Review of Probability and Statistics

1
Time Series Data

• yt b0 b1xt1 . . . bkxtk ut
• Basic Analysis

2
Time Series vs. Cross Sectional

• Time series data has a temporal ordering, unlike
  cross-section data
• Will need to alter some of our assumptions to
  take into account that we no longer have a random
  sample of individuals
• Instead, we have one realization of a stochastic
  (i.e. random) process

3
Examples of Time Series Models

• A static model relates contemporaneous
  variables yt b0 b1zt ut
• A finite distributed lag (FDL) model allows one
  or more variables to affect y with a lag
• yt a0 d0zt d1zt-1 d2zt-2 ut
• More generally, a finite distributed lag model
  of order q will include q lags of z

4
Lagged Dependent Variable Models

• Another common type of time series model is
  where one or more lags of the dependent variable
  appear, e.g.
• yt a0 d0yt-1 d1yt-2 d2yt-3 ut
• Such models are not considered in ES5611 but
  reappear in ES5622
• Here they are ruled out by assumption TS.2

5
Assumptions for Unbiasedness

• TS.1 Assume a model that is linear in
  parameters yt b0 b1xt1 . . . bkxtk ut
• TS.2 Zero conditional mean assumption E(utX)
  0, t 1, 2, , n
• Note that this implies the error term in any
  given period is uncorrelated with the explanatory
  variables in all time periods
• This assumption also called strict exogeneity

6
Assumptions (continued)

• An alternative assumption, more parallel to the
  cross-sectional case, is E(utxt) 0
• This assumption would imply the xs are
  contemporaneously exogenous
• Contemporaneous exogeneity will only be
  sufficient in large samples

7
Assumptions (continued)

• TS.3 Assume that no x is constant, and that
  there is no perfect collinearity
• Note we have skipped the assumption of a random
  sample
• The key impact of the random sample assumption
  is that each ui is independent
• Our strict exogeneity assumption takes care of
  it in this case

8
Unbiasedness of OLS

• Based on these 3 assumptions, when using
  time-series data, the OLS estimators are unbiased
• Omitted variable bias can be analyzed in the same
  manner as in the cross-section case

9
Variances of OLS Estimators

• Just as in the cross-section case, we need to
  add an assumption of homoskedasticity in order to
  be able to derive variances
• TS.4 Assume Var(utX) Var(ut) s2
• Thus, the error variance is independent of all
  the xs, and it is constant over time
• TS.5 Assume no serial correlation Cov(ut,us
  X) 0 for t ? s

10
OLS Variances (continued)

• Under these 5 assumptions, the OLS variances in
  the time-series case are the same as in the
  cross-section case. Also,
• The estimator of s2 is the same
• OLS remains BLUE (Gauss-Markov)
• With the additional assumption of normal errors,
  inference is the same

11
Example using Microfit (10.3)

• Microfit 4 available on all networked computers
• Econometric package geared towards time series
  analysis
• Mainly menu driven package
• Has some quirks
• Practice exercise and handout on Web page but
  not expecting you to become proficient - more in
  ES5622

12
Example using Microfit (contd)

• Castillo-Freeman and Freeman (1992) effect of
  minimum wage on employment in Puerto Rico
• Variables prepop - employment rate, mincov -
  importance of minimum wage,
• Simple Model
• log(prepop) b0 b1log(mincov) u
• Clear prediction from economic theory of sign of
  b1

13
Microfit output - page 1

• Ordinary Least Squares
  Estimation
• 
  
• Dependent variable is LPREPOP
• 38 observations used for estimation from 1950 to
  1987
• 
  
• Regressor Coefficient
  Standard Error T-RatioProb
• CONSTANT -1.1598
  .027281 -42.5120.000
• LMINCOV -.16296
  .019481 -8.3650.000
• 
  
• R-Squared .66029
  R-Bar-Squared .65085
• S.E. of Regression .054939 F-stat.
  F( 1, 36) 69.9728.000
• Mean of Dependent Variable -.94407 S.D. of
  Dependent Variable .092978
• Residual Sum of Squares .10866 Equation
  Log-likelihood 57.3657
• Akaike Info. Criterion 55.3657 Schwarz
  Bayesian Criterion 53.7282
• DW-statistic .34147
• 
  

