• They developed the following decision rule

16
Detection The Durbin-Watson Test

• To test for negative serial correlation the
  decision rule is

• Can use a two-tailed test if there is no strong
  prior belief about whether there is positive or
  negative serial correlationthe decision rule is

17
Serial Correlation

• Table of critical values for Durbin-Watson
  statistic (table E11, page 833 in BLK textbook)
• http//hadm.sph.sc.edu/courses/J716/Dw.html

18
Serial Correlation Example

• What is the effect of the price of oil on the
  number of wells drilled in the U.S.?

19
Serial Correlation Example

• What is the effect of the price of oil on the
  number of wells drilled in the U.S.?

20
Serial Correlation Example

• Analyze residual plots but be careful

21
Serial Correlation Example

• Remember what serial correlation is

• This plot only works if obs number is in same
  order as the unit of time

22
Serial Correlation Example

• Same graph when plot versus year

• Graphical evidence of serial correlation

23
Serial Correlation Example

• Calculate DW test statistic
• Compare to critical value at chosen sig level
• dlower or dupper for 1 X-var n 62 not in
  table
• dlower for 1 X-var n 60 is 1.55, dupper
  1.62

• Since .192 lt 1.55, reject H0 ? 0 in favor of
  H1 ? gt 0 at a5