Conditional Mean Models

1
Conditional Mean Models

• Peter Christoffersen
• McGill University
• Peter.christoffersen_at_mcgill.ca

2
Overview

• Univariate Models
• Autocorrelation
• ARMA Models
• Unit Roots
• Hurwicz Bias
• Long Memory
• Seasonality
• Tsay (2002)
• Christoffersen (2003)

• Multivariate Models
• Time Series Regression
• Spurious Regression
• Cross-Correlation
• Vector Autoregressions
• Granger Causality
• Cointegration
• Stylized Facts of Speculative Returns

3
Autocorrelation

• The sample correlation between two random
  variables, x and y, is calculated as
• The sample autocorrelation is
• Linear dynamics

4
Testing Autocorrelations

• Technical trading rules often rely on returns
  being predictable from their own past.
• Bartlett standard errors 1/root(T)
• The Ljung-Box test can be used to test that the
  autocorrelation for lags 1 through m are all
  jointly zero
• Choice of m? Rule of thumb mln(T)

5
Autoregressive (AR) Models

• We want to build time series forecasting models
  which can capture various patters in the
  autocorrelations across lags.
• The simplest and most used is the AR(1)

6
AR(1) Unconditional Moments

• If the return process is stationary
  then the unconditional mean can be had from
• The unconditional variance is similarly

7
Autocorrelation Function

• Linear time series models are characterized by
  their autocorrelation function (ACF). Assume
  w.l.o.g. that µ0 in the AR(1) model. Then
• ACF of the AR(1). Exponential decay is key

8
AR(2) Cycles

• The simples model which allows for business
  cycles is the AR(2)
• The autocorrelation function is
• By ACF symmetry we have

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Characteristic Roots

• Viewing the ACF of the AR(2) as a polynomial in ?
  we can solve for the characteristic roots
• Two AR(1) components if real roots.
• Cycles if complex roots.
• Stationarity requires that the roots are less
  than one in absolute value (or modulus).

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Partial Autocorrelation Function

• The partial autocorrelation function (PACF) gives
  the marginal contribution of an additional lagged
  term and helps determining the optimal order p in
  a general AR(p) model
• The PACF equals

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AR(p) Estimation and Diagnostics

• Estimation
• The AR(p) models can be estimated using simple
  OLS regression on observations p1 through T.
• Diagnostics
• Plot ACF of residuals and do a Ljung-Box test on
  residuals using m-p degrees of freedom.

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Conditional Mean Forecasting

• In the general AR(p) model,
• The one-step ahead forecast is simply
• The chain-rules gives the multiple step ahead
  forecast

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Moving Average (MA) Models

• In AR models The ACF die off exponentially,
  however, certain dynamic features such as bid-ask
  bounces die off abruptly and require a different
  type of model. Consider the MA(1)
• And in general the MA(q)

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ACF of MA Models

• Assume w.l.o.g. that c0 0. In the MA(1)
• Using the variance expression from before, we get
  the ACF

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Estimation of MA models

• MA models must be estimated by MLE using
  numerical optimization methods.
• Set a0 to its mean 0.
• Set parameter starting values (e.g. 0).
• Calculate shocks as function of observables
• Maximize the likelihood function

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Forecasting in MA Models

• In the MA(1) model the conditional mean forecast
  is
• In the MA(2) we get

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Combining AR and MA ARMA

• Combining AR and MA models into ARMA enables us
  to model dynamics with fewer parameters.
• Parameter parsimony is key in forecasting.
• Consider the ARMA(1,1)

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ARMA(p,q)

• The general ARMA(p,q) model is
• Forecasting using the chain rule.
• Estimation is done via MLE.
• Diagnostics on the residuals can be done via
  Ljung-Box tests with degree of freedom equal to
  m-q-p.

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Random Walk

• The random walk process is a key benchmark in
  financial forecasting. It is often used to model
  speculative (log) prices.
• The random walk is simply
• And we can write
• Past shocks have permanent effects
• Conditional mean and variance forecasts

20
Random Walk (RW) with Drift

• Equity returns typically have a small positive
  mean corresponding to a small drift in the log
  price
• Substituting back to time 0, we can write
• Constant becomes a time slope in RW
• Stochastic and deterministic elements

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Unit Roots and ARIMA Models

• A process, pt, follows an ARIMA(p,1,q) model if
  the first differences, pt pt-1, follows a
  stationary and invertible ARMA(p,q) model.
• The characteristic polynomial of an ARIMA(p,1,q)
  model has a root which is exactly equal to one,
  i.e. a unit root.
• A series could have several unit roots.