• Note where all the usual stuff is

14
Microfit output - page 2

• Diagnostic Tests
• 
  
• Test Statistics LM Version
  F Version
• 
  
• 
  
• ASerial CorrelationCHSQ( 1)
  25.8741.000F( 1, 35) 74.6828.000
• 
  
• BFunctional Form CHSQ( 1)
  2.8662.090F( 1, 35) 2.8553.100
• 
  
• CNormality CHSQ( 2)
  .071873.965 Not applicable
• 
  
• DHeteroscedasticityCHSQ( 1)
  4.3213.038F( 1, 36) 4.6192.038
• 
  
• ALagrange multiplier test of residual serial
  correlation
• BRamsey's RESET test using the square of the
  fitted values
• CBased on a test of skewness and kurtosis of
  residuals
• DBased on the regression of squared residuals
  on squared fitted values

15
Trending Time Series

• Economic time series often have a trend
• Just because 2 series are trending together, we
  cant assume that the relation is causal
• Often, both will be trending because of other
  unobserved factors - leads to spurious regression
• Even if those factors are unobserved, we can
  control for them by directly controlling for the
  trend

16
Example - trending data

• UK aggregate consumption and income 1948-85,
  annual data
• Note extremely high correlation in scatter plot
• R2 0.9974

17
Trends (continued)

• One possibility is a linear trend, which can be
  modeled as yt a0 a1t et, t 1, 2,
• Another possibility is an exponential trend,
  which can be modeled as log(yt) a0 a1t et,
  t 1, 2,
• Another possibility is a quadratic trend, which
  can be modeled as yt a0 a1t a2t2 et, t
  1, 2,

18
Detrending

• Adding a linear trend term to a regression is
  the same thing as using detrended series in a
  regression
• Detrending a series involves regressing each
  variable in the model on t
• The residuals form the detrended series
• Basically, the trend has been partialled out

19
Detrending (continued)

• An advantage to actually detrending the data
  (vs. adding a trend) involves the calculation of
  goodness of fit
• Time-series regressions tend to have very high
  R2, as the trend is well explained
• The R2 from a regression on detrended data
  better reflects how well the xts explain yt

20
Example again

• Define time trend variable, t
• Original model

• Model with trend

21
Example (contd)

• Scatter plot of detrended series
• R2 0.7868
• Still high but a more accurate measure of how
  well Y explains C
• However, these data may be highly persistent and
  simple methods not appropriate (Wooldridge,
  Ch.11, ES5622)

22
Time Series Data

• yt b0 b1xt1 . . . bkxtk ut
• Serial Correlation

23
Serial Correlation defined

• Serial correlation (autocorrelation) is where
  TS.5 does not hold
• Cov(ut,usX) ? 0 for t ? s.
• A particular form of serial correlation is
  extremely common in time series data
• This is because shocks tend to persist through
  time

24
Implications of Serial Correlation

• Unbiasedness (or consistency) of OLS does not
  depend on TS.5
• However OLS is no longer BLUE when serial
  correlation is present
• And OLS variances and standard errors are biased
• Hence usual inference procedures are not valid

25
The AR(1) Process

• First order autoregressive error process a
  useful model of serial correlation
• yt b0 b1xt1 . . . bkxtk ut
• ut rut-1 et ? lt 1
• where et are uncorrelated random variables with
  mean 0 and variance se2
• Typically expect r gt 0 in economic data
• Positive serial correlation

26
The AR(1) Process

• E(ut) 0
• Var(ut) se2/(1-r2)
• Cov(ut, utj) rjVar(ut)
• Corr(ut, utj) rj
• So the error term is zero mean, homoscedastic
  but has serial correlation which is positive if ?
  gt 0

27
Estimator variance simple regression
This is not equal to the usual formula since
Cov(ut, us) ? 0 in the presence of serial
correlation
28
Testing for AR(1) Serial Correlation

• Want to be able to test for whether the errors
  are serially correlated or not
• Want to test H0 r 0 in ut rut-1 et, t
  2,, n, where ut is the regression model error
  term
• With strictly exogenous regressors, an
  asymptotically valid test is very straightforward
  simply regress the residuals on lagged
  residuals and use a t-test

29
Testing for AR(1) Serial Correlation (continued)

• An alternative is the Durbin-Watson (DW)
  statistic, which is calculated by many packages
• If the DW statistic is around 2, then we can
  reject serial correlation, while if it is
  significantly lt 2 we cannot reject
• Critical values are in the form of bounds
  reject if DWltdL, do not reject if DWgtdU,
  inconclusive otherwise. Tables available.

