Economic Forecasting Seminar

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Economic Forecasting Seminar

• Introduction to the
• Box-Jenkins
• Forecasting Methods

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Box-Jenkins Methodology

• Univariate forecasts based on a statistical
  analysis of the past data. Differs from
  conventional regression methods in that the
  mutual dependence of the observations is of
  primary interest.
• Forecasts are linear functions of the sample
  observations.

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Box-Jenkins Methodology

• Identification of the type of model to be used is
  a critical first step. Differs from other
  univariate techniques in that a thorough study of
  the properties of a time series is carried out
  before applying a forecasting technique.

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Principle of Parsimony

• Find efficiently parameterized models. A model
  with a large number of parameters will achieve a
  good historical fit, but post-sample forecasts
  are likely to be poor.

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• By building models based on past realizations of
  a time series we are implicitly assuming that
  there is some regularity in the process
  generating the series.
• One way to view such regularity is through the
  concept of stationarity.
• The use of Box-Jenkins modeling techniques
  requires a stationary process.

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Definition 1 Stochastic Process

• A stochastic process is a collection Xt
  t1,2,,T of random variables ordered in time.
  Xt t Î T is a continuous process if - lt t
  lt . If T is finite, Xt is a discrete process.
• Example the error term in a linear regression
  model is assumed to be a stochastic process.

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• If X is a random variable, then the variance of X
  is
• which we can estimate by

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• If X and Y are two random variables having a
  joint probability density function, then the
  covariance of X and Y is
• which we can estimate by

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• Finally, the simple correlation coefficient
  between X and Y is

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Definition 2 Stationarity

• A stochastic process is covariance stationary if,
  for all values of t,

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Intuition

• A stochastic process is said to be covariance
  stationary if its statistical properties do not
  change with time.
• More precisely, a stationary series is one for
  which the mean and variance are constant across
  time and the covariance between current and
  lagged values of the series (autocovariances)
  depends only on the distance between the time
  points.

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Interpretation

• Recall that were trying to explain the future in
  terms of what is already known.
• And for stationary series
• But these two properties don't help very much in
  this regard.

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Interpretation

• However the above says, for example, that if g1 gt
  0, a high value of X today will likely be
  followed by a high value tomorrow.
• By assuming that the gk are stable, this
  information can be exploited and estimated.

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Autocorrelations

• Covariances are often difficult to interpret
  because they depend on the units of measurement
  of the data.
• Correlations, on the other hand, are scale-free.
  Thus we can obtain the same information about the
  time series by computing the autocorrelations of
  a time series.

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Autocorrelation Coefficient

• The autocorrelation coefficient between Xt and
  Xt-k is
• A graph of the autocorrelations is called a
  correlogram.
• Knowledge of the correlogram implies knowledge of
  the process model which generated the series
  and vice versa.

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Partial Autocorrelations

• Another important function in the Box-Jenkins
  methodology is the partial autocorrelation
  function.
• Partial autocorrelations measure the strength of
  the relationship between observations in a series
  controlling for the effect of intervening time
  periods.

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• If the observations of X in period t are highly
  related to the observations in, say, period t-12
  then a plot of the partial autocorrelations for
  that series (partial correlogram) should exhibit
  a spike, or relative peak, at lag 12.
• Monthly time series with a seasonal component
  should exhibit such a pattern in their partial
  correlograms.
• Monthly time series with seasonal components will
  also exhibit spikes in their correlograms at
  multiples of 12 lags.

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Autocorrelations
Partial Autocorrelations
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Nonstationarities I

• In the level of the mean. Here the mean changes
  over different segments of the data. Time series
  with a strong trend are not mean stationary.
• Nonlinear trends will cause the covariances to
  change over time.

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SP 500
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Nonstationarities II

• Seasonality. Time series with a seasonal
  component exhibit variations in the data which
  are a function of the time of the year.
• The autocovariances will depend on the location
  of the data points in addition to the distance
  between observations.
• Such nonstationarities are made worse by seasonal
  components which are not stable over time.

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Nonstationarities III

• Shocks. A drastic change in the level of a time
  series will cause it to be nonstationary.

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Removing Nonstationarities

• First differencing D(Xt) Xt - Xt-1
• Second differencing
• D(Xt) - D(Xt-1) Xt - 2Xt-1 Xt-2
• Period-to-period annualized rate of growth

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Differencing Using EViews

• Three equivalent forms of first differencing
• d(gasprice,1) d(gasprice) gasprice-gasprice(-
  1)
• Two equivalent forms of seasonal differencing
• d(gasprice,0,12) gasprice-gasprice(-12)
• Two equivalent forms of first differencing a
  seasonally differenced series
• d(gasprice,1,12) d(gasprice-gasprice(-12))
• Three equivalent forms of second differencing
• d(gasprice,2) d(gasprice-gasprice(-1))
• d(gasprice)-d(gasprice(-1))

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• Identification

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Autoregressive Processes

• Xt depends on past values of X plus some random
  shock, or disturbance. This is an AR(p) process.
• If p were known, the fact that the error term in
  the equation satisfies the assumptions of the
  classical linear regression model means that we
  could use OLS to estimate fi.

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Practical Problem What is p?

• Use OLS and traditional t-tests to determine p.
• Minimize the Akaike Information Criteria (AIC).
• Minimize the Schwarz Criteria.

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First-Order Autoregressive Model

• An autoregressive process of order 1, i.e., an
  AR(1) model can be written as
• and simply says that the value of X at time t
  depends on the value of X in the preceding period
  plus some random disturbance.

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Random Walk Model

• A widely used AR(1) process is the random walk
  for which f11.
• Let X00.
• Then
• And in general

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Properties of the Random Walk

• Q Is Xt a stationary process?
• A

No
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Xt f1 Xt-1 et

• When f11 the random walk model can be written
  as
• Q Is et a stationary process?

Yes
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White Noise Process

• If me 0 then et DXt is said to be a white
  noise series.
• et is perfectly correlated with itself
• r0 1
• The correlogram of et would have a vertical spike
  at k 0 and then truncate for all values of k gt
  0 since the autocorrelations equal 0.

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Moving Average Processes

• An MA(q) process can be written as
• A moving average forecasting model uses
  contemporaneous and lagged values of random
  shocks to represent movements in the series.
• The current value of X is expressed as a function
  of current and lagged unobservable shocks.

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ARMA Models

• A time series may be represented by a combination
  of autoregressive and moving average processes.
• In this case the series is an ARMA(p,q) process
  and takes the form

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Box-Jenkins Methodology

• 1. Identification. Determine, given a sample of
  time series observations, what the model of the
  differenced data is.
• 2. Estimation. Estimate the parameters of the
  chosen model
• 3. Forecasting. Use the resulting model to
  forecast the time series. If necessary, undo the
  differencing to restore the original time series.

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Identification

• Determining whether a series follows an AR, MA,
  or a mixture of both processes is a difficult
  task requiring judgment and experience on the
  part of the forecaster.
• By plotting the correlogram (ACF) and partial
  correlogram (PACF) for a series we can apply
  known characteristics of theoretical moving
  average and autoregressive processes to help
  identify a series as characteristic of one of
  these classes of models.

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Guidelines for B-J Identification

• PARTIAL
• MODEL CORRELOGRAM CORRELOGRAM
• AR(p) Dies Off Truncates after lag p
• MA(q) Truncates after lag q Dies Off
• ARMA(p,q) Dies Off Dies Off