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Mean Reversion versus Unit Root

• Financial time series often have a root very
  close to 1. A root .999 versus 1 have very
  different implications for longer term
  forecasting. Mean reversion or not?
• Consider AR(1) setup

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Unit Root Test

• Testing the unit root hypothesis
• Can be done using the OLS estimate to form the
  Dickey-Fuller t-test

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Unit Root Test Critical Values

• When the null hypothesis is true, the DF unit
  root test does not have the usual Students t
  distribution.
• The asymptotic Normal distribution is only valid
  when the drift is non-zero and even so it is not
  a good finite sample approximation.
• The MacKinnon (1991) asymptotic critical values
  are given below.

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Asymptotic DF Critical Values
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Hurwicz Bias

• The OLS estimator contains an important finite
  sample bias in dynamic models.
• In an AR(1) when the true AR coefficient is close
  or equal to 1, the finite sample OLS estimate
  will be biased downward.
• Keep this in mind when people try to convince you
  that they have found mean-reversion where a
  random walk is more plausible

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Introducing Long Memory

• Often neither AR(1) nor RW seem adequate.
• Introducing the backshift operator, we write a
  random walk as
• Fractional differencing allows for flexibility
• Regular differencing will ensure -.5 lt d lt .5
• Structural breaks versus long memory

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ACF with Polynomial Decay

• The key feature of fractionally integrated
  processes is that the ACF decays at polynomial
  rate rather than exponential rate as is the case
  in basic ARMA models.
• We have
• Fractional differencing can be done using a
  truncated version of the infinite sum

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Modeling Seasonality Dynamics

• Most macroeconomic time series contain an annual
  seasonal effect.
• The volatility of intraday speculative returns
  often contain an important daily seasonal effect.
  
• Energy prices Several seasonals.
• Define the difference and lag/backshift operators

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Seasonal Differencing

• Quarterly data which shows an annual seasonal
  effect can be modeled using the 4-lag difference
  operator
• Seasonal differencing of the first difference
  instead gives

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Joint Seasonal and MA Effects

• Typically some dynamics are left even after
  accounting for seasonal effects.
• Multiplicative Seasonal MA model
• Additive Seasonal MA model
• Again, use visual diagnostics (ACF plot) and
  Ljung-Box on residuals to check model.
• Parsimony is key.

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Seasonal Adjustment

• Regression analysis with dummy variables (e.g.
  for each month in a year) is used to capture
  seasonal effects. This restricts seasonal
  patterns to be deterministic.
• Highly nonlinear filters such as X11 and X12
  are often used by government agencies for macro
  data.
• They are typically impossible to invert and they
  complicate out of sample forecasting.

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Multivariate Time Series

• Overview
• Time Series Regression
• Spurious Regression
• Cross-Correlation
• Vector Autoregressions
• Granger Causality
• Cointegration

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Time Series Regression

• The relationship between two (or more) time
  series can be assessed applying the usual
  regression analysis.
• But the regression errors must be scrutinized
  carefully.
• Consider a simple bivariate regression of two
  highly correlated series, e.g. two interest rates

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Regression Error Analysis

• Always plot the ACF of the regression errors
• Ljung-Box test can be used again
• If ACF dies off only very slowly (recall Hurwicz
  bias) then first-difference each series and rerun
  regression
• Check ACF of errors again and model using ARMA
  if they appear stationary.
• Re-estimate entire model using MLE.

36
Spurious Regression

• Checking the ACF of the error term is
  particularly importance due to the so-called
  spurious regression phenomenon
• Two completely unrelated times serieseach with a
  unit rootare likely to appear related in a
  regression (that is have a non-zero coefficient).
  
• If so, then the error term will have a highly
  persistent ACF and the regression on first
  differences will not show any relationship.

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Sample Cross-Correlation Matrices

• The sample cross-correlation matrices are the
  multivariate analogues of the ACF function. The
  cross-covariance matrix is
• Note rt is k-by-1. The cross-correlations are
• Where D is a diagonal matrix of standard dev.

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Multivariate Ljung-Box

• We want to test the null that all the
  cross-correlation matrices up to a lag order m
  are jointly zero.
• The test statistic is
• Where the trace operator takes the sum of the
  diagonal elements in the matrix.

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Vector Autoregressions (VAR)

• The VAR is arguably the simplest and most used
  multivariate time series model. Consider a
  first-order VAR, VAR(1)
• The bivariate case is simply
• Contemporaneous relation via s12

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Estimation and Diagnostics

• If the variables included on the right-hand-side
  of each equation in the VAR are the same the OLS
  can be used equation-by-equation.
• The multivariate Ljung-Box test can be used on
  the VAR residuals
• Where g is the number of parameters estimated in
  the VAR coefficient matrices.
• Forecasting using the multivariate chain-rule.

