Univariate Time series - 2

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Univariate Time series - 2

• Methods of Economic Investigation
• Lecture 19

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Last Time

• Concepts that are useful
• Stationarity
• Ergodicity
• Ergodic Theorem
• Autocovariance Generating Function
• Lag Operators
• Time Series Processes
• AR(p)
• MA(q)

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Todays Class

• Building up to estimation
• Wold Decomposition
• Estimating with exogenous, serially correlated
  errors
• Testing for Lag Length

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Refresher

• Stationarity Some persistence but not too much
• Ergodicity Persistence dies out entirely over
  some finite period of time
• Square Summability (assumption for MA process)
  with parameter ? such that
• Invertibility (assumption for AR process) with
  parameter f which has roots ? such that

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ARMA process

• In general can have a process with both AR and MA
  components
• A general ARMA(p, q) process in our lag function
  notation this looks likea(L)xt b(L)et
• For example, we may have an ARMA(2, 1)
• xt (f1xt-1 f2xt-3) et ?1 et-1
• (1 - f1L f2L2) xt (1 ?1L) et
• If the process is invertible then we can rewrite
  this as xta(L)-1 b(L) et

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Why Focus on ARMA processes

• Define the range of ARMA processes (invertible AR
  lag polynomials, square summable MA lag
  polynomials) which can rely on convergence
  theorems
• any time series that is covariance stationary,
  has a linear ARMA representation.

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Information Sets

• At time t-n
• Everything for time t-n and before is known
• Everything at time t is unknown
• Information set Ot-n
• Define Et-n(et) Eet Ot-n
• Distinct from Eet because we know previous
  values of es up until t-n
• For example, suppose n 1 and et p et-1?,
• E (et) 0 for all t so its a mean zero process
• Et-1(et) p et-1

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Recalling the CEF

• Define the linear conditional expectation
  function CEF(a b) which is the linear project,
  i.e. the fitted values of a regression of a on b.
  i.e. a ßb
• This is distinct from the general expectations
  operator in that it is imposing a linear form of
  the conditional expectation function.

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Wold Decomposition Theorem - 1

• Formally the Wold Decomposition Theorem says
  that
• Any mean zero weakly stationary process xt can
  be represented in the form
• This comes with some properties for each term

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Wold Decomposition Theorem - 2

• Where
• et xt - CEF(xt xt-1, xt-2, . . . ,x0).
• Properties of et
• CEF (etxt-1, xt-2, . . . x0)0, E(etxt-j) 0,
  E(et) 0,
• E(et2) s2 for all t, and E(et es) 0 for all t
  ?s
• The MA polynomial is invertible
• The parameters ? is square summable
• ?j and es are unique.
• ?t is linearly deterministic
• i.e. ?t CEF(?txt-1, . . . .).

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A note on the Wold Decomposition

• Much of the properties come directly from our
  assumptions tha the process is weakly stationary
• While it says mean zero process, remember we can
  de-mean our data so most processes can be
  represented in this format.

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Uses of Wold Form

• This theorem is extremely useful because it
  returns time-series processes back to our
  standard OLS model.
• Notice that weve relaxed some of the conditions
  for the Gauss-Markov theorem to hold.
• the Wold MA(8) representation is unique.
• if two time series have the same Wold
  representation, then they are the same time
  series
• This on true only up to second moments in linear
  forecasting
•  

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Emphasis on Linearity

• although CEF(et xt-j) 0, can have
  E(et xt-j) ? 0 with nonlinear projections
• If the true xt is not generated by linear
  combinations of past xt plus a shock, then the
  Wold shocks (es) will be different from the true
  shocks.
• The uniqueness result only states that the Wold
  representation is the unique linear
  representation where the shocks are linear
  forecast errors.

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Estimating with Serially Correlated Errors

• Suppose that we have Yt ßXt et
• Eet xt 0, Eet2 xts2
• E et et-k ?k for k?0 and so define Eetek
  s2G
• We could consistently estimate ß but our standard
  errors would be incorrect making it difficult to
  do inference.
• Just a heteroskedasticity problem which we have
  already seen with random effects
• Use feasible GLS to estimate weights and then
  re-estimate OLS to obtain efficient standard
  errors.

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Endogenous Lagged Regressors

• May be the case that either the dependent
  variable or the regressor should enter the
  estimating equation in lag values too
• Suppose we were estimating t
• Yt ß0 Xt ß1 Xt-1 ßk Xt-k et.
• We think that these Xs are correlated with Y up
  to some lag length k
• We think these Xs are correlated with each other
  (e.g. some underlying AR process)
• but were not sure how many lags to include

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Naive Test

• Include lags going very far back r gtgt k
• test the longest lag coefficient ßr 0 and see
  if that is significant. If not, drop it and keep
  going.
• Problems
• Practically, the longer lags you take, the more
  data you make unusable because it doesnt have
  enough time periods to construct the lags.
• doesnt allow lag t-6 but exclude lag t-3.
• The theoretical issue is that we will reject the
  null 5 percent of the time, even if its true (or
  whatever the significance of the test is).

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More sophisticated testing

• Can be a bit more sophisticated comparing
  restricted and unrestricted models
• define pmax as some long lag length greater than
  the expected relevant lag length
• In general, we do not test our pmax but as
  before, as p ? pmax the sample size decreases.
• Define ej Yt ß0 Xt ß1 Xt-1 ßj Xj and
  let N be the sample size.
• We therefore could imagine trying to minimize the
  sum of squared residual

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Cost Functions

• Intuition c( . ) is a penalty for adding
  additional parameters
• thus we try to pick the best specification using
  that cost function to penalize inclusion of extra
  but irrelevant lags.
• Akaike (AIC) c(n) 2
• the AIC criterion is not well-founded in theory
  and will be biased in finite samples
• the bias will tend to overstate the true lag
  length
• Bayesian c(n) log(n)
• the BIC will converge to the true p.

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Return to Likelihood Ratio Tests

• The minimization problem is just likelihood ratio
  test
• To see this, compare lag length j to lag length
  k. We can write
• Define constant

Constant Declining in N
LR test
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Next Time

• Multivariate Time Series
• Testing for Unit Roots
• Cointegration
• Returning to Causal Effects
• Impulse Response Functions
• Forecasting