30
Testing for AR(1) Serial Correlation (continued)

• Note that the t-test is only valid
  asymptotically while DW has an exact distribution
  under classical assumptions including normality.
• If the regressors are not strictly exogenous,
  then neither the t nor DW test are valid
• Instead regress the residuals (or y) on the
  lagged residuals and all of the xs and use a
  t-test

31
Testing for Higher Order S.C.

• Can test for AR(q) serial correlation in the
  same basic manner as AR(1)
• Just include q lags of the residuals in the
  regression and test for joint significance
• Can use F test or LM test, where the LM version
  is called a Breusch-Godfrey test and is (n-q)R2
  using R2 from auxiliary (residual) regression

32
Example

• In the Puerto Rico minimum wage example, DW
  0.3415
• dL 1.535 so we can reject H0 r 0 against
  H1 r gt 0
• Assuming strict exogeneity, an auxiliary
  regression gives
• Hence reject H0
• S/C is present

33
Correcting for Serial Correlation

• Start with case of strictly exogenous
  regressors, and maintain all G-M assumptions
  except no serial correlation
• Assume errors follow AR(1) so
• ut rut-1 et, t 2,, n
• Var(ut) s2e/(1-r2)
• We need to try and transform the equation so we
  have no serial correlation in the errors

34
Correcting for S.C. (continued)

• Use a simple regression model for convenience
• Consider that since yt b0 b1xt ut , then
• yt-1 b0 b1xt-1 ut-1
• If you multiply the second equation by r, and
  subtract if from the first you get
• yt r yt-1 (1 r)b0 b1(xt r xt-1) et ,
  since et ut r ut-1
• This quasi-differencing results in a model
  without serial correlation

35
Feasible GLS Estimation

• OLS applied to the transformed model is GLS and
  is BLUE
• Problem dont know r, so we need to get an
  estimate first
• Can use estimate obtained from regressing
  residuals on lagged residuals without an
  intercept
• This procedure is called Cochrane-Orcutt
  estimation

36
Feasible GLS Estimation

• One slight problem with Cochrane-Orcutt is that
  we lose an observation (t 1)
• The Prais-Winsten method corrects this by
  multiplying the first observation by (1-?2)1/2
  and including it in the model
• Asymptotically the two methods are equivalent
• But in small sample, time-series applications
  two methods can get different answers

37
Feasible GLS (continued)

• Often both Cochrane-Orcutt and Prais-Winsten are
  implemented iteratively
• This basic method can be extended to allow for
  higher order serial correlation, AR(q)
• Most statistical packages including Microfit
  will automatically allow for the estimation of
  such models without having to do the
  quasi-differencing by hand

38
FGLS versus OLS

• In the presence of serial correlation OLS is
  unbiased, consistent but inefficient
• FGLS is consistent and more efficient than OLS
  if serial correlation is present and the
  regressors are strictly exogenous
• However OLS is consistent under a weaker set of
  assumptions than FGLS
• So choice of estimator depends on weighing up
  different criteria

39
Serial Correlation-Robust Standard Errors

• Its possible to calculate serial
  correlation-robust standard errors, along the
  same lines as heteroskedasticity robust standard
  errors
• Idea is to scale the OLS standard errors to take
  into account serial correlation
• The details of this are beyond our scope - the
  method is implemented in Microfit where it goes
  by the name of Newey-West

40
Example

• In Puerto Rico minimum wage example
• found serial correlation
• compute Cochrane-Orcutt estimates by hand
• find iterated C-O estimates
• Find Newey-West s/c robust standard errors
• Demonstrate in Microfit
• Results compared on next slide

41
Example summary of results
42
Next Week and beyond

• Next Week (8/12/03) Go through mock exam
  answers in usual lecture time and place
• KC will keep office hours (Wed 10-12) while the
  University is open
• Email for appointment outside these times
  (ken.clark_at_man.ac.uk)
• Check with tutors for their availability

43
Next Semester - ES5622

• 5 lectures on Cross Section econometrics by Dr
  Martyn Andrews - Wooldridge, Chapter 7 is good
  preliminary reading
• 5 lectures on Time Series methods by Dr Simon
  Peters, Wooldridge Chapters 10-12 good
  preliminary reading, other references mentioned
  in lectures