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Granger Causality

• How much of current r1t can be explained by past
  r2t once past r1t is accounted for?
• r2t is said to Granger cause r1t if
• r1t is said to Granger cause r2t if
• Use several lags. Null hypothesis of no Granger
  causality is tested via F-test of joint zeros.

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Cointegration

• If two series each have a unit root (they are
  integrated), but a linear combination of them do
  not, then we say they are cointegrated.
• Examples
• Spot-Futures Parity Ft,T St exp(rT)
• Pairs Trading Find two stocks whose prices tend
  to move together. If they diverge then long the
  cheap and short the dear.

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Forecasting with Cointegration

• Simple bivariate system
• System forecasts
• Univariate forecasts

44
Stylized Facts of Asset Returns

• We can consider the following list of so-called
  stylized facts which apply to most stochastic
  returns.
• Each of these facts will be discussed in detail
  in the first part of the book.
• We will use daily returns on the SP500 from
  1/1/97 to 12/31/01 to illustrate each of the
  features.

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Stylized Fact 1

• Daily returns have very little autocorrelation.
  We can write
• Returns are almost impossible to predict from
  their own past.
• Fig 1.1 shows the correlation of daily SP500
  returns with returns lagged from one to 100 days.
  
• We will take this as evidence that the
  conditional mean is roughly constant.

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Autocorrelations of Daily SP Returns for Lags 1
through 100 1/1/97-12/31/01Figure 1.1
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Stylized Fact 2

• The unconditional distribution of daily returns
  have fatter tail than the normal distribution.
• Fig.1.2 shows a histogram of the daily SP500
  return data with the normal distribution imposed.
• Notice how the histogram has longer and fatter
  tails, in particular in the left side, and how it
  is more peaked around zero than the normal
  distribution.
• Fatter tails mean a higher probability of large
  losses than the normal distribution would
  suggest.

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Histogram of Daily SP Returns Superimposed on
the Normal Distribution 1.1.97 12.31.01
Fig.1.2
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Stylized Fact 3

• The stock market exhibits occasional, very large
  drops but not equally large up-moves.
• Consequently the return distribution is
  asymmetric or negatively skewed. This is clear
  from Figure 1.2 as well.
• Other markets such as that for foreign exchange
  tend to show less evidence of skewness.

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Stylized Fact 4

• The standard deviation of returns completely
  dominates the mean of returns at short horizons
  such as daily.
• It is typically not possible to statistically
  reject a zero mean return.
• Our SP500 data has a daily mean of 0.0353 and a
  daily standard deviation of 1.2689.

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Stylized Fact 5

• Variance measured for example by squared returns,
  displays positive correlation with its own past.
• This is most evident at short horizons such as
  daily or weekly.
• Fig 1.3 shows the autocorrelation in squared
  returns for the SP500 data, that is
• Models which can capture this variance dependence
  will be presented in Chapter 2.

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Autocorrelation of Squared Daily SP500 Returns
for Lags 1 through 1001.1.97 12.31.01
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Stylized Fact 6

• Equity and equity indices display negative
  correlation between variance and returns.
• This often termed the leveraged effect, arising
  from the fact that a drop in stock price will
  increase the leverage of the firm as long as debt
  stays constant.
• This increase in leverage might explain the
  increase variance associated with the price drop.
  We will model the leverage effect in Chapter 2.

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Stylized Fact 7

• Correlation between assets appears to be time
  varying.
• Importantly, the correlation between assets
  appear to increase in highly volatile
  down-markets and extremely so during market
  crashes.
• We will model this important phenomenon in
  Chapter 3.

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Stylized Fact 8

• Even after standardizing returns by a
  time-varying volatility measure, they still have
  fatter than normal tails.
• We will refer to this as evidence of conditional
  non-normality.
• It will be modeled in Chapters 4 and 5.

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Stylized Fact 9

• As the return-horizon increases, the
  unconditional return distribution changes and
  looks increasingly like the normal distribution.
• Issues related to risk management across horizons
  will be discussed in Chapter 5.

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Asset Return Model

• Based on the above of stylized facts our model of
  individual asset returns will take the generic
  form
• The conditional mean return is thus and the
  conditional variance
• The random variable is an innovation term,
  which we assume is identically and independently
  distributed (i.i.d.) as D(0,1).

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Overview of Remaining Material

• Chapter 2 discusses methods for estimating and
  forecasting variance on an asset-by-asset basis.
• Chapter 3 presents methods for modeling the
  correlation between two or more assets.
• Chapter 4 introduces methods to model the tail
  behavior in asset returns which is not captured
  by volatility and correlation models and which is
  not captured by the normal distribution.
• Chapter 5 introduces simulation based methods in
  risk management.